Accurate and Robust Localization Techniques for Wireless Sensor Networks

Accurate and Robust Lo calization T ec hniques for Wireless S ensor Net w orks Mohamed AlHa jri Ab dulrahman Goian Muna Darw eesh Rashid AlMemari Advisor s: Prof. Raed Sh ubair, Dr . Luis W eruaga , Dr. Ahmed AlT unaiji 2015 Abstract The c oncept of wireless sens or net w ork rev olv es around a group of sensor n o des that utilize the radio signals in order to communicate among eac h other. These no d es are typicall y made, of sensors, a m emory , a multi- con troller, a transceiv er, and a p o w er sour ce to supply the energy to th ese comp onents. T here are many facto rs that restrict the design of wireless sensor net w ork suc h as the size, cost, and functionalit y . The ﬁeld of wireless sensor net work, witnessed a remark able rev olution sp ecial after the environmen tal a wa k ening in the last t wo decades. Its imp ortance emanates from its capabilit y to monitor physical and en vironmenta l conditions (suc h as sound, temp erature, and pressure) with minimal p o w er consump tion. T h e principle of passing d ata co op erativ ely though a netw ork to a main lo cation play ed a vital role in th e success of the m etho dolog y of WSN. This rep ort explores the concept of wireless sens or netw ork an d h o w it is b een made viable through th e conv ergence of wireless comm unications and m icro-elec tro-mec h anical systems (MEMS) tec hn ology together digital electronics. T he rep ort addresses some of the factors that ha v e to b e considered when choosing the lo calization algorithm. It is v ery im p ortan t to choose prop erly since the lo ca lization pr o cess ma y in v olv es int ensiv e computational load, b ased on man y diﬀerent criteria, as w ell as analysis metho d. T he rep ort also views the adv an tages and disadv anta ges of localizatio n tec h n iques. Nev ertheless, it inv estigates the c hallenges asso ciated with the Wireless Sensor Net w orks. The rep ort categorizes the algorithms, d ep ending on where the compu tational eﬀort is carried out, into central ized and d istributed algorithms. With minim al compu tational complexit y and signaling ov erhead, the pr o ject aims to d ev elop algorithms that can accurately lo calize sensor no des in real-time with lo w computational requirement s, and robustly adapt to channel an d net w ork dy n amics. The rep ort fo cuses on three areas in particular: the ﬁ rst is the Receiv ed Signal Str en gth indicator tec hn ique, Direction of Arriv al tec hniqu e, and the in tegration of t w o algorithms, RSS and DO A, in order to build a h ybrid, more robust algorithms. In the Receiv ed S ignal Strength (RSS), the un kno wn n o de lo cation is estimated using trilateratio n. This rep ort examines the p erformance of diﬀeren t estimators su ch as Least Square, W eigh ted Least Sq u are, and Hub er robustness in order to obtain the most r obust p erformance. In the direction of arriv al (DO A) metho d, the estimation is carried out using Multiple Signal Classiﬁcation (MUSIC), Ro ot-MUSIC, and Estimation of Signal P arameters Via Rotational Inv ariance T ec hniqu e (ES PRIT) algorithms. W e inv estigate multiple signal scenarios utilizing v arious antenna geometries, whic h includes u niform lin ear arra y (ULA) and uniform circular arra y (UCA). Sp eciﬁc atten tion is giv en for multipath scenarios in whic h signals b ec ome spatially correlate d (or coherent ). This requ ired the use of pre-pro cessing tec hn iques, wh ic h include ph ase mo de excitation (PME), spatial sm o othing (SS), and T o eplitz. F urther impro v emen ts of existing localization tec hniques are demonstrated th r ough the use of a hybrid approac h in wh ic h v arious com binations of RS S and DO A are explored, simulated, and analyzed. Th is h as led to t wo ma jor con tribu tions: the ﬁrst con tribution is a com bined i Abstract ii RSS/DO A metho d, based on UCA, whic h has the tolerance of d etecting b oth uncorrelated and coheren t signals sim ultaneously . The second ma j or con tribution is a combined Ro ot- MUSIC/T o epltiz metho d, based on UCA, which is outp erforms other tec hn iques in terms of increased num b er of detected signals and r educed computationally load. T able of Con ten ts Abstract i Con ten ts iii List of Sym b ols viii List of Figures x List of T ables xiv Ac knowledgemen ts xvi 1 Introduction 1 1.1 Goals and Ob jectiv es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Ob jectiv es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Health S y s tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 En vironmenta l Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Home Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.4 Infrastructur e Health Monitoring . . . . . . . . . . . . . . . . . . . . . . 7 1.3.5 In telligen t T ransp ortation . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.6 Searc h and Rescue op eration . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.7 Smart Campu s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.8 Military app lications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 iii T a ble o f Conten ts iv 1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Ac h iev emen ts and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Ac h iev emen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Ov erview of the Rep ort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Surv ey of WSN Lo calization 13 2.1 Lo cation Disco v ery T ec hniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 2.1.1 T riangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 T rilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 Multilaterati on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Ranging T ec hniqu es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Time of Ar riv al (TOA) T ec hnique . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Receiv ed Signal Strength (RSS) T ec hn ique . . . . . . . . . . . . . . . . 20 2.2.3 Radio Hop Coun t T echnique . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.4 Direction of Arriv al (DO A) T ec hnique . . . . . . . . . . . . . . . . . . . 21 2.3 Lo calizatio n Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Cen tralized Algo rithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1.1 Semi-deﬁnite Programming (SDP) . . . . . . . . . . . . . . . . 22 2.3.1.2 Multidimensional Scaling (MDS) . . . . . . . . . . . . . . . . . 23 2.3.2 Decen tralized (Distributed) Algorithm . . . . . . . . . . . . . . . . . . . 25 2.3.2.1 Anc hor-based . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2.2 Relaxatio n-based . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 2.3.2.3 Co ordinate System Stitc h ing Algorithm . . . . . . . . . . . . . 28 2.3.2.4 Hybrid Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 3 Receiv ed Signal Strength 30 3.1 RSS Mo d eling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Norm App ro ximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3.2.1 Least-squares Appr o ximation . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Hub er Robus tn ess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 T a ble o f Conten ts v 3.3.1 P erformance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 3.3.2 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Direction of Arriv al 42 4.1 Con v en tional Sensor Arra y C onﬁgurations . . . . . . . . . . . . . . . . . . . . . 42 4.1.1 Uniform Linear Ar ra y (ULA) . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.2 Uniform Circular Arra y (UCA) . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 T yp e of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.1 Uncorrelated S ignals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.2 Correlated Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 4.2.2.1 Phase Mo de Excitatio n . . . . . . . . . . . . . . . . . . . . . . 4 6 4.2.2.1 .1 Phase Mo de Excitation Principle . . . . . . . . . . . . 4 8 4.2.2.1 .2 Arra y Pa ttern of Uniform Lin ear Arra y (ULA) . . . . 48 4.2.2.1 .3 Phase of Excitation of Con tin uous Circular Arra y (CCA) . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.2.1 .4 Phase of Excitation of Uniform C ircular Array (UCA) 53 4.2.2.2 Spatial Smo ot hing T ec hn iqu es . . . . . . . . . . . . . . . . . . 55 4.2.2.2 .1 F orw ard S patial Smo othing (FSS) . . . . . . . . . . . 55 4.2.2.2 .2 F orw ard/Bac kward Spatial Smo othing (FBSS ) . . . . 57 4.2.2.3 T o eplitz Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.2.3 .1 Mo deling of S ingular Co v ariance Matrix . . . . . . . . 60 4.2.2.3 .2 Realizat ion of T oeplitz Algorithm . . . . . . . . . . . 60 4.3 DO A Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.1 MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.2 Ro ot-MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.3 UCA-Ro ot-MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 4.3.4 ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.5 UCA-ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.6 Comparison in the Pe rformance of ULA and UCA . . . . . . . . . . . . 70 4.3.7 DO A Algorithms P erformance . . . . . . . . . . . . . . . . . . . . . . . 7 1 4.3.7.1 DO A Algorithms P erformance in ULA . . . . . . . . . . . . . . 72 T a ble o f Conten ts vi 4.3.7.1 .1 Num b er of S ensor Element s . . . . . . . . . . . . . . . 72 4.3.7.1 .2 Num b er of In ciden t Signal . . . . . . . . . . . . . . . 74 4.3.7.1 .3 Angular Separation b et w een Incident S ignals . . . . . 76 4.3.7.1 .4 Num b er of S amples . . . . . . . . . . . . . . . . . . . 7 7 4.3.7.1 .5 Signal to Noise Ratio (SNR) . . . . . . . . . . . . . . 79 4.3.7.1 .6 Signal Correlation . . . . . . . . . . . . . . . . . . . . 81 4.3.7.2 DO A Algorithms P erformance in UC A . . . . . . . . . . . . . 85 4.3.7.2 .1 Num b er of S ensor Element s . . . . . . . . . . . . . . . 85 4.3.7.2 .2 Num b er of In ciden t Signal . . . . . . . . . . . . . . . 87 4.3.7.2 .3 Angular Separation b et w een Incident S ignals . . . . . 88 4.3.7.2 .4 Num b er of S amples . . . . . . . . . . . . . . . . . . . 9 0 4.3.7.2 .5 Signal to Noise Ratio (SNR) . . . . . . . . . . . . . . 92 4.3.7.2 .6 Signal Correlation . . . . . . . . . . . . . . . . . . . . 94 4.3.7.3 System Mo deling . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3.7.3 .1 Uncorrelated Signals . . . . . . . . . . . . . . . . . . . 98 4.3.7.3 .2 Correlated Signals . . . . . . . . . . . . . . . . . . . . 99 5 Hybrid T ec hniques 101 5.1 Hybrid RS S and DOA u sing one Hybrid No de . . . . . . . . . . . . . . . . . . . 101 5.1.1 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Hybrid RS S and DOA w ith Spatial S mo othing . . . . . . . . . . . . . . . . . . 104 5.2.1 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Hybrid tec hniqu e usin g Least Square Based T ec hniques . . . . . . . . . . . . . 106 5.3.1 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Hybrid T ec hn ique using T w o Lines . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.4.1 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6 Conclusion 114 6.1 Summary of w ork done . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3 Critical App raisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 T a ble o f Conten ts vii A 119 B 121 C 122 References 123 List of Sym b ols P r Receiv ed Po w er P t T ransmitted Po wer G t T ransmitter Gain G r Receiv er Gain D i Distance A System Co ord in ated Matrix p s Unkno wn no d e b Distance V ector N Num b er of Array Elements M Num b er of Receiv ed Signals s m Narro wband Signal θ m Azim u th Angle θ e Elev ation Angle A s Arra y Steering Matrix φ m Phase Shift b et ween the Elemen ts of the Sensor Arra y θ n Angular Lo cation of eac h Elemen ts of the S ensor Ar ra y ζ = 2 π r λ UCA dela y Paramet er w Beamforming W eigh ts V ector f Bea mpattern of an Arr ay f c p Normalized far-ﬁeld Patt ern of the p th phase mo de in CCA f s p Normalized far-ﬁeld Patt ern of the p th phase mo de in UCA h Highest Excited Mo d e viii List of Symbols ix a s v Steering V ector of VULA T v T ransform ation Matrix based on PME to mat UC A in to VULA x v VULA Output V ector R Co v ariance Matrix R s Signal Co v ariance Matrix R c Coheren t Co v ariance Matrix R T T o eplitz C o v ariance Matrix constructed fr om R c σ 2 n Noise V ariance V n Noise Eigen v ector V s Signal Eigen v ector P MUSIC Sp atial Sp ectrum Q Root − M U S I C Ro ot-MUSIC P olynomial z Ro ot v alue of the Ro ot-MUSIC Polynomial T w Noise-prewhitened T r ansformation Matrix Q U C A − Root − M U S I C UCA-Ro ot-MUSIC Pol ynomial ∆ x Displacemen t of ESPRIT S ubarrays Φ Diag onal Matrix Cont aining the phase sh ift b etw een ESPRIT Subarr a y s T Non-singular Matrix to r elate S teering V ectors of ES PRIT Sub arr a ys ( . ) H Hermitian Op eratio n ( . ) T T ranp ose Op eration of a Matrix ( . ) ∗ Complex Conjugate of a Matrix List of Figures 1.1 Co deBlue p ro ject conﬁguration . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Sensors for detecting vo lcanic eru ptions . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The conﬁguration for NA WMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 The eﬀect of non-line-of-sigh t and m ulti-path on r anging tec h nique . . . . . . . 10 2.1 T riangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 T rilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Iterativ e Mu ltilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Collab orativ e Multilat eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Multi-no de T DO A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 2.6 Multisignal-TDO A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 Sensor array pro cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.8 Classiﬁcation of Localization Algorithms in WSNs . . . . . . . . . . . . . . . . 23 2.9 Conﬁgurations that represen t C entralize d Algorithms . . . . . . . . . . . . . . . 24 2.10 Illus trations of Semi-deﬁnite programming . . . . . . . . . . . . . . . . . . . . . 25 2.11 Illus trations of the b ounding b ox int eraction . . . . . . . . . . . . . . . . . . . . 27 3.1 No de estimation in RSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 RMSE for d iﬀeren t v alues of SNR us ing LS . . . . . . . . . . . . . . . . . . . . 36 3.3 RMSE for d iﬀeren t v alues of SNR us ing WLS . . . . . . . . . . . . . . . . . . . 37 3.4 RMSE for d iﬀeren t v alues of SNR us ing Hub er Robu stness . . . . . . . . . . . 38 3.5 RMSE for d iﬀeren t v alues of SNR (least,w eigh ted least squ are, and h ub er ro- bustness) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 x List of Figures xi 3.6 RMSE for d iﬀeren t v alues of SNR (least,w eigh ted least squ are, and h ub er ro- bustness) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.7 Robustness of th e algorithm to c h anges to in the pathloss exp onent . . . . . . . 41 4.1 Geometry of N-element s (ULA) . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 T op view of the N-element circular arra y in x-y plane . . . . . . . . . . . . . . 4 5 4.3 Arra y geometry for UCA includ ing the elev ation angle . . . . . . . . . . . . . . 47 4.4 Arra y p attern of ULA w ith 10 elemen ts receiving a signal from θ m =0 . . . . . 50 4.5 Bessel fu nctions for J 0 ( ζ ) up to J 7 ( ζ ) where 0 ≤ ζ ≤ 2 π . . . . . . . . . . . . . 52 4.6 FSS spatial s mo othing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.7 Applying FSS on Matrix R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.8 F orw ard/Bac kward spatial smo othin g . . . . . . . . . . . . . . . . . . . . . . . 58 4.9 Illustration of Sensor Arra y Usin g ESP R I T Algorithm . . . . . . . . . . . . . . 67 4.10 DOA estimation us in g ULA and UCA . . . . . . . . . . . . . . . . . . . . . . . 71 4.11 Im p act of c hanging the n umber of elemen ts in UCA on the p erf ormance of MU- SIC algorithm with settings (M=2, θ = 10 o and − 10 o , d = 0 . 5 λ ) , SNR=10dB and K = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.12 Im p act of c hanging th e n umber of in ciden t signals imp in ging ULA on the p er- formance of MUSIC algorithm with settings ( N =6, d =0.5 λ , SNR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 4.13 Im p act of c hanging th e angular separation b et we en the incident signals im- pinging on ULA on the p erforman ce of MUSIC algorithm with settings ( N =3, d =0.5 λ , S NR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.14 Im p act of changing the num b er of samples of th e incident signals impin ging on ULA on the p erf ormance of MUSIC algorithm with s ettings ( N =5, θ =20 o and − 20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . 78 4.15 Im p act of changing SNR for ULA on the p erformance of MUSIC algorithm with settings ( N =5, θ = 20 o and − 20 o , d= 0.5 λ and K =100) . . . . . . . . . . 80 4.16 Im p lemen tation of Standard MUSI C and MUS I C w ith FSS for ULA in C orre- lated Environmen t with the settings ( N =12, θ = − 40 o , − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , S NR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . 82 List of Figures xii 4.17 Im p lemen tation of FSS and FBSS u sing MUSIC algorithm for ULA in Corre- lated Environmen t with the settings ( N = 9, θ = − 40 o , − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , S NR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.18 Im p lemen tation of FBSS and T o ep litz using MUSIC algorithm for ULA in Cor- related Environmen t with the settings ( N =7, θ = − 40 o , − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , S NR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.19 Performance Comparison b et ween FBSS and T o eplitz using MUSIC algorithm for ULA in Correlated Environmen t with the settings ( θ = − 40 o , − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , S NR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.20 Im p act of c hanging the num b er of elemen ts in UC A on the p erformance of MUSIC algo rithm with settings ( M =2, θ =20 o and − 20 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.21 Im p act of c hanging the num b er of incident signals imp inging UCA on the p er- formance of MUSIC algorithm with settings ( N =5, θ e =20 o , d =0.5 λ , S NR=10dB and K = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.22 Im p act of c hanging th e angular separation b et we en the incident signals im- pinging on UCA on the p erformance of MUSIC algorithm with settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . 89 4.23 Im p act of changing the num b er of samples of th e incident signals impin ging on UCA on the p erf ormance of MUSIC algorithm with settings ( N =5, θ = 30 o and − 30 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . 91 4.24 Im p act of c hanging SNR for UCA on the p erformance of MUSIC algorithm with settings ( N =5, θ =30 o and − 30 o , θ e =20 o , d =0.5 λ and K =100) . . . . . . . 93 4.25 Im p lemen tation of stand ard MUSIC and MUSIC with FSS for UC A in Corre- lated Environmen t with the settings ( N = 12, θ = − 140 o , − 80 o , − 20 o , 50 o , 80 o , 140 o , θ e =20 o , d =0.5 λ , SNR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . 95 4.26 Im p lemen tation of FSS and FBSS using MUSIC algorithm for UCA in Corre- lated Environmen t with the settings ( N = 9, θ = − 140 o , − 80 o , − 20 o , 50 o , 80 o , 140 o , θ e =20 o , d =0.5 λ , SNR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . 96 4.27 Lo ca tion of the no des in a p ractical en vironment . . . . . . . . . . . . . . . . . 9 7 4.28 RMSE for diﬀerent S NR for un correlated signals . . . . . . . . . . . . . . . . . 98 List of Figures xiii 4.29 RMSE for diﬀerent S NR for correlated signals . . . . . . . . . . . . . . . . . . . 99 5.1 P osition of the no des in the x-y plane u sed for hybrid testing . . . . . . . . . . 103 5.2 RMSE for d iﬀeren t v alues of SNR us ing Hybrid tec hniqu e and RSS tec hn ique . 104 5.3 RMSE for diﬀerent v alues of SNR usin g RSS an d DOA Hybrid tec hnique with Spatial Smo ot hing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4 RMSE for diﬀeren t v alues of SNR usin g Hybrid tec hnique RSS LS and RSS tec hn ique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5 RMSE for d iﬀeren t v alues of SNR usin g Hyb r id tec h nique R S S LS and ﬁr st prop osed Hybr id tec hn ique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.6 RMSE for d iﬀeren t v alues of SNR us ing Hybrid tec hniqu e RSS WLS . . . . . . 110 5.7 P osition of the no des in the x - y plane used for testing the least h ybrid . . . . 111 5.8 RMSE for diﬀerent v alues of SNR usin g tw o Hybr id tec hniques, the last hybrid and hybrid w ith one hybrid no de tec hniqu es . . . . . . . . . . . . . . . . . . . 112 5.9 RMSE for diﬀerent v alues of SNR usin g tw o Hybr id tec hniques, the last hybrid and hybrid LS and RSS techniques . . . . . . . . . . . . . . . . . . . . . . . . . 113 List of T abl es 4.1 Impact of c hanging the num b er of elemen ts in ULA on th e p erformance of Ro ot-MUSIC algorithm with settings ( M =2, θ = 10 o and − 10 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Impact of changing th e n umber of elemen ts in ULA on the p erformance of ES - PRIT algorithm with settings ( M =2, θ = 10 o and − 10 o , d=0.5 λ , SNR=10dB and K = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Impact of c hanging the num b er of incident signals imp inging ULA on the p er- formance of Ro ot-MUSIC algorithm with settings ( N =6, d =0.5 λ , SNR=20dB and K = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Impact of c hanging the num b er of incident signals imp inging ULA on the p er- formance of ES PRIT algorithm with settings ( N = 6, d =0.5 λ , SNR=20dB an d K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 4.5 Impact of changing the angular separation b et wee n the in cident signals im- pinging on ULA on the p erformance of Ro ot -MUSIC algorithm with settings ( N =3, d =0.5 λ , SNR=20dB and K =100) . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Impact of c hanging the num b er of samples of the incident signals impinging on ULA on the p erform an ce of Ro ot-MUSIC algorithm with settings ( N =5, θ =20 o and − 20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . 79 4.7 Impact of c hanging the num b er of samples of the incident signals impinging on ULA on the p erform ance of ESPRIT algorithm with settings ( N =5, θ =20 o and − 20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . 79 4.8 Impact of c hanging SNR for ULA on the p erform an ce of Ro ot-MUSIC algo- rithm with settings ( N =5, θ = 20 o and − 20 o , d= 0.5 λ and K =100) . . . . . . . 80 xiv List of T ables xv 4.9 Impact of c hanging SNR for ULA on th e p erformance of ESPRIT algorithm with settings ( N =5, θ = 20 o and − 20 o , d= 0.5 λ and K =100) . . . . . . . . . . 81 4.10 Im p act of c hanging the num b er of elemen ts in UC A on the p erformance of Ro ot-MUSIC algorithm with settings ( M =2, θ =20 o and − 20 o , θ e =20 o , d =0.5 λ , SNR= 10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.11 Im p act of c hanging the num b er of elemen ts in UC A on the p erformance of ESPRIT algorithm w ith settings ( M =2, θ =20 o and − 20 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.12 Im p act of c hanging the num b er of incident signals imp inging UCA on the p er- formance of Ro ot-MUSIC and ESPRIT algorithms with settings ( N =5, θ e =20 o , d =0.5 λ , SNR= 10dB and K =100) . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.13 Im p act of c hanging th e angular separation b et we en the incident signals im- pinging on UCA on the p erformance of Ro ot-MUSIC algo rithm with settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . 90 4.14 Im p act of changing the angular separation b etw een the incident signals im p ing- ing on UCA on the p erformance of E SPRIT algorithm with settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . . . . . . . . 90 4.15 Im p act of c hanging the num b er of samp les of the incident signals impinging on UCA on the p erformance of Ro ot-MUSIC algorithm with settings ( N =5, θ = 30 o and − 30 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . 91 4.16 Im p act of changing the num b er of samples of th e incident signals impin ging on UCA on the p erf orm ance of ESPRIT algorithm with settings ( N =5, θ = 30 o and − 30 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) . . . . . . . . . . . . . . 92 4.17 Im p act of changing SNR for UCA on the p erform ance of Ro ot-MUSIC algo- rithm with settings ( N =5, θ =30 o and − 30 o , θ e =20 o , d =0.5 λ and K =100) . . . 93 4.18 Im p act of c hanging S NR for UCA on the p erformance of ESPRIT algorithm with settings ( N =5, θ =30 o and − 30 o , θ e =20 o , d =0.5 λ and K =100) . . . . . . . 94 4.19 No d es lo cation on the environmen t . . . . . . . . . . . . . . . . . . . . . . . . . 97 Ac k no wledgemen ts W e w ould like to thank our thr ee academic su p ervisors: Prof. Raed S h ubair, Dr. Luis W eruaga, and Dr. Ahmed AlT una ji, for their tec hn ical help, v aluable advice, and constan t supp ort. Our main pro ject sup er v isor Pr of. Raed S h ubair h as b een a tremendous mentor and an exceptional m otiv ator. He kept push ing us b eyo nd th e limits of a normal senior design pro ject exp ectation. His timeless h elp and relen tless guidance were essen tial for ac hieving the new results and con tributions of this pro ject. Prof. Rae d Shubair is the one who mov ed our w ork to the inte rnational academic frontie r when he pushed us and taugh t us ho w to write eloq u en t tec h nical pap ers wh ic h s u bsequently got accepted for present ation at tw o prominent conferences namely IEEE ICTRC20 15 in Abu Dhabi, UAE and IEEE APS /URSI2015 in V ancouv er, Canada. F urthermore, Prof. Raed Shubair gu id ance and coac h ing were essentia l for us in winning the 1st place in th e IEEE ICCSP A15 P oster Comp etit ion. Our co-sup ervisor Dr. Luis W eruaga has b een in strumenta l in providing us with d eep insigh ts into statistical estimation theory and tec hniques. W e wo uld like to thank h im for h is patience whenever w e had any doubts on the theoretical asp ects of the algorithms. Our co-sup ervisor Dr. Ahmed AlT unaiji help ed u s signiﬁcan tly at the early stages of the pro ject. He engaged w ith us in numerous discussions which made us grasp very well the fundamental concepts of wireless sensor net w ork lo calizat ion. He coac hed us on the use of the professional tec h n ical writing soft w are Latex, and help ed us in dev eloping v arious building blo c ks w ithin ou r Matlab co de. This pro ject w ould not ha v e b een completed withou t the generous supp ort from Khalifa Univ ersit y whic h pr ovided us with a d istinctiv e edu cational exp erience. S p ecial thanks are extended to the senior d esign pro ject co ordinator Dr. Kahtan Mezher, as w ell as to our examiners Dr. Sh ih ab J imaa and Dr. Nazar Ali. xvi Ac knowledgement s xvii Finally , w e wo uld lik e to thank our own families, and esp ecia lly ou r paren ts eac h. Their endless lo ve and coun tless sacriﬁces we re the d riv e for all the s u ccesses and distinctions we ac h ieved in this pro ject. Chapter 1 In tro duction 1.1 Goals and Ob jectiv es 1.1.1 Goals 1. S urvey on recent adv ances in lo calizati on for WSNs. 2. I mplemen tation of selecte d algorithms in MA TLAB. 3. P rop osed changes to selected algorithm f or an im p ro v ed p erformance. 4. T esting, veriﬁcatio n, and v alidation of results. 1.1.2 Ob jectiv es 1. L earn ab out r ecen t adv ances in lo calization in WSNs. 2. Un derstand diﬀerent p ractical applications and limitations for localization. 3. L earn ab out simulation pr ograms suc h as MA TLAB. 4. S im ulate v arious RSS and DO A tec hn iques. 5. E nhance WSNs lo calization accuracy and robustness using R S S and DOA technique. 6. I n v estigate diﬀerent p r e-pro cessing sc hemes to de-correlate coheren t signals in DOA mo del. 7. Design a hybrid sys tem by com bining RSS with DOA mo del. 1 Chapter 1. Introduction 2 8. I mpro v e the total p erformance b y pro viding diﬀerent mo diﬁcatio n for the algorithm. 9. Bu ild a MA TLAB-based GUI to facilitate the sim ulation of the d ev elop ed lo calizatio n system. 1.2 Motiv ation WSNs refer to net wo rks in v olving spastically disp ersed sensor no des that comm unicate b e- t w een eac h other thr ough wireless c hannels. Generally , the senor no des ha v e limited capabilit y in terms of lo w p ow er consumption, radio transceiv er and small-size memory . Due to th e con- tin uous decrease of b oth cost and size of senor no des, it b ecomes eﬃcien t to utilize such no des to construct a large-scale WSN. A WSN can b e used for area monitoring lik e in military observ ation wh ere the senor nod es are spr ead o ver a wide area to detect an y p ossible intru- sion from enem y . In suc h a scenario, the spatial inf ormation obtained from sensors no des will b e inte rpreted w rongly unless w e kno w the resp ecti v e lo cations of these no des. F or this reason, the lo calization pr o cess b ecomes an imp ortan t asp ect of a WSN and, ov er the recen t y ears, researc hers atten tion is f o cused on ho w w e can imp ro v e the accuracy of the lo calization pro cess. Applications of WSN include healthcare monitoring, wat er qualit y monitoring and forest ﬁre d etectio n where th ese app lications are unattainable un less we are capable of routing the information in the netw ork [1]. Sp eciﬁcally , lo ca lization is deﬁned as the abilit y to assign physica l co ordinates to un- kno wn no des in a n et work. The b asic idea b ehind lo calizati on is that some nod es, so-called anc hor no des, in the net wo rk must hav e known co ordin ates. T he anchor n o des trans m it their co ordinates to u nkno wn no des to assist them in lo calizing themselves. The assignmen t of co ordinates to anc hor no d es can b e done through either hard co ding or equipp ing them with Global P ositioning System (GPS). The former metho d is impractical as it preve nt the r an - dom d isp erse of no des in an area and do es not tak e in to accoun t th at a sen sor can c h ange its location with time as in military area where wind can mov e a sensor. The latter metho d is exp ensiv e and is inaccurate in indo or areas wh ere GPS signals b ecome inaccurate due to obstacles lik e trees and buildin g. Besides, GPS demands signiﬁcant p o w er wh ic h ma y cause battery failure for no d es that do es n ot rely on extern al p o w er su p ply . Lo calization is carried out using v arious lo calization tec hn iques such as trilateration and triangulation whic h utilized Chapter 1. Introduction 3 the neigh b our anc h or n o des lo cations to determine the lo cations of u nkno wn no des [2 ]. 1.3 Applications Recen tly , Wireless sensor net w orks ha v e gathered the world’s attent ion b ecause of their di- v erse applications in military , environmen t, h ealthcare applications, habitat mon itoring and industry [3]. In these d iﬀeren t applications, th e need for the lo cation information is essentia l. The pro cess of collect ing information inv olv es placing man y diﬀeren t typ es of sensors on the sensor no d e su ch as mec hanical, biological and thermal sensors to ev aluate th e en vironment s prop erties [4]. These sensors sh ould b e p laced accurately to get the correct in formation to b e sensed [3]. 1.3.1 Health Systems Health systems based on WSNs are m ean t to provide accurate information ab out emergency situations and sen d r ecords ab out the desired data. Th is is useful esp ecially for elderly as sensors can send information ab out their h ealth condition and th u s monitor their acti vities, whic h w ould help them in their daily life [5]. Caregive rs Assistant and CareNet Displa y pro ject is an example to monitor the elderly . In this pr o ject , sensors are placed on eac h home ob ject, thus, a record of the elderlys mov ement s and actions is r ecorded [3]. F or the patien ts in th e hosp itals, sensors give cont inuous inf ormation ab out their b o d y , wh ich h elp do ctors to cure the illness in its early stage and p rev en t serious health issu es. F or ins tance, Co deBlue pro ject has man y sensors that sense the blo o d p ressure and heart rate [3]. Its conﬁ gu r ation is sho wn in Figure 1.1. Also, this p r o ject, hop efully , will help to cure patien ts with diﬀerent diseases [5]. Moreo v er, d o cto rs will b e able to lo cate th eir p atien ts and their colleagues in the hospital by using b o dy sensors net work. In this w ay , patient s will b e under full con trol an d other d o cto rs will b e able to delive r the required assistance quick er [3]. These health systems are not limited to patien ts only; infant s, also, ha v e sp ec ial systems. F or example, S leep Safe is designed to monitor th e sleeping p osition for the in fan t and alert the paren t if their baby sleeps on his stomac h. This system has tw o sensors, one is placed on the b ab y’s clothing to detect the baby’s p osit ion with resp ect to gra vit y and the other one is connected to compu ter. After collect ing the required data, the computer p ro cesses these data u p on stand ards that Chapter 1. Introduction 4 are set b y the user and, consequently , the bab ys p ositio n will b e known. Moreo v er, the Baby Glo ve is a system that will measure the bab ys temp erature, hydratio n and pulse rate. A sensor is attac hed to the swaddling wrap to send these data to a computer. This computer pro cesses this data and d etermines if there is anything, whic h is abnorm al and alerts the paren t [4]. 1.3.2 En vironmen tal Monitoring WSNs h a ve an imp ortant role in the en vironmen tal side. It can b e used to monitor envi ron- men tal prop ertie s suc h as temp erature and pressure. Many applications are in monitoring plan ts and anim als, ﬁ re d etectio n and natural c hanges suc h as v olcanic eruption and ﬂ o o ds are based on WSNs [3]. Rare sp ecies of sea birds can b e monitored by inserting sensors in their nets. T hese sensors can measure the humidit y , temp erature, and the pressu re. Thus, the researc her s will kno w the sp eciﬁc lev els that aﬀect th ese bird s. This is the aim of Great Duc k Island Pr o ject . Animals suc h as Zebr as can b e trac ked. Their m ov es and b ehaviors are monitored to stud y th e eﬀects of h uman in terference and h ow the Z ebras in teract with other sp ecies [3, 5]. Hence, aroun d the zebr a nec k, a sensor no de is placed [5]. This no d e consists from seve ral comp onents GPS u n it, long and sh ort range r adio transmitters, micro controlle r, solar array and battery [3]. This is th e main purp ose of Z ebraNet pr o ject [3 , 5]. F or m on itor- ing the p lan ts, com binations of sensors are used to sense the h umidit y , temp erat ure, r ainfall and sunlight. Also, all these no d es are equipp ed with camera to tak e images. After that th ese images are ins erted in the rare plan t sp ec ies, that to b e mon itored, and in the s urround ing area wh er e these sp eci es do es n ot exist. All th ese data will help the researc hers in PODs pro ject to unders tand the conditions that aﬀect these r are plan ts sp eci es [3]. In d etecting Figure 1.1: Co deBlue pro ject conﬁguration Chapter 1. Introduction 5 forests ﬁre, the no d es con tain envi ronmental sensor that can measure temp erature, ligh t in- tensit y , barometric pressure and h umidit y . T his will giv e indications ab ou t the p ossibilit y of ha ving ﬁre in the real time. Thus, this will sa v e p eople’s life and their prop erties [3]. F or v olcanic eruptions, WSNs plays an imp ortan t r ole b ecause it d o es not require riskin g p eo ples life. The sensors in this applicatio n should provi de reliable data at the ev en t time and with high d ata rates [4]. Combinatio ns of microphone and seismometer sensors th at collect seismic and acoustic data are used to m on itor the volca no [3]. An example of th ese sensors is shown in Figure 1.2. Flo o ds can b e predicated using WSNs. Th is will help to sa v e p eo ple and reduce losses. Sensors that can m easure the wa ter ﬂo w, rainfall and air temp erature is used giv e information ab out the environmen t [3]. 1.3.3 Home Applications WSNs in h ome applications hav e systems that can sp ot human presence, control the air conditioning and the ligh ts without the need for the human in terv en tion. Thus, it is essential in these app lications to h a ve small size sensor no de, so it can b e placed easily on h ome ob jects [3]. Non-in tru s iv e Autonomous W ater Monitoring S ystem (NA WMS) is a pr o ject that aims to iden tify the w astage in w ater usage for eac h s ou r ce alone. The b eneﬁt is that eac h pip e in the home will b e monitored prop erly and with minim um cost. The pro ject w orks on the p rop ortionalit y b et we en the vibration of the pip e and the wa ter ﬂ o w. This vibration is determined b y ins erting accelerometers to the wa ter pip es. Figure 1.3 sho ws the conﬁgur ation for NA WMS. The sensor no des are connected to the m ain w ater meter to compute the in formation ab ou t the wa ter ﬂ o w and send it to every no de. The main purp ose of these sensors is to measure the vibration for eac h pip e. Then, this vibr ation is routed to the computation no de, so the sensor will b e calibrated an d the wa stage will b e sp eciﬁed. The computation no de uses an op timized m etho d to calibrate the sensors, in the sense that the addition of the wate r ﬂo ws in ev ery non-calibrated sensor should b e equal to w hat in th e wa ter meter [3]. Also, Smart bu ilding is one of the applications for WSNs. Inside this b uilding, sensors and actuators are placed. Th ese conﬁgur ations control , monitor, and impro v e th e living conditions and allo w the r eduction in the energy consum ption, whic h can b e done b y con trolling the airﬂo w and temp erature [5]. Chapter 1. Introduction 6 Figure 1.2: Sen s ors for detecting vo lcanic eruptions Chapter 1. Introduction 7 Figure 1.3: The conﬁgur ation for NA WMS 1.3.4 Infrastructure Health Monitoring WSNs are u s ed to pr edict the h ealth of diﬀerent infrastructur e suc h as b ridge. Sens ors measure the am bien t structural vibrations w ith ou t aﬀecting the b ridges op eration. This monitors the bridge and, h ence, the app ropriate pr o ce dur es are tak en when any p roblem o ccurs [3 ]. 1.3.5 In telligen t T ranspor tation In this application, a traﬃc monitoring system pr o vid es b etter safety for road u s ers and prev en ts acciden ts. The sen sors gather data ab out the v ehicles direction and sp eed, moreo v er, the num b er of the v eh icles that are in the road. This information h elps to pr o vide safet y w arnings on th e roads [3 ]. 1.3.6 Searc h and R escue op eration Searc h and rescue op eration is one of many jobs that pu t the p erson life on the line. WSN oﬀers impr o vemen t to the Search and rescue op eration that w ill reduce the risk faced by rescuers. WSN can h elp in ﬁn ding the victims lo cation and the area of the catastrophes [3]. Chapter 1. Introduction 8 1.3.7 Smart Campus An in telligen t Campu s is called iCampu s that improv es and transforms the end-to-end life cy- cle of the kno wledge of ecosystem wh ic h can b e done b y using the WSNs. The intellig ence can app ear in t w o paths either in th e sen sor no des itself or in the co op er ative inform ation net w ork system or b oth of them. This system do es not only supp ort the students b ut it pro vides with new tea c hing style and expands th e managemen t capabilities for the admin istration. T here are many examples for th e iCampu s, such as monitoring the students by the m ean of sen s ory smart card. This card allo ws the student to use diﬀeren t services a v ailable in the universit y . Also, it h elps the managemen t to study their b ehavio rs and actions [3]. 1.3.8 Military applications Since WSNs localize, iden tify and trac k the lo cation of the desired target. Thus, it pro vides the s oldiers or m ilitary tro op with inf ormation ab out the enem y orienta tion and lo cation. This is th e main concept of the VigilN et wh ic h is surveill ance system. This is used to tac k the ob jects in the enem y’s territories. Figure 1 shows the netw ork for this system. Moreo ve r, another application is the counter snip er system whic h can detect the sho oters lo cation. This system has tw o diﬀeren t arc hitectures. The ﬁrst one is Boomerang system which is a group of microph on es that detect the ﬁre by pro cessing the audio signal from the m icroph ones. This problem in this system is that the lac k of capabilit y in the case of m ulti-path in sound detection. T o solve that, a distribu ted n etw ork comp osed of acoustic sens ors is used, sho wn in Figure 2. This works on the concept of th e centrali zed concept, discussed later, whic h uses group of sensor to get information ab out th e time and lo cation to id entify the bullet [3 ]. PinPrt is an ad hop acoustic s ensor net wo rk that lo cates th e snip er. This n et work s en ses the acoustic sho c k wa v e that results f rom the sound of gun ﬁre. T h e arriv al times at sensor no des will estimate the s nip ers lo cation [5]. Also, the mineﬁeld is monitored using a self-organizing sensor net w ork in wh ic h there is a p eer to p eer comm un ication b et w een an ti-tank mines, so react to any attac ks and c h anges the place of the mines. This imp edes the enem y mo v emen t [5 ]. Large sen s or netw ork can b e used to r ecord any attempt fr om in v aders to pass the count ry b oarders illegally . These sensors must h a ve camera to record an y m otion [3]. Chapter 1. Introduction 9 1.4 Challenges Despite the v ariet y of applications that are based on WSNs, researc hers encounte r man y issues and c hallenges suc h as physical la yer measur ement errors, compu tation constrain ts, lo w-end sensor no d es, net w ork sh ap e, no d e mobilit y , and lac k of GPS data. There are diﬀeren t tec h niques u sed in WSNs to iden tify the p osition of the unk n o wn no de based on the signal strength and signal direction, which can b e con v erted to distance and orien tation measuremen ts, resp ec tiv ely . These measuremen ts hav e some errors due to m ultipath and shado wing, as shown in Figure 1.4. F or example, if th e signal is transmitted from the anchor A to the no d e B directly , the no d e B will b e able to compute its lo cation by estimating distance b et w een A and B and this will b e A, as shown in Figure 1.4 (a). Ho wev er, the r eal path b et we en the transmitter and the receiv er will not b e clear. Man y problems arise across this path su c h as multipat h and sh ado wing [3]. In multipat h, multiple copies of the transmitted signal from v ariet y of paths combines in either constructiv e or d estru ctiv e manner whic h aﬀect the receiv ed signal. T he receiv er m ust sp ecify the ﬁrst-arrivin g p eak. This is implemen ted by measuring the time that the cross-correlation ﬁrst crosses the threshold [6]. In the shadowing, the signal is aﬀected b y diﬀeren t obs tacles from the surr ounding en vironmen t suc h as w alls, b uildings and trees. The signal d iﬀracts along diﬀerent paths b et w een the transmitter and receiv er [7]. Therefore, in the case of m ultipath and shadowing, n o de B will not b e able to get th e accurate estimation with r esp ect to no de A, th is case is sho wn in Figure 1.4 (b). This diﬀeren ce will result in high errors in the estimated p osition [7]. Hence, it is required to app ly d iﬀeren t tec h niques to compute the d istance measur emen ts and reduce the errors [6]. Other attempt is to use av eraging tec hn iques to correct th e errors, but it is not desired, so it is b etter to not tak e into acco unt th ese measurements with errors [3]. The computation constrains are the source limitations. No d es ha ve restricted amoun t of p o w er, small memory and small computations pro cessors. Th us, m ost of the r anging tec hniqu es ma y not b e done accurately . An attempt In order to solv e this problem is to use cen tralized system b ecause it can do large computation. On the other h an d , these systems increase the comm u nication o verhead. Alternativ e option is to u s e simple an d distrib uted algorithm. This algorithm sh ows reduction in the energy p o w er and pro cessing [3]. T o get the lo cation information of the no d e the GPS is u sed. Actually , the GPS is exp ensiv e and has high en ergy Chapter 1. Introduction 10 Obstacle (a) Ranging measurements with line-of-sight communication (b) Ranging measurements with non-line-of-sight communication A B X B A d Figure 1.4: Th e eﬀect of n on-line-of-sigh t and multi-path on ranging techniques consumption. Thus, it is used in the anchor no d es only [3]. Moreo ver, th e netw ork shap e pla ys an imp ortan t role in the lo calization pr o cess. F or example, if the no d e is at the b order of the net w ork or n ot in th e con v ex b o dy , th ere will b e few measurements; thus, the p ositio n w ill not b e estimated correctly [3]. No de mobilit y is one issue th at should b e considered in the lo calizatio n pro cess. The ﬁr s t scenario is w hen the anc hor n o des are mo ving and the sensor n o des are in ﬁ xed p ositio n. This is an example in military in w hic h th e anc hor no des are placed to soldiers wh o scan the place and sensor no des are static and d istributed in the battleﬁeld. This will improv e the lo cation accuracy of the sensor no des. The second scenario is when the anc hor no des are in ﬁxed p ositio ns while the sensor n o des are mo vin g. F or examp le, th e anc hor no des are at the t w o sides of the river and sensor no des are in the riv er . Th e last s cenario is when b oth types of no des are mo ving su c h as in the w ind [3]. The ﬁnal c hallenge is f or the wireless sensor n o des comp onent s is the qu alit y of s en sor n o de comp onent s. In the sensor no d es, lo w-end comp onen ts or hard w are measur emen t devices are placed b ecause they reduce the system cost. These comp onen ts allo w error in distance or orien tation estimation, which is added to the error pro du ced b y the channel. T o s olv e this issue, diﬀeren t m easur emen ts can b e tak en from v arious neigh b ors in diﬀerent time slots to get go o d lo calizatio n accuracy . I n addition, these low-end comp onen ts m a y in tro duce temp orary or p erm an ent n o de failure [3 ]. Chapter 1. Introduction 11 1.5 Ac hiev emen ts and Publications 1.5.1 Ac hiev emen ts • P articipation in R T A inno v ation week • Winning the ﬁrs t p lace in the IEEE In ternational Conference on C omm unications, Signal Pro cessing, and their Applications (IC C SP A’15) studen t p oster comp etition • P articipation in the IEEE UAE student da y 1.5.2 Publications • M. I. AlHa jri, R. M. Shubair, L. W eruaga, A. R. Kulaib, A. Goian, M. Darw eesh, R. AlMemari,Hybrid Metho d for En hanced Detection of C oheren t Signals u s ing Cir- cular Antenna Arra ys, IEE E Int ernational Symp osium on An tennas and Propagation (APS’15), 2015. • M. I. AlHa jri, A. Goian, M. Darw eesh, R. AlMemari, R. M. Shubair, L. W eruaga, A. R. Kulaib, Hyb rid RSS -DO A T ec hn ique for Enhance WSN Lo calizat ion in a Corre- lated Environmen t, IEEE In ternational Conf er en ce on In formation and Communication T ec hnology Researc h, 2015. 1.6 Ov erview of the Rep ort The rest of r ep ort is arr an ged as follo ws. A generalized surve y ab out the localization tec h- niques is explained in C hapter 2. Also, Chapter 2 discus s es th e lo calization algorithms wh ic h is divided in to cen tralized and distrib uted algorithms. C hapter 3 tac kles in depth the for- m ulation of R S S m etho d and its s im ulation results. Chapter 4 studies the form ulation of Direction of Arriv al (DO A) metho d as theory and simulati on of Uniform Lin ear Arra y (ULA) and Uniform Cir cular Arra y (UCA). In add ition, C h apter 4 studies the formulation of Direc- tion of Arriv al metho d (DO A) in case of u ncorrelated signals while in tro duces pre-pro ce ssing tec hn iques to deal with correlated signals. In add ition, C hapter 4 compares the p erformance of diﬀeren t DO A algorithms based on environmen tal and arra y-related parameters. C h apter 5 exp lains in depth th e form ulation of diﬀeren t hybrid tec hn iqu es and its sim u lation resu lts. Chapter 1. Introduction 12 Chapter 6 describ es the MA TLAB-based GUI that facilitate the simulati on of the develo p ed lo calizat ion system. Lastly , Chapter 7 concludes our rep ort with su m mary of w ork done an d the exp ected future wo rk to b e accomplished. Chapter 2 Surv ey of WSN Lo calization In this c h apter, several localization d isco very and ranging tec hn iques are discussed for esti- mating the un kno wn no de [8]. In addition, a general classiﬁcation for lo caliza tion is p resen ted. 2.1 Lo cation Disco v ery T ec hniques There are thr ee basic lo calizati on disco very tec hnique; T r iangulation, T rilaterati on, and Mu l- tilaterati on. 2.1.1 T riangulation T riangulation exp loits the geometric p r op erties of triangles in order to appr o xim ate the lo- cation of the un kno wn no de. Pr ecisely , triangulation requires at least t w o angles from th e reference no des and their resp ect iv e locations to compute the un kno wn no de p osition [9]. Figure 2.1 demonstrates the op eration of triangulatio n where the unknown no de will lie on the in tersection of the three b earing lines (the line from the anc h ors to the no de). Two Figure 2.1: T riangulation 13 Chapter 2. Survey o f WSN Lo caliza tion 14 anc hor with locations p i = [ x i , y i ] T and measured angles θ i ( x ) = [ θ 1 ( x ) θ 2 ( x )] T (expressed relativ e to a ﬁxed v ertical line as in the Figure 2.1), then, the lo cation of the unkno wn n o de p s = [ x s , y s ] T is determin ed from the relationships: tan ( θ i ( x )) = y i − y s x i − x s (2.1) F rom equation 2.1, we got a system of t w o equations and tw o un kno wns w hic h can b e solved to obtain the unkno wn n o de p osition. In realit y , the measured angles will not b e the exact angles due to the pr esen t of noise. Thus, the equation of the real angles ( B ) is giv en as: B = θ ( x s ) + δ ( θ ) (2.2) where δ ( θ ) is a noise mo d eled as Additive Gaussian White Noise (A GWN). The er r or in the measured angles will preven t the b ea ring lines from intersec ting into a single p oin t leading to o ver-determined system. Solving suc h system will require the us e of statistic approac h lik e Maxim u m-lik elihoo d or least- square metho d. 2.1.2 T rilateration Unlik e tr iangulation, trilateration lo cates a no de p osition b ased on the distances measured from three referen ce p oint s with kno w n lo cations. In t w o-dimensional s p ace, trilateration requires at least thr ee-distance measurement from n on-collinear anc hors to obtain a sin gle lo cation (the in tersection of thr ee circles) as showing in Figure 2.2. Ho wev er, in r ealit y 3-D sp ace, w e will d emand at least four non-coplanar anc hors to solve for the unknown lo cation [10]. Assu ming that we ha v e N an chor n o des with lo cations p i = [ x i , y i ] T ( i = 1 · N ) and distances b et we en the unk n o wn no des p osition p s = [ x s , y s ] T and the anc h ors are d i ( i = 1 · N ). Using these information, we can express the anc h or-no de relationship usin g this system of equ ations:          ( x 1 − x s ) 2 + ( y 1 − y s ) 2 ( x 2 − x s ) 2 + ( y 2 − y s ) 2 . . . ( x N − x s ) 2 + ( y N − y s ) 2          =          d 2 1 d 2 2 . . . d 2 N          (2.3) Chapter 2. Survey o f WSN Lo caliza tion 15 Figure 2.2: T rilaterati on After simp lifying the ab ov e system, we can express it as a linear system: Ap s = b (2.4) With co eﬃcien ts: A =          2( x N − x 1 ) 2 2( y N − y 1 ) 2 2( x N − x 2 ) 2 2( y N − y 2 ) 2 . . . . . . 2( x N − x N − 1 ) 2 2( y N − y N − 1 ) 2          (2.5) b =          d 2 1 − d 2 N − x 2 1 − y 2 1 + x 2 N + y 2 N d 2 1 − d 2 N − x 2 2 − y 2 2 + x 2 N + y 2 N . . . d 2 1 − d 2 N − x 2 N − 1 − y 2 N − 1 + x 2 N + y 2 N          (2.6) In real-time, there is an er r or in the distance estimation which prev en ts the circles fr om in tersecting in a single p oint. Therefore, in this scenario, we will hav e to solv e the linear sys tem to obtain the b est ﬁtting solution using estimati on method s like least squ are ap p roac h . Th e no de p ositio n p s = ( x s , y s ) T from the least square is d etermin ed using: p s = ( A T A ) − 1 A T b (2.7) Chapter 2. Survey o f WSN Lo caliza tion 16 The error in the anc hor p ositio n and lo cation can b e presen ted u sing Gauss ian distr ib ution with zero-mean as: w i = 1 q σ 2 position i + σ 2 distance i (2.8) where σ 2 distance i represent s the v ariance in the distance b et wee n the no de and th e anchor wh ile σ 2 position i , is th e v ariance in the anc hor locations will con tains b oth v ariance as: σ 2 position i = σ 2 x i + σ 2 y i (2.9) The new coeﬃcient of the lin ear system Ap s = b b ecomes: A =          2 w 1 ( x N − x 1 ) 2 2 w 1 ( y N − y 1 ) 2 2 w 1 ( x N − x 2 ) 2 2 w 1 ( y N − y 2 ) 2 . . . . . . 2 w 1 ( x N − x N − 1 ) 2 2 w 1 ( y N − y N − 1 ) 2          (2.10) b =          w 1 ( d 2 1 − d 2 N − x 2 1 − y 2 1 + x 2 N + y 2 N ) w 1 ( d 2 2 − d 2 N − x 2 2 − y 2 2 + x 2 N + y 2 N ) . . . w 1 ( d 2 N − 1 − d 2 N − x 2 N − 1 − y 2 N − 1 + x 2 N + y 2 N )          (2.11) 2.1.3 Multilateration Both of t wo p revious p ositioning tec hniques are limited to the need of the presence of at least 2 anc h ors (for triangulation) and 3 anc hors (for trilateration) to compute the u nknown no de lo cation. Ho w ev er, this problem can b e o v ercome b y turnin g the no de that iden tiﬁed their lo cations into anchor b r oadcasting their lo cations to the all surroun ding no des. This tec hn ique is kno w n as iterativ e m u ltilaterati on and it con tinues u n til all no d es in a n et work ha v e b ee n lo calized [11]. Figure 2.3 d emonstrate th e iterativ e multilate ration where th e grey no de uses th e blac k anc hor to estimates its lo cation. Then, the grey no d e b ecomes an anchor to assist the tw o w hite n o des to estimate their p ositions with the help of the original an chors. The d isadv an tage of this metho d is that the lo caliza tion error accum ulates with eac h iteration executed. In case a no de is n ot su rround ed by three anc h ors , then, it can still estimate its Chapter 2. Survey o f WSN Lo caliza tion 17 p osition by using a metho d called collab orativ e multilat eration [11]. Figure 2.4 illustr ates this metho d where we ha v e four anc h ors and tw o no des participating all together to determine the t w o no d es lo cations. A no de will estimate its lo cation by solving a system of o v er-determined quadratic equation, whic h relates its lo cation to the other n eigh b ors. 2.2 Ranging T ec hniqu es Sev eral lo calization tec hniqu es that are u sed to estimate the distance or an gle to localize the p osition of sensor no des ha ve b een prop osed in the literature. They include Time of Arr iv al (TO A), Receiv ed Signal Strength (RSS), Radio Hop Count, and Direction of Arriv al (DOA ). 2.2.1 Time of A rriv al (TOA) T echnique One of the sim p lest tec hniques to measur e the distance b et wee n no des is TOA. TOA w ill utilize the p rop ortionalit y relationship b et w een the time and distance to estimate separation b et w een th e t wo no des [12]. A signal will b e sent by a n o de at time t 1 and it reac h ed by the receiv er no de at t 2 , the d istance b etw een the t w o n o des is: d = s r ( t 2 − t 1 ) (2.12) where s r is the propagation sp eed of the radio signal. Ho w ev er, for this tec hnique to w ork accurately the tw o clo c ks at the transmitter and receiv er needed to b e sy n c hronized pr ecisely whic h is quite diﬃcult to ac hiev e in a p ractical environmen t. T o o vercome this limitation, Round-trip Time of Arriv al (R T oA) and Time Diﬀerence of Arr iv al (TDoA) are dev elop ed [13]. R TOA is quite similar to T OA b ut it do es not require the sync hronization b et w een the sender and receiv er n o des. It works by recording the time of transmission t t at n o de A by its Figure 2.3: Iterativ e Multilat eration Chapter 2. Survey o f WSN Lo caliza tion 18 Figure 2.4: Collab orative Multilateratio n o w n clo c ks , then the signal is receiv ed by no de B wh ic h after a s p eciﬁc time p erio d t per it will send the signal bac k and the reception time t r is recorded at no de A. Then th e distance is: d = s r t t − t r − t per 2 (2.13) TDO A tec hniques can b e classiﬁed in to tw o main typ es: multi-nod e TDO A, and multi- signal TDO A [14]. The multi-nod e TDO A is where the diﬀerence in time at which the single signal from a single no de arriv es at three or more no des resulting in tw o hyp erb oloids [15]. These t w o hyp erb oloids will intersect at a single p oin t, which is the lo cation of the unkn o wn n o de as sho wn in Figure 2.5. While th e multi-sig nal no de measures the diﬀerence in timings at whic h m ultiple signals from a sin gle no d e arr iv es at another no d e as sho wn in Figure 2.6. T o ac h iev e this, no des m u st b e equipp ed with extra hardware to send tw o signals sim ultaneously . The ﬁrst signal usually tra vels at sp eed of ligh t (3 × 10 8 m/s ) and the second signal at the sp eed of sound (340 m/s ).The distance is: d = ( s r − s s )( t 2 − t 1 − t delay ) (2.14) where s r = radio signal sp eed, s s = sound signal sp ee d, t 2 = arriv al time of sound, t 1 = arriv al time of light Chapter 2. Survey o f WSN Lo caliza tion 19 S A B C TDoA (B-A) TDoA (C-A) Figure 2.5: Multi-no d e TDO A A B R a d i o Sound t delay t delay t sound d / (s radio – s sound ) Figure 2.6: Multisignal-TDO A Chapter 2. Survey o f WSN Lo caliza tion 20 2.2.2 Receiv ed Signal Strength (R SS) T echn ique RSS is a common tec h nique in lo ca lizing sensor no des; this is d ue to the fact th at almost all no des ha ve the capabilit y to measure the strength of the r eceiv ed signal. T he distance is estimated b etw een t w o no des based on th e strength of the signal receiv ed b y the target n o de, transmitted p o wer, and the path-loss mo d el [13]. The op eration starts when an anchor no de sends a signal that is receiv ed by th e receiv er and passed to the Receiv ed Signal Strength Indicator (RSSI) to d etermin e the p ow er of the receiv ed signal [15]. Then the distance can b e calculated using the equation: P R = P T − 10 η lo g( d ij ) + X ij (2.15) where P T is a constant due to transmitted p o we r and the ante nna gains of the sensor n o des, η is the atten uation constan t, and X ij is the u ncertain t y f actor due to multipath and shadowing. The RSS measur emen t from three anc h or n o des can b e com b ined w ith their lo cations (trilateratio n) to estimate the lo cation of the n o de. 2.2.3 Radio Hop Coun t T echn ique In Radio Hop Coun t T echnique if t w o n o des can communicate with eac h other then the distance b et w een them is less than R, wher e R is the maximum range of the radios. Then to calculate the distance b et w een tw o n o des th ree main steps need to b e done [16]. First, calculate the minimum n umb er of hops b etw een th e un kno wn no d e and eac h anchor no de. Then a v erage the actual distance of one h op using the follo wing equation: H opS iz e i = P p ( x i − x j ) 2 + ( y i − y j ) 2 P h ij (2.16) where ( x i , y i ),( x j , y j ) anc hor n o des, h ij n umber of h ops b et wee n anc hor no des Then th e d is- tance b et ween the u nkno wn no d es an d anchor no d es is calculated usin g the follo wing equation: d = H opS iz e i × h ik (2.17) Chapter 2. Survey o f WSN Lo caliza tion 21 2.2.4 Direction of A rriv al (DOA) T ec hnique In this tec hnique the directions of n eigh b ouring sensors rather than the distance to neigh b our - ing sensor is estimated. DOA can b e classiﬁed in to t wo tec hniques sensor arra y an d d irectional an tenna [ ? , ? , ? , 17–52, 54–57]. The sensor array is comprised of tw o or more ind ividual sensors (microphone or ante nnas) as shown in Figure 2.7. T he DOA will estimate the diﬀerence in the arriv al timings. T he angle of arr iv al is calculated using the follo wing equation: ∆ t = δ co s( θ ) y (2.18) where ∆ t is the d iﬀerence in arr iv al time, δ is the antenna separation, θ is th e angle of arriv al, and y is the v elocity of the RF or acoustic signal In the case of d irectional antenna, it op erates b y calculating the RSS r atio b et we en sev eral directional ante nnas. Figure 2.7: Sensor arr a y pro cessing 2.3 Lo calization Classiﬁcation A lo calization algorithm is a term that refers to the pro ce ss or set of ru les to b e follo w ed in calculations or other p roblem-solving op erations, esp eci ally b y a computer to establish lo cation-based tec h n ology . The researc h comm unity p rop oses many diﬀerent classiﬁcations for the area of lo calization in WSNs. Dep end ing on the needed criteria, lo calization algorithms Chapter 2. Survey o f WSN Lo caliza tion 22 can b e categorized as sin gle-hop and m u lti-hop or as anc hor b ased and n on-anc h or b ased, or as cen tralized and decent ralized (distributed). In th is su rv ey , the classiﬁcatio n that shall b e adopted is the centrali zed and decen tralized (distribu ted), as sh own in Figure 2.8 [14, 58–60]: 2.3.1 Cen tralized A lgorithm The ﬁrs t class of algorithm is referred to as cen tr alized lo calization algorithm. Centraliz ed lo calizat ion is basically migration of int er-no de ranging and connectivit y data to a suﬃ ciently p o we rful cen tral b ase station and then the migration of resulting lo cations bac k to resp ectiv e no des. C en tralized lo calization algorithm is charact erized by its need for enormous computa- tional p o w er. T he h igh amount of computational p o we r giv es the cen tr alized lo calization its capabilit y to execute complex mathematical op erations. This s up erb adv an tage comes with the d isadv ant age of the high comm unication cost. This d isadv an tage is a result of th e pr o cess itself. As all no d es of a net work send their data to the central receiv er, the compu ted p osi- tions are sent back to resp ectiv e no de; the communicatio n cost, as a result, of su c h b ecome s considerably h igh. Cen tralized lo calization algorithm itself can b e divided in diﬀerent t yp es, as sh o wn in Figure 2.9. These t yp es dep end on the wa y they p ro cess d ata at the cen tral receiv er. There are t wo p opu lar t yp es of centraliz ed algorithms [3, 14] 1. S emi-deﬁnite Pr ogramming (SDP) 2. Mu ltidimensional Scaling (MDS) 2.3.1.1 Semi-deﬁnite Programming (SDP) One of the m ost p opular appr oac hes of cen tralized lo calization algorithms is kn o wn as Semi- deﬁnite Programming (SDP). This approac h uses Lin ear Matrix Inequalities (LMIs) to rep- resen t geomet ric constr aints among no des. After establishing the LMIs, they , all L MIs, get com b ined in order to form a single semi-deﬁnite program. S olving it pro du ces a b ounding region f or eve ry single no de of the net wo rk. T his tec hniqu e can b e used to execute enormously complex m athematica l op erations. Ho w ev er, since the SDP is b ased on the appr opriate utilit y of LMIs, SDP can only work with those geometric th at can b e represented by LMIs. In other w ords, since LMI cannot b e utilized to r ep resen t all geometric constrain ts (Precise range data, Chapter 2. Survey o f WSN Lo caliza tion 23 Localization in WSNs Centralized Localization Distributed Localization Relaxation- Based Coordinate System Stitching Anchor- Based Hybrid Localization Error Propagation Aware Localization Interferometric Ranging Based Diffusion Gradient Bounding Box Figure 2.8: Classiﬁcation of Lo calization Algorithms in WSNs e.g. rings), b y extension, SDP cannot b e used to rep r esen t those conﬁgur ations. The r ule of th umb is that LMI can only b e used to represent geometrical constrain ts that form con v ex regions, suc h as rep resen ting the h op coun t with a circle, and the angle of arriv al with a triangle, Figure 2.10. 2.3.1.2 Multidimensional Scaling (MDS) Another p opu lar approac h of centrali zed lo calizatio n algorithms is kno wn as Multidimensional Scaling (MDS). This appr oac h is also kno wn as MDS-MAP . This appr oac h utilizes th e MDS that originates from mathematical psyc hology . MDS-MAP uses La w of Cosines and linear algebra to reconstru ct the relativ e p ositions of the p oi nt s b ased on the pairwise distances. This approac h uses a set of well-kno wn steps to b e eﬃcient ly constricted. 1. T he ﬁrst step starts by the gathering of the ranging d ata. Th ese data are b eing obtained from the n et work that u s ed to obtain a matrix R , in suc h scenario d ij refers to the distance b et w een n o des j and i . 2. T he s econd step is ﬁnish ed after using the gathered data for the sak e of completing the matrix of the inter-node distances D , whic h is gotten by app lying the shortest path algorithm su c h as Dijkstras or Flo yd s on R . 3. T he third step starts after ﬁnish ing the matrix of the inter-nod e distances. It starts b y ﬁ nding the estimates of no d e p ositions p s b y administrating MDS tec hnique on D , Chapter 2. Survey o f WSN Lo caliza tion 24 Figure 2.9: Conﬁgurations th at represent Centralize d Algorithms retaining the ﬁrst 2 (or 3) largest eigen v ectors and eigen v alues to construct a 2-D (or 3-D) map. 4. T he f ou r th and the ﬁnal step is d one as a transformation of the no de p ositions p s tak e place int o the global co ordin ates by utilizing 3 or more anc hors in the case of 2-D, and 4 or more anc hors in th e case of 3-D. Comparing the ab o v e t w o cen tralized localization algorithms, MDS-MAP has a u n ique c har- acteristic of that when any imp ro v emen t adm in istered in r an ging accuracy will r esult in a noticeable improv emen t in MDS-MAP , whic h SDP lac ks . Another adv an tage of the MDS- MAP on the SDP is that MDS-MAP d o es not n eed anchor no des in the start of the p ro cess. It transf orms into absolute lo cations b y u tilizing co ordin ates using 3 or more anc hor n o des. A ma jor dr a wbac k on MDS-MAP is that it costs wa y to o h igh for higher order op erations. Moreo ver, the cost can b e red u ced partially by u sing decen tralized algorithms (i.e. coord inate system stitc hing), explained in the next section. Chapter 2. Survey o f WSN Lo caliza tion 25 (a) (b) (c) Figure 2.10: Illustrations of Semi-deﬁnite p rogramming 2.3.2 Decen tralized (Distributed) Algorithm The second class of lo calizati on algo rithms is r eferred to decentral ized lo calization algorithm, whic h is also kn o w n as distr ibuted lo calizatio n. As cen tralized lo calization is a migration of in ter-no de ranging and connectivit y data to a suﬃcientl y p o we rful cen tral base station and then th e migratio n of resulting lo cations back to resp ectiv e no d es, Distributed Lo calization Algorithm diﬀers from it since in distributed lo calization algorithm all the computations relev an t to th e no des are done by the no des themselve s; moreo ver, the no des comm unicate with eac h other in order to obtain their p ositions in the netw ork. In other w ords, the no d es of the distribu ted alg orithms use eac h n o des computational p o we r to p erform its op erations. Th is tec hn ique d emands relativ ely h igh inner-no de commination b esides the parallelism for wh ic h it needs to run tasks comparable to cen tralized systems. Unlike the cen tr alized lo calizati on algorithm, this tec hnique d o es not requir e a centre no de with signiﬁcantl y h igh computational p o we r, wh ic h r ed uces the cost of imp lementati on. An other adv an tage of the distributed lo calizat ion algorithm is that it is, generally , faster than the centraliz ed algorithm as eac h op eration is done within the relev ant no de(s). Distributed algorithms can b e divided into six main categories: 1. An c hor-based 2. R elaxation-based 3. C o ordinate sys tem stitc hing 4. Hyb rid Lo calization Chapter 2. Survey o f WSN Lo caliza tion 26 5. I n terferometric Ranging Based Lo calizat ion 6. E rror Prop agation Aware Lo calizatio n algorithms 2.3.2.1 Anc hor-based The ﬁrst categ ory of distr ibuted algo rithms is b etter kno wn as the an chor-based d istributed algorithms. Anchor-based is a type of distr ib uted algorithms that utilizes the anc hors in order to ﬁ nd the location of unkn o w n no d e(s). In suc h algorithms, the no des obtaining a d istance measuremen t of a few anc h ors, as a start. It th en determines their lo cation based on the foun d measuremen ts. Anc hor- based distribu ted algorithms has sev eral algorithmic approac hes: • The ﬁrst of these algorithmic appr oac hes is d iﬀusion. Diﬀusion algo rithm dep ends sole on the radio connectivit y data, which mak es this algorithm fairly easy . Th is algorithm is based on the follo wing w orking principle: Assuming that a no d e p s is most p robably at a cen troid of its neighb ou r s p ositions. Th ere are tw o d iﬀeren t alternativ es to th is algorithm. The ﬁ r st option a v er ages the p ositions of all anc hors that can comm unicate with the no de using radio, in order to lo calize the p ositio n of that no de. This tec h n ique w as d ev elop ed by Bulu sul [61]. The accuracy of this algorithm is lo w wh en a no de is far a wa y from the anc hor no des or anc hor density is lo w. The second option considers b oth anc hors and normal no d es in determinin g th e p osition of the n o de at the cen tre. Also in this algorithm, the accuracy is lo w when a n o de is far aw a y from the anchor no d es, no d e densit y v aries across the net w ork, or no de densit y is low [62]. The ﬁrst tec h n ique h as an adv an tage when it comes to th e needed n um b er of n o de to start with, as it requir es few er num b er than the second tec hnique. • The second anchor-based distributed algorithmic app roac h is kno wn as the Bounding b o x algorithm. This algorithm calculates the no d es p osition based on the ranges of it to n umerous anc hors. Eac h anchor co v ers a range around it, which can b e expressed as a b o x sur roundin g the anchor [63, 64]. The inte rsection b et w een the b o xes of the anchors determines the p ositi on (lo cation) of an u nkno wn no de. An illustration is s h o wn in Figure 2.11. Chapter 2. Survey o f WSN Lo caliza tion 27 Figure 2.11: Illustrations of the b ounding b o x interacti on 2.3.2.2 Relaxation-based The second catego ry of distribu ted algorithms is b etter kno wn as the r elaxation-based d is- tributed algorithms. This algorithm com bines the adv ant ages of distributed algorithms scheme that lies in the computational p o we r with the adv an tages of the cen tralized scheme that lies in the precision and accuracy . Suc h a fusion b rings the algorithm to an utmost capabilit y . The pro cess of the r elaxatio n-based distribu ted algorithm follo ws ve ry sp eciﬁc steps, which are explained as b elo w [65 , 66]: 1. T he ﬁrst step b egins by giving an estimate of the p ositio n of the no des. Th is step can b e done using an y of the distribu ted tec hniqu es ment ioned earlier. 2. T he second step starts as the p osit ions are reﬁn ed usin g the p osition estimates of the neigh b ouring anc hors. Those neigh b ourin g anc hors are considered temp orary anchors. The reﬁned step is us u ally obtained using lo cal neigh b our ho o d m ultilatertaion. In th e case of this algorithm, neigh b ourh o o d m ultilatertaio n is called the sp ring mo d el. The terminology refers to the d istances b et w een th e no d es with their resting sprin gs. 3. Usin g an optimization tec hnique will c hange the no des p osition ev ery single or a num b er of iteration(s). What determines the end of this step is that all the no des in v olv ed must Chapter 2. Survey o f WSN Lo caliza tion 28 ha v e zero forces sub jected to them. It is only then the p ro cess mo v es to the ﬁn al s tep. 4. T he ﬁnal step in v olv es f orming a global m inim um, wh ic h happ en s if and only if: • The magnitude of all the acting- on-no des forces is equal to zero. • The magnitude of the forces b et wee n no des is also zero. By follo win g the step, relaxati on-based distributed algorithm can b e obtained gracefully . Al- though th is algorithm is qu ite an eﬀectiv e one, it has a serious dra wbac k that lies in its sensitivit y to the initial p ositio ns that it starts with. The problem manifests ev en more if the initial p osition in a local minima. Ho we v er, this prob lem is solv able by utilizing a fold-free algorithm. This t yp e of algorithm can p osit ion the no des in a starting p ositio n suc h that they almost certainly nev er fall in lo cal minima. 2.3.2.3 Co ordinate System Stitc hing Algorithm The third catego ry of distribu ted algorithms is kno wn as Co ord inate System Stitc h ing algo- rithm. In this algorithm, the netw ork is divided into smaller and ov erlapping r egions, usually kno wn as sub regions. Eac h of those sub regions creates its o wn optimum lo cal map. T here are man y approac hes f or th is tec h n ique, yet the general steps to app lyin g it is as the follo wing [67]: 1. T he ﬁrst s tep is the most crucial on e, if not d one correctly the algorithm can n ev er work. This step in v olv es splitting the net work into areas. Those areas, su b regions, m u st b e o verlapping. This is usually used with a single no de and its one-hop neigh b our . Th is step d iﬀers from an algo rithm to another. 2. T he second step b egins w h en th e s p litting of the sub regions is done. A t this stage, a lo cal map must b e computed for eac h of the s ub regions. Th is step diﬀers from an algorithm to another. 3. T he ﬁ nal step s tarts after ﬁnishin g the computation of the optimum lo cal map. The third step inv olves placing the en tire sub regions into one single global co ordinate sy s tem, th us the name. This step is d one using r egistration p ro cedure. This step common for all algorithms. Chapter 2. Survey o f WSN Lo caliza tion 29 2.3.2.4 Hybrid Algorithm The fourth category of d istributed algorithms is kn o wn as Hybr id Algorithm. In suc h scheme, t w o diﬀerent lo calization tec h niques are b eing used. F or example, MDS an d th e Pro ximit y based map (PDM) or MDS with Ad- ho c p ositioning system (APS). Hybrid tec hniques and algorithms are usually used to m ak e adv an tages and to o vercome the limitatio ns of b oth tec hn iques in u s e. Overall, the p erf ormance of the h ybrid algorithm is b ett er th an eac h of the tec hn iques used individually . Let us tak e an example in details; the author in [68] com bined the MDS with the PDM in order to establish a hybrid tec h nique and so to lo calize the s ensor no des lo cation. It starts by dividin g th e no d es into 3 classes: • Primary anc hors • Secondary anc hors • Normal sensors The manifestation of the hybrid tec h nique is in the f ollo wing steps: 1. I nitially , eac h p r imary anchor sends an invita tion pac k er con tains an ID that is unique to eac h invitat ion, w hic h conta ins a v alue, ks, con trolling the num b er of neighbour ing secondary anchors. 2. At this instan t, a counter function is initialized and set to zero. The invi tation sent is then receiv ed b y the normal sensor, which by its tu rn p erf orm s a Bern oulli trial. 3. I f the outpu t of th e trail is true (p), the n ormal sensor increments the counte r and thus b ecomes a secondary anc hor. 4. T ransmitting the pac ket from n eigh b ouring sensors f r om one to another until all th e coun ters are equal. 5. F ollo wing th e steps with th e pr imary and again w ith the secondary rep eatedly can guaran tee the success of th e h ybrid tec hnique. The hybrid tec hn ique h as an adv antag e of that it can b e u sed in b oth ind o or and outdo or lo- calizat ion. Inte rferometric ranging based lo calization and error propagation a w are lo calization algorithm is not d iscussed as it is b ey ond the scop e of th is pro ject. Chapter 3 Receiv ed Si gnal Strength In this c hapter, the log-normal channel mo del is deriv ed and explained. In addition, a com- parison b et w een d iﬀeren t estimators is presented. 3.1 RSS M o deling The fr ee-space propagation mo del is us ed for predicting the receiv ed signal s trength in the line- of-sigh t (LOS) en vironment wh ere there is no obstacle b etw een the tr ansmitter and receiv er [69]. The RSS equation is derived fr om F riis T ransmission form ula [3] P r = P t G t G r λ 2 (4 π × d ij ) 2 (3.1) where P r is the receiv ed signal strength fr om sen sor no de i at n o de j at time t , P t is the transmitted p o wer, G t is the transmitter gain, G r is the receiv er gain, d ij is the distance from sensor no de i at n o de j , and λ is the wa vel ength of the signal. F rom th e equation ab ov e, th e receiv ed p o wer P r atten u ates exp onen tially w ith the d istance d ij [69]. The free-space p ath loss, P L F , is d eriv ed f rom the equation ab ov e by 10 log the ratio of the transmitted p ow er P t to the receiv ed p ow er P r and setting G t = G r =1 b eca use in most of the cases, the antennas that are used are isotropic an tennas, wh ic h radiate equally in all d ir ection, giving constan t radiation pattern [69]. Th us, the form ula is the f ollo wing: P L F ( d ij )[ dB ] = 10 log  P t P r  = 20 log  4 π × d ij λ  (3.2) 30 Chapter 3. Received Signa l Str ength 31 In the f r ee-space mo d el, the a verag e receiv ed signal in all the other actual en vironment s decreases w ith the distance b et w een the transmitter and r eceiv er d ij , in a logarithmic manner . Therefore, p ath loss mo del generalized form can b e obtained by c hanging th e free-space path loss with the path loss exp onent n d ep ending on the environmen t. T his is kno wn as the log-distance path loss mo del which will result in the follo wing form ula [69]: P L LD ( d ij )[ dB ] = P L F ( d 0 ) + 10 η log  d ij d o  (3.3) where d 0 is the r eferen ce distance at w hic h the path loss in herit the c haracteristics of free- space in equation 3.2 [69]. Th is distance is set to d iﬀerent v alue dep ending on the p r opagation en vironment ; f or example, it is 1 km for a large co v erage cellular sy s tem. In our case we w ill consider this v alue to b e 1m. T he v alue of n is 2 whic h resem bles the free-space. Diﬀeren t v alues for n resem bles diﬀerent en vironment conditions [69]. Ev ery p ath b etw een the sender and the receiv er has diﬀerent path loss since the en viron- men t c haracteristics changes as the lo cation of the receiv er c hanges. Moreo ver, the signal ma y not p enetrate in the same w a y . F or that reason, a more realistic m o delling of the transm ission is assumed w hic h is the log-normal mo d elling. P L ( d ij )[ dB ] = P L F ( d 0 ) + 10 η log  ( d ij d o  + X σ (3.4) where P L ( d i j )[ d B ] = P t [ dB ]- P r [ dB ] is the p ath loss at distance d ij from the tr an s mitter, P L F ( d 0 ) is the p ath loss mo del at th e r eference distance d 0 whic h is constan t. X σ is Gaussian random v ariable with a zero mean and stand ard deviation σ [68 ]. In order to ﬁn d the lo cation of the b lin d no de w ith resp ect to three r eference no d es ( p 1 , p 2 , p 3 ), three circles will b e used to d ra w three lines. T hese lines passes thr ough t w o p oint s at whic h the t wo circles in tersect, and they are called the line of p osition (LOP), as sho wn in Figure 3.1. T o ﬁn d th e equ ation that represents eac h line, we s tart w ith the distance equation th at d escrib es the length b et w een the i th reference no de and the blind no de, whic h is giv en b y [3]: D i = k p i − p s k = q ( x i − x s ) 2 + ( y i − y j ) 2 (3.5) where p i = ( x i , y i ) is the p osition of the reference no de, p s = ( x s , y s ) is the p osition of the Chapter 3. Received Signa l Str ength 32 unknown no d e, an d k x k is the norm of vect or x . Then, the LOP b et we en p 1 and p 2 , will b e computed b y squaring and taking the diﬀeren ce b et w een D 2 and D 1 ,th us, the result will b e as f ollo w: ( x 2 − x 1 ) x s + ( y 2 − y 1 ) y s = 1 2 ( k x 2 k 2 − k x 1 k 2 + D 2 1 − D 2 2 ) (3.6) The same scenario applies b et w een p 2 and p 3 and b et w een p 1 and p 3 , as follo ws [1] ( x 3 − x 2 ) x s + ( y 3 − y 2 ) y s = 1 2 ( k x 3 k 2 − k x 2 k 2 + D 2 2 − D 2 3 ) (3.7) ( x 3 − x 1 ) x s + ( y 3 − y 1 ) y s = 1 2 ( k x 3 k 2 − k x 1 k 2 + D 2 1 − D 2 3 ) (3.8) The b lin d no de lo cation w ill b e estimated by solving for x s and y s in the ab ov e three equations. Ho we v er, in creasing the n umber of the reference no d es will r esult in more inte rsec- tion p oints. These lines will not in tersect at the same p oint du e to the error in the distance measuremen ts. Hence, it is required to ﬁnd another tec hniqu e to p ro vide b etter estimation for the location of the u nkno wn no d e [3]. F or an y lo calization system, with N anc h or n o des, th ere will b e N − 1 in dep end en t lines represent ed in the follo wing form ula: a i, 1 x s + a i, 2 y s = b i (3.9) where the co eﬃcien ts a i and b i are known. The ab ov e equation, also, can b e w ritten as follo ws: a i p s = b i (3.10) where a i =[ a i, 1 a i, 2 ] and p s = [ x s y s ] T . All lines equations can b e describ ed in matrix format: Ap s = b (3.11) where A = [ A 1 A 2 ............ A N − 1 ] T , b = [ b 1 b 2 ............ b N − 1 ] T , This equation h as no solution, as the str aight lines do not intersect at one p oint. F or that reason, d iﬀerent estimation tec hniques ha v e b ee n u sed to ﬁ nd the p osit ion of p s . Chapter 3. Received Signa l Str ength 33 p 1 p 2 p 3 D 1 D 2 D 3 p s Figure 3.1: No de estimatio n in RS S 3.2 Norm Ap pro ximation The simplest n orm approximati on problem is an unconstrained problem of the form minimize k Ap s − b k 2 2 (3.12) where A ∈ R m 1 × n 1 and b ∈ R m 1 are pr oblem d ata, p s ∈ R n 1 is th e v ariable, and k . k is a norm on R m 1 . A solution of th e norm appro ximation problem is s ometimes called an app ro ximate solution of Ap s ≈ b ,in the norm k . k . The vec tor r residual = Ap s − b (3.13) is called the r esidual f or the pr oblem; its comp onen ts are sometimes called the individu al residuals asso ciated with p s . The norm approxima tion pr oblem in 3.12 is a conv ex p roblem, and is solv able. Its optimal v alue is zero if and only if b ∈ ℜ (A). W e can assu me without loss of generalit y that the columns of A are in dep end ent; in particular, th at m 1 ≥ n 1 . When m 1 = n 1 the op timal p oint is simply A − 1 b , so w e can assume that m 1 > n 1 [70]. Chapter 3. Received Signa l Str ength 34 3.2.1 Least-squares Appro ximation The most common norm appro ximation problem in v olv es the ℓ 2 -norm [71]. By squaring the ob jectiv e, w e obtain an equiv alen t pr oblem whic h is called the least-squares approximati on problem, minimize k Ap s − b k 2 2 (3.14) where the ob jectiv e is the sum of squares of th e errors. This p roblem ha v e an analyticall y solution by expr essing the ob jectiv e as a qu adratic f u nction: f ( p s ) = p T s A T Ap s − 2 b T Ap s + b T b (3.15) A p oint p s minimizes f if and only if ▽ f ( p s ) = 2A T Ap s − 2A T b = 0 (3.16) A T Ap s = A T b (3.17) whic h alw a ys ha v e a solution. S ince w e assume the columns of A are indep enden t, the least- squares appr o xim ation problem has th e un ique solution: p s = (A T A ) − 1 ( A T b) (3.18) The least-squares estimator is a maxim um lik eliho o d estimator of a Gaussian d istr ibution with a zero mean and v ariable v ariance (see App end ix A). Ho w ev er , in the least-squares all of the anc hor no d es will b e giv en the same w eigh t in the estimation of the n o de regardless of the lo cation of the no de, wh ic h will resu lt in a higher error in the d istance estimation. T o o vercome this limitation, more w eigh t will b e giv en to those measuremen ts corresp onding to the closer distances an d this will results in a h igher accuracy . The W eighted L east Square (WLS) formuls is: p s = (A T W A ) − 1 ( A T Wb) (3.19) Chapter 3. Received Signa l Str ength 35 where W is the w eigh tin g matrix The computation complexit y of the WLS is O(S 3 ) where S is the num b er of reference no des inside the cov erage area of the blind no de but, not the total num b er of the anc hor n o des. On the other hand, the computational complexit y in the LS is O (S) [11]. Ho w ev er, in practical deplo ymen ts the v alue of S is usu ally small, so WLS can b e executed in a resource-constrained device. 3.2.2 Hub er Robustness In the Hub er robu stness the ℓ 1 -norm is used: minimize k Ap s − b k 1 (3.20) In the ℓ 1 -norm the absolute error is minimized, whic h means the size of th e error will b e smaller than the case of ℓ 2 -norm and so will b e less sensitive to the outliers [72].Ho wev er, in the case of Hub er r obustness we ha ve an abs olute v alue and so th e deriv ativ e can not b e applied and for that reason the pr oblem is solv ed u s ing th e Iterativ e R eweighed Least Square (IRLS) tec hnique. In the ﬁrst iteration the wei gh ts will b e carried ou t u sing the WLS. After that, in the second iteratio n the absolute error ( e i ) w ill b e ev aluated and the w eigh ts will b e up d ated using the follo wing equation: w i = n X i =1 1 | e i | + ǫ | e i | 2 (3.21) After sev eral iterations th e answ er will con v erge to the min im um. Th e v alue of ǫ is add ed to a v oid h a vin g a discon tin uit y and it is critical to the p erformance of the estimator [73]. This is b ecause the v alue of ǫ resembles the size of the ℓ 2 -norm curve in the Hub er robustness. 3.3 Sim ulation Results 3.3.1 P erformance In ord er to compare the p erformance of the three RSS based lo calization app roac h es in terms of accuracy of the lo calizatio n results, s ev eral no des are placed in a t w o-dimensional region Chapter 3. Received Signa l Str ength 36 with the size of (100m × 100m). Eac h anc hor no d e contai ns one an tenna op erating at a frequency of 1GHz. T o ev aluate th e p erformance of the tw o tec hniques, Ro ot Mean Square Error is ev aluated 150 times for ev ery SNR v alue. The p erforman ce of least square is plotted on Figure 3.2. 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 X: 1 Y: 36.52 Signal to Noise Ratio (dB) RMSE (m) X: 5 Y: 2.479 X: 12 Y: 0.9457 Figure 3.2: RMSE for diﬀerent v alues of SNR using LS It is evident that b y increasing th e SNR the RMS E decrease. This is b ecause by increasing the SNR, th e error in distance measurement s will b e reduced and therefore a b ette r estimation of the un kno wn no de location. Wh en the SNR is 1 dB the RMSE w as 36.52m and as the SNR increases the RMS E decreases r eac hin g 0.9457 m at 12 dB. Comparing the tw o Figures 3.2 , 3.3 we n oticed that at the same SNR v alue, WLS ind icates lo wer RMSE v alue compared to that in LS, that is at SNR=12, RMSE in WLS is 0.4574 m and in LS 0.9457m. Actually , a closer lo ok in equ ation 3.4, w e can notice that the RSS d o es not v ary linearly on th e distance b et w een no des, th us, the same error w ill result in larger errors in pro ce ss of estimating th e distance, esp ecial ly , in long distance b et we en the n o des. More p recisely , the accuracy in the estimation relies on the d istance b et wee n the no des [11]. Chapter 3. Received Signa l Str ength 37 0 5 10 15 20 25 30 0 2 4 6 8 10 12 X: 1 Y: 10.71 Signal to Noise Ratio (dB) RMSE (m) X: 5 Y: 1.235 X: 12 Y: 0.4574 Figure 3.3: RMSE for diﬀerent v alues of SNR u s ing WLS LS treats diﬀerent distance lengths equally , w hic h can b e noticed from equation 3.18, so the error will b e high. Ho w ev er, WLS take s into account the diﬀerent lengths b y using the w eigh ting matrix, as in equation 3.19, which giv es high weigh t for the sh ort distances, th us b etter accuracy . This matc hes w ith the sim ulation results, which sho w the dramatic reduction in RMSE f or WLS compared to that in LS at the same v alue for SNR. The third estimator b eing used is the Hu b er r obustness. F r om the simulat ion results in Figure 3.4 it can b e seen that the b est p erformance achiev ed usin g the Hub er r obustness is the same as the WLS and can n ot ac hiev e a b etter p erformance. T h is is b ecause the noise is Gaussian and the b est estimator to d eal with a Gaussian noise is an ℓ 2 -norm and more sp eciﬁcally WLS. Chapter 3. Received Signa l Str ength 38 0 5 10 15 20 25 30 0 2 4 6 8 10 12 X: 1 Y: 10.54 Signal to Noise Ratio (dB) RMSE (m) X: 5 Y: 1.16 X: 12 Y: 0.4565 Figure 3.4: RMSE for diﬀerent v alues of SNR u s ing Hub er Robustn ess Chapter 3. Received Signa l Str ength 39 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 Signal to Noise Ratio (dB) RMSE (m) Weighted Least Square Huber Robustness Least Square Figure 3.5: RMSE for diﬀerent v alues of SNR (least,w eigh ted least squ are, and hub er r obust- ness) It is clear that the WLS and Hub er robustness ac hiev es the optimal p erformance in com- parison to the least square as shown in Figure 3.5. The p erformance of the WLS and Hub er robustness is almost 5 times b etter than the least squ are. This is b ecause small d istance estimations ha v e more signiﬁcan t eﬀect on the ﬁn al estimated lo cation than large d istance estimations. Ho wev er, by making the distance b etw een the an chor n o des and unkn o wn no d e almost equal the p erformance of the three estimato rs is almost th e same as sh o wn in Figure 3.6. Chapter 3. Received Signa l Str ength 40 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Signal to Noise Ratio (dB) RMSE (m) Weighted Least Square Huber Robustness Least Square Figure 3.6: RMSE for diﬀerent v alues of SNR (least,w eigh ted least squ are, and hub er r obust- ness) The p erformance is almost the same b ecause the we igh t for eac h anc hor no de will b e almo st the same b eca use the d istance b etw een no des is almost equal. Therefore, the adv anta ge of WLS and Hu b er o v er L S will v an ish in this cond ition. 3.3.2 Robustness In the previous test, the environmen t was exactly the same as the environmen t used to dev elop the pathloss mo del. How eve r, in a r eal s itu ation it is v ery likely that there will b e some c h anges in the environmen t. F or example, c h an ges in f u rniture lo cations or movi ng p eeople in the area. When c hanges happ en in the environmen t, the pathloss m o del do es not r esemble the real environmen t. Therefore, it is imp ortan t to test the p erformance of the algorithm to c h anges in the pathloss exp onen t as s ho wn in Figure 3.7. It can b e seen clearly from Figure 3.7 that for a lo we r pathloss exp onent the RMSE is larger Chapter 3. Received Signa l Str ength 41 2 2.5 3 3.5 4 4.5 5 5.5 6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Pathloss (n) RMSE (m) Weighted Least Square Huber robustness Least Square Figure 3.7: Robu s tness of the algorithm to c hanges to in th e pathloss exp onen t and this is b ecause the distance estimated will b e larger and so a higher error is asso ciated with it.Ho w ev er, for the higher p athloss exp onen t the distance estimated is smaller and so lo wer error asso ciated w ith it.Moreo ve r, it can b e seen that the Hub er robustness and the WLS is less sensitiv e to c h anges in th e pathloss exp onen t compared to the least square. Chapter 4 Direction of Arri v al In this c hapter, t wo con v en tional arr a y geometries namely Uniform Linear Array (ULA) and Uniform Circular Arra y (UCA) are analysed to b e used with DO A-based algorithms. Then , the imp ortance of receiv ed signals to b e uncorrelated is h ighligh ted in t yp e of signal s ection as w ell as ho w to de-correlate correlated signals using Phase mo de Excitation, S patial S m o othing and T eopiltz algorithm. After that, the DO A-based algorithm are introdu ced whic h includ es MUSIC, Ro ot-MUSIC, UCA-Root-MUSIC, ESPRIT and UC A-ES PRIT. Lastly , the numer- ical resu lts are provi ded to show the p erformance of DO A-based algorithm and DO A-based lo calizat ion. 4.1 Con v en tional S ensor Ar r a y Conﬁgurations F or the DOA , antenna arr a ys are recommended as they can detect m ultiple signals at the same time, whic h is a qualit y that directional antenna lac ks. Based on this t yp e of ant ennas, s everal geometrica l conv enti onal sensor arra y conﬁgurations can b e us ed to p erform the estimati on of DO A. The most adopted conﬁgurations are kn o wn as Uniform Linear Array (ULA) and the Uniform Cir cular arra y (UCA). 4.1.1 Uniform Linear A rra y (ULA) The Uniform Linear Arra y (ULA) is one of the adopted conﬁgurations of DO A, as sho wn in Figure 4.1. In it, a set of N sensors (ante nnas) is b eing scattered along a single dimensional line. The sens ors must main tain uniform an d equal distances b et we en them, often, the sens ors 42 Chapter 4. Direction o f Ar riv al 43 used in suc h conﬁguration are kno wn as omni-dir ectional sensors. In this scenario, those sensors receiv e M n umbers of narro wband signals s m ( t ), wher e 1 ≤ m ≤ M . The angle of whic h th e signal originate s from is diﬀeren t from th e others. Th ose angles are k n o wn as azim u th angles θ m . T he N -dimensional receiv ed data v ector x at an y time t is given b y [74]: x ( t ) = M X m =1 a s ( θ m ) s m ( t ) + n ( t ) = A s ( θ ) s ( t ) + n ( t ) (4.1) where n is a noise ve ctor mo delled as white and zero-mean complex Gaussian, A s ( θ ) is a ma- trix whic h consist of M steering vect ors, and a s ( θ m ) is the steering v ector, which corresp ond s to the DO A of the m th signal, deﬁn ed as: a s ( θ m ) =  1 e − j φ m e − 2 j φ m ... e − j ( N − 1) φ m  T (4.2) where φ m represent s th e phase shift b et wee n the elemen ts of the sens or arra y and is expressed as: φ m = 2 π  d λ ) sin( θ m )  (4.3) n =1 n =2 n = N ------- array normal incident plane wave d d sin n = N -1 Figure 4.1: Geometry of N-elemen ts (ULA) Chapter 4. Direction o f Ar riv al 44 4.1.2 Uniform Circular A rra y (UCA ) The Uniform Cir cu lar Arr a y (UCA) is another adopted conﬁ guration of DOA, shown in Figure 4.2. In it, a set of sensors (receiv ers) are b eing scattered in t wo dimensions, x - y plane. T o p erform this conﬁguration, let us assume a set of N num b ers of sensors. Those sensors are placed in a circular sh ap e, making a r ing that has a rad iu s of r . It is crucial to main tain a uniform an d equal distances b etw een the sensors, d , along the rin g circumference. It was established that the N -d im en sional r eceiv ed data v ector x at any time t is giv en by equation 4.4 [68, 7 5]. The steering v ector of a circular arra y is expressed by the follo wing: a s ( θ m ) =  e j ( 2 πr λ ) cos( θ m − θ n )  T ; 1 ≤ n ≤ N (4.4) where θ n is the angu lar location of eac h elemen t and is calculat ed us in g: θ n = 2 π  n − 1 N  (4.5) 4.2 T yp e of Signals In comm u n ication systems and signal pro ce ssing, a signal is deﬁn ed as th e function that carries d ata or conv eys inf ormation ab out th e attribute or b eha vior of some phenomenon. IEEE T r ansactio ns on Signal Pr o c essing stretc hes the deﬁ n ition of the word signal to in clude all video, audio, s p eec h, image, radar, sonar, communicatio n, m usical, geoph ysical or eve n medical signals [76]. Signals can b e classiﬁed based on many diﬀerent criteria. F or example, in the p hysics w orld, signals can b e categorized based on their exhibited v ariations in time or space or b oth. In this p ro ject, s in ce w e are considering pr actica l envi ronment, the classiﬁcation that is mostly relev ant is based on the correlati on of the signals. In other words, th e signals are cataloged in to correlated signals and uncorrelated s ignals. 4.2.1 Uncorrelated Signals The d eﬁnition of the u ncorrelated s ignals comes from p robabilit y theory and statistics. In statistics, t w o r eal-v alued random v ariables A and B are said to b e uncorrelated if and only Chapter 4. Direction o f Ar riv al 45 r N 1 2 N -1 n n +1 m θ x y n θ Figure 4.2: T op view of the N-eleme nt circular arr a y in x -y plane if their co v ariance is equ al to zero. i.e. E(AB) − E(A)E(B) = 0 (4.6) Consequent ly , if the t w o v ariables are considered un correlated, there cannot b e linear relation b et w een them. F or example, uncorrelated w hite noise refers to the face th at n o tw o p oint s in the time domain of the noise is asso ciated with eac h other. This also means th at the noise v alue at time ( t +1) cannot b e predicted if noise at time ( t ) is kno wn [77]. In the detection of DO A, w hen an un correlated signal is in use the metho d of d etection is q u ite straigh tforw ard to p erform. Th e standard MUSIC, Ro ot-MUSIC, ESPRIT algorithms, or an y other typ e of algorithm of d etectio n can b e p erformed and u sed. That is due to the fact that ev ery signal is b eing treated as a separate ind ividual ent it y . In other wo rds, eac h signal has its unique p atterns and trends. It will b e discu ssed with more details in late r c hapters, Chapter 4. Direction o f Ar riv al 46 but it is imp ortant to h ighligh t that MUSIC algorithm, for example, can detect u p to N -1 uncorrelated signals without problems. 4.2.2 Correlated Signals The other t yp e of s ignals that w e are considering in this pro ject is kno wn as the correlated signal. Generally sp ea king, correlation is a mathematical op erat ion that is very similar to con v olution. It can b e deﬁn e as the degree of asso ciation b et wee n tw o random v ariables. F or example, w hen sp eaking ab out t w o graphs of t w o set of data, the correlatio n b et we en them refers to the degree of resembla nce b et w een th em. Unlike the non -correlation, the co v ariance of t w o inputs m ust not n ecessarily equal to zero to b e called correlated. i.e. E(AB) − E(A)E(B) 6 = 0 (4.7) It is imp ortan t to highligh t that correlation is not the same as causalit y . Causalit y set the relation b etw een to eve nts b y making one the cause of the other (i.e. cause and eﬀect). Cross correlation is the correlation b et w een t w o diﬀeren t signals. It also can b e b et we en a signal and itself, in whic h it is called auto-correlation [78]. In the d etection of DO A, correlated signal(s) cannot b e detected by standard metho d s. Standard u sage of the metho ds, mentio ned earlier, will result in to a wrongful int erpretation of the data. T o a v oid su c h int erpretations, new metho dology must b e introdu ced to th e s tandard detection m etho ds su c h as spatial smo othing, phase m o de excitatio n, and T o eplitz algorithm. 4.2.2.1 Phase Mo de Excitation Practically , unif orm circular arra y is of a great imp orta nce when it comes to estimate the DO A of inciden t signals. C omp are to ULA, UCA provides a complete co verag e for all azim uthal angles as well as a uniform p erformance in detecting them. How ev er, UCA falls b ehind ULA in the wa y its steering ve ctor is designed. ULA has a steering vect or of V an d ermonde stru cture. This structure is uniqu e as it allo ws eﬃcien t compu tational algorithms lik e Ro ot-MUSIC and de-coheren t tec h niques to b e imp lemen ted. F or UCA, it d o es not p ossess V and ermonde structure du e to the dep endence of UCA steering vec tor with the sensor angular lo cation. T o o v ercome suc h limitation, phase mo de excitation is used. Ph ase mo de excitation is a Chapter 4. Direction o f Ar riv al 47 b eamforming tec hniqu e to con vert the steering v ector of UCA in to a virtual steering v ector c h aracterized by a V andermond e s tructure. In other w ord, the UCA is mapp ed and conv erted in to a vir tual ULA. Th us, all m etho ds applicable to ULA can now b e emp loy ed to UCA indirectly thr ough th e phase mo d e excitatio n [79]. T o compr eh end the concept b ehind the ph ase mo de excitation, w e ﬁr st need to extend our UCA mo deling from 2-D in to 3-D, where an elev ation angle θ e is in tro du ced as sho wn in the Figure 4.3. Figure 4.3: Arr ay geometry for UCA includ in g the elev ation angle In the p r evious UCA mo del discussed in section 4.1.2, the elev ation angle wa s assumed to b e ﬁxed at 90 o for all incident signals. In the new UCA mod el, w e will keep the v alue of elev ation angle constan t f or any arb itrary incident signals bu t the actual angle can v ary from 0 o up to 90 o . Consequent ly , the resulting s teering ve ctor for UCA ha ving N elemen ts with radius r and receiving a narro wband signal arriving from azim uth angle θ m ∈ [ − π , π ] and θ e ∈ [ 0 , π 2 ] is mo deled as: a s ( θ m ) =  e j ( 2 πr λ ) sin( θ e ) cos( θ m − θ n )  T , 1 ≤ n ≤ N (4.8) Chapter 4. Direction o f Ar riv al 48 The elev ation angle θ e is measured from the inciden t signal do wn to the z-axis while the azim u th angle θ m is measur ed from th e pro jected v ector of the incident signal to the x-axis in a cloc kwise direction. As θ e is assumed to b e ﬁxed for all receiv ed signals, w e can simplify equation 4.8 by deﬁnin g ζ = ( 2 π r λ ) sin( θ e ). No w , the elev ation angle θ e is dep endent throu gh ζ and th e equation 4.8 b ecome: a s ( θ m ) =  e j ζ cos( θ m − θ n )  T , 1 ≤ n ≤ N (4.9) 4.2.2.1.1 Phase Mode Excitation Principle The basic idea of phase mo d e excitation is ab out applying a b ea mforming vec tor to transform the b eam p attern of UCA into a b eam pattern similar to that of ULA and, sim u ltaneously , reserving the original steering dir ection of UCA. T o carry out this b eamforming, w e ﬁ rst n eed to pro duce and visualize the arra y p attern of ULA. This pro cedur e is done to give us an idea of ho w to syn thesis the phase mo de excitation b eamformer . Then, th e p hase mo de excitatio n b eamformer will b e deriv ed for con tin uous circular arra y (CCA) and extended to un iform circular array (UCA). In case of CCA, there is an inﬁn ite num b er of sensors arranged o v er rin g, so the eﬀect in ter- elemen t spacing in this scenario is ignored. Undoub tedly , CCA cannot b e realized practically . Ho wev er, from theoretical p oi nt of view, it considered as an ideal free-error scenario for emplo ying the phase mo de excitation b eamformer. Th en, the b eamformer will b e employ ed on the UCA whic h represents a practical m o del. A t the latter mo del, w e will u nderstand wh y a quantiza tion error arisein the UCA geomet ry [79]. 4.2.2.1.2 Arra y P at tern of Uniform Linear Array (ULA) The Arra y pattern of an array can b e obtained usin g b ea mforming tec hn ique. Beamforming is simply a spatial ﬁ ltering op eration wh er e weig ht s are assigned to the arra y outpu t follo wed b y a summing pro ce ss to obtain a b ea m. Assu me x ( t ) to b e the arra y outpu t at time t and w H = [ w 0 , w 1 , · · · , w ( N − 1) ] is the corresp ondin g b eamformin g we igh t vec tors assigned for eac h N element. The equation of b eam outpu t y ( t ) is giv en as [80]: y ( t ) = w H x ( t ) (4.10) Chapter 4. Direction o f Ar riv al 49 As we are intereste d in obtaining th e b eam pattern of an arra y , x ( t ) is replaced by a s ( θ ). That is b ecause the arra y r esp onse of an arra y to a source signal impinging fr om arbitrary angle θ is the steering vect or to that angle. The elemen ts in the s teering v ector a s ( θ ) are separated by equally phase shifts. Th us, for N elemen ts arra y , its b eam pattern equation is deﬁned as f ( θ ) = w H a s ( θ ) = N − 1 X n =0 w n a s ( θ ) − 90 o ≤ θ ≤ 90 o (4.11) The arr a y pattern f ( θ ) in equation 4.11 is giv en in complex-v alue form. By taking the absolute v alue, the arra y p attern | f ( θ ) | usu ally follo w an oscillatory function with main-lob e and side-lob es. The main-lob e corresp onds to the pass-band of the b eamformer so signals receiv ed at main-lob e p ass unatten uated. S imilarly , signals receiv ed at side-lob es will b e sev erely atten u ated. Th us, b eamformer weigh ts can b e selected to make the steered arra y lo ok in a sp eciﬁc dir ection. In cophasal b eamforming, the w eigh ts w H are assigned to steer th e arra y to the desired direction θ m [11]. This can b e simply achiev ed b y assigning w H = a s ( θ m ) H N so signal coming from θ m will b e u natten uated. The w eigh ts are divid ed by N for normalized p urp oses. Thus, the arra y pattern un der cophasal b eamformer for a signal receiv ed at azim uth angle θ m is giv en as: f ( θ ) = a s ( θ m ) a s ( θ ) N , − 90 o ≤ θ ≤ 90 o (4.12) The arr a y pattern of ULA under coph asal b eamformer is determined by substituting a s ( θ ) and a s ( θ m ) in equation 4.12 with their equ iv alen t quantit y in ULA giv en in equation 4.2. let v = (2 d ) sin ( θ ) λ , so a s ( v ) and a s ( v m ) can b e wr itten as: a s ( v ) =  1 e j πv ... e j ( N − 1) πv  T (4.13) a s ( v m ) = e j πv m (4.14) By substituting a s ( v ) and a s ( v m ) in equation 4.12, the resulting arra y pattern for ULA b ecomes: Chapter 4. Direction o f Ar riv al 50 f( v ) = e j ( N − 1) π 2 ( v − v m ) sin( N π 2 ( v − v m )) N sin( π 2 ( v − v m )) , − 1 ≤ v ≤ 1 (4.15) T o visualize the b ea m pattern of ULA, let us consider a narro wband signal impin ging on ULA from θ m = 0 o with N =10 and in ter-elemen t spacing of d =0.5 λ . Th ese v alues will result in v m = 2 dsin ( θ m ) λ . Thus, b y substituting v m in equation 4.15 and plotting the result for − 1 ≤ v ≤ 1, w e obtain the ULA arra y p attern as sh o w n in Figure 4.4. −1 −0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 v |f(v)| Figure 4.4: Arr ay pattern of ULA w ith 10 elemen t receiving a signal from θ m =0 Clearly from Figure 4.4, th e cophasal b eamformer has steered th e ULA into direction θ m = 0 o as the main-lob e of the b eam pattern is cen tered at v =0 which corresp ond to θ m = 0 o . 4.2.2.1.3 Phase of E xcitation of C on tin uous Circular Array (CCA) F rom the previous s ection, we noticed that a desired b eam pattern of ULA is obtained b y applying weigh ts directly on the in dividual elemen ts. Un fortunately , this metho dolog y cannot b e used with circular arra y d ue to the existence of angular p osition. Ho w ev er, there exists a p o we r approac h to o v ercome this problem. A b eamformer we igh t called excitatio n function is applied with angular argumen t θ n . In the excitatio n function, the phase of the w eigh t are assigned in a linear mann er based on the angular lo cation of an elemen t to a reference elemen t (arra y elemen t 0). Th e linear phase increase of the excitati on function weig hts, is done in a similar pattern compared to the cophasal excitatio n of a ULA. As θ n is p eriodic with a p eriod of 2 π , then any excitation function w ( θ n ) can b e expressed using F ourier series with eac h of Chapter 4. Direction o f Ar riv al 51 its harmonic is term ed with th e phase mo d e m . Th e general represent ation for the excitation function w ( θ n ) is giv en as [79]: w ( θ n ) = ∞ X p = −∞ c p e j pθ n (4.16) where c p is the F ourier series coeﬃcient for the corresp ondin g m th phase mo de obtained as: c p = 1 2 π 2 π Z 0 w ( θ n ) e − j pθ n dθ n (4.17) The excitation function for th e p th phase m o de is mo deled as w p = e ( j p θ n ) whic h simp ly represent s a spatial harmonic for the generalized excitation function w ( θ n ). T o obtain th e normalized far-ﬁeld of p th phase mo de, w p is multiplied with circu lar arr a y steering vect or for eac h elemen t follo wed by summation pro cess. As we are d ealing with CCA, th en the summation approac h integral take n from 0 to 2 π . Th e resulting normalized far-ﬁeld of phase mo de p is giv en as: f c p ( θ ) = 1 2 π 2 π Z 0 w ( θ n ) e − j ζ cos( θ m − θ n ) dθ n (4.18) where θ = ( ζ , θ m ) represent s the angle of receiv ed signal in term of b ot h azim uth and elev ation angles while su p erscript c donates con tin uous arra y . Using Bessel f unction prop ert y , f c p ( θ ) can b e expressed in a compact form as: f c p ( θ ) = j p J p ( ζ ) e j pθ m (4.19) where J p ( ζ ) is a ﬁrs t kind Bessel function of order p . By analyzing equ ation 4.19, the follo wing observ ation is deduced: • In the far-ﬁeld p attern f c p ( θ ), th e resulting azim u thal v ariation giv en b y e j pθ m ha v e the same form as the excit ation function e j pθ n . Th is prop ert y of phase mo de excita tion is considered as the basis for pattern synt hesis to b e emp lo yed with UCA. • The eﬀect of amplitude and elev ation in far-ﬁeld pattern f c p ( θ ) is determined through the Bessel f unction. This relationship imp oses a limit on the num b er of mo des to b e syn thesized. As the visu al r egion f or θ e is b oun ded by 0 ≤ θ e ≤ π 2 , its corresp ond ing ζ Chapter 4. Direction o f Ar riv al 52 will b e b ounded within 0 ≤ ζ ≤ 2 π r λ . F or a giv en Bessel function J p ( ζ ), its amplitude b ecomes small as the ord er p exceed the argument ζ . Th us, the highest order h to excite a mo d e with a reasonable strength is given as h ≈ 2 π r λ (4.20) This implies that the total excitation mo des for CCA is p ∈ [ − h, h ]. T he follo wing example will demonstrate the v alidit y of equation 4.20. C onsidering a CCA w ith r = λ , then equ ation 4.20 suggests that the m axim um order to b e excit ed is h = 6. This is clear from Figure 4.5 wh ere J 7 ( ζ ) corresp onds to mo de 7 is v ery small in the visible region 0 ≤ ζ ≤ 2 π and hence can b e safely ignored. Th erefore, we conclude that the phase m o des in the range p ∈ [ − 6 , 6] considered enough to b e excited b y CC A with a reasonable strength. Figure 4.5: Bessel f unctions for J 0 ( ζ ) up to J 7 ( ζ ) wh er e 0 ≤ ζ ≤ 2 π Chapter 4. Direction o f Ar riv al 53 4.2.2.1.4 Phase of E xcitation of Uniform C ircular Arra y (UCA) As UCA is a sampled version of C CA, its b eamforming vect or w m is also a sampled v ersion of that for the CCA. F or a UCA w ith N elements, its normalized b ea mforming ve ctor to excite an arra y with phase mo d e p , where p ∈ [ − h, h ], is mo deled as: w p = 1 N  1 e − j 2 πp N ... e − j 2 πp N  (4.21) The b ea m p attern of UCA is obtained by applying the n ormalized b ea mforming vecto rs on the UC A steering vec tor giv en as [79]: f s p ( θ ) = w H p a s ( θ ) = N − 1 X n =0 e j pθ n e − j ζ cos( θ m − θ n ) (4.22) where sup erscript s don ates a sampled arr a y After mathematical s impliﬁcation explained in [81], equ ation 4.22 is written as: f s p ( θ ) = j p J p ( ζ ) e j pθ m + ∞ X q =1 ( j g 1 J g ( ζ ) e − j g 2 θ m + j h J h ( ζ ) e j hθ m ) = j m J m ( ζ ) e j mθ m + ǫ m (4.23) where v ariable ǫ m represent s the indu ced error whic h is the su mmation term, w hile ind exes g 1 and g 2 corresp onds to g 1 = N q − p and g 2 = N q − p . By analyzing equ ation 4.23, the follo wing observ ation is deduced: • There are t wo main terms in equation (4.23), the ﬁrst term is calle d the prin ciple term whic h is iden tical to f ar-ﬁeld pattern obtained fr om CCA. The second term is the residu al term ǫ p generated d ue to samp lin g CCA with N element s to form the UC A. • The exact v alue of the error ǫ p is not constan t b ut it v aries d ep endin g on the DOA of the receiv ed signal. W e conclude that ǫ p follo ws an exp onen tial decrease as θ e is lo w ered from 90 o to 0 o . Ho wev er, ǫ p follo ws a sinusoidal f u nction as w ith the changes in θ m . • In order to make UCA far-ﬁ eld p attern close the ideal CC A case, ǫ p m ust b e minimized .This can b e ac hiev ed by setting N to b e far larger than the an y excitation m o de which is give n by the relationship N > 2 | p | [11]. As h is the largest p ossib le phase m o de to Chapter 4. Direction o f Ar riv al 54 b e excited, then N must meet the criteria N > 2 h to m ak e f s p ( θ ) = j p J p ( ζ ) e j pθ m (4.24) 4.2.2.1.4.1 T ransformation matrix based on Phase mo de excitation on UCA F rom our analysis on phase mod e excitatio n, the steering v ector of virtual arra y a sv ( θ ) can b e constructed to contai n all individual elemen ts e ( j p θ m ) sp eciﬁed by f s p ( θ ) giv en in equ ation 4.24 for p ∈ [ − h, h ]. Mathematic ally , th e virtual steering vecto r will ha v e the follo w in g form: a sv ( θ ) =  e − j mθ m ... 1 e j mθ m  T (4.25) where sup erscript v donates a virtual arr a y T o obtain a sv ( θ ), all the b eamformer weig ht s w H p are collected to form a matrix called F of size (2 h + 1) × N w h ic h giv en as: F = 1 N =                    1 w − h w − 2 h · · · w − ( N − 1) h . . . . . . . . . · · · . . . 1 w − 1 w − 2 · · · w − ( N − 1) 1 1 1 · · · 1 1 w 1 w 2 · · · w ( N − 1) . . . . . . . . . · · · . . . 1 w h w 2 h · · · w ( N − 1) h                    (4.26) where w = e j 2 π/ N ) . The F matrix w ill provide u s with all p ossible f s p ( θ ) for p ∈ [ − h, h ]. Ho wev er, as we are only intereste d with the azim uth angle θ m pro vided in f s p ( θ ), w e need to get rid of the term j p J p ( ζ ) b y introdu cing a (2 h + 1) × (2 h + 1) d iagonal matrix J deﬁn ed as: J = d iag( 1 j p J p ( ζ ) ) , wher e − h ≤ p ≤ h (4.27) By m ultiplying F matrix with J , w e obtain a transf ormation matrix T v to map th e elemen t of UCA into the elemen ts of the v ir tual un iform linear arr a y (VULA). The transform ation matrix T v will hav e the size of (2 h + 1) × N and it is deﬁned as: Chapter 4. Direction o f Ar riv al 55 T v = FJ (4.2 8) By applying T v on the steering matrix of UCA where N ≥ 2 h , the steering matrix of the VULA will b eco me: A sv = T v A s =       e − j hθ 1 . . . e j hθ m . . . . . . . . . e j hθ 1 · · · e j hθ m       (4.29) It is clear that A sv has the same v andermonde structure as in the ULA. When imple- men ting techniques lik e UCA-Ro ot-MUSIC and UCA-ESPRIT, w e will need to transform th e whole d ata receiv ed b y UCA into that of the VULA. In a similar manner to A sv , the receiv ed data of VULA is giv en as: x v ( t ) = T v x ( t ) (4.30) 4.2.2.2 Spatial Smo othing T ec hniques As explained earlier, signals can b e either correlated or uncorrelated. In practical environ- men t, how ev er, incident signals are mostly correlated on the s en sor arra y (i.e. signals that ha v e similar pattern or tren d du r ing the time of observ ation). As it was established earlier, the signal cov ariance matrix ( R s = s ( t ) s ( t ) H ), inﬂ uences th e p erformance of the DOA algo- rithms. That is b ecause the correlatio n matrix loses its non-singularit y . A metho d prop osed to ov ercome the eﬀect of correlation on incident signals is kno wn as Spatial Smo othing. This tec hn ique pr op oses decomp osing the sensor arra y into smaller subarrays. Th e metho dology of Spatial Smo othing is derive d for ULA but it can b e extended to UCA u sing phase mo de exci- tation. Sp eciﬁcally , this tec hnique is catego rized into tw o t y p es: F orward Sp atial Sm o ot hing (FSS), F orward/Ba c kw ard Sp atial Smo othing (FBSS) [82, 83]. 4.2.2.2.1 F orw ard Spatial Smoothing (FSS) This tec h nique pr op oses dividing th e sensor arra y into o v erlapping sub-arrays. This sh all in tro duce phase sh if ts b et we en them and so resolv es th e problem of the correlated incident signals. F or the FSS, let u s consider an arra y sensor of 6 elemen ts as shown in Figure 4.6. Chapter 4. Direction o f Ar riv al 56 Those elemen ts are to b e divided in to 4 o v erlapping sub-arrays of length 3 eac h ( L ss =4, p ss =3). The incident s ignal, th us, m o deled as the follo win g [82]. subarray 1 subarray 2 subarray 3 subarray 4 sensor array of 6 elements Figure 4.6: FSS sp atial smo othin g x F k = AD ( k − 1) s ( t ) + n k ( t ) (4.31) where ( k -1) denotes the k th p o we r of the diagonal matrix D : D = diag  e − j 2 π λ sin( θ 1 ) . . . . . . e − j 2 π λ sin( θ M )  (4.32) The spatial co v ariance matrix R can b e mo d eled as th e co v ariance matrices of the forward sub-arrays as the follo wing, Figure 4.7: R = 1 L ss L ss − 1 X K =0 R F k (4.33) The division of the sensor arra y s cannot b e done r andomly . There are some r ules to b e follo wed to obtain the optimum results of the FSS: • The ﬁrst ru le imp lies th at the num b er of th e sub-arrays must b e greater th an th e n umber of elemen ts of the d iagonal matrix D . L > M , ( N − p ss + 1) > M (4.34) • The s econd ru le implies that the num b er of elemen ts in eac h sub-arr a y must b e greater Chapter 4. Direction o f Ar riv al 57 R11 R12 R13 R14 R15 R16 R21 R22 R23 R24 R25 R26 R31 R32 R33 R34 R35 R36 R41 R42 R43 R44 R45 R46 R = R51 R52 R53 R54 R55 R56 R61 R62 R63 R64 R65 R66 Figure 4.7: App lying FSS on Matrix R than the n um b er of elements of th e diagonal matrix D y et less th an N . N > p ss > M (4.35) Com bining the rules results in the follo wing conclusion: the min im um v alue of p ss can b e obtained at M max +1. Substituting that in the equations leads to the f ollo wing conclusion: the maximum n um b er of correlated signals that ma y b e d etected by FSS metho d is equiv alent to N / 2. The num b er of un correlated signals that can b e detected by con v entional MUSIC algorithm, wh ich will b e discus sed later, is ( N − 1). It can b e obs er ved that this num b er of correlated signals that can b e detected is less than the num b er of uncorrelated signals that can b e detected b y con v entional MUSIC algorithm. 4.2.2.2.2 F orw ard/Bac kw ard Spatial Smo othing (FBSS) This tec h nique pr op oses dividing th e sensor arra y into o v erlapping sub-arrays. This sh all in tro duces phase sh ifts b et wee n them. This will solv e the problem of the correlated inciden t signals; it aims to increase the num b er of detectable correlate d signal signiﬁcan tly from N / 2 to 2 N / 3. The principle of this tec hnique is quite simple. It works by utilizing the principle of conjugate of the forw ard spatial smo othing (FSS). In other w ord s, using a set of forward sub-arrays and their conju gate,as shown in Figure 4.8 [82]. Chapter 4. Direction o f Ar riv al 58 Sensor array of 6 elements forward subarray 1 forward subarray 2 forward subarray 3 forward subarray 4 backward subarray 1 backward subarr ay 3 backward subarray 4 Forward Smoothing Backward Smoothing backward subarray 2 Figure 4.8: F orward/Bac kwa rd s p atial smo othing Figure 4.8 sh o w s the prin ciple of FBSS applied on a sensor arra y of 6 elemen ts. The 6-elemen t arra y is divided int o 4 o v erlapping forward sub-arrays and 4 o v erlapping bac kw ard sub-arrays (i.e. L F = 4 and L B = 4). Ea c h sub-array is of size 3 (i.e. p ss = 3). The receiv ed signals v ector in the case of the FBSS at the k th bac kw ard s u b-arra y is giv en as the follo win g [82, 8 4]: x B k = AD ( k − 1) [ D ( N − 1) s ( t )] ∗ + n k ( t ) (4.36) And the spatial cov ariance matrix R in this case can b e calculated b y the follo wing equa- tion: R = R F + R B 2 (4.37) where R F represent s the av erage co v ariance matrix in the case of the forw ard su b-arra y v ectors, and R B resem bles the a v erage cov ariance matrix of the bac kw ard sub-arra y v ectors. The rules of dividing the sensor arra y in to sub-array in the FSS case are also applicable in the case of the FBSS. Th at is 1. T he num b er of the su b-arra ys m u st b e greater th an th e num b er of elemen ts of the diagonal matrix D . 2. T he num b er of elemen ts in eac h sub-arr ay must b e greater than the n um b er of elemen ts of the d iagonal matrix D yet less than N . Chapter 4. Direction o f Ar riv al 59 It is imp ortant to highligh t that these rules pu t restrain ts up on the size of p ss and the n umber of su b-arra y L ss . In other w ords, for a successful detection of correlated signals, these rules and constrains must b e met or else the detection will b e in accurate. If w e go bac k to the example of the sensor arra y of elemen ts, the r ules suggest that the sys tem will fail to detect th ree signals if the an tenna arra y is divided into L ss =2 o v erlapping sub -arra y with eac h h a v in g 5 elemen ts ( p ss =5). While, it will succeed if it is divided in to th r ee sub -arra y ( L ss =3) with eac h ha ving 4 elemen ts in ( p ss =4). It is signiﬁcan t to mentio n that th e FSS may detect u p to N/ 2 correlated signals whereas the FBSS may d etect up to 2 N/ 3 correlated signals. Th is means that the FBSS w ill succeed in detecting signals that FSS will fail at if the range of th e signals is ab o v e N / 2 yet less than 2 N/ 3. None of th em though is comparable to the MUSIC algorithm when d etecting uncorrelated signal since it can detect up to N − 1 uncorrelated signal [3, 8 2]. 4.2.2.3 T o eplitz Algorithm When all receiv ed signals are correlated, th e resulting co v ariance matrix b ecome singular. That is b eca use the rank of the co v ariance matrix is diminished from N to 1. A metho d to o vercome rank deﬁcient is by applying spatial smo othin g tec hn ique on the co v ariance matrix. This metho d as pr evious explained is ju st a rank reconstruction allo wing N / 2 and 2 N / 3 of coheren t signals to b e detected using FSS and FBSS. Ho wev er, the spatial smo othing metho d is done by d ivid ing the main arra y in to subarra ys an d the smo othed co v ariance is obtained b y av eraging the individual co v ariance matrix of eac h subarrays. Th is clearly implies that the de-correlation p erform an ce of spatial smo othing is done at the cost of reducing the size of the main arra y . Unlik e spatial smo othing, the T o eplitz algorithm do es not redu ce the size of the m ain arra y . P recisely , the T oeplitz algorithm is implemented by constructing a T oeplitz matrix from th e co v ariance matrix of the correlated source. The generated T o ep litz matrix is a diagonal matrix of rank N . T herefore, T oeplitz algorithm provides a full de-correlatio n for receiv ed coherent sou r ces allo wing N -1 coheren t signals to b e detected [85 ]. T o exp lain the mo deling of T o eplitz algorithm, we w ill b egin by the mathematical mo del of the sin gu lar co v ariance matrix R c of correlated s ignals. Th en , we will sho w ho w the T o ep litz algorithm construct a non-signula r matrix R T from R c . Chapter 4. Direction o f Ar riv al 60 4.2.2.3.1 Mo deling of Singular Co v ariance Matrix In a coherent environmen t, all the signals impinging on ULA will h a ve the same ph ase. Assuming M r eceiv ed correlated signal with amplitude ρ m where m =  1 , 2 , . . . , M  . The signal co v ariance matrix is realize d as: R s =          ρ 2 1 ρ 1 ρ 2 . . . ρ 1 ρ M ρ 2 ρ 1 ρ 2 2 · · · ρ 2 ρ M . . . . . . . . . . . . ρ M ρ 1 ρ M ρ 2 · · · ρ 2 M          (4.38) Consequent ly , the co v ariance matrix R c of the ULA is giv en as R c = A H s R s A s =          R c (1 , 1) R c (1 , 2) . . . R c (1 ,N ) R c (2 , 1) R c (2 , 2) . . . R c (2 ,N ) . . . . . . . . . . . . R c ( N , 1) R c ( N , 2) . . . R c ( N ,N )          (4.39) Due to the multiplicatio n of R s with A and its complex conjugate A H , R ( c ( i,j )) will represent the complex conjugate of R ( c ( j,i )) . Hence R c can b e simpliﬁed as: R c = A H s R s A s =          R c (1 , 1) R c (1 , 2) . . . R c (1 ,N ) R ∗ c (2 , 1) R c (2 , 2) . . . R c (2 ,N ) . . . . . . . . . . . . R ∗ c ( N , 1) R ∗ c ( N , 2) . . . R c ( N ,N )          (4.40) With th e coherence b et w een receiv ed signals, R c b ecomes a singular matrix. In a singular matrix, any ro w can b e written as linear combination of other ro ws and the same thing is applied to the column . In order to d e-correlate R c , we n eed to remov e the linear com bination relationship among the ro ws and the columns. 4.2.2.3.2 Realization of T o eplitz Algorithm By using the T o ep litz algorithm, the lin ear com b ination relation is o vercome by taking the ﬁrst row and column in matrix R c , wh ile omitting the rest of the elemen ts. Then, the tak en elemen ts are used to construct a diagonal matrix that is non-singular. Du e to th e fact,t hat Chapter 4. Direction o f Ar riv al 61 the ﬁrst ro w in R c is a complex conjugate f or the ﬁ rst column, we only need to obtain the actual v alues of the ﬁr s t ro w. By usin g the steering v ector a s ( θ m ) of ULA giv en in equation 4.2 the ﬁ rst row elemen ts of R c are giv en th r ough ve ctor V c as [86]: V C =          R C (1 , 1) R C (1 , 2) . . . R C (1 ,N )          =          ρ 1 α + · · · + ρ M α ρ 1 αe − j Φ 1 + · · · + ρ M αe − j Φ m . . . ρ 1 αe − j ( N − 1)Φ 1 + · · · + ρ M αe − j ( N − 1)Φ m          (4.41) where α is a constant deﬁ n ed as α = ρ 1 + · · · + ρ M Also, v ector R c can b e r ealized a m ultiplication of t w o matrix giv en as: V C =          R C (1 , 1) R C (1 , 2) . . . R C (1 ,N )          =          1 1 . . . 1 e − j Φ 1 e − j Φ 2 · · · e − j Φ m . . . . . . . . . . . . e − j ( N − 1)Φ 1 e − j ( N − 1)Φ 2 · · · e − j ( N − 1)Φ M                   ρ 1 ρ 2 . . . ρ M          α (4.42) V C = A s S C (4.43) Lastly , based on the v alues sp ec iﬁed in vect or V c , T o eplitz matrix R T is constructed to replace the original cov ariance matrix R c and its v alue is mo deled as [85, 86]: R T =          R C (1 , 1) R C (1 , 2) · · · R C (1 ,N ) R ∗ C (1 , 2) R C (1 , 1) . . . R C (1 ,N − 1) . . . . . . . . . . . . R ∗ C (1 ,N ) R ∗ C (1 ,N − 1) . . . R C (1 , 1)          = A H s f R S A s (4.44) where f R s is a d iagonal matrix conta ining the v alues of R ( c (1 , 1)) . Hence, f R S = diag { αρ 1 , . . . , αρ M } (4.45) By analyzing R T , the follo wing observ ation is d educed: • T o eplitz algorithm can completely resolv e correlated signal as R T is a diagonal matrix of rank N . I n other w ords, T o eplitz algorithm allo ws N -1 coherent signal to b e detected Chapter 4. Direction o f Ar riv al 62 compared to N/ 2 and 2 N/ 3 detected coheren t signals usin g FSS and FBSS . • T o eplitz algorithm enh ances the p o w er of r eceiv ed s ignals from ρ 2 m to αρ m = ( ρ 1 + ρ 2 + . . . + ρ M ) ρ m . Th is will lead to a more sh arp p eaks on MUSIC angular sp ectrum compared to spatial smo othin g indicating that T o eplitz algo rithm has more robu st p erformance [85]. Due to this sup erior p erformance, the T oeplitz algorithm can even detect a mixture of correlated and uncorrelated signals up to N -1. • The computation load of T o eplitz algorithm is less compared to spatial sm o ot hing tec h- nique as it has a diagonal structure 4.3 DO A Algorithms DO A algorithm are used to pr o vide an estimation of the DOA for inciden t signals impinging on a sensor arra y . A t yp e of DOA algorithm w hic h p ro vides a high-resolution and accurate DO A estimation are kno wn as s u bspace-based algorithms. S uc h tec hn iques op erate on th e input co v ariance matrix w hic h can b e decomp osed into eigen v alues and eigen vec tors b elo ng to the signal subspace while the reset b elong to the n oise subspace. The reason that subspace- based algorithm adopted this name is due to the existence of b oth n oise and signal sub space surroun ding the sensor arr ay . In our pro ject, we will fo cus on three most eﬃcient subspace- based tec hniques f or DO A estimation whic h are MSUC, Ro ot-MSUIC and ESPR I T. The latter t w o tec hnique are only applicable to ULA so phase mo de excitation is used to obtain their equiv alen t v ers ion in UCA n amely UCA-ro ot-MUSIC and UCA-ESPRIT. 4.3.1 MUSIC The MUSIC algorithm is one of th e earlier tec h nique pr op osed by Sc hmidt [87] to oﬀer a high-resolution detectio n f or incoming signal DOA impin ging on the sensor array . The main principle of MUSIC algorithm is based on exploiting the eigenstructure of the cov ariance matrix R , whic h is practically calculated by taking K snapsh ots of the inciden t signal x( t ) follo wed by av enging pro cess ov er K . Thus, the co v ariance matrix, is expressed as: R = 1 K K X t =1 x ( t ) x ( t ) H (4.46) Chapter 4. Direction o f Ar riv al 63 T o obtain the full equ ation of co v ariance matrix R , the signal mo del x ( t ) in equation 4.1 is substituted into 4.46 , whic h yields: R = A s R s A H s + σ 2 n I (4.47) where R s is the signal co v ariance matrix, σ 2 n is the noise v ariance and I is an iden tit y matrix of dimension N × N [88]. The eigen v alues of R can b e exp ressed as { γ 1 , γ 2 , γ 3 , . . . , γ N } wh ic h are obtained u sing the equation: | R − γ i I | = 0 (4.48) Similarly , the eigen ve ctors of R is expressed as { q 1 q 2 q 3 . . . q N } and they must satisfy the follo w ing condition: ( R − γ i I ) q i = 0 (4.4 9) Using equation 4.47, equation 4.48 can b e extended as:   A s R s A H s + σ 2 n I − γ i I   = A s R s A H s − ( γ i − σ 2 n ) I (4.50) Based on the deﬁn ition of eigen v alues, we conclude that the terms A s R s A H s has γ i − σ 2 n eigen v alues. The matrix A s con tains a linearly indep endent s teering ve ctor signals with dimension N × M . Thus, to p erform eigen decomp osition on R , signal co v ariance matrix R s m ust b e non-singular wh ic h is guarantee d as long as the incident signals are uncorrelated and the incoming signals M is less than the elements N in the s ensor arr a y . Based on this condition, it is concluded that the term A s R s A H s has zero eigen v alues for ( N − M ). Thus, N eigen v alues of R can b e sorted into signal eigen v alues with largest M eigen v alues while the r emaining ( N − M ) eigen v alues corresp onds to the noise v ariance σ 2 n [74]. The key secret b ehind MUSIC algorithm is that the n oise subs pace eigen v ectors V n asso- ciated w ith the noise eigen v alues are orth ogonal to the s teering v ectors making up the matrix A s . This can b e p r o v ed b y mo d eling the noise eigen v ectors, whic h corresp ond s to ( N − M ) Chapter 4. Direction o f Ar riv al 64 eigen v alues as: ( R − σ 2 n I ) q i = A s R s A H s q i + σ 2 n Iq i − σ 2 n Iq i = 0 (4.51) A s R s A H s q i = 0 ( 4.52) As R s is non-singular, the only wa y for A s R s A H s q i = 0 to b e zero is b y setting A H s q i =0 whic h pr o ves the orth ogonalit y b et w een the noise eigen v ectors and matrix A s . This allo ws the MUSIC angular sp ectrum to b e exp ressed as: P ( θ ) = A s ( θ ) H A s ( θ ) A H s ( θ ) V n V H n A s ( θ ) (4.53) The orthogonali t y b et w een A s and V n will minimize the denominator w hic h giv es p eaks in th e MUSIC angular sp ec trum. The lo cations of the p eaks will corresp ond to the correct angle of arriv al of the incoming in ciden t signals. T o determine the righ t DO A, MUSIC requ ir e a searc h throu gh all p ossib le steering ve ctors unti l the correct steering vec tor em b o d ied on the co v ariance matrix R is foun d. Giv en a sensor arra y of N element s, the MUS I C can detect up to ( N -1) uncorrelated signals in b oth ULA and UCA geometry [87]. 4.3.2 Ro ot-MUSIC Ro ot-MUSIC was deve lop ed to reduce the computational load of MUSIC algorithm. In MSUIC algorithm, the DOA is estimated b y an exhaustive searc h thr ough all p ossible steering v ectors that are orthogonal to the noise eigen v ectors. In Ro ot-MUSIC, the DOA is estimated via the zeros of a p olynomial so the exhaustiv e sp ectral searc h emp lo yed by MSUIC is a v oided. Ro ot-MUSIC is only app licable to ULA as the steering vec tor of ULA has a v and ermonde structure allo wing th e denominator of MUSIC sp ectrum equation to b e mo d elled as a p oly- nomial [81, 89]. T o formulate the p olynomial, the notation of the steering v ector in ULA is sligh tly m o diﬁed to h av e the b elow f orm: Chapter 4. Direction o f Ar riv al 65 a s ( θ ) =          1 e 2 π ( d λ ) sin ( θ ) . . . e 2 π ( d λ )( N − 1) sin ( θ )          =          1 z . . . z N − 1          = a s ( z ) (4.54) By substituting a s (z) in 4.53, the d enominator of MUSIC algo rithm can b e mo delled by the follo w ing p olynomial: Q Root − M U S I C ( z ) = a T s (1 /z ) V n V H n a s ( z ) (4.55) The p olynomial Q Root − M U S I C ( z ) b ecome zero wh en z= z m = e j 2 π ( d λ ) sin( θ m ) whic h corre- sp ond the actual DO A. T herefore, the true DOA can b e iden tiﬁed by ﬁ nding the ro ots of Q Root − M U S I C ( z )=0 and identiﬁes the on e close to the unit circle. Also to men tion that the p olynomial Q Root − M U S I C ( z ) has a 2( N -1) ro ots that comes as r ecipro cal conjugate p air. In other w ords, if z is a ro ot of Q Root − M U S I C ( z ) , th en 1 z ∗ m ust also b e one of its ro ots. After the closet ro ot s z m to the unit circle are iden tiﬁed, the tru e DOA are giv en as: θ m = arcsin  ( λ 2 π d  arg( z m )) (4.56) 4.3.3 UCA-Ro ot-MUSIC This algorithm wa s mainly devel op ed to make Ro ot -MUSIC applicable to UCA. As we know, the stand ard Ro ot-MUSIC is implemented by expressing the d enominator of MUSIC sp ec trum as a p olynomial. Ho w ev er , constructing a p olynomia l is done on the assumption that the steering vecto r has a v andermonde stru cture whic h is true for ULA. As we pr o ceed to UCA, this assu mption is no longer tru e du e to the dep endence of the sensor an gu lar lo ca tion. T o ov ercome this pr oblem, phase mo de excitatio n is used wher e the UC A b eam p attern is transformed into a b eamspace in which the steering v ector follo w s a v andermond e structure. A transformation matrix T v , in equation 4.26, is d eriv ed based on the phase mo de excitati on to directly m ap UCA into desired VULA. The implementa tion of UCA-Ro ot-MUSIC is summarized in the follo win g s tep: • Obtain the transform ed receiv ed data of UCA giv en as x v ( t )= T w x ( t ) w h ere T w = Chapter 4. Direction o f Ar riv al 66 (T H v T v ) ( − 1 / 2) T v . Here, we can not directly app ly T v to the UCA receiv ed data as T H v T v 6 = I which, in terms, leads to a VULA having a color noise. Th is will imp ose a problem as Ro ot-MUSIC algorithm assume a while bac kground noise. T o o v ercome this pr oblem, T w is used instead which ob eys a unitary transf orm ation( T H w T w = I ). Th e unitary pr osp erit y of T w is obtained by using a prewh itening sc heme em b o died in the term (T H v T v ) ( − 1 / 2) to con v ert the color n oise bac k into a white noise [90, 91]. • Calculate the VULA cov ariance matrix as R v = 1 K K P t =1 x v ( t ) x v ( t ) H for K snapshots • P erform E igen-v alue decomp osition on R v to obtain V n noise eige nv ectors wh ic h cor- resp ond to the smallest h − M eigen v alues • T o constru ct a p olynomial , th e VULA steering v ector in 4.23 is realized as a sv ( θ ) =             e − j hθ . . . 1 . . . e j hθ             =             z − h . . . 1 . . . z h             = a sv ( z ) (4.57) where z= e j θ • Substituting a sv ( z ) in 4.57, a pr ewhitened p olynomia l is constructed as [92]: Q U C A − Root − M U S I C ( z ) = a T sv (1 /z )( T H v T v ) − 1 / 2 V n V H n ( T H v T v ) − 1 / 2 a sv ( z ) H (4.58) • Just lik e in Ro ot -MUSIC, the DOAs of UCA are estimated from the largest-magnitude ro ots z m of Q U C A − Root − M U S I C ( z ) and their estimated v alues are giv en as: θ m = arg( z m ) (4.59) • Due to the app r o ximation in 4.22, there w ill b e a b ias in the estimated θ m . T o obtain θ m with relativ ely less bias, N m ust b e selected based on the criteria N ≥ 2 h and the larger the N is, the lo w er the bias. Chapter 4. Direction o f Ar riv al 67 4.3.4 ESPRIT A diﬀeren t subsp ace-based metho d for estimating DOA was in tro duced by Ro y and Kailath [93] whic h called Estimation of s ignal parameters via rotation inv ariance tec hniqu es (ES- PRIT). S uc h tec hnique has an adv ant age ov er MUSIC that th e DOA is estimated dir ectly from the incident signal eigen v alues without going through the exhaustive searc h of all p ossible steering v ectors. This, in terms, r ed uces th e ov erall computation and storage requir emen ts as in Ro ot-MSUIC. ESPRIT op erates by exploiting the prop erty that a sensor arr a y structure can b e decomp osed into t w o iden tical su barra ys ha ving the same size called doublets [94]. These d oublets are d isp laced from eac h other by a ﬁxed distance ∆ x as s h o wing in Figure 4.9. Therefore, ESPRIT can b e applied to ULA as its s tructure can simp ly b e divided in to o verlapping s ubarray Figure 4.9: Illustr ation of Sensor Arr a y Using ESP R I T Algorithm Assuming ULA receiving M incident signal, then eac h subarra ys, as sho wn in Figure 4.9, will receiv ed a data v ector x 1 ( t ) and x 2 ( t ) for subarray-1 and subarr a y-2, resp ective ly . Th e com b ined outpu t x ( t ) of b oth su barraies is mo delled as: x ( t ) =    x 1 ( t ) x 2 ( t )    =    A s A s Φ    s ( t ) +    n 1 ( t ) n 2 ( t )    (4.60) where Φ represents a diagonal matrix conta ining the ph ase shifts b etw een the doublets for M inciden t signals and it is giv en as: Chapter 4. Direction o f Ar riv al 68 Φ = diag  e j k ∆ xsin ( θ 1 ) e j k ∆ xsin ( θ 2 ) · · · e j k ∆ xsin ( θ M )  (4.61) The Φ represents a scaling op erato r to r elates the ﬁr st su barra y measurements x 1 ( t ) to the second subarra y measurement s x 1 ( t ). Also, Φ op eration can b e analysed as tw o d im en sion rotation and hence it is called a rotational op erator [20]. F rom the receiv ed signal x ( t ), the correlation matrix R can b e formed where the largest M eigen v alues corresp ond s to the signal eigen vect ors V s . Due to the existence of Φ b et we en the subarrays, the signal eigen vect ors V s is linke d to the steering v ector matrix A s through n on -sin gular m atrix called T w h ic h deﬁned as: V s =    A s A s Φ    T (4.62) Lastly , matrix Ψ is deﬁn ed as Ψ = T − 1 ΦT where the eigenv alues of Ψ corresp onds to the diagonal elements of Φ wh ile eigenv ector of Ψ corresp onds to the column elemen ts of T . F rom equation 4.62, V s can b e d ecomp osed int o V 1 = A s T and V 2 = A s Ψ T and hence: V 2 = ΦV 1 (4.63) This sho ws that the k ey secret b ehind ESPRIT is the abilit y of Ψ eigen v alues whic h are Φ to map the signal subspace of V 1 that span signal subsp ace of V 2 . Practically , it is imp ossible to ac hiev e the relationship in equation 4.63 due to the introdu ction of n oise in the measuremen ts and the system is ov er determinan t, hen ce, Φ can b e estimated usin g statistic tec hn iques like L east S q u are (LS). After Φ is obtained, the DOA ( θ m ) is expr essed as: θ m = arcsin( arg( φ m ) k ∆ x ) (4.64) where Φ are the eigen v alues of Ψ , k = 2 π λ and ∆ x is the d isplacemen t b etw een the t wo subarrays. Chapter 4. Direction o f Ar riv al 69 4.3.5 UCA-ESPRIT This algorithm w as dev elop ed to mak e ESPRIT app licable to UCA. As discussed in s ection 4.3.4, The ESPRIT tec h nique is designed to w ork with arra y geomet ry that can b e decom- p osed into t w o id en tical o v erla yings. In other w ords, ES PRIT can b e on arr ay geometry th at has a v and er m onde structure just lik e th e ULA. Ho w ev er, the UCA do es not ha ve a v and er- monde structure. This problem is resolv ed b y tran s forming the UCA into VULA usin g the transformation matrix T v giv e in 4.26. Then, the azim u th angles are estimated from VULA using the stand ard ESPRIT. The implementa tion of UCA-ESPRIT is su mmarized in th e follo wing steps: • Obtain the observ ation ve ctor of VULA as x v ( t ) = T v x ( t ). Unlik e UCA-Ro ot-MUSIC, the noise-prewh itening tec hnique cannot b e emplo y ed h ere as it will d estro y the shift- in v ariance b et we en the t w o d ecomp osed VULAs. By repr esen ting T v in its EVD form ( T v = VΛV H ) , it is dedu ced that b oth VULAs will ha v e their noise giv en as W 1 = σ 2 I and W 2 = σ 2 n Λ 2 . Eve n though th e noise v ariance of the second VULA is pr op ortional to the square of T v eigen v alues, b oth noises are still spatially white and so we can pro ceed with the ESPRIT Algorithm [95]. • Calculate th e VULA co v ariance matrix as R v = 1 K K P t =1 Q c x v ( t ) x v ( t ) H Q H c where Q c is centro- hermitian m atrix that ob eys Q c Q H c = I . The pu rp ose of Q c is to reduce the computational load of EVD op eration. • P erform EVD on R v to obtain E s = Q c V s whic h corresp ond s to the largest M eigen- v alues. As we are in terested V s , the equation is rearranged as V s = Q H c E s . • By Decomp osing the main VULA in to tw o identic al su barra ys with one in ter-element spacing b etw een them, as in Figure 4.3.4, then, V s is also decomp ose d int o V 1 and V 2 where V 1 has the ﬁrst M -1 elemen ts of V s while V 2 has the last M -1 elemen t of V s . • Estimate Φ from the relationship V 2 = Φ V 1 using statistic tec hniques like Least Square. • Due to p hase m o de excitation, Φ of the VULA will b e in the th eoretical form give n as: Φ = diag  e j θ 1 e j θ 2 . . . e j θ N  (4.65) Chapter 4. Direction o f Ar riv al 70 Hence, the estimated DO A ( θ m ) is estimated as: θ m = arg(Φ m ) (4.66) • Due to the appro ximation in equation 4.24, Φ will b e close b ut not equal to its th eoretical form giv en in equation 4.65 and hence a bias will o ccur in the estimated θ m . T o obtain θ m with relativ ely less bias, N m ust b e selected based on the criteria N ≥ 2 h and the larger is N , th e lo w er the bias. 4.3.6 Comparison in the Performance of ULA and UCA In this section, w e compare th e p erformances of ULA and UCA usin g MUSIC algorithm. MUSIC is selected here b eca use it oﬀers a sp ectrum whic h can b e used to extract inform ation ab out the charact eristics of b oth geometries. F or a fair comparison, b ot h th e ULA and UCA will ha v e parameters (SNR = 15dB, N = 8 and K = 100) in uncorrelated environmen t. T he spacing b etw een the arra y elemen ts in b oth geometry is chosen to b e 0.5 λ whic h is r equ ired as minim um distance for MUSIC algorithm to w orks. Both geometries will receiv e three signals with DO A (85 o , 0 o , − 85 o ) transmitted o v er A W GN c h annel. The results for ULA and UCA are plotted on Figure 4.10. Fig ure 4.10 sho ws a comparison of the tests of the ULA and the UCA, it can b e seen that ULA is n ot very capable of in terpreting correctly when the steering vec tors are b eing emitted fr om wider angles, close to the endﬁ re d irection ( θ = ± 90 o ). Th is is b eca use the p o we r distribu tion gets weak er at th e edges of th e linear arra y geometry . Whereas in the case of the UCA, th e geometry makes p o we r distribu ted equally in all directions. Another drawbac k of the ULA is that it pro d uces a sym metric angular sp ectrum that w as seen in b oth tests of the ULA. In other w ords, wh en an angle of 15 o in needed, a p eak at the angle app ears as well as a p ea k at an angle of 180 o − 15 o = 165 o app ears that is undesired . This in tro duction of an und esir ed angle causes an am b iguit y in the inte rpretation of th e results. As seen, this was r esolved when using UCA, whic h only pro du ced a single p eak at the desired angle. This comparison yields the follo wing conclusion: Chapter 4. Direction o f Ar riv al 71 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm Relative Power (dB) Actual DOA Estimated DOA by ULA Estimated DOA by UCA Figure 4.10: DOA estimation using ULA and UCA 4.3.7 DO A A lgor ithms Performance In this section, we will analyse thoroughly the p erformance of all DOA algorithms w hic h explained in the rep ort n amely MUSIC, Ro ot-MUSIC and ES PRIT. T o hav e a w ell-established p erformance analysis of DO A algo rithms, all the parameters in v olv ed in DO A algorithms to detect the DOA of the inciden t s ignals will b e inv estigated separately . By this metho dology , w e will b e able to r ealize the accuracy and th e capacit y of eac h algorithm. Also, w e will b e able to realize how to eﬀectiv ely implement th em in r eal w orld with less p ossible requir ements. The parameters can b e classiﬁed und er t w o categories. The ﬁrst category is related to the sensor arra y geometry used whic h represent ed by the n umber of sensor elemen ts b een arranged on the sensor array . The second category is related to the signal environmen t impact whic h includes the num b er of incoming signals, the angular separation b et we en these them. Also, it includes the num b er of samples tak en for the receiv ed signal, their SNR ratio and their t yp es whether they are correlated or u ncorrelated. In simulati on, the noise is mo delled as Chapter 4. Direction o f Ar riv al 72 A W GN. It is crucial to note that that some algorithms, Ro ot-MSUIC and ESPRIT, in UCA require the use of phase mo d e excitation (PME) to b e imp lemen ted. Ho w ev er, the u s e of PME will result in a small quantiza tion err or ( ǫ q ). T o consider this issue in our analysis, w e will study DOA algorithms in resp ect to ULA and UCA. The true accuracy of DO A algorithms can b e dedu ced fr om ULA simulation and its results is almost stable. In UCA, the results is not stable due to ǫ q . T o reﬂect the imp act of ǫ q in the angle measurements, the maxim um p ossible error will b e measured u nder the v ariable margin error. ǫ q can b e reduced in UCA b y increasing N to allo w accurate detection. Also, as PME is f u nction of elev ation angle it is assumed to b e 20 o ﬁxed for all s ignals receiv ed b y UCA. In the sim ulation, th e results of MUSIC algorithm represent angular sp ectrum graphs while the results of Ro ot-MUSIC and ESPRIT are n umerically listed in a table. 4.3.7.1 DOA Algorithms P erformance in ULA 4.3.7.1.1 Num b er of Sensor E lemen ts In this test, the impact of ULA num b er of sensors is in v estigated on DOA algorithms b y applying t w o incoming signals ( − 10 o , 10 o ) with N =4 and N =8 on ULA. The r esults show that increasing the n umber of element s in ULA will improv e the DO A estimation in all DO A algorithms. The impro v emen t in MUSI C sp ec trum is evident by sh arp er p ea ks at the directions of in ciden t signals and lo w er noise ﬂo or in Figure 4.11. Th is is also clear in Ro ot - MUISC where the r o ots b ecome closer to the u nit circle allo wing accurate d etection as giv en in T able 4.1. T he enhancemen t in ESPRIT is in a form of more accurate estimation as sho wn in T able 4.2. Chapter 4. Direction o f Ar riv al 73 −90 −70 −50 −30 −10 10 30 50 70 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in ULA Relative Power (dB) N=4 N=8 Figure 4.11: Impact of c han ging the n umber of elemen ts in UCA on th e p erformance of MUSIC algorithm with settings (M=2, θ = 10 o and − 10 o , d = 0 . 5 λ , S NR=10dB and K =100 T able 4.1: Imp act of c hanging th e n umber of elemen ts in ULA on the p erformance of Ro ot- MUSIC algorithm with settings ( M =2, θ = 10 o and − 10 o , d=0.5 λ , SNR=10dB and K =100) Chapter 4. Direction o f Ar riv al 74 T able 4.2: Impact of c hanging the n umber of elemen ts in ULA on the p erformance of ESPRIT algorithm with settings ( M = 2, θ = 10 o and − 10 o , d= 0.5 λ , SNR=10dB and K =100) 4.3.7.1.2 Num b er of Inciden t Signal In the second test, the impact for the num b er of incoming signals on DO A algorithm is examined b y considerin g tw o scenarios with 4 in coming signals ( − 20 o , − 10 o , 0 o , 10 o ) and 2 incoming signal (0 o , 10 o ) impinging on ULA. F rom Figure 4.12, we conclude that as the inciden t signals increases, the p erformance of MUSI C will starts to degrade leading to less sharp p eaks. This eﬀect is also clear fr om T able 4.3 and 4.4 where the detection accuracy of Ro ot-MUSIC and ESPRI T is redu ced. This problem can b e resolved by increasing the n umber of sen s or elemen ts in an arr a y . Chapter 4. Direction o f Ar riv al 75 −90 −70 −50 −30 −10 10 30 50 70 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in ULA Relative Power (dB) θ = −20 ° −10 ° 0 ° 10 ° θ = 0 ° 10 ° Figure 4.12: Impact of c hanging the num b er of incident signals impinging ULA on the p er- formance of MUSIC algorithm with settings ( N =6, d =0.5 λ , S NR= 20dB and K =100) T able 4.3: Imp act of c hanging the num b er of inciden t signals im p inging ULA on the p erfor- mance of Ro ot-MUSIC algorithm with settings ( N =6, d =0.5 λ , SNR=20dB and K =100) Chapter 4. Direction o f Ar riv al 76 T able 4.4: Imp act of c hanging the num b er of inciden t signals im p inging ULA on the p erfor- mance of ES PRIT algorithm with settings ( N =6, d =0.5 λ , SNR=20dB and K =100) 4.3.7.1.3 Angular Separation b et ween Inciden t Signals In this test, the im p act of the angular separation b etw een the incoming signals on DO A algorithm is examined by consid ering t w o scenarios. I n the ﬁrst scenario, ULA will receiv e t w o inciden t signals (10 o , 20 o ) h aving a s m all angular s ep aration of 10 o . In the second scenario, the angular separation w ill b e increased to 50 o as the t wo incident signals will hav e a direction of arriv al of 10 o and 60 o . Clearly , F r om Figure 4.13, increasing the angular separation will impro v e the p erformance of MUS I C algorithm through p ro ducing sharp sp ectral p ea ks and reduce th e noise ﬂo or. Th e same conclusion is deduced from T able 4.5 where the detection accuracy of Ro ot -MUSIC and ESPRIT is increased as the angular separation is increased. Chapter 4. Direction o f Ar riv al 77 −90 −70 −50 −30 −10 10 30 50 70 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in ULA Relative Power (dB) θ = 10 ° and 20 ° (angular separation = 10 ° ) θ = 10 ° and 60 ° (angular separation = 50 ° ) Figure 4.13: Impact of c hanging the an gu lar separation b et w een the inciden t signals impinging on ULA on the p erformance of MUSIC algorithm with settings ( N = 3, d =0.5 λ , SNR=10dB and K = 100) T able 4.5: Imp act of c hanging the angular separation b et wee n the incident signals impin g- ing on ULA on the p erform an ce of Ro ot-MUSIC algorithm with settings ( N =3, d =0.5 λ , SNR=20dB and K =100) 4.3.7.1.4 Num b er of Samples In the follo wing test, the impact of samples num b er tak en f or the incoming signals on DOA algorithm is examined by applyin g tw o incoming signals ( − 20 o , 20 o ) on with K =50 and Chapter 4. Direction o f Ar riv al 78 K =500. F rom Figure 4.14 , we conclude that increasing the n umb er of samples enhances the p erformance of MUSIC algorithm as the p eak b ecomes sharp er and the noise ﬂo or is lo w ered. This conclusion is also evident from T able 4.6 and 4.7 where the detection accuracy of ro ot- MUSIC and ESP RIT has increased. The reason for this impro v emen t is b eca use increasing the num b er of s amp les will lead to a more accurate estimation of inciden t signals. Hence, a co v ariance matrix w ill b e more accurate as we ll. −90 −70 −50 −30 −10 10 30 50 70 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in ULA Relative Power (dB) K=50 K=500 Figure 4.14 : Impact of c h anging th e n um b er of samples of the incident signals impin ging on ULA on the p erf orm ance of MUSIC algorithm with settings ( N =5, θ = 20 o and − 20 o , d =0.5 λ , SNR=10dB and K =100) Chapter 4. Direction o f Ar riv al 79 T able 4.6: I m pact of c hanging the num b er of samples of the inciden t signals impinging on ULA on the p erformance of Ro ot -MUSIC algorithm with settings ( N =5, θ =20 o and − 20 o , d =0.5 λ , S NR=10dB and K =100) T able 4.7: I m pact of c hanging the num b er of samples of the inciden t signals impinging on ULA on the p erformance of ES PRIT algorithm with settings ( N =5, θ = 20 o and − 20 o , d =0.5 λ , SNR=10dB and K =100) 4.3.7.1.5 Signal to Noise Ratio (SNR) In this test, the impact of S NR on DOA algorithm is inv estigated through limiting the n oise p o we r introdu ce b y the c h annel to meet SNR=10 and SNR= 20. Bo th condition of S NR is applied for detecting t w o incident signals ( − 20 o , 20 o ) on ULA. By analysing Figure 4.15 with T able 4.8 and 4.9, w e conclude th at increasing the SNR to higher v alues will im p ro v e the MUSIC algorithm as it will pro duce sharp er p eaks with reduced noise leve l. Also, the ro ots will come closer to unit circle allo w ing accurate estimation b y Ro ot-MUSIC whereas p recise detection is evident in ESP RIT as SNR increases. Chapter 4. Direction o f Ar riv al 80 −90 −70 −50 −30 −10 10 30 50 70 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in ULA Relative Power (dB) SNR=10 dB SNR=20 dB Figure 4.15: Im pact of c hanging S NR f or ULA on th e p erforman ce of MUSIC algorithm with settings ( N =5, θ = 20 o and − 20 o , d=0.5 λ and K =100) T able 4.8: Impact of c hanging SNR for ULA on the p er f ormance of Ro ot-MUSIC algo rithm with settings ( N =5, θ = 20 o and − 20 o , d=0.5 λ and K =100) Chapter 4. Direction o f Ar riv al 81 T able 4.9: Impact of c hanging SNR for ULA on the p erformance of ESPRIT algorithm with settings ( N =5, θ = 20 o and − 20 o , d=0.5 λ and K =100) 4.3.7.1.6 Signal Correlation Unlik e the pr evious discussed parameters, this particular parameter which is signal correlation will mak e the input co v ariance matrix singular p rev en ting the DO A algorithms f r om fu nc- tioning. T o r esolv e this issue, p repro cessing tec h niques, explained in section 4.2.2, are n eeded to change th e co v ariance matrix in to non-singular. In another words, the p repro cessing tec h- niques will d e-correlate the coherent signals so th at the DOA algorithms can p rop erly works. Generally , the pr epro cessing tec h niques can op erate along all the DO A algorithms as they only mo dify the co v ariance matrix. In the f ollo wing test, w e will sho w the eﬃciency for eac h of the p repro cessing tec hniqu es as they are emplo y ed with MUSIC Algorithm in ULA. The geometry of ULA allo ws the use of three p repro cessing tec hniques namely F orw ard Spatial Smo othing (FSS), F orw ard-Bac kw ard Spatial Smo ot hing (FBSS) and T oeplitz Algorithm. In our test, w e will consider a scenario wh er e six correlated signals are impinging on ULA with angles ( − 40 o , − 30 o , − 20 o ,20 o ,30 o ,40 o ). Firstly , w e will use the normal MUSI C and MUSIC w ith FSS to d etect th ese correlated s ignals wh en they r eceiv ed b y ULA h a ving N =12. Clearly from Figure 4.16, we conclude that MUSIC wa s un able to resolv e the correlated signals alone b u t MUSIC su cceeds wh en it is used w ith FSS . Th e FSS tec hnique is theoretical ly capable of detecting N / 2 correlated signals and this is p ro v en from simulati on results as 12 / 2=6. By redu cing to N =9, FSS will fail to d etect the correlated signals but FBSS will succeed as sho wn in Figure 4.17. That is b ecause theoretically , FBSS can d etect up to 2 N / 3 signals which is in this case (2 × 9) / 3=6. By fu rther reducing the N to 7, the FBSS will b e unab le to resolve the correlated signals b ut the T o eplitz algorithm will b e as shown in Figure 4.18 . Th at is b ecause T o ep litz algorithm can restore the whole r ank of co v ariance matrix p ermitting the N -1 full range d etection of MUSIC algorithm. As N =7, the whole six Chapter 4. Direction o f Ar riv al 82 correlated signals can b e p erfectly detected with T o eplitz algorithm. Inte restingly , T o eplitz Algorithm oﬀer more rob ot p erformance with N =7 compared with FBSS w ith N =9 whic h is eviden t by the sh arp p eaks and lo wer noise ﬂo or as sh o w n in Figure 4.19. Therefore, w e conclude that T oeplitz Algorithm is the b est c hoice to b e used practically with ULA in coheren t en vironmen t due to its lo w ph ysical requirement, strong p erformance and less computational complexit y . −90 −60 −30 0 30 60 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm for ULA Relative Power (dB) Actual DOA Standard MUSIC MUSIC with FSS Figure 4.16: Implemen tation of Standard MUSIC an d MUSIC with FSS for ULA in Correlated En vironment with the settings ( N =12, θ = − 40 o − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , S NR=20dB and K = 100) Chapter 4. Direction o f Ar riv al 83 −90 −60 −30 0 30 60 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm for ULA Relative Power (dB) Actual DOA MUSIC with FSS MUSIC with FBSS Figure 4.17: Implementa tion of FSS and FBSS usin g MUSIC algorithm for ULA in Correlated En vironment with the settings ( N =9, θ = − 40 o , − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , SNR=20dB and K = 100) Chapter 4. Direction o f Ar riv al 84 −90 −60 −30 0 30 60 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm for ULA Relative Power (dB) Actual DOA MUSIC with FBSS MUSIC with Toeplitz Algorithm Figure 4.18: Implementati on of FBSS and T o eplitz u s ing MUSIC algorithm f or ULA in Correlated Envi ronment with the settings ( N =7, θ = − 40 o , − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , SNR=20dB and K =100) Chapter 4. Direction o f Ar riv al 85 −90 −60 −30 0 30 60 90 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm for ULA Relative Power (dB) Actual DOA MUSIC with FBSS for N=9 MUSIC with Toeplitz Algorithm for N=7 Figure 4.19 : P erformance Comparison b et wee n FBSS and T o eplitz using MUSI C algorithm for ULA in Correlated Environmen t with the settings ( θ = − 40 o − 30 o , − 20 o , 20 o , 30 o , 40 o , d =0.5 λ , S NR=10dB and K =100) 4.3.7.2 DOA Algorithms P erformance in UCA 4.3.7.2.1 Num b er of Sensor E lemen ts In this test, the impact of UCA num b er of sensors is inv estigated on DOA algorithms by applying t wo in coming signals ( − 10 o , 10 o ) w ith N =5 and N =9. The resu lts from Figure 4.20, show that increasing the num b er of elemen ts in UCA will imp ro v e the DO A estimation b y MUS I C algorithm as evident by the sharp er p eaks and low er noise ﬂo or in the sp ectrum. T able 4.10 sho ws that the p erformance of Ro ot-MUSIC will imp r o v e wh er e the ro ots b ecome closer to the un it circle as w ell as b ett er d etectio n. Similarly , the accuracy of ESPRIT is impro v ed as sho wn in T able 4.11. By comparin g all these results, it is noticeable to mention that MUSIC in UCA oﬀer the most accurate results at low num b er of elemen ts. That is b ecause Ro ot-MUSIC and ESPRIT h as an error margin of almost ± 1 d ue to the quan tization Chapter 4. Direction o f Ar riv al 86 error in tro duced b y PME. T o reduce the quan tization error,we n eed to rise N to allo w accurate transformation from UCA to VULA and this is clear fr om case N = 9 w ere the margin error is reduced to almost ± 0 . 3 in b oth algorithms. −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in UCA Relative Power (dB) N=5 N=9 Figure 4.20: Impact of c han ging the n umber of elemen ts in UCA on th e p erformance of MUSIC algorithm with settings ( M =2, θ =20 o and − 20 o , θ e =20 o , d =0.5 λ , SNR= 10dB and K =100) T able 4.10: Impact of changing the num b er of elemen ts in UCA on the p erformance of Ro ot- MUSIC algorithm with settings ( M =2, θ =20 o and − 20 o , θ e =20 o , d =0.5 λ , SNR= 10dB and K =100) Chapter 4. Direction o f Ar riv al 87 T able 4.11: Impact of changing the num b er of elemen ts in UCA on the p erformance of ES PRIT algorithm with settings ( M = 2, θ = 20 o and − 20 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) 4.3.7.2.2 Num b er of Inciden t Signal In the s econd test, the impact num b er of incoming signals on DOA algorithm is examined by considering tw o scenarios with 3 incoming signals ( − 40 o , 0 o , 40 o ) aga inst 1 incoming signal from (0 o ) impin ging on UC A. F rom Figure 4.21 , we conclude that as the incident signals increases, the p erform an ce of MUSIC will starts to degrade leading to a less sharp p eaks. This eﬀect is also clear from T able 4.12 where the detection accuracy of Ro ot-MUSIC and ESPRIT is reduced leading to a larger error margin. In fact the large margin err or is due to the com bined eﬀect of increasing num b er inciden t signal as w ell as the quantiza tion error due to PME. Thus, the margin error can b e further reduced b y incr easing th e num b er of elemen ts in UCA. Chapter 4. Direction o f Ar riv al 88 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in UCA Relative Power (dB) θ = −40 ° 0 ° 40 ° θ = 0 ° Figure 4.21: Imp act of changing the num b er of incident signals impinging UCA on the p er f or- mance of MUSIC algorithm with settings ( N = 5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) T able 4.12: Impact of changing the n umber of incident signals imp inging UCA on the p er- formance of Ro ot-MUSIC and ESPRIT algorithms with settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) 4.3.7.2.3 Angular Separation b et ween Inciden t Signals In this test, the im p act of the angular separation b etw een the incoming signals on DO A algorithm is examined by considering t wo scenarios. In the ﬁrst scenario, UCA will receiv e t w o incident signals (0 o , 20 o ) ha ving a sm all angular separation of 20 o . In the second scenario, Chapter 4. Direction o f Ar riv al 89 the angular separation w ill b e increased to 50 o as the t wo incident signals will hav e a direction of arriv al of 0 o and 60 o . Clearly , F rom Figure 4.22, th at increasing the angular sep aration will impro v e the p erforman ce of MUSIC algorithm through pr o ducing sharp sp ectral p eaks and reduce the noise ﬂo or. The same conclusion is deduced f rom T able 4.13 and 4.14 where the detection accuracy of Ro ot-MUSIC and ESPRI T is in creased as the angular separation is increased. How ev er, T he MUSIC algorithm oﬀers more results for smaller angular resolution. A t s uc h case, the r esolution of Ro ot-MUSIC and ESPRIT can b e increased by impr o ving the estimation of VULA whic h requir e increasing the n umber of element s. −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in UCA Relative Power (dB) θ = 0 ° and 20 ° (angular separation = 20 ° ) θ = 0 ° and 60 ° (angular separation = 60 ° ) Figure 4.22: Impact of c hanging the an gu lar separation b et w een the inciden t signals impinging on UCA on the p erformance of MUSIC algorithm with settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) Chapter 4. Direction o f Ar riv al 90 T able 4.13: I mpact of c hanging the angular separation b et we en th e inciden t signals impinging on UCA on the p erform ance of Ro ot-MUSIC algorithm with settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) T able 4.14: I mpact of c hanging the angular separation b et we en th e inciden t signals impinging on UCA on the p erformance of ESPRIT algorithm w ith settings ( N =5, θ e =20 o , d =0.5 λ , SNR=10dB and K =100) 4.3.7.2.4 Num b er of Samples In the next test, the imp act of num b er of snapshots tak en f or the in coming signals on DOA algorithm is examined by applyin g tw o incoming signals ( − 30 o , 30 o ) on with K =50 and K =500. F rom Figure 4.23, w e conclude that increasing the num b er of snapshots improv e th e p erformance of MUSIC algorithm as the p eak b ecomes sharp er and the noise ﬂo or is lo w ered. This conclusion is also clear fr om T able 4.15 and 4.16 w h ere the d etection accuracy of Ro ot- MUSIC and ESPRIT has increased leading to a low er noise m argin. Ho we v er, the do es not approac h zero due to present of qu an tizatio n err or from the P ME. Chapter 4. Direction o f Ar riv al 91 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in UCA Relative Power (dB) K=50 K=500 Figure 4.23: Imp act of c han ging the num b er of samples of the inciden t signals impinging on UCA on the p erformance of MUSIC algorithm with settings ( N =5, θ = 30 o and − 30 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) T able 4.15: Impact of c hanging the n um b er of samples of the incident signals imp inging on UCA on the p erformance of Ro ot-MUSIC algorithm with settings ( N =5, θ = 30 o and − 30 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) Chapter 4. Direction o f Ar riv al 92 T able 4.16: Impact of c hanging the n um b er of samples of the incident signals imp inging on UCA on the p erformance of ESP RIT algorithm w ith settings ( N =5, θ = 30 o and − 30 o , θ e =20 o , d =0.5 λ , SNR=10dB and K =100) 4.3.7.2.5 Signal to Noise Ratio (SNR) In Th e last test for UCA, the imp act of S NR on DOA algorithm is inv estigated through limiting the ratio b et w een the signal p o w er and noise to meet SNR=10 and SNR=20. Both condition of SNR is applied for detecting t w o incident signals ( − 30 o , 30 o ) on UCA. By ex- amining Figure 4.24 with T able 15 and 16, we conclude that in creasing the SNR to higher v alues will impro v e the MUSIC algorithm as it will pro duce sharp er p eaks with red uced the noise lev el. Also, the d etectio n of ESPRIT and Ro ot-MSUIC is increased as evident from the reduction in qu an tization error. Chapter 4. Direction o f Ar riv al 93 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm in UCA Relative Power (dB) SNR=10dB SNR=20dB Figure 4.24: Impact of c hanging SNR for UCA on the p erformance of MUSIC algorithm with settings ( N =5, θ =30 o and − 30 o , θ e =20 o , d =0.5 λ and K =100) T able 4.17: Imp act of c hanging SNR for UCA on the p erformance of Ro ot-MUSIC algorithm with settings ( N =5, θ =30 o and − 30 o , θ e =20 o , d =0.5 λ and K =100) Chapter 4. Direction o f Ar riv al 94 T able 4.18: Imp act of c hanging SNR for UCA on the p erformance of ES PRIT algorithm w ith settings ( N =5, θ =30 o and − 30 o , θ e =20 o , d =0.5 λ and K =100) 4.3.7.2.6 Signal Correlation Unlik e the previous discussed parameters, this particular parameter s ignal correlatio n will cause a pr oblem for DOA estimation. Here, we will assume th e w orst scenario where all re- ceiv ed signal are correlated. This in terms will redu ce the r ank of inp ut co v ariance matrix to 1 prev en ting DO A algorithms fr om functioning. T o resolv e this issue, pr ep ro cessing tec hniques, explained in section 4.2.2, are need to restore the co v ariance matrix rank. Generally , th e prepro cessing tec hn iques can op erate along all the DO A algorithms as they only mo d ify th e co v ariance matrix. In the follo w ing test, w e w ill sho w the eﬃcien tly eac h of th e pr epro cessing tec hn iques as they emplo y ed with MUSIC Algorithm in UCA. The geomet ry of UCA allo w s the us e of tw o prepr o ce ssing tec hniques namely F orw ard Spatial S mo othing (FSS), F orw ard- Bac kward S patial Smo othing (FBSS).Th ese tec hniques are implemen ted th rough the PME. In our test, we will consider a scenario where six correlated signals are impinging on UCA with angles ( − 140 o , − 80 o , − 20 o , 50 o , 80 o , 140 o ). Firs tly , we will use th e normal MUSIC and MUSIC with FS S to detect these correlated signals wh en they receiv ed b y UCA ha ving N =12. Clearly from Figure 4.29, we conclude that MUSIC wa s un able to resolv e the correlated signals alone but MUSIC succeeds when it u sed with FSS. T h e FSS tec hn ique is theoretically capable of detecting N / 2 corr elated signals and this is pro v en from simulatio n results as 12 / 2=6. By reducing N =9, FS S will fail to detect the correlate d signals b ut FBSS w ill succeed as shown in Figure 4.26. That is b ecause theoretically , FBSS can detect up to 2 N/ 3 signals whic h is in this case (2 × 9) / 3=6. In term of compu tational complexit y , FBSS requires more computation o ver FSS due to the fact that FBSS divides the main arra y in to subarr ays in b oth forward and bac kwa rd direction while FSS op eratio n is r estricted to the forw ard direction. Ho w ev er, for exp ected high num b er of correlated signals, FBSS is the preferred c h oice. Chapter 4. Direction o f Ar riv al 95 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm for UCA Relative Power (dB) Actual DOA Standard MUSIC MUSIC with FSS and PME Figure 4.25: Implemen tation of standard MUSIC and MUSIC with FSS for UCA in Correlated En vironment with the settings ( N =12, θ = − 140 o , − 80 o , − 20 o , 50 o , 80 o , 140 o , θ e =20 o , d =0.5 λ , SNR=20dB and K =100) Chapter 4. Direction o f Ar riv al 96 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 −80 −70 −60 −50 −40 −30 −20 −10 0 Estimated Angle of Arrival in degrees by the MUSIC algorithm for UCA Relative Power (dB) Actual DOA MUSIC with FSS and PME MUSIC with FBSS and PME Figure 4.26: Im plemen tation of FSS and FBSS u sing MUSI C algorithm for UCA in Correlated En vironment w ith the settings ( N =9, θ = − 140 o , − 80 o , − 20 o , 50 o , 80 o , 140 o , θ e =20 o , d =0.5 λ , SNR=20dB and K =100) 4.3.7.3 System Mo deling In order to conform the u sabilit y , practicalit y , and accuracy of the work d one, a virtual simu- lation using MA TLAB was u sed. In th e simulatio n, the conditions of a practical en vironment w ere mimick ed with u tmost precision p ossible. The follo wing will introd u ce one of th e simu- lated scenarios: In this scenario, a land w ith an area of 10 × 10 m is b eing considered with frequency of in terest equal to the 1 GHz and num b er of snapsh ots is 100. This scenario tries to simulate the b ehavior of the MUSIC algorithm when using UCA-geometry sensor no des with N=5. In the test carried out by this scenario, three no d es were c hosen to h a v e random lo cations and the lo cation h app ened to b e as in the follo w in g Figure: Chapter 4. Direction o f Ar riv al 97 x-axis (m) 0 1 2 3 4 5 6 7 8 9 10 y-axis (m) 0 1 2 3 4 5 6 7 8 9 10 A1 A2 Target Figure 4.27: Lo cation of the n o des in a p ractical environmen t T able 4.19 : No des lo cation on the environmen t The Figure ab o ve sho ws the lo cation of the no des in a pr actica l en vironment. Th e scenario considers tw o anc hor no de, n o des with kno wn lo cation, and a sin gle unkn o w n no de w hose lo- cation will b e determined usin g the anc h or n o des us ing triangulation method . Another aim of this scenario is to see the ho w the SNR b eha ve verse th e RMSE of the algorithm. T he algo- rithm used to determine the lo cation is MUSIC algorithm since it results in accurate r esults in UCA wher eas using UCA-ESPRIT or UCA Ro ot -MUSIC algorithms require additional s teps , suc h as ph ase mo o d excitation, in the case of the correlated signals, wh ic h int ro du ce small Chapter 4. Direction o f Ar riv al 98 error. The scenario is carried out with b oth corr elated signals and u ncorrelated signals. 4.3.7.3.1 Uncorrelated Signals The ﬁrst p art of this scenario deals with the un corr elated signals. In the case of the uncorre- lated s ignal, the implementa tion was straigh tforw ard. In this scenario, no d e #2 has an angle of the 116 . 56 o , and no de #3 has an angle of 78 . 69 o with the horizonta l line. The b ehavio r of the system is sho wn in the follo w ing Figure: 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SNR (dB) RMSE (m) Figure 4.28: RMS E for d iﬀerent SNR for uncorrelated s ignals The Figure ab o v e sh ows the RMSE of diﬀerent v alues of SNR in the practical en vironment c h osen when un correlated signals are in use. It is im p ortan t to h ighligh t th at the n umber of an tennas in the UCA geometry is ﬁve. The general b eha vior that can b e seen from the graph is that as the v alue of the SNR increases, the v alue of RMSE decreases. Moreo v er , as the v alue of the SNR gro w s greater, the RMSE shows an asymp totic b eha vior around 0. This indicates that the error as the S NR increases decreases drastically , whic h shows ho w accurate this metho d is in estimating the lo cation of the no des. Chapter 4. Direction o f Ar riv al 99 4.3.7.3.2 Correlated Signals The s econd part of this s cenario deals with the correlated s ignals. as it was established earlier, the correlated s ignals defers fr om the uncorr elated signals in its n eed for more careful imple- men tation in order not to obtain wrongful results. The same conditions w ere applied for this part of the scenario, same p ractical environmen t with same lo cations of anc hor no d es and u n - kno wn no d e. The only additional step in this part is the need for th e use of spatial sm o othing tec hn ique, sp ec iﬁcally F orwa rd/Bac kw ard spatial smo othing FBSS. In th is scenario, no de # 2 h as an angle of the 116 . 56 o , and no de # 3 has an angle of 78 . 69 o with the horizon tal line. The b eha vior of this system is sh o wn in th e follo win g Figur e: 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 SNR (dB) RMSE (m) Figure 4.29: RMSE f or diﬀerent S NR for correlated s ignals The Figure ab o v e sh ows the RMSE of diﬀerent v alues of SNR in the practical en vironment c h osen when correlated signals are in use. It is imp ortant to highlight that the num b er of an tennas in the UCA geometry is ﬁve. The general b eha vior that can b e seen from the graph is that as the v alue of the SNR increases, the v alue of RMSE decreases. Moreo v er , as the Chapter 4. Direction o f Ar riv al 100 v alue of the SNR gro w s greater, the RMSE shows an asymp totic b eha vior around 0. This indicates that the error as the S NR increases decreases drastically , whic h shows ho w accurate this metho d is in estimating the lo cation of the no des. Chapter 5 Hybrid T ec hniques In th is c hapter, d iﬀeren t hybrid tec hniques is discussed to improv e the accuracy of estimation of the lo cation of the unkn o wn no d e [3, 4, 45]. Th is is b eca use the measuremen t noise from v arious tec hniques comes fr om diﬀerent sources. Con s equen tly , th e errors in the estimation of the p ositio n using d iﬀeren t tec h niques are partially indep enden t. The indep endence, in th ese measuremen ts, allo ws creating an estimator with a b etter p erformance. On other cont rary , there is h igh complexit y in terms of time and compu tation [29, 40, 41, 53 , 96]. 5.1 Hybrid RSS and DO A using one Hyb rid No de A w ell-kno wn hybrid tec hn ique com bines RSS and DO A. As previously s tated, the RSS has lo w accuracy than DO A. The reason b ehind this is that it is diﬃcult to p erfectly mo del the signal propagation in the en vironmen t [97]. Therefore, DO A can comp ensate for this lo w accuracy and p ro vide a h ybrid system that has higher accuracy than th e RSS alone [45]. In [45 ], it wa s prop osed th at one h ybrid no de can estimate the p osition of the unkn o w n n o de. This h ybrid n o de uses the an tenna elemen ts to ﬁ nd the target direction. Th is is equiv alen t to a line th at originates f rom the array cen ter to the target. Also, these elemen ts m easure RSS and eac h of these measuremen ts is mo d elled b y a circle which its cen ter is the ante nna elemen t. The line and eac h circle inte rsect at one p oin t. T hese intersecti ons are a v eraged to ﬁnd the lo cation of the unknown no de. T o mo del th e line and circles mathematically , w e will use the p arametric of the line and implicit circle equ ations. Thus, the p arametric equation of 101 Chapter 5. Hybrid T echniques 102 the line in 2D is: Q parametri c = p + t d (5.1) where Q is th e m atric that contai ns x , y comp onents and p =( x 0 , y 0 ) is a p oin t on the line, t is the parameter and d dir is the d irection ve ctor for line. The equation ab o ve is expressed in the x and y comp onen ts f orm as the follo wing: x = x 0 + f dir t (5.2) y = y 0 + g dir t (5.3) where f dir and g dir are the comp onen ts of the direction v ector d dir . In this hybrid ap p roac h this is ev aluated by the pro duct of arr o w length and the cos and s in th e angle from DO A f or f dir and g dir , resp ectiv ely . The circle equation with cente r ( x c , y c ), other than origin is expressed as the follo wing: ( x − x c ) 2 + ( y − y c ) 2 = r 2 (5.4) where r is the radius. This line and eac h circle in tersect at one p oin t. Th us, sub stituting the equations 5.2 and 5.3 in equation 5.4, the result is the f ollo wing: ( x 0 + f dir t − x c ) 2 + ( y 0 + g dir t − y c ) 2 = r 2 (5.5) By expanding the terms and eliminating the terms that are equiv alen t to r , the equation reduces to th e follo wing: ( f dir t ) 2 + ( x 0 − x c ) f dir t = − ( g dir t ) 2 − ( y 0 − y c ) g dir t (5.6) Dividing by t and isolating it at one s id e results in the follo wing: t = ( x c − x 0 ) f dir + ( y c − y 0 ) g dir ( f dir ) 2 + ( g dir ) 2 (5.7) After ev aluating this v alue, it is sub stituted in the equations 5.2 and 5.3 and x 0 and y 0 is Chapter 5. Hybrid T echniques 103 the estimated p osition no de with resp ect to the ante nna elemen t. Consequen tly , the equations 5.2 and 5.3 is x and y coord inates of the intersectio n p oint, resp ec tiv ely . These in tersections are a v eraged to get one p oin t, whic h is the lo cation of the unkn o w n no de. 5.1.1 Sim ulation Results T o test this algorithm, 3 anc hor n o des are p laced on the x - y plane with the size of (30m × 30m) as sho wn in Figure 5.1. Eac h anc hor n o de conta ins 4 ant ennas placed in circular conﬁguration, the center frequency is 1 GHz and the RMSE is compu ted 150 times for ev ery SNR v alue. Comparing th e r esults with RSS alone, the hybrid shows b etter p erform ance than the RSS tec hn ique. This is an exp ected result b ecause the DO A h as high accuracy wh ic h impr ov es th e h ybrid tec hnique. x-axis,m 0 5 10 15 20 25 30 y-axis,m 0 5 10 15 20 25 30 Position of the nodes in the x-y plane used for testing A1 Target A2 A3 Figure 5.1: Posit ion of the n o des in the x-y plane used for hybrid testing Clearly , fr om the ﬁgure 5.2 the h ybrid tec hnique outp erform s the RSS tec h n ique. T he results in the hybrid are ab out 4 times b et ter. One hybrid no de is suﬃcient to p ro duce these results. This matc hes th e exp ecte d resu lt b eca use the presence of the DO A improv es the accuracy , lo w er RMSE at eac h S NR v alue, hence, b et ter estimation.Ho w ev er, DOA p ro vides b etter accuracy compared to the h ybrid tec hnique [45]. Chapter 5. Hybrid T echniques 104 Signal to Noise ratio (SNR), dB 0 5 10 15 20 25 30 RMSE, m 0 10 20 30 40 50 60 RMSE for different values of SNR using Hybrid technique and RSS technique Hybrid technique RSS technique X: 1 Y: 9.746 X: 1 Y: 55.26 X: 16 Y: 0.3693 X: 16 Y: 1.091 Figure 5.2: RMSE for diﬀerent v alues of SNR u s ing Hybr id tec hnique and RSS tec hnique Ho wev er, this algorithm do es not take into consider ation an en vironment that has corre- lated signals. F or this reason, th e next algorithm will address this p oint by using the spatial smo othing in DOA tec h nique. 5.2 Hybrid RSS and DO A with Spatial S mo othing When th e en vironment is correlated, the previous hybrid technique w ill no longer b e able to lo cate the unkn own no d e. That is b eca use the correlation w ill interfere with the execution of DOA algorithms whic h is needed by the hybrid tec hn ique to op erate. In fact, the DOA algorithms are designed to wo rk under the assump tion that the receiv ed signals is uncorre- lated resulting in non-sin gular co v ariance matrix. Ho we v er, the co v ariance matrix b ecome s singular when th e receiv ed signals are correlated causin g a violation in the p rinciples of DOA algorithms. Consid ering a UCA-conﬁguration for the used no des, the correlation p roblem can b e ov ercome b y usin g spatial smo othing, particularly F orward Bac kw ard Spatial Sm o ot hing FBSS. The s patial smo othing will d ivide the main arr a y into o verla ying sub arra ys and then a verage th eir co v ariance matrices to obtain a n on-singular sm o ot hed co v ariance m atrix. Ho w - ev er, FBSS is done linearly and hence the UCA m ust b e con v erted into VULA using PME b efore FBSS can b e used. Chapter 5. Hybrid T echniques 105 5.2.1 Sim ulation Results In our sim ulation for the hybrid tec hnique, w e will consider the same en vironmen t depicted in Figure 5.3. In ord er to make the en vironment correlated, the unkno wn no de will exp erience a scenario w here it receiv ed th r ee correlated signals simultaneously . The kn o w n n o de will receiv e thr ee correlated signals with the angles 53 . 13 o , 116 . 5 7 o , 32 o in resp ec t to the horizonta l line. Also, frequency of th e receiv ed signals are set to 1 GHz and the num b er of snapshot tak en for them is 100. Both the unknown and anc h or no d es are assum ed to p ossess a UCA conﬁguration with 8 antennas and RS S capabilit y . The b eha vior of the hybrid tec hniqu e is sho wing in Figure 5.3.By analyzing Figure 5.3, we observe that as the SNR increases, the v alue of RMSE decreases. In addition, as the v alue of S NR gro w s larger, the RMSE, the RMSE follo ws a p lateau of around 0. F rom this results, w e conclude that the Spatial Smo othin g was capable of restoring the op eration of h ybrid sys tem in correlated environmen t. 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 SNR (dB) RMSE (m) Hybrid with FBSS Figure 5.3: RMSE for diﬀeren t v alues of SNR usin g R S S and DOA Hybrid tec hnique with Spatial Smo ot hing Chapter 5. Hybrid T echniques 106 5.3 Hybrid tec hnique using L east S q u are Based T ec hniques Ho wev er, when the RS S tec hnique is used to estimate the p osition, the noise in th e envi- ronment aﬀect the intersecti on of the circles and the estimation will not b e in single p oint .Therefore, LS is method to minimize the eﬀect of th e noise and giv e one p oi nt. The LS is a close form and easy to compute but its b est p erf ormance app ears when the noise p o w er is small [97]. In th is algorithm, the L S approac h is u sed to ﬁ nd the lo cation of the u n kno wn no de. In this scheme, t w o RSS and one Hybr id no d es are used to ﬁnd the lo cation of the un- kno wn no de. Initially , the RSS technique is used to ﬁn d the measurements. Then, the resu lts are u sed to estimate the lo cation of the u nknown no d e using LS appr oac h as in equation 3.18. Then, with this resu lt a circle is dra wn u sing this estimated p osition using LS and its cen ter is the hybrid no de. The radius of this circle is the diﬀerence b et wee n the cent er and estimated p oin t, as sho wn in equation 5.10. Assume the hybrid no de p hy b and for the LS p oint p LS . Then, the diﬀeren ce in the x -coord inates is noted as x dif f and the equation is the follo wing: x dif f = | x hy b − x LS | (5.8) The same thing go es for the y-coord inates and the r esults are the follo win g: y dif f = | y hy b − y LS | (5.9) Using the results from equations 5.8 and 5.9 and su bstitute them in equation 5.10 wh ic h represent s the radius. r = q ( x dif f ) 2 + ( y dif f ) 2 (5.10) DO A w ill estimate the p osit ion by measurin g the angle ( θ m ) and dr awing a line. This will result in another p oint. Th e p oin t p D O A is foun d b y the follo wing equations: x D O A = x hy b + r cos ( θ m ) (5.11) Chapter 5. Hybrid T echniques 107 y D O A = y hy b + r sin( θ m ) (5.12) These t w o p oint s are a verag ed and the lo cation is estimated. p s = 1 2    x LS + x D O A y LS + y D O A    (5.13) Ho wev er, the estimated p osition using LS includes high error b ecause the least square do es not pro vide information ab out the diﬀerent d istances b et wee n d iﬀeren t anc hors to th e unknown n o de or simply the link qualit y . Thus, th e h ybrid system w ill pro d uce b etter results if this information is utilized. The link wh ic h has high noise v ariance can b e give n less w eigh t compared to that with lo w n oise. This is what basically the WLS algorithm do es; it introd u ces the w eigh ting matrix [98]. In th is hybrid algorithm, the WLS estimator is used in stead of LS. T he same m etho d that w as used in Hybr id LS is used for the fusing tec h n ique. The resu lts are represente d in the next section. 5.3.1 Sim ulation Results The environmen t that w as used in the ﬁrst hybrid tec hnique is us ed in testing the L S and WLS hybrid algorithm. T h e Hybrid anc hor n o de cont ains 4 antennas placed in circu lar conﬁguration. Eac h RSS no des con tain one antenna. The resu lt is in Figure 5.4. The Hybrid tec hn ique sho ws less R MS E v alues than RS S with LS tec hnique alone. Ho w ev er, in a p ublished pap er, it states that the hybrid tec h nique with LS h as no improv ement ov er the RSS LS tec hnique b ecause according to metho d the RSS dominates and th e estimator works only on the RSS measur emen ts [99]. Thus, by using this tec hnique we were able to o v ercome this problem. Comparing these r esults with the results in th e ﬁ r st algorithm, we concluded that the ﬁrst tec hniqu e pro duces b ette r results than the tec hnique with LS. Th is b ecause that there are more RSS no des inv olv ed in the measurements so more errors will b e in tro duced to the system, hen ce, lo w accuracy . Ho w ev er , the ﬁrst tec hn ique uses only one hybrid no d e, th us, less error in the measur emen ts compared to the Hyb r id LS tec h nique. T he r esults are shown Chapter 5. Hybrid T echniques 108 in Figure 5.5. The third comparison was conducted in Hybrid LS and WLS and RSS technique. F rom the analysis, it is concluded that the WLS w ill ou tp erform all the other tec h n ique. In Figure 5.6, the results prov e what it is in the theory . The WLS is the b est p erformance b ecause it giv es more weigh t for the shorter distance with resp ect to the u nknown no de. At higher SNR the WLS sho ws dramatic r ed uction in error compared to th e other tec hniques. Signal to Noise ratio (SNR) 0 5 10 15 20 25 30 RMSE 0 10 20 30 40 50 60 70 RMSE for different values of SNR Hybrid with RSS LS RSS technique X: 1 Y: 68.93 X: 12 Y: 1.342 X: 12 Y: 0.8903 Figure 5.4: RMSE for diﬀeren t v alues of SNR using Hybrid tec h nique RSS LS and RSS tec hn ique 5.4 Hybrid T ec hn iqu e using T w o Lines After inv estigating diﬀerent tec hniques that use the RSS tec hnique to dra w circles to cor- resp ond s to its measuremen ts and the DO A to plot lin e that indicates the compu ted angle. It is wo rth to in v estigate the RS S measurement s to b e represen ted as a line. Thus, in this algorithm, b oth RSS and DO A will b e used to dra w t w o lines and the intersecti on for these t w o lines will b e the estimate d p osition( p s ). T o appr oac h this tec hn ique, one RSS ( p 1 ) and one h ybrid no d es ( p hy p ) are used. Th ese no des form t w o circles and eac h one is a center for eac h circle. These t w o circles in tersect at t w o p oin ts to form the LOP . The equation of this line is r epresen ted as the follo wing: Chapter 5. Hybrid T echniques 109 Signal to Noise ratio (SNR) 0 5 10 15 20 25 30 RMSE 0 5 10 15 20 25 30 35 40 45 50 RMSE for different values of SNR Hybrid with RSS LS Hybrid(one hybrid node) X: 23 Y: 0.4712 X: 1 Y: 46.37 X: 12 Y: 0.9128 X: 12 Y: 0.5062 X: 23 Y: 0.2503 X: 1 Y: 9.415 Figure 5.5: RMS E for diﬀerent v alues of S NR using Hyb rid tec hnique RSS LS and ﬁrst prop osed Hybr id technique ( x hy p − x 1 ) x s + ( y hy p − y 1 ) y s = 1 2 ( k x hy p k 2 − k x 1 k 2 + D 2 1 − D 2 hy p ) (5.14) The second line is foun d from the DO A m easuremen ts which repr esen ted us in g the equa- tions 5.11 and 5.12. In this case x D O A = x s and y D O A = y s . Thus, by rearran ging the preceding t w o equations x s − x hy p cos( θ m ) = y s − y hy p sin( θ m ) (5.15) (sin( θ m )) x s − (cos( θ m )) y s = (sin( θ m )) x hy p − (cos( θ m )) y hy p (5.16) Com bing equations 5.14 and 5.16 will result in the follo wing: Chapter 5. Hybrid T echniques 110 Signal to Noise ratio (SNR) 0 5 10 15 20 25 30 RMSE 0 0.2 0.4 0.6 0.8 1 1.2 1.4 RMSE for different values of SNR using WLS X: 5 Y: 0.3834 X: 28 Y: 0.06873 X: 12 Y: 0.1728 Figure 5.6: RMSE for diﬀerent v alues of SNR u s ing Hybrid tec hnique RSS WLS    ( x hy p − x 1 ) ( y hy p − y 1 ) sin( θ m ) − cos( θ m )       x s y s    =    1 2 ( k x hy p k 2 − k x 1 k 2 + D 2 1 − D 2 hy p ) sin( θ m ) x hy p − cos( θ m ) y hy p    (5.17) By assigning name for eac h matrix, the resu lt is the f ollo wing: Cp s = D (5.18) T aking the inv erse of b oth sides th e results is th e follo wing: p s = C − 1 D (5.19) The simulati on results are p r esen ted in the f ollo wing section. 5.4.1 Sim ulation Results In this tec hniqu e, one h ybrid and one RSS n o des are u sed along with the same sp eciﬁca tion that for the previous environmen t. Th e environmen t is sho wn in Figure 5.7. F rom Figure 5.8, the p erformance of this tec hn ique is less than th e ﬁr st prop osed hybrid with one hybrid n o de at lo w S NR. Ho w ev er, at higher S NR the tw o techniques pro duce the same error, thus they Chapter 5. Hybrid T echniques 111 ha v e the same p erformance. T he reason is that with the hybrid no d e, the all measurements from an y tec hniqu e will ha v e approxi mately th e same factors that aﬀect the results. How ev er, with man y no des that hav e d iﬀeren t capabilities, the measurements m ay h a ve diﬀerent factor that aﬀect the r esults and may pro du ce higher errors. Also, comparison b etw een this metho d and the h ybrid metho d with L S and RSS technique is sho wn in Figure 5.9. This tec h nique outp erforms the Hybrid LS and RSS tec hniqu e alone b ecause it uses only one RSS and one hybrid n o des this will int ro du ce less error compare to other tec hniques. x-axis,m 0 5 10 15 20 25 30 y-axis,m 0 5 10 15 20 25 30 Position of the nodes in the x-y plane used for testing A1 A2 Target Figure 5.7: Posit ion of the no d es in the x - y plane used for testing the least hybrid Chapter 5. Hybrid T echniques 112 Signal to Noise ratio (SNR), dB 0 5 10 15 20 25 30 RM SE,m 0 5 10 15 20 25 RMSE for different values of SNR, different hybrid techniques Hybrid 2 lines(1RSS&1RSSDOA) Hybrid(one hybrid node) X: 25 Y: 0.2254 X: 1 Y: 20.5 X: 12 Y: 0.467 X: 12 Y: 0.5931 X: 1 Y: 8.811 X: 25 Y: 0.2254 Figure 5.8: RMSE for d iﬀeren t v alues of SNR u s ing tw o Hybrid tec hniqu es, the last hybrid and hybrid w ith one hybrid no de tec hniques Chapter 5. Hybrid T echniques 113 Signal to Noise ratio (SNR), dB 0 5 10 15 20 25 30 RMSE, m 0 10 20 30 40 50 60 RMSE for different values of SNR Hybrid with 1 RSS and 1 RSSDOA Hybrid with RSS LS RSS technique X: 5 Y: 2.482 X: 5 Y: 1.657 X: 5 Y: 3.795 Figure 5.9: RMSE for d iﬀeren t v alues of SNR u s ing tw o Hybrid tec hniqu es, the last hybrid and hybrid LS and RSS tec hniques Chapter 6 Conclusion 6.1 Summary of w ork done In this p ro ject, s ev er al tasks were achiev ed. These tasks are summ arized as follo ws: • Discussion regarding the metho d ologies b ehind lo calizat ion disco v ery tec h niques whic h includes trilateration, triangulation and multilate ration. • In v estigatio n of th e techniques u sed in ranging estimation whic h based on either dis- tances or angles. Some of these tec h niques includ e TOA, RSS, Radio Hop Cou nt, and DO A. • Detaile d study ab out the classiﬁcatio ns of lo calization algorithms in WSNs whic h is divided into tw o main br anc h es; cen tralized and distrib uted lo calization algorithms. • Comprehension of the RS S mo d el in th eory and simulatio n u s ing LS, WLS, and Hub er robustness. • A high apprehension of DOA m etho d of localization with a pr op er grasp and a wareness of the v arious algorithms used to sim ulate the d iﬀerence conﬁguration m o dels along with inno v ativ e solutions for problems that face the ﬁeld suc h as the problem of correlated en vironment s. • In v estigatio n and simulat ion of diﬀerent h ybrid tec hniques to rip e the b est p erformance from b ot h tec hniqu es. 114 Chapter 6. Conclusion 115 6.2 Conclusion A t the ﬁrst stage of the pro ject, we carried out a generalize d survey ab out th e metho ds , tec hn iques, and algorithms used to ac h iev e localization. This su rv ey stage was n ecessary to pro vide us with the exp erience and the bac kground in the lo calizatio n ﬁeld. T o design a high-accuracy lo calization system, t wo tec h niques were s elected, RSS and DOA , to b e fu sed in to one system called a Hyb rid system. The RSS mo d el is based on estimating th e distance b etw een the unknown no de and several reference no d es. T o estimate the lo cation of the un k n o wn no de, a minimum of 3 anc h or no d es are needed whic h will form three corresp ond ing circles and eac h anc hor is at the cen ter of its circle. T he intersec tion of these three circles r epresen ts the lo cation of the unkn o w n no de. T o impro v e the estimation of the u nkno wn no de location, estimators suc h as the L S and WLS are used. The ℓ 2 -norm is inv estigated thr ough these t wo estimato rs. Th e WLS outp erform s the LS as the squ are error is less compared to the LS. T his p erformance is clear in th e case where the anc hor no des are at diﬀerent distances with resp ec t to the unkn o w n n o des. Also, the ℓ 1 -norm or Hub er robustness sho ws the same results compared to WLS . This is b eca use the en vironment is Gaussian and the b est estimator is WLS. The DO A mo del is b ased on determinin g the DOA of incoming s ignal b et w een the unknown no de and a reference no d e. T o b e able to lo cate the unknown no d e, a minim um of t w o reference no des are needed. Th is will pr ovide u s with t w o b earing lines and the u n kno wn no d e will lies on the in tersection of these t wo b earing lines. In this mo d el, the inciden t signal is detecte d using an tenn a arr a y instead of single an tenna to allo w the capabilit y of detecting m ore than one incident signal impinging at the same time. Tw o p opular an tenna arra y geometries are in v estigated which are ULA and UCA. Simulat ion results sho ws that the p erformance of UCA exceed ULA as it pro vides equal p o w er distribu tion in all direction and resolv e am biguit y due to 180 o co verag e of ULA. Diﬀeren t t y p es of signals can b e detected, using DO A tec hniques, such as correlated and uncorrelated signals. In the case of uncorrelated signals, they can b e d etected directly us in g subspace algorithms namely MUSIC, Ro ot-MUSIC, and ESP RIT. As the signal b ecome s correlated, the standard algorithms w ill fall to detect them. T o resolv e this p roblem, pre- pro cessing schemes must b e in itially emp loy ed to r emo v e the correlation b et we en the receiv ed Chapter 6. Conclusion 116 signals. Th e pre-pro cessing sc hemes used in our p ro jects inv olve s ph ase mo d e excitation (PME), Spatial S mo othing (SS) and T o eplitz algorithm. In the PME, UCA elements is mapp ed into con v erted to virtual ULA. With th e use of PME, linear op erations like Spatial Smo othing can no w b e emp lo yed ind irectly with the UCA. In the S patial smo othing tec h nique, t wo m ain metho ds can b e used which are FSS and FBSS. FSS divides the array int o subarrays in th e forward direction allo wing N / 2 correlated signal to b e d etecte d. FBSS extends the capacit y of detected signals to 2 N/ 3 through dividing the main arr ay in to s ubarrays in b oth forwa rd and b ac kward direction. In T o eplitz algorithm, the correlated signals are fully de-correlate whic h, in terms, p ermits the full N -1 d etectio n b y s u bspace algorithms. Also, this tec h nique surp asses spatial smo othing b y oﬀering more robust p erf orm ance coup led with less compu tational load. Three main DO A subspace algorithms w ere used in this pro ject to p ro vide high accu- racy in detecting the angles of inciden t signals impin ging on an arra y . Th ese tec hn iques are MUSIC, R o ot -MUSIC and ESPRIT . MUSIC algorithm estimate the DOA b y emplo ying an exhaustiv e searc h through all p ossible steering vect ors th at are orthogonal to the noise vec tors. Ho wev er, this metho d ology made MUSIC algorithm a high computational- load metho d. In Ro ot-MUSIC, the DOA is estimated via the zeros of a p olynomial allo wing less computation compared to MUSIC algorithm. The third tec hnique is th e ESPRIT wh ere th e main s ubarray is d ivided into t w o iden tical doublets. Th e m ain feature of ESPRIT tec hnique is the fact that the corresp onding signal eigenv ectors of doublets are related by r otational matrix, w here DO A is em b o d ied in that matrix. The application Ro ot -MUSIC and ES PRIT tec hniqu es are limited to ULA as it p ossess a v andermonde structure. Ho w ev er, they can b e applicable with UCA after conv erting the UCA in to VULA us ing PME. The p erformance of DOA subsp ace algorithms was ev aluated based on d iﬀeren t parameters in b oth ULA and UCA. These parameters includ es the num b er of an tenn a elemen ts, angular separation b et ween incident singles, SNR, num b er of samples and signal correlation. In the case of the signal correlation, our simulati on results prov es the su p eriorit y T o eplitz algorithm in case of ULA while FBSS in case of UCA. Finally , the hybrid mo del is u sed to mak e the b est of b oth tec hniqu es. Diﬀeren t h ybrid metho ds are imp lemen ted to ac h ieve the b est p erf ormance. The ﬁr st metho d is to use one h y- brid no d e. This no d e provi des u s with the RS S and DO A measur emen ts. DOA measurement Chapter 6. Conclusion 117 is rep resen ted by parametric line that crosses imp licit circle whic h represen ts the RSS m ea- surements. This tec hn iqu e giv es b etter results than the RSS technique alone. T he p receding algorithm addr esses the environmen t of u ncorrelated signals. Ho wev er, to consider the case of correlated signals, sp atial smo othing sp eciﬁcally FBSS is used. The second metho d is to us e the tw o estimators w h ic h are the LS and WLS . In this tec hniqu e, one hybrid no d e and tw o RSS n o des are used to estimate the unkno wn no d e. T he fu sing metho d is av eraging the t w o p oint s that result from the RSS and DO A measurements. T h e simulatio n results for the L S Hybrid outp erf orm th e p erformance the of the RSS tec hnique. Th e same tec h nique that w as used in th e LS h ybrid is applied in WLS h ybrid. T he WLS p erformance is th e b est compared to RSS tec hnique and LS h ybrid. The Final tec hn ique is to us e one h ybrid and one RSS no des. I n which the t w o m easuremen ts are p resen ted using tw o lines and their in tersection is the unkn o w n n o de. This tec hnique sho w s a b etter p erformance th an the LS hybrid an d the RSS tec h niques. Ho wev er, the hybrid tec hn ique that uses one hybrid no de giv es b etter resu lts than the h ybrid with tw o lines tec hnique. 6.3 Critical Appraisal In this pro ject man y tec hniques are inv estigated and diﬀerent algorithms are sim u lated and tested usin g MA TLAB. Also, wide ranges of topics are discu ssed. Th e en tire survey that wa s carried out at the initial stage of the p ro ject wa s su ﬃcien t to choose the tw o tec hn iques that w e w ork ed on. Also, we in v estigated the adv ant ages and disadv an tages for man y tec hniques and w e chose the b est t wo tec hniqu es to b e able to dev elop Hybrid tec hniqu e. Also , we in v estigated d iﬀeren t estimators to impro v e the RSS accuracy . W e inv estigated the id ea of the correlated signals that to b e d etected using RSS tec hniqu e and we concluded th at it w ill b e adv anced for senior design pro ject. Also, in the DOA, w e studied d iﬀeren t tec h niques to address the tw o t yp es of signal wh ic h are correlated and u ncorrelated signals. F urth ermore, w e come u p with diﬀeren t metho ds for the fusing th e R S S and DO A tec hniqu es to form the h ybrid tec h nique. T o implemen t this pro ject, w e ev en read in v arious ﬁelds that are outside lo calizat ion and WSN ﬁ eld. Ho wev er, sometimes it wa s diﬃcult to carry out the Matlab s imulation b ecause our lap- tops cannot handle th e high computation complexit y . Also, sometimes the database th at is Chapter 6. Conclusion 118 pro vided by the library was d own and could n ot ﬁnish the required researc h. In add ition, there are s ome resources that are not av aila ble in th e un iv ersit y library or online w hic h made the researc hes slo w a little bit. If time allo ws u s, w e would lik e to imp lemen t R S S and DOA techniques u sing hardwa re. This is b ecause th e r eal implemen tation will mak e the reader to appreciate th e b eneﬁts of diﬀeren t simulators and algorithm that w as u sed in this pro ject. 6.4 Recommendations There are tw o p oten tial topics that w e come u p with dur in g our inv estigation in the S DP pro ject. Due to the limited p eriod of SDP pro ject, w e w ould like to lea v e these topics as a future tasks. The ﬁrst topic is ab out applying T o ep litz with UCA after it is conv erted into VULA. This p r o cedure will greatly impro v e th e p erformance of DO A algorithms with UCA in coherent en vir onmen t. The second topic is c hanging the c hann el mo del and applyin g the ℓ 1 -norm to achiev e the optimum p erf orm ance in a more realistic en vironmen t. App endix A The least square is one tec hn ique fr om the w ell-kno w n estimator the Maxim um Lik eliho o d tec hn iques u nder the condition of Gaussian d istribution, with mean ( µ ) equals to zero and a v ariance ( σ 2 ) [101]. Hence, assume the equ ation for the straigh t line is: y i = αx i + ǫ i (A.1) where α is the slop e of the lin e and ǫ i is the s q u are error. The Gaussian p robabilit y dens ity function (p df ) is giv en by the follo win g function: f ( x ) = 1 √ 2 π σ 2 e − ( x − µ ) 2 2 σ 2 (A.2) where x is random v ariable Supp ose that x i is ﬁxed and y i is a rand om v ariable and ǫ is indep enden t for α , m =0 and ǫ i = y i - αx i . Hence, f ( x . . . α, σ 2 ) = 1 √ 2 π σ 2 e − ( y i − αx i ) 2 2 σ 2 (A.3) L = N Y i =1 1 √ 2 π σ 2 e − ( y i − αx i ) 2 2 σ 2 = ( 1 √ 2 π σ 2 ) N N Y i =1 e − ( y i − αx i ) 2 2 σ 2 (A.4) T ak e the normal log for b oth sid es: l = N l n ( 1 √ 2 π σ 2 ) − 1 2 σ 2 N X i =1 ( y i − αx i ) 2 (A.5) Then diﬀerenti ate and set the term to zero: 119 Appendix A. 120 dl dα = 1 σ 2 N X i =1 x i ( y i − αx i ) = 0 (A.6) Hence, the estimator b α is: [ α M L = N P i =1 x i y i N P i =1 x 2 i (A.7) App endix B The Line of p osition (LOP) b et wee n p 1 and p 2 Starting fr om the f orm ula of th e distance b et w een tw o p oint s as sho wn b elo w: D i = k p i − p s k = q ( x i − x s ) 2 + ( y i − y j ) 2 (B.1) where i =1,2, No w, b y squarin g and taking the diﬀerence b et w een D 2 2 and D 2 1 : D 1 = k p 1 − p s k = q ( x 1 − x s ) 2 + ( y 1 − y j ) 2 (B.2) D 2 = k p 2 − p s k = q ( x 2 − x s ) 2 + ( y 2 − y j ) 2 (B.3) D 2 2 − D 2 1 = ( x 2 − x s ) 2 + ( y 2 − y s ) 2 − ( x 1 − x s ) 2 − ( y 1 − y s ) 2 (B.4) ( x 2 − x 1 ) x s + ( y 2 − y 1 ) y s = 1 2 ( x 2 2 + y 2 2 ) − 1 2 ( x 2 1 + y 2 1 ) + 1 2 ( D 2 1 − D 2 2 ) (B.5) It can b e simp liﬁed using the form ula k p i k 2 = x 2 i + y 2 i ( x 2 − x 1 ) x s + ( y 2 − y 1 ) y s = 1 2 ( k p 2 k 2 − k p 1 k 2 + D 2 1 − D 2 2 ) (B.6) 121 App endix C The weigh ting matrix W is the in v erse of the co v ariance matrix S of ve ctor b . Assuming that the measur ements of the distances are indep den t and x i and y i are constant s, the matrix S can b e easily calculated: S = E { bb T } =          V ar ( d 2 1 ) + V ar ( d 2 2 ) V ar ( d 2 1 ) · · · V ar ( d 2 1 ) V ar ( d 2 1 ) V ar ( d 2 1 ) + V ar ( d 2 3 ) · · · V ar ( d 2 1 ) . . . . . . . . . . . . V ar ( d 2 1 ) V ar ( d 2 1 ) . . . V ar ( d 2 1 ) + V ar ( d 2 N )          (C.1) Assuming that the c h annel is lognormal, it can b e deriv ed from equation 3.4 that the estimated d istance is a rand om v ariable deﬁned b y: e d i = d i 10 N (0 ,σ ) 10 η = 10 N (log 10 ( d i ) , σ 10 η ) = e N (log 10 ( d i ) , σ 10 η ) ln(10) = e N (ln( d i ) , σ l n(10) 10 η ) (C.2) That is e d i is a lognormal random v ariable with parameter [102] µ d = ln( d i ); σ d = σ ln(10) 10 η (C.3) The v ariance is calculated and subsituted into the co v ariance m atrix [11] V ar ( d 2 i ) = e 4 µ d ( e 8 σ 2 d − e 4 σ 2 d ) (C.4) 122 References [1] R. Kulk arni, A. 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Accurate and Robust Localization Techniques for Wireless Sensor Networks

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