sparse sensing in colocated mimo radar: a matrix...

$: Sparse Sensing in Colocated MIMO Radar: A Matrix ...meeting.xidian.edu.cn/workshop/miis2014/uploads/files/July-9_8 30a… · Y. Yu, A.P. Petropulu and R. Madan, \Compressive Sensing$
Sparse Sensing in Colocated MIMO Radar:A Matrix Completion Approach

Athina P. Petropulu

Department of Electrical and Computer EngineeringRutgers, the State University of New Jersey

AcknowledgmentsShunqiao Sun, D. Kalogerias, W. Bajwa, Rutgers UniversityOffice of Naval Research Grant ONR-N-00014-12-1-0036

July 8, 2014

A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 1 / 43

Outline

1 Motivation

2 MIMO Radars vs Phased Arrays

3 Compressive Sensing Based MIMO Radars (CS-MIMO)

4 MIMO Radars based on Matrix Completion (MC-MIMO)


Motivation

There is increasing interest in networked radars that are inexpensive andenable reliable surveillance. Unfortunately, these requirements are competingin nature.

In a networked radar the processing can be done at a fusion center, whichcollects the measurements of all receive antennas. Reliable surveillancerequires collection, communication and fusion of vast amounts of data fromvarious 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 achievesuperior resolution.

The talk presents new results on networked MIMO radars that rely onadvanced signal processing, and in particular, sparse sensing and matrixcompletion, in order to achieve an optimal tradeoff between reliability andcost (bandwidth, power).

These techniques will enable the radar to meet the same operationalobjectives with traditional MIMO radars while involving significantly fewersamples, be robust, and operate on mobile platforms.


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


MIMO Radars

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

Independent waveforms ⇒ omnidirectional beampatternCorrelated 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]


CS-MIMO radar - Signal model (1)

[Yu, Petropulu and Poor, IEEE JSTSP, 2010]

Mt transmit and Nr receive colocated nodes are employed;there are K point targets in the far field and multiple jammers;the k-th target is at azimuth angle θk , moving with constant radial speed vk

and of range dk (t);the distance between the ith TX/RX antenna and the kth target

dt/rik (t) ≈ dk (0)− vk t − ηt/r

i (θk ), where ηt/ri (θk ) = r

t/ri cos(θk − αt/r

i );the transmit nodes transmit pulses of pulse repetition interval (PRI) T .

x

k

The k-th targety

kθt/ri-α

t/rik ( )d t

The i-th TX/RX node

t/riγ


CS-MIMO Radars - Signal Model (2)

The signal reflected by the k-th target equals

yk (t) = βk

Mt∑i=1

xi (t − d tik (t)/c)e j2πf (t− dt

ik (t)

c ) (1)

The signal at the l-th receive antenna equals

zl (t) =K∑

k=1

yk (t − d rlk (t)

c) + εl (t)

=K∑

k=1

Mt∑i=1

βk xi (t − d tik (t) + d r

lk (t)

c)e j2πf (t− dt

ik (t)+drlk (t)

c )

+ εl (t) (2)

εl(t): i.i.d.Gaussian noise with zero mean and variance σ2.



The transmit waveforms, xi (t), i = 1, . . . ,Mt , are narrowband

The targets are moving slowly, i.e., it holds2vfTp

c 1.

Received baseband signal at the l-th antenna:

zl (t) ≈K∑

k=1

Mt∑i=1

βk xi (t)e j2πf (t− dtik (t)+dr

lk (t)

c ) + εl (t)

=K∑

k=1

βk e−j 2πλ 2dk (0)e j 2π

λ ηrl (θk )e j2πfk txT (t)v(θk ) + εl (t) (3)

λ: the transmitted signal wavelength

fk = 2vk f /c : the Doppler shift caused by the k-th target

v(θk ) = [e j 2πλ η

t1(θk ), ..., e j 2π

λ ηtMt

(θk )]T

x(t) = [x1(t), ..., xMt (t)]T



On letting L denote the number of snapshots and Ts the sampling period, thereceived samples collected during the m-th pulse are given by

zlm =

zl ((m − 1)T + 0Ts)...

zl ((m − 1)T + (L− 1)Ts)

=

K∑k=1

γk e j 2πλ η

rl (θk )e j2πfk (m−1)T D(fk )Xv(θk ) + elm (4)

where

γk = βk e−j 2πλ 2dk (0),D(fk ) = diag[e j2πfk 0Ts , . . . , e j2πfk (L−1)Ts ]

elm = [εl ((m − 1)T + 0Ts), . . . , εl ((m − 1)T + (L− 1)Ts)]T

X = [x(0Ts), . . . , x((L− 1)Ts)]T (L×Mt).


CS-MIMO Radars - Discretization of Angle-Speed Plane

1γ

2γ

3γ4γ( n1 n1)

( n2 n2)

( n3 n3)

( n4 n4)

Let us discretize the angle-Doppler plane on a fine grid as:

a = [(a1, b1), . . . , (aN , bN )] (5)


CS-MIMO Radars - Sparse Signal Recovery Formulation

Rewrite as

zlm =N∑

n=1

sne j 2πλ η

rl (an)e j2πbn(m−1)T D(bn)Xv(an) + elm (6)

where

sn =

γk , if the k-th target is at (an, bn)

0, otherwise.

In matrix form we have

zlm = Ψlms + elm (7)

wheres = [s1, . . . , sN ]T

Ψlm = [e j 2πλ η

rl (a1)e j2πb1(m−1)T D(b1)Xv(a1),

. . . , e j 2πλ η

rl (aN )e j2πbN (m−1)T D(bN )Xv(aN )]

The number of targets is small

⇒ the positions of targets are sparse in the angle-Doppler plane⇒ s is a sparse vector.


CS-MIMO Radars - Implementation

The lth node obtains measurements as:

rlm = Φlmzlm = ΦlmΨlms + elm, (8)

Nr receive nodes forward the compressed measurements collected during Np

pulses to a fusion center.The fusion center combines all the received data as

r = [rT11, . . . , r

T1Np, . . . , rT

Nr 1, . . . , rTNr Np

]T = Θs + E (9)

s is recovered by applying the Dantzig selector to the above convex problem[Candes and Tao 2007]

The lth receiveantenna

1 ( )l

t( )( )lmy t

lmr (1)

lmr (M)lmr (1) lmr (M)

is the continuous signal formed by the i-th row of of length L+Ll

( )l

M t( )

( )l

i t( )A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 12 / 43

CS-MIMO Radars - References

Y. Yu, A.P. Petropulu and R. Madan, “Compressive Sensing for MIMO UrbanRadar,” Chapter 12 in Compressive Sensing for Urban Radars, Edited by MoenessAmin, CRC Press, 2014.

Y. Yu, S. Sun, A. Petropulu and R. Madan, “Power Allocation and WaveformDesign for the Compressive Sensing based MIMO Radar,” IEEE Trans. onAerospace and Electronic Systems, to appear in 2014.

B. Li and A. Petropulu, “Performance Guarantees for distributed MIMO Radarbased on sparse sensing,” accepted by IEEE Radar Conference, Cincinnati OH, May19- 23, 2014.

Y. Yu, A. Petropulu and H.V. Poor, “CSSF MIMO RADAR: Low-ComplexityCompressive Sensing Based MIMO Radar That Uses Step Frequency,” IEEE Trans.on Aerospace and Electronic Systems, Vol. 48, Issue 2, pp. 1490 - 1504, 2012.

Y. Yu, A. Petropulu and H.V. Poor, “Measurement Matrix Design for CompressiveSensing-Based MIMO Radar,” IEEE Trans. on Signal Processing, Vol. 59, Issue 11,Page(s): 5338 - 5352, 2011. Also availableathttp://arxiv.org/PS_cache/arxiv/pdf/1102/1102.5079v1.pdf

Y. Yu, A. Petropulu and H.V. Poor, “Measurement Matrix Design for CompressiveSensing-Based MIMO Radar,” IEEE Trans. on Signal Processing, Vol. 59, Issue 11,Page(s): 5338 - 5352, 2011. Also availableathttp://arxiv.org/PS_cache/arxiv/pdf/1102/1102.5079v1.pdfA. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 13 / 43

http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.5079v1.pdf

http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.5079v1.pdf

MIMO Radars based on Matrix Completion (MC-MIMO) -Motivation

CS-MIMO Radars can achieve the same resolution as MIMOradars but with significantly fewer samples, or better resolutionwith the same number of samples.

CS-MIMO Radars need grid discretization.

We next present MC-MIMO Radars, which maintain theadvantages of CS-MIMO Radars but do not rely on griddiscretization.


Matrix Completion

Recover a rank r matrix M ∈ CN1×N2 based on partial knowledge of its entries.

Define Y = PΩ(M) as

[Y]ij =

[M]ij , (i , j) ∈ Ω;0, otherwise.

where Ω is the set of indices of observed entries with cardinality m.

Under certain conditions, M can be recovered by solving the followingoptimization problem:

min ‖X‖∗s.t. PΩ(X) = PΩ(M) (10)

Nuclear norm minimization is the tightest convex relaxation of the NP-hard rankminimization problem.

[Candes & Recht, 2009], [Candes & Tao, 2010], [Candes & Plan, 2010]A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 15 / 43

Matrix Completion - Coherence and Recoverability (1)

Definition

(Subspace Coherence) Let U ≡ Cr ⊆ CN be a subspace spanned by the set oforthonormal vectors

ui ∈ CN×1

i∈N+

r. Also, define the matrix

U , [u1 u2 . . . ur ] ∈ CN×r and let PU , UUH ∈ CN×N be the orthogonalprojection onto U. Then, the coherence of U with respect to the standard basiseii∈N+

Nis defined as

µ (U) ,N

rsupi∈N+

N

‖PUei‖22 ∈

[1,

N

r

]. (11)

Concerning the recoverability of M, the following assumptions regarding thesubspaces U and V are of particular importance [?].

A0 max µ (U) , µ (V ) ≤ µ0 ∈ R++.

A1∥∥∥∑i∈N+

rui vH

i

∥∥∥∞≤ µ1

√r

N1N2, µ1 ∈ R++.



(Exact MC) Let M ∈ CN1×N2 be a matrix of bounded rank r obeying theincoherence property with bounded parameter µ0 and set N , max N1,N2.Suppose we observe m entries of M with locations sampled uniformly at random.Then, there exist positive numerical constants C1 and C2 such that if

m ≥ C1µ40Nr 4 log2 N = O

(N log2 N

)or (12)

m ≥ C2µ20Nr 2 log6 N = O

(N log6 N

), (13)

the minimizer to the program (10) is unique and equal to M with probability atleast 1− N−3.



An N1 ×N2 matrix of rank r depends upon N1N2 − (N1 − r)(N2 − r) degreesof freedom. When r is small the number of d.f. is smaller than N1N2, andthis is why subsampling is possible.

In compressed sensing, the number of d.f. corresponds to the sparsity of thesignal, (i.e., number of non-zero entries)

It is remarkable that exact recovery occurs as soon as the sample size exceedsthe number of d.f. by a couple of logarithmic factors

If the sampling misses one column or one row the one cannot hope to recoverthe matrix, even if the matrix has rank one.

One needs to sample at random. It is established that one needs O(N log N)samples for this to happen (N = max(N1,N2)).


Matrix Completion

Recover a rank r matrix M ∈ CN1×N2 based on partial knowledge of its entries.Let Y = PΩ (M + E) as

[Y]ij =

[M]ij + [E]ij , (i , j) ∈ Ω

0, otherwise(14)

where E is noise, and Ω is the set of indices of observed entries with cardinality m.Under certain conditions, M can be recovered by solving the followingoptimization problem:

min ‖X‖∗s.t. ‖PΩ (X− Y)‖F ≤ δ. (15)

Assuming that the noise is zero-mean, white, δ is related to the noise variance, σ2,as δ2 = (m +

√8m)σ2.


Recovery Error in Noisy Observation

When observations are corrupted with white zero-mean Gaussian noise withvariance σ2, the recovery error is bounded as

∥∥∥M− M∥∥∥

F≤ 4

√1

p(2 + p) min (n1, n2)δ + 2δ, (16)

where p = mN1N2

is the fraction of observed entries, and δ2 = (m +√

8m)σ2.


Generic Assumptions for Colocated MIMO Radar

Transmission antennas transmit narrowband and orthogonal waveforms,that is,

1

Tp c

λ, (17)

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 ∈ Ci∈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 supportof each waveform Tp.

The Doppler spread of the received signals is much smaller than thebandwidth of the pulse, that is,

2ϑi

λ 1

Tp, ∀i ∈ N+

K (18)

where ϑi ∈ R denotes the speed of the respective target.


MIMO Radar with Matric Completion (MC-MIMO)

orthogonal transmit waveforms,K targets

Matched filterbank

Fusion center

Matched filterbank

rdReceivers

Matched filterbank

Y=

1 rM1rM

1

tM 1tM 1

tM

tM1

1

rM


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 atthe fusion center can be expressed as

Y , ∆ + Z ∈ CMr×Mt , (19)

where Z is an interference/observation noise matrix that may also describe modelmismatch due to weak correlations among the transmit waveforms and

∆ , Xr DXTt , (20)

where Xr ∈ CMr×K (respectively for Xt ∈ CMt×K ) constitutes an alternant matrixdefined as

Xr ,

γ0

0 γ01 · · · γ0

K−1

γ10 γ1

1 · · · γ1K−1

......

. . ....

γMr−10 γMr−1

1 · · · γMr−1K−1

∈ CMr×K , (21)

...A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 23 / 43

MC-MIMO Radar (3)

...with

γ lk , e j2πrT

r (l)T (θk ), (l , k) ∈ NMr−1 × NK−1 (22)

rr (l) ,1

λ[x r

l y rl ]T ∈ R2×1, l ∈ NMr−1 and (23)

T (θk ) ,

[cos (θk )sin (θk )

]∈ R2×1, k ∈ NK−1. (24)

The sets

[x rl y r

l ]T

l∈NMr−1

and θkk∈NK−1contain 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 thetarget reflection properties and the speeds.

For the simplest ULA case,[x

r(t)l y

r(t)l

]T

≡[0 ldr(t)

]T, l ∈ NMr(t)−1. (25)

and Xr and Xt degenerate to Vandermonde matrices.A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 24 / 43

MC-MIMO Radar: Sampling Scheme I

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

Random

Switch Unit

! !*

1lx d ! !"#

$ % $ %*

2lx d ! !"#

$ %lx

LPF

!,1x l

!, 2x l

!,t

x l M

max

max

max

! !*

tl Mx d ! !"#

Simple power saving Bernoulli switching with selection probability p.

RMFBs are implemented in receiver, constructing the Bernoulli subsampledversion 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.



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

Matched filterbank

Fusion center

Matched filterbank

rdReceivers

Matched filterbank

1 rM1rM

1

tM 1 tM 1 tM

Y=

tM1

1

rM



Coherence bounds - Simulations results

0 50 100 150 200 250 300 350 40010

0

100.1

100.2

100.3

Mr

max

(µ(U

),µ(

V))

M

t=10

Mt=40

Mt=80

0 10 20 30 40 50 60 70 80 9010

0

100.1

100.2

100.3

∆θ

max

(µ(U

),µ(

V))

M

r=40, M

t=10

Mr=40, M

t=40

Mr=10, M

t=40

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

of number of transmit and receive antennas, and for ∆θ = 5; (b) the averagemax (µ (U) , µ (V )) of ZMF

q as function of DOA separation.



DOA resolution - Simulations results

Scheme I: DOA resolution. The parameter are set as Mr = Mt = 20, p1 = 0.5 andSNR = 10, 25dB.



MC estimation error - Simulations results

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

m/df

Rel

ativ

e R

ecov

ery

Err

ors

Reciprocal of SNR

Relative recovery error with Δθ=0°



Scheme I, K = 2 targets: the relative recovery error for ZMFq under different

values of DOA separation. Mr = Mt = 40.A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 29 / 43

MC-MIMO Radar: Sampling Scheme II (1)

[Sun, Bajwa, Petropulu, IEEE AES 2014, IEEE ICASSP 2013], [Sun, Petropulu,ASILOMAR, 2014]

rd

rd

Receivers

Matched

filterbank

Fusion center

Suppose that the Nyquist rate samples of the signals at the RX nodescorrespond to sampling times ti = iTs , i ∈ NN−1 with N = Tp/Ts .Let the lth receive antenna sample at times τ l

j = jTs , j ∈ J l , where J l isthe output of a random number generator, containing L2 integers in theinterval [0,N − 1].The lth receive antenna forwards the L2 samples to the fusion center.A. P. Petropulu (Rutgers) Sparse Sensing in Colocated MIMO Radar: A Matrix Completion Approach July 8, 2014 30 / 43


In this case, the fully observed version of the data matrix is of the form

∆ , Xr DXTt S, (26)

where S ∈ CMt×N contains as its rows the waveform samples of thetransmission antennas.

Assuming that N ≥ Mt ≥ K , ∆ will be a rank-K matrix.

As before, 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.

However, in this case we have to make use of the assumed discreteorthogonality of S,

SSH ≡ I. (27)



MC estimation error and probability of target resolution - Simulations results

Consider ULAs with Mt = 20, Mr = 40.

The SNR is set to 25dB.

0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

p

Val

ues

Reciprocal of SNRHadamardG−Orth

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Minimum DOA separation (degree)

Pro

babi

lity

of r

esol

utio

n

Hadamard p=0.3Hadamard p=0.5G−Orth p=0.3G−Orth p=0.5


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)

Nand s2 ,

tr(M2)

N− τ 2. (28)

Then, it is true that

τ − s√

N − 1 ≤ λmin (M) ≤ τ − s√N − 1

and (29)

τ +s√

N − 1≤ λmax (M) ≤ τ + s

√N − 1. (30)

Further, equality holds on the left (right) of (29) if and only if equality holds onthe left (right) of (30) if and only if the N − 1 largest (smallest) eigenvalues areequal.


Recoverability & Performance Guarantees: Scheme I (2)

We can show that if ∆ = Xr DXTt ∈ CMr×Mt ,

µ (U) ≤ Mr

λmin (XHr Xr )

and µ (V ) ≤ Mt

λmin

(XH

t Xt

) .For a ULA, the elements of XH

t Xt (respectively for XHr Xr ) are of the form

δi,j ,Mt−1∑m=0

e j2πm(αti−α

tj ), ∀ (i , j) ∈ NK−1 × NK−1. (31)

The trace of XHt Xt is MtK .



tr((XH

t Xt)2)

=K−1∑k1=0

M2t +

K−1∑k2=0k1 6=k2

sin2(πMt

(αt

k1− αt

k2

))sin2

(π(αt

k1− αt

k2

))

≡K−1∑k1=0

M2t +

K−1∑k2=0k1 6=k2

φ2Mt

(αt

k1− αt

k2

) ≤K−1∑k1=0

M2t + (K − 1) sup

x∈[ξt ,12 ]φ2

Mt(x)

, KM2

t + K (K − 1)βξt (Mt) .

αrk ,

dr sin (θk )

λ

ξt : smallest αti − αt

j folded in[0,1

2]

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

5

10

15

20

25

30

35

φ2 M(x)

x

M=4M=5M=6



Theorem(Brief Version) (Coherence for ULAs) Consider a Uniform Linear Array (ULA) transmission - reception pairand assume that the set of target angles θkk∈NK−1

consists of almost surely distinct members. Then, for

any fixed Mt and Mr , as long as

K ≤ mini∈t,r

Mi√βξi

(Mi )

, (32)

the associated matrix ∆ obeys the assumptions A0 and A1 with

µ0 , maxi∈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 6= j .



Theorem

(Cont.) Further, if ξ , min ξr , ξt 6= 0, then, for any fixed K, as long as

mini∈t,r

Mi ≥ K√βξ ≡

K

sin(πξ)= O (K) , (33)

both βξt (Mt) and βξr (Mr ) can be replaced by the constant βξ (that is, independent ofboth Mt and Mr ). Additionally, in the limit with respect to Mt and Mr , we have

µ (V ) ≡ µ (U) ≡ 1, (34)

that is, the coherence of ∆ is asymptotically optimal.

In other words, under very reasonable assumptions, the coherence of ∆ isboth asymptotically and approximately optimal with respect to thenumber of transmission and reception antennas.



In colocated MIMO Radar systems, due to the need for unambiguous angle estimation (targetdetection), it is very common to assume that θi ∈ [−π/2, π/2] , ∀ i ∈ NK−1.

Also, especially for the case where ULAs are employed for transmission and reception, another commonassumption is to choose dr ≡ dt = λ/2.

Lemma(ULA Pairs Coherence Control) Consider the hypotheses and definitions of Theorem 2 and setdr ≡ dt = λ/2. If additionally

(θi , θj ) ∈ A, with (35)

A ,

(x, y) ∈[−π

2,π

2

]2∣∣∣∣ η ≤ |y − x| ≤ π − η

(36)

∀ (i, j) ∈ NK−1 × NK−1 with i 6= j and for some apriori chosen η ∈(

0,π

2

], then

ξ = 1− cos

(η

2

)∈(

0,2−√

2

2

](37)

and the asymptotics of Theorem 2 always hold true. In particular, the higher the value of η, the higher thevalue of ξ and the lower the coherence of ∆.


General 2D Array Case - Scheme I (1)

Theorem(Coherence for Arbitrary 2D Arrays) Let the abstract sets T and R contain all theessential information regarding the transmitter and receiver array topologies, and assumethat the target angles are distinct. Then, for any Mt and Mr , as long as

K ≤ mini∈t,r

Mi√βi

, (38)

the associated matrix ∆ obeys the assumptions A0 and A1 with

µ0 , maxi∈t,r

Mi

Mi − (K − 1)√βi

and µ1 , µ0

√K . (39)

with probability 1 and where

βt(r) , sup(x,y)∈A

∣∣ϕt(r) (x , y |T (R) )∣∣2 ∈ [0,M2

t(r)

), (40)

with

ϕt(r) (x , y |T (R) ) ,

Mt(r)−1∑m=0

exp(

j2πrTt(r) (m) (T (x)− T (y))

), (41)


Recoverability & Performance Guarantees:Scheme I - UCA Case Example (1)

Consider a MIMO Radar system equipped with identical Uniform CircularArrays (UCAs) with Mr = Mt = 20 , M antennas.

Their positions in the 2-dimensional plane are defined as

[x

t(r)l

yt(r)l

], R

cos

(2π (l − 1)

M

)sin

(2π (l − 1)

M

) , (42)

l ∈ NM−1, where R = 0.5 m.

The wavelength utilized for the communication is chosen as λ = 0.5 m.


Recoverability & Performance Guarantees:Scheme I - UCA Case Example (2)

y

x

UCAs: Mr =Mt = 20, R = 0.5, λ = 0.5

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

50

100

150

200

250

300

350

400

yx

ULAs: Mr =Mt = 20, dt/λ = 1/2

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

50

100

150

200

250

300

350

400

The function∣∣ϕt(r) (x , y |T (R) )

∣∣2 of Example 1 with respect to (x , y) ∈ [−π, π]2

for the case of (a) a symmetric UCA pair and (b) a symmetric ULA pair.


Conclusions

We have investigated the problem of reducing the volume of data typicallyrequired for accurate target detection and estimation in colocated MIMOradars.

We have presented two sparse sensing schemes (Schemes I & II) forinformation acquisition, leading to the natural formulation of a low rankmatrix completion problem, which can be efficiently solved using convexoptimization.

Numerical simulations have justified the effectiveness of our approach.

Specifically for Scheme I, we have presented theoretical results, guaranteeingnear optimal performance of the respective matrix completion problem, forthe case where ULAs are employed for transmission and reception.


Relevant Publications

D. Kalogerias and A. Petropulu, “Matrix Completion in Colocated MIMORadar: Recoverability, Bounds & Theoretical Guarantees,” IEEE Transactionson Signal Processing, Volume 62, Issue: 2, Page(s): 309 - 321, 2014.

S. Sun, W. Bajwa and A. Petropulu, “MIMO-MC Radar: A MIMO RadarApproach Based on Matrix Completion,” IEEE Trans. on Aerospace andElectronic Systems, under review in 2014.

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


sparse sensing in colocated mimo radar: a matrix...

Documents