1
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],
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-
February 28, 2008 DRAFT
3
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
February 28, 2008 DRAFT
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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,
February 28, 2008 DRAFT
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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.
February 28, 2008 DRAFT
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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
February 28, 2008 DRAFT
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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)
February 28, 2008 DRAFT
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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.
February 28, 2008 DRAFT
9
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
February 28, 2008 DRAFT
10
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
February 28, 2008 DRAFT
11
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,√
Rck is an optimal solution of EQP.
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
February 28, 2008 DRAFT
12
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,√
Rck is an optimal solution of EQP.
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)
February 28, 2008 DRAFT
13
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
February 28, 2008 DRAFT
14
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.
February 28, 2008 DRAFT
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].
February 28, 2008 DRAFT
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.
February 28, 2008 DRAFT
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
February 28, 2008 DRAFT
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.
February 28, 2008 DRAFT
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
February 28, 2008 DRAFT
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,
February 28, 2008 DRAFT
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].
February 28, 2008 DRAFT
22
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February 28, 2008 DRAFT
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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 ).
February 28, 2008 DRAFT
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.
February 28, 2008 DRAFT
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 .
February 28, 2008 DRAFT
27
February 28, 2008 DRAFT
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.
February 28, 2008 DRAFT
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 .
February 28, 2008 DRAFT
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February 28, 2008 DRAFT
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.