A CLT on the SINR of the diagonally loaded Capon/MVDR beamformer Francisco Rubio 1 joint work with Xavier Mestre 1 and Walid Hachem 2 1 Centre Tecnològic de Telecomunicacions de Catalunya 2 Télécom ParisTech and Centre National de la Recherche Scienti…que Workshop on Large Random Matrices and their Applications Télécom ParisTech, 11-13 October, 2010 Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 1 / 17
Outline Capon/MVDR beamforming (or spatial …ltering) Characterization of output SINR performance Asymptotic deterministic equivalents A Central Limit Theorem Conclusions Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 2 / 17
Capon/MVDR beamforming Signal model Consider the following set of independent observations drawn from the general Gauss-Markov linear model L ( y ( n ) ; x ( n ) s ; R ) : y ( n ) = x ( n ) s + n ( n ) 2 C M ; n = 1 ; : : : ; N where x ( n ) � signal waveform , s 2 C M � spatial signature , n ( n ) 2 C K � i+n Typical scenario in sensor array signal processing applications: We are interested in linearly …ltering the observed samples to estimate x ( n ) Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 3 / 17
Capon/MVDR beamforming Problem statement Optimal coe¢cients of M inumum V ariance D istornionless R esponse …lter: w 2 C M w H Rw subject to w H s = 1 w MVDR = arg min R � 1 s = s H R � 1 s where R is the covariance matrix of interference-plus-noise random vectors In practice, R is unknown and implementations rely on the S ample C ovariance M atrix or any other improved estimator based on regularization or shrinkage : � � R = 1 I N � 1 Y H + � R o , ^ N 1 N 1 0 N Y Y = [ y (1) ; : : : ; y ( N )] N where R o is a positive matrix and � > 0 is the diagonal loading or shrinkage intensity parameter L If � = 0 then ^ R = ^ R SCM and, under Gaussianity, ^ N R 1 = 2 XTX H R 1 = 2 1 R SCM = where the entries of X are CN (0 ; 1) , and T models either sample weighting or temporal correlation across samples Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 4 / 17
Characterization of the output SINR performance Definition The S ignal-to- I nterference-plus- N oise R atio at the output of the MVDR …lter is: � � � w H s � 2 SINR ( w ) = � 2 x w H Rw with � 2 x � signal power The optimal SINR is SINR ( w MVDR ) = s H R � 1 s � k u k 2 For the MVDR …lter implementation based on diagonal loading: � � 2 s H � � � 1 ^ R + � I M s SINR ( ^ w MVDR ) = � � � 1 � � � 1 ^ ^ s H R + � I M R R + � I M s We are interested in the properties of SINR ( ^ w MVDR ) Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 5 / 17
Characterization of the output SINR performance Known properties In the case ^ R = ^ R SCM ( T = I N and � = 0 ), the distribution of � � 2 s H ^ R � 1 s SINR ( ^ w MVDR ) SINR ( w MVDR ) = � 1 ss H R � 1 s s H ^ R � 1 R^ R is known in the array processing literature to have a density [ Reed-Mallet-Brennan,T.AES’74 ] N ! ( M � 2)! ( N + 1 � M )! (1 � � ) M � 2 � N +1 � M f � ( � ) = In particular, SINR ( ^ w MVDR ) = SINR ( w MVDR ) � Beta ( N + 2 � M; M � 1) with mean = N + 2 � M N + 1 and variance = ( M � 1) ( N + 2 � M ) ( N + 1) 2 ( N + 2) What about the general case with arbitrary positive T and � ? [ Rao-Edelman, ASAP’05 ] Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 6 / 17
Asymptotic analysis of the SINR Asymptotic Deterministic Equivalents of the SINR First-order analysis: � � 2 s H � � � 1 ^ R + � I M s SINR ( ^ w MVDR ) = � � � 1 � � � 1 ^ ^ s H R + � I M R R + � I M s � � 2 s H ( x M R + � R o ) � 1 s � = SINR ( ^ w MVDR ) 1 � s H ( x M R + � R o ) � 1 R ( x M R + � R o ) � 1 s 1 � � ~ such that h i � T ( I N + e M T ) � 1 � x M = 1 � 1 ~ N Tr N Tr E � R ( x M R + � R o ) � 1 � e M = 1 � 1 N Tr N Tr [ E ] h E 2 i � E 2 � 1 1 ~ and � = N Tr and ~ � = N Tr Asymptotics of SINR ( ^ w MVDR ) involve both the eigenvalues and also the eigenvectors of the random matrix model Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 7 / 17
Asymptotic analysis of the SINR A Random Matrix Theory result If the entries of X have 8 th -order moment and k R k and k T k are bounded, as N = N ( M ) ! 1 and 0 � lim inf c M � lim sup c M < + 1 ( c M = M=N ), a.s., [ Rubio-Mestre, submitted SPL’10 ] � H � � � 1 N R 1 = 2 XTX H R 1 = 2 � z I M � � � H ( � R o + x ( z ) R � z I M ) � 1 � A + 1 for each z 2 C � R + and an arbitrary nonrandom, unit-norm � , where � T ( I N + e ( z ) T ) � 1 � x ( z ) = 1 N Tr and e ( z ) is the unique solution in C � R + to � R ( � R o + x ( z ) R � z I M ) � 1 � e ( z ) = 1 N Tr � 1 � � 1 and note that N XTX H + � R � 1 � z I M De…ne Q M ( z ) = Q 2 @ M ( z ) = @z f Q M ( z ) gj z =0 along with � � 2 u H Q M (0) u SINR ( ^ w MVDR ) = u H Q 2 M (0) u Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 8 / 17
Asymptotic analysis of the SINR Consistent estimator of the SINR We also have the following estimate not depending on the unknown R : s H � � � 1 ^ � � � 1 ^ ^ R + � R o R R + � R o s SINR ( ^ w MVDR ) � � M ( � ) � � � 2 � � � 1 ^ s H R + � R o s where 1 � � � 1 � � M ( � ) = � 1 � 1 ^ ^ N Tr R R + � R o The previous estimate can be used to …nd the optimal diagonal loading factor or shrinkage intensity parameter for arbitrary shrinkage target R o What about the ‡uctuations of SINR ( ^ w MVDR ) ? Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 9 / 17
Second-order asymptotic analysis A Central Limit Theorem We analyze the variance � 2 M of SINR ( ^ w MVDR ) and prove the Central Limit Theorem � � L � � 1 SINR ( ^ w MVDR ) � SINR ( ^ w MVDR ) M;N !1 N (0 ; 1) ! M by applying the Delta method to the random vector 2 s H � � � 1 3 � a M � ^ R + � I M s 4 5 s H � � � 1 � � � 1 = b M ^ ^ R + � I M R R + � I M s whose distribution is obtained by using the Cramér-Wold device after managing the following computations... Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 10 / 17
Second-order asymptotic analysis Elements of the proof (1/2) � 1 N XTX H + � R � 1 � � 1 and k u k 2 = s H R � 1 s , and de…ne Recall Q M (0) = Q M = a M � a M = u H Q M u � � b M � b M = u H Q 2 M u We follow the approach by Hachem et al. in [ H-K-L-N-P, T.IT’2008 ] and show that � � � ! 2 � 2 � M ( ! ) � exp M = 2 M;N !1 0 ! where � M ( ! ) is the characteristic function of the random variable p p � � b M � � A N ( a M � � a M ) � B N b M To identify the variance, we proceed as p p �� � � @ b M � � @! � M ( ! ) = i A N E [( a M � � a M ) � M ( ! )] + i B N E b M � M ( ! ) Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 11 / 17
Second-order asymptotic analysis Elements of the proof (2/2) As in [ H-K-L-N-P, T.IT’2008 ], we make intensive use of the integration by parts formula ( Z = DX ~ D , with D ; ~ D being diagonal) � @ � ( Z ) � E [ Z ij � ( Z )] = d i ~ d j E @Z � ij and the Nash-Poincaré inequality "� 2 # � � � X M X N � � 2 � � @ � ( Z ) @ � ( Z ) d i ~ � � � � var (� ( Z )) � d j E + � � � � @Z � @Z � ij ij i =1 j =1 to compute the expectation and variance controls for the following quantities: h i �Q k Tr M � � XZ 1 X H �Q k Tr M N where k = 1 ; 2 ; 3 ; 4 and � = ab H and � = 1 N Z 2 ( a ; b unit-norm and Z 1 ; Z 2 diagonal with bounded spectral norm) Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 12 / 17
Second-order asymptotic analysis Delta method Gathering terms together as � � � N � 1 � @ A 2 � a 2 + AB� ab + BA� ba + B 2 � b 2 @! � M ( ! ) = � ! � M ( ! ) + O � N � 1 � � N � 1 � and E [ b M ] = � along with E [ a M ] = � a M + O b M + O , we get � a M � � � � � a 2 � p a M � ba L ! N ( �; � ) , N � = b M � � b M � ab � b 2 where � = 0 and � ab = � ba w MVDR ) = f ( a M ; b M ) with f ( x; y ) = x 2 =y and Since SINR ( ^ � � ( x=y ) 2 � r f = 2 x=y , then it follows by the Delta method that p � � �� a M ; � N f ( a M ; b M ) � f � b M � �� � � � � H � r f � L a M ; � a M ; � a M ; � � H r f ! N � b M ; r f � b M � b M Francisco Rubio (CTTC) CLT on SINR of DL-MVDR beamformer Paris - October 13, 2010 13 / 17
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