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Random Matrix Theory in Array Signal Processing: Application Examples

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Conventional tools in array signal processing have traditionally relied on the availability of a large number of samples acquired at each sensor or array element (antenna, hydrophone, microphone, etc.). Large sample size assumptions typically guarantee the consistency of estimators, detectors, classifiers and multiple other widely used signal processing procedures. However, practical scenario and array mobility conditions, together with the need for low latency and reduced scanning times, impose strong limits on the total number of observations that can be effectively processed. When the number of collected samples per sensor is small, conventional large sample asymptotic approaches are not relevant anymore. Recently, large random matrix theory tools have been proposed in order to address the small sample support problem in array signal processing. In fact, it has been shown that the most important and longstanding problems in this field can be reformulated and studied according to this asymptotic paradigm. By exploiting the latest advances in large random matrix theory and high dimensional statistics, a novel and unconventional methodology can be established, which provides an unprecedented treatment of the finite sample-per-sensor regime. In this talk, we will see that random matrix theory establishes a unifying framework for the study of array signal processing techniques under the constraint of a small number of observations per sensor, which has radically changed the way in which array processing methodologies have been traditionally established. We will show how this unconventional way of revisiting classical array processing has lead to major advances in the design and analysis of signal processing techniques for multidimensional observations.

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Random Matrix Theory in Array Signal Processing: Application Examples

  1. 1. Random Matrix Theory in Signal Processing Xavier Mestre xavier.mestre@cttc.cat Centre Tecnològic de Telecomunicacions de Catalunya (CTTC) Klagenfurt University (Austria) February 25, 2019
  2. 2. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Outline • Introduction to RMT: Convergence of spectral statistics of the sample covariance matrix. • First Application: Subspace-based estimation of directions-of-arrival (DoA). • Second Application: Detection tests of correlation and sphericity. • Third Application: Large multivariate time series analysis • Fourth Application: Outlier production characterization in Conditional/Unconditional Maximum Likelihood parametric estimation. Xavier Mestre: Random Matrix Theory in Signal Processing. 2/41
  3. 3. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ The sample covariance matrix We assume that we collect  independent samples (snapshots) from an array of  antennas: Consider the  ×  observation matrix Y = [y(1)     y()] and the sample covariance matrix ˆR = 1  YY  = 1  X =1 y()y () Xavier Mestre: Random Matrix Theory in Signal Processing. 3/41
  4. 4. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Problem statement and objective of the talk Typically, one expects ˆR to be a close approximation of R. Under conventional statistical assumptions, we have ˆR → R almost surely when  → ∞ for a fixed . Furthermore, if (·) is a reasonable function, we also have  ³ ˆR ´ →  (R). Unfortunately, when   have the same order of magnitude, this does not hold anymore: the Finite Sample Size effect appears. In these situations, the regime where   → ∞ but  → , 0    ∞ becomes much more relevant. RMT will help us in solving the following two problems in this regime: • To what  ³ ˆR ´ does converge to? • How do we design  (·) so that  ³ ˆR ´ →  (R). Xavier Mestre: Random Matrix Theory in Signal Processing. 4/41
  5. 5. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Conditional versus unconditional model Typically, the observations are superposition of some signals plus noise. For example, in array processing the observation consists of the contribution from  signals: Y = A (Θ) S + N where: • Matrix S ∈ C× contains the contribution of the  signals (at each of its rows). • Matrix A (Θ) ∈ C× contains, at each of its columns, the spatial signature of each source, namely A (Θ) = £ a (1) a (2) · · · a () ¤ • Matrix N ∈ C× contains the background noise samples. It is typically modeled as a matrix with i.i.d. entries following a zero mean Gaussian distribution {N} ∼ CN ¡ 0 2 ¢  Xavier Mestre: Random Matrix Theory in Signal Processing. 5/41
  6. 6. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Conditional versus unconditional model (II) Depending on how the signals are modeled, we differentiate between the conditional and the unconditional models. Let us denote S = [s(1)     s()] • Conditional Model: The entries of S are modelled as deterministic unknowns. In this case, the observation can be described as y() ∼ CN ¡ A (Θ) s() 2 I ¢  • Unconditional Model: The entries of S are modelled as random variables. Typically, we assume that the column vectors s() are independent, circularly symmetric Gaussian Random variables, i.e. s ∼ CN (0 P), P  0. In this case, we have y() ∼ CN (0 R)  R = A (Θ) PA (Θ) + 2 I Xavier Mestre: Random Matrix Theory in Signal Processing. 6/41
  7. 7. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Conditional versus unconditional model (III) Depending on the signal model, the structure of the sample covariance matrix model will be inherently different. Let X be an  × matrix of i.i.d. Gaussian standardized entries {X} ∼ CN (0 1). The two most important models can be described as: • Conditional Model (also known as Information plus Noise model), the SCM can be expressed as ˆR = 1  (V + X) (V + X) where V some deterministic matrix that contains the signal (information) contribution. • Unconditional Model (also known as Single Side Correlation model), the SCM can be expressed as ˆR = R 12  µ 1  XX  ¶ R 12  where R 12  is the positive Hermitian square root of R. In many of the results obtained by RMT, the Gaussian assumption can be dropped. Xavier Mestre: Random Matrix Theory in Signal Processing. 7/41
  8. 8. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Uncorrelated signals: the Marchenko-Pastur Law Consider the simplest case where ˆR = 1  XX , where the entries of X are zero mean i.i.d. with unit variance. Consider the eigenvalue distribution for different  , but fixed ratio . 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 Eigenvalues Numberofeigenvalues Histogram of the eigenvalues of the sample covariance matrix, M=80, N=800 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 Eigenvalues Histogram of the eigenvalues of the sample covariance matrix, M=800, N=8000 Numberofeigenvalues Xavier Mestre: Random Matrix Theory in Signal Processing. 8/41
  9. 9. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Uncorrelated signals: the Marchenko-Pastur Law (II) It turns out, that when   → ∞,  → , 0    ∞, the empirical density of eigenvalues converges to a deterministic measure. 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 Eigenvalues Histogram of the eigenvalues of the sample covariance matrix, M=800, N=8000 Numberofeigenvalues For   1 () = 1 2 q¡  − − ¢ ¡ + −  ¢ I[− + ]() − = (1 − √ ) 2  + = (1 + √ ) 2 . Xavier Mestre: Random Matrix Theory in Signal Processing. 9/41
  10. 10. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ The sample covariance matrix We assume that we collect  independent samples (snapshots) from an array of  antennas: ˆR = 1  P =1 y()y (). Example: R has 4 eigenvalues {1 2 3 7} with equal multiplicity. 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 Histogram of the eigenvalues of the sample covariance matrix, M=80, N=800 lambda Numberofeigenvalues 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 lambda Numberofeigenvalues Histogram of the eigenvalues of the sample covariance matrix, M=400, N=4000 Xavier Mestre: Random Matrix Theory in Signal Processing. 10/41
  11. 11. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ The sample covariance matrix: asymptotic properties When both   → ∞,  → , 0    ∞, the e.d.f. of the eigenvalues of ˆR tends to a deterministic density function. Example: R has 4 eigenvalues {1 2 3 7} with equal multiplicity. 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Eigenvalues Aysmptotic density of eigenvalues of the sample correlation matrix c=0.01 c=0.1 c=1 Xavier Mestre: Random Matrix Theory in Signal Processing. 11/41
  12. 12. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ 1st Application: Subspace-based estimation of directions of arrival (DoA) Xavier Mestre: Random Matrix Theory in Signal Processing. 12/41
  13. 13. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Introduction and signal model We consider DoA detection based on subspace approaches (MUSIC), that exploit the orthogonality between signal and noise subspaces. Consider a set of  sources impinging on an array of  sensors/antennas. We work with a fixed number of snapshots , {y(1)     y()} assumed i.i.d., with zero mean and covariance R. The true spatial covariance matrix can be described as R = A (Θ) ΦA (Θ) + 2 I where A (Θ) is an  × matrix that contains the steering vectors corresponding to the  different sources, A (Θ) = £ a (1) a (2) · · · a () ¤ and 2 is the background noise power. Xavier Mestre: Random Matrix Theory in Signal Processing. 13/41
  14. 14. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Introduction and signal model (II) The eigendecomposition of R allows us to differentiate between signal and noise subspaces: R = £ E E ¤ ∙ Λ 0 0 2 I− ¸ £ E E ¤  It turns out that E a () = 0,  = 1    . Since R is unknown, one must work with the sample covariance matrix ˆR = 1  X =1 y()y () The MUSIC algorithm uses the sample noise eigenvectors, and searches for the deepest local minima of the cost function MUSIC () = a () ˆE ˆE a ()  It is interesting to investigate the behavior of MUSIC () when   have the same order of magnitude. Xavier Mestre: Random Matrix Theory in Signal Processing. 14/41
  15. 15. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Asymptotic behavior of MUSIC The MUSIC algorithm suffers from the breakdown effect. The performance breaks down when the number of samples or the SNR falls below a certain threshold. Cause: ˆE is not a very good estimator of E when   have the same order of magnitude. The performance breakdown effect can be easily analyzed using random matrix theory, especially under a noise eigenvalue separation assumption: |MUSIC () − ¯MUSIC ()| → 0 ¯MUSIC () = s () Ã X =1 ()ee  ! s () () = ⎧ ⎨ ⎩ 1 − 1 − P =−+1 ³ 2 −2 − 1 −1 ´  ≤  −  2 −2 − 1 −1    −  where {  = 1     } are the solutions to 1  P =1  − = 1  Xavier Mestre: Random Matrix Theory in Signal Processing. 15/41
  16. 16. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Asymptotic behavior of MUSIC: an example We consider a scenario with two sources impinging on a ULA ( = 05,  = 20) from DoAs: 35◦ , 37◦ . −100 −80 −60 −40 −20 0 20 40 60 80 100 −35 −30 −25 −20 −15 −10 −5 0 MUSIC asymptotic pseudospectrum, M=20, DoAs=[35,37]deg Azimuth (deg) 32 34 36 38 40 −32 −30 −28 −26 −24 −22 −20 −18 N=25 N=15 SNR=12dB SNR=17dB 2 4 6 8 10 12 14 16 18 20 25 30 35 40 45 50 SNR (dB) Azimuth(deg) Position of the two deepest local minima of the asymptotic MUSIC cost function 10 12 14 35 36 37 N=15 N=25 Xavier Mestre: Random Matrix Theory in Signal Processing. 16/41
  17. 17. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/  -consistent subspace detection: G-MUSIC We can derive an  -consistent estimator of the cost function  () = s () EE s (): G-MUSIC () = s () Ã X =1 ()ˆeˆe  ! s () () = ⎧ ⎨ ⎩ 1 + P =−+1 ³ ˆ ˆ−ˆ − ˆ ˆ−ˆ ´  ≤  −  − P− =1 ³ ˆ ˆ−ˆ − ˆ ˆ−ˆ ´    −  where now ˆ1     ˆ are the solutions to the equation 1  X =1 ˆ ˆ − ˆ = 1   Xavier Mestre: Random Matrix Theory in Signal Processing. 17/41
  18. 18. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Performance evaluation MUSIC vs. G-MUSIC Comparative evaluation of MUSIC and G-MUSIC via simulations. Scenarios with four (−20◦  −10◦  35◦ , 37◦ ) and two (35◦ , 37◦ ) sources respectively, ULA ( = 20,  = 05). −80 −60 −40 −20 0 20 40 60 80 10 −4 10 −3 10 −2 10 −1 10 0 Example of MUSIC and GMUSIC cost function, SNR=18dB, M=20, N=15, DoAs=35, 37, −10, −20 deg. Angle of arrival (azimuth), degrees MUSIC GMUSIC 34 36 38 10 −4 10 −3 10 −2 5 10 15 20 25 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 SNR (dB) MSE Mean Squared Error MUSIC GMUSIC CRB M=20, N=15 M=20, N=75 Xavier Mestre: Random Matrix Theory in Signal Processing. 18/41
  19. 19. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ 2nd Application: characterization of sphericity and correlation tests Xavier Mestre: Random Matrix Theory in Signal Processing. 19/41
  20. 20. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Problem formulation We consider two very important tests in signal processing, which try to establish the structure of the covariance matrix of the received signal: • Sphericity test: seeks to establish whether the received signal is spatio-temporal white noise: H0 : R = 2 I H1 : R 6= 2 I • Correlation test: seeks to establish whether the signals received from multiple sensors is corre- lated: H0 : R = R ¯ I H1 : R 6= R ¯ I In both cases, the true covariance matrix is unknown, so one must work on the sampled version ˆR. Xavier Mestre: Random Matrix Theory in Signal Processing. 20/41
  21. 21. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Generalized Maximum Likelihood Ratio Test (GLRT) In order to address this binary hypothesis problem, one may resort to the Generalized Likelihood Ratio Test (GLRT): supR Y =1 Φ (y; R) sup2 Y =1 Φ (y; 2I) H1 ≷ H0  supR Y =1 Φ (y; R) supD Y =1 Φ (y; D) H1 ≷ H0  where Φ (y; R) is the pdf of a complex Gaussian with zero mean and covariance R. For  ≥ , the GLRT for sphericity and correlation respectively reject H0 for large values of ˆsphr  = log ∙ 1  tr ³ ˆR ´¸ − 1  log det ³ ˆR ´ ˆcorr  = log ∙ 1  tr ³ ˆC ´¸ − 1  log det ³ ˆC ´ where ˆC = ³ ˆR ¯ I ´−12 ˆR ³ ˆR ¯ I ´−12 is the sample correlation/coherence matrix. Xavier Mestre: Random Matrix Theory in Signal Processing. 21/41
  22. 22. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Frobenius norm test Other more ad-hoc tests can be constructed using a more intuitive reasoning: • Non-sphericity will manifest in ˆR being far from proportional to the identity. • Correlation will lead to high absolute values of the off-diagonal elements of ˆC . Therefore, it seems reasonable to design the test to reject H0 for large values of ˆsphr  = 1  ° ° ° ° ˆR − 1  tr h ˆR i I ° ° ° ° 2  ˆsphr  = 1  ° ° °ˆC − I ° ° ° 2   In both cases, we have ˆ  = 1  X =1 ˆ 2  − Ã 1  X =1 ˆ !2 which are LSS with () = 2 and () = . Xavier Mestre: Random Matrix Theory in Signal Processing. 22/41
  23. 23. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ General study of tests It is generally difficult to derive the distribution of these tests, so in practice the literature has focused on the case  → ∞ for fixed  We would like to know the asymptotic behavior of these tests, for   having the same order of magnitude, allowing for the possibility of    (undersampled regime). Fortunately, there is a direct relationship between LSS and Stieltjes transform: ˆ = 1  X =1  ³ ˆ ´ = 1 2 j I C− () ˆ() where ˆ() = 1  X =1 1 ˆ −  and where the contour C− enclosed all the positive eigenvalues and not zero. Xavier Mestre: Random Matrix Theory in Signal Processing. 23/41
  24. 24. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ First order convergence By replacing ˆ() with the asymptotic equivalent, we obtain the almost sure asymptotic behavior of the test ˆ, in the sense that |ˆ − ¯| → 0 where ¯ = 1 2 j I C− () ¯() Most of the times, we can carry out the integral and find a closed form for ¯. For example, for the ¯  , we can establish ¯  = (   +   + −  log ¯ ¯1 −   ¯ ¯      +   − −  log |∗| + 1  P =1  log ¯ ¯ ¯  −∗ ¯ ¯ ¯    where ∗ ≤ 0 is a solution to a certain equation and   is the large- value of the GLRT   = log " 1  X =1  # − 1  X =1 log  Xavier Mestre: Random Matrix Theory in Signal Processing. 24/41
  25. 25. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Second order convergence Using RMT tools that establish how ˆ() fluctuates around ¯(), we may establish a CLT on these tests. Under certain statistical conditions, the LSS ˆ will asymptotically fluctuate as Gaussian random variable, in the sense that −1  ( (ˆ − ¯) − ) L −→ N (0 1)  where  = 1 2 j I C−  () ()  2  = −1 42 I C−  I C−  (1)(2)2  (1 2) 12 where () is the original test function () after some change of variable, and where the mean  () and variance 2  (1 2) are different for the Sphericity and Correlation tests. These integrals can be computed in closed form, and one can generally approximate ˆ ≈ N ¡ ¯ +  2 2 ¢  Xavier Mestre: Random Matrix Theory in Signal Processing. 25/41
  26. 26. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Numerical Results: correlation test Simulations for 105 independent simulation runs (GLRT). Under H0, D takes uniform values between 0 and 1. Under H1, R = D + Ψ where {Ψ} = 09|−| . 250 300 350 400 450 500 550 600 0 0.005 0.01 0.015 Density of the statistic under H 0 300 350 400 450 500 550 600 0 0.005 0.01 0.015 Density of the statistic under H1 Simulated Theory (large M,N) Theory (large N) M=20,N=25 M=20,N=25  = 20  = 25 250 300 350 400 450 500 550 0 0.005 0.01 0.015 Density of the statistic under H 0 , M=20, N=100 350 400 450 500 550 600 650 700 0 0.005 0.01 0.015 Density of the statistic under H1 , M=20, N=100 Simulated Theory (large M,N) Theory (large N)  = 20  = 100 Xavier Mestre: Random Matrix Theory in Signal Processing. 26/41
  27. 27. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ 3rd Application: Large multi-variate time series Xavier Mestre: Random Matrix Theory in Signal Processing. 27/41
  28. 28. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Introduction: testing independence of multiple time series We consider an -variate zero-mean Gaussian time series y() = [1()     ()] where  = 1     , and ask ourselves whether the different components of the series are independent. Xavier Mestre: Random Matrix Theory in Signal Processing. 28/41
  29. 29. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Motivation Consider a certain window of  samples and the extended random vector y() = ⎡ ⎣1()     1( +  − 1) | {z }  samples      ()     ( +  − 1) | {z }  samples ⎤ ⎦   and consider the second order statistics of this vector, namely E £ y()y  () ¤ = R = ⎡ ⎢ ⎢ ⎢ ⎣ R (11)  R (12)  · · · R (1)  R (21)  R (22)  · · · R (2)  ... ... ... ... R (1)  R (2)  · · · R ()  ⎤ ⎥ ⎥ ⎥ ⎦ where R (0 )  has dimensions  × . If the different time series are independent, R becomes block diagonal and  = 1  Ã log det R − X =1 log det R ()  ! = 0 Xavier Mestre: Random Matrix Theory in Signal Processing. 29/41
  30. 30. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Spatio-temporal sample covariance matrix In practice, R has to be estimated from the observations y(). Using the above formulation, we can estimate the time covariance between series  and 0 as ˆR (0 )  = 1  YY 0 where Y = ⎡ ⎢ ⎢ ⎢ ⎣  (1)  (2) · · ·  () · · ·  ()  (2) ... · · · ... ...  ( + 1) ...  () ... ... · · · ...  () · · ·  ()  ( + 1) · · ·  ( +  − 1) ⎤ ⎥ ⎥ ⎥ ⎦ has a Hankel structure. Under the null hypothesis (uncorrelation), and assuming stationarity E £ ()∗ 0(0 ) ¤ =  ( − 0 ) =0  () = Z 1 0 S () e2i  Xavier Mestre: Random Matrix Theory in Signal Processing. 30/41
  31. 31. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Tests from the spatio-temporal sample covariance matrix We can therefore build the test ˆ = 1  Ã log det ˆR − X =1 log det ˆR ()  ! and ask ourselves how to choose  (time window parameter) to make ˆ close to zero under the uncorrelation hypothesis. There is some trade-off between choosing  small (so that ˆR is close to R in spectral norm) and testing independence in large time lags  ar large as possible. For this all this, it appears reasonable to investigate the behavior of the eigenvalues of ˆR when  → ∞ and  → ∞ at the same rate, so that  =   → , 0    +∞. We will assume that 4 → 0, the spectral densities are uniformly bounded above and away from zero, and that sup  X ∈Z Ã 1  X =1 | ()|2 !12  +∞. Xavier Mestre: Random Matrix Theory in Signal Processing. 31/41
  32. 32. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Tests from the spatio-temporal sample covariance matrix Let Q(),  ∈ C+ , denote the resolvent matrix of ˆR, that is Q() = ³ ˆR − I ´−1 . Under the above assumptions, Q() ³ T() (deterministic asymptotic equivalent), where T() is the unique solution to T() = −1  µ I +  µ −1  ¡ I + Ψ (T()) ¢ ¶¶−1 in the class of matrix valued Stieltjes transforms, where Ψ : C× → C× and Ψ : C× → C× are the operators Ψ(A) = Z 1 0 d  () Ad ()  ¡ S () ⊗ d () d  () ¢  Ψ(B) = 1  X =1 Z 1 0 S () d  () B() d ()  d () d  ()  where d () = £ 1     ei(−1) ¤ and S () = diag (S1 ()      S ()). Xavier Mestre: Random Matrix Theory in Signal Processing. 32/41
  33. 33. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ 4th Application: outlier characterization of Maximum Likelihood estimation Xavier Mestre: Random Matrix Theory in Signal Processing. 33/41
  34. 34. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Considered scenario • Consider an array of  sensors receiving the signal transmitted by  sources from parameters ¯ = £ ¯(1)     ¯() ¤ , where we assume   . • Let y() denote an  × 1 complex vector containing the received samples. We model this obser- vation vector as y() = A(¯)s() + n() where s() contains the signal transmitted by the    sources, n() contains the received noise (assumed i.i.d. and CN(0 2 )) and A(¯) = £ a ¡ ¯(1) ¢    a ¡ ¯() ¢ ¤ • We assume that a total of  snapshots are available, and that   . • Problem: estimate the parameters ¯ from the observations {y()  = 1    }. • We investigate the use of Maximum Likelihood approaches =⇒ Highest resolution at the cost of increased computational complexity (multidimensional search) Xavier Mestre: Random Matrix Theory in Signal Processing. 34/41
  35. 35. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Maximum likelihood • The estimated angles are determined as ˆ = arg min∈Θ ˆ () where ˆ () is the negative (concen- trated) log-likelihood function. • The “conditional” (or deterministic) model: assumes that the signals s() are deterministic un- knowns. In this situation, one must minimize ˆ () = 1  tr h P⊥ ()ˆR i where ˆR = 1  P =1 y()y () is the sample covariance matrix, P⊥ () = I − P(), and P() = A() ¡ A ()A() ¢−1 A () is the orthogonal projection on the column space of A(). • The “unconditional” (or stochastic) model: assumes that the source signals are random variables, typically s() ∼ CN(0 P) and i.i.d. in the time domain. In this situation, ˆ () = 1  log det h ˆ () P⊥ () + P()ˆRP() i  Xavier Mestre: Random Matrix Theory in Signal Processing. 35/41
  36. 36. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Breakdown effect in maximum likelihood (I) • General nonlinear parametric estimators exhibit a threshold effect. • At low SNR, or low , the MSE suddenly departs from the Cramér Rao Bound. The presence of outliers is the main cause for this behavior. MSE Threshold effect B CRB d SNR dB Xavier Mestre: Random Matrix Theory in Signal Processing. 36/41
  37. 37. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Breakdown effect in maximum likelihood (II) At low values of the SNR and , there exist realizations of the cost functions for which local minima corresponding to outliers become deeper than the intended one. UML, SNR=0dB, M=5, N=20, uncorrelated signals,DoA=[16,18]deg θ1 (deg)θ2 (deg) −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 UML cost function Local Minima Intended Minimum Selected Minimum Xavier Mestre: Random Matrix Theory in Signal Processing. 37/41
  38. 38. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Probability of resolution • We consider here the definition of [Athley, 05] of the resolution probability. If ˆ () is a generic cost function that fluctuates around a deterministic ¯ (), which has  + 1 local minima at the values ¯ 1     , the probability of resolution can be defined as  = P "  =1 © ˆ ()  ˆ ¡ ¯ ¢ª #  • It was shown in [Athley, 05] that this definition of  provides a very accurate description of both the breakdown effect and the expected mean squared error (MSE) of the DoA estimation process. • Unfortunately, in our ML setting,  is difficult to analyze for finite values of   due to the complicated structure of the cost functions, especially ˆ (). • We propose to use the asymptotic distributions (as   → ∞) instead of the actual ones. Very accurate description of the actual probability, even for very low  . Xavier Mestre: Random Matrix Theory in Signal Processing. 38/41
  39. 39. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ First order behavior When   → ∞, under several technical conditions, the two ML cost functions become (pointwise) asymptotically close to two deterministic counterparts, namely |ˆ () − ¯ ()| → 0 |ˆ () − ¯ ()| → 0 a.s. pointwise in  as   → ∞, where ¯ () = 1  tr £ P⊥ ()R ¤ and ¯ () = 1  log det £ 2 () P⊥ () + P()RP() ¤ +  −   log µ   −  ¶ −   respectively, where R is the true covariance matrix of the observations and 2 () = 1 − tr £ P⊥ ()R ¤ . Xavier Mestre: Random Matrix Theory in Signal Processing. 39/41
  40. 40. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Second order behavior Let 1      be a set of multidimensional points (e.g. local minima of ¯ () or ¯ ()). Let ˆη = [ˆ (1)      ˆ ()] and ¯η = [¯ (1)      ¯ ()] and take the equivalent definitions for the UML cost function. Assume that y() ∼ CN (0 R). Under certain technical conditions, as   → ∞ ,  → , 0    1, we have Γ−1  (ˆη − ¯η) → N (0 I) and Γ−1  (ˆη − ¯η) → N (0 I) for some covariance matrices Γ, Γ given by {Γ} = 1  tr £ P⊥  P⊥  ¤ and {Γ} = 1 2 2  1  tr £ P⊥  P⊥  ¤ + 1 2  1  tr £ P⊥ Q ¤ + 1 2  1  tr £ P⊥  Q ¤ − log ¯ ¯ ¯ ¯1 − 1  tr [QQ] ¯ ¯ ¯ ¯ where P = R 12  P ³  ()  ´ R 12  ,P⊥  = R − P, and Q = R 12  A £ A  RA ¤−1 A  R 12  . Xavier Mestre: Random Matrix Theory in Signal Processing. 40/41
  41. 41. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Application • The previous asymptotic result is basically stating that for sufficiently large  , we are able to approximate the finite-dimensional distributions of the functions ˆ () and ˆ () as multivariate Gaussians, namely ˆη ∼ N (¯η Γ) ˆη ∼ N (¯η Γ) • It turns out that these results are very good approximations of the finite dimensional reality, even for relatively low values of  . • This provides a tool to evaluate the resolution probability of the ML method according to  = P "  =1 © ˆ ()  ˆ ¡ ¯ ¢ª # = P ⎡ ⎣ ⎡ ⎣ −1 1 ... ... −1 1 ⎤ ⎦ ˆη  0 ⎤ ⎦ which can be evaluated by computing the cdf of the Gaussian law. Xavier Mestre: Random Matrix Theory in Signal Processing. 41/41
  42. 42. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Simulation results ULA of  = 5 elements, two sources coming from 16 and 18 degrees with respect to the broadside. −15 −10 −5 0 5 10 15 20 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB) Prob.ofRes. Prob. of res., M=5, Theta=[16,18] deg, corr=0 UML (Predicted) UML (Simulated) CML (Predicted) CML (Simulated) N=100 N=10 Uncorrelated sources −15 −10 −5 0 5 10 15 20 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB)Prob.ofRes. Prob. of res., M=5, Theta=[16,18] deg, corr=0.95 UML (Predicted) UML (Simulated) CML (Predicted) CML (Simulated) N=100 N=10 Highly correlated sources Xavier Mestre: Random Matrix Theory in Signal Processing. 42/41
  43. 43. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Concluding remarks Xavier Mestre: Random Matrix Theory in Signal Processing. 43/41
  44. 44. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Conclusions Random Matrix Theory offers the possibility of analyzing the behavior of different quantities depending on ˆR when the sample size and the number of sensors/antennas have the same order of magnitude. The objective is to describe the asymptotic behavior of a certain scalar function of ˆR, namely  ³ ˆR ´ . • Traditional Approach: Assuming that the number of samples is high, we might establish that  ³ ˆR ´ →  (R) in some stochastic sense as  → ∞ while  remains fixed. • New Approach: In order to characterize the situation where   have the same order of magnitude, one might consider the limit   → ∞,  → , 0    ∞. Results obtained under this asymptotic limit turn out to be extremely accurate, even for reasonably low  . Xavier Mestre: Random Matrix Theory in Signal Processing. 44/41
  45. 45. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/ Thank you for your attention!!! Xavier Mestre: Random Matrix Theory in Signal Processing. 45/41

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