IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019 579

Sparse Antenna and Pulse Placementfor Colocated MIMO Radar

Ehsan Tohidi , Mario Coutino, Student Member, IEEE, Sundeep Prabhakar Chepuri ,Hamid Behroozi , Member, IEEE, Mohammad Mahdi Nayebi, Senior Member, IEEE, and Geert Leus, Fellow, IEEE

Abstract—Multiple-input multiple-output (MIMO) radar isknown for its superiority over conventional radar due to its an-tenna and waveform diversity. Although higher angular resolution,improved parameter identifiability, and better target detection areachieved, the hardware costs (due to multiple transmitters andmultiple receivers) and high-energy consumption (multiple pulses)limit the usage of MIMO radars in large scale networks. On onehand, higher angle and velocity estimation accuracy is required,but on the other hand, a lower number of antennas/pulses is desir-able. To achieve such a compromise, in this paper, the Cramer–Raolower bound (CRLB) for the angle and velocity estimator is em-ployed as a performance metric to design the antenna and the pulseplacement. It is shown that the CRLB derived for two targets isa more appropriate criterion in comparison with the single-targetCRLB since the two-target CRLB takes into account both the main-lobe width and the sidelobe level of the ambiguity function. In thispaper, several algorithms for antenna and pulse selection basedon convex and submodular optimization are proposed. Numericalexperiments are provided to illustrate the developed theory.

Index Terms—Angle and velocity estimation, antenna place-ment, MIMO radar, submodularity, two-target CRLB, pulseplacement.

I. INTRODUCTION

MULTIPLE input multiple output (MIMO) radar has beengaining a lot of interest during the last decade [2]. The

main reason behind this growth is the enormous capabilities thatthis type of radar provides, e.g., higher angular resolution, im-proved parameter identifiability, and radar cross section (RCS)diversity [2], [3]. Based on the antenna configuration MIMO

Manuscript received March 27, 2018; revised August 24, 2018; acceptedOctober 23, 2018. Date of publication November 16, 2018; date of currentversion December 18, 2018. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Remy Boyer. Thiswork was supported in part by the Vice President of Research and Technologyat Sharif University of Technology under Grant QB960606, and in part by theKAUST-MIT-TUD consortium under Grant OSR-2015-Sensors-2700 and theASPIRE Project (Project 14926 within the STW OTP programme), which isfinanced by the Netherlands Organisation for Scientific Research (NWO). Thework of M. Coutino was supported by CONACYT. This paper was presentedin part at the 51st Asilomar Conference on Signals, Systems, and Computers,Pacific Grove, CA, October 2017. (Corresponding author: Ehsan Tohidi.)

E. Tohidi, H. Behroozi, and M. M. Nayebi are with the Department of Elec-trical Engineering, Sharif University of Technology, Tehran 11365-11155, Iran(e-mail:,tohidi@ee.sharif.edu; behroozi@sharif.edu; nayebi@sharif.edu).

M. Coutino, S. P. Chepuri, and G. Leus are with the Faculty of ElectricalEngineering, Mathematics and Computer Science, Delft University of Technol-ogy, Delft 2628, The Netherlands (e-mail:,M.A.CoutinoMinguez-1@tudelft.nl;s.p.chepuri@tudelft.nl; g.j.t.leus@tudelft.nl).

Digital Object Identifier 10.1109/TSP.2018.2881656

radars are categorized, into colocated and widely-separatedMIMO radars. Colocated MIMO radars have closely locatedantennas, which see the targets from the same angle. High an-gular resolution, improved target parameter identifiability, andenhanced beampattern design exploiting waveform diversity aresome of the main advantages of colocated MIMO radars [4]–[10]. The other category, widely-separated MIMO radars, havetransmitter/receiver antennas placed far from each other. Thisresults in different target angles of view for different transmitter-receiver pairs. Improving radar performance by exploiting RCSdiversity, handling slow moving targets by exploiting Dopplerestimates from multiple directions, and supporting high reso-lution target localization are among the advantages of widely-separated MIMO radars (see [11]–[15]). In this paper, the focusis on the colocated MIMO radar configuration to estimate theangle and velocity of the targets (the developed design algo-rithms can be easily adapted to widely-separated MIMO radarsas discussed later).

Angle of arrival and velocity estimation are two main tasks ofany radar system [16]. Due to the additional degrees of freedom,MIMO radars perform these tasks much better than a single radar[9], [11], [17]. In [18] and [19] the Cramer-Rao lower bound(CRLB) for a MIMO radar has been derived to prove this ad-vantage. Beside the numerous advantages of MIMO radar overconventional radars, the main drawbacks of this radar is, how-ever, the large hardware costs due to multiple transmitter andreceiver chains, the high energy consumption due to multipletransmitted pulses, and the large computational complexity in-volved in processing the transmitted pulses. To reduce thesecosts, keeping in mind the low number of targets in the regionof interest, compressive sensing (CS) based approaches haveshown promising performance [16], [20]. Although CS-basedapproaches reduce the number of measurements to be processed,the hardware costs are not reduced. This is because of the densesampling matrices used in CS that limit the number of measure-ments while requiring all the antennas and pulses. Alternatively,antenna and pulse selection (i.e., employing only a subset of allthe antennas and pulses) via sparse sensing can be performedto reduce the hardware sensing costs as well as the energy con-sumption, while achieving the desired performance. We wouldlike to stress here that pulse placement for radar has been rarelyconsidered before. A closely related topic is waveform design[2] which deals with the design and selection of transmit wave-forms with proper characteristics. However, such designs are

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580 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019

mainly concerned with statistical properties of the signal withineach pulse rather than position of the transmit pulses within thepulse sequence.

Sensor selection is the problem of choosing a subset of sen-sors out of a set of candidate sensors. Sensor selection is impor-tant to reduce the hardware costs, computational complexity,network energy consumption, and has been studied vastly asdetailed next. A knapsack problem formulation for sensor se-lection is proposed in [21], where an algorithm based on a greedyheuristic is presented. Sensor selection via convex optimizationis proposed in [22], where the problem is first relaxed to aconvex program, and then, sensors are selected through solv-ing a convex optimization problem. Similarly, [23] proposeda sparsity-enforcing sensor selection scheme for direction ofarrival estimation, where a single-target CRLB is used as objec-tive function with additional constraints on the sidelobe level.In [24] and [25], sensor selection for general non-linear modelsthrough convex and submodular optimization, respectively, isproposed.

Antenna and pulse selection can be posed as a sensor selectionproblem where a subset of antennas and pulses is selected out ofa large number of antennas and pulses. We refer to this problemas antenna and pulse placement. Antenna placement in widelyseparated MIMO radar for joint target position and velocityestimation is studied in [11], [26], [27], which are all basedon the single-target CRLB. For instance, [28] proposed a DOAestimation framework for a MIMO radar in which transmit andreceive antenna positions are drawn at random from a uniformdistribution. In a similar way, by employing the single-targetCRLB in [29], joint array and waveform optimization techniquesfor MIMO radar are investigated. The authors in [29] show thatboth local and global errors incurred by the estimator must beconsidered during the design phase. In fact, these effects occurin low SNR scenarios when the estimator exhibits a thresholdeffect due to local maxima in the ambiguity function. However,interactions between two or more targets is not considered in[29]. This is achieved in this work by considering the generalexpression of the two-target CRLB.

Typically, radars transmit several pulses with a uniform timeseparation, which is called the pulse repetition interval (PRI). Byexploiting the phase differences of the reflected pulses from thetargets, Doppler (or velocity) estimation is performed [30]. A ve-locity estimation algorithm for wide-band frequency-modulatedcontinuous-wave radar systems using the phase differences ofconsecutive uniformly separated pulses is proposed in [31]. Toreduce the network energy consumption and processing costs,we aim to have an irregular pulse transmission pattern (i.e., bytransmitting only a subset of the uniformly separated pulses).Fig. 1(a) shows an example of such an irregular pulse placement.Similar to pulse placement, the idea behind antenna placementis to perform the angle of arrival estimation task with a smallernumber of antennas. In colocated MIMO radars, transmitters andreceivers are usually placed uniformly along a line with a spac-ing of half a wavelength. However, we want to systematicallydesign the transmitter-receiver positions to obtain a nonuniformarray with a reduced number of transmit/receive elements. Inparticular, we start with a large set of candidate locations where

Fig. 1. (a) Uniform vs. irregular pulse transmission. Dashed lines indicate thatthe pulses are not transmitted in that interval. (b) Uniform vs. irregular antennaplacement. Light colored antennas indicate that the antennas are not used duringtransmission or reception.

we can place the antennas. Then, we select the best subset out ofthose locations in order to achieve a desired estimation perfor-mance. This antenna placement procedure helps to reduce thehardware costs and computational complexity, while maintain-ing a prescribed performance. Fig. 1(b) illustrates an irregulartransmitter and receiver placement in comparison with a uni-form placement.

The aim of this paper is to find the optimal antenna and pulseplacement that guarantees a desired angle of arrival and velocityestimation accuracy. It should be noted that the performance istraded off with cost when the number of antennas and/or pulsesare reduced.

A. Contributions

In this paper, we study further the joint antenna and pulseplacement for a colocated MIMO radar for angle of arrival andvelocity estimation based on sparse sensing [32] by extendingour previous work [1] by a more detailed signal model, a deriva-tion of the two-target CRLB, and a submodular optimizationframework as a fast and reliable alternative approach to convexoptimization.

The conventionally used performance measure, namely, thesingle-target CRLB only considers the mainlobe width of theambiguity function but does not take into account the sidelobelevel. Therefore, we derive the CRLB for two targets, whichtakes into account both the mainlobe width and sidelobe level(particularly the sidelobe level around the mainlobe) of the am-biguity function. Based on this two-target CRLB, we proposeseveral performance measures and develop a number of algo-rithms for designing the optimal antenna and pulse placementof colocated MIMO radar systems. Firstly, single antenna pulseplacement and single pulse MIMO radar antenna placement arepresented as two specific cases of the problem. Then, we presentthe general case of joint antenna and pulse selection. Since theantenna-pulse selection is a combinatorial optimization, and isNP-hard [22], we propose several suboptimal algorithms forsolving the selection problem. One of the proposed approachesis based on submodular optimization. We prove the submod-ularity of the employed performance measure, which enablesus to use a greedy algorithm to perform the selection with

TOHIDI et al.: SPARSE ANTENNA AND PULSE PLACEMENT FOR COLOCATED MIMO RADAR 581

near-optimality guarantees. The second proposed approach isbased on convex optimization, where by employing some re-laxations, the optimization is turned into a convex program.However, due to these relaxations, a suboptimal solution is ob-tained in general. The advantages and disadvantages of thesetwo approaches are explained in more detail in Section IV.

B. Outline and Notations

The rest of the paper is organized as follows. In Section II,the signal model is introduced. Section III provides the requiredpreliminaries for this paper. The problem formulation is dis-cussed in Section IV. Two basic examples to illustrate the con-cept are presented in Section V. The proposed algorithms forthe most general form of the antenna-pulse selection problemare presented in Section VI. Simulation results are reported inSection VII. Finally, Section VIII concludes the paper.

We adopt the notation of using boldface lower (upper) case forvectors a (matrices A). The transpose, Hermitian, and complexconjugate operators are denoted by the symbols (.)T , (.)H , and(.)∗, respectively. RN×M is the set of N ×M real matrices.diag (a) indicates the diagonal matrix formed by the compo-nents of vector a along the main diagonal. det {.} is the matrixdeterminant and tr {.} is the matrix trace operator. In addition,||A||0 and ||a||0 are the number of non-zero entries of A anda, respectively. If a and b are two vectors, then 〈a,b〉 is theinner product between a and b (i.e., 〈a,b〉 = aHb). λmax{A}and λmin{A} are the maximum and minimum eigenvalues ofthe matrix A, respectively. Given a reference set U and a subsetA ⊆ U , the absolute complement of A is denoted as Ac, i.e.,Ac = U \ A. Finally, |A| is the cardinality of the set A.

II. SIGNAL MODEL

Consider a colocated MIMO radar with R receivers placedalong a line at coordinates [d, 2d, . . . , Rd], where d is the inter-element spacing, which is assumed to be λ/2 (λ is the wave-length). In addition, I transmitters are placed along a line atcoordinates [−d,−2d, . . . ,−Id]. The ith transmitter can trans-mit a waveform si(t) for P times with PRI TP . Note that si(t)is non-zero only in the interval [0, TP ]. Assume Q targets existin the region of interest. Our aim is to estimate the position andvelocity of the targets based on the received signals. In partic-ular, the direction cosine, uq = cos ψq with ψq being the qthtarget’s angle of arrival and radial velocity, vq of the qth target,are the desired unknown parameters.

The noiseless baseband representation of the signal receivedat the rth receiver during the time interval [pTP , (p+ 1)TP ] dueto all the targets is

xr (t) =Q∑

q=1

αqh(t; vq )sTr,q (t− pTP )φr (uq ), (1)

where αq is the effect of the qth target’s RCS and h(t; vq ) =exp(j4πvq t/λ) with 2vq/λ being the Doppler frequency ofthe qth target. Due to the colocated configuration assumption,for each target, the Doppler frequency and RCS seen by alltransmitter-receiver pairs are equal. Here, the assumption is

that the RCS and propagation attenuation are constant dur-ing the observation interval (i.e., Swerling I model). Further-more, sr,q (t) = [s1(t− τ1,q ,r ), . . . , sI (t− τI ,q ,r )]T ∈ CI in-cludes the received signal from all the transmitters, where

τi,q ,r = c−1 [(Rq − di cosψq ) + (Rq − dr cosψq )]

= c−1 [2Rq − (di + dr )uq ]

is the time delay of signal propagation between the ith trans-mitter, qth target, and rth receiver, and c the speed of light. Inaddition,

φr (uq )=[exp(−j2πfcτ1,q ,r ), . . . , exp(−j2πfcτI ,q ,r )]T ∈ CI

contains the related phase shifts with fc being the carrier fre-quency. In the expressions for sr,q (t) and φr (uq ), Rq is the qthtarget distance from the center of the coordinate system, and diand dr are the positions of transmitter i and receiver r on thex-axis, respectively.

Linear frequency modulation (LFM) is selected for signaling.In fact, we adopt a set of LFM signals that have the same shape,but are slightly shifted in time which yields an efficient orthog-onal transmission scheme [33]. More specifically, we design theith transmitted waveform as si(t) = s(t− (i− 1)tsh), wheretsh is the time shift between

adjacent LFM signals to achieve orthogonality and s(t) is thebaseband LFM waveform

s(t) = exp(jπkt2

), (2)

where k is the rate of sweeping the whole bandwidth for thepulse duration TC , i.e., k = B/TC with B being the signalbandwidth. Note that to satisfy the orthogonality condition, tshshould be selected larger than the time delay of the farthesttarget of interest. In addition, since all pulses from all antennasneed to fit within a single PRI after reception, we need thecondition TC + Itsh < TP . It should be pointed out that anyset of orthogonal waveforms other than LFM signaling couldalso be employed for our model.

Employing I matched filters (de-ramping plus filtering)matched to the I transmit waveforms, the observed signal inthe time interval [pTP , (p+ 1)Tp ] from all the transmitters ofthe rth receiver zr (t) = [zr,1(t), . . . , zr,I (t)]T , after some sim-plifications is given by

zr (t) =Q∑

q=1

αqh(t; vq )βq (t− pTP )φr (uq ) + er (t), (3)

where βq (t) = exp{j4πktRqc−1}. Note that the entries of

er (t) = [er,1(t), . . . , er,I (t)]T are noise terms at the output ofthe I matched filters at the rth receiver. Since the I transmitwaveforms are orthogonal, the entries of er (t) can be assumedindependent. Thus, er,i(t), i = 1, . . . , I, are modeled as i.i.d.with distribution N (0, σ2

e ). It is worth mentioning that if weconsider correlated clutter in (3), based on the methods pre-sented in [34], [35], we can design samplers to handle clutter.To use the aforementioned methods, we have to assume thatthe clutter covariance matrix as a function of the number of

582 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019

antennas/pulses is known. In other words, for the proposed off-line design, we require knowledge of the model information.

Sampling the observed signal with sampling period Ts , Nsamples per pulse are obtained, where the nth sample of the pthpulse related to the transmitter-receiver pair (i, r), zr,i,p [n], isgiven by

zr,i,p [n] = zr,i(pTp + nTs)

=Q∑

q=1

αqh(pTP + nTs ; vq )βq (nTs)φr,i(uq )

+ er,i(pTP + nTs)

=Q∑

q=1

y(q)r,i,p [n] + er,i,p [n] = yr,i,p [n] + er,i,p [n], (4)

where φr,i(uq ) is the ith entry of φr (uq ). Collecting all themeasurements, we have a non-linear model of the form

z = y(θ) + e ∈ CNRIP , (5)

where the unknown parameters of the qth target are representedby the vector θq = [uq , vq ]T ∈ R2 . So, θ = [θT1 , . . . ,θ

TQ ]T ∈

R2Q collects all the unknown parameters.

III. PERFORMANCE METRIC

As seen in Section II, the measurements are a non-linear func-tion of the unknown parameters. As a result, the mean squarederror (MSE) does not admit a closed form expression [36]. Onthe other hand, the CRLB provides a lower bound on the varianceof any unbiased estimator and can be used to evaluate the per-formance of unbiased estimators. Since the CRLB can alwaysbe computed in closed form, it is employed as an estimationperformance criterion.

It is well known that under the regularity condition, the co-variance of any unbiased estimator θ of the unknown vector θis lower bounded by the CRLB as [24], [37]:

E{(θ − θ)(θ − θ)H } ≥ C(θ) = F−1(θ), (6)

where C is the CRLB matrix and F is the Fisher informationmatrix (FIM), which can be calculated as [24]

F(θ) = −E

{∂2 ln p(z;θ)∂θ ∂θH

}

= E

{∂ ln p(z;θ)

∂θ

∂ ln p(z;θ)∂θH

}∈ C2Q×2Q , (7)

where p(z;θ) is the probability density function (pdf) of z pa-rameterized by the unknown vector θ. Due to uncorrelated er-rors, the log-likelihood ln p(z;θ) is additive, and given by

ln p(z;θ) =R∑

r=1

I∑

i=1

P∑

p=1

N∑

n=1

ln p(zr,i,p [n];θ). (8)

Due to (8) the FIM in (7) is also additive and can be writtenas [24]

F(θ) =R∑

r=1

I∑

i=1

P∑

p=1

Fr,i,p(θ), (9)

whereFr,i,p(θ) is the FIM of the pth pulse due to the transmitter-receiver pair (i, r) for all the N samples, i.e.,

Fr,i,p(θ) =N∑

n=1

Fr,i,p,n (θ)

=4σ2e

N∑

n=1

∂yr,i,p [n]∂θ

∂yr,i,p [n]∂θH

. (10)

One of the contributions of this paper is to introduce theCRLB for two targets as a better performance measure for an-tenna/pulse selection in comparison with the CRLB for a singletarget. The reasoning is based on the fact that, for two targets, thecorrelation between the signals echoed from the targets is takeninto account and both the estimation accuracy (mainlobe widthof the ambiguity function) and the robustness against ambigu-ities (the sidelobe level around the mainlobe of the ambiguityfunction) are accounted for in the cost function that is used tooptimize the antenna/pulse placement. In contrast, in the single-target CRLB, only the estimation accuracy of one target is con-sidered, which essentially makes the mainlobe width as narrowas possible, ignoring the occurrence of ambiguities due to thesidelobes (i.e., a high sidelobe level around the mainlobe mightoccur due to the nonuniform antenna/pulse placement). This isthe main advantage of considering a multi-target CRLB insteadof a single-target CRLB. Due to these reasons, we employ thetwo-target CRLB as a performance measure in our optimizationproblems. In the following, the FIMs for the two-target case forall the 2Q unknown parameters are derived.

A. Two-target CRLB

In this scenario, two targets are considered in the region ofinterest and the CRLB is derived for these two targets. ForQ = 2, (4) simplifies to

zr,i,p [n] = yr,i,p [n] + er,i,p [n]

= α1h(pTP + nTs ; v1)β1(nTs)φr,i(u1)

+ α2h(pTP + nTs ; v2)β2(nTs)φr,i(u2)

+ er,i,p [n]. (11)

The partial derivative of the signal w.r.t. the unknowns is givenby

∂yr,i,p [n]∂θ

=j2πλ

⎛

⎜⎜⎜⎜⎜⎝

(di + dr )y(1)r,i,p [n]

2(nTs + pTP )y(1)r,i,p [n]

(di + dr )y(2)r,i,p [n]

2(nTs + pTP )y(2)r,i,p [n]

⎞

⎟⎟⎟⎟⎟⎠, (12)

where y(1)r,i,p [n] and y(2)

r,i,p [n] are the noiseless signal terms dueto the first and second target, respectively. This allows us to

TOHIDI et al.: SPARSE ANTENNA AND PULSE PLACEMENT FOR COLOCATED MIMO RADAR 583

compute the Fisher information matrix as

Fr,i,p,n =

[J(1)

1 J2

J∗2 J(2)1

], (13)

where J(q)1 is the single-target FIM for the qth target given by

J(q)1 =

32π2α2q

λ2σ2e

⎡

⎣12(di + dr )2 (di + dr )(nTs + pTP )

(di + dr )(nTs + pTP ) 2(nTs + pTP )2

⎤

⎦,

(14)

whereα2q

σ 2e

is the signal to noise ratio (SNR). The expression (14)implies that the single-target FIM is independent of θ and thusFr,i,p in (10) and F in (9) are also independent of θ for a single-target scenario. In addition, J2 ∈ C2×2 is the cross correlationbetween the signals of the two targets calculated as follows

J2 =32π2α1α

∗2

λ2σ2e

⎡

⎣12(di + dr )2 (di + dr )(nTs + pTP )

(di + dr )(nTs + pTP ) 2(nTs + pTP )2

⎤

⎦

h(pTp + nTs ; v1 − v2)φr,i(u1 − u2). (15)

It is easy to see that the unknown parameters appear only inthe cross correlation terms between the two targets. Moreover,the Fisher information matrix only depends on the differencebetween direction cosines and on the difference between thevelocities. As will be seen later, we use this characteristic toreduce the search space. The final expression of the FIM iscalculated from (9) and (10).

The calculated CRLB and Fisher information matrices areuseful when all the unknown parameters in θ have the sameunits. However, in this paper we have parameters with differentunits such as direction cosine and velocity (i.e., cosine of radi-ans andm/s). Moreover, the desired estimation accuracy for thetwo targets might be different as well. In fact, if the estimationerror of one of the parameters is much higher than that of theothers, then that parameter would play the dominant role in theoptimization problem and the selection would be based on thatparameter solely. As a result, the final design would not be sat-isfactory in terms of the other parameters’ estimation accuracy.Thus, to make a balance among the parameters, we introducecompensation weights and modify the CRLB matrix as

C′(θ) = diag (γ)C(θ)diag (γ), (16)

where γ2i is the known compensation weight for the ith un-

known parameter which depends on the application and γ =[γ1 , . . . , γ2Q ]T . Similar to the modified CRLB matrix, we candefine the modified Fisher matrix as

F′(θ) = diag−1{γ}F(θ)diag−1{γ}. (17)

As seen in (13) and (15), the two-target CRLB (unlike thesingle-target CRLB) is a function of the unknown parameters.Therefore, while optimizing the CRLB, in order to keep the op-timization problem tractable, we grid the region of interest intoa discrete set of points for which we can evaluate the CRLB.Since the two-target CRLB only depends on the difference be-tween the direction cosine and velocity [cf. (15)], we only gridthese differences in the region of interest resulting in the setD = {δθ1 , ...δθD}, where δθd denotes the dth difference be-tween the two targets’ parameters. Hence, a 1-D scan of thedifference of each parameter suffices to obtain all feasible two-target CRLB matrices. Since the CRLB is a matrix, in the nextsection, we introduce scalar measures of the CRLB as optimiza-tion criteria to design the antenna/pulse placement that shouldbe optimized over these grid points.

IV. PROBLEM FORMULATION

In this work, on one hand, we want to reduce the sensingcost, i.e., reduction in the number of transmitters, pulses, andreceivers, while guaranteeing a desired estimation error. The Rreceivers next to the I transmitters and P pulses of each trans-mitter are the parameters that affect both the estimation qualityand the sensing cost (hardware and computational complexity).Therefore, the selection problem might be posed in the follow-ing two ways. In the first problem, we minimize the sensing costwith a constraint on the estimation error. In the second problem,we minimize the estimation error with a constraint on the sens-ing cost. Since we know how many antennas are available, wefocus on the second problem. The other case can be tackled ina similar way when the desired estimation error is known.

We model the sensing framework by introducing the follow-ing sets: (i) the set of selected transmitter-pulsesA ⊆ P , whereP = {a1,1 , a1,2 , . . . , aI ,P } is the set of all the IP transmitter-pulses, and (ii) the set of selected receivers B ⊆ R, whereR = {b1 , . . . , bR} is the set of all the R receivers. In addi-tion, we further introduce the transmitter-pulse selection matrixand the receiver selection vector for easier notation. That is,A ∈M{A} is a transmitter-pulse selection matrix defined bythe set of selected transmitter-pulses A, with M{A} definingthe singleton:

M{A} = {A|A ∈ {0, 1}I×P ; [A]i,p = 1 ⇐⇒ ai,p ∈ A}.(18)

Here, the (i, p)th entry of A, denoted by [A]i,p , is equal to 1(0)if the ith transmitter transmits the pth pulse (or not). In a similarway, we introduce b ∈ V{B} as the receiver selection vectordefined by the set of selected receivers B, with V{B} definingthe singleton:

V{B} = {b|b ∈ {0, 1}R ; [b]r = 1 ⇐⇒ br ∈ B}, (19)

where the rth entry of b, denoted by [b]r , is equal to 1(0) if therth receiver is (not) selected.

Note that these definitions are in the most general form withcomplete freedom to select any of the transmitters, pulses, orreceivers. For specific purposes, which are discussed later on,one may consider only the receiver selection vector (i.e., by

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employing all the transmitters transmitting all the pulses), onlythe transmitter-pulse selection matrix (i.e., by employing all thereceivers), or the selection vectors for receivers and transmitters(where we assume that each active transmitter would transmitall the pulses). As there is a one-to-one relation between thematrix (vector) and the setA (B), from now on we employ theminterchangeably.

Using the selection variables A and b, the collected measure-ments can be written as follows

zr,i,p [n] = [b]r [A]i,p(yr,i,p [n] + er,i,p [n]), (20)

where depending on whether a transmitter-receiver-pulse is se-lected, the measurement will be collected. It is easy to show thatthe Fisher information matrix [cf. (9)] will be modified basedon (20) as

F(A,b, δθ) =R∑

r=1

[b]rI∑

i=1

P∑

p=1

[A]i,pFr,i,p(δθ). (21)

Because the two-target FIM is used, the difference of the twotargets’ parameters is considered. The most general form of theoptimization problem can be mathematically formulated as

minA⊆P,B⊆R

g(f(A,b, δθ),D)

subject to A ∈M{A}, |A| ≤ KP ,

b ∈ V{B}, |B| ≤ KR, (22)

where KP and KR are the maximum number of transmitter-pulses and receivers, respectively. Here, f(A,b, δθ) is a func-tion of the estimation error at the grid point δθ ∈ D, g(·) is a gen-eral composition of the function f(·) evaluated over all the gridpoints in D, e.g., maximization or average of f(A,b, δθ) forall δθ ∈ D, the setsA and B represent the selected transmitters-pulses and receivers, respectively. To guarantee an estimationaccuracy level over all the grid points, g(·) should be the maxfunction. To guarantee an average accuracy level, g(·) can bedefined as the average over D. Since, (22) is a combinatorialoptimization problem and NP-hard in nature [22], we use con-vex relaxation techniques to employ convex optimization andsurrogate submodular functions to employ greedy optimizationas two general approaches to solve this problem.

As convex optimization requires a convex cost function andconvex constraints, we require a function f(A,b, δθ) that isconvex and that the non-convex sets A and B are relaxed toobtain convex constraints. Both the maximum and expectedvalue for g(·) could be employed for convex optimization. Byreformulation in its epigraph form, we will use maximizationfor convex optimization which in general leads to a semidefiniteprogram (SDP) that has a cubic computational complexity.

The other approach to solve this problem is to employ sub-modular optimization which has been shown useful to solvecombinatorial optimization problems [35], [38], [39]. A setfunction f : 2|N | → R is called submodular, if and only if, forevery S1 ⊆ S2 ⊆ N and u ∈ Sc

2 , it shows the property of di-minishing returns, i.e.,

f(S1 ∪ {u})− f(S1) ≥ f(S2 ∪ {u})− f(S2). (23)

It is known that, if the function f is nondecreasing, normal-ized and submodular, then by employing a conceptually simplegreedy algorithm, which starts with an empty (full) set, and initeration i, adds the best (removes the worst) element to (from)the set, to maximize the function (not minimize), it is possi-ble to obtain an 1− 1/e approximation of the optimum valueof maxS⊆N ,|S|≤K f(S) for some cardinality K [40], where erepresents Euler’s number. Thus, if g(f(A,b,θ),D) satisfiesthis property, then we can use the greedy algorithm with near-optimality guarantees.

In essence, we could say that the advantage of the convex op-timization approach is its higher freedom in terms of objectivefunctions and constraints. On the other hand, submodular opti-mization generally leads to low computational methods whichmakes it appropriate for large-scale scenarios. In Section VI,both algorithms will be explained in more detail.

A. Scalar Measures of the CRLB

Since the CRLB is a matrix, it is not possible to employ it asan objective function for the optimization problem. Thus, in thefollowing, scalar measures of the CRLB (or the FIM) that areemployed in the proposed algorithms are introduced [24], [36],[41].

� A-optimality: minimize the trace of the CRLB, i.e.,f(A,b, δθ) = tr {C(A,b, δθ)}.

� D-optimality: maximize the determinant of the FIM, i.e.,f(A,b, δθ) = log det (F(A,b, δθ)).

� E-optimality: minimize the maximum eigenvalue of theCRLB, i.e., f(A,b, δθ) = λmax{C(A,b, δθ)}.

� Modified frame potential: The frame potential is a scalarquantity that measures the orthogonality between vectorsof a frame [36], [42]. In this paper, the elements of thesevectors are related to the measurements due to differentpulses and receivers, and orthogonality translates to howuncorrelated these measurements are. We are interested ina set of measurements that provides the maximum per-formance meaning that the measurements should be asuncorrelated as possible. In other words, when the mea-surements are statistically orthogonal or less correlated,it implies that the corresponding pulses and receivers aremore informative. Therefore, the frame potential can beconsidered as a good performance measure for designingantenna/pulse placements. Due to the non-linearity of ourmodel, we employ the first derivative of the measurements∂yr,i,p [n]/∂θ as defined in (12). For each i, r, and pwhichis selected, we have N entries in the measurement matrix.Thus, the FP for our system model would be

FP(S, δθ) =∑

y ,y ′∈Y{S}

N∑

n=1

|〈∂y[n], ∂y′[n]〉|2 , (24)

where ∂y[n] is a simplified notation for ∂y[n]/∂θ fory ∈ Y{S}, S = A ∪ B is the union set of transmitterpulses in set A, and receivers in set B, and Y{S} ={yr,i,p [n], n ∈ {1, . . . , N} | r ⇐⇒ br ∈ B, (i, p) ⇐⇒ai,p ∈ A} is the set of measurements due to the

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Fig. 2. Single antenna pulse selection based on velocity estimation error for a total of P = 12 pulses: (a) selected pulses, (b) velocity ambiguity function,(c) scalar measures performance.

transmitter-pulses and receivers of S presented in (4).Even though the dependency with respect to δθ is notexplicitly stated in (24), substituting (12) in (24), it isstraightforward to show that (24) is a function of theparameters difference vector, δθ. It has been shown thatthe FP performs the best under equal row norms andthat the minimization of the FP and the MSE is related.However, in this problem, rows have different norms. Onone hand, rows with lower norms are prioritized by the FP,but on the other hand, rows with higher norms contributemore to the estimation accuracy. Thus, we propose tonormalize the rows and call the related FP the modifiedFP (MFP), which is given by

FP(S, δθ) =

∑

y ,y ′∈Y{S}

N∑

n=1

∣∣∣∣〈∂y[n], ∂y′[n]〉

〈∂y[n], ∂y[n]〉 〈∂y′[n], ∂y′[n]〉∣∣∣∣2

. (25)

The above mentioned measures are employed as costfunctions in different algorithms, which are presented in thefollowing. It would be shown later that some are appropriatefor convex optimization, while others are good for submodularoptimization. It should be noted that each of these measureshas some advantages and disadvantages. In other words, noneof them are the best in general and based on the applicationand requirements, one may employ one or another. Finally,as mentioned before, the developed design approach can beadapted to widely-separated MIMO radars. Although therewould be some minor changes in the signal model and theCRLB derivation, the overall idea would be the same andsimilar algorithms would be applicable.

V. TWO BASIC EXAMPLES

In this section, by explaining two simple examples, the ideabehind this work is illustrated. In addition, some useful insightscan be obtained from these examples. It should be mentionedthat these are just small-scale examples for illustrating the gen-eral idea. As a result, an exhaustive search is used for solvingthe optimization problem (22). The proposed algorithms forlarge-scale problems are explained in Section VI.

A. Single Transmitter-Receiver Pair and Multiple Pulses

In this example, we consider the problem of a singletransmitter-receiver pair which is able to transmit P identicalpulses. The aim is to compare the estimation accuracy betweenemploying all the pulses or just a few pulses after an appropri-ate selection. Omitting the transmitter and receiver indices, themeasured signal for the pth pulse would be

zp [n] = [a]p(yp [n] + ep [n]), (26)

where the matrix A is now a column vector a, as a single trans-mitted is selected. In addition, as a single receiver is selected thevector b is now omitted. The optimization problem (22) thensimplifies to

mina

maxδθ∈D

f(a, δθ)

subject to ||a||0 ≤ KP ,a ∈ {0, 1}P , (27)

where f(a, δθ) is one of the aforementioned measures, δθ =v1 − v2 (since we are dealing with a single antenna pair there isno angle estimation, and the direction cosine is not considered asan unknown parameter), a is the pulse selection vector, andKP

is the constraint on the number of transmitted pulses. We solvethis problem by performing an exhaustive search over all thepossible combinations of pulses for both the single- and two-target CRLB criterion. In Fig. 2, the result of pulse selectionon the velocity estimation error is represented where P = 12pulses in total are considered. In one case 5 and in the other case8 pulses are selected. The result of pulse selection for these twocases for the single and two-target CRLB using A-optimalityas the performance measure is shown in Fig. 2(a). It is clearthat for the single-target CRLB case, the selection prioritizesthe edges. However, for the two-target CRLB case, edge pulsesare combined with intermediate pulses. This difference in pulsepattern causes the difference in the velocity ambiguity functionwhich is depicted in Fig. 2(b). It can be seen that, employingthe two-target CRLB reduces the sidelobe level (especially forthe sidelobes close to the mainlobe) at the price of a widerbeamwidth. Finally, Fig. 2(c) shows the trace of the two-targetCRLB for different cost functions. Definitely, A-optimality isperforming better than the others, because both the optimizationcost function and the evaluation measure are the same. However,

586 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019

Fig. 3. MIMO radar antenna selection based on angle estimation error for a total of 8 transmitters and 4 receivers. (a) Selected antennas,(b) beampatterns for 4 transmitters and 3 receivers, (c) beampatterns for 6 transmitters and 2 receivers.

based on the plot, it turns out that all measures are performingsimilarly. In addition, the MSE of the maximum likelihood es-timator (MLE) is also plotted for the optimal subset of pulsesbased on A-optimality, which shows the introduced measure isa good representative for the MSE. Note that although the op-timization of the MSE of the estimator was the original aim,the MSE does not admit a closed form which makes it difficultto optimize. In contrast, calculating the surrogate measures wementioned before is straightforward and based on Fig. 2(c), weobserve that they are consistent with the MSE.

B. MIMO Radar and Single Pulse

In this scenario, we are investigating another phenomenon,which is the effect of the antenna positions on the target angleestimation error. Since the number of pulses does not play a rolein this example, a single pulse is considered for simplicity. Theaim is to find the optimal antenna placement for a maximumangle estimation accuracy using different numbers of antennas.The optimization problem is as follows

mina,b

maxδθ∈D

f(a,b, δθ)

subject to ||a||0 ≤ KI ,a ∈ {0, 1}I ,||b||0 ≤ KR,b ∈ {0, 1}R , (28)

where f(a,b, δθ) is one of the different measures, δθ =u1 − u2 , a and b are the transmitter and receiver selectionvectors, respectively, and KI and KR are the total number ofselected transmitters and receivers, respectively. Similar to theprevious example, the optimization is solved by performing anexhaustive search over all possible combinations of transmittersand receivers.

As an example, we perform the optimization for a total of8 transmitters and 4 receivers and only consider A-optimality.Fig. 3 represents the result for two cases: 4 transmitters com-bined with 3 receivers and 6 transmitters combined with 2 re-ceivers for the single and two-target CRLB. The selected an-tennas are depicted in Fig. 3(a) for these four cases. As for thesingle antenna pulse selection example, the selected antennasfor the single-target CRLB have a tendency to appear at theedges. However, for the two-target CRLB, antennas from both

the edges and the middle of the array are selected. In addition,Figs. 3(b) and 3(c) compare the beampatterns for the single andtwo-target CRLB. For both patterns, the sidelobe levels close tothe mainlobe are reduced when the two-target CRLB is used incomparison with the examples obtained using the single-targetCRLB. However, in Fig. 3(c), higher sidelobes appear furtheraway from the mainlobe. This effect is due to the fact thatthe sidelobes close to the mainlobe cause an ambiguity in dis-tinguishing the two targets whereas the other sidelobes do not.Thus, the antenna selection focuses more on this issue. Note thatit is possible to apply different weights to different u-coordinatesin order to emphasize some specific regions in the beampattern.

C. Discussion

Based on the above two simple examples, it seems reason-able to seek the optimum sparse sensing scheme (both spatialand temporal) for different numbers of antennas and pulsesand compare the estimation accuracy with full sensing. It maybe possible to significantly reduce the number of samples atthe price of only a small reduction in estimation accuracy. Inthe following sections, the general problem is stated, algorithmsare proposed, and simulation results are presented.

VI. TRANSMITTER-RECEIVER SELECTION

Let us now study the most general case of transmitter-receiver-pulse selection. In other words, we would try to solvethe original problem stated in (22). It should be noted that,a transmitter is selected if and only if, it transmits at leastone pulse. In the following, we would propose two gen-eral approaches to solve the problem: convex and submodularoptimization.

A. Convex Optimization - E-optimality

In this subsection, we try to solve the problem employing con-vex optimization. In principle, all scalar measures could be usedsince they are all convex, however we only consider E-optimalityhere because it is the easiest to formulate. By restricting g(·)to be the maximum value and f(·) to be the maximum eigen-value of the CRLB, and by relaxing the Boolean constraints in(22), the optimization problem can be written in the epigraph

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form as

maxA ,b,γ

γ

subject to F(A,b, δθ) � γI4×4 ,∀δθ ∈ DI∑

i=1

P∑

p=1

[A]i,p ≤ KP ,

R∑

r=1

[b]r ≤ KR,

0 ≤ [A]i,p ≤ 1, 1 ≤ i ≤ I, 1 ≤ p ≤ P,0 ≤ [b]r ≤ 1, 1 ≤ r ≤ R. (29)

where F(A,b, δθ) is the Fisher information matrix, A and bare the selection matrix and selection vector defined in (21),respectively, and KP and KR are the number of selected pulsesand receivers, respectively. Due to the presence of the productsof unknowns (i.e., see (21)), the optimization problem in (29)is not convex. Therefore, a convexifying process is introducedin several steps. First we define a pulse selection vector byvectorizing the selection matrix (i.e., a = vec (A)). Then, weintroduce the total selection vector w by concatenating both thepulse and receiver selection vectors as

w = [aT ,bT ]T . (30)

Finally, we introduce the total selection matrix as W = wwT .Employing this new selection vector and matrix, the multiplica-tion of the unknowns can be eliminated, and W = wwT is theonly remaining non-convex term. We relax the rank one con-straint to the inequality form W � wwT which by employingthe Schur complement [43], leads to the following inequality

[W w

wT 1

]� 0. (31)

Therefore, the relaxed convex optimization problem can bestated as

maxW ,w ,γ

γ

subject to F(W, δθ) � γI4×4 ,∀δθ ∈ D,[

W w

wT 1

]� 0,

[W]i,j = [W]j,i , 1 ≤ i, j ≤ IP +R,

[W]i,i = [w]i , 1 ≤ i ≤ IP +R,

IP∑

i=1

[w]i ≤ KP ,IP +R∑

r=IP +1

[w]r ≤ KR,

0 ≤ [w]i ≤ 1, 1 ≤ i ≤ I × P +R, (32)

where now the Fisher information matrix, F(W, δθ), is ex-pressed in terms of W and is given by

F(W, δθ) =R∑

r=1

[W]IP +r,IP +r

I∑

i=1

P∑

p=1

[W]I (p−1)+i,I (p−1)+iFr,i,p(δθ). (33)

The optimization problem in (32) is a standard semidefiniteprogramming problem in the inequality form which can be effi-ciently solved in polynomial time using interior-point methods.We can solve (32) with any of the off-the-shelf solvers. Thesolution of the relaxed optimization problem is used to computethe suboptimal Boolean solution for the selection problem. Astraightforward technique that is often used is based on a simplesorting technique, in which the KP pulses corresponding to thelargest values in A and the KR receivers corresponding to thelargest values in b are selected as the transmitted pulses andreceivers, respectively (A and b are obtained from the selectionvector w and considering (30)). However, randomized roundingis employed here which selects the antennas and pulses with aprobability equal to the output of the convex problem. At theoutput of the convex optimization problem, we have a selectionmatrix and a selection vector whose entries are between 0 and1 for transmitter pulses and receivers, respectively. To obtain asuboptimal Boolean solution employing randomized rounding,we repeat the following procedure L times: generate a Booleanmatrix and a Boolean vector where the entries are set to 1 withprobability given by the corresponding entries in the selectionmatrix and vector, respectively. Due to the constraints on thenumber of pulses and receivers, some of these rounding realiza-tions may not be feasible. Therefore, among theseL realizations,the one that is feasible and achieves the highest maximum ob-jective value is selected. More details of randomized roundingare explained in [24].

B. Submodular Optimization—MFP

Although convex optimization is an efficient method, in thissection, greedy submodular optimization is considered as a so-lution approach. The reason is the computational complexitywhich is much lower for greedy algorithms in comparison withconvex optimization algorithms. This issue is especially impor-tant when dealing with large-scale scenarios.

Let us recall P , the set of all transmitters-pulses, and R, theset of all receivers. Furthermore, let us consider A ⊆ P andB ⊆ R as the set of selected pulses and receivers, respectively,and S = A ∪ B as the union set of transmitter-pulses in A andreceivers in B. Finally, we define the ground set U = P ∪R asthe union set of all the transmitter-pulses and receivers.

Now, we introduce a set functionG : 2|U| → R+ , defined overthe subsets of the ground set U , as the performance measurewhich is defined based on the modified frame potential as

G(X ) = FP(U)− FP(U \ X ) for X ⊆ U , (34)

588 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019

where FP(U) and FP(X ) are the MFPs due to the set of mea-surements Y(U) and Y(X ), respectively. It is clear from thedefinition (34) that aiming to maximize G(Sc), where Sc isthe complementary set of S, i.e., Sc = U \ S, is tantamount tominimizing the MFP for the selected set of measurements S.Therefore, it is possible to use (34) as a performance metric toselect the set of transmitter-pulses S by first identifying whichelements should be discarded. In the following, the next theoremguarantees the submodularity of the performance measure andthus gives near-optimal guarantees when the greedy algorithmis employed.

Theorem VI.1. For transmitter-receiver selection,G : 2|U| →R+ is a normalized, monotone, submodular set function.

Proof: The proof is derived in Appendix A. �The transmit pulse-receiver selection problem, using the per-

formance metric defined in (34) and the union set S, can be nowformally introduced as

maxSc ⊆ U

G(Sc)

subject to Sc ∈ Ip(IP −KP ,R−KR ), (35)

where Ip(IP −KP ,R−KR ) is a partition matroid [44]whose independent sets are defined as

Ip(IP −KP ,R−KR ) = {X : |X ∩ P| ≤ IP −KP ,

|X ∩ R| ≤ R−KR |}, (36)

leveraging the fact that {P,R} is a proper partition of U . Dueto the monotonicity of G the maximum is achieved when theinequalities in the definition of the partition matroid are metwith equality [cf. (36)]. Therefore, the complementary set ofthe solution set of (35) will meet the following properties:

|S ∩ P| = KP , |S ∩ R| = KR, (37)

which are desired cardinality conditions for the set of selectedtransmitter-pulses. The following greedy algorithm is proposedfor transmit pulse-receiver selection. At the starting point, allpulses and receivers are selected, i.e., S = P ∪R. That is, weinitialize the algorithm with Sc = Ø. Then, in each step, thegreedy algorithm selects the element, either a receiver or trans-mit pulse, providing the highest cost function value and adds it tothe setSc . This procedure continues until the constraints are met.It should be noted that, if one of the constraints of the partitionmatroid is met with equality while the other is not, the proposedmethod continues adding elements (receivers or transmit pulses)until the desired cardinality is achieved. Fortunately, due to thestructure of the ground set, and its partition, the independenceoracle is easily implemented, i.e., routine for checking if a givenset S is contained in a given matroid. Therefore, no overhead isincurred due to this procedure. The pseudocode of the algorithmis presented in Algorithm 1. The set returned by Algorithm 1achieves 1/2 near-optimality guarantee [45]. In the case that thematroid (36) is substituted by a cardinality constraint on the setSc , the greedy heuristic returns a 1− 1/e near-optimal set. Thissituation can arise in instances when instead of having sepa-rated budget for transmit pulses and receivers, a joint budget isconsidered.

Algorithm 1: Transmitter-Pulse-Receiver Greedy SelectionBased on MFP.

Initialization:Sc = ØV ← UGreedy algorithm:while V �= Ø dou∗ = argmax

u ∈ VG(Sc ∪ {u})

if Sc ∪ {u∗} ∈ Ip thenSc ← Sc ∪ {u∗}

end ifV ← V \ {u∗}

end whileS ← U \ Sc

S ∩ P and S ∩R are the selected transmitters-pulsesand receivers, respectively.

C. Fixed Receivers - Selection of Pulses and Transmitters

The most general form of the optimization problem was stud-ied in the previous section. In this scenario, we want to introducethe special case of fixed receivers. Proposing this special case isworthwhile for two reasons. First, in some applications we mayhave the freedom to only select pulses while the transmitters-receivers are fixed. The second reason is that this is a simplerversion of the general case that helps to clarify part of the gen-eral case, i.e., in the procedure to solve the general problem, ifthe constraint is met for one of the parameters (i.e., pulses or re-ceivers), the algorithm continues for the other parameter whichis similar to this special case. In this case, while the receivers areconsidered to be fixed, we are trying to optimize the selection ofpulses and transmitters to minimize the target’s angle-velocityestimation error (the other case of fixed transmitter-pulses andthe selection of receivers is similar). Since convex optimizationbased on E-optimality and submodular optimization based onthe MFP for the general case have been already covered, wewill not repeat these discussions here for this special case sincethey are similar and even simpler. However, we show here thatin this case the log determinant is also a submodular functionand it is possible to employ the greedy heuristic as an alternativeoptimization algorithm to solve the transmitter-pulse selectionproblem near optimally. It should be pointed out that the logdeterminant is not a submodular function for the general opti-mization problem and thus we only employ it as an objectivefunction for this special case.

Submodular optimization – D-optimality: In this case, weconsider the log determinant set function be defined as

h(S) =

{0 S = Ø

log det (F S) otherwise, (38)

where F S is the Fisher information matrix [cf. (9)] obtainedby employing all the pulses in S ⊆ P . The set function (38) isemployed as a performance measure (D-optimality). The greedyalgorithm goes as follows. We start with all pulses and all trans-mitters (i.e., S = P). In each step, we remove the pulse that re-duces the goal function the least. This procedure is repeated until

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TABLE ICOMPUTATIONAL COMPLEXITY OF THE PROPOSED ALGORITHMS

Fig. 4. MSE versus number of transmitted pulses for (a) 16 pulses in total, (b) 48 pulses in total. (c) MSE versus number of transmit pulses for differentapproaches.

Algorithm 2: Transmit Pulse Greedy Selection Based onLog Determinant.

Initialization:S = P;Greedy algorithm:while |S| > KP dop = argmax

p∈Sh(S \ {p}) [cf. (38)]

S = S \ {p};end whileS is the set of selected transmit pulses.

we achieve the required number of pulses. The pseudocode ofthe greedy algorithm is presented in Algorithm 2. Submodularityof this cost function is proven in the following theorem, whichensures the 1− 1/e performance bound of the greedy algorithm.

Theorem VI.2: For pulse and transmitter selection, the setfunction h : 2|P| → R+ [cf. (38)] is a normalized, monotone,submodular function.

Proof: The proof is derived in Appendix B. �

D. Computational Complexity of the Proposed Algorithms

In the following, we present a brief analysis of the com-putational complexity of the proposed algorithms. First, thegreedy algorithm in Algorithm 1 requires |U| iterations atmost. For each iteration, we find the element which maxi-mizes the objective function. Simply performing the calculationimplies a O(|U|2 |Y{U}|N) complexity which leads to a totalcomputational complexity of O(|U|3 |Y{U}|N). However, per-forming the procedure using a proper data structure such thateach operation is calculated only once, the total computationalcomplexity of the greedy algorithm in Algorithm 1 becomesO(|U|2 + |Y{U}|2N). In the case that parallel processing isemployed, the total computational complexity can be reducedto O(|U||Y{U}|N).

Fig. 5. MFP-submodular optimization result for 24 pulses: (a) angle-velocityambigutiy function, (b) selected transmitters-pulses.

For Algorithm 2, we require to perform |P| iterations at most.Each iteration takes O(|P|) computations. This iteration com-plexity comes from the fact that evaluating the cost function foreach new element requires a constant time, e.g., independent ofthe size of the input. Hence, the total computational complexityof Algorithm 2 isO(|P|2). Similar to Algorithm 1, incorporatingparallel processing in each iteration of Algorithm 2 for findingthe maximum, we can achieve a computational complexity ofO(|P|) for Algorithm 2.

Finally, we discuss the complexity of the convex method.Since the convex optimization problem in (32) is in SDP form,each iteration requires O(|U|3) computations, where |U| is thenumber of rows (or equivalently columns) in W [43]. In prac-tice, most of the conventional SDP problems, require just a fewiterations for achieving convergence. Therefore, we can considerthe iteration complexity as the total complexity of the method.It should be noted that this complexity order is calculated forthe simplest form of SDP and considering that the number ofiterations is not dependent on the parameters values.

In summary, as mentioned earlier, using proper structure incalculations, employing a greedy algorithm has a computa-tional advantage, especially for Algorithm 2. The computationalcomplexity of all the proposed algorithms is summarized inTable I.

590 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019

Fig. 6. MFP-submodular optimization for 12 pulses and 3 receivers: (a) angle-velocity ambiguity function, (b) selected transmitters-pulses-receivers. (c) MSEversus number of transmit pulses for a large scenario.

VII. SIMULATION RESULTS

In this section, we study the performance of the proposedalgorithms through numerical simulations. The simulations areperformed for a radar using a 77-GHz frequency band with a100 MHz bandwidth which is typically used for automotiveradar systems [31]. For the following simulations, we employedCVX to solve the convex optimization problems, and we useSNR = 10 dB, γv = 0.1, and γu = 1.

A. Fixed Receivers

In this part, we test the performance of the proposed algo-rithms for the fixed receivers case. For the first scenario, threereceivers, four transmitters, and four pulses are considered intotal. All the receivers are assumed to be fixed. In Fig. 4(a),the results of the different algorithms are represented. The MSEfor both angle and velocity estimation of all these three opti-mization algorithms in addition to the optimum MSE versus thenumber of pulses are presented in Fig. 4(a). This plot shows theperformance of each algorithm and that their results are close tothe optimum value.

In another scenario, we consider two fixed receivers, six trans-mitters, and eight pulses in total. Fig. 4(b) depicts the MSE forboth angle and velocity estimation of the three proposed algo-rithms versus the number of transmitted pulses. All of themhave a very close performance in terms of the MSE. In addition,Fig. 5(a) shows the ambiguity function for the result obtainedby the submodular algorithm for the MFP when 24 pulses areselected. Here, a low sidelobe level and narrow beamwidth forboth the direction cosine and velocity is achieved. The set of se-lected pulses is presented in Fig. 5(b). As it is shown in Fig. 5(b),pulses are selected from all the transmitters. Although, there isa tendency of selecting pulses towards the edges, the selectedset includes different pulse numbers.

B. Transmitter-Receiver-Pulse Selection

Simulation results for the most general case of selectingtransmitters-receivers-pulses is studied in this section. In total,we consider four receivers, four transmitters, and four pulses.Fig. 4(c) presents the MSE for both angle and velocity esti-mation of the two optimization algorithms in addition to theoptimum MSE. This plot shows again that the results are very

close to the optimum value. Note that the results are plottedfor two different cases. In the first case, one out of four re-ceivers is selected and in the second case, three out of fourreceivers are selected. It is clear that the MSE is lower for thelast case. Moreover, Fig. 6 presents the result of the submodu-lar algorithm for the MFP when 12 pulses and 3 receivers areselected. The resulting ambiguity function and selected trans-mitters, receivers, and pulses are depicted in Figs. 6(a) and 6(b),respectively.

Finally, we consider a large-scale scenario with 20 receivers,20 transmitters, and 10 pulses in total (i.e., the total number oftransmit pulses is 200). It should be noted that due to the largenumber of parameters, the greedy algorithm is the only tractableoptimization method. This is one of the advantages of submodu-lar optimization over convex optimization. Fig. 6(c) presents theMSE of the submodular algorithm for the MFP versus the num-bers of selected transmit pulses for different number of selectedreceivers. As expected, the MSE decreases by increasing thenumber of transmit-pulses and receivers. However, it is shownthat this improvement is saturated after a certain point. We couldfind some operating points in this figure such that by decreas-ing the performance slightly, a huge reduction in the numberof transmit-pulses and receivers is achieved. For instance, theMSE for 80 transmit-pulses and 8 receivers is less than twicethat of the full case, but with a much lower cost.

VIII. CONCLUSIONS

In this paper, we presented algorithms to perform antennaplacement and pulse selection that guarantees a desired esti-mation error. In particular, the placement obtained from theproposed greedy algorithm with the submodular approach isguaranteed to be 1/2-optimal. It turned out that a significantreduction in the number of pulses and antennas with a smallreduction in estimation accuracy is possible. Beside reducinghardware complexity (the number of antennas) and energy con-sumption (the number of pulses), the computational complexityis also reduced hugely due to the lower number of total samples.The one- and two-target CRLB for multiple antennas and mul-tiple pulses were derived and it was shown that the two-targetCRLB is a better measure for antennas and pulses selection op-timization problem. Even though, several performance metrics

TOHIDI et al.: SPARSE ANTENNA AND PULSE PLACEMENT FOR COLOCATED MIMO RADAR 591

were proposed, it should be stated that, there is no best solutionfor all problem instances and the appropriate performance met-ric should be selected based on the specific application. Convexand submodular optimization as the two different optimizationapproaches to antenna and pulse selection were introduced. Itwas shown that convex optimization provides more degrees offreedom in the optimization problem, i.e., it enables min-maxoptimization. On the other hand, the greedy submodular op-timization obtains a near optimal solution with a low compu-tational complexity which is desired especially in large-scalescenarios.

APPENDIX

A. Proof of Theorem VI.1

Proof: First, we show that the function is normalized. Thatis, G(Ø) = 0. This can be proved by noting

G(Ø) = FP(U)− FP(U \Ø) = FP(U)− FP(U) = 0. (39)

Now, we show the monotonicity ofG. Without loss of generality,we focus on the case that a new transmit pulse, x ∈ P , is addedas the proof for the other case (new receiver) can be constructedin a similar way. To show this, we require to show the following

G(Sc ∪ {x})−G(Sc) ≥ 0. (40)

First, we recall the definition (34), and expand the left-hand-sideof the above inequality as

G(Sc ∪ {x})−G(Sc) = FP(U \ Sc)− FP(U \ {Sc , x}).(41)

Using the fact that U \ Sc = S (by definition) we canrewrite (41) as

G(Sc ∪ {x})−G(Sc) = FP(S ∪ {x})− FP(S), (42)

where S = S \ {x}. Substituting in (41) the identity

FP(S ∪ {x}) = FP(S) + FP({x} ∪ B) + FP(S, {x} ∪ B),(43)

where B = S ∩ R, and we have defined the MFP for two sets(last term in the above expression) as

FP(S1 ,S2) =

∑

y∈Y(S1 ),y ′∈Y(S2 )

N∑

n=1

∣∣∣∣〈∂y[n], ∂y′[n]〉

〈∂y[n], ∂y[n]〉 〈∂y′[n], ∂y′[n]〉∣∣∣∣2

, (44)

we can show that

G(Sc ∪ {x})−G(Sc) = FP(B ∪ {x})+ FP(S,B ∪ {x}) ≥ 0, (45)

which proves the monotonicity of G.Finally, we show the submodularity of the set function. To do

so, we restrict the proof to the general case, i.e., the elementsinvolved in the proof are a transmit pulse, x ∈ P , and a receiver,y ∈ R. As for the case of monotonicity, this general proof canbe particularized for the case in which both elements are of thesame kind, i.e., two receivers, or two transmit pulses.

To show submodularity we need to prove that

G(Sc ∪ {x})−G(Sc) ≥ G(Sc ∪ {x, y})−G(Sc ∪ {y}).(46)

Expanding both sides of the inequality, we obtain

FP(U \ Sc)− FP(U \ {Sc , x}) ≥ FP(U \ {Sc , y})− FP(U \ {Sc , x, y}). (47)

Using the identity S = U \ {Sc , x, y} we can express the in-equality as

FP(S ∪ {x, y})− FP(S ∪ {y}) ≥ FP(S ∪ {x})− FP(S). (48)

Finally, using the identity (43) and the monotonicity of the MFP,we can show that

FP(B ∪ {x, y}) + FP(S ∪ {y}, B ∪ {x, y}) ≥ (49)

FP(B ∪ {x}) + FP(S, B ∪ {x})FP(B ∪ {x, y})− FP(B ∪ {x})

+ FP(S ∪ {y}, B ∪ {x, y})− FP(S, B ∪ {x}) ≥ 0, (50)

which proves the submodularity of the set function. In (50) wehave defined B = S ∩ R for readability. �

B. Proof of Theorem VI.2

Proof: First, let us recall P as the set of pulses of all thetransmitters and S ⊂ P and FS as the Fisher information matrixobtained by employing all the pulses in S. Now, let p1 , p2 ∈P \ S, then to prove submodularity we need to show

log det (FS∪{p1 })− log det (FS) ≥log det (FS∪{p1 ,p2 })− log det (FS∪{p2 }). (51)

Noting that

FS∪{p1 } = FS + F{p1 }

FS∪{p2 } = FS + F{p2 }

FS∪{p1 ,p2 } = FS + F{p1 } + F{p2 }, (52)

inequality (51) can be rewritten as

log det (FS + F{p1 })− log det (FS) ≥log det (FS + F{p1 } + F{p2 })− log det (FS + F{p2 }). (53)

Thus, (53) implies

det (FS + F{p1 })det (FS + F{p2 })det (FS)det (FS + F{p1 } + F{p2 })

≥ 1. (54)

Considering F{p1 } = UVT and employing the matrix determi-nant lemma in

det (FS)det (I + VT F−1S U)det (FS + F{p2 })

det (FS)det (FS + F{p2 })det (I + VT (FS+F{p2 })−1U)≥1,

(55)

592 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 3, FEBRUARY 1, 2019

leads to

det (I + VT F−1S U)

det (I + VT (FS + F{p2 })−1U)≥ 1, (56)

which is clear to be true since F{p2 } is a positive semi-definite matrix, i.e., F−1

S � (FS + F{p2 })−1 as FS � FS +

F{p2 }. Therefore, this function is submodular. In addi-tion, monotonicity and normalization are clear from thedefinition. �

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TOHIDI et al.: SPARSE ANTENNA AND PULSE PLACEMENT FOR COLOCATED MIMO RADAR 593

Ehsan Tohidi received the B.Sc., M.Sc., and Ph.D.degrees in electrical engineering (communicationsand signal processing) from the Sharif University ofTechnology, Tehran, Iran, in 2011, 2013, and 2018,respectively. From September 2016 to March 2017,he was a visiting Ph.D. student with the Departmentof Electrical Engineering, Mathematics, and Com-puter Science, Delft University of Technology, Delft,The Netherlands. His research interests include sig-nal processing, MIMO radar, optimization, machinelearning, and resource allocation.

Mario Coutino (S’17) received the M.Sc. degree(cum laude) in electrical engineering from the DelftUniversity of Technology, Delft, The Netherlands, inJuly 2016, where he is working toward the Ph.D. de-gree in signal processing since October 2016. He hasheld positions at Thales Netherlands, during summer2015, and at Bang and Olufsen during 2015–2016.His research interests include signal processing onnetworks, submodular and convex optimization, andnumerical lineal algebra. He was the recipient of theBest Student Paper Award for his publication at theCAMSAP 2017 conference in Curacao.

Sundeep Prabhakar Chepuri received the M.Sc.degree (cum laude) in electrical engineering andPh.D. degree (cum laude) from the Delft Universityof Technology, Delft, The Netherlands, in July 2011and January 2016, respectively. He was a PostdoctoralResearcher (September 2015 to December 2018) withthe Delft University of Technology, The Netherlands,a Visiting Researcher with the University of Min-nesota, USA, and a Visiting Lecturer with Aalto Uni-versity, Finland. He has held positions with RobertBosch, India, during 2007–2009, Holst Centre/imec-

nl, The Netherlands, during 2010–2011. His general research interest lies inthe field of mathematical signal processing, learning, and statistical infer-ence applied to computational imaging, communication systems, and networksciences.

Dr. Chepuri was the recipient of the Best Student Paper Award for hispublication at the International Conference Acoustics, Speech, and Signal Pro-cessing 2015 conference in Australia. He is currently an Associate Editor forthe EURASIP Journal on Advances in Signal Processing.

Hamid Behroozi (S’04–M’08) received the B.Sc.degree in electrical engineering from the Universityof Tehran, Tehran, Iran, in 2000, the M.Sc. degreein electrical engineering from the Sharif Universityof Technology, Tehran, Iran, in 2003, and the Ph.D.degree in electrical engineering from Concordia Uni-versity, Montreal, QC, Canada, in 2007. From 2007to 2010, he was a Postdoctoral Fellow with the De-partment of Mathematics and Statistics, Queens Uni-versity, Kingston, ON, Canada. He is currently anAssociate Professor with the Department of Electri-

cal Engineering, Sharif University of Technology, Tehran. His research interestsinclude information theory, joint source-channel coding, cooperative commu-nications, MIMO radar signal processing, machine learning, and physical layersecurity.

Dr. Behroozi was the recipient of several academic Awards, including On-tario Postdoctoral Fellowship awarded by the Ontario Ministry of Research andInnovation, Quebec Doctoral Research Scholarship awarded by the Governmentof Quebec (FQRNT), Hydro Quebec Graduate Award, and Concordia Univer-sity Graduate Fellowship.

Mohammad Mahdi Nayebi (SM’05) received theB.S. and M.S. degrees in analysis and design ofnonuniformly-sampled digital filters for applicationin MTI Radars from the Sharif University of Tech-nology, Tehran, Iran, in 1988 and 1990, respectively,and the Ph.D. degree in electrical engineering sub-ject on detection of radar targets in clutter plus noisefrom Tarbiat Modarres University, Tehran, Iran, in1994. He joined the Sharif University of Technologyin 1994. He is currently working on MIMO radar sig-nal processing. His main research interests include

detection and estimation theory, radar signal processing, pattern recognition,electromagnetic compatibility, and communications.

Geert Leus (M’01–SM’05–F’12) received the M.Sc.and Ph.D. degree in electrical engineering from theKU Leuven, Leuven, Belgium, in June 1996 and May2000, respectively. He is currently an “Antoni vanLeeuwenhoek” Full Professor with the Faculty ofElectrical Engineering, Mathematics and ComputerScience, Delft University of Technology, Delft, TheNetherlands. His research interests are in the broadarea of signal processing, with a specific focus onwireless communications, array processing, sensornetworks, and graph signal processing. He was the

recipient of the 2002 IEEE Signal Processing Society Young Author Best PaperAward and the 2005 IEEE Signal Processing Society Best Paper Award. He wasa Member-at-Large of the Board of Governors of the IEEE Signal ProcessingSociety, the Chair of the IEEE Signal Processing for Communications and Net-working Technical Committee, and an Editor-in-Chief of the EURASIP Journalon Advances in Signal Processing. He was also on the Editorial Boards of theIEEE TRANSACTIONS ON SIGNAL PROCESSING, the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS, the IEEE SIGNAL PROCESSING LETTERS, and theEURASIP Journal on Advances in Signal Processing. He is a Member of theIEEE Sensor Array and Multichannel Technical Committee, an Associate Editorfor the Foundations and Trends in Signal Processing, and an Editor-in-Chief ofEURASIP Signal Processing. He is a Fellow of EURASIP.