Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa da Costa University of Brasília (UnB) Department of Electrical Engineering (ENE) Laboratory of Array Signal Processing PO Box 4386 Zip Code , Brasília - DF Homepage:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (1)   Linearly Constraint Minimum Variance (LCMV) Beamformer   only one constraint 2   We can generalize the LCMV by introducing L multiple linear constraints.   We assume that the constraints are linearly independent.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (2) 3   Given that M > L, then the remaining subspace with M – L vectors is orthogonal to vector of C.   Therefore, the total signal subspace can be represented as   The weight vector w is given by   Replacing the signal subspace using the matrices C and C a

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (3) 4   We can observe that q has a component multiplied by C and another one multiplied by C a.   Therefore, we can rewrite q as   Hence,   We have to find v and w a vectors in order to determine w.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (4) 5   In order to find v, we replace w expression in our primal equation.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (5) 6   Note that the signal subspace which does not belong to the constraints may change with time. Therefore, this is the adaptive part of w.   The output of the linear filter is given by

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (6) 7   We desire to minimize the power of y(n) by removing the information which is not related to the constraints. Removing the information not related to the constraints. Therefore, the trivial solution does not work.   The expression to be minimized is given by

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (7) 8   We can minimize the expression below by applying the gradient with respect to w a *   Hence, we have that   Therefore:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Generalized Sidelobe Canceler – GSC (8) 9   The weight vector is given by

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 10 Channel sounding 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)

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 11 Channel sounding 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

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 12 Existing approaches 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] (many more) standard R-D Tensor-ESPRIT Unitary R-D Tensor-ESPRIT

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos DOA Estimation (1)   Deterministic Expectation Maximization 13   Incomplete Maximum Likelihood Function   Complete Maximum Likelihood Function   Expectation Step

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos DOA Estimation (2)   Deterministic Expectation Maximization 14   Maximization Step   An example how the algorithm works is shown on the board.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos DOA Estimation (3)   Algorithm working…. 15 After some iterations d = 4

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos DOA Estimation (4)   Estimation of Signal Parameters via Rotation Invariance Techniques ESPRIT 16  d Shift Invariant Equation

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos DOA Estimation (5)   Estimation of Signal Parameters via Rotation Invariance Techniques ESPRIT 17   The steering matrix A and the first d eigenvectors U of the covariance matrix generate the same subspace   Note that U s is related to the low-rank approximation.   Spatial frequencies

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos DOA Estimation (6)   Estimation of Signal Parameters via Rotation Invariance Techniques ESPRIT 18   Advantages No parameter searches are necessary Lower computation complexity Assuming that SVD is easier to compute than parameter searches.   Drawbacks The array should obey the shift invariant equation.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Tensors The term “tensor” Here: Intuitive definition: Here: Intuitive definition: ”A tensor of order p is a collection of elements referenced by p incides“ ”A tensor of order p is a collection of elements referenced by p incides“ Mathematics: 1846: W. Voigt Mathematics: 1846: W. Voigt Physics: 1915: M. Grossmann and A. Einstein Physics: 1915: M. Grossmann and A. Einstein  very abstract definition (multilinear maps)  describe physical quantities ScalarsVectorsMatricesOrder-3-tensorsOrder-4-tensors ? … (requires covariant and contravariant indices  [2,0],[1,1],[0,2] tensors)  “multi-way arrays”

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 20 Why tensors?  Well, why even matrices?  Matrix equations are usually more compact  insights, manipulations  Example: DFT  Not a different data model but a more compact representation  More than two dimensions: tensors even more compact  new insights d n = ? d n = ?? 

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 21 Operations on tensors Outer product Outer product Inner product Inner product Higher-order norm Higher-order norm Unfoldings Unfoldings Concatenation Concatenation

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 22 Operations on tensors and matrices n-mode product n-mode product i.e., all the n-mode vectors multiplied from the left hand side by 1 2 3

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Operations on Tensors Outer product Outer product Higher-order norm Higher-order norm For example:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 24 The tensor rank  Definition of the tensor rank  A tensor is rank one, iff  A tensor is rank r iff it can be decomposed into a sum of r and not less than r rank one tensors  (Only) connection to the n-ranks:  The rank of a tensor can exceed its size (which is a good thing and a bad thing) 2 x (maximum rank, cf. [Kolda08])

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 25 Extension to the HOSVD of tensors “Full HOSVD” Low-rank approximation “Economy size HOSVD”

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 26 Data model (1) Classical approach: Matrix R-Dimensional measurements New approach: Tensor  More natural representation of sampled R-D signal

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 27 Data model (2) Example: m2m2m2m m1m1m1m1 n m1m1m1m m2m2m2m2123 n More natural representation of sampled R-D signal

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 28 Data model (3) Classical approach: Matrix New approach: Tensor

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 29 Signal subspace estimation Matrix case signal part noise part rank d Basis for the signal subspace  and span the same column space Tensor case signal part noise part rank d Basis for the signal subspace  spaces spanned by the -mode vectors are equal for

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 30 Shift invariance Equations  shift invariance equations in terms of the estimated signal subspace  can be solved for spatial frequencies

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 31 Least Squares solution Tensor – LS Matrix – LS Enhanced signal subspace estimate If the number of sources Limitation: then: i.e., signal subspace estimates are equal.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 32 Summary of R-D standard Tensor-ESPRIT 1. Signal subspace estimation: Compute the tensor from the HOSVD- based low-rank approximation of the measurement tensor 2. Shift invariance equations: Solve the shift invariance equations given by for the matrices. 3. Spatial frequency estimation: Compute the eigenvalues of the matrices jointly for The spatial frequencies are given by

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 33 Correlated sources RMSE SNR [dB]  Significant gain in the estimation error for both methods

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 34 Note  Unitary ESPRIT  Inclusion of the Forward Backward Averaging (FBA), which virtually doubles the number of samples (already in the literature)  Inclusion of the Real Value Transformation (already in the literature)  The RVT degrades the parameter estimation for low SNR regimes.  Fancy formulation (new)

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos PARAFAC  Decomposition of a tensor into a sum of rank 1 terms (PARAllel FACtors)  An extension of what the SVD does in the matrix case …  ICA/PCA !

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos PARAFAC  To generalize it to a 3-D signal we need 3-D rank one terms  The PARAFAC model the minimum r: the tensor rank! a 3-D rank-1 tensor

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos PARAFAC - notation  Define the matrices  Expressions for the model M1M1M1M1 M2M2M2M2 M3M3M3M3