Sebastian Semper1 and Florian Roemer1,2 ADMM for ND Line Spectral Estimation using Grid-Free Compressive Sensing from Multiple Measurements with Applications to DOA Estimation Sebastian Semper1 and Florian Roemer1,2 1Ilmenau University of Technology, Germany 2Fraunhofer Institute for Nondestructive Testing IZFP

Overview: Line Spectral Estimation Given a (noisy) superposition of sampled harmonics, estimate their frequencies Application: Harmonic retrieval, e.g., for Radar, Image Processing, Signal Reconstruction, Localization, Channel Sounding, DOA estimation Extensions to the classical LSE problem considered in this work Multidimensional harmonics Needed for, e.g., 2-D DOA estimation, bistatic MIMO-Radar Many methods do not generalize easily (e.g., Root-MUSIC/Prony) Generalized line spectral estimation (observe linear mixtures) Facilitates use of realistic antenna arrays (EADF) and Compressed Sensing Most simple methods fail (e.g., ESPRIT, Root-MUSIC)

Line Spectral Estimation Given a (noisy) superposition of sampled harmonics, estimate their frequencies Example DOA: 1-D uniform linear array with isotropic antenna elements where is a Vandermonde matrix

R-D Line Spectral Estimation Given a (noisy) superposition of sampled harmonics, estimate their frequencies Example DOA: 2-D uniform rectangular array with isotropic antenna elements where is a Vandermonde matrix

Generalized Line Spectral Estimation Given a (noisy) superposition of sampled harmonics, estimate their frequencies Example DOA: 1-D arbitrary array with arbitrary antenna patterns (EADF) Effective Aperture Distribution Function (EADF) Let be the array response. Naturally, is 2π-periodic. Also is smooth. Therefore

Generalized R-D Line Spectral Estimation Given a (noisy) superposition of sampled harmonics, estimate their frequencies Example DOA: 2-D arbitrary array with arbitrary antenna patterns (2-D EADF) 2-D Effective Aperture Distribution Function (EADF)

State of the art (excerpt) LSE model only Allows generalized LSE Root-MUSIC / Prony [HS17] Subspace-based Grid-based l1 Bound to 1-D [BTR13] Atomic Norm based grid-free MUSIC [MCW05] ESPRIT, RARE [CC15] (2-D) This paper: Efficient R-D generalized LSE via ADMM Allows R-D [MCW05] Malioutov, Cetin, and Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays”, 2005. [BTR13] Bhaskar, Tang, and Recht, “Atomic Norm Denoising With Applications to Line Spectral Estimation”, 2013. [CC15] Chi, Chen, “Compressive Two-Dimensional Harmonic Retrieval via Atomic Norm Minimization“, 2015 [HS17] Heckel and Soltanolkotabi, “Generalized Line Spectral Estimation via Convex Optimization”, 2017. [MCW05] Malioutov, Cetin, and Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays”, 2005.

Step 1: Atomic Norm Minimization Approach Let the atomic set be defined as Define the atomic norm of a given Then the R-D generalized LSE problem can be formulated as Convex but infinite-dimensional problem  cannot be solved directly.

Step 2: Convex reformulation Based on [Tang et. al. 2013, Candés et. al. 2013], it can be shown that … which is an SDP. However, it is huge! For R-D, computations are prohibitive. ADMM to the rescue! where is a Hermitian multi-level Toeplitz matrix e.g., two-level:

Step 3: ADMM formulation The SDP can be formulated via the augmented Lagrangian and then solved iteratively via ADMM ADMM Main difficulty: finding derivatives, especially with respect to u in light of the multi-level Toeplitz structure.

Resulting algorithm: ADMM for R-D Generalized LSE Repeat until convergence: where sums elements along positions in A corresponding to elements in u („diagonals“) projects onto the set of positive semi-definite matrices (truncate negative eigenvalues)

Numerical results: 3-D Line Spectral Estimation N = 3 x 3 x 3 = 27 (no compression) K = 100 snapshots S = 3 sources

Numerical results: 3-D Generalized Line Spectral Estimation N = 3 x 3 x 3 = 27 M = 20 (25% compression with a random Gaussian kernel) K = 100 snapshots S = 3 sources

Numerical results Example 3: 2-D DOA estimation with a stacked uniform circular array (3 stacks with 12 elements each) K = 100 snapshots, noise variance = 10-3.

Conclusions Line Spectral Estimation problem, incorporating: R-D case: Multilevel Toeplitz formulation Generalized LSE: observe linear projections Allows DOA estimation with arbitrary antenna arrays Allows to incorporate Compressed Sensing Continuous, grid-free formulation Solutions Atomic Norm Minimization  SDP reformulation ADMM as a low-complexity solution

BACKUP

Computational complexity? (N1 x 2N2 x … x 2NR) (N x N) (N x N) (N x T) (N x N) (N+T x N+T) where sums elements along positions in A corresponding to elements in u („diagonals“) projects onto the set of positive semi-definite matrices (truncate negative eigenvalues)

1-D DOA Estimation: Stacked UCA N = 24 (2x12) M = 12 S = 2 sources

Reconstruction guarantee (1-D)