Grid-FreeDirection-of-Arrival ... · Grid-Free Direction-of-Arrival Estimation with Compressed...
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Grid-Free Direction-of-ArrivalEstimation with Compressed Sensing
and Arbitrary Antenna ArraysSebastian Semper – [email protected]
Florian Roemer, Thomas Hotz, Giovanni Del Galdo
April 13, 2018
Technische Universität Ilmenau
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Direction of Arrival Estimation
Reconstruction Schemes
Main Results
Simulations
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Direction of Arrival Estimation
Reconstruction Schemes
Main Results
Simulations
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Preliminaries & Motivation
. . .θ
Ï Given an array of M antennasÏ Planar (narrowband) waves in the far
field impinging from unknown directionsÏ We take noisy measurementsÏ Goal: Estimate directions of arrival
(DOA) for each waveÏ Motivation: direction finding [1], mas-
sive MIMO and 5G [2] and radar [3]
[1] Valaee, Champagne, and Kabal, “Parametric localization of distributed sources”, 1995.[2] Chen, Yao, and Hudson, “Source localization and beamforming”, 2002.[3] Blair and Brandt-Pearce, “Monopulse DOA estimation of two unresolved Rayleigh targets”, 2001.
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State of the Art
Ï Grid-free subspace based DOA (MUSIC, ESPRIT)Ï Grid-bound Sparse Recovery based DOA [4]Ï Grid-free sparsity based line spectral estimation [5]Ï Grid-free Sparse Recovery based DOA for (randomly subselected) ULAs [6]Ï Grid-free compressed sensing based line spectral estimation [7]
[4] Malioutov, Cetin, and Willsky, “A sparse signal reconstruction perspective for source localization with sensorarrays”, 2005.[5] Bhaskar, Tang, and Recht, “Atomic Norm Denoising With Applications to Line Spectral Estimation”, 2013.[6] Xenaki and Gerstoft, “Grid-free compressive beamforming”, 2015.[7] Heckel and Soltanolkotabi, “Generalized Line Spectral Estimation via Convex Optimization”, 2017.
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Main Contributions
Ï Grid-free sparsity based DOA estimation via ANMÏ Arbitary antenna arrays via Effective Aperture Distribution Function (EADF)Ï Spatial compression with an analog combining network using generalized line
spectral estimation
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Beampatterns
0◦
30◦
60◦90◦
120◦
150◦
180◦
210◦
240◦270◦
300◦
330◦
a1a2a3a4
Figure. Beampatterns of M= 4 elements.
Ï Let a(θ) : [0,2π)→CM modelresponse of antenna arraycomprising ofM antennas for aplanar wave impinging fromazimuth angle θ
Ï Can be measured in practiceÏ Can model arbitrary, thus more
realistic antennasÏ Formal model for beam patterns?
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Fourier Series to the Rescue Effective Aperture Distribution Function
Ï Antenna patterns a(θ) are periodic in θ and smoothÏ Good approximation via truncated Fourier series element wise
am(θ)≈L−12∑
ℓ=− L−12
gm,ℓe ȷθℓ
Ï For G ∈CM×L containing the gm,ℓ and f(θ)= [e− ȷθ(L−1)/2, . . . ,e ȷθ(L−1)/2] we can
summarizea(θ)≈G · f(θ)
Ï Concise and efficient description of the whole antenna behaviour [8]Ï We observe measurements of S waves impinging with unknown amplitudes cs
x=S∑
s=1cs ·a(θs)=G ·
S∑s=1
cs · f(θs)
[8] Landmann and Galdo, “Efficient antenna description for MIMO channel modelling and estimation”, 2004.
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Spice things up with Compression
Ï Motivation: reduction ofhardware complexity
Ï We currently have G ∈CM×L and
x=G ·S∑
s=1cs · f(θs)
Ï Pick a matrix Φ ∈CK×M for someK ∈N and we get
y=Φ ·x=Φ ·G ·S∑
s=1cs · f(θs)+n
Ï Action of Φ is realized via analogcombining
Φ
Figure. Spatial compression scheme from5 antenna ports down to 3 outputs
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Direction of Arrival Estimation
Reconstruction Schemes
Main Results
Simulations
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Atomic Norm Minimization for Line Spectral Estimation
Ï For the much simpler model
z=S∑
s=1csf(θs)+n
with f(θ)= [e− ȷθ(L−1)/2, . . . ,e ȷθ(L−1)/2]
Ï Define an atomic setA = {
f(θ) ∈CM | θ ∈ [0,2π)}
Ï Define the atomic norm
x 7→ ∥x∥A = inf {t> 0 | x ∈ t ·conv(A )}
Ï Pose following convex optimization problem for some suitably chosen ϵ
minx
∥x∥A subject to ∥z−x∥2 ≤ ϵ
⇝ Atomic Norm Minimization (ANM)Slide 7 of 15Reconstruction Schemes
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The Semidefinite Reformulation
Ï Problem: How to get a solution to the atomic norm minimization?Ï Solution:
Ï Reformulate it as a semidefinite program according to [9] using duality:
min(x,u,t)∈CL×CL×R
12n
trToep(u)+ 12t
subject to(Toep(u) xxH t
)⪰ 0,
∥z−x∥2 É ϵ.
(1)
Ï Extract the frequencies, i.e. DOAs θ1, . . . ,θS, from Toep(u) using covariance basedspectral estimation methods, like MUSIC or ESPRIT.
[9] Megretski, “Positivity of trigonometric polynomials”, 2003.
Slide 8 of 15Reconstruction Schemes
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Atomic Norm Minimization for DOA
Ï Incorporate the model for compression and EADF in optimizationÏ Modify atomic norm minimization to
minx
∥x∥A subject to ∥z−Φ ·G·x∥2 ≤ ϵ
Ï Modify the semidefinite program to
min(x,u,t)∈CL×CL×R
12n
trToep(u)+ 12t
subject to(Toep(u) xxH t
)⪰ 0,
∥z−Φ ·G·x∥2 É ϵ.
(2)
Ï Still we can use Standard ESPRIT to estimate the θ1, . . . ,θSÏ Conditions on the θ1, . . . ,θS, Φ and G such that ANM has a unique solution?
Slide 9 of 15Reconstruction Schemes
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Atomic Norm Minimization for DOA How to design the combining network?
Ï Requirements on ΦÏ Good reconstruction propertiesÏ Fairly easy to implement with
phase shiftersÏ Draw Φ randomly
Definition (sub-Gaussian)A real valued random variable X iscalled sub-Gaussian with variancefactor c, if
Eexp(λX)Éλ2c2
for all λ ∈R.
Ï Examples: Gaussian, Binomial,Rademacher
Ï Conditions on the θ1, . . . ,θS, Φand G such that ANM has aunique solution with highprobability?
Ï Matrices can be sub-Gaussian aswell
Ï ⇝ Rademacher
Slide 10 of 15Reconstruction Schemes
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Direction of Arrival Estimation
Reconstruction Schemes
Main Results
Simulations
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Ï Combine results in [10] with our model and measurement setup
Theorem (S. S.)If
|θi−θj| > 1L
for all i = j, then using the Rademacher measurement setup, the matrix Φ ·G isb-sub-Gaussian with b−1 = max
1ÉℓÉL∥gℓ∥22 and Σ=GHG. Thus the ANM has a unique
solution with high probability, if the number of measurements K obeys
KÊ c ·S · log(M) · max1ÉℓÉL
∥gℓ∥22 ·σ(GHG),
where σ(GHG) denotes the condition number of GHG.
[10] Heckel and Soltanolkotabi, “Generalized Line Spectral Estimation via Convex Optimization”, 2017.
Slide 11 of 15Main Results
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Direction of Arrival Estimation
Reconstruction Schemes
Main Results
Simulations
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Simulations using genuine antenna measurements
Figure. The simulated SPUCPA setup
Ï Stacked polarimetric uniformcircular patch array (SPUCPA)with 58 ports.
Ï Used only the portscorresponding to the two stackedcircular arrays with 12 elementsper ring
Ï ANM to estimate covarianceToep(u) done via cvx using theSDPT3 solver
Ï DOA estimation based oncovariance via Standard ESPRIT
Slide 12 of 15Simulations
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Figure. Estimation error vs. SNR for S= 2 sources, M= 24 elements, L= 25 Fouriercoefficients and K= 12 measurements
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Conclusion & Future Extensions
ConclusionÏ Arbitrary AntennasÏ CompressionÏ Grid-free
OutlookÏ Estimate multidimensional
frequencies / directionsÏ Bistatic Tx / Rx setups for
channel sounding / radar
Slide 14 of 15Simulations
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Thank You!Questions?
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BACKUP SLIDES
Slide 15 of 15Simulations
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Figure. Estimation error vs. SNR for S= 3 sources, M= 24 elements, L= 25 Fouriercoefficitent and K= 15 measurements
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0 1 2 3 4 5
x/
1
2
3
4
y/
-10 -5 0 5 10 15 20 25 30
SNR [dB]
10-6
10-4
10-2
100
MS
E
Median, 25/75iles
CRB
Figure. Randomly drawn virtual (toy) array geometry (M= 29)
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5 10 15 20
Number of measurements (K)
1
2
3
4
5
6
Nu
mb
er
of
sou
rces
(S)
-30
-25
-20
-15
-10
-5
K = S
K = 2S
Figure. Estimation error (logarithmic scale) vs. K, S for the noise-free case and Rademacherdistributed compression matrices.
Slide 15 of 15Simulations
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C(θ)= σ2
2tr
([ℜ(DHΠ⊥AD⊙ (ccH)T)
]−1), (3)
where A=GF(θ), D= ȷGdiag(µ)F(θ) and Π⊥A = I−A(AHA)−1AH. Here
F(θ)= [f(θ1), . . . ,f(θS)], the vector µ=−(L−1)/2, . . . ,+(L−1)/2 and ℜ denotes the realpart of a complex number. If we incorporate compression, it changes to
C(θ)= σ2
2tr
([ℜ(DHΠ⊥
AD⊙ (ccH)T)]−1)
, (4)
where A=ΦA=ΓF(θ) and D=ΦD= ȷΓdiag(µ)F(θ).
Slide 15 of 15Simulations