1 code design to optimize radar detection …zhangs/reports/2008_ddhzf.pdf1 code design to optimize...

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Code Design to Optimize Radar Detection

Performance Under Accuracy and Similarity

Constraints

A. De Maio, S. De Nicola, Y. Huang, S. Zhang, A. Farina

Abstract

This paper deals with the design of coded waveforms which optimize radar performances in the

presence of colored Gaussian disturbance. We focus on the class of linearly coded pulse trains and

determine the optimum radar code according to the followingcriterion: maximization of the detection

performance under a control on the region of achievable Doppler estimation accuracies, and imposing

a similarity constraint with a pre-fixed radar code. This last constraint is tantamount to requiring a

similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with

the pre-fixed sequence. The resulting optimization problemis non-convex and quadratic. In order to

solve it, we propose a technique (with polynomial computational complexity) based on the relaxation

of the original problem into a semidefinite program. Thus thebest code is determined through a rank-

one decomposition of an optimal solution of the relaxed problem. At the analysis stage, we assess the

performance of the new encoding technique in terms of detection performance, region of achievable

Doppler estimation accuracies, and ambiguity function.

Index Terms

Radar Signal Processing, Non-Convex Quadratic Optimization, Semidefinite Programming Relax-

ation.

A. De Maio (Corresponding Author) and S. De Nicola are with Universita degli Studi di Napoli “Federico II”, Dipartimento

di Ingegneria Elettronica e delle Telecomunicazioni, Via Claudio 21, I-80125 Napoli, Italy. E-mail: [email protected],

[email protected].

Y. Huang and S. Zhang are with Department of Systems Engineering and Engineering Management, The Chinese University

of Hong Kong, Shatin, Hong Kong. E-mail: [email protected], [email protected].

A. Farina is with SELEX - Sistemi Integrati via Tiburtina Km.12.4, 00131, Roma, Italy. E-mail: [email protected].

February 28, 2008 DRAFT

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I. INTRODUCTION

The huge advances in high speed signal processing hardware,digital array radar technology,

and the requirement of better and better radar performanceshas promoted, during the last two

decades, the development of very sophisticated algorithmsfor radar waveform design [1].

The synthesis of narrowband waveforms with a specified ambiguity function is considered

in [2]. In [3], the theory of wavelets is exploited for the design of radar signals capable of

operating in a dense target environment. The use of information theory to devise waveforms

for the detection of extended radar targets, exhibiting resonance phenomena, is studied in [4].

Therein, two design problems with constraints on signal energy and duration are solved.

The concept of matched-illumination for optimized target detection and identification has also

received a considerable attention [5], [6], [7, and references therein] due to the advent of adaptive

radar transmitters which permit the use of advanced pulse shaping techniques. This theory derives

the optimized transmission waveform and the correspondingreceiver response which maximizes

the Signal to Noise Ratio (SNR). The case of single polarimetric channel is considered in [6],

whereas the most general situation of full polarization radar transmission is handled in [7]. The

concept is also further investigated in [8], with referenceto Gaussian point-target and stationary

Gaussian clutter. The results show that the optimum procedure places the signal energy in the

noise band having minimum power.

Waveform optimization in the presence of colored disturbance with known covariance matrix

has been addressed in [9]. Three techniques based on the maximization of the SNR are introduced

and analyzed. Two of them also exploit the degrees of freedomprovided by a rank deficient

disturbance covariance matrix. In [10], a signal design algorithm relying on the maximization

of the SNR under a similarity constraint with a given waveform is proposed and assessed.

The solution potentially emphasizes the target contribution and de-emphasizes the disturbance.

Moreover, it also preserves some characteristics of the desired waveform. In [11], a signal

subspace framework, which allows the derivation of the optimal radar waveform (in the sense

of maximizing the SNR at the output of the detector) for a given scenario, is presented under

the Gaussian assumption for the statistics of both the target and the clutter.

A quite different signal design approach relies on the modulation of a pulse train parameters

(amplitude, phase, and frequency) in order to synthesize waveforms with some specified prop-


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erties. This technique is known as the radar coding and a substantial bulk of work is nowadays

available in open literature about this topic. Here we mention Barker, Frank, and Costas codes

which lead to waveforms whose ambiguity functions share good resolution properties both in

range and Doppler. This list is not exhaustive and a comprehensive treatment can be found in

[12], [13]. It is, however, worth pointing out that the ambiguity function is not the only relevant

objective function for code design in operating situationswhere the disturbance is not white.

This might be the case of optimum radar detection in the presence of colored disturbance, where

the standard matched filter is no longer optimum and the most powerful radar receiver requires

whitening operations.

In this paper, we attack the problem of code optimization in the presence of colored Gaussian

disturbance. At the design stage, we focus on the class of coded pulse trains and propose a code

selection algorithm which is optimum according to the following criterion: maximization of the

detection performance under a control both on the region of achievable values for the Doppler

estimation accuracy and on the degree of similarity with a pre-fixed radar code. Actually, this

last constraint is equivalent to force a similarity betweenthe ambiguity functions of the devised

waveform and of the pulse train encoded with the pre-fixed sequence. The resulting optimization

problem belongs to the family of non-convex quadratic programs [14], [15]. In order to solve

it, we first resort to a relaxation of the original problem into a convex one which belongs to the

Semidefinite Program (SDP) class. Then an optimum code is constructed through the rank-one

decomposition of [16] applied to an optimal solution of the relaxed problem. Remarkably the

entire code search algorithm possesses a polynomial computational complexity.

At the analysis stage, we assess the performance of the new encoding algorithm in terms of

detection performance, region of estimation accuracies that an estimator of the Doppler frequency

can theoretically achieve, and ambiguity function. The results show that it is possible to realize

a trade off among the three aforementioned performance metrics. In other words, detection

capabilities can be swaped for desirable properties of the waveform ambiguity function and/or

for an enlarged region of achievable Doppler estimation accuracies.

The paper is organized as follows. In Section II, we present the model for both the transmitted

and the received coded signal. In Section III, we discuss some relevant guidelines for code design.

In Section IV, we introduce the code design optimization problem and the algorithm which solves

it exploiting SDP relaxation. In Section V, we assess the performance of the proposed encoding


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method also in comparison with standard radar codes. Finally, in Section VI, we draw conclusions

and outline possible future research tracks.

II. SYSTEM MODEL

We consider a radar system which transmits a coherent burst of pulses

s(t) = atu(t) exp[j(2πf0t + φ)] ,

whereat is the transmit signal amplitude,j =√−1,

u(t) =N−1∑

i=0

a(i)p(t − iTr) ,

is the signal’s complex envelope (see Figure 1),p(t) is the signature of the transmitted pulse,Tr

is the Pulse Repetition Time (PRT),[a(0), a(1), . . . , a(N − 1)] ∈ CN is the radar code (assumed

without loss of generality with unit norm),f0 is the carrier frequency, andφ is a random phase.

Moreover, the pulse waveformp(t) is of durationTp ≤ Tr and has unit energy, i.e.∫ Tp

0

|p(t)|2dt = 1 ,

where| · | denotes the modulus of a complex number. The signal backscattered by a target with

a two-way time delayτ and received by the radar is

r(t) = αrej2π(f0+fd)(t−τ)u(t − τ) + n(t) ,

whereαr is the complex echo amplitude (accounting for the transmit amplitude, phase, target

reflectivity, and channels propagation effects),fd is the target Doppler frequency, andn(t) is

additive disturbance due to clutter and thermal noise.

This signal is down-converted to baseband and filtered through a linear system with impulse

responseh(t) = p∗(−t), where(·)∗ denotes complex conjugate. Let the filter output be

v(t) = αre−j2πf0τ

N−1∑

i=0

a(i)ej2πifdTrχp(t − iTr − τ, fd) + w(t) ,

whereχp(λ, f) is the pulse waveform ambiguity function [12], i.e.

χp(λ, f) =

∫ +∞

−∞

p(β)p∗(β − λ)ej2πfβdβ,


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andw(t) is the down-converted and filtered disturbance component. The signalv(t) is sampled

at tk = τ + kTr, k = 0, . . . , N − 1, providing the observables1

v(tk) = αa(k)ej2πkfdTrχp(0, fd) + w(tk), k = 0, . . . , N − 1 ,

whereα = αre−j2πf0τ . Assuming that the pulse waveform time-bandwidth product and the ex-

pected range of target Doppler frequencies are such that thesingle pulse waveform is insensitive

to target Doppler shift2, namelyχp(0, fd) ∼ χp(0, 0) = 1, we can rewrite the samplesv(tk) as

v(tk) = αa(k)ej2πkfdTr + w(tk), k = 0, . . . , N − 1 .

Moreover, denoting byc = [a(0), a(1), . . . , a(N − 1)]T the N-dimensional column vector

containing the code elements3, p = [1, ej2πfdTr , . . . , ej2π(N−1)fdTr ]T the temporal steering vector,

v = [v(t0), v(t1), . . . , v(tN−1)]T , and w = [w(t0), w(t1), . . . , w(tN−1)]

T , we get the following

vectorial model for the backscattered signal

v = αc ⊙ p + w , (1)

where⊙ denotes the Hadamard element-wise product [17].

III. RELEVANT PERFORMANCE MEASURES FORCODE DESIGN

In this section, we introduce some key performance measuresto be optimized or controlled

during the selection of the radar code. As it will be shown, they permit to formulate the design

of the code as a constrained optimization problem. The metrics considered in this paper are:

A. Detection Probability.

This is one of the most important performance measures whichradar engineers attempt to

optimize. We just remind that the problem of detecting a target in the presence of observables

described by the model (1) can be formulated in terms of the following binary hypotheses test

H0 : v = w

H1 : v = αc ⊙ p + w

(2)

1We neglect range straddling losses and also assume that there are no target range ambiguities.

2Notice that this assumption might be restrictive for the cases of very fast moving targets such as fighters and ballistic missiles.

3(·)T is the transpose operator.


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Assuming that the disturbance vectorw is a zero-mean complex circular Gaussian vector with

known positive definite covariance matrix

E[ww†] = M

(E[·] denotes statistical expectation and(·)† conjugate transpose), the Generalized Likelihood

Ratio Test (GLRT) detector for (2), which coincides with theoptimum test (according to the

Neyman-Pearson criterion) if the phase ofα is uniformly distributed in[0, 2π[ [18], is given by

|v†M−1(c ⊙ p)|2H1><H0

G , (3)

whereG is the detection threshold set according to a desired value of the false alarm Probability

(Pfa). An analytical expression of the detection Probability (Pd), for a given value ofPfa, is

available both for the cases of non-fluctuating and fluctuating target. In the former case (NFT),

Pd = Q

(

√

2|α|2(c ⊙ p)†M−1(c ⊙ p),√

−2 ln Pfa

)

, (4)

while, for the case of Rayleigh fluctuating target (RFT) withE[|α|2] = σ2a,

Pd = exp

(

ln Pfa

1 + σ2a(c ⊙ p)†M−1(c ⊙ p)

)

, (5)

whereQ(·, ·) denotes the MarcumQ function of order1. These last expressions show that, given

Pfa, Pd depends on the radar code, the disturbance covariance matrix and the temporal steering

vector only through the SNR, defined as

SNR=

|α|2(c ⊙ p)†M−1(c ⊙ p) NFT

σ2a(c ⊙ p)†M−1(c ⊙ p) RFT

(6)

Moreover,Pd is an increasing function of SNR and, as a consequence, the maximization of Pd

can be obtained maximizing the SNR over the radar code.

B. Doppler Frequency Estimation Accuracy.

The Doppler accuracy is bounded below by Cramer-Rao Bound (CRB) and CRB-like tech-

niques which provide lower bounds for the variances of unbiased estimates. Constraining the

CRB is tantamount to controlling the region of achievable Doppler estimation accuracies, referred

to in the following asA. We just highlight that a reliable measurement of the Doppler frequency


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is very important in radar signal processing because it is directly related to the target radial

velocity useful to speed the track initiation, to improve the track accuracy [19], and to classify

the dangerousness of the target. In this subsection, we introduce the CRB for the case of known

α, the Modified CRB (MCRB) [20] and the Miller-Chang Bound (MCB) [21] for randomα,

and the Hybrid CRB (HCRB) [22, pp. 931-932] for the case of a random zero-meanα.

Proposition I. The CRB for knownα (Case 1), the MCRB and the MCB for randomα (Case

2 and Case 3 respectively), and the HCRB for a random zero-mean α (Case 4) are given by

∆CR(fd) =Ψ

2∂h†

∂fd

M−1 ∂h

∂fd

, (7)

whereh = c ⊙ p,

Ψ =

1

|α|2 Case 1

1

E[|α|2] Cases 2 and 4

E

[

1

|α|2]

Case 3

(8)

Proof. See Appendix A.

Notice that∂h

∂fd

= Tr c ⊙ p ⊙ u ,

with u = [0, j2π, . . . , j2π(N − 1)]T , and (7) can be rewritten as

∆CR(fd) =Ψ

2T 2r (c ⊙ p ⊙ u)†M−1(c ⊙ p ⊙ u)

. (9)

As already stated, forcing an upper bound to CRB, for a specified Ψ value, results in a lower

bound on the size ofA. Hence, according to this guideline, we focus on the class ofradar codes

complying with the condition

∆CR(fd) ≤Ψ

2T 2r δa

, (10)

which can be equivalently written as

(c ⊙ p ⊙ u)†M−1(c ⊙ p ⊙ u) ≥ δa , (11)


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where the parameterδa rules the lower bound on the size ofA (see Figure 2 for a pictorial

description). Otherwise stated, suitably increasingδa, we ensure that new points fall in the

regionA, namely new smaller values for the estimation variance can be theoretically reached

by estimators of the target Doppler frequency.

C. Similarity Constraint.

Designing a code which optimizes the detection performancedoes not provide any kind of

control to the shape of the resulting coded waveform. Precisely, the unconstrained optimization

of Pd can lead to signals with significant modulus variations, poor range resolution, high peak

sidelobe levels, and more in general with an undesired ambiguity function behavior. These

drawbacks can be partially circumvented imposing a furtherconstraint to the sought radar code.

Precisely, it is required the solution to be similar to a known codec0 (‖c0‖2 = 1)4, which shares

constant modulus, reasonable range resolution and peak sidelobe level. This is tantamount to

imposing that [10]

‖c − c0‖2 ≤ ǫ , (12)

where the parameterǫ ≥ 0 rules the size of the similarity region. In other words, (12)permits

to indirectly control the ambiguity function of the considered coded pulse train: the smallerǫ,

the higher the degree of similarity between the ambiguity functions of the designed radar code

and ofc0.

IV. CODE DESIGN

In this section, we propose a technique for the selection of the radar code which attempts to

maximize the detection performance but, at the same time, provides a control both on the target

Doppler estimation accuracy and on the similarity with a given radar code. To this end, we first

observe that

(c ⊙ p)†M−1(c ⊙ p) = c†[

M−1 ⊙ (pp†)∗]

c = c†Rc (13)

and

(c ⊙ p ⊙ u)†M−1(c ⊙ p ⊙ u) = c†[

M−1 ⊙ (pp†)∗ ⊙ (uu†)∗]

c = c†R1c , (14)

4‖ · ‖ denotes the Euclidean norm of a complex vector.


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whereR = M−1 ⊙ (pp†)∗ andR1 = M−1 ⊙ (pp†)∗ ⊙ (uu†)∗ are positive semidefinite [22, p.

1352, A.77]. It follows thatPd and∆CR(fd) are competing with respect to the radar code. In

fact, their values are ruled by two different quadratic forms, (13) and (14) respectively, with two

different matricesR andR1. Hence, only when they share the same eigenvector corresponding

to the maximum eigenvalue, (13) and (14) are maximized by thesame code. In other words,

maximizingPd results in a penalization of∆CR(fd) and vice versa.

Exploiting (13) and (14), the code design problem can be formulated as a non-convex opti-

mization Quadratic Problem (QP)

maxc c†Rc

s.t. c†c = 1

c†R1c ≥ δa

‖c − c0‖2 ≤ ǫ

which can be equivalently written as

(QP)

maxc c†Rc

s.t. c†c = 1

c†R1c ≥ δa

Re(

c†c0

)

≥ 1 − ǫ/2

(15)

The feasibility of the problem, which not only depends on theparametersδa and ǫ but also on

the pre-fixed codec0, is discussed in Appendix B.

A. Solution to the Optimization Problem

In this subsection, we show that an optimal solution of (15) can be obtained from an optimal

solution of the following Enlarged Quadratic Problem (EQP):

(EQP)

maxc c†Rc

s.t. c†c = 1

c†R1c ≥ δa

Re2(

c†c0

)

+ Im2(

c†c0

)

= c†c0c0†c ≥ δǫ

(16)

where δǫ = (1 − ǫ/2)2. Since the feasibility region of EQP is larger than that of QP, every

optimal solution of EQP, which is feasible for QP, is also an optimal solution for QP [14]. Thus,

assume thatc is an optimal solution of EQP and letφ = arg (c†c0). It is easily seen thatcejφ


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is still an optimal solution of EQP. Now, observing that(cejφ)†c0 = |c†c0|, cejφ is a feasible

solution of QP. In other words,cej arg (c†c0) is optimal for both QP and EQP.

Now, we have to find an optimal solution of EQP and, to this end,we exploit the equivalent

matrix formulation

(EQP)

maxC tr(CR)

s.t. tr(C) = 1

tr(CR1) ≥ δa

tr(CC0) ≥ δǫ

C = cc†

(17)

whereC0 = c0c0† and tr(·) is the trace of a square matrix [17].

Problem (17) can be relaxed into a SDP neglecting the rank-one constraint [24]. By doing so

we obtain an Enlarged Quadratic Problem Relaxed (EQPR)

(EQPR)

maxC tr(CR)

s.t. tr(C) = 1

tr(CR1) ≥ δa

tr(CC0) ≥ δǫ

C � 0

(18)

where the last constraint means thatC is to be Hermitian positive semidefinite [14]. The dual

problem of (18), EQPR Dual (EQPRD), is

(EQPRD)

miny1,y2,y3y1 − y2δa − y3δǫ

s.t. y1I − y2R1 − y3C0 � R

y2 ≥ 0, y3 ≥ 0

whereI denotes the identity matrix. This problem is bounded below and is strictly feasible, so

the optimal value is the same as the primal [15] and the complementary conditions are satisfied

at the optimal point, due to the strict feasibility of the primal problem (see Appendix B).

In the following, we prove that a solution of EQP can be obtained from a solution of EQPR

C, and from a solution of EQPRD(y1, y2, y3). Precisely, we show how to obtain a rank-one


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feasible solution of EQPR that satisfies optimality conditions (complementary conditions)

tr[

(y1I − y2R1 − y3C0 − R) C]

= 0 (19)[

tr(CR1) − δa

]

y2 = 0 (20)[

tr(CC0) − δǫ

]

y3 = 0 (21)

Such rank-one solution is also optimal for EQP. The proof, wepropose, is based on the following

Proposition II. Suppose thatX is aN×N complex Hermitian positive semidefinite matrix of

rankR, andA, B are twoN ×N given Hermitian matrices. There is a rank-one decomposition

of X (synthetically denoted asD(X, A, B)),

X =R∑

r=1

xrx†r (22)

such that

x†rAxr =

tr(XA)

Randx†

rBxr =tr(XB)

R(23)

Proof. See [16].

Moreover, we have to distinguish four possible cases:

1) tr(

CR1

)

− δa > 0 and tr(

CC0

)

− δǫ > 0

2) tr(

CR1

)

− δa = 0 and tr(

CC0

)

− δǫ > 0

3) tr(

CR1

)

− δa > 0 and tr(

CC0

)

− δǫ = 0

4) tr(

CR1

)

− δa = 0 and tr(

CC0

)

− δǫ = 0

Case 1: Using the decompositionD(C, I, R1) of Proposition II, we can expressC as

C =R∑

r=1

crcr†

Now, we show that there exists ak ∈ {1, . . . , R} such that√

Rck is an optimal solution of EQP.

Specifically, we first prove that(√

Rck)(√

Rck)† is a feasible solution of EQPR, and then that

(√

Rck)(√

Rck)† and(y1, y2, y3) comply with the optimality conditions, i.e.(

√Rck)(

√Rck)

† is

a rank-one optimal solution of EQPR and, hence,√


The decompositionD(C, I, R1) implies that every(√

Rcr)(√

Rcr)†, r = 1, . . . , R satisfies

the first and the second constraints in EQPR. Moreover, theremust be ak ∈ {1, . . . , R} such


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that (√

Rck)†C0(

√Rck) ≥ δǫ. In fact, if (

√Rcr)

†C0(

√Rcr) < δǫ for everyr, then

R∑

r=1

(√

Rcr)†C0(

√Rcr) < Rδǫ

tr

[(

R∑

r=1

√Rcrc

†r

√R

)

C0

]

< Rδǫ

tr(CC0) < δǫ

which is in contrast with the feasibility ofC. This proves that there exists at least onek ∈{1, . . . , R} for which (

√Rck)(

√Rck)

† is feasible for EQPR. As to fulfillment of the optimality

conditions, tr(

CR1

)

−δa > 0 and tr(

CC0

)

−δǫ > 0 imply y2 = 0 andy3 = 0, namely (20) and

(21) are verified for every(√

Rcr)(√

Rcr)†, with r = 1, . . . , R. Therefore, (19) can be recast as

tr[

(y1I − R)C]

= tr

[

(y1I − R)

(

R∑

r=0

crc†r

)]

= 0

which, sincecrc†r � 0, r = 1, . . . , R, and y2I − R � 0 (from the first constraint of EQPRD),

implies

tr[

(y2I − R)(√

Rcrcr†√

R)]

= 0

It follows that there exists onek ∈ {1, . . . , R} such that(√

Rck)(√

Rck)† is an optimal solution

of EQPR, and thus,√


Cases 2 and 3: The proof is very similar to Case 1, hence we omit it.

Case 4: In this case, all the constraints of EQPR are active, namely tr(C) = 1, tr(CR1) = δa,

and tr(CC0) = δǫ. It follows that

tr[C (R1/δa − I)] = 0

and

tr[C (C0/δǫ − I)] = 0

According toD(C, R1/δa − I, C0/δǫ − I), we decomposeC as

C =R∑

r=1

crcr† ,

and observe that

tr(

crcr†)

=1

γr

, r = 1, . . . , R , (24)


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with γr > 1 such that∑R

r=1 1/γr = 1.

We now prove that each(√

γrcr)(√

γrcr)† is an optimal solution of EQPR. Precisely, we

first show that(√

γrcr)(√

γrcr)† is in the feasible region of EQPR and then we prove that

(√

γrcr)(√

γrcr)† satisfies the optimality conditions. Equation (24) impliesthat the first constraint

in EQPR is satisfied. From the feasibility ofC and from the used decomposition, we can also

claim that(√

γrcr)(√

γrcr)† satisfies the second and the third constraints of EQPR. In fact, with

reference to the second constraint we have

tr[C (R1/δa − I)] = 0

tr[C (R1/δa − I)]

R= 0

cr† (R1/δa − I) cr = 0

cr†(R1/δa)cr = cr

†cr

cr†(R1/δa)cr = tr

(

crcr†)

cr†(R1/δa)cr = 1/γr

√γrcr

†(R1/δa)√

γrcr = 1

√γrcr

†R1√

γrcr = δa

As to the third constraint, we observe that

tr[C (C0/δǫ − I)] = 0

tr[C (C0/δǫ − I)]

R= 0

cr† (C0/δǫ − I) cr = 0

cr†(C0/δǫ)cr = cr

†cr

cr†(C0/δǫ)cr = tr

(

crcr†)

cr†(C0/δǫ)cr = 1/γr

√γrcr

†(C0/δǫ)√

γrcr = 1

√γrcr

†C0√

γrcr = δǫ

It remains to prove that(√

γrcr)(√

γrcr)† complies with the three optimality conditions. As to


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the first, we note that

tr[

(y1I − y2R1 − y3C0 − R) C]

=

tr

[

(y1I − y2R1 − y3C0 − R)R∑

r=1

crcr†

]

= 0

which, sincecrcr† � 0 and y1I − y2R1 − y3C0 − R � 0, implies that

tr[

(y1I − y2R1 − y3C0 − R) (√

γrcr

√γrcr

†)]

= 0 ,

proving the first optimality condition. The compliance withthe second optimality condition can

be shown as follows

cr† (R1/δa − I) cr = 0

(√

γrcr)† (R1/δa − I)

√γrcr = 0

[√γrcr

† (R1/δa − I)√

γrcr

]

y2 = 0

tr[

(R1/δa − I)(√

γrcrc†r

√γr

)

y2

]

= 0

As to the third optimality condition, we have

cr† (C0/δǫ − I) cr = 0

√γrcr

† (C0/δǫ − I)√

γrcr = 0

[√γrcr

† (C0/δǫ − I)√

γrcr

]

y3 = 0

tr[

(C0/δǫ − I)(√

γrcr

√γrc

†r

)

y3

]

= 0

and the proof is completed.

In conclusion, using the decomposition of Proposition II, we have shown how to construct a

rank-one optimal solution of EQPR, which is tantamount to finding an optimal solution of EQP.

Summarizing, the optimum code can be constructed accordingto the following procedure.


15

Code Construction Procedure

1) Enlarge the original QP into the EQP (i.e. substitute Re(c†c0) ≥ 1−ǫ/2 with |c†c0|2 ≥δǫ);

2) Relax EQP into EQPR (i.e. relaxC = cc† into C � 0);

3) Solve the SDP problem EQPR finding an optimal solutionC;

4) Evaluate tr(CR1) − δa and tr(CC0) − δǫ: if both are equal to0 go to 5), else go to

7);

5) Using Proposition II, evaluateD(C, R1/δa−I, C0/δǫ−I), obtainingC =∑R

r=1 crc†r;

6) Computec =√

γ1c1, with γ1 = 1/‖c1‖2, and go to 9);

7) Using Proposition II, evaluateD(C, R1, I) obtainingC =∑R

r=1 crc†r;

8) Find k such thatc†kC0ck ≥ δǫ/R and computec =

√Rck;

9) Evaluate the optimal solution of the original QP ascejφ, with φ = arg(c†c0).

The computational complexity connected with the implementation of the algorithm is polyno-

mial as both the SDP problem5 and the decomposition of Proposition II can be performed in

polynomial time. In fact, the amount of operations, involved in solving the SDP problem, is

O(

N3.5 log 1ζ

)

[15, p. 250] and the rank-one decomposition requiresO(N3) operations.

V. PERFORMANCE ANALYSIS

The present section is aimed at analyzing the performance ofthe proposed encoding scheme.

To this end, we assume that the disturbance covariance matrix is exponentially shaped with

one-lag correlation coefficientρ = 0.8, i.e.

M(i, j) = ρ|i−j| ,

and fix Pfa of the receiver (3) to10−6. The analysis is conducted in terms ofPd, region of

achievable Doppler estimation accuracies, and ambiguity function of the coded pulse train which

results exploiting the proposed algorithm, i.e.

χ(λ, f) =

∫ ∞

−∞

u(β)u∗(β − λ)ej2πfβdβ =

N−1∑

l=0

N−1∑

m=0

a(l)a∗(m)χp[λ − (l − m)Tr, f ] ,

5An SDP problem can be efficiently solved in polynomial time throughinterior point methods[14], namely iterative algorithms

which terminate once a pre-specified accuracyζ is reached. The number of iterations necessary to achieve convergence usually

ranges between 10 and 100[14].


16

where [a(0), . . . , a(N − 1)] is an optimum code. As to the temporal steering vectorp, we

set the normalized Doppler frequency6 fdTr = 0. The convex optimization MATLAB toolbox

SElf-DUal-MInimization (SeDuMi) [25] is exploited for solving the SDP relaxation. The decom-

positionD(·, ·, ·) of the SeDuMi solution is performed using the technique described in [16].

Finally, the MATLAB toolbox of [26] is used to plot the ambiguity functions of the coded pulse

trains. In the following, two distinct situations corresponding to different similarity codes are

considered: generalized Barker sequence [12, pp. 109-113]and Px code [12, pp. 118-122].

A. Generalized Barker Code

The generalized Barker codes are polyphase sequences whoseautocorrelation function has

minimal peak-to-sidelobe ratio excluding the outermost sidelobe. Examples of such sequences

were found for allN ≤ 45 [27], [28] using numerical optimization techniques. In thesimulations

of this subsection, we assumeN = 7 and set the similarity code equal to the generalized Barker

sequencec0 = [0.3780, 0.3780,−0.1072−0.3624j,−0.0202−0.3774j, 0.2752+0.2591j, 0.1855−0.3293j, 0.0057 + 0.3779j]T .

In Figure 3a, we plotPd of the optimum code (according to the proposed criterion) versus

|α|2 for several values ofδa, δǫ = 0.01, and for non-fluctuating target. In the same figure, we

also represent both thePd of the similarity code as well as the benchmark performance,namely

the maximum achievable detection rate (over the radar code), given by

Pd = Q(

√

2|α|2λmax (R),√

−2 ln Pfa

)

, (25)

whereλmax (·) denotes the maximum eigenvalue of the argument.

The curves show that increasingδa we get lower and lower values ofPd for a given |α|2

value. This was expected since the higherδa the smaller the feasibility region of the optimization

problem to be solved for finding the code. Nevertheless the proposed encoding algorithm usually

ensures a better detection performance than the original generalized Barker code.

In Figure 3b, the normalized CRB (CRBn = T 2r CRB) is plotted versus|α|2 for the same

values ofδa as in Figure 3a. The best value of CRBn is plotted too, i.e.

CRBn =1

2|α|2λmax (R1). (26)

6We have also considered other values for the target normalized Doppler frequency. The results, not reported here, confirm

the performance behavior showed in the next two subsections.


17

The curves highlight that increasingδa better and better CRB values can be achieved. This is in

accordance with the considered criterion, because the higherδa the larger the size of the regionA.

Summarizing, the joint analysis of Figures 3a-3b shows thata trade off can be realized between

the detection performance and the estimation accuracy. Moreover, there exist codes capable of

outperforming the generalized Barker code both in terms ofPd and size ofA.

The effects of the similarity constraint are analyzed in Figure 3c. Therein, we setδa = 10−6

and consider several values ofδǫ. The plots show that increasingδǫ worse and worsePd values

are obtained; this behavior can be explained observing thatthe smallerδǫ the larger the size of

the similarity region. However, this detection loss is compensated for an improvement of the

coded pulse train ambiguity function. This is shown in Figures 4b-4e, where such function is

plotted assuming rectangular pulses,Tr = 5Tp and the same values ofδa andδǫ as in Figure 3c.

Moreover, for comparison purposes, the ambiguity functionof c0 is plotted too (Figure 4a). The

plots highlight that the closerδǫ to 1 the higher the degree of similarity between the ambiguity

functions of the devised and of the pre-fixed codes. This is due to the fact that increasingδǫ is

tantamount to reducing the size of the similarity region. Inother words, we force the devised

code to be similar and similar to the pre-fixed one and, as a consequence, we get similar and

similar ambiguity functions.

Finally, Table I provides the average number of iterationsNit and CPU time (in seconds)

which are required to solve the SDP problem (18). The computer used to get these results is

equipped with a3 GHz Intel XEON processor.

B. Px Code

A Px code is a modified version of a Frank code [29]. It shares the same aperiodic peak

sidelobe as the Frank code but a lower integrated sidelobe level. The mathematical definition of

the code elements can be also found in [12]. In the numerical examples presented here, we assume

N = 9 and set the similarity code equal to the Px sequencec0 = [0.1667 − 0.2887j, 0.1667 +

0.2887j,−0.3333, 0.3333, 0.3333, 0.3333, 0.1667+ 0.2887j, 0.1667− 0.2887j,−0.3333]T .

In Figure 5a, we plotPd of the optimum code versus|α|2 for several values ofδa, δǫ = 0.01,

and for non-fluctuating target. For comparison purposes, wealso represent both thePd of the

similarity code as well as the benchmark performance. The curves confirm the behavior of

Figure 3a. Specifically, they show that increasingδa, worse and worse values ofPd are obtained


18

TABLE I

AVERAGE Nit AND CPUTIME IN SECONDS REQUIRED TO SOLVE PROBLEM(18). GENERALIZED BARKER CODE AS

SIMILARITY SEQUENCE.

δa δǫ AverageNit Average CPU time

10−6 0.01 21 0.30

6165.5 0.01 11 0.15

6792.6 0.01 11 0.15

7293.9 0.01 16 0.19

10−6 0.6239 22 0.28

10−6 0.8997 19 0.24

10−6 0.9994 17 0.23

for a given|α|2.In Figure 5b, the CRBn is plotted versus|α|2 in correspondence of the same values ofδa

andδǫ considered in Figure 5a. As expected, increasingδa, lower and lower values of the CRB

can be achieved. Otherwise stated, while an increase ofδa is detrimental forPd, it leads to

improvements in the actual CRB. Moreover, the new encoding technique leads to some codes

which behave better that the chosen similarity sequence both in terms of detection capabilities

and in terms of achievable region of Doppler estimation accuracies.

In Figure 5c, the impact of the similarity constraint on the detection performance is studied.

Therein, we plotPd versus|α|2 for δa = 10−6 and several values ofδǫ. The performance behavior

is similar to that of Figure 3c, namelyPd is a decreasing function ofδǫ. However, the detection

loss is compensated for a more regular behavior of the coded pulse train ambiguity function.

This can be observed in Figures 6b-6e, where the aforementioned function is plotted for the

considered values ofδǫ along with the ambiguity of the pulse train encoded with the similarity

sequence (Figure 6a).

Finally, the averageNit and CPU time (in seconds), required to solve the SDP problem (18)

are reported in Table II for several values of theδa andδǫ.


19

TABLE II

AVERAGE Nit AND CPUTIME (IN SECONDS) REQUIRED TO SOLVE PROBLEM(18). PX CODE AS SIMILARITY SEQUENCE.

δa δǫ AverageNit Average CPU time

10−6 0.01 22 0.30

10940 0.01 12 0.17

13245 0.01 13 0.17

14000 0.01 14 0.21

10−6 0.5115 24 0.32

10−6 0.8359 22 0.29

10−6 0.9954 18 0.25

VI. CONCLUSIONS

In this paper, we have considered the design of coded waveforms in the presence of colored

Gaussian disturbance. We have devised and assessed an algorithm which attempts to maximize

the detection performance under a control both on the regionof achievable values for the Doppler

estimation accuracy, and on the similarity with a given radar code. The proposed technique,

whose implementation requires a polynomial computationalcomplexity, is based on the SDP

relaxation of non-convex quadratic problems and on a suitable rank-one decomposition of a

positive semidefinite Hermitian matrix. The analysis of thealgorithm has been conducted in

terms of the following performance metrics

• detection performance;

• region of achievable Doppler estimation accuracies;

• ambiguity function of the coded pulse waveform;

under two setups for the similarity code. Hence, the trade off among the three considered

performance measures has been thoroughly studied and commented.

Possible future research tracks might concern the possibility to make the algorithm adaptive

with respect to the disturbance covariance matrix, namely to devise techniques which jointly

estimate the code and the covariance. Moreover, it should beinvestigated the introduction in

the code design optimization problem of knowledge-based constraints, ruled by the a-priori

information that the radar has about the surrounding environment. It might also be of interest to

consider the cases of non-Gaussian clutter and of a system equipped with multiple transmitters


20

and/or receivers. Finally, the extension of the framework to the target classification problem

might represent another area for future researches.

VII. A PPENDIX A: PROOF OFPROPOSITION I

Cases 1, 2, and 3 follow respectively from [22, p. 928, Eq. 8.37], the definitions [20, Eq. 4],

and [23, Eq. 20]. Case 4 is based on the division of the parameter vectorθ into a non-random

and a random component, i.e.θ = [θ1, θT2 ]T , whereθ1 = fd, θ2 = [αR, αI ]

T , αR andαI are the

real and the imaginary part ofα. Then, according to the definition [22, pp. 931-932, Section

8.2.3.2], the Fisher Information Matrix (FIM) for the HCRB can be written as

JH = JD + JP , (27)

whereJD andJP are given by [22, pp. 931-932, Eq. 8.50 and 8.59]. Standard calculus implies

that

JD = 2diag

(

E[|α|2]∂h†

∂fd

M−1 ∂h

∂fd

, h†M−1h, h†M−1h

)

, (28)

and

JP =

0 0T

0 K

, (29)

where diag(·) denotes a diagonal matrix,0 = [0, 0]T , andK is a 2 × 2 matrix whose elements

are related to the a-priori probability density function ofα through [22, pp. 930, Eq. 8.51]. The

HCRB for Doppler frequency estimation can be evaluated as[J−1H ](1, 1) which provides (7).

VIII. A PPENDIX B: FEASIBILITY OF QP, EQPR,AND EQPRD

A. Feasibility of QP

The feasibility of QP depends on the three parametersδa, δǫ andc0.

Accuracy parameterδa. This parameter rules the constraintc†R1c ≥ δa. Sincec has unitary

norm, c†R1c ranges between the minimum(λmin(R1)) (which is zero in this case) and the

maximum (λmax(R1)) eigenvalue ofR1. As a consequence, a necessary condition onδa in

order to ensure feasibility isδa ∈ [0, λmax].

Similarity parameterδǫ. This parameter rules the similarity between the sought code and the

pre-fixed code, through the constraint‖c − c0‖2 ≤ ǫ. If ǫ < 0 then QP is infeasible. Moreover,


21

0 ≤ ǫ ≤ 2 − 2Re(c†c0) ≤ 2, where the last inequality stems from the observation that we

choose the phase ofc such that Re(c†c0) = |c†c0| ≥ 0. It follows that a necessary condition on

δǫ = (1 − ǫ/2)2 in order to ensure feasibility isδǫ ∈ [0, 1].

Pre-fixed codec0. Choosing0 ≤ δa ≤ λmax and0 ≤ δǫ ≤ 1 is not sufficient in order to ensure

the feasibility of QP. A possible way to construct a strict feasible QP problem is to reduce the

range ofδa according to value ofc0. In fact, denoting byδmax = c†0R1c0, the problem QP

is strictly feasible, assuming that0 ≤ δa < δmax and 0 ≤ δǫ < 1. Moreover, a strict feasible

solution of QP isc0 itself.

B. Feasibility of EQPR

The primal problem is strictly feasible due to the strict feasibility of QP. Precisely, assume that

cs is a strictly feasible solution of QP, i.e.c†scs = 1, c†

sR1cs > δa and|c†sc0|2 ≥ Re2(c†

sc0) > δǫ.

Then there areu1, u2, . . . , uN−1 such thatU = [cs, u1, u2, . . . , uN−1] is a unitary matrix [17],

and for a sufficient smallη > 0 the matrixCs

(1 − η)Ue1e†1U

† +η

N − 1U(

I − e1e†1

)

U † , (30)

with e1 = [1, 0, . . . , 0]T , is a strictly feasible solution of the SDP problem EQPR. In fact,

tr(Cs) = 1 (31)

tr(CsR1) = (1 − η)c†sR1cs +

η

N − 1

N−1∑

n=1

u†nR1un (32)

tr(CsR1) = (1 − η)c†sC0cs +

η

N − 1

N−1∑

n=1

u†nC0un (33)

which highlight that, ifη is suitable choosen, thenCs is a strictly feasible solution, i.e. tr(Cs) =

1, tr(CsR1) > δa and tr(CsC0) > δǫ.

C. Feasibility of EQPRD

The dual problem is strictly feasible because, for every finite value ofy2 and y3, say y2, y3,

we can choose ay1, such thaty1I − y2R1 − y3C0 ≻ R. It follows that (y1, y2, y3) is a strictly

feasible solution of EQPRD [15].


22

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[14] S. Boyd and L. Vandenberghe,Convex Optimization, Cambridge University Press, 2003.

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23

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24

Figure 1: Coded pulse trainu(t) for N = 5 andp(t) with rectangular shape.

Figure 2: Lower bound to the size of the regionA for two different values ofδa (δ′a < δ′′a ).


25

−4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|α|2 [dB]

Pd

increasing δa

Figure 3a: Pd versus|α|2 for Pfa = 10−6, N = 7, δǫ = 0.01, non-fluctuating target, and several values of

δa ∈ {10−6, 6165.5, 6792.6, 7293.9}. Generalized Barker code (dashed curve). Code which maximizes the SNR

for a givenδa (solid curve). Benchmark code (dotted-marked curve). Notice that the curve forδa = 10−6 perfectly

overlaps with the benchmarkPd.

−4 −2 0 2 4 6 8−52

−50

−48

−46

−44

−42

−40

−38

−36

−34

−32

|α|2 [dB]

CR

Bn [d

B]

increasing δa

Figure 3b: CRBn versus|α|2 for N = 7, δǫ = 0.01 and several values ofδa ∈ {10−6, 6165.5, 6792.6, 7293.9}.

Generalized Barker code (dashed curve). Code which maximizes the SNR for a givenδa (solid curve). Benchmark

code (dotted-marked curve). Notice that the curve forδa = 7293.9 perfectly overlaps with the benchmark CRBn.


26

−4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|α|2 [dB]

Pd

increasing δε

Figure 3c: Pd versus|α|2 for Pfa = 10−6, N = 7, δa = 10−6, non-fluctuating target, and several values of

δǫ ∈ {0.01, 0.6239, 0.8997, 0.9994}. Generalized Barker code (dashed curve). Code which maximizes the SNR for

a given δǫ (solid curve). Benchmark code (dotted-marked curve). Notice that the curve forδǫ = 0.01 perfectly

overlaps with the benchmarkPd.

Figure 4a: Ambiguity function modulus of the generalized Barker codec0 = [0.3780, 0.3780,−0.1072 −

0.3624j,−0.0202− 0.3774j, 0.2752+ 0.2591j, 0.1855− 0.3293j, 0.0057 + 0.3779j]T .


27


Figures 4b-4e: Ambiguity function modulus of code which maximizes the SNRfor N = 7, δa = 10−6, c0

generalized Barker code, and several values ofδǫ: (b) δǫ = 0.9994, (c) δǫ = 0.8997, (d) δǫ = 0.6239, (e) δǫ = 0.01.

28

−4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|α|2 [dB]

Pd

increasing δa

Figure 5a: Pd versus|α|2 for Pfa = 10−6, N = 9, δǫ = 0.01, non-fluctuating target, and several values of

δa ∈ {10−6, 10940, 13245, 14000}. Px code (dashed curve). Code which maximizes the SNR for a given δa (solid

curve). Benchmark code (dotted-marked curve). Notice thatthe curve forδa = 10−6 perfectly overlaps with the

benchmarkPd.

−4 −2 0 2 4 6 8−55

−50

−45

−40

−35

−30

|α|2 [dB]

CR

Bn [d

B]

increasing δa

Figure 5b: CRBn versus|α|2 for N = 9, δǫ = 0.01 and several values ofδa ∈ {10−6, 10940, 13245, 14000}. Px

code (dashed curve). Code which maximizes the SNR for a givenδa (solid curve). Benchmark code (dotted-marked

curve). Notice that the curve forδa = 14000 perfectly overlaps with the benchmark CRBn.


29

−4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pd

|α|2 [dB]

increasing δε

Figure 5c: Pd versus|α|2 for Pfa = 10−6, N = 9, δa = 10−6, non-fluctuating target, and several values of

δǫ ∈ {0.01, 0.5115, 0.8359, 0.9954}. Px code (dashed curve). Code which maximizes the SNR for a given δǫ (solid

curve). Benchmark code (dotted-marked curve). Notice thatthe curve forδǫ = 0.01 perfectly overlaps with the

benchmarkPd.

Figure 6a: Ambiguity function modulus of the Px codec0 = [0.1667 − 0.2887j, 0.1667 +

0.2887j,−0.3333, 0.3333, 0.3333, 0.3333, 0.1667+ 0.2887j, 0.1667− 0.2887j,−0.3333]T .


30


Figures 6b-6e: Ambiguity function modulus of code which maximizes the SNRfor N = 9, δa = 10−6, c0 Px

code, and several values ofδǫ: (b) δǫ = 0.9954, (c) δǫ = 0.8359, (d) δǫ = 0.5115, (e) δǫ = 0.01.

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