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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 1
MIMO Radar Waveform Design Based on Mutual
Information and Minimum Mean-Square Error
Estimation
Yang Yang, Student Member, IEEE, and Rick S. Blum, Fellow, IEEE
Abstract—This paper addresses the problem of radar wave-
form design for target identification and classification. Both
the ordinary radar with a single transmitter and receiver and
the recently proposed multiple-input multiple-output (MIMO)
radar are considered. A random target impulse response is
used to model the scattering characteristics of the extended
(nonpoint) target, and two radar waveform design problems with
constraints on waveform power have been investigated. The first
one is to design waveforms that maximize the conditional mutual
information (MI) between the random target impulse response
and the reflected waveforms given the knowledge of transmitted
waveforms. The second one is to find transmitted waveforms that
minimize the mean-square error (MSE) in estimating the target
impulse response. Our analysis indicates that under the same
total power constraint, these two criteria lead to the same solution
for a matrix which specifies the essential part of the optimum
waveform design. The solution employs water-filling to allocate
the limited power appropriately. We also present an asymptotic
formulation which requires less knowledge of the statistical model
of the target.
Index Terms— Multiple-input multiple-output (MIMO)
radar, radar waveform design, identification, classification, ex-
tended radar targets, mutual information (MI), minimum mean-
square error (MMSE), waveform diversity.
I. INTRODUCTION
MOST radar systems operate by radiating a specific
electromagnetic signal into a region and detecting
the echo returned from the reflecting targets. The nature of
the echo signal provides information about the target, such
as range, radial velocity, angular direction, size, shape and
so on [1]. This signal is usually referred to as the radar
waveform, and plays a key role in the accuracy, resolution,
and ambiguity for radar in performing the above mentioned
tasks [2]. The design of radar waveforms has been under long
and intensive study to optimize a radar’s performance in target
detection and information extraction. For example, in [3], the
author considers the problem of radar waveform design for
maximizing the signal-to-noise ratio (SNR) at the output of
the receiver filter for deterministic targets in Gaussian noise.
The research in [4] considers the design of a transmit pulse
and a receiver filter jointly maximizing the output signal-to-
interference-plus-noise ratio for the optimal detection of a
This work was supported by the Air Force Research Laboratory under Grant
No. FA9550-06-1-0041 and by the National Science Foundation under Grant
No. CCR-0112501.
Y. Yang and R. S. Blum are with the Department of Electrical and Computer
Engineering, Lehigh University, 19 Memorial Drive West, Bethlehem, PA
18015, USA(email: yay204@lehigh.edu; rblum@eecs.lehigh.edu).
target return contaminated by signal-dependent interference.
The work in [5] investigates optimal radar waveform design for
the problems of radar target identification and classification,
which is to find signals that discriminate between a collection
of targets of interest after observing the backscatter from an
illuminated unknown target. Here we also focus on waveform
design for radar target identification and classification.
There is also work pertaining to the use of information
theory for optimum radar waveform design. In fact, the ap-
plication of information theory to radar can be traced to the
early 1950’s when Woodward and Davies [7]-[10] examined
the use of information-theoretic principles to obtain the a
posteriori radar receiver, shortly after the publication of Shan-
non’s milestone work in information theory [6]. Woodward
and Davies [7]-[10] give an excellent example of how one
can use information theory to benefit radar system design.
Unfortunately, Woodward and Davies did not extend this
information-theoretic rationale to the problem of radar wave-
form design. Since their work, few researchers had consid-
ered the connection between information theory and radar
waveform design problems until in 1993, Bell published his
paper [3] that suggested maximizing the mutual information
(MI) between the target impulse response and the reflected
radar signal to design radar waveforms. The result obtained in
[3] is interesting, and provides a unique information-theoretic
view into radar waveform design. We notice that it is implied
in [3] that the greater the MI between the target impulse
response (the target reflection) and the measurement (the
reflected signals), the better capability of a radar to estimate
the parameters describing the target. However, we have not
seen quantitative analysis demonstrating this. This deficiency
partly motivated us to reconsider this previously reported
research, but for the first time, to formulate and simultaneously
solve the optimal waveform design problem in terms of both
information theoretic and estimation theoretic criteria while
concurrently handling cases with both multiple transmit and
receive antennas (called MIMO radar as described next). Since
in [3], the target identification and classification problems
are formulated as problems where one wants to estimate
the target impulse response, considering estimation theoretic
criteria seems very important.
Recently, motivated by the latest developments in com-
munication theory [11], [12], several publications have advo-
cated the concept of MIMO radar [13]-[22]. The proposed
MIMO radar enjoys similar benefits to those enjoyed by
existing MIMO communication systems. Specifically, MIMO
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 2
radar, as shown in [13]-[16], overcomes target radar cross
section (RCS) scintillations by transmitting different signals
from several uncorrelated transmitters. The received signal
is a superposition of independently “faded” signals, and the
average SNR of the received signal is approximately constant,
i.e., it does not scintillate as in conventional radar systems.
MIMO radar can exploit the spatial diversity of target scatters,
and enables a variety of new techniques that can improve the
performance in many aspects [13]. However, as far as we
are aware, waveform design for MIMO radar has not been
considered thus far, and it is our aim in this paper to formulate
this problem and to describe methods for finding optimum
strategies. In the process, we relate information theoretic and
estimation theoretic criteria in an important and interesting
way. Our results apply to single transmit and receive antenna
radars also and appear to be novel for this case also.
In this paper, we consider waveform design for MIMO
radar for estimation of extended targets modeled using an
impulse response as in [3] (point targets are a special case).
Assuming that the radar transmitter has knowledge of the
target’s second-order statistics1and that the transmitted power
is constrained, we investigate the optimal radar waveform
design based on the following two criteria:
•maximizing the conditional MI between the random tar-
get impulse response and the reflected radar waveforms;
•minimizing the value of minimum mean-square error
(MMSE) in estimating the target impulse response.
We are especially interested in quantifying any equivalence
between these information theoretic and estimation theoretic
criteria.
Our findings are interesting, as they indicate that these
two criteria lead to the same optimum solution for a par-
ticular matrix which is the fundamental quantity specifying
the waveform design. In terms of this fundamental quantity,
maximizing the MI is equivalent to minimizing the MMSE.
Further, the optimum solution employs water-filling, which
allocates the transmitted power in proportion to the quality of
the particular mode in question. We also provide an asymptotic
formulation in this paper that lessens the required knowledge
about the statistical model to just a few samples of the power
spectral density (PSD) which would be much more suitable in
practice.
In the remainder of this paper, we firstly consider the
optimum waveform design for MIMO radar, and later discuss
some special cases based on the MIMO radar model. In
Section II, we present the statistical model for MIMO radar,
and then formulate the problem of waveform design. This
includes a discussion of the rationale behind the optimality
criteria used in this formulation which include both estimation
theoretic and information theoretic criteria. In Section III we
investigate the waveform design problems for MIMO radar
by considering first the more general non-asymptotic case and
then its simplification under the asymptotic assumption. In
Section IV, we discuss two special cases: one arises from
the situation when the coefficients of target impulse response
1This could be obtained through field measurements, or by some feedback
mechanism, which is often referred to as covariance feedback.
are independently Gaussian distributed, and the other one
is the optimal waveform sequence design for a traditional
radar equipped with a single transmit and a single receive
antenna. Numerical examples illustrating the optimum radar
waveform design are given in Section V. Conclusions are given
in Section VI.
Notation: Throughout this paper, we use bold upper case
letters to denote matrices, and bold lower case letters to signify
column vectors. Superscripts {·}H,{·}∗and {·}Twill be
used to denote the complex conjugate transpose, conjugate,
and transpose of a matrix, respectively. We use det{·} and
tr{·} for the determinant and trace of a matrix, and E{·} for
expectation with respect to all the random variables within the
brackets. The symbol k · k denotes the Euclidean norm of a
vector, ℜ{·} stands for the real part of a complex number,
⊗for Kronecker product, and diag{a}for a diagonal matrix
with its diagonal given by the vector a. We let IKdenote the
identity matrix of size K×K, and 0denote a zero vector with
appropriate length. Finally, (a)+denotes the positive part of
a, i.e., (a)+= max [0, a].
II. PROBLEM FORMULATION
7UDQVPLW$UUD\5HFHLYH$UUD\(OHPHQWV(OHPHQWV
P
Q
6FDWWHUV
Fig. 1. Illustration of MIMO radar for a bistatic radar scenario
A bistatic MIMO radar scenario is depicted in Fig. 1
when this radar is equipped with Ptransmit elements and
Qreceive elements. Our model, appropriate for both bistatic
and monostatic scenarios, is a slight generalization of that
in [13] for the case of extended targets. For simplicity we
consider this model in discrete-time and in baseband. For an
extended target, we model the reflection from the signal sent
from the pth transmit element and captured at the qth receive
element using a finite impulse response (FIR) linear system
with order ν, whose impulse response is g(p,q )(l), l ∈[0, v].
Then the received waveform at the qth receive element can
be formulated as
yq(k) =
P
X
p=1
ν
X
l=0
g(p,q)(l)xp(k−l) + nq(k)(1)
where xp(k)is the waveform transmitted from the pth transmit
element and nq(k)is the additive complex Gaussian noise
measured at the qth receive element.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 3
Let Lcorrespond to the duration of the observed
signals. Here we assume L > ν. Denoting g(p,q)=
[g(p,q)(0),···, g(p,q )(ν)]T,yq= [yq(k),···, yq(k+L−
1)]Tand nq= [nq(k),···, nq(k+L−1)]T, we can write
(1) using matrix-vector notation as
yq=
P
X
p=1
Xpg(p,q)+nq(2)
where Xpis an L×M(M=ν+ 1) Toeplitz matrix which
contains the waveforms transmitted from the pth transmit
element, i.e.,
Xp=
xp(k)··· xp(k−ν)
.
.
.....
.
.
xp(k+L−1) ··· xp(k+L−1−ν)
(3)
We further define ¯
X= [X1,···,XP]and ¯gq=
[(g(1,q))T,···,(g(P,q))T]T, so that (2) could be reformulated
into
yq=¯
X ¯gq+nq.(4)
Collecting the received waveforms from all the Qreceive
elements to create ¯y = [y1T,···,yQT]Tand defining
X=IQ⊗¯
X, we obtain
¯y =X¯g +¯n,(5)
where ¯g = [¯gT
1,···,¯gT
Q]Tand ¯n = [n1T,···,nQT]T.
To facilitate our ensuing analysis on the waveform design
of MIMO radar for extended targets, we assume the model in
(5) with the following assumptions:
A1) The target impulse response vector ¯g is a Gaussian
random vector with zero mean and full rank (for
simplicity) covariance matrix Σ¯g.
A2) The components of the noise vector ¯n are assumed
to be independently and identically distributed (i.i.d)
and complex Gaussian, with zero mean and variance
σ2
n.
Note that Σ¯g can be diagonalized through its eigenvalue
decomposition, i.e.,
Σ¯g =UΛUH(6)
where Uis a unitary matrix whose columns are eigenvectors
and Λ= diag{Λ11,...,ΛP QM ,P QM }is a diagonal matrix
with each diagonal entry given by a real and nonnegative
eigenvalue.
A. Conditional MI
Let us consider the conditional MI between ¯y and ¯g given
the knowledge of ¯
X, which is hereafter referred to as MI. We
have [23] (h(·)denotes differential entropy)
I(¯y;¯g|¯
X) = h(¯y|¯
X)−h(¯y|¯g,¯
X)
=h(¯y|¯
X)−h(¯n).(7)
Conditioned on ¯
X(equivalent to conditioning on X), we can
easily find that ¯y is Gaussian distributed with mean 0and
covariance (XΣ¯gXH+σ2
nIQL), i.e.,
¯y|X∼ CN 0,XΣ¯gXH+σ2
nIQL.
Using (7), the MI is [23]
I(¯y;¯g|¯
X)
= log hdet(XΣ¯gXH+σ2
nIQL)i−log det(σ2
nIQL)
= log hdet(σ−2
nXΣ¯g XH+IQL)i
= log hdet(σ−2
nΣ¯g XHX+IP QM )i(8)
where (8) follows from
det(Ir+AB) = det(It+BA).(9)
Our goal is to find those X(transmitted waveform) that
maximize the MI I(¯y;¯g|¯
X)between the random target im-
pulse response ¯g and the received (reflected) radar waveform
¯y under the constraint tr{XHX} ≤ LQP0which effectively
limits the total transmit power. Therefore we can express the
problem of waveform design based on MI as:
max
Xdet(σ−2
nΣ¯g XHX+IP QM )
s.t.tr{XHX} ≤ LQP0.
It is noted that, since X=IQ⊗¯
X, finding ¯
Xwill determine
X. Similarly, the power constraint of tr{XHX} ≤ LQP0
can be written as tr{¯
XH¯
X} ≤ LP0.
B. MMSE Estimation
Now we consider the problem of radar waveform design
from the viewpoint of estimation. It is easy to verify that,
conditioned on X,¯g and ¯y are jointly Gaussian distributed as
¯g
¯y ∼ CN 0
0,Σ¯g Σ¯g XH
XΣ¯g XΣ¯g XH+σ2
nIQL .
The conditional distribution of ¯g given ¯y and Xis also Gaus-
sian, with conditional mean ˆµ¯g given by (using Eq. (IV.B.53)
and (IV.B.55) on pp. 156 of [24])
ˆµ¯g =Σ¯g XH(XΣ¯g XH+σ2
nIQL)−1¯y
= (XHX+σ2
nΣ−1
¯g )−1XH¯y (10)
and conditional covariance matrix ˆ
Σ¯g given by (using
Eq. (IV.B.54) on pp. 156 of [24])
ˆ
Σ¯g = E{(¯g −ˆµ¯g )(¯g −ˆµ¯g)H}
=Σ¯g −Σ¯g XH(XΣ¯g XH+σ2
nIQL)−1XΣ¯g
= (σ−2
nXHX+Σ−1
¯g )−1,(11)
where the matrix inversion lemma2was employed in obtaining
both (10) and (11).
Let ˆ
¯g denote the Bayes estimate of ¯g. When the cost
function is defined as k¯g−ˆ
¯g k2, the Bayes estimate ˆ
¯g will be a
linear MMSE estimator [24] which is given by the conditional
mean of ¯g given ¯y (for a given value of X), i.e.,
ˆ
¯g =ˆµ¯g = (XHX+σ2
nΣ¯g
−1)−1XH¯y.(12)
In this case, the Bayes risk for a given Xwill be
MMSE =E{k ¯g −ˆ
¯g k2}= tr{ˆ
Σ¯g }
= tr{(σ−2
nXHX+Σ¯g
−1)−1}.(13)
2(A+BCD)−1=A−1−A−1B(DA−1B+C−1)−1DA−1.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 4
Our goal is to find those X(transmitted waveform) that
minimize the value of MMSE under a constraint on the total
transmit power of tr{XHX} ≤ LQP0. Or equivalently we
can express the problem of waveform design based on MMSE
estimation as:
min
Xtr{(σ−2
nXHX+Σ¯g
−1)−1}
s.t.tr{XHX} ≤ LQP0.
III. OPTIMAL WAVEFORM DESIGN FOR MIMO RADAR
The following lemma will be useful in developing equa-
tions specifying optimal waveforms.
Lemma 1: Let Abe an N×Npositive semidefinite
Hermitian matrix with (i, j )th entry aij. Then the following
inequalities,
det(A)≤
N
Y
i=1
aii (14)
and
tr(A−1)≥
N
X
i=1
1
aii
(15)
hold respectively, where both equalities are achieved if and
only if Ais diagonal.
Proof: See Appendix I for some comments.
As indicated by Lemma 1, the maximum value of
I(¯y;¯g|¯
X)will be achieved when (σ−2
nΣ¯g XHX+IP QM )
is diagonal, and the minimum value of MMSE will be attained
when (σ−2
nXHX+Σ¯g
−1)is diagonal, also. The following
theorem addresses optimum Xby building on Lemma 1.
Theorem 1: Let assumptions A1) and A2) hold true.
Then, under the constraint tr{XHX} ≤ LQP0, only Xthat
satisfy
X=Ψ diag "η−σ2
n
Λ11 +
, . . . ,
η−σ2
n
ΛP QM,P QM +#!1/2
UH,(16)
can be optimum in the sense of either maximizing
the MI or minimizing the MMSE. In (16), Λ=
diag[Λ11,...,ΛP QM ,P QM ]and Uare defined in (6), Ψis
an LQ ×P QM matrix with orthonormal columns, and the
scalar constant ηis chosen to satisfy
P QM
X
i=1 η−σ2
n
Λii +
=LQP0.
The resulting maximum value of MI is
Imax(¯y;¯g|¯
X) =
P QM
X
i=1 log(σ−2
nΛii η)+,
and the corresponding minimum value of MMSE is
MMSE =
P QM
X
i=1
Λii
(Λiiσ−2
nη−1)++ 1 .
Further, provided X=IQ⊗¯
Xfor some ¯
X, then any Xfrom
(16) is optimum for both MI and MMSE.
Proof: Inserting the eigenvalue decomposition from (6)
into (8) and using (9), we obtain
I(¯y;¯g|¯
X) = log hdet(σ−2
nΛUHXHXU+IP QM )i
= log det(σ−2
nΛZHZ+IP QM )(17)
where Z=XUis a LQ ×P QM matrix. Since Uis unitary,
it follows that tr{XHX}= tr{ZHZ}.
According to Lemma 1, the maximum value of (17) is
achieved when (σ−2
nΛZHZ+IP QM )is diagonal. Thus, Q=
ZHZmust be a diagonal matrix with positive elements Qii ≥
0,∀i∈[1, P QM ]. Consequently the optimal waveform design
based on the criterion of mutual information can be rewritten
as
max
Q
P QM
X
i=1
log(σ−2
nΛiiQii + 1)
s.t.
P QM
X
i=1
Qii ≤LQP0
Qii ≥0,∀i∈[1, P QM ].
This is a constrained optimization problem which can be
solved using the method of Lagrange multipliers [32]. Towards
this goal we form
J=
P QM
X
i=1
log(σ−2
nΛiiQii + 1) + λ(
P QM
X
i=1
Qii).
Differentiating Jwith respect to Qii and setting equal to 0,
we have Λii
ΛiiQii +σ2
n
+λ= 0.
Solving for Qii , setting η=−1/λ, and applying the Karush-
Kuhn-Tucker (KKT) conditions [32, Ch. 5.5.3], the optimum
solution must satisfy
Qii =η−σ2
n
Λii +
,∀i∈[1, P QM ],(18)
where ηcan be found by solving
P QM
X
i=1 η−σ2
n
Λii +
=LQP0.(19)
The MI result from Theorem 1 then follows from the concavity
of log det(σ−2
nΛQ+IP QM )as a function of Q[32]. The
resulting maximum value of MI is
Imax(¯y;¯g|¯
X) =
P QM
X
i=1
log "σ−2
nΛii η−σ2
n
Λii +
+ 1#
=
P QM
X
i=1 log(σ−2
nΛii η)+.
For the optimum design based on the estimation error
criterion, we use (6) to rewrite (13) as
MMSE = tr{(σ−2
nZHZ+Λ−1)−1}.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 5
Also based on Lemma 1,(σ−2
nZHZ+Λ−1)−1should be
diagonal to achieve the minimum value of MMSE. Therefore,
ZHZis still required to be diagonal, and hence can also
be represented by the diagonal matrix Q. Then the optimal
waveform design based on the criterion of minimizing MMSE
can be similarly reformulated into
min
Q
P QM
X
i=1
Λii
ΛiiQii σ−2
n+ 1
s.t.
P QM
X
i=1
Qii ≤LQP0
Qii ≥0,∀i∈[1, P QM ],
which can also be solved by the method of Lagrange multi-
pliers. As before, we construct the functional
J=
P QM
X
i=1
Λii
ΛiiQii σ−2
n+ 1 +λ(
P QM
X
i=1
Qii),
and differentiate it with respect to Qii to obtain
−(σnΛii)2
(ΛiiQii +σ2
n)2+λ= 0,
where Qii is given by
Qii =σn
√λ−σ2
n
Λii
.
Denote η=σn/√λand apply the KKT conditions, we finally
obtain
Qii =η−σ2
n
Λii +
,∀i∈[1, P QM ],
which is the same as (18). Once again ηmust be chosen to
solve (19). The MMSE result from Theorem 1 follows since
tr{(σ−2
nQ+Λ−1)−1}can be shown to be convex function
of Q[32]. The resulting value of MMSE can be obtained by
plugging in this optimum solution to obtain
MMSE =
P QM
X
i=1
Λii
(Λiiσ−2
nη−1)++ 1 .
Next we note that ZHZis invariant3to post multiplication
of Zby an appropriate matrix so a more general solution can
be composed as
Z=ΨQ1/2,
where Ψis a LQ ×P QM matrix with its columns forming
an orthonormal basis4. Since Z=XUwe have
X=ΨQ1/2UH
=Ψ diag "η−σ2
n
Λ11 +
, . . . ,
η−σ2
n
ΛP QM,P QM +#!1/2
UH,
3A similar issue arises in [34] with respect to a communication system
problem.
4Ψwill be unitary when it is a square matrix, or equivalently L=P M .
which is (16).
Remark 1: Based on Theorem 1, we see that any XHX
meeting the given power constraint which maximizes MI
will also minimize MMSE, and this XHXis unique for a
given Σ¯g. Further (16) describes the solution. There are some
interesting relationships between the radar research discussed
here and research on communication systems. For example,
the mathematical optimization problems formulated in [34]-
[37] can be seen to be special cases of that treated in Theorem
1.
A. Asymptotic Simplification
A review of Theorem 1 indicates that knowledge of the
full matrix Σ¯g is needed to perform an eigenvalue decom-
position. We provide an asymptotic formulation here that
lessens the required information to just a few samples of the
PSD which would be much more suitable in practice. The
asymptotic approach employs the additional assumption:
A3) The covariance matrix Σ¯g of ¯g has a Toeplitz struc-
ture [25]. The Toeplitz structure just requires that the
correlation between different elements of ¯g depends
only on the relative position of the elements in the
vector. This assumption would be necessary for ¯g to
be wide sense stationary (WSS) if the length of ¯g
were to approach infinity.
It is known that Toeplitz matrices can be approximated by
their associated circulant matrices, and asymptotically (in
the dimension of these matrices), they are equivalent [26],
[27]. The asymptotic equality of two matrices implies that
their eigenvalues and inverses (and certain products) behave
similarly [28]. Therefore, in many applications, approximat-
ing Toeplitz matrices with their associated circulant matrices
has led to considerable simplification of the problems to be
tackled5.
The first task is to describe the circulant matrix approx-
imation of Σ¯g . Denote the (i, j )th entry of the covariance
matrix Σ¯g as Σ¯g (i, j), then we have Σ¯g (i, j ) = ri−j
and r−k=r∗
k, where we assume the sequence {rk;k=
0,±1,···,±(P QM −1)}is absolutely summable, and its
2π-periodic truncated Fourier spectrum is given by
f(ω) =
P QM−1
X
k=1−P QM
rke−jkω .
Here f(ω)is real valued, finite and nonnegative, and represents
the PSD of the random process ¯g when P QM approaches
infinity [28], [29]. Then the circulant matrix ˜
Σ¯g that is
asymptotically equivalent to Σ¯g could be constructed by [28,
pp. 34]: ˜
Σ¯g =FP QM VFH
P QM ,(20)
where FP QM is the P QM ×P QM unitary discrete Fourier
transform (DFT) matrix with its (k, l)th entry given by:
1
√P QM exp −j2π(k−1)(l−1)
P QM ,∀k, l ∈[1, P QM ];
5For completeness we note proper care must be taken, e.g., [30] and [31],
but these issues do not apply here.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 6
and the diagonal matrix Vcontains samples of the PSD of ¯g
along its diagonal, or equivalently
Vii =f(2π(i−1)/(P QM ))
=r0+ 2 ℜ(P QM −1
X
k=1
rke−j2πk(i−1)/(P QM ))
where i= 1,···, P QM .
The following theorem gives the modification of Theo-
rem 1 for the asymptotic case. Note the essential change is
that FP QM replaces Uand Vreplaces Λ.
Theorem 2: Suppose assumptions A1),A2) and A3) hold
true, and the entries in the diagonal matrix Vare available.
Then under a constraint tr{XHX} ≤ LQP0, only Xthat
satisfy
X=Ψ diag "η−σ2
n
V11 +
, . . . ,
η−σ2
n
VP QM,P QM +#!1/2
FH
P QM ,(21)
can be asymptotically optimum in the sense of either max-
imizing the MI or minimizing the MMSE, where Ψis an
arbitrary LQ ×P QM matrix with orthonormal columns, and
the constant ηis chosen to satisfy
P QM
X
i=1 η−σ2
n
Vii +
=LQP0.
Further, provided X=IQ⊗¯
Xfor some ¯
X, then any Xfrom
(21) is asymptotically optimum for both MI and MMSE.
Proof: Substituting ˜
Σ¯g for Σ¯g in (8) and (13), respec-
tively, we have
I(¯y;¯g|¯
X) = log det(σ−2
nVZHZ+IP QM ),
and
MMSE = tr{(σ−2
nZHZ+V−1)−1},
where Zis now defined as Z=XFP QM . Then, following
the same procedure as in the proof for Theorem 1, we obtain
(21).
Note that to produce Xin (21), the full covariance
matrix Σ¯g is not needed, only samples of the PSD for Vare
necessary. This is a more reasonable assumption in practice.
It is also noted that the solution for Xis not unique, but
as we pointed out previously, XHXis unique for a given
Σ¯g . Further, this XHXwill simultaneously maximize MI and
minimize MMSE.
IV. SPECIAL CASES AND DISCUSSION
In this section, we will consider two special cases for
optimum radar waveform design. The first special case arises
when the covariance matrix Σ¯g is strictly diagonal. The second
special case considers a traditional radar equipped with a
single transmit and a single receive antenna.
A. Σ¯g Strictly Diagonal
Here we modify assumption A1) to obtain:
A1’) The elements of ¯g are independently Gaussian dis-
tributed with zero mean. The diagonal covariance
matrices Σ¯gq= E{¯gq¯gH
q}are the same ∀q∈[1, Q]
and hereby denoted as Σ. Thus, Σ¯g is diagonal and
Σ¯g =IQ⊗Σ.
Based on A1’), we can rewrite the MI (8) and MMSE (13) as
I(¯y;¯g|¯
X) = Q·log det(σ−2
nΣ¯
XH¯
X+IP M ),(22)
and
MMSE =Q·tr (σ−2
n¯
XH¯
X+Σ−1)−1,(23)
respectively. The power constraint remains the same, i.e.,
tr{¯
XH¯
X} ≤ LP0.
For brevity, we present only the main results in the
following corollary.
Corollary 1: Under A1’) and A2), the block waveform ¯
X
that maximizes the MI and minimizes the MMSE for MIMO
radar requires that ¯
XH¯
Xis diagonal, which insists that
XH
iXj=0,∀i, j ∈[1, P ] & i6=j,
and that XH
iXiis diagonal ∀i∈[1, P ]. Further, with the
power constraint tr{¯
XH¯
X} ≤ LP0, the optimal block wave-
form ¯
Xmust satisfy
¯
X=Ψ diag "η−σ2
n
Σ11 +
,...,η−σ2
n
ΣP QM,P QM +#!1/2
,
(24)
where Ψis an arbitrary L×P M matrix with orthonormal
columns and the constant ηis chosen according to
P M
X
i=1 η−σ2
n
Σii +
=LP0.
In the next subsection, we focus on cases with P=Q= 1.
B. Traditional Radar with Single Antenna under Asymptoti-
cally Large L
For simplicity, we omit the antenna index por qin our
ensuing analysis and assume Lis asymptotically large. Then
accordingly, (21) in Theorem 2 becomes
X=¯
X=X
=Ψ diag "η−σ2
n
V11 +
, . . . ,
η−σ2
n
VM,M +#!1/2
FH
M(25)
where Ψis now an L×Mmatrix, Xis Toeplitz as defined in
(3) and Vis now an M×Mdiagonal matrix with its diagonal
containing samples of the PSD of the target impulse response
g.For convenience, we rewrite the expression for MI as:
I(y;g|X) = log det(σ−2
nΣgXHX+IM)(26)
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 7
Denoting ˆ
gas the Bayes estimate of g, then when the cost
function is defined as kg−ˆ
gk2, the Bayes risk will be
MMSE = tr{(σ−2
nXHX+Σ−1
g)−1}.(27)
While if the cost function is defined as kXg −Xˆ
gk2instead,
it is not hard to show that the Bayes estimate is still equal to
the conditional mean, but in this case, the Bayes risk, or the
weighted MMSE6is given by
WMMSE = tr{XHX(σ−2
nXHX+Σ−1
g)−1}.(28)
Recall the data matrix Xdefined in (3), and note we can
rewrite it as
XT= [x(1),x(2),···,x(L)].
where the M-by-1input waveform vector x(i)is given by
x(i) = [x(k+i−1) ,···, x(k+i−1−ν)]T(i= 1,···, L).
Then by defining Φ∗=XHX, we obtain
Φ=XTX∗=
L
X
i=1
x(i)xH(i),(29)
which is referred to as the time-average correlation matrix or
simply the correlation matrix7[25]. Clearly the M×Mmatrix
Φis Hermitian, but in general, non-Toeplitz. Invoking the
WSS characteristic of the waveform sequence in assumption
A3) and the assumption of asymptotically large L, we have
that Φis Toeplitz. Thus, we can construct a circulant matrix
asymptotically equivalent to Φby8
˜
Φ=FH
MDFM,
where Dis a diagonal matrix, whose diagonal entries contain
samples of the PSD of the waveform sequence. Due to the
Hermitian constraint on Φ, the diagonal entries in the diagonal
matrix Dare real. Hence we have
˜
Φ∗=FMDFH
M.
The main results arising from the solution of single
antenna optimal waveform sequence design based on maxi-
mization of MI and minimization of MMSE are described in
the the following theorem.
Theorem 3: Under the constraint tr{XHX} ≤ LP0and
assumptions A1) -A3), only a waveform sequence that has
Dii =η−σ2
n
Vii +
,∀i∈[1, M ](30)
can be asymptotically optimum in terms of either maximizing
the MI I(y;g|X), or minimizing the value of MMSE (MMSE)
6This formulation will be shown later to lead to an interesting relationship
between the MI and the MMSE (see Corollary 2).
7In fact, Φshould be divided by the scaling factor Lto be a true time
average. The scaled form of Φis referred to as the sample correlation matrix
[25]. For large Lwhere the law of large numbers applies, the scaled version
of Φwould converge to the expectated value of the correlation matrix. A
similar technique has been employed in [43].
8The choice of an appropriate circulant matrix to approximate a Toeplitz
matrix is not unique [28], and here the circulant matrix ˜
Φis constructed
through the inverse Fourier transform [30].
or weighted MMSE (WMMSE), where ηcan be found by
solving
M
X
i=1 η−σ2
n
Vii +
=LP0.(31)
The corresponding maximum value of conditional MI is given
by
Imax(y;g|X) =
M
X
i=1 log(σ−2
nVii η)+,(32)
and the resulting minimum values of MMSE and WMMSE are
MMSE =
M
X
i=1
Vii
(Viiσ−2
nη−1)++ 1 (33)
WMMSE =
M
X
i=1
(ηVii −σ2
n)+
(Viiσ−2
nη−1)++ 1.(34)
Further, any waveform sequence that has Dii from (30) is
simultaneously asymptotically optimum for MI, MMSE and
WMMSE.
Proof: See Appendix II.
Relying on the above theorem and its proof, and follow-
ing a similar procedure to that taken in [38], we immediately
have the following corollary.
Corollary 2: Define snri=Dii/σ2
nas the transmitted
SNR at the ith frequency bin. Then, when the cost function
kg−ˆ
gk2is chosen, the value of MMSE is related with the
MI through
M
X
i=1
d
dsnri
I(y;g|X) = MMSE.(35)
Alternatively, define snr = 1/σ2
n. Then, when the cost
function kXg −Xˆ
gk2is chosen, the relationship between
this weighted MMSE and the MI is expressed by
d
dsnrI(y;g|X) = WMMSE.(36)
This corollary presents an interesting relationship that
also relates the two criteria we used in optimum radar wave-
form design. It is noteworthy that this fundamental relationship
was recently unveiled in [38] and [39], demonstrating that the
derivative of the MI (nats) with respect to the SNR is equal to
the MMSE, regardless of the input statistics. It is not hard to
show that such a relationship also holds for the MIMO radar
case, but we omit this discussion.
Remark 2: The optimum solution for the waveform se-
quence given in Theorem 3 can be viewed as a discrete-time
sequence which is similar to the continuous-time sequence
obtained in [3].
V. NUMERICAL EXAMPLES
In this section, we present numerical examples that illus-
trate the waveform design solution derived in this paper.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 8
A. Radar with Single Transmit/Receive Antenna (Asymptotic)
First we study the special case of a radar with a single
transmit antenna and a single receive antenna. In this case we
will employ the asymptotic results in Theorem 3 to generate
Xand show that this Xyields near optimum performance.
We consider an extended target with a finite-energy, complex
Gaussian impulse response gwith memory ν= 19 or M=
20. According to Theorem 3, we assume samples of the PSD
of g,Vii (i= 1,···,20), are available and take on the values
given in Fig. 2. In all the simulations, the noise is assumed to
be complex and i.i.d Gaussian distributed with σ2
nnormalized
to 1, i.e., it has real and imaginary components with variance
1/2each.
4 8 12 16 20
0
0.2
0.4
0.6
0.8
1
Vii
i
Fig. 2. Samples of the PSD of g
Fig. 3 illustrates samples of the PSD of the transmitted
waveform, Dii (i= 1,···,20), which are obtained through
(30) and (31) with L= 20 and P0= 10. Additionally, we also
4 8 12 16 20
0
5
10
15
4 8 12 16 20
0
5
10
15
σ2
n/Vii
Dii
i
i
Fig. 3. Illustration of water-filling type power allocation
plot the values of σ2
n/Vii (i= 1,···,20) for the purpose of
comparison. It is clearly seen that this waterfilling solution
allocates the transmitted power to those frequency bins where
the PSD of gindicates the presence of significant scattering of
the extended target; while, for those frequency bins where the
target PSD is small as compared with the background noise
(large values of σ2
n/Vii), less transmitted power is allocated.
This optimum solution is reasonable and makes intuitive and
logical sense. If one wishes to measure a target, one should
concentrate the power where there is least uncertainty about
the target in order to overcome the noise. On the other hand,
it is inefficient to focus the power where the noise hides the
target image.
10 20 30 40 50 60 70 80 90 100
55
65
75
85
95
105
115
125
135
Imax (y;g|X) in nats
P0
Asymptotic analytical results
Nonasymptotic results
L=80
L=160
L=320
Fig. 4. Imax(y;g|X)as a function of Land P0
10 20 30 40 50 60 70 80 90 100
10−2
10−1
100
MMSE
P0
Asymptotic analytical results
Nonasymptotic results
L=320
L=160
L=80
Fig. 5. MMSE as a function of Land P0
Based on Vii, we can calculate the asymptotic max-
imum value of the MI Imax (y;g|X)and the asymptotic
minimum value of the MMSE directly through (32) and (33),
respectively. In our curves these results are labelled as the
“asymptotic analytical results”. On the other hand, we can
compute Imax(y;g|X)and MMSE through (26) and (27) using
a Gaussian distributed random sequence xto fill the matrix X
and label these results as “nonasymptotic results”. We generate
data for the matrix Xto have the PSD Dii (i= 1,...,20). For
simplicity, covariance matrices of gand xare assumed to be
circulant, and can be obtained from Dii and Vii , respectively.
In Fig. 4, we plot the asymptotic maximum MI
Imax(y;g|X)for different values of Land P0. Fig. 4 shows
that when the value of either Lor P0increases, the value of
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 9
the maximum MI Imax (y;g|X)increases accordingly. That
increasing P0, i.e., the available power, can increase the MI
is reasonable. Increasing Lcorresponds to increasing the time
we observe the radar targets, which should also improve the
performance. Fig. 4 also indicates that the difference between
the asymptotic and the nonasymptotic results decreases as L
increases. This gives some indication that these numerical
results are reasonable since as we increase L, we see Φin
(29) approaches a scaled version of the covariance matrix of
x, which was assumed to be circulant. Thus for large Lthe
results in Theorem 3 apply exactly and so for sufficiently large
L, the curves for both analytical and simulated results will
completely overlap each other. Additionally, we plot in Fig. 5
the values of MMSE as a function of both Land P0.
B. MIMO Radar with P= 2 and Q= 1 (Non-asymptotic)
We consider here a simple MIMO radar case with P= 2
and Q= 1. The target impulse response between the pth
transmit element and the single receiver element is assumed
to have a memory of ν= 9, or equivalently M= 10.
We assume the Hermitian symmetric covariance matrix Σ¯g
is known, and for simplicity we set its eigenvalues, Λii =
Vii ,∀i∈[1,20], as given in Fig. 2. We assume the noise is
complex and Gaussian distributed, and its power σ2
nis fixed
at 1. Thus, we depict in Fig. 6 the power allocation for several
typical eigenmodes, where we choose i= 7,9,16 for the
modes illustrated. We also plot the equal power assignment for
comparison. The total transmitted power LP0and the power
assigned per mode are both given in dB, and Lis always
fixed at 20. When the total transmitted power is low, only those
strongest or dominant eigenmodes are used by the power water
fill, e.g., the 7th mode. As the transmitted power increases,
more and more eigenmodes are allocated power, e.g., the 9th
and 16th eigenmodes. When the total transmitted power is very
large, the power assigned to each mode will be approximately
the same, as shown in Fig. 6. Thus, the power values computed
through the optimal and the equal power allocation schemes
become indistinguishable.
8 12 16 20 24 28 32 36 40
0
4
8
12
16
20
24
28
LP0(L=20) in dB
Totoal transmitted power
Power allocated to per mode in dB
Optimal power allocation
Equal power allocation
the 7th eigenmode
the 9th eigenmode
the 16th eigenmode
Fig. 6. Optimal power allocation for selected eigenmodes
0 4 8 12 16 20 24 28
0
8
16
24
32
40
48
56
Total transmitted power
Mutual informaton in nats
LP0(L=20)indB
Optimal Power Allocation
Equal Power Allocation
Fig. 7. Mutual information for optimal vs. equal power allocation
0 4 8 12 16 20 24 28
0
1
2
3
4
5
6
7
8
9
10
Total transmitted power
MMSE
LP0(L=20)indB
Optimal power allocation
Equal power allocation
Fig. 8. MMSE for optimal vs. equal power allocation
We plot in Fig. 7 the MI and in Fig. 8 the MMSE versus
the total transmitted power LP0, under both the optimal and
equal power allocation schemes. It is seen that the optimal
allocation outperforms the equal power assignment counterpart
in both MI and MMSE. It should be noted that Fig. 7 and Fig. 8
apply for one specific example. In other cases, the difference
in MI or MMSE arising from the choice of different power
allocation schemes might be even larger, and the superiority of
optimal power allocation over equal power assignment could
be much more pronounced.
VI. CONCLUSION
The second-order statistics, i.e. the covariance matrix,
of the extended target impulse response contains important
information regarding the target characteristics and could be
efficiently exploited for optimum radar waveform design for
single transmitter/receiver radar and MIMO radar. Capitalizing
on the knowledge of such statistics, we have investigated radar
waveform design for optimizing two criteria: maximization of
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 10
the MI and minimization of the MMSE. In both cases, we
imposed a constraint on the transmitted power. Our findings
indicate that these two criteria lead to the same optimum solu-
tion for XHXwhich is the fundamental quantity specifying
the waveform design. In terms of this fundamental quantity,
maximizing the MI is equivalent to minimizing the MMSE.
The same solution for XHX, which we described in this
paper, optimizes both criteria. Further, the optimum solution
employs water-filling, which allocates the transmitted power in
proportion to the quality of the particular mode in question.
In particular, more transmitted waveform power is allocated
to modes that have higher power, indicating the presence
of significant target scattering, and modes with low power
deserve excitation with lower transmitted waveform power.
We also provided an asymptotic formulation in this paper
that lessens the required knowledge about the statistical model
to just a few samples of the power spectral density which
would be much more suitable in practice. Our solution for
waveform design will be asymptotically optimal with an
increase in the duration of target impulse response or with
an increase in the number of transmit or receive antennas.
The system model in (5) is linear. This model was
chosen since it is the most well accepted model for this
application. We expect this is partially due to the simplicity
of this model and also to some extent to the generality of
this model. For example, the class of linear systems of the
form of (5) is a very broad one and this class of models
provides a useful approximation to a large portion of the
systems encountered in practice. Further, the analysis of linear
systems is a well developed area and the advantages of using
linear approximations are well documented. For example, we
are able to perform the analysis and optimization for the whole
class of linear systems of the form of (5), regardless of the
exact sizes or properties of the matrices involved. The model
in (5) ignores nonlinearities which may be present in reality,
but the appropriate models for these nonlinearities are much
less well accepted and the exact form of the models chosen
for these nonlinearities would influence the analysis. Also, the
methods of optimization and analysis for nonlinear systems
are far less developed than those of linear systems. None the
less, extensions to nonlinear models would be interesting.
The purpose of this paper is to analytically find the form
of the best waveforms and to analytically demonstrate the
equivalence of two popular criteria. We have chosen what we
feel to be the most basic and the best accepted target model for
this purpose. Of course, it would be interesting to extend the
analysis to other models also, including new improved models.
APPENDIX I
COMMENTS ON PROOF OF LEMMA 1
There are several methods that can be used to obtain
either inequality (14) 9or (15). For example, for a positive
definite matrix A, a proof for (14) has been obtained in [33]
and [40], and a proof for (15) has been given in [41]-[43].
The theory of majorization can be applied even for positive
semidefinite Hermitian matrix A. Readers are referred to [36]
9The inequality (14) is usually referred to as Hadamard’s inequality.
or [44] for the basic concepts and some important results, or
to [45] for a complete reference on the subject.
APPENDIX II
PROOF OF THEOREM 3
As stated in [28, Th. 4.4], products of Toeplitz matrices
are also asymptotically equivalent to the products of their as-
sociated circulant matrices. Thus, we use the ciculant matrices
˜
Φand ˜
Σgto replace Φand Σgrespectively in
det(σ−2
nΣgΦ∗+IM) = det(σ−2
n˜
Σg˜
Φ∗+IM)
= det(σ−2
nVD +IM)
=
M
Y
i=1
(σ−2
nViiDii + 1)
and
tr{˜
Φ∗}= tr{FMDFH
M}=
M
X
i=1
Dii ≤LP0.
Then the MI in (26) can be approximately calculated by
I(y;g|X) =
M
X
i=1
log σ−2
nViiDii + 1.
As a consequence, the problem of waveform design based on
MI is simplified to
max
D
M
X
i=1
log(σ−2
nViiDii + 1)
s.t.
M
X
i=1
Dii ≤LP0
Dii ≥0,∀i∈[1, M ]
This turns out to be a problem of constrained optimization, and
can be solved by the method of Lagrange multipliers [32]. The
optimum solution can be expressed as
Dii =η−σ2
n
Vii +
,∀i∈[1, M ]
where ηcan be chosen by solving
M
X
i=1
Dii =
M
X
i=1 η−σ2
n
Vii +
=LP0.
The corresponding maximum MI can be calculated by
Imax(y;g|X) =
M
X
i=1
log "σ−2
nVii η−σ2
n
Vii +
+ 1#
=
M
X
i=1
log max σ−2
nViiη, 1
=
M
X
i=1 log(σ−2
nVii η)+.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, JANUARY 2007 11
The inverse of a Hermitian Toeplitz matrix is shown to
be asymptotically equivalent to the inverse of its associated
circulant matrix [28, Th. 4.3]. As a consequence, (27) becomes
tr{(σ−2
nΦ∗+Σ−1
g)−1}= tr{(σ−2
n˜
Φ∗+˜
Σ−1
g)−1}
= tr{(σ−2
nD+V−1)−1}
=
M
X
i=1 Dii
σ2
n
+1
Vii −1
.
Invoking [28, Th. 4.4] and following similar procedures for
(28), we readily obtain
tr{˜
Φ∗(σ−2
n˜
Φ∗+˜
Σ−1
g)−1}=
M
X
i=1
Dii Dii
σ2
n
+1
Vii −1
.
Therefore, the MMSE and weighted MMSE can be approxi-
mately computed, respectively, by
MMSE =
M
X
i=1
Vii
ViiDii σ−2
n+ 1,
and
WMMSE =
M
X
i=1
ViiDii
ViiDii σ−2
n+ 1.
Accordingly, the problems of waveform design based on
MMSE estimation are reduced to:
min
D
M
X
i=1
Vii
ViiDii σ−2
n+ 1,if kg−ˆ
gk2is chosen
M
X
i=1
ViiDii
ViiDii σ−2
n+ 1,if kXg −Xˆ
gk2is chosen
s.t.
M
X
i=1
Dii ≤LP0
Dii ≥0,∀i∈[1, M ].
These two optimization problems can also be solved by
the method of Lagrange multipliers [32]. It is easily shown
that the optimum solution (for both problems) is the same as
that for optimizing the MI, which also represents water-filling
in the spectral domain.
The resulting values of MMSE and weighted MMSE can
be obtained by inserting the optimum solution, and are written
as
MMSE =
M
X
i=1
Vii
(Viiσ−2
nη−1)++ 1
and
WMMSE =
M
X
i=1
(ηVii −σ2
n)+
(Viiσ−2
nη−1)++ 1 ,
respectively.
ACKNOWLEDGMENT
The authors would like to thank Professor Leonard J.
Cimini, Jr. of University of Delaware, Professor Alexander
Haimovich of New Jersey Institute of Technology and Pro-
fessor Sergio Verd´u of Princeton University for their helpful
comments and suggestions.
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