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Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach

Athina P. Petropulu

Department of Electrical and Computer Engineering Rutgers, the State University of New Jersey Acknowledgments Shunqiao Sun, D. Kalogerias, W. Bajwa, Rutgers University Office of Naval Research Grant ONR-N-00014-12-1-0036 National Science Foundation ECCS 1408437

December 11, 2014

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 1 / 22

Motivation

There is increasing interest in networked radars that are inexpensive and enable reliable surveillance. Unfortunately, these requirements are competing in nature. In a networked radar the processing can be done at a fusion center, which collects the measurements of all receive antennas. Reliable surveillance requires collection, communication and fusion of vast amounts of data from various antennas, which is a bandwidth and power intensive task. The communication with the fusion center could occur via a wireless link (radar on a wireless sensor network). MIMO radars have received considerable recent attention as they can achieve superior resolution. The talk presents new results on networked MIMO radars that rely on advanced signal processing, and in particular, sparse sensing and matrix completion, in order to achieve an optimal tradeoff between reliability and cost (bandwidth, power). These techniques will enable the radar to meet the same operational

• bjectives with traditional MIMO radars while involving significantly fewer

samples, be robust, and operate on mobile platforms.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 2 / 22

Phased Array Radars

A phased array radar is composed of many closely spaced antennas all antennas transmit the same waveform is capable of cohering and steering the transmit energy

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 3 / 22

MIMO Radars

Multiple input multiple output (MIMO) radar employs colocated TX/RX antennas or widely separated TX/RX antennas; uses multiple waveforms:

Independent waveforms ⇒ omnidirectional beampattern Correlated waveforms ⇒ desired beampattern

[Xu et. al. 2006, Li et. al. 2007, Li et. al. 2008] [Fisher et. al. 2004, Lehmann et. al. 2006, Haimovich et. al. 2008]

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 4 / 22

Matrix Completion

[Candes & Recht, 2009],[Candes & Tao, 2010],[Candes & Plan, 2010]

Matrix completion is done by solving a relaxed nuclear norm

• ptimization problem

min X∗ s.t. PΩ (X) = PΩ (M) (1) where where Ω is the set of indices of observed entries with cardinality m, and the observation operation is defined as [Y]ij = [M]ij, (i, j) ∈ Ω 0,

• therwise

(2)

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 5 / 22

Matrix Completion

[Candes & Recht, 2009],[Candes & Tao, 2010],[Candes & Plan, 2010]

Matrix completion is done by solving a relaxed nuclear norm

• ptimization problem

min X∗ s.t. PΩ (X) = PΩ (M) (1) where where Ω is the set of indices of observed entries with cardinality m, and the observation operation is defined as [Y]ij = [M]ij, (i, j) ∈ Ω 0,

• therwise

(2) For noisy observations: [Y]ij = [M]ij + [E]ij, (i, j) ∈ Ω min X∗ s.t. PΩ (X − Y)F ≤ δ, (3)

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 5 / 22

Matrix Coherence and Recovery Guarantee

Definition

Let U be a subspace of Cn1 of dimension r that is spanned by the set of orthogonal vectors {ui ∈ Cn1}i=1,...,r, PU the orthogonal projection onto U, i.e., PU =

• 1≤i≤r

uiuH

i ,

and ei the standard basis vector whose i-th element is 1. The coherence of U is defined as µ (U) = n1 r max

1≤i≤n1 PUei2

∈

• 1, n1

r

• ≡ n1

r max

1≤i≤n1

• U(i)
• 2

; U = [u1, ..., ur] (4) U(i): i − th row of U Consider the compact SVD of M, i.e., M =

r

• k=1

ρkukvH

k = UΛVH

Matrix M has coherence with parameters µ0 and µ1 if (A0) max (µ (U) , µ (V )) ≤ µ0 for some positive µ0. (A1) The maximum element of

• 1≤i≤r

uivH

i is bounded by µ1

• r/(n1n2) in absolute

value.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 6 / 22

Matrix Coherence and Recovery Guarantee, Continued

Suppose that matrix M ∈ Cn1×n2 satisfies (A0) and (A1). The following lemma gives a probabilistic bound for the number of entries, m, needed to estimate M.

Theorem

[Cand` es & Recht 2009] Suppose that we observe m entries of the rank−r matrix M ∈ Cn1×n2, with matrix coordinates sampled uniformly at random. Let n = max{n1, n2}. There exist constants C and c such that if m ≥ C max

• µ2

1, µ1/2

µ1, µ0n1/4 nrβ log n for some β > 2, the minimizer of the nuclear norm problem is unique and equal to M with probability at least 1 − cn−β. For r ≤ µ−1

0 n1/5 the bound can be improved to

m ≥ Cµ0n6/5rβ log n, without affecting the probability of success.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 7 / 22

Generic Assumptions for Colocated MIMO Radar

Transmission antennas transmit narrowband and orthogonal waveforms, that is, 1 Tp ≪ c λ, (5) where Tp ∈ R, λ ∈ R and c ≡ 3 · 108 m/s denotes the waveform duration, the communication wavelength and the speed of light, respectively. The target reflection coefficients {βi ∈ C}i∈N+

K (K is the number of targets in

the far field) remain constant during a number of pulses Q. The delay spread in the received signals is smaller that the temporal support

• f each waveform Tp.

The Doppler spread of the received signals is much smaller than the bandwidth of the pulse, that is, 2ϑi λ ≪ 1 Tp , ∀i ∈ N+

K

(6) where ϑi ∈ R denotes the speed of the respective target.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 8 / 22

MIMO Radar with Matric Completion (MC-MIMO)

• rthogonal transmit waveforms,

K targets

Matched filterbank Fusion center



Matched filterbank



r

d

Receivers Matched filterbank



Y= 1

r

M 1

r

M  1



t

M



1

t

M



1

t

M



t

M 1 

  

1

r

M

  



• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 9 / 22

MC-MIMO Radar (2)

[Kalogerias, Petropulu, IEEE TSP 2014, GLOBALSIP 2013], [Sun, Bajwa, Petropulu, IEEE AES 2014, IEEE ICASSP 2013] It can be shown that the fully observed version of the data matrix formulated at the fusion center can be expressed as Y ∆ + Z ∈ CMr ×Mt, (7) where Z is an interference/observation noise matrix that may also describe model mismatch due to weak correlations among the transmit waveforms and ∆ XrDX❚

t ,

(8) where Xr ∈ CMr×K (respectively for Xt ∈ CMt×K) constitutes an alternant matrix defined as Xr      γ0 γ0

1

· · · γ0

K−1

γ1 γ1

1

· · · γ1

K−1

. . . . . . ... . . . γMr −1 γMr −1

1

· · · γMr −1

K−1

     ∈ CMr ×K, (9) ...

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 10 / 22

MC-MIMO Radar (3)

...with γl

k ej2πr❚

r (l)T (θk),

(l, k) ∈ NMr −1 × NK−1 (10) rr (l) 1 λ [xr

l y r l ]❚ ∈ R2×1,

l ∈ NMr −1 and (11) T (θk) cos (θk) sin (θk)

• ∈ R2×1,

k ∈ NK−1. (12) The sets

• [xr

l y r l ]❚ l∈NMr −1

and {θk}k∈NK−1 contain the 2-dimensional antenna coordinates of the reception array and the target angles, respectively, λ ∈ R++ denotes the carrier wavelength, and D ∈ CK×K is a non-zero diagonal matrix whose elements depend on the target reflection properties and the speeds. For the simplest ULA case,

• xr(t)

l

y r(t)

l

❚ ≡

• 0 ldr(t)

❚ , l ∈ NMr(t)−1. (13) and Xr and Xt degenerate to Vandermonde matrices.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 11 / 22

MC-MIMO Radar: Sampling Scheme I

Sparse Sensing is implemented through the following Random Matched Filter Bank (RMFB) architecture.

Random Switch Unit

! !

* 1 l

x d ! ! "

#

$ % $ %

* 2 l

x d ! ! "

# $ %

l

x

LPF

!

,1 x l

!

,2 x l

!

,

t

x l M max max max ! !

*

t

l M

x d ! ! "

#

Simple power saving Bernoulli switching with selection probability p. RMFBs are implemented in receiver, constructing the Bernoulli subsampled version of ∆, P (∆). Matrix Completion is applied for the stable recovery of ∆. The recovered matrix is fed into standard array processing methods (e.g. MUSIC) for extracting target information.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 12 / 22

MC-MIMO Radar: Sampling Scheme I

[Kalogerias, Petropulu, IEEE TSP 2014, GLOBALSIP 2013], [Sun, Bajwa, Petropulu, IEEE AES 2014, IEEE ICASSP 2013]

Matched filterbank Fusion center



Matched filterbank



r

d

Receivers Matched filterbank



1

r

M 1

r

M  1



t

M



1

t

M



1

t

M



Y=

t

M 1  1

r

M

  

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 13 / 22

MC-MIMO Radar: Sampling Scheme I

Coherence bounds - Simulations results

50 100 150 200 250 300 350 400 10 10

0.1

10

0.2

10

0.3

Mr max(µ(U),µ(V)) Mt=10 Mt=40 Mt=80 10 20 30 40 50 60 70 80 90 10 10

0.1

10

0.2

10

0.3

∆θ max(µ(U),µ(V)) Mr=40, Mt=10 Mr=40, Mt=40 Mr=10, Mt=40

Scheme I, K = 2 targets: (a) the average max (µ (U) , µ (V )) of ZMF

q

as function

• f number of transmit and receive antennas, and for ∆θ = 5◦; (b) the average

max (µ (U) , µ (V )) of ZMF

q

as function of DOA separation.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 14 / 22

MC-MIMO Radar: Sampling Scheme I

DOA resolution - Simulations results Scheme I: DOA resolution. The parameter are set as Mr = Mt = 20, p1 = 0.5 and SNR = 10, 25dB.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 15 / 22

Recoverability & Performance Guarantees: Scheme I [Kalogerias & Petropulu, 2013, 2014]

Useful results:

Theorem

[Wolkowicz 1980] Let M ∈ CN×N be a matrix with real eigenvalues. Define τ tr (M) N and s2 tr

• M2

N − τ 2. (14) Then, it is true that τ − s √ N − 1 ≤ λmin (M) ≤ τ − s √ N − 1 and (15) τ + s √ N − 1 ≤ λmax (M) ≤ τ + s √ N − 1. (16) Further, equality holds on the left (right) of (15) if and only if equality holds on the left (right) of (16) if and only if the N − 1 largest (smallest) eigenvalues are equal.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 16 / 22

Recoverability & Performance Guarantees: Scheme I (2)

We can show that if ∆ = XrDX❚

t ∈ CMr ×Mt,

µ (U) ≤ Mr λmin (XH

r Xr)

and µ (V ) ≤ Mt λmin

• XH

t Xt

. For a ULA, the elements of XH

t Xt (respectively for XH r Xr) are of the form

δi,j

Mt−1

• m=0

ej2πm(αt

i −αt j), ∀ (i, j) ∈ NK−1 × NK−1.

(17) The trace of XH

t Xt is MtK.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 17 / 22

Recoverability & Performance Guarantees: Scheme I (3)

tr

• (XH

t Xt)2

=

K−1

• k1=0

       M2

t + K−1

• k2=0

k1=k2

sin2 πMt

• αt

k1 − αt k2

• sin2

π

• αt

k1 − αt k2

• 

      ≡

K−1

• k1=0

       M2

t + K−1

• k2=0

k1=k2

φ2

Mt

• αt

k1 − αt k2

• 

      ≤

K−1

• k1=0

  M2

t + (K − 1)

sup

x∈[ξt, 1

2]

φ2

Mt (x)

   KM2

t + K (K − 1) βξt (Mt) .

αr

k dr sin (θk)

λ ξt : smallest αt

i − αt j folded in[0, 1

2]

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 5 10 15 20 25 30 35

φ2

M(x)

x M=4 M=5 M=6

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 18 / 22

Recoverability & Performance Guarantees: Scheme I (4)

Theorem

(Brief Version) (Coherence for ULAs) Consider a Uniform Linear Array (ULA) transmission - reception pair and assume that the set of target angles {θk}k∈NK−1 consists of almost surely distinct members. Then, for any fixed Mt and Mr , as long as K ≤ min

i∈{t,r}

     Mi

• βξi (Mi)

     , (18) the associated matrix ∆ obeys the assumptions A0 and A1 with µ0 max

i∈{t,r}

     Mi Mi − (K − 1)

• βξi (Mi)

     and µ1 µ0 √ K. In the above

• βξi (Mi) denotes a constant dependent on Mi. ξi, i ∈ {t, r}, mostly depends on the pairwise

differences |sin (θi) − sin (θj)| , (i, j) ∈ NK−1 × NK−1, i = j.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 19 / 22

Conclusions

We have investigated the problem of reducing the volume of data typically required for accurate target detection and estimation in colocated MIMO radars. We have presented a sparse sensing scheme for information acquisition, leading to the natural formulation of a low rank matrix completion problem, which can be efficiently solved using convex optimization. Numerical simulations have justified the effectiveness of our approach. We have presented theoretical results, guaranteeing near optimal performance

• f the respective matrix completion problem, for the case where ULAs are

employed for transmission and reception.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 20 / 22

Relevant Publications

• D. Kalogerias and A. Petropulu, “Matrix Completion in Colocated MIMO

Radar: Recoverability, Bounds & Theoretical Guarantees,” IEEE Transactions

• n Signal Processing, Volume 62, Issue: 2, Page(s): 309 - 321, 2014.
• S. Sun, W. Bajwa and A. Petropulu, “MIMO-MC Radar: A MIMO Radar

Approach Based on Matrix Completion,” IEEE Trans. on Aerospace and Electronic Systems, under review in 2014.

• S. Sun, A. P. Petropulu and W. U. Bajwa, “High-resolution networked MIMO

radar based on sub-Nyquist observations,” Signal Processing with Adaptive Sparse Structured Representations Workshop (SPARSE), EPFL, Lausanne, Switzerland, July 8-11, 2013. item S. Sun, A. P. Petropulu and W. U. Bajwa, “Target estimation in colocated MIMO radar via matrix completion,” in Proc. of International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, Canada, May 26-31, 2013.

• A. P. Petropulu (Rutgers)

Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach December 11, 2014 21 / 22