1. 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

2. 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)

3. 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

4. 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

5. 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

6. 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)

7. 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.

8. 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.

9. 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:

10. 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.

11. 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)

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

13. 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

14. 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.

15. 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

16. BACKUP

17. 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)

18. 1-D DOA Estimation: Stacked UCA
N = 24 (2x12) M = 12 S = 2 sources

19. Reconstruction guarantee (1-D)