06.06.2018 Estimation Techniques for High Resolution and Multi-Dimensional Array Signal Processing DVT Research Group – Fh IIS and TU IL Wireless Distribution Systems / Digital Broadcasting LASP – UnB Laboratory of Array Signal Processing Prof. João Paulo C. Lustosa da Costa joaopaulo.dacosta@ene.unb.br

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 2 2 2

High resolution array signal processing: data model (1) 06.06.2018 High resolution array signal processing: data model (1) Assumptions about the signals Planar wave fronts Narrowband How many signals are received? Model order The amount of temporal samples (snapshots) N is greater than the amount of antennas M. Overdetermined problem: the amount of antennas M is greater than the amount of sources d. Uniform Linear Array (ULA) 3 3

High resolution array signal processing: data model (2) 06.06.2018 High resolution array signal processing: data model (2) Complex valued data Sensors always return real values. Examples of complex valued representation of real valued data. 1) Separation of a signal into in phase and quadrature components 2) Signal represented using the analytical form via the Hilbert transform 3) Time Frequency Analysis Short-Time Fourier Transform (STFT) 4) Antenna polarization: vertical and horizontal 4 4

High resolution array signal processing: data model (3) 06.06.2018 High resolution array signal processing: data model (3) Circular wave fronts: mapping into two variables Planar wave fronts: mapping into a single variable Higher distance between sources and antenna array Spacing between the antennas and the amount of antennas 5 5

High resolution array signal processing: data model (4) 06.06.2018 High resolution array signal processing: data model (4) The real-valued received high frequency signals at each antenna are given by T is the sampling period while  is related to the carrier frequency. 6 6

High resolution array signal processing: data model (5) 06.06.2018 High resolution array signal processing: data model (5) Converting them to baseband (complex-valued) 7 7

High resolution array signal processing: data model (6) 06.06.2018 High resolution array signal processing: data model (6) Applying the narrowband approximation, i.e. T is omitted for the sake of simplicity. 8 8

High resolution array signal processing: data model (7) 06.06.2018 High resolution array signal processing: data model (7) In case of a single wave front, i.e., d = 1.  M-1 3 2 1 Relationship between the direction of arrival  and the spatial frequency  c is the speed of the electromagnetic wave and f is the frequency of the narrowband signal. If the signal has  = 0, then the phase shifts  are equal to zero. 9 9

High resolution array signal processing: data model (8) 06.06.2018 High resolution array signal processing: data model (8) Vandermonde structure 10 10

High resolution array signal processing: data model (9) 06.06.2018 High resolution array signal processing: data model (9) In case of N temporal samples Data matrix Vector with signal samples Steering vector Rank of the data matrix is one. All columns and rows are linearly dependent. 11 11

High resolution array signal processing: data model (10) 06.06.2018 High resolution array signal processing: data model (10) Superposition of d sources In the figure of the example, d = 4. Being N > M = 5, the rank of X is equal to 4. 12 12

High resolution array signal processing: data model (11) 06.06.2018 High resolution array signal processing: data model (11) Symbol matrix Steering matrix Being N > M > d, the rank of the matrix X is equal to d. A is a Vandermonde matrix. 13 13

= Noiseless case: not realistic In the presence of noise: 06.06.2018 High resolution array signal processing: data model (12) Noiseless case: not realistic = + + In the presence of noise: : measurements : noise 14 14

Blind Source Separation (BSS) for antenna arrays 06.06.2018 High resolution array signal processing: data model (13) Blind Source Separation (BSS) for antenna arrays Given only the noisy measurements , we desire and . Step 1) To estimate the model order, i.e., d, via model order selection techniques Step 2) To estimate the spatial ferquencies , for i = 1, …, d and then the directions of arrival , for i = 1, …, d Step 3) To reconstruct the steering matrix A given the Vandermonde structure and the estimated spatial frequencies Step 4) To estimate the symbol matrix S via Moore Penrose pseudoinverse 15 15

The correlation function is defined as 06.06.2018 High resolution array signal processing: data model (14) Data model The correlation function is defined as We assume d uncorrelated signals and the additive noise is zero mean i.i.d.. is the expected value operator. 16 16

Signals and noise are uncorrelated! 06.06.2018 High resolution array signal processing: data model (15) Covariance matrix Signals and noise are uncorrelated! 17 17

Noise covariance matrix 06.06.2018 High resolution array signal processing: data model (16) Covariance matrix Noise covariance matrix 18 18

Signal covariance matrix 06.06.2018 High resolution array signal processing: data model (17) Covariance matrix Signal covariance matrix 19 19

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 20 20 20

Eigenvalues of the sample covariance matrix 06.06.2018 High resolution array signal processing: MOS (1) Eigenvalues of the sample covariance matrix d = 2, M = 8 SNR  ∞, N  ∞ M-d eigenvalues equal to zero d nonzero eigenvalues 21 21

Eigenvalues of the sample covariance matrix 06.06.2018 High resolution array signal processing: MOS (2) Eigenvalues of the sample covariance matrix d = 2, M = 8, SNR = 0 dB Finite SNR, N  ∞ M - d noise eigenvalues d signal eigenvalues Asymptotic behaviour of the noise eigenvalues 22 22

Sample covariance matrix 06.06.2018 High resolution array signal processing: MOS (3) Sample covariance matrix In practice, there is a limited amount of samples. Note that: 23 23

Autovalores da matriz de covariância de amostras 06.06.2018 High resolution array signal processing: MOS (4) Autovalores da matriz de covariância de amostras d = 2, M = 8, SNR = 0 dB, N = 10 Finite SNR, N = 10 M - d noise eigenvalues d signal eigenvalues 24 24

Why is important to know the model order? 06.06.2018 High resolution array signal processing: MOS (5) Why is important to know the model order? - Case of underestimation of the model order: Signals are modeled as noise. Then, low SNR! 25 25

Why is important to know the model order? 06.06.2018 High resolution array signal processing: MOS (6) Why is important to know the model order? - Case of overestimation of the model order: Noise is modeled as signals. In this case, extracted information and parameters have no meaning. 26 26

- Estimation of the model order d 06.06.2018 High resolution array signal processing: MOS (7) Eigenvalues profile - Estimation of the model order d Visual inspection: subjective and not automated - Esimation of the model order in na automated fashion Akaike Information Criterion Original version using directly the ML function Version based on the eigenvalues H. Akaike, “A new look at the statistical model identification,” IEEE Transactions on Automatic Control 19 (6): 716–723 , 1974 M. Wax, and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoustics, Speech and Signal Processing, vol. 33, pp. 387-392, Apr. 1985 27 27

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (8) Akaike Information Criterion (AIC) Objective: To find the best model that fits to the data Model family: parameterized family of probability density functions (PDF) Model parameters Measurements In order to find the best model, the following function should be minimized varying the candidate value of the model order Degrees of freedom or amount of free parameters Log maximum likelihood function 28 28

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (9) Akaike Information Criterion (AIC) Covariance matrix for the candidate model order k where Rank k Applying the EVD on the covariance matrix where Note that: Moreover, due to the mixture matrix , for 29 29

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (10) Akaike Information Criterion (AIC) 30 30

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (11) Akaike Information Criterion (AIC) Assuming N a temporal i.i.d. Gaussian samples whose model order is given by k 31 31

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (12) Akaike Information Criterion (AIC) Applying the log function on the Maximum Likelihood (ML) function Removing the constant term: The trace tr{ } of a scalar is the scalar itself. 32 32

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (13) Akaike Information Criterion (AIC) The trace opeartor allows the commutation of terms in the argument. The sum of the traces is the trace of the sum. Definition of the sample covariance matrix: 33 33

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (14) Akaike Information Criterion (AIC) The eigenvalues of the matrix are given by Since , the second term is approximately constant. 34 34

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (15) Akaike Information Criterion (AIC) After removing the second constant term: Replacing by the eigenvalues: Adding a constant to the expression: Constant 35 35

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (16) Model Order Selection Akaike Information Criterion (AIC) Noise eigenvalues: Final expression of the log ML: 36 36

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (17) Akaike Information Criterion (AIC) Computation of the degrees of freedom: k eigenvalues with M complex-valued elements: k real eigenvalues : Eigenvectors are unitary: Eigenvalues are orthogonal: 37 37

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (18) Akaike Information Criterion (AIC) AIC expression using only the eigenvalues: Non realistic assumption: Despite the assumption of asymptotic behavior of the noise eigevalues, AIC can be easily applied in several applications only by using the eigenvalues. 38 38

Akaike Information Criterion (AIC) 06.06.2018 High resolution array signal processing: MOS (19) Akaike Information Criterion (AIC) AIC expression using only eigenvalues: 39 39

Exponential Fitting Test (EFT) 06.06.2018 High resolution array signal processing: MOS (20) Exponential Fitting Test (EFT) where 40 40

Exponential Fitting Test (EFT) 06.06.2018 High resolution array signal processing: MOS (21) Exponential Fitting Test (EFT) White noise eigenvalues exhibit an exponential profile. Using the eigenvalues prediction, the “the breaking point” can be found. Let P be the amount of noise eigenvalues. Choose the greatest P that fits to the exponential decaying. d = 3, M = 8, SNR = 20 dB, N = 10 41 41

Outperformed approaches 06.06.2018 High resolution array signal processing: MOS (22) Approach Classification Tested scenario Gaussian noise Outperformed approaches EDC 1986 Eigenvalue based - White AIC, MDL ESTER 2004 Subspace based M = 128; N = 128 White/ Colored EDC, AIC, MDL RADOI 2004 M = 4; N = 16 White/ Colored Greschgörin Disk Estimator (GDE), AIC, MDL EFT 2007 M = 5; N = 6 AIC, MDL, MDLB, PDL SAMOS 2007 M = 65; N = 65 ESTER NEMO 2008 N = 8*M (various) 42 42

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 43 43

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 44 44

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 45 45 45

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 46 46 46

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 47 47 47

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 48 48 48

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 49 49 49

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 50 50 50

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 51 51 51

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 52 52 52

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 53 53 53

High resolution array signal processing: MOS (23) 06.06.2018 High resolution array signal processing: MOS (23) Comparing MOS schemes via Probability of Correct Detection (PoD) 54 54 54

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 55 55 55

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 56 56 56

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 57 57 57

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 58 58 58

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 59 59 59

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 60 60 60

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 61 61 61

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 62 62 62

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 63 63 63

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 64 64 64

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 65 65 65

High resolution array signal processing: MOS (24) 06.06.2018 High resolution array signal processing: MOS (24) Comparing MOS schemes via Probability of Correct Detection (PoD) 66 66 66

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 67 67 67

Which criterion? Minimum variance? Maximum sum? 06.06.2018 High resolution array signal processing: Beamforming (1) How to choose ? Which criterion? Minimum variance? Maximum sum? How to filter out the signal in a certain direction? Here we assume that the desired DOA is known. For the EVD based beamforming, the model order is known. 68 68

- Noiseless case with single source - Choosing the case that: 06.06.2018 High resolution array signal processing: Beamforming (2) Delay and Sum - Noiseless case with single source - Choosing the case that: Constructive interference In practice, there are noise and signals coming from other directions. 69 69

Capon: Minimum Variance Distortionless Response (MVDR) 06.06.2018 High resolution array signal processing: Beamforming (3) Capon: Minimum Variance Distortionless Response (MVDR) First high resolution signal processing scheme and not based on the eigenvalues decomposition (EVD) Given the filter output: The power output of the antenna array is given by: Note that: J. Capon, “High-Resolution Frequency-Wavenumber Spectrum Analysis”, Proc. IEEE, Vol. 57, 1408-1418, 1969 70 70

How to minimize the quadratic functionn with a constraint function? 06.06.2018 High resolution array signal processing: Beamforming (4) Capon: Minimum Variance Distortionless Response (MVDR) We desire to minimize the power variance - independent on the signal, i.e., statistically - Constraint: How to minimize the quadratic functionn with a constraint function? 71 71

Method of the Lagrange multipliers - Lagrange equation 06.06.2018 High resolution array signal processing: Beamforming (5) Method of the Lagrange multipliers - Lagrange equation Lagrange multiplier Constraint Function to be minimized 72 72

Method of the Lagrange multipliers 06.06.2018 High resolution array signal processing: Beamforming (6) Method of the Lagrange multipliers - Lagrange equation: 73 73

Method of the Lagrange multipliers - Lagrange equation: 06.06.2018 High resolution array signal processing: Beamforming (7) Method of the Lagrange multipliers - Lagrange equation: 74 74

Method of the Lagrange multipliers - Lagrange equation: 06.06.2018 High resolution array signal processing: Beamforming (8) Method of the Lagrange multipliers - Lagrange equation: 75 75

Spatial Power Density: varying 06.06.2018 High resolution array signal processing: Beamforming (9) Spatial Power Density: varying 76 76

06.06.2018 High resolution array signal processing: Beamforming (10) SPD via DS: scenario 1 77 77

SPD via Capon: scenario 1 06.06.2018 High resolution array signal processing: Beamforming (11) SPD via Capon: scenario 1 78 78

06.06.2018 High resolution array signal processing: Beamforming (12) SPD via DS: scenario 2 79 79

SPD via Capon: scenario 2 06.06.2018 High resolution array signal processing: Beamforming (13) SPD via Capon: scenario 2 80 80

06.06.2018 High resolution array signal processing: Beamforming (14) SPD via DS: scenario 3 81 81

SPD via Capon: scenario 3 06.06.2018 High resolution array signal processing: Beamforming (15) SPD via Capon: scenario 3 82 82

Four subspaces of the data 06.06.2018 High resolution array signal processing: Beamforming (16) Four subspaces of the data Two nonnormalized covariance matrices: row and column 83 83

Four subspaces of the data 06.06.2018 High resolution array signal processing: Beamforming (17) Four subspaces of the data EVD of the two nonnormalized covariance matrices Assuming that M < N, then is rank defficient. Moreover, 84 84

Four subspaces of the data 06.06.2018 High resolution array signal processing: Beamforming (18) Four subspaces of the data Using the row and column subspaces, we obtain the SVD: where Replacing the SVD matriz on the definition of the covariace matrix: 85 85

Four subspaces of the data 06.06.2018 High resolution array signal processing: Beamforming (20) Four subspaces of the data Since: Then: 86 86

SVD plus model order information Four subspaces of the data 06.06.2018 High resolution array signal processing: Beamforming (21) SVD plus model order information Four subspaces of the data Row subspace: Row subspace composed of signals and noise subspace using the model order: Column subspace: Column subspace composed of signals and noise subspace using the model order : 87 87

SVD plus model order information 06.06.2018 High resolution array signal processing: Beamforming (22) SVD plus model order information Low rank approximation , rank r and MO d “Complete SVD” “Economic SVD” Low rank approximation 88 88

SVD plus model order information Denoising via low rank approximation 06.06.2018 High resolution array signal processing: Beamforming (23) SVD plus model order information Denoising via low rank approximation Denoising from the data by using the column and row signal subspaces and the signal singular values Filtering based on the orthogonality between the signal and noise subspaces 89 89

The DOA information is not necessary to find the beamforming vector . 06.06.2018 High resolution array signal processing: Beamforming (24) Eigenfilter The DOA information is not necessary to find the beamforming vector . The signal and noise subspaces are assumed to be orthogonal. The antenna array power output is given by The power in case of only noise is given by: Hence, SNR is given by: 90 90

The filter should satisfy: Hence, the SNR is given by: 06.06.2018 High resolution array signal processing: Beamforming (25) Eigenfilter The filter should satisfy: Hence, the SNR is given by: The SNR is maximized when the filter power output is equal to the greatest eigenvalue. Therefore, 91 91

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 92 92 92

High resolution array signal processing: DOA estimation (1) 06.06.2018 High resolution array signal processing: DOA estimation (1) Direction of Arrival (DOA) Delay Doppler shift Direction of Departure (DOD) Receive array: 1-D or 2-D Frequency Time Transmit array: 1-D or 2-D 93

High resolution array signal processing: DOA estimation (2) 06.06.2018 High resolution array signal processing: DOA estimation (2) R-D parameter estimation Spatial dimensions RX  Direction of Arrival Model: superposition of d undamped exponentials sampled on a R-dimensional grid and observed at N subsequent time instances. Spatial dimensions TX  Direction of Departure Frequency  Delay Time  Doppler shift R-Dimensional measurements Spatial frequencies  One to one mapping to physical parameters 94

High resolution array signal processing: DOA estimation (3) 06.06.2018 High resolution array signal processing: DOA estimation (3) High resolution parameter estimation Maximum-Likelihood EM SAGE [Fessler et. al. 1994] Extensions [Fleury et. al. 1999, Pederson et. al. 2000, Thomä et. al. 2004] Subspace-based MUSIC [Schmidt 1979], Root MUSIC [Barabell 1983] ESPRIT [Roy 1986], R-D Unitary ESPRIT [Haardt et. al. 1998] RARE (Rank reduction estimator) [Pesavento et. al. 2004] MDF (Multidimensional folding) [Mokios el. al. 1994] ... 95

Multiple Signal Classification (MUSIC) 06.06.2018 High resolution array signal processing: DOA estimation (4) Multiple Signal Classification (MUSIC) Using the model order and applying the EVD, the signal and noise subspaces can be decomposed. where The SPD is given by: 96 96

DS, CAPON and MUSIC can be estimated via the SPD. 06.06.2018 High resolution array signal processing: DOA estimation (5) Estimation of signal parameters via rotational invariant techniques (ESPRIT) DS, CAPON and MUSIC can be estimated via the SPD. In general, the DOA estimation requires a high computational complexity for searching the peaks of the SPD. ESPRIT is a closed-form solution for DOA estimation and requies no peak searches. 97 97

Shift invariance equation 06.06.2018 High resolution array signal processing: DOA estimation (6) Estimation of signal parameters via rotational invariant techniques (ESPRIT) Shift invariance equation  d where J1 and J2 are the selection matrices. 98 98

06.06.2018 High resolution array signal processing: DOA estimation (7) Estimation of signal parameters via rotational invariant techniques (ESPRIT) The steering matrix A and the d eigenvectors of E corresponding to the d greatest eigenvalues of the covariance matrix generate the same subspace. Note that Es is obtained via low rank approximation. We apply again the EVD on the matrix 99 99

High resolution array signal processing: DOA estimation (8) 06.06.2018 High resolution array signal processing: DOA estimation (8) Deterministic Expectation Maximization Incomplete Maximum Likelihood Function Complete Maximum Likelihood Function Expectation Step 100 100

High resolution array signal processing: DOA estimation (9) 06.06.2018 High resolution array signal processing: DOA estimation (9) Deterministic Expectation Maximization Maximization Step An example how the algorithm works is shown on the board. 101 101

High resolution array signal processing: DOA estimation (10) 06.06.2018 High resolution array signal processing: DOA estimation (10) Algorithm working…. After some iterations d = 4 102 102

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 103 103 103

Signal reconstruction via the Moore Penrose pseudo inverse 06.06.2018 High resolution array signal processing: signal reconstruction (1) Signal reconstruction via the Moore Penrose pseudo inverse Given the DOA estimates, an estimate of the steering matrix can be reconstructed exploiting its Vandermonde structure. Given and given , we desire 104 104

Signal reconstruction via the Moore Penrose pseudo inverse 06.06.2018 High resolution array signal processing: signal reconstruction (2) Signal reconstruction via the Moore Penrose pseudo inverse The matrix pseudo-inverse is given by: Note that in order to compute the pseudo-inverse d < M. 105 105

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 106 106 106

06.06.2018 High resolution array signal processing: Prewhitening (1) Matrix data model Colored noise model 107

Stochastic prewhitening schemes Deterministic prewhitening scheme 06.06.2018 High resolution array signal processing: Prewhitening (2) Analysis via SVD Stochastic prewhitening schemes With colored noise the d main components are more affected. Deterministic prewhitening scheme 108

GSVD and GEVD can be applied instead of matrix inversion. 06.06.2018 High resolution array signal processing: Prewhitening (3) Estimation of the prewhitening matrix via only noise samples Estimating the prewhitening subspace via matrix inversion SVD of the prewhitened data matrix Recovering the subspace (low rank approximation) GSVD and GEVD can be applied instead of matrix inversion. The recovered subspace can be applied directly to the Standard ESPRIT to obtain the estimated spatial frequencies. 109

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 110 110 110

High resolution array signal processing: instantaneous ICA (1) 06.06.2018 High resolution array signal processing: instantaneous ICA (1) The cocktail party problem: is the mixture matrix (unknown); is the symbol matrix or signal matrix (unknown); is the measurement matrix (known); Given , we desire to find . Assumptions: The signals follow a non Gaussian distribution; The signals are independent. and are instantaneous mixed. Note that, in practice, audio signals are convolutively mixed. 111 111

High resolution array signal processing: instantaneous ICA (2) 06.06.2018 High resolution array signal processing: instantaneous ICA (2) Independence versus orthogonality versus uncorrelation Let x and y be two non zero vectors then: x and y are linearly independent if and only if the following relationship is not satisfied for any constant a x and y are orthogonal if and only if x and y are uncorrelated if and only if J. L. Rodgers, W. A. Nicewander, and L. Toothaker, „Linearly Independent, Orthogonal, and Uncorrelated Variables“, The American Statistician, Vol. 38, 1984. 112 112

High resolution array signal processing: instantaneous ICA (3) 06.06.2018 High resolution array signal processing: instantaneous ICA (3) Data preparation for the Independent Component Analysis (ICA) Estimating the mean for each line of the matrix X where is one element of the matrix X at position (m,n). Estimating the variance of each line for the matrix Transformation of the rows of the matrix X into zero mean and unitary variance for m = 1, ..., M After this procedure, we obtain the normalized matrix X´ 113 113

High resolution array signal processing: instantaneous ICA (3) 06.06.2018 High resolution array signal processing: instantaneous ICA (3) Whitening the matrix X´ Computation of the sample covariance matrix Applying the EVD The whitening matrix is defined as Applying the whitening matrix, we obtain: or Z corresponds to a rotated and normalized matrix S´. 114 114

High resolution array signal processing: instantaneous ICA (4) 06.06.2018 High resolution array signal processing: instantaneous ICA (4) Maximizing the non Gaussianity From the central limit theorem: the arithmetic mean of a sufficiently high amount of independent random variable can be approximated to a normal distribution. In order to separate the mixed signals, we use a criterion the maximization of the non Gaussianity. Hence, ICA cannot be applied to signals that are Gaussian distributed! For a zero mean Gaussian signal, we have that the fourth moment is given by Kurtosis Therefore, the Gaussianity of a random variable can be measured by computing If the random variable is Gaussian then: If the variable s is non Gaussian and has unitary variance, then 115 115

High resolution array signal processing: instantaneous ICA (5) 06.06.2018 High resolution array signal processing: instantaneous ICA (5) Maximizing the non Gaussianity Example of the function 116 116

High resolution array signal processing: instantaneous ICA (6) 06.06.2018 High resolution array signal processing: instantaneous ICA (6) Maximizing the non Gaussianity Once we have a function to be maximized, we can apply the steepest descent. The filter is iteratively computed using the cost function and the gradient method as follows where corresponds to the step of convergence, J(n) is the cost function to be maximized and is the filter function to be found. Note that if is too small then the convergence can take several iterations, while if too high the function cannot converge. Note that the cost function is given by 117 117

High resolution array signal processing: instantaneous ICA (7) 06.06.2018 High resolution array signal processing: instantaneous ICA (7) Maximizing the non Gaussianity Applying the gradient method: where is 1 if the argument is positive and -1 if the argument is negative, Note that z has zero mean and unitary and the same is valid for y, that is the filter output. 118 118

High resolution array signal processing: instantaneous ICA (8) 06.06.2018 High resolution array signal processing: instantaneous ICA (8) Example of ICA application d = 3 signals Original signals (before mixing) are given by 119 119

High resolution array signal processing: instantaneous ICA (9) 06.06.2018 High resolution array signal processing: instantaneous ICA (9) Example of ICA application d = 3 signals Mixed signals (after mixing) are given by 120 120

High resolution array signal processing: instantaneous ICA (10) 06.06.2018 High resolution array signal processing: instantaneous ICA (10) Example of ICA application d = 3 signals Whitened signals are given by 121 121

High resolution array signal processing: instantaneous ICA (10) 06.06.2018 High resolution array signal processing: instantaneous ICA (10) Example of ICA application d = 3 signals Demixed signals, after the whole ICA process, are given by 122 122

High resolution array signal processing: instantaneous ICA (10) 06.06.2018 High resolution array signal processing: instantaneous ICA (10) Example of ICA application The approximation of the PDF of the signal by using histograms. Note that no signals have a Gaussian distribution! Curtose: -1.5000 .: Subgaussian (sinusoid) 0.7692 .: Supergaussiana -1.2000 .: Subgaussiana (triangular wave) 123 123

High resolution array signal processing: instantaneous ICA (10) 06.06.2018 High resolution array signal processing: instantaneous ICA (10) Example of ICA application Computation of the Kurtosis cost function for each iteration. Triangular Sinusoid 124 124

Content of the intensive course (1) 06.06.2018 Content of the intensive course (1) Introduction to multi-channel systems Mathematical background High resolution array signal processing Data model Model order selection Beamforming Direction of arrival (DOA) estimation Signal reconstruction via pseudo inverse Prewhitening Independent Component Analysis (ICA) for instantaneous mixtures ICA for convolutive mixtures 125 125 125

High resolution array signal processing: convolutive ICA (1) 06.06.2018 High resolution array signal processing: convolutive ICA (1) Mixture of audio signals The mathematical representation is given by the convolutive mixture. Source 1 Mixed signal 1 Source 2 Mixed signal 2 Source 3 Mixed signal 3 126 126

High resolution array signal processing: convolutive ICA (2) 06.06.2018 High resolution array signal processing: convolutive ICA (2) Mixed audio signals Convolutive model Superposition of Q sources 127 127

High resolution array signal processing: convolutive ICA (3) 06.06.2018 High resolution array signal processing: convolutive ICA (3) Mistura de sinais sonoros 128 128

High resolution array signal processing: convolutive ICA (4) 06.06.2018 High resolution array signal processing: convolutive ICA (4) Convolutive ICA Instantaneous ICA can be applied only for instantaneous mixtures, in the context of telecommunications means narrowband signals. Audio signals are extremely broadband, since they are on the base band signals. Hence, audio signals are convolutively mixed. The short-time Fourier transform (STFT) allows to transform the time samples (snaptshots) in two dimensions which are time (frames) and frequency. The STFT is a Time–Frequency Analysis (TFA) scheme. After the STFT, a signle audio signal (broadband) is transformed into F narrowband signals. Hence, after the STFT, F instantaneous ICA can be applied to the F narrowband signals. 129 129

High resolution array signal processing: convolutive ICA (5) 06.06.2018 High resolution array signal processing: convolutive ICA (5) Step 1 of the STFT: Segmentation with superposition (overlap) of the narrowband signal into frames Amplitude Snapshots Hanning window Step 2 of the STFT: Windowing of the frame - Product of the chosen window and frame is performed point to point in real-valued domain. 130 130

High resolution array signal processing: convolutive ICA (6) 06.06.2018 High resolution array signal processing: convolutive ICA (6) The windowing significantly reduces the spectral leakage as follows Amplitude Amplitude Time (s) Frequency (Hz) Amplitude Amplitude Time (s) Frequency (Hz) Amplitude Amplitude Time (s) Frequency (Hz) 131 131

High resolution array signal processing: convolutive ICA (6) 06.06.2018 High resolution array signal processing: convolutive ICA (6) Step 3 of the STFT: Frames are transformed to the frequency domain by applying the Discrete Fourier Transform in each frame. The convolutive ICA convolutiva can be repesented in the following fashion: Segmentation and windowing Instantaneous ICA for f = 1 FFT IFFT Adjusting the permutation of the d signals for F frequencies Segmentation and windowing Instantaneous ICA for f = F FFT IFFT STFT for each microphone After IFFT, the inverse transformation should perform the inverse operation with respect to the segmentation and windowing. 132 132

High resolution array signal processing: convolutive ICA (6) 06.06.2018 High resolution array signal processing: convolutive ICA (6) Testing environment with two sound sources and a microphone array 133 133

High resolution array signal processing: convolutive ICA (6) 06.06.2018 High resolution array signal processing: convolutive ICA (6) Example of the two mixed signals Original signals Mixed signal Separated signals applying TRINICON 134 134