Fourier Methods in Array Processing

Transcription

1 Cambridge, Massachusetts! Fourier Methods in Array Processing Petros Boufounos MERL 2/8/23

2 Array Processing Problem Sources Sensors MERL 2/8/23 A number of sensors are sensing a scene. A number of sources transmit in a scene. Can we localize and reconstruct sources in the scene? This talk: overview of basic models and methods

3 (Linearized) Wave Propagation Sensor (array element) m Distance: d k,m ((((((((( Propagation delay: τ k,m =d k,m /c Source k Signal delayed according to distance and speed of wave propagation in medium y m (t) =x k (t τ k,m ) Y m (ω) =e iωτ k,m X k (ω) Superposition: Signal at receiver sum of all transmitted signals Narrowband approximation: Phase delay same for all ω Free space assumption: No secondary reflections MERL 2/8/23

4 Wave Propagation, Far-field Approximation Sensor array Source Propagating waves are circular: Same delay for same distance from source Propagating waves Far field approximation Sources located far relative to array size Propagating waves become flat (planar) Sensor array MERL 2/8/23

5 Linear Array, Far Field Approximation Sensor m Source k Delay to origin unknown: T=D/c Signal at origin: X k (ω) Position p m θ k d θ,m =p m cosθ k Signal at position p due to source k: Y p (ω) =e iω p cos θ k c X k (ω) Frequency: ω=2πf, Wavelength: λ=c/f Total signal at position p Y p (ω) = k e i2π p λ cos θ k X k (ω) MERL 2/8/23

6 Linear Array, Far Field Approximation Y p (ω) = k e i2π p λ cos θ k X k (ω) Drop ω from notation Substitute variable u k =cosθ k We get a (spatial, inverse) Fourier transform! Y p = k e i2π p λ u k X k Received Signal MERL 2/8/23 Sampling point In wavelengths Source Frequency (cosine of angle) Source Signal

7 Discretizing the (inverse) Fourier Transform Sensor m Y p = k e i2π p λ u k X k Position p m Uniform linear array: p m =mp, m=,,m Spacing p θ k d θ,m =p m cosθ k u k =cosθ k Y m = n e i2π mp λ Set u [-,] on a grid u n =-+2n/N ( + 2n K-Sparse ) N Xn = e i2π mp λ e i 2πmn N 2p λ Xn MERL 2/8/23 n

8 Discretizing the (inverse) Fourier Transform Sensor m Position p m Y m = e i2π mp λ n e i 2πmn N 2p λ Xn Set this to one. We get the DFT! Spacing p θ k d θ,m =p m cosθ k u k =cosθ k Half wavelength spacing: P m =λ/2 M=N array elements. For other spacing p, use DFT manipulations: Zero padding and aliasing (folding) Y=FX MERL 2/8/23

9 Inversion Problem MITSUBISHI ELECTRIC RESEARCH LABORATORIES! Y=FX Inversion Problem: What X generated Y? Classical approach: X = F Y = F H Y Common names: Beamforming, Backprojection, Matched Filter Main design issue: Given target at certain angle, what does the inversion look like? MERL 2/8/23

10 Inversion Problem MITSUBISHI ELECTRIC RESEARCH LABORATORIES! Y=FX Inversion Problem: What X generated Y? X = arg min X (possible) Sparse approach: Y FX 2 s.t. X K Also uses F H Y for most algorithms: coherence is important Main design issue: Given target at certain angle, what does the coherence look like? MERL 2/8/23

11 Beampattern/coherence Given target at certain angle, what does inversion look like? X = F Y = F H Y Narrow beampattern Grating lobes Wider beampattern No grating lobes Too wide beampattern No grating lobes MERL 2/8/23 p > λ/2 p = λ/2 p < λ/2 F= e j 2π N.. e j 2πm N. e j 2πM N 2p λ e j 2πn 2p N λ e j 2πN 2p N λ p λ e j 2πmn 2p N λ e j 2πmN 2p N λ.... 2p λ e j 2πMn 2p N λ e j 2πMN 2p N λ u=cosθ

12 MERL 2/8/23 MITSUBISHI ELECTRIC RESEARCH LABORATORIES! Beampattern/coherence Given target at certain angle, what does inversion look like? p > λ/2 p = λ/2 p < λ/ u=cosθ Larger aperture Narrower main lobe Large element spacing Grating lobes Narrow main lobe, no grating lobes Many array elements?

13 Random Element Spacing.8.6 p =4λ/ p =4λ/ u=cos! MERL 2/8/23 Solves the grating lobes problem!

14 Remaining Problem: Grid! All this analysis has an implied angle (frequency) grid Critically sampled Grid.5.5 On grid frequency disc. cont..5.5 Off grid frequency disc. cont..5.5 Off grid mixture disc. cont. Oversampled Grid.5.5 Normalized frequency MERL 2/8/23 disc. cont..5.5 disc. cont. Normalized frequency.5.5 disc. cont. Normalized frequency But sources are not always on the grid!!!

15 Solution (?) Make grid very fine Actual source closer to a grid point, leakage is smaller. Big problem: Computational complexity Application of F is O(NlogN) Sparse FFT could (maybe) help Bigger problem: Coherence!!! u=cos! u=cos! MERL 2/8/23

16 Solution: Previous talk Off-the grid sampling Previous talk (Yi Li) Goal: identify continuous frequency components Look ma no grid! Advantages: Very efficient No grid Nice guarantees (robustness) Did we solve the coherence problem? Partly: no leaking problem with off-grid frequencies Partly NOT: sources should be separated by O(beamwidth) MERL 2/8/23

17 Other solutions Finite Rate of Innovation (Vetterli et al.) Advantages: Computationally very efficient No robustness guarantees Not very robust in practice Newer results improving robustness (Eldar et al.) Atomic norm minimization (Recht et al.) Advantages: Optimization-based principled approach, nice guarantees Computationally very expensive Also provides reconstruction guarantees for sparse minimization on a fine grid (less expensive than atomic norm minimization!) Grid guarantees better than coherence/rip-derived ones Grid guarantees only in 2 sense (not on support estimation) MERL 2/8/23

18 Broadband Processing [w/ Smaragdis, Raj] F(ω)= e j 2π N.. e j 2πm N. e j 2πM N 2p λ e j 2πn 2p N λ e j 2πN 2p N λ p λ e j 2πmn 2p N λ e j 2πmN 2p N λ.... 2p λ e j 2πMn 2p N λ e j 2πMN 2p N λ vs. θ=-π/2 vs. θ= High resolution, Ambiguity High Frequency (Large ω) Just right d=λ/2=c/4πω Low resolution, No ambiguity Low Frequency (Small ω) Sensor location is fixed. Can we exploit bandwidth? MERL 2/8/23

19 Broadband Processing True Broadband Source ω Aliased images in other bands ω 2 ω 3 ω 4 ω 5 Joint sparsity across bands selects correct location! MERL 2/8/23

20 Localization vs. Recovery Signal recovery: Invert system on detected locations Broadband Source ω ω 2 ω 3 ω 4 ω 5 What if we have a second source? MERL 2/8/23

21 Localization vs. Recovery Signal recovery: Invert system on detected locations Broadband Source ω ω 2 We can localize the sources Can not invert in all frequencies ω 3 ω 4 ω 5 What if we have a second source? MERL 2/8/23

22 Simulation Examples MITSUBISHI ELECTRIC RESEARCH LABORATORIES! Sensor Actual Source Estimated Source SNR ~5dB Average Success Rate F=248 F=24 F= Number of sources MERL 2/8/23

23 MERL 2/8/23 MITSUBISHI ELECTRIC RESEARCH LABORATORIES! Discussion and open questions Relationship with FRI and Atomic Norm Can we identify sources closer than O(beamwidth)? There is a resolution limit (can be proven by the nullspace of F) but can we improve the constant in O(.) Related issue: remember that we are operating in u=cosθ.8 Ambiguity in θ different on sides Can we resolve that? (maybe not) 2D-versions? Off-grid joint sparsity? Aperture size Resolution limit Can it be improved with signal models? u=cos! !

24 Questions/Comments? More info: MERL 2/8/23

25 MERL 2/8/23