772 IEEE TRANSACTIONS ON SIGNAL … IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 4, FEBRUARY...

772 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 4, FEBRUARY 15, 2014

A Doppler Robust Design of Transmit Sequence andReceive Filter in the Presence ofSignal-Dependent Interference

Mohammad Mahdi Naghsh, Student Member, IEEE, Mojtaba Soltanalian, Student Member, IEEE,Petre Stoica, Fellow, IEEE, Mahmoud Modarres-Hashemi, Antonio De Maio, Fellow, IEEE, and

Augusto Aubry, Member, IEEE

Abstract—In this paper, we study the joint design of Dopplerrobust transmit sequence and receive filter to improve the perfor-mance of an active sensing system dealing with signal-dependentinterference. The signal-to-noise-plus-interference (SINR) of thefilter output is considered as the performance measure of thesystem. The design problem is cast as a max-min optimizationproblem to robustify the system SINRwith respect to the unknownDoppler shifts of the targets. To tackle the design problem, whichbelongs to a class of NP-hard problems, we devise a novel method(which we call DESIDE) to obtain optimized pairs of transmit se-quence and receive filter sharing the desired robustness property.The proposed method is based on a cyclic maximization of SINRexpressions with relaxed rank-one constraints, and is followed bya novel synthesis stage. We devise synthesis algorithms to obtainhigh quality pairs of transmit sequence and receive filter that wellapproximate the behavior of the optimal SINR (of the relaxedproblem) with respect to target Doppler shift. Several numericalexamples are provided to analyze the performance obtained byDESIDE.

Index Terms—Code design, Doppler shift, interference, receivefilter, robust design, synthesis, transmit sequence.

I. INTRODUCTION

T HE performance of an active sensing system can be signif-icantly improved by judiciously designing its transmit se-

quence and receive filter. Such a design usually deals with sev-eral challenges including the fact that Doppler shifts of movingtargets are often unknown at the transmit side, the existence of

Manuscript received May 08, 2013; revised August 25, 2013; accepted Oc-tober 10, 2013. Date of publication October 30, 2013; date of current versionJanuary 16, 2014. The associate editor coordinating the review of this manu-script and approving it for publication was Dr. Branko Ristic. This work wassupported in part by the European Research Council (ERC) by Grant #228044and the Swedish Research Council.M. M. Naghsh and M. Modarres-Hashemi are with the Department of Elec-

trical and Computer Engineering, Isfahan University of Technology, Isfahan84156-83111, Iran (e-mail: [email protected]).M. Soltanalian and P. Stoica are with the Department of Information Tech-

nology, Uppsala University, Uppsala, SE 75105, Sweden.A. De Maio is with the Dipartimento di Ingegneria Biomedica Elettronica

e delle Telecomunicazioni, Universitá degli Studi di Napoli “Federico II”,I-80125 Napoli, Italy.A. Aubry is with CNR, IREA, I-80124 Napoli, Italy.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.2013.2288082

signal-dependent interference as well as signal-independent in-terference at the receive side, and practical constraints such assimilarity to a given code.Joint design of the transmit sequence and the receive filter

has been considered in a large number of studies during thelast decades. Most of the works have been concerned with ei-ther stationary targets or targets with known Doppler shifts (seee.g. [1]–[8]). In [9], considering a stationary target, a frequencydomain approach has been employed to obtain an optimal re-ceive filter and corresponding optimal energy spectral densityof the transmit signal; then a synthesis procedure has been usedto approximately provide the time domain signal. The works of[10] and [11] consider a related problem to that of [9] under apeak-to-average power ratio (PAR) constraint. The [12] dealswith joint design of transmit sequence and receive filter undera similarity constraint in cases where the Doppler shift of thetarget is known. In [13], constant-modulus transmit sequencesare considered in a framework similar to that of [12]. Several re-searches consider signal-independent clutter scenarios (see e.g.[14]–[18]). The unknown Doppler shift of the target has beentaken into account in [16] and [18]. The [16] considers Dopplerrobust code design problem for signal-independent clutter casesunder a similarity constraint. The ideas of [16] are generalizedin [18] where the PAR constraint is also imposed.In this paper, we devise a novel method for Doppler robust

joint design of transmit sequence and receive filter (which wecall DESIDE) in the presence of clutter. We focus on radar sys-tems but the design methodology can be useful for other activesensing systems such as sonar, seismic exploration, etc. We con-sider the SINR at the output of the receive filter as the perfor-mance measure. Besides an energy constraint, a similarity con-straint is imposed on the transmit sequence to control certaincharacteristics of the transmit waveform. The design problem iscast as a max-min optimization and shown to belong to a classof NP-hard problems.We devise a cyclic maximization to tacklea relaxed version of the design problem. Furthermore, we pro-pose a synthesis stage to obtain optimized pairs of transmit se-quences and receive filters which possess the desired Dopplerrobustness.The rest of this paper is organized as follows. The data

modeling and problem formulation are presented in Section II.Section III contains the steps for the derivation of the cyclicapproach to tackle the relaxed problem. The required synthesis

1053-587X © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

NAGHSH et al.: DOPPLER ROBUST DESIGN OF TRANSMIT SEQUENCE AND RECEIVE FILTER 773

stage is discussed in Section IV. Numerical results are providedin Section V. Finally, conclusions are drawn in Section VI.Notation: We use bold lowercase letters for vectors and bold

uppercase letters for matrices. , and denote thevector/matrix transpose, the complex conjugate, and the Her-mitian transpose, respectively. represents the identity matrixin . and are the all-one and the all-zero vectors/ma-trices. is the th standard basis vector in . The -normof a vector is denoted by . The symbol stands for theHadamard (element-wise) product of matrices. is the traceof a squarematrix argument. The notations andindicate the principal and the minor eigenvalues of a Hermi-tian matrix, respectively. denotes the diagonal matrixformed by the entries of the vector argument, whereasdenotes the vector formed by collecting the diagonal entries ofthe matrix argument. We write iff is positivesemi-definite, and iff is positive-definite.and denote the real-part and the phase angle (in radians)of the complex-valued argument. Finally, , and representthe set of natural, real and complex numbers, respectively.

II. PROBLEM FORMULATION

We consider a radar system with (slow-time) transmit se-quence and receive filter . The discrete-time received signal backscattered from a moving target cor-responding to the range-azimuth cell under the test can be mod-eled as (see, e.g. [12], [13], and [15]):

(1)

where is a complex parameter associated with backscat-tering effects of the target as well as propagation effects,

with being the normalizedtarget Doppler shift (expressed in radians), is the -di-mensional column vector containing clutter (signal-dependentinterference) samples, and is the -dimensional columnvector of (signal-independent) interference samples. The vectoris the superposition of the returns from different uncorrelatedscatterers located at various range-azimuth bins and can beexpressed as [12]

where is the number of range rings1 that interferewith the range-azimuth bin of interest (0,0), is the number ofdiscrete azimuth sectors, and denote the echo andthe normalized Doppler shift, respectively, of the scatterer in therange-azimuth bin , and denotes the aperiodic shiftingmatrix for , viz.

ifif

with .

1Note that the model considers the general case of range ambiguous clutterand reduces to unambiguous range scenario for . See [12] and [15] forjustifications of the employed model and several examples of scenes that can bemodeled in this way.

The SINR at the output of the receive filter can be formulatedas

(2)

where and is the covariance matrix ofgiven by [12]

(3)

with being the mean interfering power as-sociated with the clutter patch located at the th range-az-imuth bin whose Doppler shift is supposed to be uniformly dis-tributed in the interval

[15]. Herein where

is the covariance matrix of [12], viz.

if

if

(4)

Note that the expression for can be modified toconsider cases with an arbitrary statistical distribution of theDoppler shifts of the clutter scatterers.In this study we assume that the parameters of clutter and

signal-independent interference are known at the transmit sideby using cognitive (knowledge-aided) methods [12], [19]. Weconsider the SINR in (2) as the performance measure of thesystem [12], [15] and aim to find a robust design of the transmitsequence and the receive filter with respect to the unknownDoppler shift of the target.2 In addition to an energy constraint,a similarity constraint is imposed on the transmit sequence [12],[21], [22]:

(5)

where the parameter rules the size of the similarity re-gion and is a given sequence. There are several reasons thatjustify the use of a similarity constraint in the design of a radarsequence. The unconstrained optimization of SINR can lead tosignals with significant modulus variations, poor range resolu-tion, high peak sidelobe levels, and more generally with an un-desired ambiguity function behavior. These drawbacks can bepartially circumvented imposing the similarity constraint (5) onthe sought radar code [12], [21], [22]. Comprehensive simula-tions have been performed in [12], [16], [21] and [23] to illus-trate how the properties of the ambiguity function (e.g. rangeresolution, sidelobe levels, etc.) and modulus variations asso-ciated with the optimized code can be controlled via the valueof in the similarity constraint. By doing so, it is required thatthe solution be similar to a known sequence which has some

2The target Doppler shift can be estimated at the receiver, e.g. via a bank of fil-ters matched to different Doppler frequencies [20]; however, the Doppler shiftsof the targets are usually unknown at the transmit side and hence we considera robust design with respect to the target Doppler shift. The design approachcan also be useful for a robust confirmation process, so as to account for targetDoppler estimation errors.


good properties such as constant modulus, reasonable range res-olution, and peak sidelobe level.The problem of Doppler robust joint design of transmit se-

quence and receive filter under the similarity constraint canbe cast as the following max-min optimization problem

(6)

where denotes a given interval of thetarget Doppler shift and denotes the maximum availabletransmit energy. Note that for a priori known target Dopplershift (i.e. ), the problem boils down to the con-sidered problem in [12].Remark 1: Note that a similar discrete-time data modeling

and problem formulation applies to fast-time coding systems.In that case, the entries of denote (complex) weights of thesub-pulses within a transmit pulse. Moreover, the normalizedtarget Doppler shift is proportional to the system bandwidth(as opposed to the slow-time scheme for which is proportionalto the pulse repetition frequency of the system); hence in such acase, the Doppler robust design would be concerned with highspeed moving targets. As to the expressions, the formulation ofthe covariance matrix in (3) should be modified. Moreprecisely, for fast-time coding scenarios, the summation overin (3) should be performed for . We referinterested readers to the [10] and [20] for more details on thisaspect.To realize the hardness of the above problem, let and

denote a slack variable and an optimal solution to the problem, respectively. The optimal is obtained via solving the fol-

lowing optimization problem:

(7)

The above quadratic fractional program can be recast equiva-lently as (see Lemma 2 below and [24]):

(8)

The optimization problem is a quadratically constrainedquadratic program (QCQP) with infinitely many non-convexconstraints. This class of QCQPs is known to be NP-hard ingeneral [16], [25, Chapter 4], [26]. Note that solving the opti-mization problem with respect to is at least as hardas solving the problem .

The following lemma helps tackling the optimizationproblem via providing two alternative expressions for theobjective function in problem .Lemma 1: Let and . The

can be alternatively expressed with respect to and asfollows:

(9)

(10)

where , and

(11)

(12)

Proof: See Appendix A.To deal with the design problem , consider the following

optimization problem:

(13)

where and . Let de-note an optimal solution to the above problem. Using Lemma 1and the results of [21], it can be easily verified that an optimalsolution to is given by withand .Now observe that both the objective function and the rank

constraints in are non-convex. In addition, belongs to anon-convex set for . In the sequel, we relax the rank-oneconstraints on and in to obtain the relaxed problem :

(14)

The expression for rank-oneand (i.e., and ) is equal to

(see Lemma 1). When the rank constraints are


omitted (i.e., for arbitrary and ), the expressionmay be used in lieu of

and it will be denoted by in the following.is the restriction of over the space

of the rank-one positive semi-definite matrices and (dueto the relaxation of the rank-one constraints on and ).The optimization problem is still non-convex and will bediscussed in the next section.

III. THE PROPOSED METHOD TO TACKLETHE RELAXED PROBLEM

In this section, we devise a novel cyclic algorithm (which wecall DESIDE-R as it deals with the relaxed version of the orig-inal problem) to tackle the non-convex optimization problem. In a cyclic algorithm, the optimization variables are parti-

tioned into two parts; then, by starting from an initial point, opti-mization is cyclically performed with respect to each part (whilethe another part is fixed) [27]. In the following, we consider themaximization problem with respect to where andare the two partitions. The obtained pair which

maximizes will be used later to synthesize theoptimized transmit sequence/receive filter pair . Thesynthesis stage is addressed in Section IV.

Optimal for Fixed : Let denote a slackvariable. For fixed , the optimization in (14) is equivalent tothe following maximization problem:

(15)Note that the above problem is feasible and has a finite-valuedobjective function over the constraint set (see (29)). Moreover,problem is a linear-fractional maximization problem withinfinitely many constraints (see the first constraint in (15)). In-spired by Charnes-Cooper transform for tackling linear frac-tional programs [28], we let , for an auxil-iary variable , and consider the following optimizationproblem:

(16)Lemma 2: The optimization problems and are equiv-

alent. More precisely, they share the same optimal values and

their corresponding solutions can be uniquely obtained fromeach other.

Proof: Let and denote an optimal solutionand the optimal value of the problem , respectively. Notethat because . It isstraightforward to verify that

(17)

is feasible for the problem . Also observe that the value ofthe objective function of for in (17) is given by

(18)

and note that (18) is equal to . Therefore, for the optimalvalue of the problem , i.e. , we have

(19)

Next let denote an optimal solution to theproblem . Note that because leads to

(a contradiction, see the first constraint in ). Onecan check that is feasible for the problemwith corresponding objective value equal to . Owing to thefact that , the following inequality holds between

and :

(20)

Finally, (19) and (20) yield and the proof isconcluded.Now observe that is a convex problem with infinitely

many constraints. To deal with the constraint set, we note thatthe constraint impliesthe non-negativity of a trigonometric polynomial of over theinterval . More specifically, let

(21)

and with . It is straight-forward to verify that for any , the aforementioned con-straint is equivalent to

(22)

Interestingly, a semidefinite representation of the constraint (22)can be obtained via Theorem 3.4 in [29] which we quote below.Theorem 1: The trigonometric polynomial

is non-negative for any


TABLE IDESIDE-R METHOD FOR SOLVING THE RELAXED PROBLEM

(with ) iff there exist an Hermitian ma-trix and an Hermitian matrixsuch that

(23)

where ,with ,

and in which

with , and.

Note that an SDP representation of (22) is immediate by em-ploying the above results with , ,

, and . Consequently, is equivalentto the following SDP:

(24)Remark 2: The derivation of can be extended to

deal with cases where is a union of several (non-overlap-ping) sub-intervals of . More precisely, for each of suchsub-intervals, the SDP representation associated with the corre-sponding constraint (obtained via Theorem 1) can be added tothe constraint set of .Let denote an optimal solution to .

The corresponding optimal (i.e., an optimal solution to )for fixed is given by (see Lemma 2).

Optimal for Fixed : Using Lemma 1, can be recastinto the following equivalent form for fixed :

(25)where denotes a slack variable. The above problem can betackled in a way similar to the case of obtaining for fixed .

In particular, using Lemma 2 as well as Theorem 1, we obtainthe following SDP:

(26)

where is given by

(27)

with .Remark 3: It might be interesting in practice to control the

shape of the cross-ambiguity function of the transmit sequenceand the receive filter . An approach would then be to require

that the variables and are sufficiently similar to givenand , respectively, which possess desirable cross-ambiguityproperties. The Doppler robust design for controlling the shapeof the cross-ambiguity function could therefore be cast as thefollowing optimization problem:

(28)

where rules the size of the similarity region for the receivefilter. The problem can be tackled in a way similar to(6).The steps of DESIDE-R are summarized in Table I. Each iter-

ation of the proposed method is handled via solving two SDPs,i.e., and . The complexity of solving the SDPswith accuracy of is given by [30]. A syn-thesis stage is proposed in the next section to compute highquality transmit sequence/receive filter pairs from thesolutions obtained herein.

Convergence and the Metric: By cycli-cally solving and in DESIDE-R, it can be

easily verified that the resultingis a monotonically increasing sequence [27]. Furthermore,

is bounded from above; indeed wehave that

(29)


The third inequality above holds true because; and for the last inequality we have used the fact

that [31]. Eq. (29) along with

the increasing property of en-sure the convergence of the sequence of the objective functionvalues.

IV. THE SYNTHESIS STAGE

As discussed earlier, a judicious synthesis of the optimizedtransmit sequence and receive filter from the obtained

is required to maintain the Doppler robustness. Ifis rank-one, is available via considering ;

whereas if , for we have .The rank behavior of SDP solutions, tightness of the semidef-inite relaxation, and synthesis methods have been investigatedin the literature (see e.g. [33]–[35], and references therein). Forexample, it is known that for a solvable3 SDP with con-straints, there exists an optimal solution of rank at most equalto [33]. However, the result does not ensure the existenceof rank-one solutions for the case considered in this paper dueto the fact that and have andconstraints, respectively. Herein we remark on the fact that theranks of and depend on the employed starting point inaddition to the parameters of the design problem.In cases where the rank of either or is larger than one,

the synthesis of or is more complicated. To tackle thisproblem, this section initially considers the rank-one decompo-sition method [36]. Then we devise novel synthesis algorithmsto design pairs of possessing the desired robustness.

A. The Rank-One Decomposition Method

The main result of the rank-one decomposition method canbe summarized as follows [36]:Theorem 2: Let be a non-zero complex Her-

mitian positive semidefinite matrix (with ) andbe Hermitian matrices. Suppose that

forany non-zero complex Hermitian positive semidefinite matrixof size . Then,• if , one can find, in polynomial time, arank-one matrix such that (synthetically denotedas ) is in the , and

• if , for any not in the range space of , onecan find a rank-one matrix such that (syntheticallydenoted as ) is in the linearsubspace spanned by , and

Proof: See [36, Theorem 2.3].Let denote an optimal solution to , and let

(30)

3Meaning that the SDP is feasible, bounded, and its optimal value is attained(see [30] for more details).

Considering Theorem 2 and the problem , a suitablerank-one matrix can be found such that

(31)

and that

(32)

where and are two arbitrary Doppler shifts in .The equations in (32) have been written using the identity

.Note that Theorem 2 lays the ground for considering two moreDoppler frequencies (i.e. and ) other than . This leadsto a receive filter with a behavior more similar to that of

with respect to target Doppler shift . Consequently,is obtained via or

for cases whereor , respectively. Note that the condition

, onthe matrices , , , and in Theorem 2 is satisfied;more precisely, there exists such that

(see [37]).The can be obtained in a similar way; particularly,

( 1 or 2 depending on the rank of) where

(33)

(34)

B. New Algorithms for Synthesis Stage

As explained earlier, the rank-one decomposition method candeal with at most four trace equalities for the synthesis of thereceive filter and the transmit sequence. This ability allows forconsidering the values of the at three Dopplershifts for the receive filter synthesis (and one Doppler shift forsynthesis of the transmit sequence). However, the pairobtained by applying Theorem 2 can lead to whosebehavior with respect to target Doppler shift is not “sufficiently”close to the behavior of the . It means that the

can have significantly lower minimum value withrespect to . In this subsection, we devise novel algorithms tosynthesize high quality and from the solutions to theproblem , i.e. and . The idea is to consider the valuesof as the optimal energy spectral density


(ESD) associated with the transmit sequence and the receivefilter. The sought receive filter and transmit sequenceare obtained to approximate well the behavior of the optimalESD with respect to . Due to the fact that (of

) is a scaled version of the optimal ESD (see (9)), wedeal with the denominator of the by imposingconstraints in the synthesis problems.Let denote a discrete set of target Doppler

shifts “uniformly distributed” over , and consider the fol-lowing quantities:

(35)

where . We define the vectoras the optimal ESD.Receive Filter Synthesis: Herein the aim is to synthesize the

optimized receive filter for given . Observe that

(36)

Note also that for all and so that there must exist(of rank ) such that . Moreover, considering

, we can write

(37)

where all are unit-norm. Therefore, the receive filtercan be found as the minimizer of the following metric:

(38)

where , , and. Note that the optimal

(corresponding to ) is a scaled ver-sion of the optimal ESD for given (see (9)). As aresult, to obtain the receive filter that yields valuesclose to , we should also take into account thedenominator of the , viz.

(39)

Consequently, we consider the following optimization problemto obtain :

(40)

with . In the sequel, we propose a cyclicminimization to tackle the non-convex problem .For fixed , the problem boils down to the non-convex

problem:

(41)

where contains the entries of corresponding to for. The above minimization can be decoupled into

minimization problems of the following form:

(42)

The solution to the above nearest-vector problem is simplygiven by

(43)

For fixed , the problem is equivalent to the followingQCQP:

(44)

Note that the positive definiteness of the matrices andensures the convexity of the above QCQP. As a result, this

QCQP can be solved efficiently via interior point methods orLagrange multipliers [38].Transmit Sequence Synthesis: A technique similar to the

above one can be used for transmit sequence synthesis. Moreprecisely, we have

(45)

Therefore, minimization of the following metric can be em-ployed for transmit sequence synthesis:

(46)

where , andare defined similarly to the case of receive filter design (

with being the rank of ). Note that for transmit se-quence synthesis, the similarity and energy constraints shouldbe taken into account in addition to the denominator of the

. Consequently, we consider the following op-timization problem to synthesize the sought transmit sequence:

(47)

where and . Let denote theoptimal solution to the above problem. Note thatis such that , and . Therefore, one canconsider as the optimized transmit sequence which lies inthe desired similarity region and is feasible for the problem .The non-convex optimization problem can be dealt

with via a cyclic minimization similar to that used for . Forfixed , the solution to the th resulting nearest-vector problemis given by

(48)


TABLE IITHE DESIDE METHOD FOR OBTAINING DOPPLER ROBUST PAIR OF

TRANSMIT SEQUENCE AND RECEIVE FILTER

where includes the entries of corresponding to for. On the other hand, the case of fixed is handled

by solving the following convex QCQP:

(49)

Remark 4: Note that the synthesized via the rank-one de-composition is a feasible point for the above convex QCQP.Indeed, the output of the rank-one decomposition procedure inSection IV-A can be used as a feasible starting point for the pro-posed cyclic minimization to obtain the transmit sequence. Thiscan also be done for the receive filter synthesis.The steps of DESIDE can be summarized as in Table II. The

first step consists of applying DESIDE-R to the relaxed problem(see Table I). Steps 2 and 3 aim to synthesize high quality

receive filters and transmit sequences, respectively. The cyclicminimizations in step 2 is terminated when a pre-defined stopcriterion is satisfied; e.g. for a givenwhere denotes the iteration number. A similar criterion can beused to terminate the algorithm in the step 3. Note that the ob-tained after satisfying the stop criterion in the step 3 is scaledto obtain with energy . The complexity of DESIDE can beaddressed considering DESIDE-R and the synthesis stage. Thecomplexity of each iteration of DESIDE-R is (see thediscussion above Table I). The complexity of each iteration ofthe proposed synthesis stage is determined by the complexityof solving the QCQPs in (44) and (49). These QCQPs can besolved via described methods in [30] with complexity.

V. NUMERICAL EXAMPLES

In this section we provide several numerical examples to ex-amine the effectiveness of DESIDE method. Throughout thesimulations, unless otherwise explicitly stated, we consider a

code length4 , number of interfering range rings, and number azimuth sectors . The interfering sig-nals backscattered from various azimuth sectors are weightedaccording to the azimuth beam-pattern characteristic of a typicallinear array (see [12] for details). A uniformly distributed clutteris assumed with for all . In addition, welet the Doppler shifts of the clutter scatterers be uniformly dis-tributed over the interval[40]. As to the target, we set . Concerning the covari-ance matrix of the signal-independent interference, it is as-sumed that with parameter . Regarding thesimilarity constraint, the generalized Barker code is used for se-quence [41]. This is a constant modulus sequence which hasgood correlation properties [12]. The size of the similarity re-gion is controlled by . The total transmit energy issupposed to be equal to the sequence length . The convex op-timization problems are solved via the CVX toolbox [42].

A. The Effect of the Design Parameters

The Width of and the Correlations Between the Inter-ference Samples: The performance of the system generally de-pends on the width of target Doppler shift interval and thecorrelations between the interference samples (controlled by theparameter ). Herein the non-robust design (i.e., with a prioriknown target Doppler shift ) of the transmit sequence and re-ceive filter (with a similarity constraint) [12], i.e. the solution tothe following problem:

(50)

is considered as a benchmark for comparisons. The effects ofthe width of interval and the value of are investigated inFig. 1, where the values of obtained by DESIDE(with and ) are compared with those of thenon-robust design for two intervals ,and for . For the non-robust design, we reason-ably set equal to with . In all examples, itis observed that DESIDE provides a robust over theconsidered interval of target Doppler shifts. The minimumvalue of obtained by DESIDE outperforms that ofthe non-robust design significantly. The superiority of DESIDEis highlighted by observing that for a considerable range of thetarget Doppler shift , the obtained by DESIDE isaround 10 dB larger than that of the non-robust design. Further-more, for any fixed , the minimum value of in theinterval is less than that for . As ex-pected, the wider range of target Doppler shift leads to a morerestricted design. Another observation is that for a fixed interval, the minimum values of increase as increases.The observation is compatible with the behavior of the upperbound on the in (29)-by increasing the

4It is expected that the output SINR of the receive filter increases by in-creasing due to the increase in the number of degrees of freedom for thedesign problem (see e.g. [39]) and the longer coherent processing interval [40].


Fig. 1. Design examples for various target Doppler shift intervals and various : (a) and , (b) and , (c)and , (d) and , (e) and , (f) and .

value of , the value of decreases and the upper boundon the becomes larger. Note that in these exam-ples, the ranks of the optimal and were equal to oneand hence the obtained pairs of the transmit sequence and thereceive filter are optimal for the problem .

Size of the Similarity Region: Examples for the robust de-sign of transmit sequences and receive filters with various sizes

of similarity region are now provided. The values ofobtained by DESIDE for different in {0.01,0.2,0.4,0.8} aredepicted in Fig. 3. The robustness property with respect to thetarget Doppler shift is observed in all examples. As expected,the larger the , the larger the worst value of the .This is due to a larger feasibility set for the optimization problem

and the fact that the optimal and are rank-one.


Fig. 2. The values of obtained at different iterationsof DESIDE-R for , as well as two intervals and

.

Fig. 3. Design examples for various sizes of the similarity region:obtained by DESIDE versus target Doppler shift for , 0.2, 0.4, and0.8.

B. Convergence of DESIDE-R

Examples of the convergence of DESIDE-R are depicted inFig. 2. This figure shows the values of the objective function (inthe maximization problem ) obtained through the iterations

(with denoting the iteration number) for ,, as well as two intervals and . As

expected, the cyclic maximization approach which is devised totackle leads to a monotonically increasing objective function

. The values of the objective functionfor are larger than those for (seethe discussions associated with Fig. 1). Note that both and

are rank-one here, and as a result, the obtained pairs of thetransmit sequence and receive filter are optimal for the originaldesign problem .

C. A Fast-Time Coding Example

As mentioned earlier (see Remark 1), the problem formu-lation and the design method can also be applied to fast-timecoding systems. We present an example of such an application

Fig. 4. Design example for a fast-time coding system.

by considering and . The target Doppler shiftis assumed to be in the interval . The consid-

ered maximum target Doppler shift corresponds to a target withan approximate velocity of Mach 4 illuminating by an L-bandradar of sampling frequency 1 MHz. Owing to the fact that nor-malized Doppler shift in this case is proportional to the systembandwidth, we neglect the effect of the Doppler shifts of clutterscatterers. Fig. 4 shows the obtained by DESIDEas well as the results for the non-robust design, for ,

, and . It is observed that employing DESIDE leadsto performance robustness of the system. In this example, theresult obtained by DESIDE outperforms that of the non-robustdesign for . Moreover, the obtained andwere rank-one too, similar to the examples presented earlier.

D. The Synthesis Algorithms

The performance analysis of the synthesis algorithms is per-formed by considering cases where the ranks of the solutionsto the relaxed problem are larger than one. We consider20 random starting points for the synthesis algorithms (with

) and report the best result. In the first example, we as-sume , , , . Fora random initialization, DESIDE-R provides with

and (it was numerically ob-served that as long as , the rank of is equalto one for most of the employed random initial points). Theoptimal ESD corresponding to the pair is shown inFig. 5(a). The values of for the synthesized andare shown in Fig. 5(b). This figure also includes the op-

timal (corresponding to ) and the re-sult of applying the rank-one decomposition method. For thelatter method, the best result is obtained with ,, and . It is observed that using the proposed syn-

thesis algorithm leads to values of that are close tothe optimal ones. Fig. 5(c) shows the optimal ,

for synthesized via the proposed algorithmand the result of rank-one decomposition method for anothercase in which . The performance of the rank-onedecomposition method is degraded considerably whereas the


Fig. 5. Results obtained with the proposed synthesis algorithms: (a) an optimal ESD, (b) results of the filter synthesis corresponding to part (a), (c) another filtersynthesis example, (d) absolute value of the loss metric of the transmit sequence synthesis versus the number of ESD samples for the considered transmitsequence synthesis example. The zoomed areas in (b) and (c) show the values of the optimal and of the proposed synthesis methodin the neighborhoods of their minimum values.

difference between the results of the proposed algorithm andis minor. This can be explained by noting that

the rank-one decomposition method can consider the values ofthe optimal at up to three points, i.e., and. On the other hand, the proposed method considers a con-

strained synthesis problem to approximate the values of optimalfor an arbitrary set of discrete . To measure the

goodness of the synthesis algorithms, we define the followingloss metric:

(51)

In this example, the loss metric for the proposed method andthe rank-one decomposition method are equal to 0.25 dB and4.1 dB, respectively. Next we study the effect of the number

of optimal ESD samples, i.e. , on the performance of the pro-posed synthesis stage. The results for a transmit sequence syn-thesis example are illustrated in Fig. 5(d). For this example, wehave , , and . Notethat it was numerically observed that the rank of is equal

to one as long as . The figure shows the absolutevalues of loss metric versus . It is seen that the performanceimprovement for is negligible. Another observation isthat there is about 2 dB loss even for sufficiently large valuesof . This might be due to imposing more constraints in thesequence synthesis as compared to the case of filter synthesis.In the example of Fig. 5(d), the loss of the rank-decompositionmethod is around 13 dB; here the latter method can take intoaccount just one point of the optimal , i.e., .

VI. CONCLUDING REMARKS

A joint robust design of the transmit sequence and receivefilter was considered for cases where the Doppler shift of thetarget is unknown. A novel method (called DESIDE) was pro-posed to tackle the design problem under the similarity con-straint. The main results can be summarized as follows:• The robust design problem was cast as a max-min problemby using the model which considers the effects of the inter-fering clutter scatterers at various range-azimuth bins andinternal Doppler shifts of these scatterers. It was shown


that for a given optimal transmit sequence, the problem canequivalently be written as a QCQP with infinitely manynon-convex constraints and hence the design problem ingeneral belongs to a class of NP-hard problems.

• DESIDE was devised to tackle the design problem. Themethod consists of solving a relaxed version of the designproblem (via DESIDE-R) as well as a synthesis stage:— DESIDE-R was based on a reformulation ofby considering and , relaxationof the rank-one constraints on the aforementioned ma-trices, and cyclic maximization of the relaxed problem.For fixed receivefilter, the relaxed optimization problemwas equivalently expressed as an SDP by using a trans-formation (inspired by Charnes-Cooper transform) andan SDP representation of the infinitely many affine con-straints. Using a similar technique, an SDPwas obtainedin the fixed transmit sequence case.

— New algorithms were devised to synthesize the receivefilters and transmit sequences from the solutions to therelaxed problem. The synthesis algorithms aim to fitthe values associated with the solutionsprovided by DESIDE-R. The synthesis stage is castas constrained non-convex problems which were dealtwith via cyclic minimization.

• The effectiveness of the devisedmethods was illustrated byproviding several numerical examples. It was shown thatthe DESIDE system performance possesses a considerablerobustness with respect to the target Doppler shift. The nu-merical analysis of the proposed synthesis algorithms con-firms that high quality pairs of receive filter and transmitsequence can be synthesized from the solutions to the re-laxed problem.

The design problem considered in this paper is based onknown parameters of clutter and signal-independent interfer-ence. Robust design of transmit sequences and receive filterswith respect to uncertainties in clutter and interference param-eters in addition to the target Doppler shift is a possible topicfor future research.

APPENDIX A

PROOF OF LEMMA 1

Note that the numerator of in (2) can be rewrittenas

(52)

where in the last equality we have used standard properties ofthe Hadamard product [31]. As to the denominator ofin (2), it is straightforward to verify that, for all ,

(53)

Using the matrix variable and substituting the aboveidentity in (3) we obtain that

(54)As a result, (52) and (54) yield the expression of in(9).To derive the alternative expression of in (10), we

begin by considering the result of the Lemma 3.1 in [12] whichimplies

(55)

Note also that for all

(56)

Therefore, using (56) as well as the fact that the covariance ma-trix , we can write

(57)

where and

(58)Now let , and observe that

(59)The above identity and (52) prove the validity of the alternativeexpression of in (10).

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Mohammad Mahdi Naghsh (S’09) received theB.Sc. and the M.Sc. degrees both in electricalengineering from Isfahan University of Technology,Isfahan, Iran.He is currently pursuing the Ph.D. degree at the

Department of Electrical and Computer Engineering,Isfahan University of Technology, Isfahan, Iran.From May 2012 to May 2013, he was a visitingresearcher with the Department of InformationTechnology, Uppsala University, Sweden. Hisresearch interests include statistical and array signal

processing with emphasis on active sensing signal processing, detection andestimation, multicarrier modulations, and cognitive radio.

Mojtaba Soltanalian (S’08) received the B.Sc. de-gree in electrical engineering from Sharif Universityof Technology, Tehran, Iran, in 2009.He is currently working toward the Ph.D. degree

in electrical engineering with applications in signalprocessing at the Department of Information Tech-nology, Uppsala University, Sweden. His research in-terests include different aspects of sequence designfor active sensing and communications.

Petre Stoica (SM’91–F’94) is currently a Professor of systems modeling at theDepartment of Information Technology, Uppsala University, Sweden. More de-tails about him can be found at http://www.it.uu.se/katalog/ps.

Mahmoud Modarres-Hashemi received the B.Sand M.S. degrees in electrical engineering in 1990and 1992, respectively, from the Electrical and Com-puter Engineering Department, Isfahan Universityof Technology (IUT), Isfahan, Iran. He receivedthe Ph.D. degree in 2000 from the Departmentof Electrical Engineering of Sharif University ofTechnology, Tehran, Iran.He was an assistant professor with the ECE De-

partment, IUT, until 2008, when he became an asso-ciate professor. His research interests are signal pro-

cessing in radar and communication, detection theory, and electronic warfare.


Antonio De Maio (S’01–A’02–M’03–SM’07–F’13)was born in Sorrento, Italy, on June 20, 1974. Hereceived the Dr.Eng. degree (with honors) and thePh.D. degree in information engineering, both fromthe University of Naples Federico II, Naples, Italy, in1998 and 2002, respectively.FromOctober to December 2004, he was a Visiting

Researcher with the U.S. Air Force Research Lab-oratory, Rome, NY. From November to December2007, he was a Visiting Researcher with the ChineseUniversity of Hong Kong, Hong Kong. Currently, he

is an Associate Professor with the University of Naples Federico II. His re-search interest lies in the field of statistical signal processing, with emphasison radar detection, optimization theory applied to radar signal processing, andmultiple-access communications.Dr. De Maio is the recipient of the 2010 IEEE Fred Nathanson Memorial

Award as the young (less than 40 years of age) AESS Radar Engineer 2010whose performance is particularly noteworthy as evidenced by contributions to

the radar art over a period of several years, with the following citation for “ro-bust CFAR detection, knowledge-based radar signal processing, and waveformdesign and diversity.”

Augusto Aubry (M’12) received the Dr.Eng. degree(with honors) and the Ph.D. degree in information en-gineering, both from the University of Naples Fed-erico II, Naples, Italy, in 2007 and 2011, respectively.He is currently under a research agreement with the

CNR-IREA. His research interests are in statisticalsignal processing and optimization theory, with em-phasis on MIMO communications and radar signalprocessing.

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