Estimation Techniques for High Resolution and Multi-Dimensional Array Signal Processing.

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Estimation Techniques for High Resolution and Multi-Dimensional Array Signal Processing EMS Group – Fh IIS and TU IL Electronic Measurements and Signal Processing Group (EMS) LASP – UnB Laboratory of Array Signal Processing Prof. João Paulo C. Lustosa da Costa joaopaulo.dacosta@ene.unb.br

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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) High resolution array signal processing: data model (1) 3

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: data model (2) 4

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: data model (3) 5

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  The real-valued received high frequency signals at each antenna are given by High resolution array signal processing: data model (4)  T is the sampling period while  is related to the carrier frequency. 6

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Converting them to baseband (complex-valued) High resolution array signal processing: data model (5) 7

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Applying the narrowband approximation, i.e. High resolution array signal processing: data model (6)  T is omitted for the sake of simplicity. 8

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  M-1 3 2 1 0   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.  In case of a single wave front, i.e., d = 1. High resolution array signal processing: data model (7) 9

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Vandermonde structure High resolution array signal processing: data model (8) 10

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  In case of N temporal samples  Data matrix  Rank of the data matrix is one. All columns and rows are linearly dependent. Vector with signal samples Steering vector High resolution array signal processing: data model (9) 11

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Superposition of d sources In the figure of the example, d = 4.  Being N > M = 5, the rank of X is equal to 4. High resolution array signal processing: data model (10) 12

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Symbol matrix Steering matrix  Being N > M > d, the rank of the matrix X is equal to d.  A is a Vandermonde matrix. High resolution array signal processing: data model (11) 13

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Noiseless case: not realistic ++ = In the presence of noise: High resolution array signal processing: data model (12) : measurements : noise 14

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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 frequencies, 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 High resolution array signal processing: data model (13) 15

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Data model The correlation function is defined as We assume d uncorrelated signals and the additive noise is zero mean i.i.d.. High resolution array signal processing: data model (14) is the expected value operator. 16

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Covariance matrix Signals and noise are uncorrelated! High resolution array signal processing: data model (15) 17

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Covariance matrix Noise covariance matrix High resolution array signal processing: data model (16) 18

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Covariance matrix Signal covariance matrix High resolution array signal processing: data model (17) 19

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Eigenvalues of the sample covariance matrix SNR  ∞, N  ∞ – M-d eigenvalues equal to zero – d nonzero eigenvalues d = 2, M = 8 High resolution array signal processing: MOS (1) 21

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Eigenvalues of the sample covariance matrix Finite SNR, N  ∞ –M - d noise eigenvalues –d signal eigenvalues –Asymptotic behaviour of the noise eigenvalues d = 2, M = 8, SNR = 0 dB High resolution array signal processing: MOS (2) 22

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Sample covariance matrix – In practice, there is a limited amount of samples. – Note that: High resolution array signal processing: MOS (3) 23

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Eigenvalues of the sample covariance matrix d = 2, M = 8, SNR = 0 dB, N = 10 Finite SNR, N = 10 – M - d noise eigenvalues – d signal eigenvalues High resolution array signal processing: MOS (4) 24

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Why is important to know the model order? - Case of underestimation of the model order: Signals are modeled as noise. Then, low SNR! High resolution array signal processing: MOS (5) 25

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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. High resolution array signal processing: MOS (6) 26

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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 High resolution array signal processing: MOS (7) 27

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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 High resolution array signal processing: MOS (8) 28

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Akaike Information Criterion (AIC) Covariance matrix for the candidate model order k where Applying the EVD on the covariance matrix where Note that: Moreover, due to the mixture matrix, for Rank k High resolution array signal processing: MOS (9) 29

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Akaike Information Criterion (AIC) High resolution array signal processing: MOS (10) 30

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Akaike Information Criterion (AIC) Assuming N a temporal i.i.d. Gaussian samples whose model order is given by k High resolution array signal processing: MOS (11) 31

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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. High resolution array signal processing: MOS (12) 32

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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: High resolution array signal processing: MOS (13) 33

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Akaike Information Criterion (AIC) The eigenvalues of the matrix are given by Since, the second term is approximately constant. High resolution array signal processing: MOS (14) 34

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Akaike Information Criterion (AIC) After removing the second constant term: Replacing by the eigenvalues: Adding a constant to the expression: High resolution array signal processing: MOS (15) Constant 35

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Model Order Selection Akaike Information Criterion (AIC) Noise eigenvalues: Final expression of the log ML: High resolution array signal processing: MOS (16) 36

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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: High resolution array signal processing: MOS (17) 37

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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. High resolution array signal processing: MOS (18) 38

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Akaike Information Criterion (AIC) AIC expression using only eigenvalues: High resolution array signal processing: MOS (19) 39

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Exponential Fitting Test (EFT) High resolution array signal processing: MOS (20) where 40

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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 High resolution array signal processing: MOS (21) 41

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ ApproachClassification Tested scenario Gaussian noise Outperformed approaches EDC 1986Eigenvalue based - WhiteAIC, MDL ESTER 2004Subspace based M = 128; N = 128 White/ Colored EDC, AIC, MDL RADOI 2004Eigenvalue based M = 4; N = 16 White/ Colored Greschgörin Disk Estimator (GDE), AIC, MDL EFT 2007Eigenvalue based M = 5; N = 6 WhiteAIC, MDL, MDLB, PDL SAMOS 2007Subspace based M = 65; N = 65 White/ Colored ESTER NEMO 2008Eigenvalue based N = 8*M (various) ColoredAIC, MDL High resolution array signal processing: MOS (22) 42

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Comparing MOS schemes via Probability of Correct Detection (PoD) High resolution array signal processing: MOS (23) 43

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ High resolution array signal processing: MOS (23)  Comparing MOS schemes via Probability of Correct Detection (PoD) 44

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 45 High resolution array signal processing: MOS (23)  Comparing MOS schemes via Probability of Correct Detection (PoD) 45

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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. High resolution array signal processing: Beamforming (1) 68

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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. High resolution array signal processing: Beamforming (2) 69

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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 High resolution array signal processing: Beamforming (3) 70

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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? High resolution array signal processing: Beamforming (4) 71

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Method of the Lagrange multipliers - Lagrange equation Constraint Function to be minimized Lagrange multiplier High resolution array signal processing: Beamforming (5) 72

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Method of the Lagrange multipliers - Lagrange equation: High resolution array signal processing: Beamforming (6) 73

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Spatial Power Density: varying High resolution array signal processing: Beamforming (9) 76

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ SPD via DS: scenario 1 High resolution array signal processing: Beamforming (10) 77

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ SPD via Capon: scenario 1 High resolution array signal processing: Beamforming (11) 78

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Four subspaces of the data Two nonnormalized covariance matrices: row and column High resolution array signal processing: Beamforming (16) 83

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Four subspaces of the data EVD of the two nonnormalized covariance matrices Assuming that M < N, then is rank defficient. Moreover, High resolution array signal processing: Beamforming (17) 84

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Four subspaces of the data Using the row and column subspaces, we obtain the SVD: where Replacing the SVD matrix on the definition of the covariace matrix: High resolution array signal processing: Beamforming (18) 85

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Four subspaces of the data Since: Then: High resolution array signal processing: Beamforming (20) 86

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 : High resolution array signal processing: Beamforming (21) 87

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ “Complete SVD” “Economic SVD” Low rank approximation  SVD plus model order information Low rank approximation, rank r and MO d High resolution array signal processing: Beamforming (22) 88

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: Beamforming (23) 89

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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: High resolution array signal processing: Beamforming (24) 90

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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, High resolution array signal processing: Beamforming (25) 91

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Receive array: 1-D or 2-D Frequency Time Transmit array: 1-D or 2-D Direction of Arrival (DOA) Delay Doppler shift Direction of Departure (DOD) High resolution array signal processing: DOA estimation (1) 93

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

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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]... High resolution array signal processing: DOA estimation (3) 95

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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: High resolution array signal processing: DOA estimation (4) 96

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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. High resolution array signal processing: DOA estimation (5) 97

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Estimation of signal parameters via rotational invariant techniques (ESPRIT) Shift invariance equation  d where J 1 and J 2 are the selection matrices. High resolution array signal processing: DOA estimation (6) 98

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 E s is obtained via low rank approximation.  We apply again the EVD on the matrix High resolution array signal processing: DOA estimation (7) 99

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Deterministic Expectation Maximization  Incomplete Maximum Likelihood Function  Complete Maximum Likelihood Function  Expectation Step High resolution array signal processing: DOA estimation (8) 100

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Deterministic Expectation Maximization  Maximization Step  An example how the algorithm works is shown on the board. High resolution array signal processing: DOA estimation (9) 101

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Algorithm working…. After some iterations d = 4 High resolution array signal processing: DOA estimation (10) 102

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: signal reconstruction (1) 104

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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. High resolution array signal processing: signal reconstruction (2) 105

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Matrix data model  Colored noise model High resolution array signal processing: Prewhitening (1) 107

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Stochastic prewhitening schemes With colored noise the d main components are more affected. Analysis via SVD Deterministic prewhitening scheme 108 High resolution array signal processing: Prewhitening (2)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Estimation of the prewhitening matrix via only noise samples 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.  Estimating the prewhitening subspace via matrix inversion  SVD of the prewhitened data matrix  Recovering the subspace (low rank approximation) 109 High resolution array signal processing: Prewhitening (3)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Given, we desire to find.  The cocktail party problem:  Assumptions:  The signals follow a non Gaussian distribution;  The signals are independent.  and are instantaneous mixed. o Note that, in practice, audio signals are convolutively mixed. is the mixture matrix (unknown); is the symbol matrix or signal matrix (unknown); is the measurement matrix (known); 111 High resolution array signal processing: instantaneous ICA (1)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ J. L. Rodgers, W. A. Nicewander, and L. Toothaker, „Linearly Independent, Orthogonal, and Uncorrelated Variables“, The American Statistician, Vol. 38, 1984.  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 112 High resolution array signal processing: instantaneous ICA (2)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: instantaneous ICA (3)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: instantaneous ICA (3)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: instantaneous ICA (4)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Maximizing the non Gaussianity  Example of the function 116 High resolution array signal processing: instantaneous ICA (5)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: instantaneous ICA (6)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: instantaneous ICA (7)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Example of ICA application  d = 3 signals  Original signals (before mixing) are given by 119 High resolution array signal processing: instantaneous ICA (8)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 120 High resolution array signal processing: instantaneous ICA (9)  Example of ICA application  d = 3 signals  Mixed signals (after mixing) are given by

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 121 High resolution array signal processing: instantaneous ICA (10)  Example of ICA application  d = 3 signals  Whitened signals are given by

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Curtose: -1.5000.: Subgaussian (sinusoid) 0.7692.: Supergaussiana -1.2000.: Subgaussiana (triangular wave) 123 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!

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Sinusoid Triangular 124 High resolution array signal processing: instantaneous ICA (10)  Example of ICA application  Computation of the Kurtosis cost function for each iteration.

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Content of the intensive course (1) 125 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

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  The mathematical representation is given by the convolutive mixture. Mixed signal 1 Source 1 Source 2 Source 3 Mixed signal 2 Mixed signal 3  Mixture of audio signals 126 High resolution array signal processing: convolutive ICA (1)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Convolutive model Superposition of Q sources  Mixed audio signals 127 High resolution array signal processing: convolutive ICA (2)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Mistura de sinais sonoros 128 High resolution array signal processing: convolutive ICA (3)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  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 High resolution array signal processing: convolutive ICA (4)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ 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. Step 1 of the STFT: Segmentation with superposition (overlap) of the narrowband signal into frames Hanning window 130 High resolution array signal processing: convolutive ICA (5) Snapshots Amplitude

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  The windowing significantly reduces the spectral leakage as follows 131 High resolution array signal processing: convolutive ICA (6) Time (s) Frequency (Hz) Amplitude

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Instantaneous ICA for f = 1 Segmentation and windowing  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 FFT Instantaneous ICA for f = F Adjusting the permutation of the d signals for F frequencies IFFT  After IFFT, the inverse transformation should perform the inverse operation with respect to the segmentation and windowing. STFT for each microphone 132 High resolution array signal processing: convolutive ICA (6)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/  Testing environment with two sound sources and a microphone array 133 High resolution array signal processing: convolutive ICA (6)

http://www.tu-ilmenau.de/it_dvt/ http://www.redes.unb.br/lasp/ Separated signals applying TRINICON Original signals  Example of the two mixed signals Mixed signal 134 High resolution array signal processing: convolutive ICA (6)

Estimation Techniques for High Resolution and Multi-Dimensional Array Signal Processing.

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Estimation Techniques for High Resolution and Multi-Dimensional Array Signal Processing.

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Presentation on theme: "Estimation Techniques for High Resolution and Multi-Dimensional Array Signal Processing."— Presentation transcript:

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