MANIFOLD SPARSE BEAMFORMING VOLKAN CEVHER JOINT WORK WITH

MANIFOLD SPARSE BEAMFORMING VOLKAN CEVHER JOINT WORK WITH: BARAN GÖZCÜ, AFSANEH ASAEI

OUTLINE Ø Array acquisition model Ø Spatial linear prediction Ø Minimum variance distortion-less response (MVDR) Ø Regularization Ø Manifold Sparse Beamforming Ø Atomic Norm Minimization Ø Numerical results 2 Ø Concluding remarks

ACQUISITION MODEL Sensor array acquisition forward model for input signal θs x. M x. x 2 x 1 Δ=λ/2 3 where x.

SPATIAL FILTERING Objective: Prediction of s given the observation x and through spatial linear filter θs x. M Δ x. x. x 2 x 1 W. W. W 2 W 1 The steering vectors are obtained via minimization of the WM w? expected prediction risk: 4 Σ

OPTIMUM WEIGHTS Assumptions: Signal is uncorrelated with the interferers and noise This is MVDR ! (Minimum variance distortionless (c=1) response) 5 Prediction risk minimization

DIAGONAL LOADING MVDR in practice: Sample covariance matrix: ISSUE: Array covariance matrix is rank-deficient diagonal loading This corresponds to Tikhonov regularization: 6 Unique solution !

PRIOR ART Ø H. Cox, R. M. Zeskind, and M. H. Owen, “Robust adaptive beamforming” IEEE Transactions on Acoustic, Speech, and Signal Processing, vol. 35, 1987. Ø Diagonal Loading … Ø S. A. Vorobyov, “Principles of minimum variance robust adaptive beamforming design” Signal Processing, Special Issue: Advances in Sensor Array Processing, 2013. 7 Ø Eigen-space projection Ø Worst case optimization over an uncertainty bound Ø Iterative refinement using sequential quadratic programming

A NEW REGULARIZATION: MANIFOLD SPARSITY We claim that the optimum weights accept an S-sparse linear sum of manifold vectors: A heuristic justification: Hence this set of equations has a solution when K+1≤M-K+1 i. e. when K≤M/2. 8 Therefore if then error is minimized. Plugging in the S-sparse linear combination above, we have the following linear system: If interferer angles are random, optimum weights lies in S=M-K+1 dimensional subspace.

ATOMIC NORM To enforce a sparse linear combination of manifold vectors, we introduce the following norm: where A is the infinite set of manifold vectors : 9 grid-free !

ATOMIC NORM WHEN ATOMS ARE SINUSOIDS Application to line spectral estimation: Bhaskar, Badri Narayan, and Benjamin Recht. "Atomic norm denoising with applications to line spectral estimation. " Communication, Control, and Computing (Allerton), 2011 49 th Annual Allerton Conference on. IEEE, 2011. 10 Dual problem

MANIFOLD SPARSE BEAMFORMING Proposed regularization is equivalent to where t is a real number and T is the map that makes a Hermitian Toeplitz matrix out of its input vector 11 SDP!

NUMERICAL EXPERIMENTS We would like to compare the two optimization problems: (Diagonal Loading) versus (Manifold Sparse) Number of snaphots: T=80 SIR levels= -10, 0, 10, and 20 d. B Number of sensors: M=8 SNR levels= 20 db and no noise 12 Parameters:

SIMULATION PROCEDURE FOR 8 sources ✕ 4 SIRs ✕ 2 SNRs ✕ 2500 runs q Choose random DOAs and take T snapshots on the sensor array q Select a lambda in [1, 2] for oracle tuning 1 - Find a good initial point by grid search 2 - Use fmincon to find a finer optimum lambda q Solve the two optimization problem by CVX using that lambda and evaluate the errors 13 END

NUMERICAL RESULTS 14 Error gain versus number of interferers:

CONCLUSIONS Ø Novel sparsity regularized beamforming Ø Solution obtained by semi-definite optimization of atomic norm Manifold sparsity assumption yields up to 2 -d. B gain in signal estimation over diagonal loading Ø Semidefinite formulation enables grid-free estimation Ø Further extensions Ø Ø Ø Regularization parameter selection Learning theoretic study of regularized beamforming Gridless source localization 15 Ø

THANK YOU! 16 Questions?