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Blind extraction of cyclostationary signal from convolutional mixtures

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Blind extraction of cyclostationary signal from convolutional mixtures
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  Blind Extraction of Cyclostationary Signal fromConvolutional Mixtures Yong Xiang, Indivarie Ubhayaratne, Zuyuan Yang School of Information TechnologyDeakin UniversityBurwood CampusBurwood, VIC 3125, AustraliaEmail:  { yxiang, kubhayar, zuyuan.yang } @deakin.edu.au Bernard Rolfe School of EngineeringDeakin UniversityWaurn Ponds CampusGeelong, VIC 3220, AustraliaEmail: bernard.rolfe@deakin.edu.au  Abstract —Extracting a signal of interest from available mea-surements is a challenging problem. One property which canbe utilized to extract the signal is cyclostationarity, which existsin many signals. Various blind source separation methods basedon cyclostationarity have been reported in the literature but theyassume that the mixing system is instantaneous. In this paper, wepropose a method for blind extraction of cyclostationary signalfrom convolutional mixtures. Given that the signal of interest hasa unique cyclostationary frequency and the sensors are placedclose to the concerned signal, we show that the signal of interestcan be estimated from the measured data. Simulations resultsshow the effectiveness of our method. I. I NTRODUCTION Blind source separation (BSS) is a fundamental problemencountered in many applications [1]-[4]. It considers practicalmultiple-input multiple-output (MIMO) scenarios and aims toestimate the unknown source signals from the measured databy sensors. To achieve BSS, the sources should have some sortof differences, e.g., they are mutually independent/uncorrelated[1], cyclostationary [5], or have constant modulus [3], [6].In some man-made systems such as wireless communicationsystems, the source signals can be pre-processed such that theypossess certain diversity [4], [7]. Among these diversity prop-erties, cyclostationarity exists in many practical applicationssuch as communication, telemetry, radar, sonar and mechanicalsystems. For example, cyclostationarity is shared by mostcommunication signals as a result of periodic switching, gat-ing, or mixing operations at the transmitter [8]. Many signalsgenerated by mechanical systems are also cyclostationary dueto the periodical rotation or movement of some parts [2].While the higher-order cyclostationary statistics of sig-nals can be employed for blind source separation [9], it isknown that the higher-order statistics based methods normallyrequire a larger number of data samples for accurate time-average approximations of higher-order statistics to achievegood statistical performance. This would significantly increasecomputation cost. In contrast, the methods based on second-order statistics are often more efficient in computation. So, thispaper restricts its attention to methods based on second-ordercyclostationary statistics. In [5], Abed-Meraim  et al.  showsthat blind source separation can be achieved using the outputcyclic correlation matrices if there do not exist two distinctsource signals whose cycle frequencies are the same and whosecyclic autocorrelation vectors are linearly dependent. A set of algorithms are proposed to perform BSS. The method in [10]exploits the phase and frequency redundancy of cyclostationarysignals in a complementary way. However, both the methodsin [5] and [10] assume that the mixing system is instantaneous.In this paper, we relax the assumption on the mixingsystem to allow it to be convolutional, which can result fromapplications where multi-path propagation exists. Specifically,we assume that the MIMO mixing system is of finite impulseresponse (FIR). Our aim is to extract a cyclostationary signalof interest from the outputs of the MIMO FIR mixing system.Extracting a cyclostationary signal from convolutional mix-tures is very challenging if not impossible. We show that thiscan be achieved if the signal of interest has a distinct cyclicfrequency and the sensors are placed close to the concernedsignal. A second-order statistics based algorithm is proposedto perform signal extraction. The effectiveness of the proposedmethod is verified by simulations.The remainder of the paper is as follows. Section IIintroduces the problem of blind signal extraction together withrelevant assumptions. The new signal extraction method isproposed in Section III. Simulation results are provided inSection IV to illustrate the performance of our method. SectionV concludes the paper.II.  PROBLEM  F ORMULATION The mixing model we consider is as follows: y ( n ) =  H ( z ) x ( n ) + w ( n )  (1)where  x ( n ) = [ x 1 ( n ) ,x 2 ( n ) ,...,x I  ( n )] T  is the source sig-nal vector,  y ( n ) = [ y 1 ( n ) ,y 2 ( n ) ,...,y J  ( n )] T  is the systemoutput vector,  w ( n ) = [ w 1 ( n ) ,w 2 ( n ) ,...,w J  ( n )] T  is theadditive noise vector,  H ( z )  is the  J   ×  I   FIR mixing matrix,and the superscript  T  denotes transpose. Assume thatA1) The source signals  x 1 ( n ) ,x 2 ( n ) ,...,x I  ( n )  are mutuallyindependent with zero mean and temporally white, and  x 1 ( n ) is cyclostationary with a unique nonzero cyclic frequency  β  1 .A2) The entries in the first column of   H ( z )  are constants andthe rest entries are polynomials, and  J > I  . Besides,  H ( z )  isirreducible and column-reduced.A3) The noise signals  w 1 ( n ) ,w 2 ( n ) ,...,w I  ( n )  are mutually  independent with zero mean and equal variance  σ 2 w . They arealso independent of the source signals.From the assumptions A1) and A3), it follows:  x 1 ( n ) x ∗ 1 ( n ) e β 1 n   >  0  (2)  x 1 ( n ) x ∗ 1 ( n  − τ  ) e β 1 n   = 0 ,  if   τ    = 0  (3)  x i ( n ) x ∗ i ( n ) e β 1 n   = 0 ,  if   i   = 1  (4)  x i ( n ) x ∗ j ( n ) e β 1 n   = 0 ,  if   i   =  j  (5)  x i ( n ) w ∗ j ( n ) e β 1 n   = 0 ,  ∀ i,j  (6)  w i ( n ) w ∗ j ( n ) e β 1 n   = 0 ,  ∀ i,j  (7)where    = √ − 1 , the superscript  ∗ is the complex conjugateoperator, and  <  ·  >  denotes the time averaging operatordefined as follows:  x i ( n ) x ∗ i ( n − τ  ) e β 1 n   = lim N  →∞ 1 N  N  − 1  n =0 x i ( n ) x ∗ i ( n − τ  ) e β 1 n (8)where  N   is the number of samples. Here, the objective isto extract the signal of interest,  x 1 ( n ) , from the mixtures y 1 ( n ) ,y 2 ( n ) ,...,y J  ( n ) .Assumption A1) and A3) are common in the context of BSS. Assumption A2) holds if the sensors are placed closeto the source generating  x 1 ( n ) , such that we can neglectthe multi-path contributions from  x 1 ( n )  to all sensor mea-surements. In some applications such as certain mechanicalsystems, placing sensors close to a source is achievable.III. P ROPOSED METHOD Let H ( z ) = L  l =0 H ( l ) z − l (9)where  L  is the order of the MIMO FIR channel matrix  H ( z ) ,and denote the order of the  j th column of   H ( z )  by  L j . Clearly, L 1  = 0  and  L  = max( L 1 ,L 2 ,...,L I  ) . Based on the  i thsystem output  y i ( n ) , we define ˜ y i ( n ) = [ y i ( n ) ,y i ( n  − 1) ,...,y i ( n  − W   + 1)] T  (10)where the slide-window width  W   is chosen to satisfy W >  ¯ L  = I   i =1 L i .  (11)Also, denote the  ( i,j ) th entry of   H ( l )  by  H i,j ( l ) . Then, from(1), (9) and (10), it follows ˜ y i ( n ) = I   j =1 H i,j ˜ x j ( n ) + ˜ w i ( n )  (12)where ˜ x j ( n ) = [ x j ( n ) ,x j ( n − 1) ,...,x j ( n − W   − L j  + 1)] T  (13) ˜ w i ( n ) = [ w i ( n ) ,w i ( n − 1) ,...,w i ( n − W   + 1)] T  (14)and H i,j  is shown at the top of the next page. Since  L 1  = 0 ,it is obvious that H i, 1  is a  W   × W   matrix with the followingform: H i, 1  =  H i, 1 (0) 0 0  ···  0  ···  00  H i, 1 (0) 0  ···  0  ···  0 ... ... 0  ···  0  H i, 1 (0) 0  ···  0  (16)where  i  = 1 , 2 , ···  ,J  .Furthermore, define the cyclic autocorrelation function ρ β 1 ii  ( τ  )  as ρ β 1 ii  ( τ  ) =  x i ( n ) x ∗ i ( n − τ  ) e β 1 n   (17)and the cyclic autocorrelation matrix  R β 1 ˜ x j ˜ x j ( τ  )  as R β 1 ˜ x j ˜ x j ( τ  ) =  ˜ x j ( n )˜ x H j  ( n − τ  ) e β 1 n   (18)where the superscript  H  stands for complex conjugate trans-pose. From (2)-(7), (13), (14), (17) and (18), we obtain R β 1 ˜ x 1 ˜ x 1 (0) =  ρ β 1 11 (0) 0 0  ···  0 00  ρ β 1 11 (0) 0  ···  0 0 ............ 0 0 0  ···  0  ρ β 1 11 (0)  =  ρ β 1 11 (0) · I W  + L 1  (19) R β 1 ˜ x i ˜ x i ( τ  ) =  0 ,  if   i   = 1  (20) R β 1 ˜ x i ˜ x j ( τ  ) =  0 ,  if   i   =  j  (21) R β 1 ˜ x i  ˜ w j ( τ  ) =  0 ,  ∀ i,j  (22) R β 1 ˜ w i  ˜ w j ( τ  ) =  0 ,  ∀ i,j  (23)where  I i  stands for the  i × i  identity matrix. Also, define ρ ii (0) =   x i ( n ) x ∗ i ( n )  .  (24)It follows R ˜ x i ˜ x i (0) =  ρ ii (0) 0 0  ···  0 00  ρ ii (0) 0  ···  0 0 .................. 0 0 0  ···  0  ρ ii (0)  =  ρ ii (0) · I W  + L i .  (25)Denoting ˜ y ( n ) = [˜ y 1 ( n ) ,  ˜ y 2 ( n ) ,...,  ˜ y J  ( n )]  (26)we have ˜ y ( n ) = H ˜ x ( n ) + ˜ w ( n )  (27)where ˜ x ( n ) = [˜ x 1 ( n ) ,  ˜ x 2 ( n ) ,...,  ˜ x I  ( n )]  (28) ˜ w ( n ) = [˜ w 1 ( n ) ,  ˜ w 2 ( n ) ,...,  ˜ w J  ( n )]  (29)  H i,j  =  H i,j (0)  ··· ···  H i,j ( L j ) 0  ···  00  H i,j (0)  ··· ···  H i,j ( L j )  ···  0 ... ... 0  ···  0  H i,j (0)  ··· ···  H i,j ( L j )  W  × ( W  + L j ) (15) H =  H 1 , 1  H 1 , 2  ···  H 1 ,I  H 2 , 1  H 2 , 2  ···  H 2 ,I  ............ H J, 1  H J, 2  ···  H J,I   JW  × ( IW  +¯ L ) .  (30)Since  J > I  , the matrix H is a tall matrix, i.e., it has morerows than columns. Also, under the assumption A1) and theinequality in (11), it is shown in [11] that the matrix H hasthe full column rank   IW   + ¯ L . Let  u 1  be a  JW  × 1  vector anddefine the compound vector c 1  =  u H  1 H  H  .  (31)Clearly, since  L 1  = 0 , the first  W   elements of   c 1  are related tothe first source signal  x 1 ( n )  and its delayed versions  x 1 ( n − 1) ,x 1 ( n − 2) , ···  ,x 1 ( n − W   +1) , respectively. So, to estimate x 1 ( n − τ  ) ,  0 ≤ τ   ≤ W   − 1  from  ˜ y ( n ) , we need to find such u 1  that one of the first  W   elements of   c 1  is nonzero and therest elements of   c 1  are zero.From (19)-(30), we have R β 1 ˜ y ˜ y (0) =  H R β 1 ˜ x ˜ x (0) H H  + H R β 1 ˜ x ˜ w (0)+( H R β 1 ˜ x ˜ w (0)) H  + R β 1 ˜ w ˜ w (0)=  H R β 1 ˜ x ˜ x (0) H H  =  H  ρ β 1 11 (0) I W  + L 1  0  ···  00 0  ···  0 ............ 0 0  ···  0  H H  . (32)From (16) and (30), we can see that the first  W   columns of the  JW   × ( IW   + ¯ L )  matrix  H  are orthogonal. Thus, from(32), the singular vectors corresponding to the  W   − 1  largestsingular values of   R β 1 ˜ y ˜ y (0)  are orthogonal to one of the first W   columns of   H . Denoting these singular vectors as  v 1 ,j ,  j  = 1 , 2 ,...,W   − 1 , we define V 1  = [ v 1 , 1 , v 1 , 2 ,..., v 1 ,W  − 1 ] .  (33)Furthermore, we can obtain the autocorrelation matrix of  ˜ y ( n )  by R ˜ y ˜ y (0) =  H R ˜ x ˜ x (0) H H  + H R ˜ x ˜ w (0)+( H R ˜ x ˜ w (0)) H  + R ˜ w ˜ w (0)=  H R ˜ x ˜ x (0) H H  + R ˜ w ˜ w (0)=  H R ˜ x ˜ x (0) H H  +  σ 2 w I JW  .  (34)Since H is a tall matrix, (34) implies that  σ 2 w  is the smallesteigenvalue of   R ˜ y ˜ y (0)  and thus can be estimated from  R ˜ y ˜ y (0) .Consequently, we can remove  σ 2 w I JW   from  R ˜ y ˜ y (0)  to obtain ¯ R ˜ y ˜ y (0)=  R ˜ y ˜ y (0) − σ 2 w I JW  =  H R ˜ x ˜ x (0) H H  =  H  R ˜ x 1 ˜ x 1 (0)  0  ···  00 R ˜ x 2 ˜ x 2 (0)  ···  0 ............ 0 0  ···  R ˜ x J  ˜ x J  (0)  H H  (35)where  R ˜ x i ˜ x i (0) =  ρ ii (0)  ·  I W  + L i ,  i  = 1 , 2 ,...,J  . Basedon the mechanism from which the cyclostationary signal isgenerated, there exists certain relationship between  ρ β 1 11 (0)  and ρ 11 (0) , i.e.,  ρ 11 (0) =  ξ  · ρ β 1 11 (0) , where  ξ   is known. Thus, itfollows from (32) and (35) that ¯ R ˜ y ˜ y (0) − ξ  · R β 1 ˜ y ˜ y (0)=  H  0 0  ···  00 R ˜ x 2 ˜ x 2 (0)  ···  0 ............ 0 0  ···  R ˜ x J  ˜ x J  (0)  H H  .  (36)We can see from (36) that the rank ok of the matrix ¯ R ˜ y ˜ y (0) − ξ   · R β 1 ˜ y ˜ y (0)  is  ( I   − 1) W   + ¯ L − L 1 . Denoting thesingular vectors corresponding to the  ( I   −  1) W   + ¯ L  −  L 1 largest singular values of the matrix  ¯ R ˜ y ˜ y (0) − ξ  · R β 1 ˜ y ˜ y (0)  as v 2 ,j ,  j  = 1 , 2 ,..., ( I  − 1) W   + ¯ L − L 1 , we define V 2  =  v 2 , 1 , v 2 , 2 ,..., v 2 , ( I  − 1) W  +¯ L − L 1  .  (37)Based on (33) and (37), we define V  = [ V 1 , V 2 ] . Then, the singular vector associated with the largest singularvalue of   V  is proportional to one of the first  W   columns of  H but orthogonal to the other columns of  H . Thus, this singularvector can be chosen as the desired extraction vector  u 1 .IV. S IMULATION  R ESULTS In this section, we present simulation examples to illustratethe performance of the proposed method. In the simulations,we consider a mixing system with three inputs, which aregenerated as follows: x 1 ( n ) =  s 1 ( n ) ∗ cos( α 1 · n ) x 2 ( n ) =  s 2 ( n ) x 3 ( n ) =  s 3 ( n )  (38)  0 1000 3000 5000 7000 9000 11000 13000 15,000−33−32.5−32−31.5−31−30.5−30−29.5−29Samples    M   I   R   L   (   d   B   ) Figure 1. MIRL versus sample size, where  I   = 3 ,  J   = 4 ,  L  = 3  andSNR = 20 dB. where  s 1 ( n ) ,  s 2 ( n )  and  s 3 ( n )  are randomly generated tem-porarily white sequences. The signal of interest,  x 1 ( n ) , iscyclostationary with cyclic frequency  β  1  = 2 α 1  and  ρ 11 (0) =2 ρ β 1 11 (0) . The MIMO FIR mixing matrix  H ( z )  is randomlygenerated in each simulation run. Randomly generated whiteGaussian noise is added to the signal mixtures and the signal-to-noise ratio (SNR) is defined as  SNR =  − 10log 10  σ 2 w  .Ideally, for the compound vector  c 1  defined in (31), onlyone of its first  W   elements is nonzero and the rest elementsshould be zero. However, this is not practically achievable dueto finite sample size and computational inaccuracy. Therefore,the performance of our method is measured by means of theMean Interference Rejection Level (MIRL) of the extractionvector. Let  c H  1  = [ c H  1 , 1 ,c H  2 , 1 ,...,c H k, 1 ,...,c H IW  +¯ L, 1 ]  and as-sume  c k, 1  is the maximum element out of the first  W   elementsof   c 1 . The MIRL index is defined as follows: MIRL(dB) = 20log 10  1  IW   + ¯ L  −  1  · | c k, 1 | IW  +¯ L  i =1 ,i  = k | c i, 1 |  . We compute MIRL by averaging 200 independent runs.Clearly, the smaller MIRL, the better signal extraction per-formance.Given a mixing system of 3 inputs, 4 outputs and order3, Fig. 1 shows MIRL versus sample size where SNR = 20 dBand Fig. 2 shows MIRL versus SNR where the sample size is 5000 . It can be seen that satisfactory performance is achievedfor all tested sample sizes and SNRs, including small samplesizes and low SNR levels. Also, with the increase of samplesize or SNR, MIRL decreases accordingly.Fig. 3 shows MIRL against different number of sensors,where  I   = 3 ,  L  = 3 ,  N   = 5000  and SNR = 20 dB. As we cansee, when 3 sensors are used, the performance is not satisfac-tory. This is expected as the assumption A2) is not satisfiedin this case. In contract, when the number of sensors used is4 or above, the MIRL is lower than  − 30 dB, indicating good 0 5 10 15 20 25 30−32−30−28−26−24−22−20SNR (dB)    M   I   R   L   (   d   B   ) Figure 2. MIRL versus SNR, where  I   = 3 ,  J   = 4 ,  L  = 3  and  N   = 5000 . 3 4 5 6 7 8−34−32−30−28−26−24−22−20−18−16−14Number of Sensors    M   I   R   L   (   d   B   ) Figure 3. MIRL versus the number of sensors, where  I   = 3 ,  L  = 3 , N   = 5000  and SNR = 20 dB. extraction performance. Finally, the MIRL is evaluated underdifferent system orders, where the other simulation parametersare  I   = 3 ,  J   = 4 ,  N   = 5000  and SNR = 20 dB. It is shown inFig. 4 that the MIRL increases (i.e., performance decreases)with the rise of system order. For the considered system orders,the proposed method demonstrate good performance.V. C ONCLUSION This paper deals with the problem of extracting a cyclosta-tionary signal from the outputs of a MIMO FIR system. Underthe assumption that the sensors are placed close to the cy-clostationary signal of interest, a second-order statistics basedmethod is proposed to extract the cyclostationary signal byusing its unique cyclic frequency. Several simulation examplesare provided to illustrate the validity of our method.  1 2 3 4 5−35.5−35−34.5−34−33.5−33−32.5−32−31.5−31Order of mixing system    M   I   R   L   (   d   B   ) Figure 4. MIRL versus channel order, where  I   = 3 ,  J   = 4 ,  N   = 5000  andSNR = 20 dB. R EFERENCES[1] A. Belouchrani, K. Abed-Meraim, J.-F. Cardoso, and E. Moulines, “Ablind source separation technique using second-order statistics,”  IEEE Trans. on Signal Processing , vol. 45, no. 2, pp. 434–444, 1997.[2] N. Bouguerriou, M. Haritopoulos, C. Capdessus, and L. Allam, “Novelcyclostationarity-based blind source separation algorithm using secondorder statistical properties: Theory and application to the bearing defectdiagnosis,”  Mechanical Systems and Signal Processing , vol. 19, no. 6,pp. 1260–1281, Nov. 2005.[3] Y. Xiang, “Blind source separation based on constant modulus criterionand signal mutual information,”  Computers  &  Electrical Engineering ,vol. 34, no. 5, pp. 416–422, Sept. 2008.[4] Y. Xiang, D. Peng, Y. Xiang, and S. Guo, “Novel Z-domain precodingmethod for blind separation of spatially correlated signals,”  IEEE Trans.on Neural Networks and Learning Systems , vol. 24, no. 1, pp. 94–105,Jan. 2013.[5] K. Abed-Meraim, Y. Xiang, J. H. Manton, and Y. Hua, “Blind sourceseparation using second-order cyclostationary statistics,”  IEEE Trans.on Signal Processing , vol. 49, no. 4, pp. 694–701, Apr. 2001.[6] Y. Xiang, V. K. Nguyen, and N. 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