Performance analysis in wireless communications and large - PowerPoint PPT Presentation

Performance analysis in wireless communications and large deviations of extreme eigenvalues of deformed random matrices Myl` ene Ma¨ ıda LM Orsay, Universit´ e Paris-Sud Joint work with P. Bianchi, M. Debbah, J. Najim and F. Benaych-Georges, A. Guionnet 1

Outline of the talk 2

Outline of the talk ◮ Performance analysis of a test in wireless communications 2

Outline of the talk ◮ Performance analysis of a test in wireless communications ◮ Presentation of the source detection problem ◮ Performances of the GLRT ◮ Study of the largest eigenvalue in a one spike model 2

Outline of the talk ◮ Performance analysis of a test in wireless communications ◮ Presentation of the source detection problem ◮ Performances of the GLRT ◮ Study of the largest eigenvalue in a one spike model ◮ Large deviations of extreme eigenvalues of some deformed models 2

Outline of the talk ◮ Performance analysis of a test in wireless communications ◮ Presentation of the source detection problem ◮ Performances of the GLRT ◮ Study of the largest eigenvalue in a one spike model ◮ Large deviations of extreme eigenvalues of some deformed models ◮ Presentation of the models ◮ General results ◮ Application to some classical models 2

Outline of the talk ◮ Performance analysis of a test in wireless communications ◮ Presentation of the source detection problem ◮ Performances of the GLRT ◮ Study of the largest eigenvalue in a one spike model ◮ Large deviations of extreme eigenvalues of some deformed models ◮ Presentation of the models ◮ General results ◮ Application to some classical models ◮ Conclusion 2

Source detection in cooperative spectrum sensing Secondary sensors try to find a bandwidth to occupy. Those K sensors can share information, each of them receiving N samples of the signal. 3

Modelisation of the statistical test 4

Modelisation of the statistical test We want to test ◮ Hypothesis H0 : No signal. Secondary sensor number k receives a series of data y k ( n ) of length N of the form : y k ( n ) = w k ( n ) , n = 1 . . . N where w k ( n ) ∼ CN (0 , σ 2 ) is a white noise. 4

Modelisation of the statistical test We want to test ◮ Hypothesis H0 : No signal. Secondary sensor number k receives a series of data y k ( n ) of length N of the form : y k ( n ) = w k ( n ) , n = 1 . . . N where w k ( n ) ∼ CN (0 , σ 2 ) is a white noise. ◮ Hypothesis H1 : Presence of a signal. The data received by sensor number k is now of the form : y k ( n ) = h k s ( n ) + w k ( n ) , n = 1 . . . N where s ( n ) is a Gaussian primary signal and h k the fading coefficient associated to the secondary sensor k . 4

Modelisation of the statistical test We want to test ◮ Hypothesis H0 : No signal. Secondary sensor number k receives a series of data y k ( n ) of length N of the form : y k ( n ) = w k ( n ) , n = 1 . . . N where w k ( n ) ∼ CN (0 , σ 2 ) is a white noise. ◮ Hypothesis H1 : Presence of a signal. The data received by sensor number k is now of the form : y k ( n ) = h k s ( n ) + w k ( n ) , n = 1 . . . N where s ( n ) is a Gaussian primary signal and h k the fading coefficient associated to the secondary sensor k . As σ and h are unknown, the Neyman-Pearson test cannot be 4 implemented.

Generalized Likelihood Ratio Test (GLRT) 5

Generalized Likelihood Ratio Test (GLRT) We gather the observation in the matrix Y = [ y k ( n )] k =1: K , n =1: N 5

Generalized Likelihood Ratio Test (GLRT) We gather the observation in the matrix Y = [ y k ( n )] k =1: K , n =1: N ◮ Under H0 , the entries of Y are i.i.d. CN (0 , σ 2 ). The likelihood writes : � � − N p 0 ( Y ; σ 2 ) = ( πσ 2 ) − NK exp σ 2 tr R . N YY ∗ is the empirical covariance matrix. where R = 1 5

Generalized Likelihood Ratio Test (GLRT) We gather the observation in the matrix Y = [ y k ( n )] k =1: K , n =1: N ◮ Under H0 , the entries of Y are i.i.d. CN (0 , σ 2 ). The likelihood writes : � � − N p 0 ( Y ; σ 2 ) = ( πσ 2 ) − NK exp σ 2 tr R . N YY ∗ is the empirical covariance matrix. where R = 1 ◮ Under H1 , the column vectors of Y are i.i.d. CN (0 , hh ∗ + σ 2 I K ) where h = [ h 1 , . . . , h K ] T is the fading vector corresponding to the K secondary sensors. The likelihood writes : � � p 1 ( Y ; h , σ 2 ) = ( π K det( hh ∗ + σ 2 I K )) − N exp − N tr ( R ( hh ∗ + σ 2 I K ) − 1 ) . 5

Generalized Likelihood Ratio Test (GLRT) 6

Generalized Likelihood Ratio Test (GLRT) Recall that σ 2 , h are unknown. The GLRT will reject H0 for high values of the statistics : sup h ,σ 2 p 1 ( Y ; h , σ 2 ) sup σ 2 p 0 ( Y ; σ 2 ) 6

Generalized Likelihood Ratio Test (GLRT) Recall that σ 2 , h are unknown. The GLRT will reject H0 for high values of the statistics : sup h ,σ 2 p 1 ( Y ; h , σ 2 ) sup σ 2 p 0 ( Y ; σ 2 ) 6

Generalized Likelihood Ratio Test (GLRT) Recall that σ 2 , h are unknown. The GLRT will reject H0 for high values of the statistics : sup h ,σ 2 p 1 ( Y ; h , σ 2 ) sup σ 2 p 0 ( Y ; σ 2 ) After some standard computations, we get the following test : Reject H0 whenever the statistics : T N := λ max 1 K tr R is above the threshold γ N YY ∗ . where λ max is the largest eigenvalue of R := 1 6

Performance analysis of the GLRT 7

Performance analysis of the GLRT For a given threshold γ , we define : ◮ the type I Error (probability of false alarm) P 0 [ T N > γ ] is the probability of deciding H1 when H0 holds, ◮ the type II Error P 1 [ T N < γ ] is the probability of deciding H0 when H1 holds (N.B. Type II Error depends on h and σ 2 ) 7

Performance analysis of the GLRT For a given threshold γ , we define : ◮ the type I Error (probability of false alarm) P 0 [ T N > γ ] is the probability of deciding H1 when H0 holds, ◮ the type II Error P 1 [ T N < γ ] is the probability of deciding H0 when H1 holds (N.B. Type II Error depends on h and σ 2 ) The Receiver Operating Characterictic (ROC curve) is the set of points (Type I Error, Type II Error) for all possible thresholds. ROC := { ( P 0 [ T N > γ ] , P 1 [ T N < γ ]) : γ ∈ R + } . 7

Performance analysis of the GLRT For a given threshold γ , we define : ◮ the type I Error (probability of false alarm) P 0 [ T N > γ ] is the probability of deciding H1 when H0 holds, ◮ the type II Error P 1 [ T N < γ ] is the probability of deciding H0 when H1 holds (N.B. Type II Error depends on h and σ 2 ) The Receiver Operating Characterictic (ROC curve) is the set of points (Type I Error, Type II Error) for all possible thresholds. ROC := { ( P 0 [ T N > γ ] , P 1 [ T N < γ ]) : γ ∈ R + } . ⇒ We study the ROC curve in the asymptotic regime : K → ∞ , N → ∞ , K N → c ∈ (0 , 1) 7

Asymptotic behavior of T N under H0 8

Asymptotic behavior of T N under H0 N YY ∗ with Y having i.i.d. entries CN (0 , σ 2 ). Recall that R := 1 8

Asymptotic behavior of T N under H0 N YY ∗ with Y having i.i.d. entries CN (0 , σ 2 ). Recall that R := 1 ◮ By the law of large numbers, 1 ( H 0) N →∞ σ 2 K tr R − − − − → 8

Asymptotic behavior of T N under H0 N YY ∗ with Y having i.i.d. entries CN (0 , σ 2 ). Recall that R := 1 ◮ By the law of large numbers, 1 ( H 0) N →∞ σ 2 K tr R − − − − → N →∞ σ 2 (1 + √ c ) 2 the right edge of the Marcenko-Pastur ( H 0) ◮ λ max − − − − → distribution and has Tracy-Widom fluctuations. 8

Asymptotic behavior of T N under H0 N YY ∗ with Y having i.i.d. entries CN (0 , σ 2 ). Recall that R := 1 ◮ By the law of large numbers, 1 ( H 0) N →∞ σ 2 K tr R − − − − → N →∞ σ 2 (1 + √ c ) 2 the right edge of the Marcenko-Pastur ( H 0) ◮ λ max − − − − → distribution and has Tracy-Widom fluctuations. K tr R and c N = K λ max ◮ We get that, if T N = N , 1 T N − (1 + √ c N ) 2 ˜ T N = N 2 / 3 (1 + √ c N )(1 + 1 √ c N ) 1 / 3 converges in distribution to a Tracy-Widom distribution. 8

Asymptotic behavior of T N under H0 N YY ∗ with Y having i.i.d. entries CN (0 , σ 2 ). Recall that R := 1 ◮ By the law of large numbers, 1 ( H 0) N →∞ σ 2 K tr R − − − − → N →∞ σ 2 (1 + √ c ) 2 the right edge of the Marcenko-Pastur ( H 0) ◮ λ max − − − − → distribution and has Tracy-Widom fluctuations. K tr R and c N = K λ max ◮ We get that, if T N = N , 1 T N − (1 + √ c N ) 2 ˜ T N = N 2 / 3 (1 + √ c N )(1 + 1 √ c N ) 1 / 3 converges in distribution to a Tracy-Widom distribution. ⇒ This determines the asymptotic threshold γ for a fixed PFA. 8

Asymptotic behavior of T N under H1 9

Asymptotic behavior of T N under H1 N YY ∗ with Recall that R := 1 � � 1 / 2 X K × N hh ∗ + σ 2 I K iid ∼ CN (0 , 1) Y = with X i , j 9

Asymptotic behavior of T N under H1 N YY ∗ with Recall that R := 1 � � 1 / 2 X K × N hh ∗ + σ 2 I K iid ∼ CN (0 , 1) Y = with X i , j > √ c Hypothesis : ρ := � h � 2 σ 2 9

Asymptotic behavior of T N under H1 N YY ∗ with Recall that R := 1 � � 1 / 2 X K × N hh ∗ + σ 2 I K iid ∼ CN (0 , 1) Y = with X i , j > √ c Hypothesis : ρ := � h � 2 σ 2 ◮ λ max converges out of the bulk de MP [Baik-Silv-06] � � 1 + c ( H 1) N →∞ σ 2 (1 + ρ ) λ max − − − − → . ρ 9