Study of Robust Distributed Beamforming Based on Cross-Correlation and Subspace Projection Techniques

In this work, we present a novel robust distributed beamforming (RDB) approach to mitigate the effects of channel errors on wireless networks equipped with relays based on the exploitation of the cross-correlation between the received data from the r…

Authors: H. Ruan, R. C. de Lamare

Study of Robust Distributed Beamforming Based on Cross-Correlation and   Subspace Projection Techniques
1 Study of Rob ust Distrib uted Beamforming Based on Cross-Correlation and Subspace Projection T echniques Hang Ruan * and Rodrigo C. de Lamare* # ∗ Dep artment of Electronics, The University of Y ork, Eng la n d, Y O10 5BB # CETUC, Pontifical Catho lic University of Rio de Jan eiro, Brazil Emails: hr648@yo rk.ac.uk, delamare@cetuc .puc-rio.br Abstract —In this work, we present a nov el robust distributed beamf orming (RDB) approach to mitigate the effects of ch annel errors on wireless n etworks equ ipped with relays b ased on the exploitation of the cross-corr elation between th e receiv ed data from the relays at the destination and the system outpu t. The proposed RDB method, denoted cross-co rrelation and subspace projection (CCSP ) RDB, considers a total re lay transmit power constraint in the system and the objectiv e of maximizing the output si gnal-to-interference-plus-noise ratio (SINR). The relay nodes are equipped with an amplify-and-forward (AF) protoco l and we assume that the channel state information (CSI) is imperfectly known at the relays and there is no d irect link between the sources and the d estination. The CCSP does not require any costly optimization procedure and simulations show an excellent performance as compared to previously reported algorithms. I . I N T RO D U C T I O N Distributed beam formin g has been wid ely in vestigated in wireless commun ic a tions and sensor array signal proc e ssing in recent years [1], [2], [ 3] [ 2 2], [23], [24], [25], [2 6], [27], [28], [2 9], [30], [31], [32], [3 3]. Such alg orithms are key f o r situations in which the channels between the sources and the destination hav e poor quality so that d evices canno t comm u- nicate directly and employ relays that rece ive and forward the sign als. In [2], relay network problems are described as optimization pro blems a n d r elated transfo r mations an d impli- cations are provided and discussed. The w ork in [7 ] fo rmulates an optimizatio n pr o blem that m a x imizes the outpu t signa l-to- interferen ce-plus-n o ise ratio (SINR) un d er total relay tran smit power constraints, by computing th e be amform in g we ig ht vector with only local inform ation. The work in [4] focuses on multiple scenarios with d ifferent optimization p roblem formu latio ns, in order to o ptimize the b eamform ing weight vector and in crease the system signal-to-n oise ratio (SNR), with the assumption that the glo bal chann e l state infor mation (CSI) is p e r fectly known. Other works like in [ 5], [6] an alyze power contr o l method s based on chann el magnitu de, whereas the powers of each relay are adaptively adjusted accor d ing to the qu alities o f their associated channels. Howe ver , in most scenarios en counter ed, th e chann els ob- served by the relays may lead to perfo rmance degra d ation be- cause of inevitable measureme n t, estimation an d quantization errors in CSI [11] as well as prop agation effects. These imp air- ments result in imper f ect CSI that can affect most distributed beamfor ming meth ods [3 4], [35], [36], [37], [ 38], [39], [40], [41], [42], [4 3], [44], [45], [5 6], [4 6], [ 47], [48], [4 9], [ 50], [51], [52], [5 3], [58], [55], [5 7], [5 8], [ 59], [60], [6 1], [ 62], [63], [64], [65], [ 66], [67], [68], [6 9], [ 70], [71], [7 2], [ 73], [74], [75], [76], [77], [78], [79], [80], [81], [82], wh ich either fail or canno t provide satisfactory p erform ance. In th is con text, robust distributed bea mformin g (RDB) tec h niques are h ence in dem and to mitigate the cha nnel errors or un certainties and preserve the relay sy stem p erform a n ce. The studies in [9], [10], [11], [2 0] minimize the to tal relay tra n smit power un der an overall quality o f service ( QoS) co nstraint, using e ith er a conv ex sem i-definite progr amme (SDP) r elaxation method or a c o n vex second-ord er cone pr ogramm e (SOCP). Th e works in [9], [1 1] consider the cha nnel errors as Gaussian random vectors with known statistical distributions b etween th e sou rce to th e re lay nod es and the r elay nodes to the destination , whereas [10] mod els the chan nel e r rors with their cov arian ce matrices as a ty p e of matrix pe r turbation . The work in [10], [12], [1 8] p r esents a robust de sign, which ensures tha t the SNR co nstraint is satisfied for imperfect CSI by ado pting a worst-case d e sig n a nd formulates th e p roblem as a co n vex optimization pro blem that can be solved efficiently . In this work, we pr opose an RDB tech nique tha t ach iev es very high estimation accuracy in terms of chann el mismatch with r e d uced comp utational complexity , in scenarios wh ere the global CSI is im perfect and local com munication is u nav ail- able. Un like existing RDB ap proach es, we aim to maximize the sy stem o u tput SINR subject to a total re lay transm it power constraint using an app roach th a t exploits the cross- correlation between the beamform ing weight vector and th e system o utput a nd the n proje c ts the obtain ed cross-cor r elation vector onto su bspaces compute d fro m the statistics of second- order imperfect channels, namely , the cross-co rrelation and subspace p rojection ( CCSP) RDB techn ique. Unlike our p re- vious work on centralized beamf orming [1 4], th e CCSP RDB technique is distributed an d has ma rked differences in the way the subspace processing is carried out. I n th e CCSP RDB method, the covariance ma trices of the channel e rrors are modeled by a certain type of add itive matrix p e rturbation methods [8], wh ich ensures that the covariance matrices ar e always positi ve-defin ite. W e consid er multiple sour ce signals and assume that there is no d ir ect link between them and the destination. The propo sed CCSP RDB techniq ue shows outstanding SINR perfo rmance as com pared to the existing distributed b eamfor ming tech niques, which f ocus on transmit power minimiza tio n over input SNR values. The re st of th is work is organize d as follows: Section II presents the system mo del. Section II I devises the pr oposed CCSP RDB m ethod. Section IV illustrates a nd discusses the 2 simulation results. Section V states the con clusion. I I . S Y S T E M M O D E L W e consider a wireless communicatio n network consisting of K signal sources (one desired signa l source with the others as interferer s) , M distributed single-antenna relays and a de stina tion. W e assume th at that direct link s are not reliable and ther efore not consider ed. Th e M relays r e ceiv e data transmitted b y th e sign al sou rces and then retr ansmit to the destination by employing beamfo rming, in which a two-step amplify- and-fo rward (AF) protoc o l is considered for cooper a tive commun ications. In the first step, the k sources tran sm it the signals to th e M single-anten na relays according to the mo d el given by x = Fs + ν , (1) where the vector s = [ s 1 , s 2 , · · · , s K ] T ∈ C K × 1 contains signals with zero mean denoted by s k = p P s,k b k for k = 1 , 2 , · · · , K , where E [ | b k | 2 ] = σ 2 b k , P s,k and b k are the transmit power and the information sy m bol o f the k th signal source, respe ctiv ely . W e assume tha t s 1 is the desired signa l while the rem a ining sour c e signals are treated as interferers. The matrix F = [ f 1 , f 2 , · · · , f K ] ∈ C M × K is the chan nel matrix b etween the signal sources and the r elays, f k = [ f 1 ,k , f 2 ,k , · · · , f M ,k ] T ∈ C M × 1 , f m,k denotes the c h annel between the m th relay and the k th source ( m = 1 , 2 , · · · , M , k = 1 , 2 , · · · , K ) . ν = [ ν 1 , ν 2 , · · · , ν M ] T ∈ C M × 1 is the complex Gaussian no ise vector at the relays and σ 2 ν is the noise variance at each relay ( ν m ˜ C N (0 , σ 2 ν ) , which refer s to the complex Gau ssian distribution with zero mean and variance σ 2 ν ). Th e vector x ∈ C M × 1 represents the received da ta at the r e lays. In the second step, the relays transmit y ∈ C M × 1 , which is an amplified and phase-steered version of x that can be written as y = Wx , (2) where W = dia g([ w 1 , w 2 , · · · , w M ]) ∈ C M × M is a diag onal matrix who se entries denote the beamfor ming weights, where diag( . ) d enote th e diag onal en try of a m atrix. T h en the signal received at the destinatio n is given by z = g T y + n, (3) where z is a scalar, g = [ g 1 , g 2 , · · · , g M ] T ∈ C M × 1 is the complex Gaussian channel vector b etween th e relay s and the destination, n ( n ˜ C N (0 , σ 2 n ) ) is the n o ise at th e destinatio n and z is the received signa l at th e destina tio n. Here we assum e that the no ise sam p les at each relay and th e destination have the same power, wh ich means we h av e P n = σ 2 n = σ 2 ν . Both ch annel m atrices F and g are mode led as Rayleigh distributed random variables, i.e., d istan ce based large- scale channel propag ation effects th a t include distance based fadin g and shadowing are c o nsidered. An expo nential based path lo ss model is de scribed by [13] γ = √ L √ d ρ , (4) where γ is the distance- based path loss, L is the known path loss at the destination, d is the distance of inter est re lati ve to the destination and ρ is th e path loss expo nent, which can vary due to dif fer ent en vir o nments and is typica lly set within 2 to 5 , with a lo wer value represen tin g a clear and uncluttered en vironm ent, which has a slow attenuation an d a higher value describ ing a clutter ed a n d highly attenuating en viro nment. Shadow fading can be descr ib ed as a rand om variable with a p r obability distribution f o r th e case o f large scale fading given by β = 10 ( σ s N (0 , 1) 10 ) , (5) where β is the shad owing parameter, N (0 , 1 ) means the Gaussian distribution with zero mean and unit variance, σ s is the shadowing spre a d in d B. The shadowing spread reflec ts the sev erity of the atten uation c a used by shadowing, and is giv en between 0 dB to 9 dB [13]. The chann els mode le d with both path-loss and sh a dowing can b e represented as: F = γ β F 0 , (6) g = γ β g 0 , (7) where F 0 and g 0 denote the Rayleigh distrib uted c h annels without large-scale propa g ation effects [1 3]. The received signal at th e m th relay c a n be expre ssed as: x m = K X k =1 p P s,k b k | {z } s k f m,k + ν m , (8) then the transmitted signal at the m th r elay is g iv en by y m = w m x m . (9) The transmit power at the m th relay is equivalent to E [ | y m | 2 ] so th at can be wr itten as P M m =1 E [ | y m | 2 ] = P M m =1 E [ | w m x m | 2 ] or in m atrix form as w H Dw where D = diag  P K k =1 P s,k σ 2 b k  E [ | f 1 ,k | 2 ] , E [ | f 2 ,k | 2 ] , · · · , E [ | f M ,k | 2 ]  + P n  is a fu ll-rank matrix, wh ere ( . ) H denotes the Herm itian transpose op e rator . The signa l received at the d estination c a n be expanded by substituting ( 8) and (9) in (3), which y ie ld s z = M X m =1 w m g m p P s, 1 f m, 1 b 1 | {z } desired signal + M X m =1 w m g m K X k =2 p P s,k f m,k b k | {z } interferers + M X m =1 w m g m ν m + n | {z } noise . (10) By tak in g expectation of the compo nents of (10), we can compute the desired sign a l power P z , 1 , the interf e r ence power P z ,i and the no ise power P z ,n at the destination a s follows: P z , 1 = E h M X m =1 ( w m g m p P s, 1 f m, 1 b 1 ) 2 i = P s, 1 σ 2 b 1 M X m =1 E h w ∗ m ( f m, 1 g m )( f m, 1 g m ) ∗ w m i | {z } w H E [( f 1 ⊙ g )( f 1 ⊙ g ) H ] w , (11) 3 P z ,i = E h M X m =1 ( w m g m K X k =2 p P s,k f m,k b k ) 2 i = K X k =2 P s,k σ 2 b k M X m =1 E h w ∗ m ( f m,k g m )( f m,k g m ) ∗ w m i | {z } w H E [( f k ⊙ g )( f k ⊙ g ) H ] w (12) P z ,n = E h M X m =1 ( w m g m ν m + n ) 2 i = P n (1+ M X m =1 E h w ∗ m g m g ∗ m w m i | {z } w H E [ gg H ] w ) , (13) where ∗ den otes complex conjugatio n. By definin g R k , P s,k σ 2 b k E [( f k ⊙ g )( f k ⊙ g ) H ] , where ⊙ is the Schur-Hadamard produ ct for k = 1 , 2 , · · · , K , Q , P n E [ gg H ] , and the SINR is compu ted as: S I N R = P z , 1 P z ,i + P z ,n = w H R 1 w P n + w H ( Q + P K k =2 R k ) w . (14) It sho uld be noted tha t in (14), the quan tities R k , k = 1 , · · · , K and Q only con sist of the seco n d-ord er statistics of the ch annels, wh ich means th at if the chan n els have no mismatches, tho se quantities d escribe the perfect k nowledge of CSI. At this poin t, in order to intro duce errors describ ed b y E = [ e 1 , · · · , e K ] ∈ C M × K and e ∈ C M × 1 to the channels ˆ F and ˆ g , we have ˆ f k = f k + e k , k = 1 , 2 , · · · , K , (15) ˆ g = g + e , k = 1 , 2 , · · · , K, (16) where ˆ f k is the k th mismatched ch annel comp onent o f F . T he elements of e k for any k = 1 , · · · , K and e are assumed to be for simplicity indepen d ent and iden tica lly distributed (i.i.d) Gaussian variables so th at the cov arian ce matrices R e k = E [ e k e H k ] and R e = E [ ee H ] are diagonal, in which case we can d irectly impose the effects of the un c e rtainties to all the ma trices associated with f k and g in (14). By assuming that the chann el e r rors are uncor related with the channels so that E [ e k ⊙ g ] = 0 , E [ e ⊙ f k ] = 0 , E [ e ⊙ g ] = 0 and E [ e k ⊙ f k ] = 0 , then we can use an additiv e Frobeniu s nor m matrix pertu rbation method as in troduce d in [ 8], thus we can have the following: ˆ R k = R k + R e k = R k + ǫ || R k || F I M , k = 1 , · · · , K , (17 ) ˆ Q = Q + R e = R k + ǫ || Q || F I M , k = 1 , · · · , K , (18) ˆ D = D + ǫ || D || F I M , (19) where ˆ R k , ˆ Q an d ˆ D a re th e m a trices per turbed after the chan- nel mismatche s are taken into account, ǫ is th e perturb ation parameter uniform ly distributed within (0 , ǫ max ] where ǫ max is a predefined constant which describes th e m ismatch level. The matr ix I M represents th e identity m a trix of dimension M and it is clear th at ˆ R k , ˆ Q and ˆ D ar e p ositiv e definite, i.e. ˆ R k ≻ 0 ( k = 1 , · · · , K ) , ˆ Q ≻ 0 and ˆ D ≻ 0 . According to (14), th e r o bust optimiza tio n problem th a t maximizes the output SINR with a total relay transmit power co n straint is written as max w w H ˆ R 1 w P n + w H ( ˆ Q + P K k =2 ˆ R k ) w sub ject to w H ˆ Dw ≤ P T . (20) The optimizatio n pro blem (20) has a similar form to the optimization problem in [4] and hence can be solved in a closed for m using an eige n -decom position method tha t only requires q uantities or parameter s with known secon d-ord e r statistics. I I I . P RO P O S E D C C S P R D B A L G O R I T H M In this section, the proposed CCSP RDB algorithm is introdu c ed. Th e algorithm is con sidered for a system with imperfect CSI, works iter ativ ely to estimate and obtain the channel statistics over sna pshots. Th e alg orithm is based on the exploitation of cross-corr ection vector b etween the relay received data and the system outp ut, as well as the construction o f eigen- su bspaces. By projecting the so o btained cross-corre lation vector onto the subspaces at the relays, the channel e r rors can be mitigated and the result leads to a precise estimate of the beamform ers. T o this en d, the sample cross-corre lation vector (SCV) ˆ q ( i ) in the i th iteration can be estimated by ˆ q ( i ) = 1 i i X j =1 x ( j ) z ∗ ( j ) , (21) which u ses sample av erag es that take into accoun t all the data observations from snapsho t on e to the curr e nt snapsho t, where x ( i ) and z ∗ ( i ) refer to the data observation vector in the i th snapshot at th e r e la y s and th e system outp ut in the i th snapshot at the destinatio n, respe ctiv ely , in th e pr esence of chan nel errors. Then, we de compose the mismatched channel matrix ˆ F ( i ) into K compon ents as ˆ F ( i ) = [ ˆ f 1 ( i ) , ˆ f 2 ( i ) , · · · , ˆ f K ( i )] and fo r each of them we co nstruct a separate projection m atrix. For the k th ( 1 ≤ k ≤ K ) comp onent, we comp ute th e covariance matrix for ˆ f k ( i ) and use it as an estimate of the tr ue channel c ovariance matrix instead of the mismatched c h annel covariance m atrices: ˆ R ˆ f k ( i ) = 1 i i X j =1 ˆ f k ( j ) ˆ f H k ( j ) . (22) ˆ R ˆ g ( i ) = 1 i i X j =1 ˆ g ( j ) ˆ g H ( j ) . (23) Here we take an approx imation for the time-averaged estimate of the cov arian ce m atrices so that we have R f k ( i ) = 1 i i P j =1 f k ( j ) f H k ( j ) ≈ 1 i i P j =1 ˆ f k ( j ) ˆ f H k ( j ) and R g ( i ) = 1 i i P j =1 g ( j ) g H ( j ) ≈ 1 i i P j =1 ˆ g ( j ) ˆ g H ( j ) . Then the error covari- ance matrices R e k ( i ) and R e ( i ) can be computed as R e k ( i ) = ǫ || R f k ( i ) || F I M . (24) 4 R e ( i ) = ǫ || R g ( i ) || F I M . (25) In order to eliminate or red uce the error s e k ( i ) from ˆ f k ( i ) and e fro m ˆ g ( i ) , the SCV o btained in (2 1) can be p rojected on to the subsp a ces described by P k ( i ) = [ c 1 ,k ( i ) , · · · , c N ,k ( i )][ c 1 ,k ( i ) , · · · , c N ,k ( i )] H , (2 6) and P ( i ) = [ c 1 ( i ) , · · · , c N ( i )][ c 1 ( i ) , · · · , c N ( i )] H , (27) where c 1 ,k ( i ) , c 2 ,k ( i ) , · · · , c N ,k ( i ) and c 1 ( i ) , c 2 ( i ) , · · · , c N ( i ) are the N pr incipal eigen vectors of the error spe c tr um matrix C k ( i ) and C ( i ) , respecti vely , defined by C k ( i ) , ǫ max Z ǫ → 0 + E [ ˆ f k ( i ) ˆ f H k ( i )] dǫ (28) and C ( i ) , ǫ max Z ǫ → 0 + E [ ˆ g ( i ) ˆ g H ( i )] dǫ. (29) Since we have already assume d that e k ( i ) and e ( i ) are uncor- related with f k ( i ) an d g ( i ) , if ǫ follows a unifo rm distribution over the sector (0 , ǫ max ] , by ap proxim ating E [ f k ( i ) f H k ( i )] ≈ R f k ( i ) , E [ e k ( i ) e H k ( i )] ≈ R e k ( i ) , E [ g ( i ) g H ( i )] ≈ R g ( i ) and E [ e ( i ) e H ( i )] ≈ R e ( i ) , (28) and (29) can be simp lified as C k ( i ) = ǫ max R f k ( i ) + ǫ 2 max 2 || R f k ( i ) || F I M , (30) and C ( i ) = ǫ max R g ( i ) + ǫ 2 max 2 || R g ( i ) || F I M , (31) Then the m ismatched channel compon ents ar e then estimated by ˆ f k ( i ) = P k ( i ) ˆ q ( i ) k P k ( i ) ˆ q ( i ) k 2 , (32) ˆ g ( i ) = P ( i ) ˆ q ( i ) k P ( i ) ˆ q ( i ) k 2 . (33) T o this point, all the K channel com ponen ts of ˆ f k ( i ) are obtained so that we ha ve ˆ F k ( i ) = [ ˆ f 1 ( i ) , ˆ f 2 ( i ) , · · · , ˆ f K ( i )] . In the n ext step, we u se the so ob tained chann e l com po- nents to p rovide estimates f or the matrix quantities ˆ R k ( i ) ( k = 1 , · · · , K ), ˆ Q ( i ) and ˆ D ( i ) in ( 2 0) as fo llows: ˆ R k ( i ) = P s,k E [( ˆ f k ( i ) ⊙ ˆ g ( i ))( ˆ f k ( i ) ⊙ ˆ g ( i )) H ] , (34) ˆ Q ( i ) = P n E [ ˆ g ( i ) ˆ g H ( i )] , (35) ˆ D ( i ) = diag  K X k =1 P s,k [ E [ | ˆ f 1 ,k ( i ) | 2 ] , · · · , E [ ˆ f M ,k ( i ) | 2 ]]+ P n  . (36) T o procee d f u rther, w e define ˆ U ( i ) = ˆ Q ( i ) + P K k =2 ˆ R k ( i ) so that (20) can be written as max w ( i ) w H ( i ) ˆ R 1 ( i ) w ( i ) P n + w H ( i ) ˆ U ( i ) w ( i ) sub ject to w H ( i ) ˆ D ( i ) w ( i ) ≤ P T . (37) T o so lve the optimization problem in ( 37), the weig ht vector is rewritten as w ( i ) = √ p D − 1 / 2 ( i ) ˜ w ( i ) , (38) where ˜ w ( i ) satisfies ˜ w H ( i ) ˜ w ( i ) = 1 . Th en (37) can be rewritten as max p, ˜ w ( i ) p ˜ w H ( i ) ˜ R 1 ( i ) ˜ w ( i ) p ˜ w H ( i ) ˜ U ( i ) ˜ w ( i ) + P n sub ject to || ˜ w ( i ) || 2 = 1 , p ≤ P T , (39) where ˜ R 1 ( i ) = ˆ D − 1 / 2 ( i ) ˆ R 1 ( i ) D − 1 / 2 ( i ) an d ˜ U ( i ) = ˆ D − 1 / 2 ( i ) ˆ U ( i ) ˆ D − 1 / 2 ( i ) . As th e objective functio n in (39) increases monoto nically with p regardless of ˜ w ( i ) , wh ich means the objective function is m aximized whe n p = P T , hence (39) can b e simplified to max ˜ w ( i ) P T ˜ w H ( i ) ˜ R 1 ( i ) ˜ w ( i ) P T ˜ w H ( i ) ˜ U ( i ) ˜ w ( i ) + P n sub ject to || ˜ w ( i ) || 2 = 1 , (40) or equivalently as max ˜ w ( i ) P T ˜ w H ( i ) ˜ R 1 ( i ) ˜ w ( i ) ˜ w H ( i )( P n I M + P T ˜ U ( i )) ˜ w ( i ) sub ject to || ˜ w ( i ) || 2 = 1 , (41) in which the ob jectiv e fun ction is maximized when ˜ w ( i ) is chosen as the princip a l eigenvector of ( P n I M + P T ˜ U ( i )) − 1 ˜ R 1 ( i ) [4], which lea d s to the solution for th e weight vector of the distributed bea mforme r with cha n nel errors given by w ( i ) = p P T ˆ D − 1 / 2 ( i ) P { ( P n I M + ˆ D − 1 / 2 ( i ) ˆ U ( i ) ˆ D − 1 / 2 ( i )) − 1 ˆ D − 1 / 2 ( i ) ˆ R 1 ( i ) ˆ D − 1 / 2 ( i ) } , (42) where P { . } denotes the principal eigenvector corre sp onding to the largest eig e n value. T h en the max imum achiev able SINR of the system in the presence o f channel error s is given by SINR max = P T λ max { ( P n I M + ˆ D − 1 / 2 ( i ) ˆ U ( i ) ˆ D − 1 / 2 ( i )) − 1 ˆ D − 1 / 2 ( i ) ˆ R 1 ( i ) ˆ D − 1 / 2 ( i ) } , (43) where λ max is the ma x imum eigenv alue. In order to rep roduce the propo sed CCSP RDB algo r ithm, we use (2 1)-(23), (3 0)- (36), (42) and ( 43) for each iteration. I V . S I M U L AT I O N S In the simulation s, we co mpare the pro posed CCSP RDB algorithm with several existing robust approach es [10], [1 1], [12], [17], [18], [ 20] (i.e. worst-case SDP on line programm ing) in the presence of impe r fect CSI. Th e simulation metrics considered include the system outp ut SI NR versus in put SNR, snapshots as well a s the maximu m allowable total tran smit power P T . In som e scenarios, we co nsider that the interfer ers are strong enou gh as compare d to the d esired signal and th e noise. In all simulations, the system inp ut SNR is k nown and can be co n trolled by adjusting only the n o ise p ower . Both channels F and g follow the Ray leigh distribution, wher eas 5 the m ismatch is o n ly con sidered for F . The sha d owing an d path loss par ameters em ploy ρ = 2 , the source-to -destination power path lo ss is L = 1 0 dB and the sha d owing spread is σ s = 3 dB. The distanc es of the source- to-relay links d s,r m ( m = 1 , · · · , M ) are mod eled as pseudo-r andom in an a r ea defined by a rang e of relative distances b ased on the sou rce- to-destination distance d s,d which is set to 1 , so as the source- to-relay link distances d s,r m are decided b y a set o f uniform random variables d istributed between 0 . 5 to 0 . 9 , with cor- respond in g relay-so u rce-destina tio n ang le s θ r m ,s,d random ly chosen f rom a n angu lar ran g e of − π / 2 to π / 2 . Therefor e, the relay-to- d estination distance s d r m ,d can be calculated usin g the trigo nometric identity giv en b y d r m ,d = q d 2 s,r m + 1 − 2 d s,r m cos θ r m ,s,d . The total nu m ber of re lays and signal sources are set to M = 8 and K = 3 , respe c ti vely . Th e system interf erence- to-noise ratio (INR) is specified in each scenario an d 10 0 snapshots are considere d. The numb er of prin cipal compo n ents is m anually selected to optimize the perf ormanc e for the CCSP RDB algo r ithm. W e fir st examine the SINR p erform ance versus a variation of max imum allowable total transmit power P T (i.e. 1 d BW to 5 dBW) by setting bo th SNR and INR to 10 d B. W e consider that all interferer s have th e same p ower . W e also set the matrix perturb ation par ameter to ǫ max = 0 . 5 for all algorithm s. Fig. 1 shows that the ou tput SINR increases as we lift u p the limit for the max imum allowable tr ansmit power . The proposed CCSP RDB method o utperfo rms the worst-case SDP alg orithm an d perfor m clo se to the case with perfect CSI. 1 1.5 2 2.5 3 3.5 4 4.5 5 2 3 4 5 6 7 8 9 10 11 P T (dBW) SINR (dB) perfect CSI imperfect CSI worst−case SDP proposed method Fig. 1. SINR versus P T , SNR= 10 dB, ǫ max = 0 . 5 , INR= 10 dB In the second example, we incr e ase the system INR from 10 dB to 20 dB, consider K = 3 u sers but rearra n ge the powers of the interferer s so that one of the m is much stronger than the o ther . W e then examine the algorith ms in an incohere nt scenario and set the power ratio o f the stronger inter f erer over the weaker to 10 . The maximum allowable total transmit power P T and the per turbation parameter ǫ max are fixed to 1 dBW and 0 . 2 , respec tively . W e observe the SINR pe rforma n ce versus SNR fo r these algorithms and illustrate the results in Fig. 2. Then we set the system SNR to 10 d B an d ob serve the output SINR perform ance versus snapsh ots as in Fig. 3. It can b e seen th at all a lgorithms h av e perfor mance degradation due to strong interferers as well as th eir power distribution. Howe ver , th e CCSP RDB algorithm has excellent robustness in terms of SIN R a g ainst the presen c es o f stro ng interfere r s with unbalan ced p ower distribution. −10 −5 0 5 10 15 20 −40 −30 −20 −10 0 10 20 SNR (dB) SINR (dB) perfect CSI imperfect CSI worst−case SDP proposed method Fig. 2. SINR versus SNR, P T = 1 dBW , ǫ max = 0 . 2 , INR= 20 dB 0 20 40 60 80 100 −10 −8 −6 −4 −2 0 2 4 snapshots SINR (dB) perfect CSI imperfect CSI worst−case SDP proposed method Fig. 3. SINR versus snapshots, P T = 1 dBW , ǫ max = 0 . 2 , SNR= 10 dB, INR= 20 dB V . 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