1

Frequency Diverse Coprime Arrays with CoprimeFrequency Offsets for Multi-Target Localization

Si Qin, Student Member, IEEE, Yimin D. Zhang, Senior Member, IEEE, Moeness G. Amin, Fellow, IEEE,and Fulvio Gini, Fellow, IEEE

Abstract—Different from conventional phased-array radars,the frequency diverse array (FDA) radar offers a range-dependent beampattern capability that is attractive in various ap-plications. The spatial and range resolutions of an FDA radar arefundamentally limited by the array geometry and the frequencyoffset. In this paper, we overcome this limitation by introducinga novel sparsity-based multi-target localization approach incor-porating both coprime arrays and coprime frequency offsets.The covariance matrix of the received signals corresponding toall sensors and employed frequencies is formulated to generate aspace-frequency virtual difference coarrays. By using O(M+N)antennas and O(M + N) frequencies, the proposed coprimearrays with coprime frequency offsets enables the localizationof up to O(M2N2) targets with a resolution of O(1/(MN)) inangle and range domains, where M and N are coprime integers.The joint direction-of-arrival (DOA) and range estimation is castas a two-dimensional sparse reconstruction problem and is solvedwithin the Bayesian compressive sensing framework. We alsodevelop a fast algorithm with a lower computational complexitybased on the multitask Bayesian compressive sensing approach.Simulations results demonstrate the superiority of the proposedapproach in terms of DOA-range resolution, localization accu-racy, and the number of resolvable targets.

Index Terms—Target localization, frequency diverse arrayradar, coprime array, coprime frequency offset, Bayesian com-pressvie sensing

I. INTRODUCTION

Target localization finds a variety of applications in radar,sonar, communications, and navigation [2]–[5]. The phased ar-ray radars are known for their capability to electronically steera beam for target detection and tracking in the angular domain[6]–[9]. To localize targets in both angle and range, beam-steering should be achieved across a signal bandwidth. Thisgenerally leads to a complicated waveform design and signalprocessing algorithms. Recently, the frequency diverse array(FDA) framework was introduced as an attractive multiple-input multiple-output (MIMO) structure that performs beamsteering over a signal bandwidth and achieves joint estimation

Copyright (c) 2016 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

The work of S. Qin, Y. D. Zhang, and M. G. Amin was supported in partby the Office of Naval Research under Grant No. N00014-13-1-0061. Part ofthe results was presented at the IEEE Radar Conference, Philadelphia, PA,May 2016 [1].

S. Qin and M. G. Amin are with the Center for Advanced Communications,Villanova University, Villanova, PA 19085, USA.

Y. D. Zhang is with the Department of Electrical and Computer Engineer-ing, Temple University, Philadelphia, PA 19122, USA.

F. Gini is with Department of Information Engineering of the Universityof Pisa, 56122 Pisa, Italy.

The corresponding author’s email address is ydzhang@temple.edu.

of targets direction-of-arrival (DOA) and range information[10]–[20]. As compared with conventional arrays that as-sume a fixed carrier frequency, FDA radars use a smallfrequency increment across array elements and thus achievebeam steering as a function of the angle and range in thefar field. In FDA radars, the spatial and range resolutions arefundamentally limited by the array aperture and maximumfrequency increment. In addition, the number of degrees-of-freedom (DOFs) offered by the array sensors and frequencyincrements determines the maximum number of detectabletargets.

The traditional FDA exploits a uniform linear array witha uniform frequency offset. The range and DOA estimationproblem using such FDA radar has been discussed in [21]–[23]. In [21], [22], the target ranges and DOAs are jointlyestimated by the minimum variance distortionless response(MVDR) and the MUSIC methods, respectively. Unlike [21],[22], an FDA utilizing coherent double pulse respectivelywith zero and non-zero frequency increments is consideredin [23], where the ranges and DOAs are estimated in twosteps. In the zero frequency increment case, the DOAs are firstestimated using a non-adaptive beamformer. The estimatedDOA information is then used as the prior knowledge byadaptive beamforming to obtain the range information in theother pulse. It is important to note that the above methods[21]–[23] use the traditional FDA radar and are discussedin the physical sensor framework rather than the virtualdifference coarray. That is, for an array with Nt sensors,there are only O(Nt) DOFs with a resolution O(1/Nt) inboth the range and angle domains. While the angular andrange resolutions can be improved by exploiting a largeinterelement spacing and a large frequency increment, suchstructure generally requires a large number of array sensors,or otherwise yields undesirable aliasing problems, i.e., causesambiguous estimations in angular and range dimensions.

Compared with uniform linear arrays (ULAs), sparse arraysuse the same number of sensors to achieve a larger arrayaperture. A properly designed non-uniform array can achievea desired trade-off between meanbeam width and sidelobelevels and, thereby, provide enhanced performance in terms ofDOA accuracy and resolution. These attributes are achievedwithout changes in size, weight, power consumption, or cost.More importantly, sparse arrays offer a higher number ofDOFs through the exploitation of the coarray concept [24]and, as such, significantly increases the number of detectabletargets. Likewise, non-uniform frequency offsets can be usedto achieve improved target identifiability and resolution in the

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range dimension [25]. Among different techniques that areavailable for sparse signal structures and array aperture synthe-sis, the recent proposed nested [26] and coprime configurations[27] offer systematical design capability and DOF analysisinvolving sensors, samples, or frequencies [28]–[41].

In [42], a nested array is employed to generate a coarraywhere the MUSIC algorithm together with spatial smoothingis applied. As a result, the number of the DOFs in theangular domain is increased to O(N2

t ). In [43], a sparsity-based method using the nested array is proposed. It achievesimproved resolution and estimation accuracy when comparedwith the conventional covariance based methods. However, thenumber of the DOFs in the range domain is still O(Nt) sincea uniform frequency offset is used. In addition, due to thelarge dimension of the joint range and angle dictionary, thismethod results in a prohibitive computational complexity thatlimits its practical applicability, particularly when the numberof antennas is large.

In this paper, we propose a novel configuration for theFDA radar, which incorporates both coprime array structureand coprime frequency offsets. In the proposed approach, theoffsets of carrier frequencies assume a coprime relationship tofurther increase the number of DOFs beyond that achieved byonly implementing the sparse arrays with uniform frequencyincrements. As a result, by using O(Nt) antennas and O(Nt)frequencies, the proposed approach achieves O(N2

t ) DOFswith a resolution of O(1/N2

t ) in both angular and rangedomains.

In this paper, we consider point-like targets and we exploittheir sparsity in both range and angular domains. We proposeboth joint and sequential estimation methods based on thespace-frequency coarray structure. For the joint estimation, thecovariance matrix of the received signals corresponding to allsensors and employed frequencies is formulated to generate avirtual difference coarray structure in the joint space-frequencydomain. Then, a joint-variable sparse reconstruction problemin the range and angular domain is presented as a singlemeasurement vector (SMV) model. We further develop anovel sequential two-step algorithm in the context of groupsparsity for reduced complexity. The cross-covariance matricesbetween the signals received at all sensors corresponding todifferent frequency pairs form space-only coarrays. Observa-tions in these coarrays exhibit a group sparsity across all fre-quency pairs, since their sparse angular domain vectors sharethe same non-zero entry positions associated with the sametarget DOAs. Therefore, the DOAs can be first solved undera multiple measurement vector (MMV) model. The valuesof nonzero entries contain the range information, and theirestimates across all frequency pairs are utilized to formulatea sparse reconstruction model with respect to the range. In sodoing, the joint DOA and range estimation problem is recastas two sequential one-dimensional (1-D) estimation problemswith a significantly reduced computational complexity.

The above sparse learning problems can be solved withinthe compressive sensing (CS) framework [44] and various CSmethods can be used for this purpose. As a preferred approach,we exploit the algorithms developed in the sparse Bayesianlearning context as they achieve superior performance and are

insensitive to the coherence of dictionary entries [45]–[51]. Inparticular, the complex multitask Bayesian compressive sens-ing (BCS) method [45], which effectively handles complex-value observations in the underlying problem, is used in thispaper.

The main contribution of this work is threefold: (a) Weachieve a significantly increased number of DOFs and improveboth angular and range resolutions by exploiting both coprimearray and coprime frequency offsets under the coarray andfrequency difference equivalence. (b) We employ a sparsity-based method to solve the joint DOA and range estimationproblem which, when compared to conventional MUSIC-basedapproach, enables more effective utilization of the availablecoarray aperture and frequency differences to resolve a highernumber of targets and improve the localization accuracy. (c)We further develop a group-sparsity based algorithm which, bycasting the joint DOA and range estimation as two sequential1-D estimation problems, significantly reduces the computa-tional complexity and processing time.

The rest of the paper is organized as follows. In Section II,the signal model of the traditional FDA radar is described. InSection III, we present a new FDA structure using coprimearrays and coprime frequency offsets. By effectively utilizingthe available coarray aperture and frequency differences, twosparsity-based multi-target localization methods are proposedin Sections IV and V that resolve a higher number of targetsand improve the localization accuracy. More specifically, inSection IV, the DOA and range are jointly estimated by a two-dimensional (2-D) sparse reconstruction algorithm, whereasa low-complexity algorithm through sequential 1-D sparsereconstruction is presented in Section V. Simulation results areprovided in Section VI to numerically compare the localizationperformance of the proposed approach with other methodsin terms of the number of resolvable targets, DOA-rangeresolution, and localization accuracy. Such results reaffirmand demonstrate the effectiveness of the proposed approach.Section VII concludes the paper.

Notations: We use lower-case (upper-case) bold charactersto denote vectors (matrices). In particular, IN denotes theN × N identity matrix. (.)∗ implies complex conjugation,whereas (.)T and (.)H respectively denote the transpose andconjugate transpose of a matrix or vector. vec(·) denotesthe vectorization operator that turns a matrix into a vectorby stacking all columns on top of the another, and diag(x)denotes a diagonal matrix that uses the elements of x as itsdiagonal elements. E(·) is the statistical expectation operatorand ⊗ denotes the Kronecker product. Pr(·) denotes theprobability density function (pdf), and N (x|a, b) denotes thatrandom variable x follows a Gaussian distribution with meana and variance b. Similarly, CN (a, b) denotes joint complexGaussian distribution with mean a and variance b. Γ(·) isthe Gamma function operator. δq,p is a delta function thatreturns the value of 1 when p = q and 0 otherwise. N andN+ respectively denote the set of non-negative integers andpositive integers, whereas R+ denotes the set of positive realnumbers. | · | denotes the determinant operation, whereas ‖ ·‖2and ‖ · ‖F represent the Euclidean (l2) norm and Frobeniousnorm, respectively. Tr(A) returns the trace of matrix A.

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𝑝1𝑑 𝑝2𝑑 𝑝3𝑑 𝑝𝑁𝑡𝑑

𝑓1 𝑓2 𝑓3 𝑓𝑁𝑡

Fig. 1. The FDA configuration.

II. FREQUENCY DIVERSE ARRAY RADAR

Without loss of generality, we limit our discussion to far-field targets in the 2-D space where the DOA is described bythe azimuth angle only. Extension to three-dimensional (3-D)space is straightforward.

A. Signal Model

As shown in Fig. 1, an FDA radar utilizes a linear array withNt antennas. Note that the array spacing can be either uniformor non-uniform. Denote p = [p1d, ..., pNtd]T as the positionsof the array sensors where pk ∈ N, k = 1, ..., Nt. The firstsensor, located at p1 = 0, is used as the reference. To avoidspatial ambiguity, d is typically taken as half wavelength, i.e.,d = λ0/2 = c/(2f0), where c is the velocity of electromag-netic wave propagation and f0 is the base carrier frequency.Different from the conventional phased-array radar where allantennas transmit the same signal with carrier frequency f0,each FDA element radiates a signal with an incremental carrierfrequency. That is, a continuous-wave (CW) signal transmittedfrom the kth element is expressed as

sk(t) = Akexp(j2πfkt), (1)

where Ak is the amplitude and the radiation frequency fk =f0 + ξk∆f is exploited with a unit frequency increment ∆f ,and ξk ∈ N is an integer coefficient of the frequency offsetapplied at the kth element, k = 1, ..., Nt. The maximumincrement is assumed to satisfy ξNt∆f � f0 so as toguarantee that the FDA radar works in a narrowband platform.Also, the frequency offsets are not necessary uniform.

An important objective of this paper is to improve theparameter identifiability using the FDA radar. Since the targetsin different bins can be simple identified, we consider ascene with Q far-field targets within the same Doppler bin.Without loss of generality, the Doppler frequency is assumedto be 0. The locations of the targets are modeled as (θq, Rq),q = 1, 2, · · · , Q. Then, the received signal at the lth sensor ismodeled as

xl(t) =

Nt∑k=1

Q∑q=1

ρq(t)exp(j2πfkt)e−j 4π

λkRqe−j 2πpld

λksin(θq)

+ nl(t), l = 1, . . . , Nt, (2)

where ρq(t), q = 1, . . . , Q, are complex scattering coefficientsof the targets, which are assumed to be uncorrelated zero-mean random variables with E[ρ∗qρp] = σ2

qδq,p, 1 ≤ q, p ≤Q, due to, e.g., the radar cross section (RCS) fluctuations. Inaddition, λk = c/fk denotes the wavelength corresponding to

carrier frequency fk. Furthermore, nl(t) is the additive noise,which is assumed to be spatially and temporally white, and isindependent of target signals.

By implementing the pass-band filtering, the received signalis converted to the signals corresponding to the respective fre-quencies. For a CW waveform with frequency fk transmittedfrom the kth sensor, the baseband signal received at the lthsensor can be expressed as

xk,l(t) =

Q∑q=1

ρq(t)e−j 4π

λkRqe−j 2πpld

λksin(θq) + nk,l(t)

=

Q∑q=1

ρq(t)e−j 4πfk

c Rqe−jπpl(f0+ξk∆f)

f0sin(θq) + nk,l(t),

(3)

where nk,l(t) is the noise at the filter output with a varianceσ2n. Because ξk∆f � f0, the above expression can be

simplified as

xk,l(t) =

Q∑q=1

ρq(t)e−j 4πfk

c Rqe−jπpl sin(θq) + nk,l(t). (4)

Stacking xk,l(t) for all k, l = 1, ..., Nt yields an N2t ×1 vector,

x(t) =

Q∑q=1

ρq(t)ap,f (θq, Rq) + n(t)

= Ap,fd(t) + n(t), (5)

where ap,f (θq, Rq) = ap(θq)⊗af (Rq) represents the steeringvector associated with the angle-range pair (θq, Rq). Herein,ap(θq) and af (Rq) are steering vectors corresponding to θqand Rq , respectively, expressed as

ap(θq) =[1, e−jπp2 sin(θq), · · · , e−jπpNt sin(θq)

]T, (6)

af (Rq) =

[e−j

4πf1c Rq , e−j

4πf2c Rq , · · · , e−j

4πfNtc Rq

]T. (7)

In addition, Ap,f = [ap,f (θ1), · · · ,ap,f (θQ)], d(t) =[ρ1(t), · · · , ρQ(t)]T , and nk(t) is the noise vector followingthe joint complex Gaussian distribution CN (0, σ2

nIN2t).

The N2t × N2

t covariance matrix of data vector x(t) isobtained as

Rx = E[x(t)xH(t)] = Ap,fRddAHp,f + σ2

nIN2t

=

Q∑q=1

σ2qap,f (θq, Rq)a

Hp,f (θq, Rq) + σ2

nIN2t, (8)

where Rdd = E[d(t)dH(t)] = diag([σ21 , . . . , σ

2Q]) represents

the target scattering power. Note that we assume the targetscattering coefficients to be frequency-independent for theemitting signals since the frequency offsets are relativelysmall. In practice, the covariance matrix is estimated usingT available samples, i.e.,

Rx =1

T

T∑t=1

x(t)xH(t). (9)

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Existing covariance matrix based techniques can then beapplied to estimate the DOA and range of the targets, e.g.,the Fourier-based power spectrum density (PSD) [52] and 2-D MUSIC [53].

B. Unambiguous Range

For each target, the DOA and range information are re-spectively determined by φθq and φRq , which are definedas the minimum phase difference in angle and range di-mensions, respectively, i.e., the phase terms of e−jπ sin(θq)

and e−j4π∆fRq/c. In reality, however, phase observations arewrapped within [−π, π). Therefore, the true phase can beexpressed as

φ(true)θq

= φθq + 2mθqπ, (10)

φ(true)Rq

= φRq + 2mRqπ, (11)

where mθq and mRq are unknown integers. As a result, therange estimate is subject to range ambiguity [54], i.e.,

Rq =cφRq4π∆f

+cmRq

2∆f. (12)

The latter term in (12) implies ambiguity in range due tophase wrapping. Thus, the range can be assumed as infinitevalues separated by Rmax = c/(2∆f), which is referred toas the maximum unambiguous range. Therefore, the use ofa large value of ∆f will reduce the maximum unambiguousrange. As a large frequency bandwidth is required to achieveproper range resolution, uniform frequency offsets must tradeoff between the range resolution and unambiguous rangeestimation. On other other hand, coprime frequency offsetsallows the use of small ∆f while collectively spanning a largesignal bandwidth.

III. FREQUENCY DIVERSE COPRIME ARRAYS WITHCOPRIME FREQUENCY OFFSETS

For the traditional FDA radar with Nt-element ULA anduniform frequency increment, it can localize up to N2

t − 1targets, with a resolution O(1/Nt) in the angle and rangedomains, respectively. Compared with the uniform case, sparsearrays and sparse frequency offsets use the same number ofsensors and frequencies to achieve a larger array aperture andfrequency bandwidth. As a result, they improve the resolutionand estimation accuracy. However, the number of resolvabletargets using sparse arrays and sparse frequency offsets isstill upper bounded by N2

t − 1, if those covariance matrixbased approaches are used directly. Such the limitation canbe overcome by the improvement of DOFs under the coarrayequivalence.

A. Coarray Equivalence

By vectorizing the matrix Rx, we obtain the following N4t ×

1 virtual measurement vector:

z = vec(Rx) = Ap,fbp,f + σ2n i, (13)

with

Ap,f = [ap,f (θ1, R1), · · · , ap,f (θQ, RQ)], (14)

bp,f = [σ21 , · · · , σ2

Q]T , (15)

i = vec(IN2

t

), (16)

where

ap,f (θq, Rq) = a∗p,f (θq, Rq)⊗ ap,f (θq, Rq)

= a∗p(θq)⊗ a∗f (Rq)⊗ ap(θq)⊗ af (Rq)

= (a∗p(θq)⊗ ap(θq))⊗ (a∗f (Rq)⊗ af (Rq))

= ap(θq)⊗ af (Rq) (17)

for 1 ≤ q ≤ Q. Benefiting from the Vandermonde structure ofap(θq) and af (Rq), the entries in ap(θq) and af (Rq) are stillin the forms of e−jπ(pi−pj) sin(θq) and e−j4π(ξi−ξj)∆fRq/c, fori, j = 1, · · · , Nt. As such, we can regard z as a receivedsignal vector from a single-snapshot signal vector bp,f , andthe matrix Ap,f corresponds to the virtual array sensorsand virtual frequency offsets which are respectively locatedat the sensor-lags between all sensor pairs and frequency-offsets between all frequency pairs. The targets can thus belocalized by using the space-frequency coarray, in lieu of theoriginal antennas and frequencies. Note that the number ofelements in the space-frequency coarray structure are directlydetermined by the distinct values of (pi − pj) and (ξi − ξj)for i, j = 1, · · · , Nt. Non-uniform arrays can substantiallyincrease the number of DOFs by reducing the number ofredundant elements in the coarray. In other words, the numberof DOFs would be reduced if different pairs of sensors orfrequency offsets yield same lags when the uniform arrays areexploited.

B. Coprime Arrays with Coprime Frequency Offsets

Among the different choices that are available for sparsearray and frequency offset designs, the recently proposed co-prime configurations [27] offer a systematical design capabilityand DOF analysis involving sensors, samples, or frequencies.In this paper, we use the extended coprime structure whichis proposed in [55] as an example. Extensions to othergeneralized coprime structures that achieve higher DOFs arestraightforward [34].

As shown in Fig. 2, the extended coprime array structureutilizes a coprime pair of uniform integers. The coprime arrayconsists of a 2M -sensor uniform linear subsarray with an

#0 #1 #𝑁 − 1 #2

𝑀

#0 #1 #2𝑀 − 1 #2

𝑁

Fig. 2. The extended coprime structure.

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interelement spacing of N , and an N -sensor uniform linearsubarray with an interelement spacing of M . The two integersM and N are chosen to be coprime, i.e., their greatest commondivisor is one. In addition, M < N is assumed. Define

P(M,N) = {Mn|0 ≤ n ≤ N − 1}⋃{Nm|0 ≤ m ≤ 2M − 1}

(18)as the union of two sparsely sampled integer subsets withrespect to the pair of coprime integers (M,N). As such, theyielding correlation terms have the positions

L(M,N) = {±(Mn−Nm)|0 ≤ m ≤ 2M−1, 0 ≤ n ≤ N−1}.(19)

An example is illustrated in Fig. 3, where M = 2 and N =3. Fig. 3(a) shows the physical elements of extended coprimestructure, and the positions of the corresponding correlationterms are depicted in Fig. 3(b). Notice that “holes”, e.g., ±8in this case, still exist in the virtual domain and are indicatedby × in the figure. It is proved in [56] that L(M,N) can achieveat least MN (up to (3MN +M −N +1)/2) DOFs with only2M + N − 1 (two subsets share the first element) entries inP(M,N).

0 1 2 3 4 5 6 7 8 9

(a)

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

(b)

Fig. 3. An example for the extended coprime structure. (a) The physicalelements in P(M,N) (4: Subset 1; ∇: Subset 2); (b) The correspondingcorrelation term positions in L(M,N).

When coprime arrays and coprime frequency offsets withpairs of coprime integers (M,N) is exploited, there are at leastMN available DOFs in each ap(θq) and af (Rq). That is, theresulting virtual array elements and virtual frequency offsetsenable estimation of at least MN distinct DOAs and MNdistinct ranges of targets. Benefitting from the sparse structure,the proposed coprime array with coprime frequency offsetsoffers a larger aperture and frequency span, thus resulting in animproved resolution in both angular and range domains. Fur-ther, it has less redundant entries in the covariance matrix Rx,implying that the resulting coarray structure and frequency lagsets provide a higher number of DOFs that can be used toidentify more targets using the CS based methods.

The localization problem in (13) is similar to handling mul-tiple targets that are fully coherent. In this case, the covariancematrix constructed from the virtual signal vector is rank-1and, as a result, subspace-based localization approaches failto function. A well-known approach that restores the rank ofthe covariance matrix is spatial smoothing [57], [58]. A majordisadvantage of such approach is that only consecutive lagsin the virtual observations can be used so that every subarrayhas a similar manifold (e.g., [−7, 7] in Fig. 3(b)), whereasthe virtual sensors that are separated by any holes have tobe discarded. Alternatively, this problem can be solved byusing sparse reconstruction methods (e.g., [34], [59]) which,

2 4 6 8 10 12 14 16 18Number of Physical Antennas Nt

0

500

1000

1500

2000

2500

TheNumber

ofDOFs

Uniform

Coprime

Fig. 4. The number of DOFs versus Nt.

by taking advantages of the fact that the targets are sparsein the angle-range domain, utilize all consecutive and non-consecutive lags (e.g., ±9 and [−7, 7] in Fig. 3(b)) in thecoarray so as to fully utilize the available DOFs offered bythe coarray configurations.

Provided that sufficient snapshots are available for reliablecovariance matrix estimation, at least O(MN) targets (nosame DOA and no same range), up to O(M2N2) targets (eachof MN DOAs has MN distinct ranges), can thus be localizedby using Nt = 2M +N − 1 antennas and Nt = 2M +N − 1frequencies. For a given number of Nt, the maximum numberof DOFs can be further optimized by

Maximize M2N2

subject to 2M +N − 1 = Nt, (20)

M < N, M,N ∈ N+.

It is demonstrated in [31] that the valid optimal coprime pairto maximize MN is the one that has 2M and N as close aspossible. This is satisfied by choosing N = 2M − 1. In thiscase, more than

[N2t (Nt + 2)2

]/64 DOFs can be obtained.

Therefore, the frequency diverse coprime arrays with coprimefrequencies can resolve more targets than that of conventionalFDA with ULA and uniform frequency increment (i.e., N2

t −1)when Nt ≥ 6, as shown in Fig. 4.

IV. TARGET LOCALIZATION USING MULTITASK BCSIn the following, we perform multi-target localization in

the sparse reconstruction framework. The general focus ofproposed methods is to resolve a higher number of targetsand improve the localization accuracy by fully utilizing allthe virtual observations achieved from lags in both sensorpositions and frequencies. For the simplicity and clarity ofthe presentation, we assume the targets to be placed on a pre-defined grid. Direct application of the proposed method inthe presence of dictionary mismatch would yield performancedegradation. However, various techniques, such as those citedin [32], [33], [60], [61], can be used to overcome this problemby exploring the joint sparsity between signals and the gridmismatch variables.

The virtual signal vector z in (13) can be sparsely repre-sented over the entire discretized angular grids as

z = Φb + ε, (21)

6

where Φ =[Φs, i

]. Herein, Φs is defined as the col-

lection of steering vectors ap,f (θg1, Rg2

) over all possiblegrids θg1

and Rg2, g1 = 1, . . . , G1, g2 = 1, . . . , G2, with

G = G1G2 � N4t > Q, and bs is the sparse vector whose

non-zero entry positions correspond to the DOAs and rangesof the targets, i.e., (θq, Rq), q = 1, . . . , Q. The term i inthe dictionary accounts for noise variance terms that haveunequal values in the vectorized entries. In addition, an errorvector ε is included to represent the discrepancies between thestatistical expectation and the sample average in computing thecovariance matrix R. The discrepancies are modelled as i.i.d.complex Gaussian as a result of a sufficiently large number ofsamples employed in the averaging.

In this paper, we elect to perform the sparse signal re-construction within the BCS framework [45]–[51] stemmingfrom their superior performance and robustness to dictionarycoherence. In particular, the complex multitask BCS approachdeveloped in [45] is used to deal with all the sparse reconstruc-tion problems. Thus, the following sparse Bayesian model ispresented as an MMV model with P tasks (measurements),whereas the SMV problem in (21) can be considered as aspecial case with a single task, i.e., P = 1.

A. Sparse Bayesian Formulation

The MMV model is expressed as

Z = ΦB, (22)

where Z = [z1, · · · , zP ] and B = [b1, · · · ,bP ]. The matrixB is jointly sparse (or row sparse), i.e., all columns of B arespares and share the same support.

Assume that the entries in jointly sparse matrix B aredrawn from the product of the following zero-mean complexGaussian distributions:

Pr(B|α) =

P∏p=1

CN (bp|0,Λ), (23)

where α = [α1, . . . , αG]T and Λ = diag(α). It is noted thatthe gth row of B trends to be zero when αg, g = 1, · · · , Gis set to zero [46]. In addition, α is placed on a complexvariable directly. As such, it achieves improved sparse signalreconstruction because by utilizing the group sparsity of thereal and imaginary components than the methods that simplydecomposing them into independent real and imaginary com-ponents.

To encourage the sparsity, a Gamma prior is placed on αg ,which is conjugate to the Gaussian distribution,

αg ∼ Γ(αg|1, ρ), g ∈ [1, · · · , G], (24)

where ρ ∈ R+ is a fixed priori. It has been demonstrated in[62] that a proper choice of ρ encourages a sparse represen-tation for the coefficients. Then, we have

Pr(α|a, b) =

G∏g=1

Γ(αg|1, ρ). (25)

All columns of B share the same prior due to the group sparseproperty. Base on [63], both of the real and image parts of bp,

p = 1, · · · , P , are Laplace distributed and share the same pdfthat is strongly peaked at the origin. As such, this two-stagehierarchical prior is a sparse prior that favors most rows of Bbeing zeros.

A Gaussian prior CN (0, β−10 I2) is also placed on the error

vector ε. Then, we have,

Pr(Z|B, β0) =

P∏p=1

CN (zp|Φbp, β−10 I), (26)

Likewise, the Gamma prior is placed on β0 with hyper-parameters c and d, expressed as

Pr(β0|c, d) = Γ(β−10 |c, d), (27)

where Gamma(β−10 |a, b) = Γ(a)−1baβ

−(a−1)0 e−

bβ0 .

By combining the stages of the hierarchical Bayesian model,the joint pdf becomes

Pr(Z,B,α, β0) = Pr(Z|B, β0)Pr(B|α)Pr(α|1, ρ)Pr(β0|c, d).(28)

To make this Gamma prior non-information, we set c = d = 0in this paper as in [46]–[51].

B. Bayesian Inference

Assuming α and β0 are known, given the measurement Zand the corresponding dictionary Φ, the posterior for B canbe obtained analytically using Bayes’s rule, expressed as aGaussian distribution with mean µ and variance Σ

Pr(B|Z,α, β0) =

P∏p=1

CN (bp|µp,Σ), (29)

where

µp = β−10 ΣΦHzp, (30)

Σ =[β−1

0 ΦHΦ + F−1]−1

. (31)

The associated learning problem, in the context of relevancevector machine (RVM), thus becomes the search for the αand β0. In RVM, the values of α and β0 are estimated fromthe data by performing a type-II maximum likelihood (ML)procedure [62]. Specially, by marginalizing over the B, themarginal likelihood for α and β0, or equivalently, its logarithmL(α, β0) can be expressed analytically as

L(α, β0) =

P∑p=1

log Pr(bp|α, β0)

=

P∑p=1

log

∫Pr(zp|bp, β0)Pr(bp|α)dbp

= const− 1

2

P∑p=1

log |C|+ (zp)H

C−1zp (32)

withC = β0I + ΦFΦH . (33)

Denote U = [µ1, · · · ,µP ] = β−10 ΣΦHZ, B = B/

√P , Z =

Z/√P , U = U/

√P , and ρ = ρ/P . An ML approximation

employs the point estimates for α and β0 to maximize (32),

7

which can be implemented via the expectation maximization(EM) algorithm to yield

α(new)g =

√1 + 4ρ(‖µg‖22 + Σg,g)− 1

2ρ, g ∈ [1, · · · , G],

(34)

β(new)0 =

E{‖Z−ΦB‖2F }NΦ

, (35)

where µg is the gth row of matrix U and Σg,g is the (g, g)thentry of matrix Σ. In addition, NΦ is the number of rows ofΦ.

It is noted that, because α(new) and β(new)0 are a function

of µp and Σ, while µp and Σ are a function of α and β0, thissuggests an iterative algorithm that iterates between (30)–(31)and (34)–(35), until a convergence criterion is satisfied or themaximum number of iterations is reached. In each iteration,the computational complexity is O(max(NΦG

2, NΦGP ))with an NΦ ×G dictionary Φ [48].

C. Complexity Analysis

For the case of 2-D BCS, the corresponding joint angle-range of targets, (θq, Rq), q = 1, · · · , Q, can be estimated bypositions of the nonzero entries in b in (21). In the sequel, weanalyze its computational complexity, which can be dividedinto the following three stages:

S1: Compute the N2t ×N2

t covariance matrix Rx with (9).S2: Generate the N4

t × 1 virtual array data z with (13) byvectorizing the covariance matrix.

S3: Perform target localization to obtain (θq, Rq), q =1, · · · , Q using (30)–(31) and (34)–(35), based on theBCS (P = 1) with an N4

t ×G1G2 dictionary.

In S1, there are O(N4t T ) complex multiplications, whereas

no multiplication operation is needed for vectorization in S2.For the BCS, we might need O(κN4

t G21G

22) complex multipli-

cation operations, where κ is the number of iterations. There-fore, the total computational load, i.e., O(N4

t T +κN4t G

21G

22),

is very huge because the exhaustive 2-D searching process,which motivates the development of fast algorithms.

V. A FAST ALGORITHM FOR TARGET LOCALIZATION

In this section, we develop an algorithm based on themultitask BCS, wherein the 2-D sparse reconstruction prob-lem is cast as separate 1-D sparse reconstruction problems.Therefore, the computational complexity can be reduced.

Stacking xk,l(t) for all l = 1, ..., Nt yields the followingNt × 1 vector,

xk(t) =

Q∑q=1

ρq(t)e−j 4πfk

c Rqap(θq) + nk(t). (36)

As such, the vector xk(t) behaves as the received signal ofthe array, corresponding to the frequency fk, k = 1, · · · , Nt.

The cross-covariance matrix between data vectors xk(t) andxk′(t), respectively corresponding to frequencies fk and fk′ ,1 ≤ k, k′ ≤ Nt, is obtained as

Rxkk′ = E[xk(t)xHk′(t)] =

Q∑q=1

σ2qe−j

4π∆fkk′c Rqap(θq)a

Hp (θq),

(37)where ∆fkk′ = fk − fk′ = (ξk − ξk′)∆f . Note that thedimension of Rxkk′ is reduced to Nt ×Nt, compared to theN2t ×N2

t matrix Rx in (8). In practice, the cross-covariancematrix is estimated by using T available samples, i.e.,

Rxkk′ =1

T

T∑t=1

xk(t)xHk′(t), 1 ≤ k, k′ ≤ Nt. (38)

Vectorizing this matrix yields the following N2t × 1 vector

zkk′ = vec(Rxkk′ ) = Abfkk′ , (39)

where

A = [ap(θ1), · · · , ap(θQ)], (40)

bfkk′ = [σ21e−j

4π∆fkk′c R1 , · · · , σ2

Qe−j

4π∆fkk′c RQ ]T . (41)

Similarly, (39) can be sparsely represented over the entireangular grids as

zkk′ = Φbkk′ , (42)

where the N2t × G1 dictionary Φ is defined as the collec-

tion of steering vectors ap(θg) over all possible grids θg1 ,g1 = 1, . . . , G1, with G1 � Q. As such, the DOAs θq ,q = 1, · · · , Q, are indicated by the nonzero entries in thesparse vector bkk′ , whose values describe the correspondingcoefficients σ2

qe−j

4π(ξk−ξk′ )∆fc Rq . Note that the nonzero entries

corresponding to different frequency pairs share the samepositions as they are associated with the same DOAs of the Qtargets. However, their values differ for each frequency pair.Therefore, zkk′ exhibits a group sparsity across all frequencypairs and the problem described in (42) can be solved in theMMV sparse reconstruction context.

Denote Z = [z1, · · · , zP ] as the collection of vectors zkk′ ,corresponding to all P = N2

t frequency pairs. Then, the MMVsparse reconstruction problem is expressed as

Z = ΦB, (43)

where B = [b1, · · · , bP ] is the sparse matrix that can bereconstructed by the multitask BCS.

Denote Q as the number of distinct DOAs of Q targets, thenq , as the index of those nonzero positions in B correspondingto θq , q = 1, · · · , Q. In addition, the P × 1 vector bnqis denoted as the nqth column of BT . Then, the range canbe estimated by solving the following sparse reconstructionproblem:

bnq = ΨRnq , q = 1, · · · , Q, (44)

where Ψ is the N2t ×G2 dictionary, whose g2th column, g2 =

1, . . . , G2, is

Ψg2=

[1, · · · , e−j

4π(ξk−ξk′ )c Rg2 , · · · , 1

]T, (45)

8

with 1 ≤ k, k′ ≤ Nt. Then, the range on θq , q = 1, · · · , Q canbe indicated by positions of nonzero entries in sparse vectorRnq .

As a summary, the proposed approach can be divided intothe following four stages:

S1: Compute all Nt×Nt covariance matrix Rxkk′ using (38),1 ≤ k, k′ ≤ Nt.

S2: Generate all the N2t × 1 virtual array data zkk′ with (39)

by vectorizing the covariance matrix, 1 ≤ k, k′ ≤ Nt.S3: Perform DOA estimation of the targets, based on the

multitask sparse reconstruction(P = N2

t

)model in (43)

with an N2t ×G1 dictionary.

S4: Perform range estimation of the targets, based on thesparse reconstruction model in (44) with an N2

t × G2

dictionary.

In S1, there are O(N4t T ) multiplication operations. The

complexity in S3 and S4 is O(κ1N2t G

21) and O(κ2N

2t G

22),

respectively, where κ1 and κ2 are the corresponding num-ber of iterations. Thus, the total computation load isO(N4t T + κ1N

2t G

21 + κ2QN

2t G

22

), which is much lower than

O(N4t T + κN4

t G21G

22) in Section IV.

Remarks: The following observations can be made re-garding the relationship between the joint and the two-stepestimation methods:

(1) Both estimation methods achieve the same number ofDOFs from the coarray;

(2) The two-step estimation method requires a significantlyreduced complexity. However, the corresponding performancebecomes sub-optimal due to error propagation. i.e., errors inthe DOA estimation stage may yield additional perturbationsin the range estimation.

VI. SIMULATION RESULTS

For illustrative purposes, we consider an FDA radar exploit-ing coprime array and coprime frequency offset, where M = 2and N = 3 are assumed. The extended coprime structureconsist of Nt = (2M+N−1) = 6 physical elements, and has(3MN +M −N + 1)/2 = 9 DOFs in the virtual domain. Assuch, the increased number of DOFs enables to localize morethan M2N2 = 36 targets with only 6 antennas exploiting 6frequencies.

The unit interelement spacing is d = λ0/2, where λ0 isthe wavelength with respect to the carrier frequency f0 = 1GHz. We choose the unit frequency increment to be ∆f =30 KHz, resulting maximum unambiguous range Rmax =c/(2∆f) = 5000 m. In all simulations, Q far-field targetswith identical target scattering powers are considered. Theqth target is assumed to be on angle-range plane (θq, Rq),where θq ∈ [−60◦, 60◦] and Rq ∈ [1000, 5000] m, forq = 1, · · · , Q. The localization performance for the coprimearray and coprime frequency offset (CA-CFO) is examined interms of the resolution, accuracy, and the maximum number ofresolvable targets. The average root mean square error (RMSE)

of the estimated DOAs and ranges, expressed as

RMSEθ =

√√√√ 1

IQ

I∑i=1

Q∑q=1

(θq(i)− θq)2,

RMSER =

√√√√ 1

IQ

I∑i=1

Q∑q=1

(Rq(i)−Rq)2, (46)

are used as the metric for estimation accuracy, where θq(i)and Rq(i) are the estimates of θq and Rq for the ith MonteCarlo trial, i = 1, . . . , I . We use I = 500 independent trialsin simulations.

A. Joint Estimation Method versus Two-step EstimationMethod

We first compare the performance of the joint estimationmethod and two-step estimation method. Q = 1 target with(10◦, 1000m) is considered. The dictionary matrices Φ andΨ are assumed to contain all possible grid entries within(5◦, 15◦) and (1250 m, 1350 m) with uniform intervals θg1

=0.2◦ and Rg2 = 1 m, respectively. Fig. 5 compares the RMSEperformance of DOA and range estimations with respect tothe input signal-to-noise ratio (SNR), where 500 snapshotsare used. In Fig. 6, we compare the RMSE performance withrespect to the number of snapshots, where the input SNR is setto −5 dB. It is clear that the joint estimation method achievesslightly better estimation accuracy at the cost of much highercomputation complexity.

-10 -8 -6 -4 -2 0 2 4 6 8 10SNR (dB)

10-3

10-2

10-1

RMSE

(deg)

JointTwo-step

(a)

-10 -8 -6 -4 -2 0 2 4 6 8 10SNR (dB)

100

RMSE

(m)

JointTwo-step

(b)

Fig. 5. RMSE versus SNR using the joint and two-step estimation methods(Q = 1 and T = 500). (a) RMSEθ ; (b) RMSER

9

B. CA-CFO versus Other Array Configuration and FrequencyOffset Designs

Next, we examine the localization performance for differentarray configuration and frequency offset designs. Particularly,the proposed CA-CFO is compared with uniform linear arrayand uniform frequency offset (ULA-UFO). Uniform lineararray with coprime frequency offset (ULA-CFO), and coprimearray and uniform frequency offset (CA-UFO) are also consid-ered. In order to reduce the computational load, we use the fastalgorithm in section V for target localization in simulations.

In Fig. 7, we compare the resolution performance of differ-ent schemes. Q = 8 targets whose true positions are shownin Fig. 7(a) are considered. The dictionary matrices Φ and Ψcontain steering vectors over all possible grids in (−60◦, 60◦)and (1000 m, 5000 m) with uniform intervals θg1

= 1◦ andRg2

= 100 m, respectively. Note that the number of targets islarger than the number of antennas, and the traditional phasedarray radar does not have sufficient DOFs to resolve all targets.The covariance matrix are obtained by using 500 snapshots inthe presence of noise with a 0 dB SNR, and the correspondinglocalization performance are illustrated in Figs. 7(b)–(e). It isevident that only the case of CA-CFO can identify targetscorrectly because the increased DOFs in both virtual arrayand frequency can estimate more DOAs than the number ofantennas, and more ranges than the number of frequencies. Inaddition, the corresponding larger apertures in both angle andrange domains enable the CA-CFO case to resolve the closelyspaced targets. The conventional FDA with ULA-UFO failsto separate both pairs of the targets with closely spaced angleand closely spaced range. However, the scenario of CA-UFOcan resolve the pair of targets with closely spaced angle andthe ULA-CFO case can identify targets with closely spacedrange, benefitting from the increased DOFs in the angle andrange domains, respectively.

We further compare the estimation accuracy through MonteCarlo simulations. To proceed with the comparison, we con-sider Q = 2 targets with (10◦, 1300 m) and (25◦, 1700m),which can be separated for all cases. The dictionary matricesΦ and Ψ are assumed to contain entries corresponding toall possible grids in (10◦, 30◦) and (1000 m, 2000 m) withuniform intervals θg1 = 0.2◦ and Rg2 = 10 m, respec-tively. Fig. 8 compares the RMSE performance of DOAand range estimations with respect to the input SNR fordifferent array configurations and frequency offset structures,where 500 snapshots are used. In Fig. 9, we compare theRMSE performance with respect to the number of snapshots,where the input SNR is set to −5 dB. It is evident that theaccuracy of both DOA and range estimates is improved as theSNR and the number of snapshots increase. In comparisonwith the uniform array/offset case, the coprime array/offsetstructure benefits from more independent measurements underthe CS framework. It is shown that the CA-UFO and ULA-CFO respectively achieve improved estimation accuracy inthe angular and range domains than that of the ULA-UFOowing to the coprime structure in the sensor positions andfrequency offsets. In particular, the CA-CFO achieves thebest performance as the advantages of coprime structure are

presented in both angular and range domains.In Fig. 10, we consider Q = 56 targets. Note that this

number is more than the available DOFs obtained from thecases of ULA-UFO (the conventional FDA radar), ULA-CFO,and CA-UFO. As the virtual array and virtual frequency offsetare obtained from the estimated covariance matrix based on thereceived data samples, the virtual steering matrix is sensitive tothe noise contamination. To clearly demonstrate the sufficientDOFs for localization of a large number of targets, we use2000 snapshots in presence of noise with a 10 dB input SNR. Itis evident that all 56 signals can be identified correctly, whichdemonstrates the effectiveness of the CA-CFO in resolvingmore targets.

C. Sparsity-based Method versus Subspace-based Method

In Figs. 11–13, we compare the sparsity-based method andthe MUSIC algorithm with spatial smoothing (MUSIC-SS)applied to the CA-CFO configuration. Note that the spatialsmoothing technique is applied to the covariance matrix of thevirtual measurement vector z so that its rank can be restoredbefore the MUSIC algorithm is applied. In this case, onlyconsecutive lags, i.e., [−7, 7], can be used so that every sub-matrix has a similar manifold. The corresponding number ofavailable DOFs is less than that of the proposed sparsity-basedapproach, which utilizes all unique lags [34]. In Fig. 11, weexamine their resolution for Q = 5 closely spaced targets,whose true positions are shown in Fig. 11(a). The localizationresults, depicted in Figs. 11(b) and 11(c), are obtained byusing 500 snapshots with a 0 dB SNR. It is clear that thesparsity-based method outperforms the MUSIC-SS approachfor resolving the closely spaced targets, since it exploitsall distinct lags to form a virtual space-frequency structure,thus yielding a larger array aperture and frequency spancompared to the corresponding MUSIC-SS technique whichonly uses consecutive lags. The respective RMSE performanceis compared in Figs. 12 and 13 under the same target scenarioconsidered in Figs. 8 and 9, whereas Q = 2 targets located at(10◦, 1300 m) and (25◦, 1700 m) are present. In Fig. 12, 500snapshots are used, while a −5 dB SNR is assumed in Fig. 13.It is evident that the proposed sparsity-based method achievesa lower RMSE than the MUSIC-SS due to the higher numberDOFs in both angular and range domains. This simulationexample shows that the sparsity-based method achieves betterperformance than the MUSIC-SS counterparts do.

D. Proposed Method versus Existing Methods

In Figs. 14–16, we compare the performance of the pro-posed method with the existing methods using sparse arrays.The methods in [42] and [43], which are referred to as theNested-MUSIC and Nested-CS, respectively, employ a nestedarray configuration but with the uniform time-delayer andfrequency increment. As a consequence, it has only O(Nt)frequency DOFs with a smaller spectral span for a coarserange resolution, although it has the same O(N2

t ) spatialDOFs as the proposed coprime FDA radar configuration.

10

The same target scenario considered in Figs. 11–13 is usedfor performance comparison. Fig. 14 depicts the angle-rangeresolution, wherein the true positions and results obtainedfrom the proposed method are reproduced from Fig. 11(a)and Fig. 11(c) as Fig. 14(a) and Fig. 14(b) for the conve-nience of comparison. The corresponding results using theNested-MUSIC and Nested-CS methods are presented in Figs.14(c) and 14(d), respectively. It is evident that only theproposed method can resolve these closely spaced targets inthe range. Furthermore, the Nested-MUSIC method producesmore blurry spectra than the Nested-CS for targets with asmall angulare separation. The RMSE is compared in Figs.15 and 16. It is clear that the Nested-MUSIC and Nested-CSmethods suffer from significant performance degradation inthe range domain due to the reduced spectral span and range-domain DOFs. Accordingly, the Nested-CS outperforms theNested-MUSIC owing to its utilizations of all distinct lags inthe coarray structure.

VII. CONCLUSIONS

In this paper, we proposed a novel sparsity-based multi-target localization algorithm, which incorporates both coprimearrays and coprime frequency offsets in an FDA radar plat-form. By exploiting the sensor position lags and frequencydifferences, the proposed technique achieved a high numberof DOFs representing a larger array aperture and increasedfrequency increments compared to conventional approaches.These attributes enable high-resolution target localization ofsignificantly more targets than the number of physical sensors.A fast algorithm was developed that cast the 2-D sparsereconstruction problem as separate 1-D sparse reconstruc-tion problems, thus effectively reducing the computationalcomplexity. The offerings of the proposed technique weredemonstrated by simulation results.

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16

Si Qin received the B.S. and M.S. degrees in Electri-cal Engineering from Nanjing University of Scienceand Technology, Nanjing, China, in 2010 and 2013,respectively. Currently, he is a Research Assistant atthe Center for Advanced Communications, VillanovaUniversity, Villanova, PA, working toward his Ph.D.degree in Electrical Engineering. His research in-terests include direction-of-arrival estimation, sparsearray and signal processing, and radar signal pro-cessing.

Mr. Si Qin received the First Prize of StudentPaper Competition at 2014 IEEE Benjamin Franklin Symposium on Mi-crowave and Antenna Sub-systems (BenMAS) and Best Student Paper Awardat 2012 IEEE International Conference on Microwave and Millimeter WaveTechnology (ICMMT).

Yimin D. Zhang (SM’01) received his Ph.D. degreefrom the University of Tsukuba, Tsukuba, Japan, in1988.

Dr. Zhang joined the faculty of the Department ofRadio Engineering, Southeast University, Nanjing,China, in 1988. He served as a Director and Tech-nical Manager at the Oriental Science Laboratory,Yokohama, Japan, from 1989 to 1995, a Senior Tech-nical Manager at the Communication LaboratoryJapan, Kawasaki, Japan, from 1995 to 1997, and aVisiting Researcher at the ATR Adaptive Commu-

nications Research Laboratories, Kyoto, Japan, from 1997 to 1998. He waswith the Villanova University, Villanova, PA, USA, from 1998 to 2015, wherehe was a Research Professor with the Center for Advanced Communications.Since 2015, he has been with the Department of Electrical and ComputerEngineering, College of Engineering, Temple University, Philadelphia, PA,USA, where he is currently an Associate Professor. His general researchinterests lie in the areas of statistical signal and array processing applied toradar, communications, and navigation, including compressive sensing, convexoptimization, time-frequency analysis, MIMO system, radar imaging, targetlocalization and tracking, wireless networks, and jammer suppression. He haspublished more than 300 journal articles and conference papers and 12 bookchapters. Dr. Zhang received the 2016 Premium Award from the Institutionof Engineering and Technology (IET) for Best Paper published in IET Radar,Sonar & Navigation.

Dr. Zhang is a member of the Sensor Array and Multichannel (SAM)Technical Committee of the IEEE Signal Processing Society. He is anAssociate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING,and serves on the Editorial Board of the Signal Processing journal. He wasan Associate Editor for the IEEE SIGNAL PROCESSING LETTERS during2006–2010, and an Associate Editor for the Journal of the Franklin Instituteduring 2007–2013.

17

Moeness G. Amin (F’01) received the Ph.D. de-gree in electrical engineering from the Universityof Colorado, Boulder, CO, USA, in 1984. Since1985, he has been with the Faculty of the De-partment of Electrical and Computer Engineering,Villanova University, Villanova, PA, USA, where hebecame the Director of the Center for AdvancedCommunications, College of Engineering, in 2002.Dr. Amin is a Fellow of the Institute of Electricaland Electronics Engineers; Fellow of the Interna-tional Society of Optical Engineering; Fellow of

the Institute of Engineering and Technology (IET), and a Fellow of theEuropean Association for Signal Processing (EURASIP). Dr. Amin is theRecipient of the 2016 Alexander von Humboldt Research Award, the 2016IET Achievement Medal, the 2014 IEEE Signal Processing Society TechnicalAchievement Award, the 2009 Individual Technical Achievement Award fromthe European Association for Signal Processing, the 2015 IEEE Aerospace andElectronic Systems Society Warren D. White Award for Excellence in RadarEngineering, the IEEE Third Millennium Medal, the 2010 NATO ScientificAchievement Award, and the 2010 Chief of Naval Research Challenge Award.He is the recipient of the 1997 Villanova University Outstanding FacultyResearch Award and the 1997 IEEE Philadelphia Section Award. He was aDistinguished Lecturer of the IEEE Signal Processing Society, 2003-2004,and is presently the Chair of the Electrical Cluster of the Franklin InstituteCommittee on Science and the Arts. Dr. Amin has over 700 journal andconference publications in signal processing theory and applications. He co-authored 20 book chapters and is the Editor of the three books Throughthe Wall Radar Imaging, Compressive Sensing for Urban Radar, and Radarfor Indoor Monitoring published by CRC Press in 2011, 2014, and 2017,respectively.

Fulvio Gini (F) received the Doctor Engineer (cumlaude) and the Research Doctor degrees in electronicengineering from the University of Pisa, Italy, in1990 and 1995, respectively. In 1993 he joinedthe Department of Ingegneria dellInformazione ofthe University of Pisa, where he become AssociateProfessor in 2000 and he is Full Professor since2006. From July 1996 through January 1997, he wasa visiting researcher at the Department of ElectricalEngineering, University of Virginia, Charlottesville.He is an Associate Editor for the IEEE Transactions

on Aerospace and Electronic Systems since January 2007 and for the ElsevierSignal Processing journal since December 2006. He has been AE for the IEEETransactions on Signal Processing (2000-2006) and a Senior AE of the sameTransactions since February 2016. He was a Member of the EURASIP JASPEditorial Board. He was co-founder and 1st co-Editor-in-Chief of the HindawiInternational Journal on Navigation and Observation (2007–2011). He is theArea Editor for the Special issues of the IEEE Signal Processing Magazine.He was co-recipient of the 2001 IEEE AES Societys Barry Carlton Awardfor Best Paper. He was recipient of the 2003 IEE Achievement Award foroutstanding contribution in signal processing and of the 2003 IEEE AESSociety Nathanson Award to the Young Engineer of the Year. He is Memberof the Radar System Panel of the IEEE Aerospace and Electronic SystemsSociety (AESS) (2008–present). He is a member of the IEEE SPS AwardsBoard. He has been a Member of the Signal Processing Theory and Methods(SPTM) Technical Committee (TC) of the IEEE Signal Processing Societyand of the Sensor Array and Multichannel (SAM) TC for many years. Heis a Member of the Board of Directors (BoD) of the EURASIP Society, theAward Chair (2006-2012) and the EURASIP President for the years 2013-2016. He was the Technical co-Chair of the 2006 EURASIP Signal andImage Processing Conference (EUSIPCO 2006), Florence (Italy), of the 2008Radar Conference, Rome (Italy), and of the 2015 IEEE CAMSAP workshop,Cancun (Mexico). He was the General co-Chair of the 2nd Workshop onCognitive Information Processing (CIP2010), of the IEEE ICASSP 2014, heldin Florence (Italy), and of the 2nd, 3rd and 4th editions of the workshop onCompressive Sensing in Radar (CoSeRa). He was the guest co-editor of thespecial section of the Journal of the IEEE SP Society on Special Topicsin Signal Processing on “Adaptive Waveform Design for Agile Sensing andCommunication” (2007), guest editor of the special section of the IEEE SignalProcessing Magazine on “Knowledge Based Systems for Adaptive RadarDetection, Tracking and Classification” (2006), guest co-editor of the twospecial issues of the EURASIP Signal Processing journal on “New trends andfindings in antenna array processing for radar” (2004) and on “Advances inSensor Array Processing (in memory of Alex Gershman)” (2013). He is co-editor and author of the book Knowledge Based Radar Detection, Trackingand Classification (2008) and of the book Waveform Diversity and Design(2012). He authored or co-authored 9 book chapters, more than 120 journalpapers and more than 155 conference papers.