3022 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

Frequency-Hopping Code Design for MIMO RadarEstimation Using Sparse ModelingSandeep Gogineni, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We consider the problem of multiple-target estima-tion using a colocated multiple-input multiple-output (MIMO)radar system. We employ sparse modeling to estimate the un-known target parameters (delay, Doppler) using a MIMO radarsystem that transmits frequency-hopping waveforms. We formu-late the measurement model using a block sparse representation.We adaptively design the transmit waveform parameters (fre-quencies, amplitudes) to improve the estimation performance.Firstly, we derive analytical expressions for the correlationsbetween the different blocks of columns of the sensing matrix.Using these expressions, we compute the block coherence measureof the dictionary. We use this measure to optimally design thesensing matrix by selecting the hopping frequencies for all thetransmitters. Secondly, we adaptively design the amplitudes of thetransmitted waveforms during each hopping interval to improvethe estimation performance. To perform this amplitude design,we initialize it by transmitting constant-modulus waveforms ofthe selected frequencies to estimate the radar cross section (RCS)values of all the targets. Next, we make use of these RCS estimatesto optimally select the waveform amplitudes. We demonstrate theperformance improvement due to the optimal design of waveformparameters using numerical simulations. Further, we employcompressive sensing to conduct accurate estimation from farfewer samples than the Nyquist rate.

Index Terms—Adaptive, colocated, frequency-hopping codes,multiple-input multiple-output (MIMO) radar, multiple targets,optimal design, sparse modeling.

I. INTRODUCTION

C ONVENTIONAL monostatic single-input single-output (SISO) radar transmits an electro-magnetic (EM)

wave from the transmitter [1]. The properties of this waveare altered while reflecting from the surfaces of the targetstowards the receiver. The altered properties of the wave enableestimation of unknown target parameters like range, Doppler,and attenuation. However, such systems offer limited degreesof freedom. Multiple-input multiple-output (MIMO) radarsystems have attracted much attention in the recent past due tothe additional degrees of freedom they offer [2]–[7].

Manuscript received June 11, 2011; revised September 20, 2011 and De-cember 27, 2011; accepted February 17, 2012. Date of publication March 08,2012; date of current version May 11, 2012. The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof. PiotrIndyk. This work was supported by ONR Grant N000140810849, NSF GrantCCF-1014908, and AFOSR Grant FA9550-11-1-0210.The authors are with the Department of Electrical and Systems Engineering,

Washington University in St. Louis, St. Louis, MO 63130 USA (e-mail: sgogi-neni@ese.wustl.edu).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2012.2189106

MIMO radar is commonly used in two different antenna con-figurations: widely-separated (distributed) and colocated. Dis-tributed MIMO radar exploits spatial diversity [8] by utilizingmultiple uncorrelated looks of the target [2], [3], [9]. Colo-cated MIMO radar systems offer performance improvement byexploiting waveform diversity [4]–[7]. Each antenna has thefreedom to transmit a waveform that is different from the wave-forms of the other transmitters. In this paper, MIMO radar refersto colocated MIMO radar. In [10], the authors exploit frequencydiversity using MIMO radar. In [11] and [12], the authors showthat frequency-hopping codes can be used to exploit waveformdiversity for colocated MIMO radar. They use the MIMO radarambiguity functions [13] of these waveforms to analyze theperformance.Sparse modeling and compressive sensing have been a

hot research topic as they enable accurate estimation fromsub-Nyquist rates [14], [15]. Since most real-world systemshave sparsity in some basis representation, these tools havebeen used in many fields, such as engineering and medicine[16]–[18]. Also, there has been recent interest in applyingthem to the field of radar by exploiting sparsity in the targetdelay-Doppler space [19]–[21], [22]. In [22], we presentedsparse modeling in the context of MIMO radar with widelyseparated antennas. Further, we presented a scheme to adap-tively select the transmitted energies from different antennas tooptimize the sparse recovery performance.In this paper, we employ sparse modeling to estimate the un-

known target parameters using a pulsed MIMO radar systemthat transmits frequency-hopping waveforms (see Fig. 1). Morespecifically, we formulate the measurement model using a blocksparse representation. Further, we adaptively design the param-eters of the transmitted waveforms to achieve improved perfor-mance. First, we derive analytical expressions for the correla-tions between the different columns of the sensing matrix. Next,we use this result for optimal design by computing the blockcoherence measure of the sensing matrix and selecting the hop-ping frequencies of all the transmitters. Finally, we transmit con-stant modulus waveforms using these selected frequencies toestimate the radar cross section (RCS) values of all the targets.We use these RCS estimates to adaptively design the amplitudesof the transmitted waveforms during each hopping interval forachieving improved sparse recovery performance.The rest of the paper is organized as follows. In Section II,

we present the radar signal model for the proposed MIMOradar system. In Section III, we present this model usingsparse representation in an appropriate basis. In Section IV, wepresent the concept of block coherence measure followed by anoptimal hopping-frequency design mechanism in Section V. In

1053-587X/$31.00 © 2012 IEEE

GOGINENI AND NEHORAI: FREQUENCY-HOPPING CODE DESIGN FOR MIMO RADAR ESTIMATION USING SPARSE MODELING 3023

Fig. 1. Example of a frequency hopping waveform with three hopping inter-vals.

Section VI, we present a sparse recovery algorithm to performthe target parameter estimation. We use these estimates inSection VII to optimally design the transmit amplitudes. InSection VIII, we present compressive sensing for accurateestimation from fewer samples. In Section IX, we presentnumerical simulations to demonstrate the performance im-provement due to optimal waveform design (code matrix andamplitudes). We also present estimation results while em-ploying compressive sensing. Finally, we provide concludingremarks in Section X.

II. SIGNAL MODEL

We consider the problem of target estimation using a colo-cated MIMO radar system operating in a monostatic configura-tion. We assume there are transmit antennas and re-ceive antennas arranged in linear arrays (see Fig. 2). The com-ponents of the transmit and receive arrays are separated by adistance of and , respectively. Further, we assume thatthese arrays form an angle with the target. The trans-mitter emits frequency hopping waveform (see Fig. 1).These waveforms are a generalization of linear frequency-mod-ulated (LFM) waveforms. LFM is a special case of frequencyhopping waveforms. In LFM, the frequency changes at the samelinear rate, whereas for these codes the rate need not neces-sarily be linear as depicted in Fig. 1. In [11], the authors demon-strate the performance improvement offered by these codes overLFM. Further, we consider a pulsed radar system in this paper.Assuming pulses make up a waveform, the signal from thetransmitter is given as

(1)

where

(2)

and

ifotherwise.

(3)

and denote the pulse repetition interval and hopping in-terval duration, respectively. and denote the hopping indexand the total number of hopping intervals, respectively.

Fig. 2. Transmit/receive antenna array.

Design of the transmit waveforms amounts to choosingand for all the transmitters and all the hopping intervals.

specifies the frequency of the transmitted signal during eachhopping interval and gives the corresponding amplitude ofthe transmitted sinusoid. We assume that each takes a valuefrom the set , where is a positive integer. We as-sume . Further, to ensure orthogonality of the wave-forms for zero lag, we assume that for every hopping interval ,

(4)

We can arrange into an dimensional code matrix .This code matrix describes all the transmitted frequencies. Fur-ther, we constrain the amplitudes to satisfyfor all transmitters and frequencies. This requirement ensurescontrol over the peak-to-average-power ratio of all the trans-mitted radar waveforms. Further, we normalize the transmittedenergy for each waveform by assuming .Define

(5)

(6)

where is the wavelength of the carrier. We assume that thetarget is made up of multiple individual isotropic scatterers.But, because of signal bandwidth constraints, these individualscatterers cannot be resolved. Therefore, we express this col-lection of scatterers as one point scatterer representing the RCScenter of gravity [2], [23]. Further, we assume that different scat-tering centers of the target resonate at different frequencies [24].Therefore, the target has an RCS that varies with the frequenciesof the waveforms. Note that unlike distributed MIMO radar, theRCS does not vary with the antenna index for colocated MIMOradar.The received signal at each receiver is a linear combination of

the target-reflected waveforms from all the transmitters. There-fore, we can express the received signal at the receiver as

(7)

where and represent the delay andDoppler, respectively, anddenotes the additive noise at the receiver. The target

RCS is given by . Note that we consider transmit waveforms

3024 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

whose bandwidth is much smaller when compared with the car-rier frequency. Equation (7) gives the measurement model whena single target is present in the region illuminated by the MIMOradar system.Now, we consider the presence of multiple targets. Considertargets in the scene illuminated by the radar. Here we assume

that all the targets are present in the far-field. Therefore, eachof them makes an angle approximately equal to with the radararrays. Then, the received signal at the receiver is a sum-mation of the reflections from all the targets. We sample the re-ceived signal to obtain

where denotes the total number of samples at each receiverduring one processing interval and denotes the corre-sponding sampling interval. Further, and represent thedelay and Doppler of the target, respectively.

III. SPARSE REPRESENTATION

Recently, sparse modeling is being used increasingly forsolving radar problems by exploiting sparsity in the targetdelay-Doppler space [19]–[22]. In this section, we will usesparse modeling to represent the radar measurements givenin the previous section. These measurements can be capturedusing a block sparse model. For each of the targets, theunknown parameters are the attenuation, delay, and Doppler.We shall discretize the delay-Doppler space into uniformlyspaced grid points. Only of these grid points correspond tothe true target parameters, and the goal is to estimate the correctgrid points. Let and represent the delay and Dopplercorresponding to the grid point.For each grid point , we define

(8)

We stack into an dimensional columnvector

(9)

where denotes the transpose of .Similarly, we stack into an dimen-

sional column vector . Each of these column vectorscorresponds to a different transmitter and hopping interval,and we stack the columns corresponding to the same hoppinginterval together. Now, for each grid point , we stack thecolumn vectors into an dimensional matrix

. Further, we arrange into andimensional matrix . This is the dictionary matrix that definesthe basis elements of our sparse representation.Stacking and corresponding to different transmit-

ters and hopping intervals, we obtain dimensional column

vectors and , respectively. Next, we define sparse vectorsand whose support set and entries are given as

ifotherwise,

(10)

ifotherwise.

(11)

Finally, we stack these vectors and correspondingto all the grid points to obtain a dimensional block-sparse vectors:

(12)

(13)

These sparse vectors contains only non-zero blocks, each cor-responding to a different target. Further, each block contains

entries. Therefore entries of are zeros.We stack the measurements and the additive noise samples at

each receiver to obtain the vectors

(14)

(15)

In addition, stacking the measurement and noise vectors at allthe receivers, we obtain

(16)

(17)

Then, our measurement model reduces to

(18)

This is a familiar linear model used in most applications ofsparse modeling.The estimation of attenuation, delay, and Doppler for all

the targets reduces to recovering the non-zero entries and thesupport set of the sparse vector from the measurement vector. In Section VI, we will present a sparse support recoveryalgorithm.

IV. BLOCK COHERENCE MEASURE

In this section, we will analyze the performance of thesparsity-based estimation approaches as a function of thesensing matrix . The correlations between the columns of thedictionary matrix determine the accuracy of sparse-recoveryalgorithms. More specifically, when the non-zero entries ofthe sparse vector appear in blocks (as in our radar estimationproblem), a major factor affecting the performance of thesystem is the block coherence measure [25], [26]. This conceptis an extension of the well-known coherence measure [14]used to block sparse signals. It can be used to derive sufficientconditions for guaranteed sparse support recovery.Let and denote the and blocks of the

dictionary, respectively. Each block contains columns.Each column corresponds to a different transmitter and hoppinginterval. Since the columns corresponding to different hoppingintervals do not overlap and, further, we imposed the condition

GOGINENI AND NEHORAI: FREQUENCY-HOPPING CODE DESIGN FOR MIMO RADAR ESTIMATION USING SPARSE MODELING 3025

in (4) to ensure orthogonality across all the transmitters for zerolag, all the columns within a block are orthogonal.If any columns of are exactly the same as the corre-

sponding columns in , we can remove them, since theywill not contribute to the sparse recovery problem while com-paring these two blocks. Therefore, we define

(19)

where denotes the number of columns of that areexactly the same as the corresponding columns of . Let usdefine the correlation matrix for each pair of blocks ofthe dictionary matrix as

(20)

Each entry of this matrix contains the auto-correlation be-tween the different columns of the selected blocks. Using thesenotations, the authors in [25] defined the block coherence mea-sure of the basis matrix as

(21)

where denotes the spectral norm [27] of :

(22)

where denotes the largest eigenvalue of .The block coherence measure provides a sufficiency mea-

sure for ensuring sparse support recovery [25]. Therefore, min-imizing the block coherence measure ensures theoretical guar-antee for sparse support recovery of signals with potentiallyhigher sparsity level. In the next section, we will use this con-cept to select the hopping frequencies of all the transmitters.

V. OPTIMAL HOPPING-FREQUENCY DESIGN

In this section, we present a mechanism for designing optimalhopping frequencies. The expression for the block coherencemeasure given in (21) depends on the transmitted code ma-trix through the correlation matrices . First, we willformulate the frequency-selection problem using the theory de-veloped in the previous sections. Next, we develop a solutionmechanism for this problem to obtain the code matrix.

A. Problem Formulation

In order to compute the optimal code matrix, we need to min-imize the block coherence measure by solving the following op-timization problem:

(23)

(24)

(25)

The correlation matrices are obtained from the basis matrixusing (20). Substituting this relation into the above expression,we obtain

(26)

B. Correlation Matrix Entries

Since directly computing the block coherence measure isdifficult, we first compute the entries of the correlation ma-trix . Let represent the element of

such that and , whereand . Note that

there is always a unique mapping between and ; similarlybetween and . Therefore, we will alternatively use thenotation instead of . Let grid pointcorrespond to the delay-Doppler pair . Further, let gridpoint correspond to the delay-Doppler pair . Below,we state the assumptions made for performing the subsequentderivations.We assume that the difference between the delays of any two

grid points is always a multiple of the duration of thehopping interval . In addition, we assume that is the sizeof the delay grid. Therefore, it gives us the range resolution ofthe sparsity-based radar estimation. Further, the target velocitycomponents that are orthogonal to the radial direction (radararray to the target) do not produce a Doppler shift. The radialspeeds of the targets are much smaller than the speed of wavepropagation in the medium. We assume that the sampling rateis at least as big as the Nyquist rate corresponding to the largestpossible hopping frequency:

(27)

Therefore, for all choices of coding matrices, we meet theNyquist sampling criterion.Then, we obtain the following expressions for the auto-cor-

relations between the different columns of the blocks corre-sponding to and :

Each column of the dictionary contains delay-Doppler shiftedversions of the transmitted waveforms. Since we chose radar

3026 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

waveforms that have a bounded temporal support (rectangularpulses multiplied by sinusoids), the columns have only a fewnon-zero samples. All the other column entries are zero. The ex-pression inside the summation will be non-zero only when thecorresponding entries of both the columns are non-zero. There-fore, we can express the term inside the summation as (28),shown at the bottom of the page, only when

(29)

and

(30)

All other entries of the summation will be zero.These conditions ensure that the rectangular pulses corre-

sponding to both columns overlap at the given temporal index.For a given we denote as and the sets con-taining all and satisfying the conditions in (29) and (30). Notethat for each pulse index, the sample indices that give non-zeroentries are different. Then, we can express the entries of thecorrelation matrix as (31), shown at the bottom of the page.The exponential term on the right side, , is constantand does not depend on the time index; hence, it can be movedout of the summation. After removing this term, each entryof matrix is a summation of the product of complexexponentials,and .Let be the number of samples per hopping interval. In

other words, . Since the radial speeds of the tar-gets are much smaller than the speed of wave propagation in themedium, the Doppler shift is measurable only between pulsesand is negligible within the pulse duration. Therefore, we canexpress the correlation terms as a product of separate summa-tions:

(32)

(33)

The above equation contains three product terms. The firstterm is independent of the temporal index. The second term rep-resents the contribution between the different pulses, and thethird term represents the contribution from within a hopping in-terval. The dependence of the third term on the code matrix isevident from the exponential. Note that the second term dependson the Doppler shift, which in turn depends on the frequency of

the complex exponential. This frequency is a sum of the carrierfrequency and the hopping frequency. Therefore, we concludethat the second and third terms in (32) depend on the code ma-trix (hopping frequencies).Now, we give expressions for these terms as a function of the

code matrix. Define as the carrier frequency, as the speedof wave propagation in the medium, and as the radialspeeds corresponding to the grid points and , respectively.Then, we have

(34)

Even though the term in (34) depends on the code matrix, thedependence is negligible since it is absorbed by the carrier fre-quency term that is much larger when compared with the base-band code frequencies:

(35)

where denotes the maximum hopping frequency. Thesummation of the samples of a complex exponential is zero forall satisfying (27). Hence, we have

ifotherwise.

(36)Finally we express the entries of the correlation

matrix corresponding to the blocks and as

if

otherwise.(37)

Note that the auto-correlation matrix need not be aHermitian matrix since need not be equal to

for all . Therefore, the spectral normand spectral radius of are not the same. Thus, we needto compute the eigenvalues of to evaluatethe spectral norm of .

C. Correlation Matrix Structure

We now partition into submatrices

each of dimensions such that

(38)

(28)

(31)

GOGINENI AND NEHORAI: FREQUENCY-HOPPING CODE DESIGN FOR MIMO RADAR ESTIMATION USING SPARSE MODELING 3027

where is the element of . We usethis notation to study the structure of for different pairsof grid points.Without loss of generality, assume . Then, we com-

bine the conditions in (29) and (30) to obtain the followingconditions:

(39)

and

(40)

Since is a multiple of , the above conditions yieldonly a maximum of one possible positive integer value forsuch that is a non-zero matrix:

(41)

When , then for everychoice of valid . In such a scenario the entire columnof blocks is filled with zero submatrices. Therefore,can be partitioned into a special structure of submatrices. It isa block-lower-triangular matrix whose non-zero blocks appearin a single diagonal line parallel to the principal diagonal. Thedistance between this line and the principal diagonal is given by

.For example, consider the difference between the delays

, when the number of hopping intervals. can be expressed as

(42)Here, the distance between the principal diagonal and the diag-onal line of non-zero blocks is 2.Whenwe are comparing blockswhose grid points have the same delay but different Doppler,

will be a block-diagonal matrix.Computing for matrices following this

structure yields block-diagonal matrices whose non-zero diag-onal blocks are given by the non-zero blocks in the diagonalline of the original matrix . In the above example, weobtain

Only when will all the diagonal blocks ofbe non-zero. All the diagonal blocks ofwill be zero when

(43)

This result is a consequence of the fact that for all delays thatexceed , the radar waveforms do not have overlappingtime intervals and hence they will be orthogonal.

D. Optimal Code Matrix Selection

We know from the properties of block-diagonal matrices thattheir largest eigenvalue can be expressed as the largest of theeigenvalues of each of the individual blocks. Using this prop-erty, we have

(44)

Next, we substitute the above expression into (25). Then, thecode design problem reduces to

(45)

Let us define

(46)

Using the definition of , we compute the el-

ement of the Hermitian matrix as

(47)

Therefore,

ifotherwise

(48)

where

(49)

and denotes the number of elements in the column ofcode matrix that have the same value as .Since we assumed orthogonality for zero lag in (4),if and only if . Therefore, (48) can be reduced to

ifotherwise.

(50)

Therefore, is a diagonal matrix. Further, (4) also im-plies that can take values only from the set . There-fore,

(51)

depends only on the difference in the Doppler shifts corre-sponding to grid points and , i.e., . Since doesnot depend on the entries of the code matrix, it does not affectthe code selection problem.

3028 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

Also,

(52)

Note that the summation is carried out only among columns thatsatisfy the condition in (41). Therefore, this summation varieswith respect to the difference of delays, . Finally, sub-stituting (51) and (52) into (45), the optimal code selection sim-plifies to the following:

(53)

(54)

(55)

(56)

Define

(57)

Since governs the performance of any code matrix , wewill use it in Section IX to show the improvement due to theoptimal code design.

E. Iterative Exhaustive Search Algorithm for Code Selection

We observe that (53) is a combinatorial optimizationproblem, and these do not yield easily to direct solution. Fur-ther, the solution to this problem need not be unique. Any ofthe optimal solutions is equally good for our purpose. Thus, wewill use an iterative approach to obtain an optimal code matrix.First, we notice from (53) that for any code matrix, the objectivefunction is a non-negative integer. Therefore, we start witha desired objective function value of 0 (corresponding to nooverlaps between the columns satisfying (41) for all differencesin delays) and search for availability of codes satisfying thisobjective. If no such codes exist, we increment the objectivefunction and follow the same procedure iteratively. We describethe steps in detail below.This algorithm is implemented in two major loops. The outer

loop corresponds to the desired objective value and the innerloop corresponds to the code column. Let denote a setcontaining all column vectors of size whose entries aretaken from . Further, we avoid the repetition of en-tries within these columns to ensure orthogonality at zero lag.Let and denote the iteration indices of the outer and innerloops, respectively. For the first outer iteration, and thecorresponding objective is . For the inner loop, we ini-tialize by selecting any arbitrary column from the set of columns

as the first column of our code matrix.

Fig. 3. Flowchart of code selection algorithm.

In every subsequent iteration, we increment the column indexand add a column from that satisfies the following condi-tion with regard to the already existing columns:

(58)

If no such column exists, we decrement the column index andreplace the existing column of the previous iteration with an-other alternative that satisfies (58). If we exhaust the inner loopwithout obtaining sufficient columns to complete the code ma-trix, we know that an objective of cannot be attained by anycode matrix. Therefore, we increment the objective

for the next outer iteration and reset the inner loopindex to .We terminate the algorithm when we obtain a full (

columns) code matrix from the inner loop satisfying the objec-tive given by the outer-loop index. The code matrix obtainedusing this algorithm will always have the optimal objectivefunction. However, the convergence times depend on , , and

. Fig. 3 shows the major blocks used in the implementationof this algorithm. The column-selection block is very critical,as it controls the inner loop of the algorithm. It searches for acolumn in that satisfies (58). Depending on the result ofthis search, we increment or decrement the column index.Note that there may be other efficient algorithms to solve (53)

to obtain an optimal code matrix using combinatorial optimiza-tion. However, it is beyond the scope of this paper to analyze thecomputational complexity and present the theory of combinato-rial optimization for developing these alternate algorithms. Wewill explore these approaches as a future extension to our work.Further, this hopping-frequency (code matrix) design is done of-fline, whereas the amplitude design given later in the paper is an

GOGINENI AND NEHORAI: FREQUENCY-HOPPING CODE DESIGN FOR MIMO RADAR ESTIMATION USING SPARSE MODELING 3029

online design procedure. Therefore, computational complexityis not a very critical issue when designing the code matrix.

VI. SPARSE RECONSTRUCTION

In this section, we present a reconstruction algorithm to re-cover the sparse vector from the noisy measurement vector .Ideally, in a noiseless scenario, we need to solve the followingoptimization problem to recover the sparse vector

(59)

However, this problem is NP hard. Therefore, this problemis relaxed to one that involves the norm, and several ap-proaches have been proposed in the literature to solve it. In [28],a heuristic iterative approach called matching pursuit (MP) ispresented. Further, [29], formulates the problem such that itcan be solved using convex programming. Approaches suchas basis pursuit (BP) and basis pursuit denoising (BPDN) arepopular in this category.However, these algorithms do not exploit the fact that the

non-zero entries of the sparse vector appear in blocks. Usingthe knowledge of block sparsity will improve recovery perfor-mance. In [25], the authors present block extension of matchingpursuit algorithm known as blockmatching pursuit (BMP). Thisalgorithm is a direct extension of the conventional MP, and isused when the columns within the blocks of the dictionary ma-trix are orthogonal. We observed in Section V that the columnsof are orthogonal since . We startwith an initial estimate of . Let denote the compo-nents of the estimate corresponding to the block. Further,we initialize the residue to be . In each subsequent it-eration , we project the residue onto each block of and pickthe block that gives the maximum correlation with the residue:

(60)

We update the residue as

(61)

Finally, we update the block of the estimate vector as

(62)

In a noiseless scenario, after iterations, the estimate vectorwill converge to the true sparse vector . Further iterations

will not result in a change in the residue or the estimate. In thepresence of noise, some of the incorrect blocks may also containnon-zero entries.Note that in the above expressions for sparse support re-

covery, we assumed that all the columns of have unit norm.When all of them are scaled by the same constant factor(non-unit norm), the update equations change by an appropriatescale factor corresponding to this norm. We will use BMP inSection IX to perform sparse support recovery.

VII. ADAPTIVE WAVEFORM AMPLITUDE DESIGN

A. Design

After we select hopping frequencies using the block coher-ence measure mentioned earlier, the transmitters emit constantmodulus waveforms; i.e., . We use sparserecovery algorithm (BMP) to estimate the unknown delay,Doppler, and RCS of the targets. We perform the amplitudedesign for all the transmitters. We use the target RCS estimatesto adaptively design the amplitudes of the sinusoids duringeach hopping interval of the subsequent pulses. Since the RCSof the targets are frequency dependent, the optimal amplitudesneed not be the same for all hopping intervals. As we shall seelater, this problem can be divided into independent optimizationproblems for each transmitter.Let denote the sparse vector reconstructed using the al-

gorithm given in the previous section. If gives the supportset corresponding to the highest reconstruction energies in ,then define as an -dimensional vector containing only theestimates corresponding to the indices in . During the initial-ization step, since , the non-zero entries of the sparsevector depend only on the attenuations . Hence, we obtain

as the estimates of the target attenuations after sparse sup-port recovery.For all subsequent steps, the entries of contain the product

of the transmitted amplitude and the target RCS . Wecompute the summation of the energies of these estimates foreach transmitter over all the hopping interval indices to obtain

(63)

Further, let denote the optimal amplitude for the trans-mitter and frequency hop. We vectorize and for thetransmitter into , , respectively.Define the vector

(64)

This vector contains the estimates of the returns from all thetargets. Note that here we assume that the indices in cor-

respond to the true target entries; i.e., delay-Doppler estimatesusing sparse reconstruction are exact. Otherwise, incorrect in-dices will impact the amplitude design and degrade the perfor-mance. Using these definitions, the amplitude design problemfor each transmitter can be expressed as

(65)

under the constraints

(66)

where denotes the minimum entry of the vector.

3030 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

We will solve this optimization problem using , aMATLAB package for specifying and solving convex pro-grams [30], [31] after appropriate convex transformation (seealso [32]). Note that we need to solve this optimization problemfor each transmitter separately. Since the dimensions involvedin solving these problems (number of targets and number oftransmitters) are typically small, we can compute the optimalenergies in quick time and implement the design online.

B. Metric

Next, we present a performance metric to analyze the accu-racy of sparse reconstruction (see also [22], [32] for more detailson this metric). Let and denote the support sets of the cor-rect and incorrect target indices, respectively. Then, we definethe performance metric as

(67)

The numerator of denotes the weakest target reconstructionand the denominator denotes the strongest reconstruction of theincorrect target indices. Therefore, guarantees that thecorrect target indices dominate the others, thereby resultingin exact estimates of the target delays and Dopplers. The exactvalue of gives the accuracy in the estimates of the target RCSvalues.In Section IX, we will demonstrate the improvement in per-

formance as a result of the optimal transmit amplitude designwhen compared with the constant modulus waveforms by usingthis performance metric.

VIII. COMPRESSIVE SENSING

In this section, we use compressive sensing to accurately re-construct the sparse vector from far fewer samples when com-pared with the Nyquist rate. The theory of compressive sensingsays that this is possible when the sensing matrix has minimalcoherence with the dictionary matrix. Since random matriceshave been shown in the literature [14] to give a low coherencemeasure, we will generate the entries of the sensing matrix asrealizations of independent and identically distributed (i.i.d.)Gaussian random variables.Let denote an dimensional random Gaussian

sensing matrix, where . Define as the mea-surement vector after compressive sensing. Then, the measure-ment model in (18) changes to

(68)

The sensors receive continuous data across all the pulses. Thisdata is projected onto a finite lower dimensional space spannedby random continuous Gaussian noise sequences. The dimen-sions of this space are much smaller than the Nyquist rate.Therefore, we are actually sampling directly at a reduced rate.The above equation is just an equivalent way of representingthe signal processing involved in this procedure.Nowwe need to recover from the compressedmeasurement

vector . The reconstruction algorithm and design schemespresented in the earlier sections of the paper are also valid for

compressive sensing. We define the percentage of compressionas

(69)

The performance of the system degrades as the value of re-duces. We will show this dependence in Section IX for differentvalues of .

IX. NUMERICAL SIMULATIONS

In this section, we present numerical simulations to demon-strate the performance of our proposed radar system.

A. Code Matrix Design

First, we will present examples for the code matrix selection.Let the number of transmitters be and the number ofhopping intervals be . In addition, we chose .Therefore, the code matrix contains 15 entries, each chosenfrom . We ran the iterative algorithm for code selec-tion and obtained the following code matrix as an optimal code:

(70)

For the first three iterations of the outer loop (i.e., ,, and ), the objective is not met. An objective ofis met by the code matrix in (70). Note that other code

matrices may also give the same objective and provide equalperformance. However, no other code matrix will give betterperformance. The block coherence measure corresponding tothe following code is the same as that of the code matrix in (70):

(71)

Both are equally good for selecting theMIMO radar waveforms,and there is not any particular advantage in choosing one ofthem over the other for performing the target parameter esti-mation.Now, we demonstrate the improvement in performance due

to the hopping-frequency design by plotting as a func-tion of the number of hopping intervals . Note that we defined

in (57). Fig. 4 compares the curves for the optimal codematrix and a random code matrix whose columns are chosenuniformly from the set of possible columns. We average across10 000 Monte Carlo runs to obtain the curve for the randomcode matrix. is a multiple of the block coherence mea-sure. Therefore, we intend to have as low a as possible.From Fig. 4, we observe that the optimal code matrix has muchlower block coherence when compared with the average blockcoherence of the random code matrix. Having a lower en-sures theoretical guarantee for sparse support recovery of sig-nals with potentially higher sparsity level [25]. Therefore, Fig. 4essentially states that while using the random code matrix, wecannot guarantee sparse recovery for the same level of sparsityas we can for the optimal code word but for specific examples,

GOGINENI AND NEHORAI: FREQUENCY-HOPPING CODE DESIGN FOR MIMO RADAR ESTIMATION USING SPARSE MODELING 3031

Fig. 4. as a function of the number of hopping intervals.

it might reconstruct the targets correctly. However, it is not reli-able as we do not have any guarantee on the performance at thehigher levels of sparsity.

B. Sparse Support Recovery

In this section, we simulated a radar system consisting ofreceive antennas. Choose 30 and ,

obtaining and . Each processing interval con-sists of 10 pulses (i.e., ). The time interval in betweenthe pulses was chosen to be 3 mS. Let the chip duration be

. Therefore, the width of each pulse 5 .1 MHz is the minimum frequency of the waveform in-

side a hopping interval. Since we chose , the maximumhopping frequency is 7 MHz. Therefore, we sampledat a Nyquist rate of samples per second. During eachchip duration, we have 14 samples.Three targets are present in the illuminated space. Each target

resonates differently at different frequencies. Therefore, wespecify the amplitudes of attenuations for each target:

(72)

(73)

(74)

Using these attenuations, the target RCS corresponding to dif-ferent hopping frequencies and transmitters can be found. Notethat we use (70) as our choice of code matrix.Now, we discretize the target delay-Doppler space. As we

mentioned earlier, we assume that the grid size in the delay di-mension is 1 . The grid points lie uniformly in theinterval . Note that this is just an example and the pro-posed approach can be applied to any arbitrary grid. In a livetracking system, the grid will be adjusted to center around thedelay estimate from the previous tracking interval. The Dopplerspace is uniformly divided in the interval Hz witha separation of 25 Hz between adjacent grid points. Therefore,

Fig. 5. Target estimates using BMP at an SNR of 2.6574 dB.

we have a total of grid points, with onlythree corresponding to the true targets.We assume the true delays and Doppler shifts of the targets

are given as

(75)

Hz (76)

Next, we perform sparse support recovery using the BMP algo-rithm to estimate the target parameters. We define the signal-to-noise-ratio (SNR) as

SNR dB (77)

where denotes the expected value of .First we show the reconstructed target parameters in Fig. 5 at

an SNR of 2.6574 dB. We observe that the delays and Dopplershifts of all three targets are exactly reconstructed. Since thetrue target indices dominate the incorrect target indices in therecovered vector, the value of the performance metric will begreater than unity. We used 30 iterations for the BMP algorithm.We have assumed the target will lie exactly on the grid points.However, in reality it may lie in between two grid points. Whensuch modeling errors occur, we have demonstrated in [32] thatthe reconstruction algorithm BMP will map the estimates to thegrid point that is closest to the true target parameter. The sameholds true even for the results in this paper as we are using BMP.

C. Adaptive Waveform Amplitude Design

After selecting the hopping frequencies using the code ma-trices mentioned earlier in the section, we consider waveformamplitude selection. We need to solve optimizationproblems. For each transmitter, we need to design 5 amplitudes,each corresponding to a different hopping interval.We constrainthese amplitudes to lie in the interval .Further, the sum of squares of these amplitudes is constrained to

3032 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

Fig. 6. Amplitudes of waveforms from transmitters.

be unity. Using CVX to solve the amplitude selection problem,we obtain the following optimal transmit amplitudes:

(78)

(79)

(80)

In Fig. 6, we plot these amplitudes as a function of the hop-ping interval. We observe that the maximum energy for eachtransmitter need not be present during the same hopping in-terval. Transmitters 1 and 2 emit their maximum energy duringthe third and fourth hopping intervals, respectively. However,transmitter 3 emits its maximum energy during the first and fifthhopping intervals. It transmits equals energy during both theseintervals, since the corresponding frequency entries of the codematrix in (70) are the same.During each hopping interval, these waveforms are multi-

plied by exponential waveforms whose frequencies are givenby the entries of the code matrix in (70). If we constrain the am-plitudes such that

(81)

then we will obtain constant-modulus waveforms. Such wave-forms are useful when the variations in the amplitudes of theradar waveforms are not desired because of hardware con-straints of the radar transmit antennas.In Fig. 7, we plot the performance metric to demonstrate

the improvement offered by the adaptive amplitude designmechanism. Recall that we defined the performance metric as

(82)

We would like to be as high as possible. assures exactreconstruction of the target delays and Dopplers. The exactvalue of gives the accuracy in the estimates of the target RCSvalues. We observe that for all SNR, the adaptive amplitudedesign provides significant improvement in performance. This

Fig. 7. Curves demonstrating the improvement in performance due to adaptiveamplitude design.

improvement is a result of maximizing the minimum targetreturns.Now, we will demonstrate the improvement due to the adap-

tive design for a completely different choice of attenuations forthe three targets:

(83)

(84)

(85)

Note that these attenuations are used only for the results inFigs. 8–10. We perform the sparse support recovery under thisscenario and plot the performance metric as a function of theSNR in Fig. 8. In this example, we demonstrate the performanceat very low SNR to investigate the situation when the sparse re-construction fails to estimate all the target parameters correctly.We observe clearly from Fig. 8 that the adaptive amplitude

design outperforms constant modulus waveforms even underthis scenario. More specifically, we observe that the value offalls below 1 for constant modulus waveform approximately

at an SNR 2.5 dB higher than for adaptive amplitude design.Therefore, constant modulus waveforms fail to estimate the truetarget parameters at an SNR of 21 dB, whereas employingadaptive design enables exact reconstruction even at this lowSNR.In Fig. 9, we plot the reconstructed estimates while using

constant modulus waveform at an SNR of 21 dB. Note thatwe plotted on a 2-D plane and used the color map to repre-sent the intensity for better understanding of these results. Thedarker the intensity, the higher the reconstruction energy cor-responding to that grid point. We observe that constant mod-ulus waveform fails to estimate the locations of all the targetscorrectly. More specifically, the target that has a Doppler of1200 Hz is wrongly estimated. However, at the same SNR, weobserve from Fig. 10 that the adaptive amplitude design man-ages to distribute the highest reconstruction energy among thethree actual targets. The three grid points that have the highestintensity correspond to the three targets. Therefore, this example

GOGINENI AND NEHORAI: FREQUENCY-HOPPING CODE DESIGN FOR MIMO RADAR ESTIMATION USING SPARSE MODELING 3033

Fig. 8. Curves demonstrating the improvement in performance due to adaptiveamplitude design in the low SNR region.

Fig. 9. Target estimates using BMP at an SNR of 21 dB.

Fig. 10. Target estimates using BMP at an SNR of while employingadaptive amplitude design.

Fig. 11. Target estimates using BMP at an SNR of 2.6574 dB with 20 .

Fig. 12. Performance metric as a function of SNR for different levels ofcompression.

clearly demonstrates the motivation for employing the adaptivedesign scheme.

D. Compressive Sensing

We employ compressive sensing to observe the performanceof the system while using far fewer samples when comparedwith the Nyquist rate. In Fig. 11, we plot the reconstructedvector at an SNR of 2.6574 dB when the percentage of com-pression is only 20 . We can clearly see a degradationin performance when compared with Fig. 5, since a lot ofenergy in the reconstructed vector is now distributed amongthe incorrect grid points. However, the three most significantcomponents of the estimated vector still correspond to the truetarget grid points, thereby leading to exact reconstruction of thedelay and Doppler.In Fig. 12, we plot the performance metric for different

values of SNRwhile employing different levels of compression.We notice the decline in performance with the increase in thelevel of compression. However, even at a low SNR of 3.08 dB,with a 10 percentage of compression, the value of theperformance metric . Since , we can exactly

3034 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

Fig. 13. Curves demonstrating the improvement in performance due to adap-tive amplitude design when 10 .

estimate the delay and Doppler of all three targets. However,there will be a reduction in the estimation accuracy of the targetRCS values. This reduction shows up in the actual value ofin the curves in Fig. 12.As we mentioned earlier, the adaptive amplitude design is ap-

plicable even when employing compressive sensing. Therefore,in Fig. 13 we show the performance improvement due to theadaptive amplitude design. We notice that even while 10 ,the adaptive design improves the performance.

X. CONCLUDING REMARKS

We proposed a sparsity-based colocated MIMO radarsystem using frequency-hopping waveforms. We estimatedthe unknown target parameters using sparse support recoveryalgorithm. We derived an analytical expression for the blockcoherence measure of the dictionary matrix and, hence, studiedthe problem of selecting the hopping frequencies. We presentedan iterative algorithm for designing an optimal code matrix.Further, we proposed an approach to optimally design theamplitudes of the transmitted waveforms during each hoppinginterval using the estimates of the target returns. We demon-strated the performance improvement due to the optimal designusing numerical examples. Further, we showed that accurateestimation can be performed from far fewer samples than theNyquist rate by employing compressive sensing.In future work, we will consider non-uniform grid spacing

to reduce the computational complexity. In addition, we willinclude the presence of clutter in the measurement model. Wewill develop more efficient algorithms for solving (53) usingthe theory of combinatorial optimization. We will use multi-ob-jective optimization techniques to jointly solve for the optimalcode frequencies and amplitudes. We aim to validate our resultsusing real radar data.

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Sandeep Gogineni (S’08) received the B.Tech de-gree in electronics and communications engineering(with Hons. in signal processing and communica-tions) from the International Institute of InformationTechnology, Hyderabad, India, in 2007, and the M.S.degree in electrical engineering from WashingtonUniversity in St. Louis, MO, in 2009.He is currently working towards the Ph.D. degree

in electrical engineering from WUSTL. His researchinterests are in statistical signal processing, radar, andcommunications systems.

Mr. Gogineni won the Best Paper Award (First Prize) in the Student PaperCompetition at the 2012 International Waveform Diversity and Design (WDD)Conference. Further, he was selected as a Finalist in the Student Paper Compe-titions at the 2010 International Waveform Diversity and Design (WDD) Con-ference and the 2011 IEEE Digital Signal Processing and Signal Processing Ed-ucation Workshop.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees from the Technion,Haifa, Israel, and the Ph.D. degree from StanfordUniversity, Stanford, CA.Previously, he was a faculty member at Yale

University and the University of Illinois at Chicago.He is currently the Eugene and Martha LohmanProfessor and Chair of the Preston M. Green Depart-ment of Electrical and Systems Engineering (ESE) atWashington University in St. Louis (WUSTL), MO.Under his leadership as ESE chair, undergraduate

enrollment has more than doubled in the last three years. He is also Professorin the Division of Biology and Biomedical Studies (DBBS) and Director of theCenter for Sensor Signal and Information Processing at WUSTL.Dr. Nehorai served as Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING from 2000 to 2002. From 2003 to 2005, he was the Vice-President(Publications) of the IEEE Signal Processing Society (SPS), the Chair of thePublications Board, and a member of the Executive Committee of this Society.He was the founding editor of the special columns on Leadership Reflections inthe IEEE Signal Processing Magazine from 2003 to 2006.Dr. Nehorai received the 2006 IEEE SPS Technical Achievement Award and

the 2010 IEEE SPS Meritorious Service Award. He was elected DistinguishedLecturer of the IEEE SPS for a term lasting from 2004 to 2005. He was a core-cipient of the IEEE SPS 1989 Senior Award for Best Paper, a coauthor of the2003 Young Author Best Paper Award, and a corecipient of the 2004 Maga-zine Paper Award. In 2001, he was named University Scholar of the Universityof Illinois. He was the Principal Investigator of the Multidisciplinary Univer-sity Research Initiative (MURI) project titled Adaptive Waveform Diversity forFull Spectral Dominance from 2005 to 2010. He has been a Fellow of the RoyalStatistical Society since 1996.