An Automated Window Selection Procedure For DFT Based Detection Schemes To Reduce Windowing SNR Loss

The classical spectrum analysis methods utilize window functions to reduce the masking effect of a strong spectral component over weaker components. The main cost of side-lobe reduction is the reduction of signal-to-noise ratio (SNR) level of the out…

Authors: Cagatay C, an

An Automated Window Selection Procedure For DFT Based Detection Schemes   To Reduce Windowing SNR Loss
An Automa ted Window Selectio n Pro ced ure F or DFT Based Detectio n Sc hemes T o Reduce Windowing SNR Loss C ¸ a˘ gata y Candan a a Dep art ment of Ele ctric al and Ele ctr onics Engine ering, Midd le East T e chnic al Uni versity (METU), 06800, Ank ar a, T urkey. Abstract The cl assical sp ectrum analysis metho ds utilize windo w functions to reduce the masking effect of a strong sp ectral comp onen t ov er weak er comp onents. The main cost of side-lob e reduction is the red uction of signal-to-noise ratio (SNR) lev el of the outpu t sp ectrum. W e pr esen t a single snapshot metho d whic h optimizes the selectio n of most suitable window function among a finite set of candidate windo ws, sa y rectangle, Hamming, Blac kman w indo ws, for eac h sp ectral bin. T he main goal i s to utilize differen t windo w functions at eac h sp ectral output dep endin g on the int erference leve l encounte red at that sp ectral b in so as to red uce the SNR loss asso ciated with the w indo wing op eration. Stated differently , the windo ws with stron g in terference su ppression capabilities are utilized only when a sufficien tly p o werful in terferer is corrupting the s p ectral bin of interest is p r esen t, i.e. o nly wh en th is window is needed. The ac h iev ed r eduction in the win do wing SNR loss can b e imp ortant for the detection of lo w SNR targets. Keywor ds: Sp ectral Analysis, Windo w F unction, Pulse-Doppler Radar, T arget Detect ion. 1. In tro duction A common app roac h, if n ot th e most common, in the sp ectral analysis of signals is the windo we d F ourier transformation, whic h is the w ell kno wn p erio dogram approac h. In man y applications, including signal-to-noise ratio (SNR) sensitiv e detection applications, a nominal windo w is p re-selecte d and applied ir r esp ectiv e of the op erational SNR thr ou gh ou t the de- plo ymen t of the d etecti on system. Y et, it is kn o w n that the desired feature of wind o w fu nction (side-lob e sup pression) comes at the cost of output SNR loss. As an example, th e frequent Email addr ess: ccandan@metu.edu .tr (C ¸ a˘ gata y Candan) Pr eprint submitte d to Si gnal Pr o c essing July 31, 2018 c hoice of Hamming windo w r esults in an S NR loss of 1.35 dB, which can b e imp ortant for th e detection of targets at lo w SNR or equ iv alen tly for the extension of th e in strument ed range of a radar system. The m ain goal of this study is to p ose the win d o w selecti on pr oblem as a h yp othesis test and present an automated pro cedure that k eeps the SNR loss du e to win- do wing op eration at a min im um. T o d o that, w e assume that the system has a finite n umber of windo w functions at its disp osal (sa y rectangle, Hamming, Blac kman windows) and aims to select the m ost su itable window function for eac h sp ectral output, i.e. for eac h d iscrete F ourier transformation (DFT) bin, through th e Ba y esian hyp othesis testing. The pr op osed metho d is based on a sin gle sn ap s hot of data and is applicable in all con v en tional detection sc hemes u tilizing win do w ed DFT op eration. The sp ectral analysis is a well established topic of statistic al signal pro cessing closely link ed with sev eral applications in sp eec h pro cessing, radar signal pro cessing, remote sensing. The main goal of sp ectral analysis is to detect and accurately estimate the p o w er of eac h sp ectral comp onent of the input. The metho d s to this aim can b e categorize d as single and m ultiple snapsh ot metho d s as illustrated in Figure 1. The single snapshots metho ds are, in general, data-indep end en t metho ds b ased on the window ed F ourier transformation. Multiple snaps hot metho ds, suc h as the Cap on’s metho d, are d ata-depen d en t metho ds and based on the estimation and m in imization (ensem ble) av erage v alue in terference at the outpu t. The m ultiple sn apshot metho ds use more information, such as the interference/signal auto- correlation function and yield sup erior resu lts, in general. In certain applications, a-priori information on the signal and in terf er en ce statistic s ma y not b e a v ailable or the sensing scheme ma y n ot suitable for an ensemble c h aracterizat ion. F or the con v en tional pu lse-Doppler radar systems, a single snapshot v ector, comp osed of slo w-time samp les from a range cell, is a v ailable to d etect the presence of a mo ving target in a range cell wh ic h can b e con taminated with clutter, j ammers and p oten tially other targets. The conv ent ional pr o cessing c h ain t ypically includes a stage of windo we d DFT. A s u itable wind o w , sa y Hamming wind ow h a vin g a go o d side-lob e suppression (43 d B f or the Hamming window) is selected to reduce the shado wing effect of stron g un desired compon ent ov er the target comp onen t. Unfortunately , th is choi ce brings an S NR loss, w h ic h is 1.35 dB for the Hamming wind ow that can b e comp ens ated by increasing the transmit p o wer. This requires a factor of 10 (1 . 35 / 10) ≈ 1 . 36 fold in crease in the n umber of transm it elements of a phase arr a y system op erating at p eak p o wer limitation. SNR 2                             ! " # $ % & ' ( ) * + , - . / 0 1 2 3 456 7 8 9 : ; < = >? @ AB C D E F G H I J K L MN O P Q R S T U V WXY Z [ \ ]^ _ ` a b c d e f g h i j k l m n o p q r s tu v w x y z{ | } ~                          Figure 1: Conv entional and Modern S pectral Analysis Metho ds losses due to pro cessing, called signal pro cessing loss in th e radar signal p ro cessing literature, should b e av oided as muc h as p ossible; since their comp ound effect on th e system design, say on the p ow er b udget, can b e a significant factor affecting the m on etary cost of the system. The selectio n of a suitable window function is an ap p licatio n or s cenario sp ecific c hoice based on some tr ade-offs. As noted b efore, the main b enefit of win do wing is the reduction of the signal sidelob es in the output sp ectrum at the exp ense of w idened mainlob e (resolution loss) and SNR loss, as do cumented in [1, C h.6]. The main goal of this study is to optimize the windo w selectio n suc h that the r esulting av erage SNR loss due to windo wing is negligibly small. T o this aim, we prop ose to use a set of conv enti onal wind o w s, say rectangle, Hamming and a Chebyshev win do ws pro viding 13, 43, 120 dB side-lob e sup pression r atios, p ose the windo w selection as a hyp othesis testing p roblem and apply the prin ciples of the Ba y esian h yp othesis testing. Up on an analysis of the resulting hyp othesis testing based metho d, we suggest a reduced complexit y windo w selection metho d and examine its p erformance. In the literature, the metho d known as the dual-ap o dization method (for t w o w indo ws) and its extension to the m u ltiple wind o w s (m u lti-ap o dization) resem b les the pr op osed line of study , [2]. In dual-ap o dization, the DFT of the inpu t is calculated t wice with t w o different window functions. F or a sp ecific DFT bin, the d ual ap o d izatio n output is the ou tp ut sp ectrum sample of t w o wind ows having the min im um magnitud e. The motiv ation of dual-ap o dization metho d can b e most easily seen for the noise-free op eration. In the absence of noise, the windo w with the b etter side-lob e supp ression is selected for th e side-lob e bins by th e mentioned calculati on of minim um DFT magnitude. F or the bins whic h are lo cated in the main-lob e, the wind o w with th e smaller b eam width, the higher r esolution window, is s elected. Hence, the inclus ion of a non-linearit y in the pro cessing c hain, join tly enables b oth high resolution an d s mall side-lob es, wh ic h is not p ossible with linear pro cessing metho ds. Th is metho d has fou n d 3 applications mainly in the imaging applications [3, 4] w here S NR is not the most ma jor issue of concern as in the detection applications. T o the b est of our kn o w ledge, th e p erformance of dual-ap o dization is n ot examined from the d etectio n theoretic p oin t of view except the stud y of [5]. In this work, w e present the comparison of the suggested h yp othesis testing based windo w s electio n metho d with the multi-apo d ization metho d. 2. Bac kground W e consider th e follo wing signal mo del, r [ n ] = √ γ s e j ( ω s n + φ s ) | {z } s [ n ] + √ γ j e j ( ω j n + φ j ) | {z } j [ n ] + v [ n ] , n = { 0 , 1 , . . . , N − 1 } . (1) Here s [ n ] denotes the signal of interest, a complex exp onen tial signal w ith f r equency ω s ; j [ n ] denotes the in ten tional or uninten tional j amming signal corrupting the observ ations r [ n ]; v [ n ] is zero mean, unit v ariance complex v alued white noise with circularly symm etric Gaussian distribution, v [ n ] ∼ CN(0 , 1). T he phase v alues of signal and jammer comp onents, φ s and φ j , are assumed to b e ind ep endent random v ariables w ith a u niform d istribution in [0 , 2 π ). Th e parameters γ s and γ j are ind ep endent, exp onent ially d istributed random v ariables with mean v alues ¯ γ s and ¯ γ j , resp ective ly . The input S NR and j ammer-to-noise ratio (JNR) is defined as SNR = E {| s [ n ] | 2 } E {| v [ n ] | 2 } = ¯ γ s , JNR = E {| j [ n ] | 2 } E {| v [ n ] | 2 } = ¯ γ j . (The mo del giv en in (1) corresp on d s to the Ra yleigh f ad ed target signal observe d under Ra yleigh faded rank-1 inte rference (jammer) and white Gauss ian noise.) Equation (1) can b e written in v ector form as r = √ γ s e j φ s s + √ γ j e j φ j j + v , wh ere s is an N × 1 column v ector with the en tries e j ω s n , n = { 0 , 1 , . . . , N − 1 } . (Th e v ector j is defined similarly .) T hroughout this wo rk, w e pr efer to express the frequency v ariables, ω s or ω j , with the units of DFT b ins, ω s = 2 π f s / N , where N is th e n umber of observ ations. With this definition, f s b ecomes a real v alued parameter in the in terv al [0 , N ). Th is defin ition simp lifies the description and p erception of the numerical v alues asso ciated with main-lob e w idth, the frequency d ifference b et wee n signals etc. Figure 2 sho ws the magnitude sp ectrum of rectangle, Hamming and Chebyshev win do ws of length 16. In the presence of white noise and abs ence of jamming, the optimal d etecto r, maximizing the outpu t S NR, is the detector matc hed to the signal vect or w = s and the 4 optimal decision statistics for the detection application is | w H r | 2 , [6]. Th is corresp onds to the p ro cessing of the inpu t data w ith the rectangle w indo w. Wh en jamming is p resen t, S NR maximizing filter is the whitened matc hed filter detector, that is w = ( I + γ j jj H ) − 1 s and the decision statistics is | s H ( I + γ j jj H ) − 1 r | 2 . In this w ork, w e assume that the user do es not ha ve the capacit y to implemen t the optimal filter du e to the lac k of statistical information on the jammer. Instead, user selects the most suitable windo w fr om a set of cand id ate win d o ws to reduce th e effect of jamming signal. The decision statistic s b ecomes | s H D win r | 2 where D win is a diagonal m atrix wh ose diagonal ent ries are th e samples of th e window function selected. T o understand the factors effecting the windo w c hoice, assume that the target is lo cated at th e DC bin . F or this target, a very strong jammer in the Region 2 of Figure 2 requires th e application of Chebyshev window. Y et, th e Hamming windo w, w hic h h as a b etter SNR loss than Chebyshev window, can b e su ffi cien t for a wea k er jammer in the same region. On th e other hand, ev en for v er y strong jammer s in Region 1 of Figure 2, the Chebyshev wind ow is not suitable, since there is no side-lob e s u ppression due to large main-lob e widen in g of this windo w. Hence for Region 1 jammers, the p ossible windows of choice are limited to rectangle or Hamming w indo w. In this study , our goal is to automate suc h reasoning pro cedures and present a window selecti on metho d with a negligible SNR loss. In Figure 2, th e multi-apo d ization output of three windows is also illustrated. F or th is case, the m ulti-ap o dization output m agnitude is simply X M A ( e j ω ) = min {| X R ( e j ω ) | , | X H ( e j ω ) | , | X C h ( e j ω ) |} (2) where X R ( · ) , X R ( · ) , X C h ( · ) is the sp ectrum of rectangle, Hamming and Chebyshev wind o w s and X M A ( e j ω ) is the multi-apo d ization outpu t, [3 , 5 ]. It can b e seen fr om (2) that the m ulti-ap o dization outpu t is simply the selection of th e magnitude sp ectrum with the sm allest magnitude for eac h sp ectral comp onent, [2]. In the absence of noise, the multi-a p o d ization yields j oin tly high resolution (main-lob e width of rectangle window) and large side-lob e sup- pression, as sho wn in Figure 2. F or the noisy scenarios, the p er f ormance of multi-apo d ization, in terms of SNR loss, is not immed iately clear and a topic of in v estigation in this study . The m ulti-ap o dization idea h as found some applications in the syn thetic ap er tu re imaging (SAR) applications, [4, 7] wh ere SNR loss is n ot the main concern. The situation is qu ite different in the pulse-dopp ler rad ar systems where m uc h few er observ ations (slo w time samples) are 5 frequency (in DFT bins) 0 2 4 6 8 10 12 14 16 dB -160 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude spectrum of windows having length N = 16 Rectangle Hamming Chebyshev (120 dB) Multi-apodization Region 2 Region 1 Region 1 X: 1.392 Y: -13.72 X: 14.62 Y: -13.72 X: 3.175 Y: -40.59 X: 12.84 Y: -40.59 Figure 2: Magnitud e sp ectrum of Rectangle, Hamming, Chebyshev window s and the m ulti-ap o dization sp ec- trum. a v ailable for the target detection. An insightful explanation for the multi- ap o dization metho d giv en in [7] states that multi-apo d ization is, in principle, equiv alent to the single snapshot v ersion of Cap on’s metho d . In this work, our goal can b e more am b itiously stated as to dev elop a b etter alternativ e f or the op en title of s ingle snapsh ot Cap on’s metho d. 3. Prop osed Approac h The p rop osed approac h is describ ed with the windo w f unctions illustrated in Figure 2. Without any loss of generalit y , we m a y assume that the signal of inte rest is at DC bin and the j amm in g p o we r is lo calized either in Region 1 or Region 2 of Figure 2. Our goal is to determine the jammer activit y thr ough M-ary hyp othesis testing and apply a suitable win do w b ased on the jamming activi t y level. The k’th h yp othesis can b e written as follo ws: H k : r = √ γ s e j φ s s + √ γ j k e j φ j j + v , k = { 1 , . . . , M } . (3) The v ectors s , j , v refer to signal, jammer and noise v ectors as defined in S ection 2. The γ s is nuisance p arameter of the test app earing in all hyp otheses. The critical parameter of the 6 k’th hyp othesis is the JNR, ¯ γ j k . W e assume that ¯ γ j 1 < ¯ γ j 2 < . . . < ¯ γ j M , that is the w eak est jammer case is r epresen ted with H 1 . F or any observ ation vec tor r , the index of the h yp othesis with the highest a-p osteriori probabilit y can b e expressed as b k = arg m ax 1 ≤ k ≤ M P ( H k | r ) . (4) W e treat the selected h yp othesis as an in dicator of the jammin g activit y lev el and asso ciate a w indo w function for eac h hyp othesis. N ote that (4) can also b e written in terms of the lik elihoo d ratios as f ollo ws, b k = arg max 1 ≤ k ≤ M P ( H k | r ) P ( H 1 | r ) = arg max 1 ≤ k ≤ M P ( H k ) P ( H 1 ) f R ( r | H k ) f R ( r | H 1 ) , (5) where P ( H k ) is the a-priori probabilit y of the k’th h yp othesis and the r igh tmost term is the lik elihoo d ratio of th e k th and fir st h yp othesis. In many scenarios, the a-priori probabilities of hypothesis are unkn o w n and are tak en as P ( H k ) = 1 / M for k = { 1 , 2 , . . . , M } . T hen, b k reduces to arg m ax 1 ≤ k ≤ M f R ( r | H k ) f R ( r | H 1 ) . 3.1. Window Sele ction T est Base d on the Likeliho o d R atio With the d efi nitions giv en Section 2, f R ( r | H k ) is Gaussian density with zero m ean and co v ariance matrix R k = ¯ γ s ss H + ¯ γ j k R n j + I , where R n j is the normalized jammer co v ariance matrix with unit elements on its diagonal. T he pro duct of normalized jammer cov ariance matrix and a v erage jammer p o we r f or the k’th hyp othesis is denoted as R j k = ¯ γ j k R n j . W e express the eigendecomposition of R j k as R j k = E j Λ j k E H j . Here, the columns of E j matrix are the eigen v ectors spann ing the jamming sp ace. The matrix Λ j k is a diagonal matrix ha ving the eigenv alues R j k on its diagonal. T he eigen v alues of R j k is ¯ γ j k × λ n i , that is the pro du ct of ¯ γ j k and the eigen v alues of n ormalized jammer co v ariance matrix R n j . The log-lik eliho o d ratio of the k’th and first h yp othesis can b e written as: log  f R ( r | H k ) f R ( r | H 1 )  = log | R 1 | − log | R k | + r H ( R − 1 1 − R − 1 k ) r . (6) T emp orarily , we call jammer plus noise co v ariance matrix for the k th hypothesis as R j k + n = ¯ γ j k R j + I = R j k + I , then R k app earing in the lik eliho o d ratio b ecomes R k = ¯ γ s ss H + R j k + n . With this definition, the qu adratic pro d uct in (6) reduces to r H R − 1 k r = r H R − 1 j k + n r − 1 1 + s H R − 1 j k + n s | r H R − 1 j k + n s | 2 . (7) 7 The jammer plus noise cov ariance matrix, R j k + n = R j k + I , can b e eigendecomp osed as R j k + n = E j Λ j k + n E H j + E n E H n . It shou ld b e n oted that jammer plu s noise co v ariance matrix is a full rank m atrix; therefore, the columns of E n matrix, whose span is the n oise sp ace, is orthogonal to the jammer sp ace, the column space of E j matrix. Using the eigendecomp osition of R j k + n , w e can express s H R − 1 j k + n s term app earing in (7) as follo ws: s H R − 1 j k + n s = s H E j Λ − 1 j k + n E H j s | {z } ≈ 0 + s H E n E H n s ≈ | E H n s | 2 . (8) Here, it is assu med that the signal vect or has negligible p o wer in the jammer sp ace. W e ma y consider that the jammer is in Region 2 of Figure 2 ; th erefore the p ro jection of signal p ow er to jammer sp ace is negligible. (The leak age of signal p o wer can b e considered as n egligible un less the S NR is extremely is high wh ic h is a case of little concern for the detection problems.) Similarly , we can express the term r H R − 1 j k + n s in (7) as r H R − 1 j k + n s = r H E j Λ − 1 j k + n E H j s | {z } ≈ 0 + r H E n E H n s ≈ r H E n E H n s . (9) Finally , the term r H R − 1 j k + n r = r H ( R j k + I ) − 1 r H in (7) b ecomes r H R − 1 j k + n r = r H E j Λ − 1 j k + n E H j r + r H E n E H n r = N j X i =1 1 ¯ γ j,k λ n i + 1 | e H i r | 2 + | E H n r | 2 , (10) where N j is rank of R j k = ¯ γ j k R n j and e i is the eigen v ector of R j k with the eigen v alue ¯ γ j,k λ n i . Using equations (8), (9 ) and (10), we can simplify the qu adratic term app earing on the left side of (7 ), r H R − 1 k r , as r H R − 1 k r ≈ (11) r H E j Λ − 1 j k + n E H j r + r H E n E H n r − 1 1 + | E H n s | 2 | r H E n E H n s | 2 . This concludes the simp lification of the qu adratic term in th e log-lik eliho o d ratio expr ession in (6). The remaining term to b e simplified in this equ ation is th e determinant of R k = 8        ¡ ¢ £ ¤¥ ¦ § ¨ © ª « ¬  ® ¯ ° ±² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Figure 3: An illustration for the partitioning of the real line into th ree parts. ¯ γ s ss H + R j k + n matrix: det( R k ) = (1 + ¯ γ s s H R − 1 j k + n s ) d et( R j k + n ) ( a ) = (1 + ¯ γ s | E H n s | 2 ) det( R j k + n ) ( b ) = (1 + ¯ γ s | E H n s | 2 ) N Y i =1 ( ¯ γ j,k λ n i + 1) (12) In line-(a) of equation (12), the relation s H R − 1 j k + n s ≈ | E H n s | 2 is utilized one more time (see equation (8)). In line-(b), the determinan t is computed via the pr o d uct of the eigen v alues. With the su b stitution of relations giv en b y equations (10) and (12) into the log-lik eliho o d ratio given in (6 ), we get the follo w ing expr ession, log  f R ( r | H k ) f R ( r | H 1 )  ≈ (13) N j X i =1 log  ¯ γ j, 1 λ n i + 1 ¯ γ j,k λ n i + 1  + N j X i =1  1 ¯ γ j, 1 λ n i + 1 − 1 ¯ γ j,k λ n i + 1  | e H i r | 2 . In the last expression e i and λ n i are the eigen v ector of norm alized jammer co v ariance matrix R n j . Next, we need to construct a hyp othesis for eac h candidate wind ow function with a prop er selection of h yp othesis parameters. On the selection of R n j matrix: W e assu me that th e jammer frequency is assu med to b e un iformly distribu ted in th e side-lob e region of the w indo w. As a concrete example, w e ma y consider the s id e-lob e region of N=16 p oint Cheb yshev window, sh o w n as the Region 2 of Figure 2 . This region ranges fr om 3.175 to 12.84 DFT bin as can b e seen from Figure 2 . (The same region with unit of radians p er sample corresp onds to Θ 1 = 2 π / N × 3 . 175 and Θ 2 = 2 π / N × 12 . 84, where N = 16.) The jammer with the frequen cy θ corresp onds to j θ = [1 e j θ e j 2 θ . . . e j ( N − 1) θ ] T . T he jammer frequency is assu med to b e uniform ly distribu ted in the in terv al [ θ 1 , θ 2 ], whic h is [3.175 , 12.84] DFT bins for the Region 2 shown in Figure 2. The auto-correlation matrix 9 for the j ammer is E { j θ j H θ } , wh ere the exp ectation op erator E {·} is o ver th e random v ariable corresp onding to the frequency Θ. The m th ro w, n th column entry of the matrix R n j can b e expressed as:  R n j  mn = θ 2 − θ 1 2 π  E Θ { j θ j H θ }  mn = 1 2 π Z θ 2 θ 1 e j ( m − n ) θ dθ =    ( θ 2 − θ 1 ) / 2 π , m = n exp( j ( m − n ) θ 2 ) − exp( j ( m − n ) θ 1 ) j 2 π ( m − n ) , m 6 = n . (14) On the selection of ¯ γ j k parameter: The candidate windows should ha v e significantly differen t sid e-lobe su ppression ratios. The r ectangle, Hamming and C heb yshev wind o w s of Figure 2 ha v e the p eak side-lob e suppression ratios of 13, 40 and 120 dB. F or eac h window, w e suggest to u se half the p eak sid e-lob e ratio in dB as ¯ γ j k parameter. F or th e m en tioned case, this resu lts in ¯ γ j 1 = 6 . 5 d B, ¯ γ j 2 = 20 d B and ¯ γ j 3 = 60 d B. This c h oice stems from the consideration that for a jammer of 15 dB JNR, w e ma y almost equally prefer to use either Hamming or rectangle w indo w, bu t not the Chebyshev window. 3.2. A Si mple Window Sele ction T est Base d On Likeliho o d R atio Appr oximation W e present a simpler test based on an appro ximation to the log-lik eliho o d r atio giv en in (14). F rom the defi n ition of normalized jammer co v ariance matrix giv en b y (14), it can b e verified that the eigen v alues of this matrix, d enoted as λ n i , are clustered around 0 and 1. (This fact can b e chec ke d b y noticing that R n j matrix giv en in (14) is the similarit y transform of the discrete prolate spheroidal sequence (DPSS) generating matrix ha ving the m th r o w , n th column en tr ies sin( θ 2 ( m − n )) / ( π ( m − n ) whose eigen v alues are kno wn to ha v e a sharp transition b et ween 1 and 0, [8, p.213].) By su bstituting λ n i ≈ 1 in (14), we get the follo wing expression log  f R ( r | H k ) f R ( r | H 1 )  ≈ N j log  ¯ γ j, 1 + 1 ¯ γ j,k + 1  +  1 ¯ γ j, 1 + 1 − 1 ¯ γ j,k + 1  N j X i =1 | e H i r | 2 , = α k + β k r H R n j r . (15) where P N j i =1 | e H i r | 2 = r H R n j r is an estimate of the jammer p o w er. Hence, the log-lik eliho o d ratio reduces to an affine function of r H R n j r , α k + β k r H R n j r . Hence, the asso ciation of window 10 functions to the hyp otheses can b e simp ly d one by partitioning the real line to M d isjoin t sets as shown in Figure 3 . The decision b ound aries, the p oin ts A and B , in Figure 3 are the JNR lev els for the win do w sw itching indicating the b ound aries b et w een lo w , m ed ium, high JNR lev els. On the determination of decision b oundaries: W e assume that the window functions are ordered in the increasing ord er of int erference suppr essing capabilities. (F or the case sho wn in Figure 2, the order is r ectangle , Hamming and Chebyshev windo ws, resp ectiv ely .) The goal is to sequent ially set the decision b oun daries, that is to d etermine the b oundary for the rectangle and Hamming windo ws first (p oin t A in Figure 3) and then the b ound ary for the Hamming and Chebyshev wind o w (p oin t B in Figure 3). It can b e said that for a candid ate windo w set with M w indo ws, w e need to determine M − 1 b ound aries b y pairwise comparin g consecutiv e windo ws ordered in th e increasing ord er of PSL. Figure 4a shows the receiv er op erating curv e (R OC) for the rectangle and Hamming win- do w ed pro cessed receiv ed vect or r w hose signal m o del is give n in (1 ). Since the signal, in- terference and noise are assumed to b e join tly Gaussian distribu ted, the detection pr oblem reduces to the problem kno wn as the detection of Gaussian signals in Gaussian noise, [9, Ch.9]. The relation b et we en probabilit y of detection and probability of false alarm, as depicted in Figure 4 is P d = P 1 / (1+SJNR) F A , wh ere S JNR refers to the signal-to- jammer-plus-n oise-ratio . SJNR at the win do w ed DFT detector output is written as follo ws: SJNR k = ¯ γ s | w H k s | 2 ¯ γ j w H k R n j w k + k w k k 2 . (16) Here w k is the k th window and without any loss of generalit y , the v ector s can b e tak en as the vec tor of all ones, that is the signal can b e assum ed at the DC bin. (If the signal is not at the DC b in, w k should b e windo w f u nction mo du lated to the frequency of the signal.) F rom the ROC curve giv en in (4)a, it can b e concluded that wh en JNR is less than 8.4 dB, which is the int ersection p oin t of ROC cur v es for t wo windows, the r ectangle window is a b etter c h oice yielding h igher probability of detection v alue. T he inte rsection p oint of ROC curv es for tw o windows can b e easily calculated by equating SJNR v alues of t w o w indo ws. If the windows are indexed as 1 and 2, the relation SJNR 1 = SJNR 2 , yields the J NR v alue for the intersectio n p oin t as ¯ γ j = | w H 1 s | 2 k w 2 k 2 − | w H 2 s | 2 k w 1 k 2 | w H 2 s | 2 w H 1 R n j w 1 − | w H 1 s | 2 w H 2 R n j w 2 . (17) 11 JNR (dB) -40 -20 0 20 40 60 80 P d 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR = 5 dB, N = 16, Jammer is in Region 1 union Region 2 Rectangle Hamming X: 8.4 Y: 0.8732 (a) Recta ngle vs. Hamming JNR (dB) -40 -20 0 20 40 60 80 P d 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR = 5 dB, N = 16, Jammer is in Region 2 Hamming Chebyshev 120 dB X: 30.5 Y: 0.8385 (b) Hamming vs. Chebyshev (120 dB) Figure 4: The receiver op erating curves (ROC) for window ed DFT detectors, P F A = 10 − 2 . In (17), the matrix R n j is the normalized jammer co v ariance m atrix defin ed in (14). The parameters θ 1 and θ 2 of this matrix is d etermin ed by considering the side-lob e sup pression regions of b oth windows. As sh o wn in Figure2, th e side-lob e sup pression region for b oth rectangle and Hamming windo ws range from 1.392 to 14.62 in terms of DFT bins. I t sh ou ld b e noted that the decision b ound aries giv en b y (17) is in v ariant to SNR and pr obabilit y of false alarm. Hence, the b oundary p oin t can b e calculated once and can b e utilized for all R OC cur v es. As a summary , the win do w selection on the appr oximate log- lik eliho o d relation is based on the qu an tizatio n of the metric d = r H R n j r / N . F or th e case shown in Figure 2, if the d v alue is smaller than 8.4 dB, then rectangle window is selected; for d v alues in b et w een 8 . 4 and 30 . 5 Hamming wind o w is selected and for d > 30 . 5 results in the s election of Ch eb yshev windo w. In the develo pment of the test based on the lik elihoo d r atio (giv en in Section 3.1) and its appro ximate ve rsion (giv en in this sectio n), the in terference is assu med to b e in the sup pression band of all windo ws, whic h is the part of sp ectrum sh own as Region 2 of Figur e 2. If the in terference lies in the transition band of a particular window (sh o w n as Region 1 of Figure 2 ), then that window should b e discarded. As an example, f or the in terference lying in Region 12 1 of Figure 2, th e Chebyshev windo w sh ould not b e utilized at all; since the Cheb yshev windo w do es not provide any impr o vemen t in interference sup pression in comparison to the Hamming windo w f or this jammer. On the cont rary , the Chebyshev window amp lifies the in terference p o w er in comparison to the Hamming window. Hence, in spite of its sup erior side-lob e su ppression capacit y for the Region 2 in terferers, the C heb yshev w indo w sh ould b e a voided for the Region 1 jammers. W e int egrate the idea of disabling some of the wind o w s based on the d etected jammer region to th e existing scheme and present the suggested metho d in T able 1. F or fu rther clarification, w e also present a ready-to-use MA TLAB co de with application examples in [10]. The suggested algorithm can b e b riefly explained as follo ws: Step 0 is the in itializ ation step wh ere the stop bands of the windows, the decision b oundaries are calculate d. In Step 1, the normalized cov ariance matrix for an interferer lying in the stop-band of eac h wind o w is calculated. In Step 2, the j amm in g p o we r residing in the stop b and of eac h windo w is calculate d. Step 3 implements th e idea of windo w disabling based on the jammer sp ectral lo cation. Step 4 is the quantiz ation of estimated jamming p o wer f or th e selection of the windo w f unction, as illustrated in Figure 3. 4. Numerical Comparisons W e p resen t a set of numerical comparisons of the suggested metho d with the con v en tional windo we d DFT detectors and multi-apo d izatio n m etho d. Case 1: Interference in the suppression band of all windo ws: T o ease the pr e- sen tation, w e con tin ue with the detectors u tilizing th e r ectangle , Hamming and Ch eb yshev windo ws whose magnitude sp ectrum is illustrated in Figure 2. F or th is case, it is assumed that the interference lies in Region 2 of Figure 2. More sp ecifically , the inte rference lies in the b and where all three windo ws hav e the abilit y of interference supp ression at different capacities. Figures 5a-5b sh o w the v ariation of the p robabilit y of detection v ersus jammer-to-noise ratio (JNR) for the scenario parameters of SNR = 0 d B, the probab ility of false alarm of 10 − 2 , the num b er of observ ations of 16, ( N = 16). The frequ ency of the interfering signal is 4 to 6 DFT bins a w a y from the frequen cy of the signal to b e d etected and the inte rfering frequency is uniformly distributed in this inte rv al guarante eing that th e in terference is in Region 2 of Figure 2. 13 JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 7 0 Pd 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 SNR = 0 dB; N = 16; Jammer in Region 2 Rectangle Hamming Chebyshev, 120 dB Multi-apodization Method 1 (Exact Likelihood) Suggested Method (a) P d vs. JNR JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 70 Pd 0.55 0.6 0.65 0.7 0.75 SNR = 0 dB; N = 16; Jammer in Region 2 Rectangle Hamming Chebyshev, 120 dB Multi-apodization Method 1 (Exact Likelihood) Suggested Method (b) Pd vs. JNR (zoo med) JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 7 0 Probability of Window Selection 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Multi-apodization Rectangle Hamming Chebyshev, 120 dB (c) Multi-ap o dizatio n Metho d JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 70 Probability of Window Selection 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Method 1 - Exact Likelihood (solid), Suggested Method (dashed) Rectangle Hamming Chebyshev, 120 dB (d) Pr op osed Metho ds Figure 5: Case 1. Subfigures (a)-(b): ROC curves for th e window ed D FT detectors, multi-apo dization metho d and p roposed detectors, P F A = 10 − 2 . Sub figures (c)-(d): The windo w selection probabilities of multi-apo dization and p roposed metho ds vs. JNR. F rom Figures 5 a-5b, it can b e immediately r ead that the rectangle window p r esen ts the b est p erformance (in the sense of detectio n pr obabilit y) for su fficien tly low JNR v alues. On the other h and, the p erformance of rectangle w in do w ed F ourier tr ansformation detector degrades rapidly , once JNR is ab o v e 20 dB. It should b e clear from these figures that the windows pro viding h igher int erference sup pression capabilities should b e utilized at high JNR lev els. In many practical s y s tems, a nominal windo w function, suc h as the Hamming windo w, 14 JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 7 0 Pd 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 SNR = 0 dB; N = 16; Jammer in Region 1 Rectangle Hamming Chebyshev, 120 dB Multi-apodization Method 1 (Exact Likelihood) Suggested Method (a) P d vs. JNR JNR (dB) -30 -25 -20 -15 -10 -5 0 5 10 15 Pd 0.55 0.6 0.65 0.7 0.75 SNR = 0 dB; N = 16; Jammer in Region 1 Rectangle Hamming Chebyshev, 120 dB Multi-apodization Method 1 (Exact Likelihood) Suggested Method (b) Pd vs. JNR (zoo med) JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 7 0 Probability of Window Selection 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Multi-apodization Rectangle Hamming Chebyshev, 120 dB (c) Multi-ap o dizatio n Metho d JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 70 Probability of Window Selection 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Method 1 - Exact Likelihood (solid), Suggested Method (dashed) Rectangle Hamming Chebyshev, 120 dB (d) Pr op osed Metho ds Figure 6: Case 2. Subfigures (a)-(b): ROC curves for th e window ed D FT detectors, multi-apo dization metho d and p roposed detectors, P F A = 10 − 2 . Sub figures (c)-(d): The windo w selection probabilities of multi-apo dization and p roposed metho ds vs. JNR. is selected b y the system designer and th is window is utilized irresp ective of the JNR v alue. Figure 5a-5b illustrate that the choi ce of Hamming w in do w brings a sub-optimal p erformance at lo w JNR and furthermore do es not p resen t a significant gain at excessiv ely high JNR v alues. Typically , the system designer r ules out the p ossib ilit y of excessiv e JNR v alues through another jammer detection mec hanism, say a side-lob e blank er, and jus tifi es the SNR loss due to the app licatio n of Hamming windo w as a necessary trade-off b et w een lo w JNR and medium 15 JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 7 0 Pd 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 SNR = 0 dB; N = 16 Jammer in the interval of [2.35, 2.45] of Region 1 Rectangle Hamming Chebyshev, 120 dB Multi-apodization Method 1 (Exact Likelihood) Suggested Method (a) P d vs. JNR JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 Pd 0.55 0.6 0.65 0.7 0.75 SNR = 0 dB; N = 16 Jammer in the interval of [2.35, 2.45] of Region 1 Rectangle Hamming Chebyshev, 120 dB Multi-apodization Method 1 (Exact Likelihood) Suggested Method (b) Pd vs. JNR (zoo med) JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 7 0 Probability of Window Selection 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Multi-apodization Rectangle Hamming Chebyshev, 120 dB (c) Multi-ap o dizatio n Metho d JNR (dB) -30 -20 -10 0 10 20 30 40 50 60 70 Probability of Window Selection 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Method 1 - Exact Likelihood (solid), Suggested Method (dashed) Rectangle Hamming Chebyshev, 120 dB (d) Pr op osed Metho ds Figure 7: Case 3. Subfigures (a)-(b): ROC curves for th e window ed D FT detectors, multi-apo dization metho d and p roposed detectors, P F A = 10 − 2 . Sub figures (c)-(d): The windo w selection probabilities of multi-apo dization and p roposed metho ds vs. JNR. JNR case. This study aims to presen t an alternativ e metho d b ased on a data-adaptiv e wind o w selection pr o cedure in whic h SNR loss d ue to windo wing is effectiv ely eliminated. The dashed line in Figure 5a-5b sho ws the p erform ance of the m ulti-ap o dization metho d. The wind o w selection pro cedur e of this metho d is established through the r elation giv en in (2). This metho d can b e summarized as follo ws. The windo w ed DFT of the inp ut is calculated for all thr ee w in do ws. F or a fixed DFT bin of in terest, the windo w ed DFT outpu t with the 16 least m agnitud e is selected. (It is imp ortan t to note that the win do w ed DFT outputs should ha v e the same resp onse w h en the incoming signal frequency p erfectly m atc hes the DFT b in frequency in the absence of n oise. Th is is easily established by equating the DC resp onse of the windows to 1 (equiv alen tly 0 dB) by scaling eac h windo w, as s h o wn in Figure 2.) When the dashed line in Figure 5 a-5b (m ulti-ap o dization m etho d ) is compared with the solid lines (con ven tional window ed DFT d etecto rs), we see that at lo w JNR v alues, the m ulti-ap o dization metho d yields a b etter p erformance than b oth Hamming and Cheb yshev windo ws. Figure 5c sho ws, the probabilit y of window selection for the multi-apo dization metho d . This figure sho ws that at low JNR v alues, ab out 50% of the time rectangle window is selected. This explains the sup eriority of multi-apo d ization metho d ov er Hamming and Cheb yshev w in do ws at lo w JNR. Similarly , when JNR is extremely high, sa y 70 dB, the p erf ormance of the m ulti-ap o dization metho d approac hes Chebyshev wind o w , whic h is the b est choice among the windo ws examined f or excessivel y high JNR v alues. F igure 5c illustrates that as JNR increases, the pr obabilit y of Chebyshev win do w selection also increases, exp laining high JNR b eha vior of m ulti-ap o dization metho d in Figure 5a-5b. F or medium-to-high JNR v alues, the m ulti-ap o dization method p resen ts an acceptable p erformance, as seen f rom Figure 5b. The main of reason of less-than-impressive p erformance is the sluggish c hange of window selection probabilities as sho wn in Figure 5c. Stated differently , the multi-apo d ization metho d d o es not r eact fast enou gh to J NR increase. The p erformance of the prop osed metho d with exact and appro ximate lik elihoo d metric is sho wn with the dash ed lines and triangle mark ers in Figure 5a/b. It can b e immediately noted that the p er f ormance with the exact and appro ximate like liho o d metric are almost identica l. The p erform ance of suggested metho d is sup erior to the wind o w ed based approac h at all JNR v alues and is almost identi cal to the p oin t wise maxim um of three detectors sho wn in Figure 5a-5b at all J NR lev els. T he sup erior p erformance in the detection prob ab ility can also b e explained by the rapid v ariation of the wind o w s electi on probabilities giv en in Figure 5d. Case 2: In terference in the transition band of a w indow: Here we assume that in terference is in the transition band of one of the windows. F or this case, the frequen cy of the in terfering s ignal is assu med to b e uniformly distribu ted 1.5 to 3 DFT b ins a wa y from the frequency of the s ignal to b e detected, i.e. in Region 1 of Figure 2 . The other sim ulation parameters are identica l to on es in C ase 1. 17 Figure 6 presents the resu lts for this scenario. It can b e noted the m ulti-ap o dization detec- tor and prop osed detecto r app roac h th e p erformance of Hamming window as JNR increases. F rom Figure 6c, it can b e noticed that the multi -ap o dization output is dominated by the selection of the Hamming windo w at high JNR. Similarly , as the dashed red line with the cross mark er indicates the suggested metho d also utilizes Hammin g wind o w with increasing JNR, but the prop osed metho d adopts the Hamming windo w at m uc h smaller J NR v alues in comparison to the multi-apo d ization metho d. The Chebyshev windo w is not selected at all with the p rop osed metho d. I t should b e in tu itiv ely clear that the c hoice of Chebyshev windo w should b e a voided f or this scenario. The next case furth er elab orates th is p oint. Case 3: Interference in the transition band of a window: Figure 7 pr esen ts the results w hen the in terfering frequency is 2.35 to 2.45 bins a wa y from th e frequency of the signal to b e detected. The int erv al for the int erfering frequen cy is sp ecially selected to illustr ate the disadv anta ges arisin g when the interference is in the transition b an d of a w indo w. F rom Figure 2, it can b e n oted that Hamming w indo w p resen ts aroun d 80 d B in the interv al of [2.35,2 .45] b ins; while the Chebyshev wind o w presen ts a suppression ration of 20 dB. I d eally , the Hamming w in do w should b e the m ost suitable c hoice for the supp r ession of a strong in terferer in this frequency in terv al. It should b e noted this situation, i.e. the p reference of a windo w ha ving a p o orer p eak-sidelob e s u ppression ratio, o ccurs only when the in terfering frequency is in the transition band of one of the windo ws. Figure 7 presents the p erf ormance of su ggested m etho d. The dashed lines in Fig 7d indicate that the Chebyshev wind o w is not selected at all JNR v alues with the suggested m etho d . It can b e said that the suggested metho d, in effect, eliminates the Chebyshev w indo w c hoice when the int erference is in the tr an s ition band of this window as desired. It can also b e seen from Figures 7a-7b th at high JNR p erformance of the suggested metho d is identic al to the p erformance of the Hamming window, as desir ed . W e also note that the metho d utilizing the exact like liho o d (whic h is not equipp ed by windo w d isabling feature) only works w ell when the interferer is in the stop-band of all windo w f unctions, as in C ase 1. Study of Window Selection Probabilit y: As a fi n al n umerical study , w e examine the probabilit y of w in do w selection in more depth. Th e inpu t for this study is assumed to b e r [ n ] = √ γ 1 e j ( ω 1 n + φ s ) + √ γ 2 e j ( ω 2 n + φ j ) + v [ n ] , n = { 0 , 1 , . . . , 15 } 18 where ω 1 = 2 π 16 0 . 1, ω 2 = 2 π 16 6 . 25 and v [ n ] is unit v ariance, zero-mean white noise. W e con tinue to u s e the SNR d efi nition in S ection 2. Under this setting, T able 2 shows the window select ion probabilities for the p r op osed metho d wh en signal comp onents forming r [ n ] has different SNR v alues. It can b e immed iately observe d that the w indo w selection probabilities v ary with the DFT bin of interest. Stated different ly , if we are inte rested in the frequen cy con ten t of the 11th DFT bin, b oth comp onen ts forming r [ n ], i.e. signals at the bins 0.1 and 6.25, act like jammers corrupting the frequ ency con tent of this bin. Case A of T ab le 2 illustrates the case for weak signal comp onents. It can b e noticed that the rectangle windo w (whose results are sho wn in the rows indicated by ’R’ letter) is s electe d with a significan t ma jorit y for the frequency b ins close to the signal comp onents (0’th, 6’th and 7’th bin s). The Hamming windo w is s elected 14.8 % of the time for the DC bin . This is essen tially d u e to in terference generated by the s ignal comp onen t at th e 6.25 DFT b in. The selection of the Hammin g windo w for 6th or 7th DFT bins is m uc h low er, close to 0.3%, since the signal at DC bin imp oses h as a 5 d B lo w er SNR and therefore causes less in terference. F or the w eak jamming signal case, it is assuring to observ e that the windo w with the highest SNR loss, i.e. the Chebyshev window, is not utilized at all. Case B of T able 2 giv es the wind o w selection probabilities when the s econd comp onen t with the sp ectral lo cation of 6.25 DFT bin has an S NR of 15 dB. It can b e observ ed that the w eak signal ha ving th e f requency of 0.1 DFT bins is pro cessed with the Hamming windo w 82.8% of th e time. The p ercen tage in crease fr om 14.8% (Case A) to 82.8% (Case B) is essen tially due to p o wer increase of the second comp onen t. One can also notice that the Hamming windo w utilization for th e 6th bin also increases fr om 0.3% to 13.4% when Cases A and B are compared. The p o w er of the fi rst signal comp onent is ident ical in b oth cases; hence this increase can not b e explained with the increase of jamming activit y due to the first comp onent. The increase in the Hamming windo w utilization probabilit y is due to the frequency mismatc h of the second comp onent (6.25 DFT bin ) to frequencies of DFT bin s, i.e. in teger v alued bins. Due to the m en tioned frequ ency mism atc h, the signal energy in the side-lob es of this signal acts as an int erference, inhibiting its detection. This problem is known as the p arameter mism atch pr ob lem (the mismatc h of assum ed fr equ encies with the actual signal frequency) and can b e reduced b y ev aluating zero-padded DFT instead of N-p oin t DFT. 19 W e d o not fu rther elab orate on th is topic in order n ot to d istract th e readers from the m ain message of this stud y . Cases C and D of T able 2 present the results as the second signal comp onen t S NR is further increased. On ly in the highest SNR 2 case (Case D), the wind o w preference sw itc hes to the Chebyshev win do w. It should b e n oted that this c hange of preferen ce occur s for the bins that the second signal comp onent lies in the stop-band of the Chebyshev windo w. Stated differen tly , the bin s 4 to 10 do n ot use Chebyshev at all, since these b ins constitute the main lob e and transition region of this windo w. Hence, for these bins the Chebyshev win do w can not provi de an y inte rference supp ression capabilit y . A brief of time of reflection on T able 2 can convince the readers that the suggested m etho d works as if lik e an exp er ienced op er ator with the knowledge of jammer sp ecific p arameters. 5. Conclusions W e present a metho d without an y application sp ecific parameter tuning that automate s the win d o w selection pro cedur e for the discrete F our ier transformation (DFT) based detectors. The windowing is, in general, an essenti al op eration to redu ce the masking effect of a strong sp ectral comp onen t o v er the wea k er one. Y et, it ma y come at a signifi can t SNR loss, t ypically in b etw een 1.5 to 3 dB. T h e suggested metho d aims to select the wind ows w ith strong side- lob e supp ression capabilities, or equiv alently the wind o w s with a high SNR loss, only when the situation arises, that is only when it is ind eed needed. The numerical r esults sh o w that the metho d is capable of switc hing window fun ction d ep ending on op erational JNR leve l suc h that the b est probabilit y of d etecti on among all detectors is achiev ed at all JNR v alues, in spite of the lac k of JNR kno wledge at th e r eceiving end (see Figures 5 to 7). W e b eliev e that the su ggested metho d is a promising con tender for the op en title of s in gle snapsh ot v ers ion of the C ap on’s metho d. An implement ation of the metho d is given in [10]. References [1] B. Porat, A C ourse in Digital S ignal Pro cessing, John Wiley & S ons, 1996. [2] H. C. Stankwitz, R. J. Dallaire, J. R. Fien up, Nonlinear ap o dization for sidelob e con trol in SAR imagery , IEEE T rans. Aerosp. Electron. Syst. 31 (1995) 267–279. 20 [3] J. A. C. Lee, D. C. Munson, Spatially v arian t ap o d ization for image reconstruction from partial Fourier data, I EEE T rans. Image Pr o cess. 9 (2000) 1914–19 25. [4] D. P astina, F. Colone, P . Lombardo, Effect of ap o dization on S AR image understandin g, IEEE T rans. Geoscience and Remote Sensing 45 (2007) 3533–3 551. [5] B. Balcı, On the detection of sin usoidal signals u nder the effect of sin usoidal in terfer- ence, Master’s thesis, Departmen t of Electrical and E lectronics En gineering, Middle East T ec hnical Unive rsit y , Advisor: Cagata y Candan, 2010. [6] M. A. Richards, F un damen tals of Radar Signal P r o cessing, McGra w-Hill, New Y ork, 2005. [7] J. A. C. Lee, J. D.C. Munson, Effectiv eness of sp atially-v ariant ap o dization, I m age Pro cessing, I nternational Conference on 1 (1995) 147–1 50. [8] A. P ap oulis, Signal Analysis, McGra w-Hill, 1977 . [9] H. L. V. T rees, Detection, Estimation and Mo du lation Theory , part 3, Joh n Wiley - Sons, 1971. [10] C. Candan, An Automated Windo w Selection Pro cedure F or Sp ectral An al- ysis T o Re duce Windowing SNR Loss, (MA TLAB Co d e), 2017 . URL: http://u sers.metu .edu.tr/ccandan/pub.htm . 21 T able 1: The suggested window selection metho d. Inputs are r : N × 1 input vector, L window functions, α : DFT bin ind ex of interest. O utput is k select where k select is th e ind ex of the windo w function to b e used for the sp ectral analysis of α ’th DFT bin. (A Matlab implementation is av ailable at [10]. Step 0: (Init.) a) Order L window f unctions in the increasing ord er sidelob e su ppression ratio. (Examp le: Rectangle, Hamming, Chebyshev windows s hould b e ordered as the first, second and thir d windo ws, r esp ectiv ely .) b) Calculate the L − 1 d ecision b oundaries for L windows, denoted as ¯ γ j [ k ], by setting w 1 as the k th window function and w 2 as the ( k + 1)th window function, k = { 1 , 2 , . . . , L − 1 } , in (17). Also set ¯ γ j [0] = 0. c) Determine the stop-band of eac h window with the in d ex k ≥ 2. (Examp le: In figure 2, the stop-band for the Hamming and Chebyshev wind o ws are 1.392 to 14.62 DFT bins an d 3.175 to 12.84 DFT bins, r esp ectiv ely .) d) T o select the wind ow for the output DFT bin, α ’th DFT bin, 0 ≤ α ≤ N − 1, mo dulate the inp ut v ector r to the zero’th DFT bin, i.e. r ← D α r wh ere D α is a diagonal matrix with the diagonal entries exp( − j 2 π αn N ), 0 ≤ n ≤ N − 1. Step 1: F or the k’th window with the stop-band θ 1 to θ 2 , calculate M k = R n j , 2 ≤ k ≤ L from (14). Step 2: Calculate d k = r H M k r / N for 2 ≤ k ≤ L , where r is the mo du lated inp ut v ector of dimensions N × 1. Step 3: F or k = { 2 , 3 , . . . , L − 1 } , chec k whether d k > 2 d k +1 condition is satisfied or not. If the condition is n ot satisfied for an y k , set k max = L ; else set k max as the low est k max v alue for wh ich the cond ition is s atisfied. (wind o w disabling) Step 4: Return k select suc h that ¯ γ j [ k select − 1] ≤ d 2 ≤ ¯ γ j [ k select ]. , where 1 ≤ k select ≤ k max . 22 T able 2: Rectangle (R), Hamming (H) and Chebyshev (C) windo w selection probabilities for eac h DFT bin when inpu t is the sup erposition of tw o complex exp onential signals of length N = 16 with the frequencies 2 π N 0 . 1 and 2 π N 6 . 25 radians p er sample, i.e. frequencies of 0.1 and 6.25 with the units of DFT b in s. DFT Bin Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Case A R 85.2 84.8 79.9 79.6 80.2 97.2 99.7 99.8 84.3 79.7 79 79.1 79.1 79.6 79.4 84.1 SNR 1 =0 d B, H 14.8 15.2 20.1 20.4 19.8 2.8 0.3 0.2 15.7 20.3 21 20.9 20.9 20.4 20.6 15.9 SNR 2 =5 d B C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Case B R 17.8 17.7 15.4 15.2 15.5 33.6 86.6 80 17.6 15.5 15.4 15.4 15.3 15.4 15.3 17.5 SNR 1 =0 d B, H 82.2 82.3 84.6 84.8 84.5 66.4 13.4 20 82.4 84.5 84.6 84.6 84.7 84.6 84.7 82.5 SNR 2 =15 dB C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Case C R 1.8 1.8 1.6 1.6 1.7 3.9 18 .8 15.7 1.9 1.6 1.6 1.6 1.5 1.5 1.5 1.7 SNR 1 =0 d B, H 95.7 95.7 96 96 98.3 96.1 81.2 84.3 98.1 98.4 96 96 96 96 96 95.8 SNR 2 =25 dB C 2.5 2.5 2.4 2.4 0 0 0 0 0 0 2.4 2.4 2.5 2.5 2.5 2 .5 Case D R 0.4 0.4 0.1 0.1 0.1 0.6 2.1 1 .9 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.4 SNR 1 =0 d B, H 30.4 30.5 30.9 31.2 99.9 99.4 97.9 98.1 99.8 99.9 31 30.9 30.7 30.6 30.6 30.3 SNR 2 =35 dB C 69.2 69.1 69 68.7 0 0 0 0 0 0 68.9 69 69.2 69.3 69.3 69.3 23

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