Signal Dependent Clutter Waveform Design forRadar STAP

Pawan SetlurWright State Research Institute

Beavercreek, OH 45431

Muralidhar RangaswamyUS Air Force Research Laboratory

Sensors DirectorateWPAFB, OH 45433

Abstract—Waveform design is a pivotal component of the fullyadaptive radar construct. In this paper we consider waveformdesign for radar space time adaptive processing (STAP), ac-counting for the waveform dependence of the clutter correlationmatrix. Due to this dependence, the joint problem of receiverweight vector optimization and radar waveform design becomesan intractable optimization problem. Therefore, we considerconstrained versions of the alternating minimization algorithm,which, at each step, optimizes the STAP weight vector andwaveform independently. Our simulations reveal a non-increasingerror variance at the output of the filter.

I. INTRODUCTION

In this paper we addresses waveform design for radar STAP[1]–[3] considering the important problem of the dependenceof the waveform on the clutter correlation matrix. The noiseand interference are also considered but unlike the clutter,are not dependent on the waveform. For convenience, anairborne calibrated uniform linear array is assumed in themodeling but other array geometries could be considered ina straightforward manner. Other assumptions made in thispaper are identical to those made in [4] and can be seenthere. A minimum variance distortionless response (MVDR)optimization objective is formulated and is a function of thetraditional STAP weight vector as well as the waveform.

An optimal STAP weight vector solution dependent onthe waveform is immediate. However, after substituting thisweight vector back into the objective yields an optimizationproblem purely with respect to the waveform, and whosesolution is not readily apparent. Further, our extensive sim-ulations leads us to believe that the original objective functionis not jointly convex in the weight vector and the waveform,attributed to the clutter covariance matrix dependence on thewaveform. Nonetheless, analytically proving /disproving jointconvexity of the objective function is neither straightforwardand nor is the focus of this paper. It is however demonstratedhere that the objective function is independently convex w.r.tthe STAP weight vector and the waveform. Therefore, theproblem calls for an iterative solution, which at each step,solves a convex optimization problem, iteratively optimizingfirst, the weight vector for a fixed waveform, and second,optimizing the waveform for the fixed weight vector designedin the first step. To addresses practical considerations for radarwaveform design, we invoke the constant modulus constrainton this objective and use a similar iterative approach to solvethe resulting optimization problems.

Literature: Clutter dependence on the waveform was rec-ognized early on in [5] and references therein. However theiterative solution proposed in [5] is for a single sensor radarand is not readily extended to a multi-sensor radar frameworksuch as STAP. Other iterative solutions were proposed butnot for STAP in [6] and [7] for the joint optimization of thereceive filter and the waveform, and for a monostatic radarand MIMO radar architecture, respectively. Signal dependentclutter waveform design with the objective of maximizing de-tection was formulated in [8] and gave rise to a waterfilling likesolution [9]. Waveforms designed from maximizing the mutualinformation and considering the signal independent /dependentclutter were the subject of [9] and [10], respectively. In [4],STAP radar waveform design ignoring the signal dependenceof clutter was treated, giving rise to the well known minimumeigenvector solution.

Organization: The paper is organized as follows, in SectionII, the model is presented and is similar to the one derived in[4]. The waveform design is considered in Section III, first forthe general unconstrained waveform, and next for waveformconstrained to a constant modulus design. Supporting simula-tions are presented in section IV, and conclusions are drawnin Section V.

II. STAP MODEL

The radar consists of an air-borne calibrated uniform lineararray comprising M sensor elements, which transmits a burstof L pulses in a coherent processing interval (CPI). Thewaveform transmitted is assumed to be discretized and consistsof N samples.

At the considered range gate, the contaminated snapshotfrom the STAP data cube is modeled as:

y =y + yi + yc + yn (1)=y + yu

where yi,yc,yn are the contributions from the interference,clutter and noise, respectively, and are assumed to be statis-tically uncorrelated with one another. The target response is

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denoted as y and is given by 1

y ∈ CNML = ρtv(fd)⊗ s⊗ a(θt, φt) (2)

where ρt is the target reflectivity assumed to be random butidentical for the entire CPI. In (2), the discretized waveformvector is s = [s(0), s(1), . . . , s(N − 1)]T ∈ CN . Similarly,the targets spatial and temporal steering vectors are denotedas a(θt, φt) and v(fd), respectively, where (θt, φt) are thetargets azimuth and elevation angles, and fd is the Dopplerfrequency of the target. More details are readily seen in [4].The contribution of the undesired returns are treated next,starting with the noise as it is the simplest.

Noise: The noise is assumed to be zero mean, identicallydistributed across the sensors, across pulses, and in the fasttime samples. The correlation matrix of yn is denoted as Rn ∈CNML×NML.

Interference: The interference consists of jammers. Let usassume that there are K interference sources,the correlationmatrix for the interference is modeled as [4]:

Ri = A(θ, φ)RαA(θ, φ)H (3)

where

Rα := Diag{R1α,R

2α, . . . ,R

Kα } ∈ CNMLK×NMLK

A(θ, φ) ∈ CNML×NLMK

: = [INL ⊗ a(θ1, φ1), INL ⊗ a(θ2, φ2), . . . , INL ⊗ a(θK , φK)],

for INL the identity matrix of size NL × NL, andDiag{·, ·, . . . , ·} the matrix diagonal operator which convertsthe matrix arguments into a bigger diagonal matrix. Forexample, Diag{A,B,C} =

[A 0 00 B 00 0 C

].

Clutter: The ground is a major source of clutter in air-borneradar applications and is present in all range gates upto thegate corresponding to the platform horizon. As in [4], othersources of clutter are ignored. Let us assume that there areQ ground clutter patches indexed by parameter q comprisingof P scatterers in each patch, indexed by p. The radar returnfrom the p-th scatterer in the q-th clutter patch is given by

γpq ⊗ v(fcq)⊗ s⊗ a(θq, φq)

where γpq is its random complex reflectivity, and fcq isthe Doppler shift observed from the q-th clutter patch, and(θq, φq) are the azimuth and elevation angles of the q-thclutter patch. It is implicitly assumed that the scatterers ina particular clutter patch have identical Doppler as theyare in the same range gate. It is assumed that the spatialresponses of scatterers in the same clutter patch are identicalto one another, and thus the clutter correlation matrix forthe q-th patch is written as: Rq

γ := BqRpqγ Bq

H where,Bq = [v(fcq)⊗ s⊗ a(θq, φq), . . . ,v(fcq)⊗ s⊗ a(θq, φq)] ∈CNML×P and Rpq

γ ∈ CP×P is the correlation matrix ofthe random vector, [γ1q, γ2q, . . . , γPq]

T . The matrix Bq can

1If x,y, z are column vectors of arbitrary sizes, then trivially T (x⊗ y⊗z) = y ⊗ x ⊗ z. The (re-ordering) transformation T was used but notmentioned in [4]. Versions of the target and clutter responses here and in [4]are identical upto a transformation matrix.

further be simplified as Bq := Bq(IP ⊗ s), where Bq :=[v(fcq) ⊗ Aq, . . . ,v(fcq) ⊗ Aq] ∈ CNML×PN , and thestructure of the matrix Aq ∈ CNM×N (not shown here) isdefined such that s⊗a(θq, φq) = Aqs. Therefore, using thesesimplifications, the clutter correlation matrix for the q-th patchis expressed as:

Rqγ = Bq(IP ⊗ s)Rpq

γ (IP ⊗ sH)BHq . (4)

Assuming that a particular scatterer from one clutter patchis uncorrelated to any other scatterer belonging to any otherclutter patch, the resulting clutter correlation matrix is givenby

Rc =

Q∑q=1

Rqγ . (5)

III. WAVEFORM DESIGN

The radar return at the considered range gate is processedby a filter characterized by a weight vector, w, whose outputis given by wH y. Since the vector s prominently figures in thesteering vectors, the objective is to jointly obtain the desiredweight vector, w and waveform vector, s. Mathematically, wemay formulate this problem as:

minw,s

E{|wHyu|2} (6)

s. t wH(v(fd)⊗ s⊗ a(θt, φt)) = κ

sHs ≤ Po

where, we may rewrite E{|wHyu|2} = wHRu(s)w. In (6),a power constraint is enforced via the second constraint toaddresses hardware limitations. Optimizing (6) w.r.t w first,the solution to (6) is well known, and expressed as

wo =κR−1u (s)(v(fd)⊗ s⊗ a(θt, φt))

(v(fd)⊗ s⊗ a(θt, φt))HR−1u (s)(v(fd)⊗ s⊗ a(θt, φt))(7)

where Ru(s) = Ri +Rc(s) +Rn. We further emphasize thatthe the weight vector is an explicit function of the waveform.Now substituting wo back into the cost function in (6), theminimization is purely w.r.t s, and cast as,

mins

κ2

(v(fd)⊗ s⊗ a(θt, φt))HR−1u (s)(v(fd)⊗ s⊗ a(θt, φt))

s. t. sHs ≤ Po (8)

A solution to (8) is not immediate, given the dependence ofRu on the waveform vector s. Further, from our extensivesimulations, we noticed that the original cost function in (6)is not jointly convex in w and s. Nevertheless, it is notstraightforward to prove / or disprove joint convexity w.r.t bothw and s analytically.

Consider, then, the following propositions:

Proposition 1. The objective function in (6) is individuallyconvex w.r.t s, for any fixed but arbitrary w

Proof. The definition of convexity (see [11]) cannot be directlyinvoked as the objective g(s) = wHRu(s)w : CN → R

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depends on the waveform s, which is complex. Considerthe following transformation, s = Ds where s ∈ R2N =[Re{s}T , Im{s}T ]T and D = [IN , jIN ] ∈ CN×2N . Now, wemay define an equivalent g(s) : R2N → R to invoke thedefinition of convexity. We have to prove that,

wH

Rn + Ri

+

Q∑q=1

Bq

(IP ⊗D(ts1 + (1− t)s2))Rpqγ

(IP ⊗ (ts1 + (1− t)s2)TDH)BHq

w

≤ twH

Rn + Ri

+

Q∑q=1

Bq(IP ⊗Ds1)Rpqγ (IP ⊗ sT1 DH)BH

q

w

+ (1− t)wH

Rn + Ri

+

Q∑q=1

Bq(IP ⊗Ds2)Rpqγ (IP ⊗ sT2 DH)BH

q

w

(9)

where t ∈ [0, 1) and si ∈ dom{g(s)}, i = 1, 2. After ele-mentary algebra, the convexity requirement in (9) transformsto:

Q∑q=1

xHq(Rpqγ ⊗D(s1 − s2)(s1 − s2)TDH

)xq ≥ 0 (10)

where xq ∈ CNP := BHq w. In other words, it is suf-

ficient to show that iff (10) is true then (9) is also trueand therefore convex. We notice immediately that (10) isa sum of Hermitian quadratic forms. Consider the matrixRpqγ ⊗D(s1−s2)(s1−s2)TDH , we know that Rpq

γ � 0,2 sinceit is a covariance matrix and by definition atleast positive semi-definite (PSD). The other matrix, i.e. D(s1− s2)(s1− s2)TDH

is of course rank-1 Hermitian, and is clearly PSD. Fromelementary Kronecker product properties, it is straightforwardto show that Rpq

γ ⊗D(s1 − s2)(s1 − s2)TDH � 0,∀q. Thenfrom the definition of positive semi-definiteness, each of the QHermitian quadratic forms in (10) is greater than zero, henceproved.

Proposition 2. The objective function in (6) is individuallyconvex w.r.t w, for any fixed but arbitrary s.

Proof. The proof is straightforward to demonstrate by invok-ing the convexity definition on the vector consisting of the realand imaginary parts of w.

A. Alternating minimization

Motivated from Propositions 1,2, we propose an alternatingminimization technique which is iterative. Before we presentdetails on the alternating minimization technique, consider thefollowing minimization problem, which optimizes s, but for a

2Here � is the Lowner partial order [12]

fixed and arbitrary w :

mins

wHRu(s)w

s. t. wH(v(fd)⊗ s⊗ a(θt, φt)) = κ (11)

sHs ≤ Po

In (11), the objective function could be rewritten as,

wHRu(s)w =wH(Rn + Ri)w (12)

+

Q∑q=1

Tr{Rpqγ (IP ⊗ sH)xqxq

H(IP ⊗ s)}.

In (12), the trace operation is further simplified as:

Tr{Rpqγ (IP ⊗ sH)xqxq

H(IP ⊗ s)}

= vec((

Rpqγ (IP ⊗ sH)xqxq

H)T)T

vec(IP ⊗ s)

= sHHT (Rpqγ ⊗ xqxq

H)Hs

= sHZq(w)s (13)

where vec(IP ⊗ s) = Hs, with H ∈ RP2N×N =

[H1T ,H2

T , . . . ,HP]T . The matrix Hk ∈ RPN×N , k =1, 2, . . . , P is further decomposed into P , N × N matrices,and is defined such that the k-th N ×N matrix is IN and theother (N − 1), N ×N matrices are all equal to zero matrices.Using (12) and (13), the Lagrangian of (11) is readily cast as,

L(s, λ1, λ2) = wH(Ri + Rn)w +

Q∑q=1

sHZq(w)s (14)

+ Re{λ∗1(wHGs− κ)}+ λ2sHINs− λ2Po

where λ1 ∈ C and λ2 ∈ R+ are the complex and realLagrange parameters, and G is defined such that v(fd) ⊗s ⊗ a(θt, φt) = Gs = (v(fd) ⊗ At)s, with At havingidentical structure as the previously defined matrix Aq usedin constructing the matrix Bq. The Lagrange dual, denotedas H(λ1, λ2) = inf

sL(s, λ1, λ2). Since (14) consists of

Hermitian quadratic forms and other linear terms of s, wehave H(λ1, λ2) = L(so(λ1, λ2), λ1, λ2), where so(λ1, λ2) isobtained by solving the first order optimality conditions, i.e.

∂L(s, λ1, λ2)

∂s= 0 (15)

where, 0 is a column vector of size N and consists of all zeros.Further, in (15), while taking the derivative the usual rules ofcomplex vector differentiation apply, i.e. treat sH independentof s. The solution to (15) is readily obtained by differentiating(14), and expressed as:

so(λ1, λ2) = −λ12

( Q∑q=1

Zq(w) + λ2IN

)−1GHw. (16)

Using (16), the dual H(λ1, λ2) is given by:

H(λ1, λ2) = wH(Ri + Rn)w − κRe{λ∗1} − λ2Po

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−|λ1|2

4wHG

( Q∑q=1

Zq(w) + λ2IN

)−1GHw. (17)

Equation (17) is further simplified by decomposing, λ1 =λ1r + jλ1i. In which case we notice that (17) is quadraticin λ1r, λ1i,and purely linear in λ2. The Lagrange dual opti-mization is therefore,

maxλ1r,λ1i,λ2

H(λ1r, λ1i, λ2)

s. t λ2 ≥ 0. (18)

Maximizing first w.r.t λ1r, λ1i, we have the solutions,

λ1r =−2κ

wHG( Q∑q=1

Zq(w) + λ2IN

)−1GHw

, λ1i = 0.

Substituting the above solutions into (17), the Lagrange dualoptimization problem and after ignoring an unnecessary addi-tive constant, takes the form,

maxλ2

κ2(wHG

( Q∑q=1

Zq(w) + λ2IN

)−1GHw

)−1− λ2Po

s. t. λ2 ≥ 0 (19)

The first order optimality conditions for the above optimizationare given by:

∂

∂λ2

( κ2

wHGF−1GHw

)− Po − γ = 0

or−κ2

(wHGF−1GHw)2wHG

∂F−1

∂λ2GHw − Po − γ = 0

orκ2

(wHGF−1GHw)2wHG

(F−1

∂F

∂λ2F−1

)GHw − Po − γ = 0

where γ is the Lagrange multiplier associated with (19), F :=Q∑q=1

Zq(w)+λ2IN , and therefore we also have ∂F∂λ2

= IN . The

complementary slackness and constraint qualifier for (19) i.e.γλ2 = 0 and λ2 ≥ 0 form the rest of the equations comprisingthe KKT conditions. It is now readily shown that the solutionto (19) is given by

λ2 = max[0, λ2] (20)

λ2 solves λ2(κ2wHGF−2GHw − Po(wHGF−1GHw)2

)= 0.

Now assuming that Slater’s conditions are satisfied, and notingthat the objective and constraints are all convex in (11). Then,it is readily seen that the waveform design solution is unique,a function of w and expressed as,

so(w) =

κ2( Q∑q=1

Zq(w) + λ2IN

)−1GHw

wHG( Q∑q=1

Zq(w) + λ2IN

)−1GHw

. (21)

Although not stated explicitly, it is noted that the Lagrangemultipliers, λi, i = 1, 2 are functions of the weight vector,

w. The alternating minimization algorithm is now succinctlystated in Table I. Properties of this algorithm such as mono-tonic decease in the cost function etc are not analyzed herebut will be shown via simulations.

TABLE IALTERNATING MINIMIZATION FOR WAVEFORM ADAPTIVE RADAR STAP

1) Initialize: Start with an initial waveform design, definedas s

(0)o , set counter k = 1

2) Filter design: Design the optimal filter weight vector,w

(k)o = wo(s

(k−1)o ), where (7) is used to compute

wo(·).3) Waveform design: Design the updated waveform s

(k)o =

so(w(k)o ), where (20),(21) are used to compute so(·).

4) Check: If convergence is achieved, exit, else k = k+1,go back to step-2.

B. Constant modulus waveform design

Thus far, the waveform design problem had no constraintson the waveform, constant modulus is a desirable propertyto have in a waveform. The optimum weight vector for theconstant modulus constraint is unchanged, from the previouscase and is given (6). However, the optimization for a fixed sis necessary, and is formulated as

mins

wHRu(s)w

s. t. wH(v(fd)⊗ s⊗ a(θt, φt)) = κ (22)|s(i)| = ρ, i = 0, 1 . . . , N − 1.

Unlike say (11), notice that in (22), constraining the powerof the waveform is unnecessary since ρ is fixed but could bechosen arbitrarily to scale up /down the waveforms energy tosatisfy hardware limitations. Therefore, the last N constraintsin (22) implicitly impose the power requirements, but moreimportantly also impose the constant modulus constraint.

One may reformulate the optimization (22) by eliminatingthe last N constraints, by imposing a structure on s, namely,s(i) = ρ exp(jαi). The new optimization problem is now w.r.tα = [α0, α1, . . . , αN−1]T ∈ RN , expressed as

minα

wHRu(s)w

s. t. wH(v(fd)⊗ s⊗ a(θt, φt)) = κ (23)

where in, s = ρ[exp(jα0), exp(jα1), . . . , exp(jαN−1)]T . Thesolution to (23) is readily derived via the the KKTs, and isgiven by

Im

{(Q∑q=1

Zq(w)so +η12

GHw

)� s∗o

}= 0

wHGso = κ (24)

where 0 is a column vector of all zeros and of dimension N ,η1 is the corresponding Lagrange parameter, so is the optimal

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constant modulus waveform and is actually a function of theoptimal αo. This relationship although evident from (23) isnot explicitly stressed in (24) for notational succinctness. TheLagrange parameter as well as so are solved for numericallyfrom (24).

IV. SIMULATIONS

A. Non-constant modulus waveform design

To reduce computation complexity while inverting largematrices and for their eigen-decompositions, we consideredM = 5, L = 32, N = 5. The element spacing i.e. d = λo/2.The noise correlation matrix was assumed to have a correlationfunction given by exp(−|0.005n|), n = 0, 1, . . . , NML. Thecarrier frequency was chosen to be 1GHz, and the radarbandwidth was 50MHz. Two interference sources were con-sidered at (θ = 0.3941, φ = 0.3) and at (−0.4941, 0.3).Both these interference sources had identical discrete corre-lation functions given by 0.2|n|, n = ±0,±1, . . .. To simulateclutter we considered two clutter patches, consisting of fivescatters each. The clutter correlation functions correspondingto the two patches were exp(−0.2|p|) and exp(−0.1|p|), p =±0,±1, . . . ,±P . The rest of the parameters are identical tothose used in [4].

In Fig. 1, the STAP beamformer cost function vs iterationsare shown for 25 independent initializations of the waveform.Before the start of the alternating minimization, the waveformvectors were all initialized randomly. The elements of therandom waveform initialization were selected from a standardcomplex Gaussian distribution, independently. It is observedfrom Fig. 1 that for these 25 trials, convergence is achievedwithin 5 iterations and at the most 16 iterations. More impor-tantly, we see that the final cost for each trial are differentfrom one another, attributed to the joint non-convexity of theobjective w.r.t w and s. The non increasing STAP beamformeroutput is also seen for each iteration in all the 25 trials.

B. Constant Modulus design

We compared the waveforms designed by the constant mod-ulus to the waveforms designed by the non constant modulusdesign, and the results are shown in Fig. 2. The alternatingminimization technique was used for both the designs. Theradar, target, noise, interference and clutter parameters areidentical to the ones used in the previous simulation, as in Fig.1. Constant modulus waveforms are a subset of the generalnon constant modulus waveforms. Therefore to be fair in ourcomparison we initialized both these designs with a randomconstant modulus waveform, whose phase is chosen uniformlyfrom [−π, π]. After termination of the algorithm, the constantmodulus optimal waveform was scaled to have the sameenergy as its non constant modulus counterpart. This procedurewas repeated for 100 independent Monte-Carlo (MC) trials.

The ratio assumed by the final objective values for the con-stant modulus design to the non constant design are shown inFig. 2. To a larger extent the primary need to avoid additionalexpensive hardware in radar transmitters and to a lesser extentthe ability to arbitrary scale up / down such waveforms to

specific RF architectures are the two main drivers for constantmodulus design in practical radar implementations. Nonethe-less for a small number of waveform samples (N = 5), weobserve from Fig. 2 that the non constant modulus designperforms better than the practically preferred constant modulusdesign. This may not be surprising because the amplitudeis constrained temporally in the constant modulus design,while the phase is allowed to be optimized. Comparing thisto its counterpart, we readily see that both the phase andthe amplitude are allowed to be optimized to suppress theeffects of noise, interference, and clutter. Moreover, recallthe minimum eigenvector waveform design in the absence ofclutter (see [3], [4]), this a non constant modulus design andis optimal. Additional Montecarlo trials and simulations forlarge and varying N are desired to obtain conclusive insightsinto this behavior.

C. Adapted patterns

The adapted pattern for the waveform dependent STAPobjective function is expressed as

P(fd, θ) = |wHo (v(fd)⊗ so ⊗ a(θ, φ))|2, for a fixed φ.

(25)

The adapted pattern in (25) is a function of angle, Doppler, theoptimal weight and the waveform vectors, wo, so, respectively.A particular adapted pattern is shown in Fig. 3, normalized byits maximum value. Two interferers at (θ = −0.2, φ = π/3)and at (−0.2, π/3) were chosen. We modeled the clutterdiscretely from all azimuth angles from −π/2 to π/2 indiscrete increments of −0.005π/2 radians. The clutter patcheswere fixed at an elevation angle of π/4 radians. the target wasassumed to be at θt = 0.7, φt = π/4 with normalized Dopplerequal to 0.31. The adapted pattern in Fig. 3 is identical (uptoa scaling) to the one obtained by the classical STAP adaptedpattern. This is not a surprise but is rather reassuring sincethe waveform in (25) affects all the Doppler frequencies andthe azimuths identically. Moreover, we can always considerso⊗a(θ, φ) as a new /modified spatial steering vector. Henceas expected the inclusion of the optimal waveform will notalter the shape of the classical STAP adapted pattern.

V. CONCLUSIONS

Waveform design in STAP was the focus of this paperassuming the dependence of the clutter response on the trans-mitted waveform. Our preliminary simulations indicated thatthe objective function was jointly non-convex in the weightand waveform vectors. However, we showed analytically thatthe objective function is individually convex in the waveformand the weight vector. This motivated a constrained alternatingminimization technique which iteratively optimizes one vectorwhile keeping the other fixed. To addresses practical designconstraints we incorporated constant modulus constraints inour alternating minimization formulation. Simulations werechosen to demonstrate the monotonic decrease of the MVDRobjective function using this alternating minimization algo-rithm. Comparisons between the constant modulus and non

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constant modulus waveform designs, and the adapted patternswere also shown in the simulations.

Classical or waveform independent STAP by itself is com-putationally expensive. It is indisputable that in general, wave-form design will enable performance improvements. However,the inclusion of the waveform in the STAP architecture posesadditional complexity. Further, due to the inclusion of thewaveform the number of range cells required to estimatethe correlation matrices also increases when compared toits classical counterpart, perhaps, making waveform adaptiveSTAP prohibitive for certain applications. Nonetheless, weare optimistic that in the near and immediate future thesepractical challenges could be overcome opening up excitingavenues for future research not just in STAP, but in generalcognitive waveform adaptive radar for multi-antennae multi-static systems.

0 2 4 6 8 10 12 14 16−50

−40

−30

−20

−10

0

10

20

30

40

Iterations

STA

P be

amfo

rmer

cos

t fun

ctio

n (d

B)

Fig. 1. Non constant modulus waveform design: objective function costs vsiterations for 25 random initializations.

ACKNOWLEDGEMENT

This work was sponsored by US AFOSR under project13RY10COR. All views and opinions expressed here are theauthors own and does not constitute endorsement from theDepartment of Defense. We acknowledge the initial commentsfrom Dr. A. DeMaio and Dr. Y. Abramovich which helpedimproved this paper.

REFERENCES

[1] R. Klemm, Principles of Space-Time Adaptive Processing. Institutionof Electrical Engineers, 2002.

[2] J. Ward, Space-time Adaptive Processing for Airborne Radar, ser.Technical report (Lincoln Laboratory). Massachusetts Institute ofTechnology, Lincoln Laboratory, 1994.

[3] J. Guerci, Space-Time Adaptive Processing for Radar. Artech House,2003.

[4] P. Setlur, N. Devroye, and M. Rangaswamy, “Waveform design andscheduling in space-time adaptive radar,” in IEEE Radar Conference,2013.

[5] D. DeLong and E. Hofstetter, “On the design of optimum radar wave-forms for clutter rejection,” IEEE Trans. Inf. Theory, vol. 13, no. 3, pp.454–463, 1967.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

MC Trials

Rat

io: c

on. m

od. c

ost/n

on c

onst

. mod

. cos

t (dB

)

Final costs

Fig. 2. Comparison between constant modulus and non constant modulusdesign. The ratio (in dB) of the final objective function for the constantmodulus design to the final objective function of the non constant designare shown for 100 Montecarlo (MC) trials.

s i n( ! )

Dop

pler

(Nor

mal

ized

)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

dB−80

−70

−60

−50

−40

−30

−20

−10

0

Fig. 3. STAP waveform adapted pattern. The dashed line (magenta) is thetheoretical clutter Doppler vs sin(θ) ridge and overlaps with the clutter nullgenerated by the STAP waveform adaptive algorithm.

[6] S. Pillai, H. Oh, D. Youla, and J. Guerci, “Optimal transmit-receiverdesign in the presence of signal-dependent interference and channelnoise,” IEEE Trans. Inf. Theory, vol. 46, no. 2, pp. 577–584, 2000.

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