text:
UFDC Home | UF Institutional Repository | UF Theses & Dissertations myUFDC Home | Help
Knowledge-Aided Signal Processing
https://ufdc.ufl.edu/UFE0022491/00001
Zhu, Xumin
[Gainesville, Fla.]
1 online resource (137 p.)
Thesis/Dissertation Information
Doctorate ( Ph.D.)
Degree Disciplines:
Li, Jian
Slatton, Kenneth C.
Zmuda, Henry
Shabanov, Sergei
A priori knowledge ( jstor )
Covariance ( jstor )
Data ranges ( jstor )
Matrices ( jstor )
Photographs ( jstor )
Prior learning ( jstor )
Radar ( jstor )
Signals ( jstor )
Spacetime ( jstor )
Waveforms ( jstor )
Electrical and Computer Engineering -- Dissertations, Academic -- UF
beamforming, knowledge, mimo, radar, stap, waveform
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Electrical and Computer Engineering thesis, Ph.D.
The concept of Knowledge-aided (KA) radar was first proposed about twenty years ago, where the a priori knowledge can be used to enhance radar signal processing. Recently, KA signal processing has been attracting increasing attention of researchers and practitioners alike. In this dissertation, we focus on its applications to space-time adaptive processing (STAP) for surveillance, to array processing for adaptive beamforming, and also to multiple-input-multiple-output (MIMO) radar for waveform syntehsis. We first study KA-STAP algorithms used in airborne radar for wide area surveillance. In STAP, the clutter covariance matrix is routinely estimated from secondary 'target-free' data. Because this type of data is, more often than not, rather scarce, the so-obtained estimates of the clutter covariance matrix are typically rather poor. In KA-STAP, an a priori guess of the clutter covariance matrix (e.g., derived from the knowledge of the terrain probed by the radar) is available. The problem we focus on is to 'combine' this a priori clutter covariance matrix and the sample covariance matrix estimated from the secondary data into a preferably more accurate estimate of the true clutter covariance matrix. We consider two shrinkage methods, called the general linear combination (GLC) and the convex combination (CC), as well as a maximum likelihood-based approach to solve this problem. Next, we consider applying KA signal processing algorithms for adaptive beamforming. In array processing, when the available snapshot number is comparable with or even smaller than the sensor number, the sample covariance matrix is a poor estimate of the true covariance matrix. In KA adaptive beamforming, the a priori covariance matrix usually represents prior knowledge on dominant sources or interferences. Since the noise power level is unknown, and thus cannot be included into the a priori covariance matrix, the prior covariance matrix is often rank deficient. We consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix, the sample covariance matrix, and an identity matrix to get an enhanced estimate of the true one. Both approaches are fully automatic and they can be formulated into convex optimization problems that can be e?ciently solved. Finally, we introduce KA waveform synthesis for MIMO radar in which the a priori knowledge is manifested as knowing a covariance matrix of the waveforms. We propose a cyclic algorithm (CA) to synthesize a set of waveforms that realize the given covariance matrix under practically motivated constraints, and which also have good auto- and cross-correlation properties in time, if desired. ( en )
In the series University of Florida Digital Collections.
Includes vita.
Source of Description:
Description based on online resource; title from PDF title page.
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2008.
Adviser: Li, Jian.
by Xumin Zhu.
UFRGP
Copyright Zhu, Xumin. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
LD1780 2008 ( lcc )
Institutional Repository at the University of Florida (IR@UF)
University of Florida Theses & Dissertations
INTERNAL UF RETRO ETDS
zhu_x.pdf
then the low PAR constraint is described by the equations:
L IX(12 (7 4)
1X. (1)|~2 = 1, < L.
In what follows, we use C to denote the set of signal matrices X that satisfy these con-
straints.
7.2.1 Without Time Correlation Considerations
R A X*X (7-5)
be the (sample) covariance matrix of the transmitted waveforms, where (.)* denotes the
conjugate transpose. The class of (unconstrained) signal waveform matrices X that realize
a given covariance matrix R is given by
X* = R1/2U, (7-6)
where U* is an arbitrary semi-unitary N x L matrix (U*U = I), and R1/2 is a Hermitian
square root of the matrix R. Then a possible mathematical formulation of the problem
of .;1,'i,//. -:.;:,1 the probing
follows [48]:
min X- LUR1/2 (7 7)
XEC;U
where || || denotes the Frobenius norm.
7.2.2 With Time Correlation Considerations
In i] ,ni real applications such as least-squares (LS) (i.e., matched filtering) based
MIMO radar in i-_- an important requirement on the synthesized waveforms is that
they have good correlation properties. The synthesis of constant-modulus transmit signals
with good auto- and cross-correlation properties and R = I is considered in [93] for
MIMO synthetic aperture radar (SAR) imaging. We now extend the problem in [93] to
synthesize a waveform matrix that realizes an arbitrary covariance matrix R under both
(a) (b)
PFA PFA
(c) (d)
Figure 3-3. Comparison of the ROC curves corresponding to the "ideal" detector (with
R R), the "initial" detector (with R Ro), the SMI detector, and the GLC
detector, for four different values of the perturbation level applied to R to
obtain Ro: (a) a 0, (b) a = 0.01, (c) a = 0.1, (d) a = 0.5.
Exactly as in the CC case, we can also derive alternative estimates for the above
quantities. To see how this can be done, note first that:
7 +p IRo R112+ E{iR- R1|2} E{|Ri- VRo12}, (333)
an estimate of which is given by I|R Ro l2. Consequently, we can also estimate co as:
ao" = (3-34)
IIR- yRo012'
and 3o as
o" 1 (3-35)
However, the estimate of jo in (3-35) is not guaranteed to be positive, unlike (3-29). To
guarantee that 3o" > 0, we can either replace (3-34) by min(&o", v) or use max(3o", 0) in
lieu of (3-35). Following a sIt---- -1 i. o in [34], we will use the first of these two alternatives
in what follows.
Remark: We have studied the behavior of the above four methods for estimating
R, viz. CC' (see (3-19)), CC" (see (3-20)), GLC' (see (3-29), (3-30)), and GLC" (see
(3-34), (3-35), and the subsequent discussion), in a number of numerical examples based
on the KASSPER data set -see the next section for details on this data set. In these
examples, GLC" (after appropriate scaling) provided slightly more accurate estimates of R
than the other methods, but the detection results (which were the ones of more interest,
see the next section) obtained with the four methods were quite similar to one another.
Consequently, for the sake of conciseness, in the following we present only the results
obtained with GLC" -which we will call GLC for short.
3.3 Application to KASSPER Data
The KASSPER data set has been generated, as described in [35], using a uniform
linear array with half-wavelength inter-element spacing and S = 11 sensors, and a number
of P = 32 pulses in each coherent processing interval. The -I ... --time steering; vector
corresponding to a target with spatial frequency w, and (normalized) Doppler frequency
The a priori covariance matrix, Ro(i), is constructed as a perturbed version of
the true R(i) (to mimic the uncertainty in the a priori knowledge that alv--, exists in
practice):
Ro(i) R(i) O tit, (4-27)
where 0 denotes the Hadamard matrix product, and ti is a vector of i.i.d. Gaussian ran-
dom variables with mean 1 and variance af. Note that the perturbed matrices generated
in this way remain positive definite.
The "nominal" space-time steering vector corresponding to a target with spatial
frequency uw and (normalized) Doppler frequency CD can be expressed as follows:
a(,, D) a(ws) a(wD), (4-28)
F iT
a(a;,) 1 w ... te(s-t1>) (4-29)
a(wD) WD ... j(P-)WD (4-30)
with (.)T denoting the transpose. The spatial frequency us has a known value for
a given azimuth angle, but since the target velocity is unknown, the Doppler fre-
quency UD is only known to belong to the following set of 32 possible Doppler values
{-327/32, -307/32, ... ,307/32}. Thus, there are totally L x K = 1000 x 32 = 32000
range-Doppler bins.
We use the same scheme described in [24] for target detection. A series of test targets
are inserted in all the range bins that span from 35 km to 50 km for a fixed Doppler
bin. The target power is the highest at the closest range bin, and reduces with range,
proportional to a fourth power of the reciprocal of the range. Let a~(i) denote the power
of the test target at the ith range bin. Define the signal-to-clutter-and-noise ratio (SCNR)
0.2F
E -0.1
I I I
-50 0 50
Angle (degree)
Figure 6-3. Differences between the beampatterns obtained from optimal R and the CA
synthesized waveforms under (a) PAR = 1, (b) PAR < 1.1, and (c) PAR < 2.
-YI r -11 --1
v -27
n) -28
-_I-
Sample Number L
Figure 6-4. MSE of the difference between R and R (CA synthesized constant modulus
waveforms) as a function of sample number L obtained with 100 Monte-Carlo
trials. R is obtained from the CA synthesized constant modulus waveforms.
15- ---
10- -----
0- -
-5- -
-- ---SCB
-10- .-..MGLC2 w/A=0
MGLC2
-1o
-SINRopt
-2 I I I I
Snapshot Number
/ -SCB
-10- ..-.MGLC2 w/A=0
-15- MGLC2
-SINRo
nfl Iopt
20 -- --------
15 -- -- -- -- -- ------" "
10- "'' ... --..
.- -
-5- ,
---SCB
-10- .-.-.MGLC2 w/A=0
> MGLC2
r_ __ I I I I
Figure 5-6.
SINR versus the snapshot number N when a~ = 20 dB, Ia = 70 dB and
o = 60 dB, and the a prior covariance matrix is given by (a) Ro = Ro (see
(5-42)), (b) Ro = R02 (see (5-43)), (c) Ro = R03 (see (5-44)) and (d)
Ro = R04 (see (5-45)).
2008 Xumin Zhu
-0.2 -
-0.2-
than the general linear combination in (3-2), which is probably the reason why (3-3) is
commonly used in the literature (see, e.g., [8] and the references there).
The first goal of this note is to obtain the a and f that minimize the mean-squared
error (\! S I) of R:
MSE= E{|R- RI2}, (3-4)
for both (3-2) and (3-3); hereafter, || || denotes the Frobenius matrix norm or the
Euclidean vector norm, depending on the context. We stress that this is a constrained
estimation problem: the two classes of covariance matrix estimates in (3-3) and (3-2)
have only 1 and, respectively, 2 free parameters. This means that the estimate R takes
values in a restricted set that in general does not contain R; therefore, the trivial but
problematic minimizer R = R of (3-4) is in general avoided.
Let ao and /0 denote the optimal values of a and 3 that minimize (3-4). The second
goal of this note is to discuss how to obtain estimates, do and /o, of ao and /0 from the
available data (as we will see shortly, and as expected, both ao and /o depend on the
unknown matrix R). Finally, we will explain how to use the proposed estimates of R,
viz. R in (3-2) or (3-3) with a = do and / = )o, in a KA-STAP exercise based on the
KASSPER data set [8, 35] (KASSPER stands for knowledge-aided sensor signal processing
and expert reasoning).
Estimation of a large-dimension covariance matrix from a limited number of samples
in the manner outlined above (and detailed in the next section) has been originally pro-
posed in [34] and its main references (with an emphasis on applications in economics) and
later on considered in several other papers, for example in [36] (with a focus on applica-
tions in bioinformatics) and in [37] (with an emphasis on array processing applications).
However, the cited papers considered only the case of real-valued data. Our approach in
this note is an extension of that in [34] to the complex-valued data case, as well as to a
general Ro matrix ([34] considers the case of Ro = I only). Also our proofs are more
explicit than those in [34], despite the more general, complex-valued data case we consider
KNOWLEDGE-AIDED ADAPTIVE BEAMFORMING
Given the true array covariance matrix R and the steering vector ao for the signal
of interest (SOI), the standard Capon beamformer (SCB) [19] can be used to maximize
the array output signal-to-interference-plus-noise ratio (SINR) adaptively. Since its first
introduction almost forty years ago, SCB and its robustified versions [50] have been
extensively studied and widely used in many applications, such as radar, sonar, wireless
communications, and biomedical imaging.
Let y(n) denote the nth output snapshot of an array comprising of M sensors. In
practice, the true array covariance matrix R, where
R= E{y(n)y*(n)} (5-1)
is unknown, and so it is usually replaced by the sample covariance matrix R, where
1 N
R 8=y()y*(n), (5-2)
with N denoting the snapshot number. However, when N is comparable with or even
smaller than M, R usually is a poor estimate of R.
To obtain an improved estimate of R when the snapshot number N is limited,
we can make use of prior environmental knowledge. The concept of knowledge-aided
(KA) signal processing was first proposed by Vannicola et al. in [1, 2] and by Haykin
in [3]. In a KA system, the a priori knowledge is manifested as having an initial guess
of the true array covariance matrix R, denoted as Ro [10, 11, 8]. When Ro has full
rank, we have considered two shrinkage approaches in [51], called the general linear
combination (GLC) and the convex combination (CC) methods, as well as a maximum
likelihood based approach in [38] to obtain an improved estimate of R based on R and
Ro. In this paper, we consider the case of Ro being rank deficient in a general adaptive
-0.5- 1 i--i---- I ---I- -I
1 ...-- --------------------
n-
--C
20-.
1.51- ""-... .-
-- -- -- -- -- -- -- -- -----
0.5-
fi._
80 1(
The averaged MGLC1 estimates of A, B and C versus the snapshot number N
when ao = = o- = 20 dB, and the a prior covariance matrix is given by (a)
Ro = Rol (see (5-42)), (b) Ro = Ro2 (see (5-43)), (c) Ro = R03 (see (5-44))
and (d) Ro Ro4 (see (5-45)).
Hence Bo can be obtained as
BD I211D112 2 tr2(D*D2)
Bo = (5-56)
pllD2 2 + IDi121D2112 tr2(DD2)
Given (5-56) and by using the Cauchy-Schwartz inequality, we have that 0 < Bo < 1.
--A
---B
("
I I ~I
L--B.
'''. --C
where C(i) E C MX is a square root of R(i), and Q(i) E CMXM is an arbitrary unitary
matrix satisfying Q(i)Q*(i) Q*(i)Q(i) IM. Consequently the whitening matrix can be
R-1/2() Q*()C-1(i). (4 8)
Since Q(i) is an arbitrary unitary matrix, R-1/2(i) in (4-8) is not unique. The non-
uniqueness of the whitening matrix will not affect the cost function in (4-5) when I =1
or when R(i) is independent of i. For example, when I =1 (without loss of generality, we
consider the range bin index i = 1),
Zi(a) Q*(1)C- (1)yly* (C*())- Q(1) -Q*(1)Q(1)
C- (1)yly(C*(1))-1 iM (4-9)
and therefore the value of the cost function does not depend on the choice of Q(1).
However, when I > 1, which is usually required for v! 1i. iii::, and R(i) is dependent on
i, the cost function in (4-5) becomes
Z,(a) = Q(i)*C- (i)yy*(C*(i)) Q(i) IM (4-10)
Note that the value of the cost function ZI(a) depends on the choice of {Q(i)}. Conse-
quently, different choices of the square root matrix may yield rather different values for a,
and there is no guideline as to which one to choose for good performance.
Next, we introduce an effective weight determination procedure based on the ML
approach. We also review the related approach recently proposed in [48].
4.3.1 Convex Combination
In the CC approach [48], a in (4-3) is allowed to vary with the range bin index. Be-
low, we omit the dependence of R, Ro, R, R and a on the range bin index for notational
simplicity. The CC approach considers a convex combination of R and Ro (see (4-3)) to
Figure 3-1. The GLC estimates of ao and Po as functions of range bin index, for four
different values of the perturbation level applied to R to obtain Ro: (a)
a2 0, (b) a = 0.01, (c) 2 = 0.1, (d) 2 = 0.5.
.4-
The CC and ML estimates of a as functions of the range bin index, for four
different values of the perturbation level of the a prior knowledge: (a) 2 = 0,
(b) af2 0.01, (c) 2 = 0.1, and (d) 2 = 0.5.
I I II
set A = B = C = 1, and hence R = Ro + R + I (we refer to this as "equal v i!il ).
Observe that by adaptively choosing the combination weights, the performance of MGLC2
is alv--i- better than that of equal weighting.
We now consider the Cases (v) and (vi). Figures 5-4(a) and 5-4(b), respectively,
show the averaged estimates of A(1)/J~, A(2)/a, B and C versus the snapshot number N
when {R } 2 (see (5-46) and (5-47)) and {R0} 1 (see (5-48) and (5-49)) are used as
the a priori covariance matrices for MGLC2. Observe that when a2 = a2 = 20 dB and
both DOAs of the two interference are accurately known, A(1)/a7 and A(2)/a2 obtained
by MGLC2 are almost identical. They have relatively large values when N is small and
decrease as N increases, as desired. Observe also that for the case where the a priori
covariance matrices {R } are given by {R?} } MGLC2 can give a proper weight on
the accurate component, i.e., RW(, and at the same time suppress the wrong component,
i.e., R) in the a priori knowledge.
In Figure 5-5, we display the averaged array output SINR as a function of the
snapshot number N for the two cases. By making use of the a priori covariance matrices
{R 1} MGLC2 outperforms MGLC2 with A) = A) = 0, especially when N is small
(see Figure 5-5(a)). When {R6 } 1 are used as the a prior covariance matrices, the
performance of MGLC2 is still slightly better than that of MGLC2 with A1 = A(2) = 0, as
shown in Figure 5-5(b).
5.5.2 Relatively Strong Interferences
We now consider a scenario where the powers of the interference are much stronger
than that of the SOI. Assume that a2 20 dB, a2 70 dB, and a = 60 dB. The
constraint in (5-34) is now active due to the presence of strong interference.
We first consider the Cases (i) (iv) when Roi, i = 1, 4, is used as the a priori co-
variance matrix. A's, B's and C's obtained by MGLC2 for this case of strong interference
have similar properties as those shown in Figures 5-1 and 5-2, except that C's now take
on larger values, which are mainly determined by the constraint (5-34). Figure 5-6 shows
R to obtain Ro, and the results were similar to those corresponding to (3-39) and thus
they will not be presented).
Let y denote the space-time snapshot corresponding to the range bin of current
interest. We will use the adaptive matched filter (AMF) detector [39] for target detection:
a* (s, oD)Ru -ly H1
a* (o, WD)R -la( ,W) Ho
where Ho is the null hypothesis (i.e., no target), H1 is the alternative hypothesis (i.e.,
Ho is false), is a threshold, and R is an estimate of the clutter-and-noise covariance
matrix, R, for the range bin in question. The threshold will be varied to generate ROC
(receiver operating characteristic) curves in the usual way (this time, using the fact that
we know which ones of the 64,000 range-Doppler bins contain targets). We will use the
GLC method, described in the previous section, to obtain R. Note that the GLC method
is fully automatic and computationally efficient (in particular, the estimate R of R
obtained with it has a closed-form expression), which are appealing features for practical
In order to make use of the GLC method, we need an (unbiased) estimate R of R,
for each range bin. We obtain this estimate from secondary data as follows: in a first step
(which is done once forever), we use the "initial detector" (3-40) with R = Ro to derive
a set, let us call it B, of range bins that are thought of being target free (some of these
bins might contain targets, but since they passed the initial detection step, these targets
must be weak compared with the clutter-and-noise level); then, for each given range bin,
we pick up N = 50 range bins from B whose covariance matrices Ro's are the closest (in
the Frobenius norm sense) to the covariance matrix Ro associated with the range bin of
current interest; let {y(n)}j 1 denote the secondary space-time snapshots corresponding to
the selected range bins from B; then, we obtain R using the standard formula in (3-1). (It
is interesting to observe that the above procedure for computing R makes also use of the a
prior information embedded in Ro's).
Because co above belongs evidently to the interval (0, 1), it is also the solution to the
constrained minimization of the MSE. Observe the intuitive character of the expression for
co in (3-7): when R is much closer to R than to Ro (in a MSE sense), then co is close to
0, and viceversa.
To estimate co from the available data, we need an estimate of E{IR R |2}. Let ,r
and r, denote the mth columns of R and R, respectively. Consequently, we have
rm = Ny ()i(),
rm= E{y(n)y*(n)},
where ym(n) denotes the mth element of y(n). To simplify the notation, in what follows
we omit the index m of some variables, such as rm and r, above (we will reinstate this
index later on, when needed). Let
It follows from the assumptions made on {y(n)}j that {x(n)} ,1 are i.i.d. random
vectors with mean t = r. Because:
E{||R R||2}
our generic problem is to estimate E{IIr
variance of the sample mean of {x(n)}
S N 2
2 N 2}
- r|2} = E Yx(n) (i.e., the
n=
1). A simple calculation gives:
t N 2
1NN
NE E {[x(n) i]*[x() p}
n=l n=l 1
NE{ x(n) A112}. (3-12)
ZE{m
r, II- ,
x(n) = y(n)y*{(n).
MAXIMUM LIKELIHOOD-BASED KA-STAP
4.1 Introduction and Preliminaries
Space-time adpative processing (STAP) is widely used in airborne radars for ground
moving target detection [4, 5, 6]. In STAP, I range bins are sampled during a coherent
processing interval (CPI), and for each range bin, the responses to P pulses are collected
from each of the S elements of the receive antenna array. This S x P data matrix for a
certain range bin is then stacked column-wise to form an SP x 1 vector, which is called
a space-time snapshot. It is well-known that the optimal weight vector, w E CMX1, with
M = SP being the degrees of freedom (DOFs) of STAP, used to maximize the signal-
to-clutter-and-noise ratio (SCNR) of the beamformer output of this snapshot is given by
[4]:
w = R-la(w, ,D), (4 1)
where R E CMXM is the true clutter-and-noise covariance matrix for that particular
range bin of current interest, and a(w ~D) E CMxl denotes the steering vector, which
is a function of the spatial frequency ua and the (normalized) Doppler frequency wD of
the target. Since a scaled version of w above does not change the SCNR, we simplify our
notation by considering the form in (4-1) only.
In practice, R is not known, and therefore must be estimated from the available
data. In standard STAP, the sample covariance matrix R is estimated from the training
or secondary data, {y(n)}j i, associated with the range bins close to the range bin of
interest (ROI), under the assumption that the training data have the same clutter and
noise statistics as the ROI, i.e., they are target free and homogeneous. Note that we
will use yi below to denote the snapshot for the ith range bin, and y(n) to denote the
nth seco,:J.li snapshot for a particular ROI. The sample covariance matrix can then be
is given by T*yi, which we still refer to as yi, for notational simplicity. Then all of our
previous discussions apply to the RD case.
Note that the optimal &ML in the ML approach for KA-STAP is constant for all
range bins. We have also tried to find optimal &ML(i) as a function of the range bin index
i by using an ML approach based on a leave-one-out cross-validation method, but the
performance was similar to that of using a constant &ML, whereas the computational
complexity was greatly increased. Consequently, the case in which &ML(i) is a function of i
will not be presented in this paper.
4.4 Numerical Results
In this section, we demonstrate the performance of the ML approach, as compared
to the CC approach, based on simulated clutter-only data for KA-STAP. Similar to [24],
in this paper, we focus on solving the problem of small sample support number in a
heterogeneous clutter environment. In our simulations, we generate the clutter data based
on the KASSPER '02 dataset [35]. The simulated airborne radar system has P = 32
pulses and S = 11 spatial channels, yielding M = PS = 352 DOFs. The mainbeam of the
radar is steered to an azimuth of 1950 and an elevation of -5 (the azimuth and elevation
are measured clockwise from the true north and the local horizon, respectively). In each
CPI, a total of I = 1000 range bins are sampled covering a range swath of interest from 35
km to 50 km. The KASSPER data simulates a severe heterogeneous clutter environment
and the true clutter-and-noise covariance matrix, R(i), is specified for each range bin. We
simulate the clutter data for the ith range bin as:
Yi R1/2(i)Vi, i 1,-. ,I, (4-26)
where (.)1/2 denotes the Hermitian square root of a matrix, and {vl} E CMxl are i.i.d.
circularly symmetric complex Gaussian random vectors with mean 0 and covariance
matrix I.
Next, we consider applying KA signal processing algorithms for adaptive beamform-
ing. In array processing, when the available snapshot number is comparable with or even
smaller than the sensor number, the sample covariance matrix R is a poor estimate of the
true covariance matrix R. In KA adaptive b, ,ivnr. i in-ir,- the a priori covariance matrix
Ro usually represents prior knowledge on dominant sources or interference. Since the
noise power level is unknown, and thus cannot be included into the a priori covariance
matrix, Ro is often rank deficient. We consider both modified general linear combinations
(ILGLC) and modified convex combinations (\!CC) of the a priori covariance matrix
Ro, the sample covariance matrix R, and an identity matrix I to get an enhanced esti-
mate of R. Both approaches are fully automatic and they can be formulated into convex
optimization problems that can be efficiently solved.
Finally, we introduce KA waveform synthesis for MIMO radar in which the a prior
knowledge is manifested as knowing a covariance matrix of the waveforms. We propose a
cyclic algorithm (CA) to synthesize a set of waveforms that realize the given covariance
matrix under practically motivated constraints, and which also have good auto- and
cross-correlation properties in time, if desired.
Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. The probing signals are
synthesized for N = 10, L = 512 and P = 10 by using CA under (a) PAR = 1,
(b) PAR < 1.1, and (c) PAR < 2 (the two curves corresponding to "Optimal
R" and "CA: PAR < 2" coincide with each other).
(constant-modulus), PAR < 1.1, and PAR < 2 with P = 1, respectively. For comparison
purposes, we also show the desired beampattern 0(0) scaled by the optimal value of a
as well as the beampattern obtained from the optimal R. Note that the beampattern
obtained using the CA synthesized waveforms is close to the desired one even under the
constant-modulus constraint, and it approaches the one obtained from the optimal R as
the PAR value increases.
Figure 7-2 shows the correlation levels (normalized values of (7-21)) versus p for
the CA synthesized waveforms with P = 1. Recall that for P = 1, the time correlation
property of the waveforms is not considered, and hence the resulting waveforms under
different constraints (i.e. PAR = 1, PAR < 1.1 and PAR < 1.2) all have relatively high
correlation levels.
We next consider the same waveform synthesis problem but with P = 10. By using
the same initial values in Step 0 of the CA algorithm (the initial value for X when P = 10
is constructed from the initial value for X when P = 1 by using (7-10)), the resulting
beampatterns are similar to those shown in Figure 7-1, as shown in Figure 7-3. Figure
7-4 shows the corresponding correlation levels (normalized values of (7-21)) versus p. The
correlation levels decrease as the PAR value increases, as expected. As we can see, the
correlation levels of the waveforms shown in Figure 7-4 are (much) lower than those shown
in Figure 7-2 due to optimizing the auto- and cross-correlations of the waveforms.
Figures 7-5 7-8 are similar to Figures 7-1 7-4, except that the sample number L
is set to be 512. As compared to the case where L = 256, improvements on the resulting
beampatterns as well as on the time correlations of the waveforms can be observed when L
is large.
Finally we consider an example with R = I and P = 10. The array is the same as
before. The corresponding correlation levels versus p with the sample number L = 256
and L = 512 are shown in Figures 7-9 and 7-10, respectively. We note from Figures 7-9
and 7-10 that even under the constant-modulus constraint, the synthesized orthogonal
beamforming application. This case occurs frequently in practice when we only have
prior knowledge on dominant sources or interference. Unlike in some literature, for
example, see [20, 21, 22], where the noise power level is treated as prior knowledge, we
now consider a more practical case where the noise power level is assumed to be unknown,
and thus cannot be included into Ro to make it full rank. We consider both modified
general linear combinations (\!GLC) and modified convex combinations (\ CC) of the
a prior covariance matrix Ro, the sample covariance matrix R, and the identity matrix
I to get an enhanced estimate of R, denoted as R. MGLC and MCC, respectively, are
the modifications of the GLC and CC methods proposed in [51]. Both MGLC and MCC
can choose the combination weights fully automatically. Moreover, both the MGLC and
MCC methods can be extended to deal with linear combinations of an arbitrary number of
positive semidefinite matrices. Furthermore, both approaches can be formulated as convex
optimization problems that can be solved efficiently to obtain globally optimal solutions.
The remainder of the paper is organized as follows. Section 5.2 formulates the
problem of interest. In Section 5.3, we discuss how to obtain enhanced covariance matrix
estimates by using MGLC and MCC via convex optimization formulations. In Section 5.4,
we use the enhanced covariance matrices instead of the sample covariance matrix in SCB
to improve the array output SINR. In Section 5.5 numerical examples are presented to
demonstrate the effectiveness of the proposed algorithms. Finally, conclusions are given in
Section 5.6.
Notation. Vectors are denoted by boldface lowercase letters and matrices by boldface
uppercase letters. The nth component of a vector x is written as xn. The inverse of a
matrix R is denoted as R- We use (.)T to denote the transpose, and (.)* the conjugate
transpose. The Frobenius norm is denoted as I|| |. The real part of an argument is denoted
as Re(-). The expectation operator is denoted as E(.) and the trace operator is denoted as
tr(-). The notation R > 0 means that R is positive semidefinite.
[14] P. R. Gurram and N. A. Goodman, "Spectral-domain covariance estimation with a
priori knowledge," IEEE Transactions on Aerospace and Electronic Sit 1ii vol. 42,
pp. 1010-1020, July 2006.
[15] S. D. Blunt, K. Gerlach, and M. R ii.- "--, ii:, "STAP using knowledge-aided
covariance estimation and the FRACTA algorithm," IEEE Transactions on Aerospace
and Electronic Sl-1. mi vol. 42, pp. 1043-1057, July 2006.
[16] J. G. Verly, F. D. Lapierre, B. Himed, R. Klemm, and M. Lesturgie (Guest Editors),
"Special issue on radar space-time adaptive processing," EURASIP Journal on
Applied S'.:j,11r Processing, vol. 2006, 2006.
[17] M. C. Wicks, M. Rangaswamy, R. Adve, and T. B. Hale, "Space-time adaptive pro-
cessing: A knowledge based perspective of airborne radar," IEEE S.:,'j l Processing
1.1 .j ..:. vol. 23, pp. 51-61, January 2006.
[18] R. Adve, T. B. Hale, and M. C. Wicks, "Knowledge based adaptive processing for
ground moving target indication," Digital S.:,jrl Processing, vol. 17, pp. 495-514,
March 2007.
[19] J. Capon, "High resolution freiqui" r--vi--enumber spectrum analysis," Proceedings of
the IEEE, vol. 57, pp. 1408-1418, August 1969.
[20] G. Bienvenu and L. Kopp, "Optimality of high resolution array processing using
the K.-, ii-,-I. ill approach," IEEE Transactions on Acoustics, Speech, and S.:j,.,r
Processing, vol. 31, pp. 1235-1247, M li- 1983.
[21] M. I. Skolnik, Radar Handbook. New York, NY: McGraw-Hill Book Co., 1990.
[22] M. Steiner and K. Gerlach, 1 I-I converging adaptive processor or a structured
covariance matrix," IEEE Transactions on Aerospace and Electronic S;,/-/. i vol. 36,
pp. 1115-1126, October 2000.
[23] J. S. Bergin, C. M. Teixeira, P. M. Techau, and J. R. Guerci, "STAP with knowledge-
aided data pre-whitening," IEEE Radar Conference, PA, April 26-28 2004.
[24] C. M. Teixeira, J. S. Bergin, and P. M. Techau, "Reduced degree-of-freedom STAP
with knowledge-aided data pre-whil Ini,.- KASSPER Wordshop, Las Vegas, April
[25] J. S. Bergin and P. M. Techau, \!hI i resolution signal processing techniques for
ground moving target detection using airborne radar," EURASIP Journal on Applied
S.:j,Irl Processing, vol. 2006, Article ID 47534 2006.
[26] N. A. Goodman and P. R. Gurram, "STAP training through KA predictive model-
ing," In Proceedings 2004 IEEE Radar Conference, Philadelphia, PA, April 2004.
The log-likelihood function of {y,},,s is hence proportional to:
C(i = {n |R(i) + yA '(i)y,}. (4-46)
nEiB
Consequently, the optimal ai for the ith range bin can be determined by maximizing
the cost function C(i) in (4-46). Note that the same fast computation methods as in
Appendix A can be applied here to improve computation efficiency.
x 105 1 x105
-1.2 -1.2-
-1.4 -1.4
-1.6- -1.6
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
The log-likelihood function as a function of a, for four different values of the
perturbation level of the a prior knowledge: (a) of 0, (b) f = 0.01, (c)
tf 0.1, and (d) of 0.5.
where {zp} are given numbers. The solution to (7-17) is evidently given by
= arg Yz, (7-18)
(P-i 1
Therefore, under the constant-modulus constraint, both steps of the cyclic algorithm have
closed-form solutions.
Under the low PAR constraint in (7-4), the generic problem in this case is:
mm |x zp2
p=1
= min const 2Re(Z )z, x + Px2
S(Pp=i ) X
m in 1- z 2 (7 19)
x P
where x is an element of the waveform matrix X. Let x, denote the nth column of X.
Then given (7-19), the column-wise minimization problem in Step 1 of the CA algorithm
under the low PAR constraint is as follows:
min I|x, z||2
s.t. ftx(1) 2
Ix,(1)2 < y, = 1,.. ,L, (7-20)
where the elements of z can be calculated from {zp} in (7-19). While the above mini-
mization problem is not convex, an efficient ild.,rithm for solving it has been proposed in
[86, 87].
7.4 Numerical Examples
We present several numerical examples to demonstrate the effectiveness of CA for
signal synthesis.
ACKNOW LEDGMENTS ................................. 4
LIST OF FIGURES .................................... 7
A BSTRA CT . . . . . . . . . . 10
1 INTRODUCTION ...................... .......... 12
1.1 Knowledge-Aided Space-Time Adaptive Processing .... ........ 12
1.2 Knowledge-Aided Adaptive Beamforming ............... 15
1.3 Knowledge-Aided Waveform Synthesis ........ ............ 16
2 LITERATURE REVIEW FOR KA-STAP ............ .......... 18
2.1 D irect U se . . . . . . . .. . 18
2.2 Indirect U se . .. . . . . . . .. 20
2.3 Contributions of Our Work ......... .......... ...... 21
3 ON USING A PRIORI KNOWLEDGE IN STAP ............... ..23
3.1 Introduction and Preliminaries .................. ..... .. 23
3.2 Minimum MSE Covariance Matrix Estimation .. . . ..... 25
3.2.1 Convex Combination (CC) .................. .. 25
3.2.2 General Linear Combination (GLC) ................. .. 28
3.3 Application to KASSPER Data .................. ... .. 30
3.4 Conclusions .................. ................ .. 34
4 MAXIMUM LIKELIHOOD-BASED KA-STAP ................. .. 38
4.2 Problem Formulation .................. ........... .. 40
4.3 Weight Determination of Prior Knowledge ............. .. 41
4.3.1 Convex Combination ............... ..... .. 42
4.3.2 Maximum Likelihood ............... ..... .. 44
4.4 Numerical Results ............... ............ .. 47
4.5 Conclusions ............... .............. .. 51
5 KNOWLEDGE-AIDED ADAPTIVE BEAMFORMING . . 58
5.1 Introduction .................. ................ .. 58
5.2 Problem Formulation ............ . . . ...... 60
5.3 Knowledge-Aided Covariance Matrix Estimation . . 61
where R = R I with & denoting the Kronecker matrix product.
Consequently, a mathematical formulation of the problem of -;, ./I,,. ..:.', the probing
'.:.,il matrix X with time correlation considerations can be written as follows:
min X VUR1/2 2 (7-12)
where U* is an arbitrary semi-unitary NP x (P + L 1) matrix (U*U = I).
We observe that (7-7) is a special case of (7-12) with P = 1 and hence we only
consider the minimization problem in (7-12) hereafter. Note that (7-12) is non-convex
due to the non-convexity of the constraint U*U = I and possibly of the set C, too. we
present in the next section a computationally attractive cyclic (or alternating) optimiza-
tion algorithm for solving (7-12), as sI--.-, -1. 1 in a related context in [86, 87]. We refer to
the cited papers for more details on this type of algorithm and its properties.
7.3 Cyclic Algorithm for Signal Synthesis
We first summarize the steps of the cyclic minimization algorithm for solving (7-12)
and then describe each step in detail.
StepO: Set U to an initial value; alternatively we can start with an initial value for X
(e.g., the elements of U or X can be independently drawn from a complex Gaussian
distribution with mean 0 and standard deviation 1), in which case the sequence of
the next steps should be inverted.
Stepl: Obtain the matrix X E C that minimizes (7-12) for U fixed at its most recent
Step2: Determine the matrix U (U*U = I) that minimizes (7-12) for X fixed at its
most recent value.
Iter.: Iterate Steps 1 and 2 until a given stop criterion is satisfied. In the numerical
examples presented later, we terminate the iteration when the Frobenius norm of the
difference between the U matrices at two consecutive iterations is less than or equal
to 10-4.
Figure page
1-1 Angle-Doppler interference image for a single range bin . ...... 13
1-2 A general block diagram for a space-time processor. .............. 13
1-3 MIMO radar versus phased array .................. ....... .. 16
1-4 Reflected pulses from different range bins can overlap significantly. . ... 17
3-1 GLC estimates of ao and /0 as functions of range bin index. . 35
3-2 Estimation error ratio. ...................... . .... 36
3-3 Comparison of the ROC curves corresponding to the "ideal" detector, the "ini-
tial" detector, and the GLC detector. ............... .. .. 37
4-1 Log-likelihood function as a function of a. ............... .. .. 55
4-2 CC and ML estimates of a as functions of the range bin index. . ... 56
4-3 Comparison of the ROC curves corresponding to the "ideal" detector, the "prior"
detector, the "CC" detector, the \!I," detector, the S; l" detector, and the
"equal-. i;,ii detector. . . . . .. .......... .. 57
5-1 Averaged MGLC1 estimates of A, B and C versus the snapshot number N. 74
5-3 SINR versus the snapshot number N when weak interference present and for
cases i iv .................. ................... .. 76
5-4 Averaged MGLC2 estimates of A(l/l2a, A(2)/oJ, B and C versus the snapshot
number N .................. ................... .. 77
cases v vi .................. ................... .. 77
5-6 SINR versus the snapshot number N when strong interference present and for
6-1 Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. ................. 98
6-2 PAR values for CA synthesized waveforms with optimal R and for colorized
Hadamard code. .................. ................ .. 98
[77] I. Bekkerman and J. Tabrikian, "Space-time coding for active arrays," IEEE Transac-
tions on S.I:i'al Processing, to appear.
[78] J. Li, P. Stoica, L. Xu, and W. Roberts, "On parameter identifiability of MIMO
radar," IEEE S.:,.Irl Processing Letters, vol. 14, pp. 968-971, December 2007.
[79] D. R. Fuhrmann and G. San Antonio, "Transmit beamforming for MIMO radar sys-
tems using signal cross-correlation," IEEE Transactions on Aerospace and Electronic
Si-1,. ,i- to appear.
[80] L. Xu, J. Li, and P. Stoica, "Target detection and parameter estimation for MIMO
radar systems," IEEE Transactions on Aerospace and Electronic Si-/. ,- to appear.
[81] J. Li, L. Xu, P. Stoica, D. Bliss, and K. Forsythe, "Range compression and waveform
optimization for MIMO radar: A Cram6r-Rao bound based study," IEEE Transac-
tions on S.:',.1 Processing, vol. 56, pp. 218-232, January 2008.
[82] J. Li and P. Stoica, \! \!O radar with colocated antennas: Review of some recent
work," IEEE S.:,.1rl Processing Ml.'.'..:,. vol. 24, pp. 106-114, September 2007.
[83] J. Li and P. Stoica, eds., MIMO Radar S.i,. Processing. New York, NY: John Wiley
& Sons, Inc., to appear in 2008.
[84] B. Guo and J. Li, \V i,. lorm diversity based ultrasound system for hyperthermia
treatment of breast cancer," IEEE Transactions on Biomedical Engineering, to
appear.
[85] A. Edelman, T. Arias, and S. Smith, "The geometry of algorithms with orthogonality
constraints," SIAM Journal on Matrix A,.il -.:. and Applications, vol. 20, no. 2,
pp. 303-353, 1998.
[86] J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, "CDMA signature se-
quences with low peak-to-average-power ratio via alternating projection," 37th
Asilomar Conference on S.:gn'il Sii-., ;,- and Computers, Pacific Grove, CA, vol. 1,
pp. 475-479, November 2003.
[87] J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, "Designing structured tight
frames via an alternating projection method," IEEE Transactions on Information
Th(.-,;. vol. 51, pp. 188-209, January 2005.
[88] C. R. Rao, \! I11 :: approximations and reduction of dimensionality in multivariate
statistical ,i: 1,-i- In Multivariate A,../l,.-.:- P. R. Krishnaiah (ed.), vol. 5, pp. 3-22.
Amsterdam: North-Holland Publishing Company, 1980.
[89] R. A. Horn and C. R. Johnson, Matrix A,,l;-.:.- Cambridge, U.K.: Cambridge
University Press, 1985.
[90] C. C. Vernon, J. W. Hand, S. B. Field, D. Machin, J. B. Whaley, J. Van Der Zee,
W. L. J. Van Putten, G. C. Van Rhoon, J. D. P. Van Dijk, D. G. Gonzalez, F. Liu,
numerical examples have been provided to demonstrate that the proposed algorithm for
waveform synthesis is quite effective.
Appendix: On Enforcing Solely the PAR Constraint
Consider the following generic form of the problem:
min Is z112 s.t. PAR(s) < p, (6-29)
where z is given and PAR(s) is as defined in (6-16). Hence we have omitted the power
constraint (6-17), which should lead to a smaller matching error. Because PAR(s) is
insensitive to the scaling of s, let us parameterize s as
s = ex; l||x|2 = 1; where c > 0 is a variable. (6-30)
Using (6-30) in (6-29) yields:
||s z||2 = cx z 12 C2 2cRe(x*z) + const. (6-31)
If Re(x*z) < 0, then the minimum value of (6-31) with respect to c > 0 occurs at c = 0. If
Re(x*z) > 0, then the minimization of (6-31) with respect to c > 0 gives:
c Re(x*z), (6-32)
and the value of (6-31) corresponding to (6-32) is smaller than the value associated with
c = 0. Because PAR(s) = PAR(x) does not depend on the phases of the elements of x,
we can ah--,v- choose x such that Re(x*z) > 0 -so that we achieve a smaller value of
(6-31). Consequently, the minimizing value c > 0 of (6-31) is alv--,v- given by (6-32). The
remaining problem is:
maxRe(x*z) s.t. l||x|2 = 1 and PAR(x) < p, (6-33)
or equivalently
min |x z|2 s.t. IlX| 2 = 1 and PAR(x) < p, (6-34)
Finally, note that even when the initial detector has a performance close to that of "coin
-.1-- ;_, (see Figure 3-3(d)), the GLC detector has a reasonable performance -a fact
which -i--.- -r that the GLC detector is robust to the selection of the set B of I i -' .-
free" range bins.
In this ('!i Ipter, which extends the approach of [34] to complex-valued data and to
arbitrary "initial guess" matrices Ro, we have described a computationally simple and
fully automatic method that obtains an enhanced estimate of a covariance matrix R by
linearly combining in an optimal MSE manner the sample covariance matrix R and Ro.
While the said method is ,-..i. l /. general, here the focus was on its use for KA-STAP.
In an application of this method to the KASSPER data set, we showed that the method
can provide (much) more accurate estimates R of R than R by carefully tuning the
weights of Ro and R in R = cRo + /3R according to the i I ive distances" of Ro and
R to R. The use of the so-obtained enhanced estimate of R in an STAP detector was
shown to outperform significantly the SMI detector, as well as the Ro-based detector in
the practically interesting case of a relatively unreliable Ro.
Figure 6-9. Breast model.
Acoustic transducer array
..... .. ...............
-4U -2U
x (mm)
2U 4U bu
Figure 6-10. PAR
-r.
Index of Acoustic Transducer
values for CA synthesized waveforms with optimal R.
r w^''^
SINR versus the snapshot number N when ao 20 dB, 2 = 70 dB and
a 60 dB, and the a prior covariance matrices are given by (a)
{R } {R 1 (see (5-46) and (5-47)), and (b) {RIO} 1- J {R 1
(see (5-48) and (5-49)).
Figure 6-8. Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and a null from -550 to -450, under the relaxed elemental power
constraint. The power gain difference between 0 and the null is constrained to
be less than or equal to 30 dB. The probing signals are synthesized for N = 10
and L = 256 by using CA under (a) PAR = 1 and (b) PAR < 1.1.
CONCLUSIONS AND FUTURE WORK
In this dissertation, we have studied several knowledge-aided (KA) signal processing
applications. The contributions of our work are listed as follows:
First, we have investigated KA space-time adaptive processing (STAP) algorithms in
ground moving target indication radar for wide area surveillance. We have proposed two
shrinkage methods and an ML approach to determine the optimal weight that should be
applied on the a priori knowledge. All of the proposed approaches are fully automatic,
and we have shown that they can adaptively adjust the weighting factor according to the
degree of accuracy of the a priori knowledge based on collected data. We have also shown
that the a priori knowledge can be used both directly or indirectly to improve target
detection performance of the radar.
Second, we have considered KA signal processing in a general adaptive beamforming
application. We have extended the shrinkage methods to their modified versions, in which
the array covariance matrix is considered to be a linear combination of multiple positive
semidefinite matrices. These modified approaches are still fully automatic, and they can be
formulated as convex optimization problems which can be efficiently solved with globally
optimal solution guaranteed. We have also shown that providing (partially) accurate a
prior knowledge can improve the performance of the adaptive beamformers, sometimes
significantly.
Third, we have proposed a simple and efficient cyclic algorithm for KA MIMO
radar waveform synthesis. Given the covariance matrix of the waveforms as the a priori
knowledge, which can be obtained in a previous (optimization) stage or simply prescribed,
we can synthesize a set of waveforms that, under the constant modulus or low PAR
constraints, realize (at least approximately) the given covariance matrix, and also have
good auto- and cross-correlation properties.
P. Goodman, and M. Sherar, "Radiotherapy with or without hyperthermia in
the treatment of superficial localized breast cancer: Results from five randomized
controlled trials," International Journal of Radiation Or.' .. ..,; B.:. ...i ; Phf, ..
vol. 35, pp. 731-744, July 1996.
[91] M. H. Falk and R. D. Issels, "Hyperthermia in oncology," International Journal of
Hyperthermia, vol. 17, pp. 1-18, January 2001.
[92] E. L. Jones, J. R. Oleson, L. R. Prosnitz, T. V. Samulski, Z. Vui i-1. vic, D. Yu, L. L.
Sanders, and M. W. Dewhirst, "Randomized trial of hyperthermia and radiation for
superficial tumors," Journal of C'.-,...' i Or. .. ..,; vol. 23, pp. 3079-3085, 2005.
[93] J. Li, X. Z1, i.- and P. Stoica, \!I\lO SAR imaging: Signal synthesis and receiver
design," The '.I International Workshop on Computational Advances in Multi-
Sensor Adaptive Processing (invited), St. Thomas, U.S. Virgin Islands, 2007.
[94] C. Y. C'!. i: and P. P. Vaidyanathan, "A subspace method for MIMO radar space-time
adaptive processing," IEEE International Conference on Acoustics, Speech, and S.:,.,1rl
Processing, vol. 2, pp. 925-928, April 2007.
[95] T. W. Anderson, "Asymptotically efficient estimation of covariance matrices with
linear structure," The Annals of Statistics, vol. 1, pp. 135-141, January 1973.
8.2 Future Work
In what follows, we discuss some further possible directions that are closely related to
8.2.1 KA Waveform Optimization for MIMO STAP
We can consider how the emerging MIMO radar techniques can be used with STAP
to advance the stat-of-the-art of STAP. The concept of MIMO radar STAP is first
introduced in [55]. MIMO radar STAP for multipath clutter mitigation is considered
in [70]. Design issues of the transmit waveforms for MIMO STAP based on the MIMO
ambiguity function are discussed in [94].
One interesting research topic in MIMO STAP would be the optimal transmit
waveform design. Given the a priori interference information, we can study waveform
optimization in MIMO radar STAP for improved target parameter estimation based on
various performance metrics (for example, the Cramer-Rao bound (CRB) matrix for target
parameters) by using the prior knowledge.
8.2.2 Asymptotic ML Method for Optimal Weight Determination
In C'! lpter 4, we have considered a linear combination of two covariance matrices
(the a priori covariance matrix Ro and the sample covariance matrix R) into an enhanced
estimate of the true clutter-and-noise covariance matrix in KA-STAP. And we have
presented an ML approach to determine the combination coefficients for the two terms.
We can also consider a linear combination of multiple semi-definite matrices instead of just
two terms:
R= A(s)R ) + BR + CI. (8-1)
s=l 1
We can again consider ML estimation of these unknown combination weights in the above
equation. The problem of estimating covariance matrices with a linear structure has been
studied in [95], and an .i-vmptotically efficient ML approach is proposed to determine the
combination coefficients in an iterative manner. One needs to investigate if it is possible to
apply the method proposed in [95] to optimal weight determination in (8-1).
the true clutter-and-noise covariance matrix R(i) for the ith range bin of interest (ROI).
As already mentioned, a is assumed to be independent of i here.
Note that the a priori knowledge can be used for secondary data selection (SDS) for
each ROI, which corresponds to an indirect exploitation of the a priori knowledge. Let
I be the number of the range bins sampled by the radar during a CPI and let {yj}i 1
denote the corresponding clutter data set. Then for the ith ROI, we pick up N range
bins (excluding the ith range bin which is the primary data) from the range bin set whose
a prior covariance matrices Ro's are the closest (in the Frobenius norm sense) to the
covariance matrix Ro(i); let Bi denote the range bin set containing the selected bins for
the ith ROI; and let E7 index the set Bi; then, the secondary training snapshots {y(n)}N 1
for the ith ROI in (4-2) can be chosen as {yn}ne,; consequently, R(i) obtained with SDS
can be computed as:
R(i) Z yny* (4-22)
Note that this SDS procedure is also used in the CC approach to obtain the sample
covariance matrix.
Next, we introduce the ML approach to determine the optimal value of a. Assume
that {y}$ 1, are independent identically distributed (i.i.d.) with the following distribution:
yi ~ Cn'(0, R(i)); R(i) aRo(i) + (1 a)R(i), (4-23)
where CA'(0, R(i)) denotes the circularly symmetric complex Gaussian distribution with
mean zero and covariance matrix R(i). The log-likelihood function of {yi}i is then
proportional to:
C = {ln 1 (i)(i )yi}
Iln aRo(i) + (1 )R(i) + y aRo(i) +(1 )R(i)] yi (4-24)
i= 1( -
The variance Ef{|x(n) p, 2} in (3-12) can be estimated as:
1 N 1 N
N N (313)
n~l n~l
It follows that we can estimate E{||rm r., I } as
-N2 y(n)yr (n) r|-. (3-14)
n=l
p E{|R- RI2}. (3-15)
Using (3-14) as an estimate of E{||rm r, I||- leads to the following estimate for p:
N2 y(n)y*(yn)- R12 (3 16)
The above expression for p has a certain theoretical appeal, but its direct calculation may
be somewhat slow. A more attractive expression for p, from a computational standpoint,
can be obtained by observing that
1 M 1 N
1P N N ly(n)Y)( n) .1
rnm= 1 n= 1
N Y N I [ i y(n) (l ) In 12 l, K1 (3 17)
and therefore that
1 Rly12 (3-18)
N2 y
Using p as an estimate of p and I|R Ro||2 as an estimate for I|R Ro 2 yields the
following estimate of co in (3-7):
o' = (319)
p+ R -Ro1 2
here. Finally, to conclude this brief review of the relevant literature, we refer the reader
interested in KA-STAP and in the problem described in this note, to the more detailed pa-
per [38], also by the present authors, which presents a maximum likelihood based approach
to KA-STAP.
3.2 Minimum MSE Covariance Matrix Estimation
We will consider the MSE minimization problem first for (3 3) and then for (3-2).
We assume that the secondary data {y(n)}j 1 are i.i.d. random vectors with mean zero
and covariance matrix R. However, we do not make any distributional assumptions on
{y(n)}j 1, such as the usual Gaussian assumption that would not be warranted in STAP.
Also, we assume that Ro is a fixed (non-random) matrix. However, if we have reasons to
assume that Ro is random, then we can modify the following results in a straightforward
manner to accommodate a random R doing so can be useful if, and likely only if, we
have some knowledge on the error I|Ro RI| or the MSE E{I|Ro R1|2}; because in KA-
STAP such a knowledge is usually unavailable, we will focus on the case of non-random Ro
3.2.1 Convex Combination (CC)
For (3-3), a simple calculation yields:
E{||R- R|2} = E{Ia(Ro-R)+(R-R)|2}
const + 2E{||Ro R||2} 2Re {tr LE{(R R)(R Ro)* }
const + a2E{||Ro RI|2} 2aE{|R R1|2}, (3-5)
where tr denotes the trace, Re stands for the real part, and where we have used the fact
that R is an unbiased estimate:
E{R} = R. (3-6)
The unconstrained minimization of (3-5) with respect to a gives:
E{I|R- R|2} E{|| R- |2}
CQ0 = 2 E{ R2 (3-7)
E{||lR-R R } E{||R-R R }+||2 IR-Rol2
(b) Minimize the cross-correlation between the probing signals at a number of given
locations (a reason for this requirement is explained in [72, 76]); the cross-correlation
between the probing signals at locations 0 and 0 is given by a*(0)Ra(0).
Assume that we are given a desired transmit beampattern 0(0) defined over a region
of interest Q. Let {pg}2 1 be a fine grid of points that cover Q2. As indicated above, our
goal is to choose R such that the transmit beampattern, a*(0)Ra(0), matches or rather
approximates (in a least squares (LS) sense) the desired transmit beampattern, ((0),
over the region of interest f, and also such that the cross-correlation (beam)pattern,
a*(0)Ra(0) (for 0 / 0), is minimized (once again, in a LS sense) over a given set {Ok}kI
Mathematically, we therefore want to solve the following problem:
m in "'K' [o(pg) a*(pg)Ra(pg)]2
a,R G
K-1 K
K a*(0k)Ra(P)12}
K2 K |( >2
k l1 p k+l
s.t. R,, N' n 1,- ,N
R > 0, (622)
where a is a scaling factor, i,., > 0, g = 1, ,G, is the weight for the gth grid point and
t > 0 is the weight for the cross-correlation term. Note that by choosing nm-:, i'., > i,
we can give more weight to the first term in the design criterion above, and vice versa for
nri::., i,,, < w,. We have shown in [67, 72, 76] that this design problem is a semi-definite
quadratic program (SQP) that can be efficiently solved in polynomial time. Once the
optimal R has been determined, we can use CA to synthesize the waveform matrix X.
As mentioned in Section 6.2, the CA solution to (6-3) may be chosen to realize R
exactly or only approximately. When the signal waveforms are synthesized as /L-iR1/2,
where U is the solution to (6-3) obtained via CA, then they realize R exactly, but satisfy
the PAR constraints only approximately. We refer to the so-synthesized waveforms as the
CA -;,'Il, -..,. waveforms with optimal R (abbreviated as optimal R). When we use the
[64] J. Tabrikian and I. Bekkerman, "Transmission diversity smoothing for multi-target
localization," The 2005 IEEE International Conference on Acoustics, Speech, and
S.:,I.rl Processing, Philadelphia, PA, vol. 4, pp. iv/1041-iv/1044, March 2005.
[65] K. W. Forsythe and D. W. Bliss, \\V ,, lorm correlation and optimization issues for
MIMO radar," 39th Asilomar Conference on S.:,.1l- S,-i, ii- and Computers, Pacific
Grove, CA, pp. 1306-1310, November 2005.
[66] E. Fisher, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela,
"Spatial diversity in radars models and detection performance," IEEE Transactions
on S.:,1,l Processing, vol. 54, pp. 823-838, March 2006.
[67] J. Li and P. Stoica, \! \!O radar -diversity means superiority," The Fourteenth
Annual Workshop on Adaptive Sensor A ,,.;, Processing (invited), MIT Lincoln
Laboratory, Lexington, MA, June 2006.
[68] J. Tabrikian, "Barankin bounds for target localization by MIMO radars," 4th IEEE
Workshop on Sensor A ,,.,; and Multi-channel Processing, Waltham, MA, July 2006.
[69] L. Xu, J. Li, and P. Stoica, "Adaptive techniques for MIMO radar," 4th IEEE
[70] V. F. Mecca, D. Ramakrishnan, and J. L. Krolik, \!I \lO radar space-time adaptive
processing for multipath clutter mitigation," 4th IEEE Workshop on Sensor A ,,.,;
and Multi-channel Processing, Waltham, MA, July 2006.
[71] L. Xu, J. Li, and P. Stoica, "Radar imaging via adaptive MIMO techniques," 14th
European S.:j,.rl1 Processing Conference, (invited), Florence, Italy, September 2006.
[72] J. Li, P. Stoica, and Y. Xie, "On probing signal design for MIMO radar," 40th
Asilomar Conference on S.:g,.l- S,-.i, ii and Computers (invited), Pacific Grove,
CA, October 2006.
[73] Y. Yang and R. S. Blum, \!i\!O radar waveform design based on mutual informa-
tion and minimum mean-square error estimation," IEEE Transactions on Aerospace
and Electronic Si-. ',- vol. 43, pp. 330-343, January 2007.
[74] T. Aittomaki and V. Koivunen, "Low-complexity method for transmit beamforming
in MIMO radars," The 2007 IEEE International Conference on Acoustics, Speech,
and S.:j,.rl1 Processing, Honolulu, Hawaii, USA, April 2007.
[75] L. Xu, J. Li, P. Stoica, K. W. Forsythe, and D. W. Bliss, \V\ i, lorm optimization
for MIMO radar: A Cramer-Rao bound based study," 2007 IEEE International
Conference on Acoustics, Speech, and S.:j,.rl1 Processing, Honolulu, Hawaii, April
[76] P. Stoica, J. Li, and Y. Xie, "On probing signal design for MIMO radar," IEEE
Transactions on S'.:j,11 Processing, vol. 55, pp. 4151-4161, August 2007.
the averaged SINR versus the snapshot number N for the four cases. It is interesting to
note that as compared to Figure 5-5, providing accurate or partially accurate a prior
knowledge gives greater SINR improvement in the presence of strong interference.
We next consider the Cases (v) and (vi) for this case of strong interference. Fig-
ure 5-7 shows the averaged SINR versus the snapshot number N. Again, an obvious
improvement of MGLC2 over the other methods can be observed.
Finally, we comment that our numerical examples (not presented herein) show that
for both cases of weak and strong interference, as N oo, all SINR curves (including
those of MGLC1) approach the optimal SINR curve, as expected. Moreover, the SINR
curves of MGLC2 ahv-- stay above those of MGLC with A = 0 as well as those of SCB.
In knowledge-aided adaptive beamforming, a priori environmental knowledge can be
utilized to improve the performance of adaptive arrays. In practice, the a priori knowledge
usually consists of (partial) information about the array covariance matrix. Consequently,
we can obtain an "initial guess" of the array covariance matrix, denoted as Ro, which
is usually rank deficient. In this paper, we have presented two fully automatic methods,
namely MGLC and MCC, for combining the sample covariance matrix R with the a
prior covariance matrix Ro and the identity matrix I to get an enhanced estimate of R
in the optimal mean squared error sense. We have also extended the MGLC and MCC
methods to deal with linear combinations of an arbitrary number of positive semidefinite
matrices. It has been shown that both MGLC and MCC can be formulated as convex
optimization problems, which can be solved efficiently in polynomial time with global
optimality guaranteed. When the MGLC and MCC techniques are used for knowledge-
aided adaptive b, ,il-r. .ii nii.r we have shown via numerical examples that MGLC can
choose the combination weights adaptively (based on the data) according to the accuracy
of the a priori knowledge. We have also shown that providing accurate or partially
(iii). Only the DOA of the first interference is accurately known:
R03 ala. (5-44)
(iv). Inaccurate a priori knowledge. We consider the case where the a prior knowl-
edge on the DOAs of the interference is totally vA. ni:. i.e.,
R a4 72 + a* a (5-45)
R04 3a3a3 + 4a4a4, (545)
where a = a a4 a and a3 and a4 are the steering vectors for two uncorrelated
signals impinging on the array from -55 and 60.
(v). The DOAs of the interference are accurately known:
R1 alat, (5-46)
R =- a2al. (5-47)
(vi). The DOA of the first interference is accurately known, but we assume wrongly
that the DOA of the second interference is 60:
RI = alat, (5-48)
RS = a4a. (5-49)
5.5.1 Relatively Weak Interferences
We first consider a scenario in which the power of the SOI and the powers of the
two interference are all equal to 20 dB, i.e., a = 20 dB, k = 0, 1,2. In the presence of
relatively weak interference, tr (R) is small and hence the constraint in (5-34) is inactive
for this case.
Figures 5-1 5-2 show the averaged estimates of A, B and C from 200 Monte-Carlo
realizations when the a priori covariance matrices {Roi} 1 (see (5-42)-(5-45)) are used
expected, the difference decreases as the PAR value increases. For the case of PAR = 2,
the difference is essentially zero. The mean-squared error (\!Sl) of R (i.e., the average
value of 62), obtained under PAR = 1 and estimated via 100 Monte-Carlo trials, is shown
in Figure 6-4 as a function of the sample number L. Note that, as also expected, the MSE
decreases as L increases. Figures 6-3(a) 6-3(c) show the corresponding beampattern
differences as a function of 0, as an ensemble of realizations obtained from the 100 Monte-
Carlo trials. In each Monte-Carlo trial, the initial value for U in Step 0 of CA was chosen
independently. Among other things, Figure 6-3 shows that CA is not very sensitive to the
initial value of U used, and that this sensitivity decreases as p increases.
Figure 6-2 shows the actual PAR values of the CA -; ,,Ih .:.../, a..r. I., ,,m- with optimal
R corresponding to Figure 6-1. These PAR values are also compared to those associated
with the waveform matrix obtained by pre-multiplying R1/2 with a 256 x 10 matrix
whose columns contain orthogonal Hadamard code sequences of length 256. The colored
Hadamard sequences also have the optimal R as their sample covariance matrix. Note
that the CA -;;u,.1 f,../ ;. .., f., -, with optimal R have much lower PAR values than
the colored Hadamard code sequences. Note also that the actual PAR values of the CA
..;.,, ..,/ ., [. -i, with optimal R obtained under PAR < 1.1 are slightly lower than
the PAR values obtained under PAR = 1.
Next, we consider a scenario where the desired beampattern has three pulses centered
at 01 = -400, 02 = 0, and 03 = 400, each with a width of 200. The same mesh grid is used
as before, and we choose the weighting factors as "'., = 1 and w, = 1. Figure 6-5 shows the
corresponding beampatterns. Remarks similar to those on Figure 6-1 can be made for this
example as well.
6.4.2 Minimum Sidelobe Beampattern Design
The minimum sidelobe beampattern design problem we consider here (see [72, 76] for
more details) is to choose R, under the uniform elemental power constraint in (6-21) or
rather a relaxed version of it (see later on), to achieve the following goals:
MIMO RADAR WAVEFORM SYNTHESIS
MIMO radar is an emerging technology that has significant potential for advanc-
ing the state-of-the-art of modern radar. Unlike a standard phased-array radar, which
transmits scaled versions of a single waveform, a MIMO radar system can transmit via
its antennas multiple probing signals that may be chosen at will. An interesting current
research topic in MIMO radar is the optimal synthesis of the transmitted waveforms.
For MIMO radar with widely separated antennas, waveform designs without practical
constraint (such as the constant-modulus constraint) have been considered in [73]. For
MIMO systems with colocated antennas, on the other hand, the recently proposed tech-
niques for transmit beampattern design or for enhanced target parameter estimation and
imaging have focused on the optimization of the covariance matrix R of the waveforms
[61, 65, 67, 72, 75, 76, 79, 81].
Now, instead of designing R, as in the cited references, we might think of designing
directly the probing signals by optimizing a given performance measure with respect to
the matrix X of the signal waveforms. However, compared with optimizing the same per-
formance measure with respect to the covariance matrix R of the transmitted waveforms,
optimizing directly with respect to X is a more complicated problem. This is so because
X has more unknowns than R and the dependence of various performance measures on X
is more intricate than the dependence on R (as R is a quadratic function of X). In effect,
there are several recent methods, as mentioned above, that can be used to efficiently com-
pute an optimal covariance matrix R, with respect to several performance metrics; yet the
same cannot be said about determining an optimal signal waveform matrix X, which is the
ultimate I..'l of the designing exercise. Furthermore, in some cases, the desired covariance
matrix is given (e.g., a scaled identity matrix), and therefore there is no optimization with
respect to R involved (directly or indirectly).
-- Ideal (Prior)
M- ML
---CC
Equal-weight
-. SMI
. . I . . .
Comparison of the ROC curves corresponding to the "ideal" detector, the
"prior" detector, the "CC" detector, the \!I/" detector, the S\!I" detector,
and the "e,1iu !-v. i!,ii detector, for four different values of the perturbation
level of the a prior knowledge: (a) a 0, (b) f = 0.01, (c) f = 0.1, and (d)
a2 0.5.
[ATci = s,-Mcc = 1, 2, (5-31)
0McC, MCCi MCCi C
be the solutions to the extended MGLCi and MCCi problems, i 1, 2, respectively. Given
(5-30) and (5-31), the resulting R can be expressed as:
RMGLC i ( LCi + MGLCR + MLCi, 1,2, (5-32)
ItMCC, = () MCCRO + ^MCCR + CMccI, i = 1,2. (5-33)
S= 1
Remark: We note from the unconstrained solution of 0 (see (5-16) with A and
b given in (5-26) and (5-27), respectively) that the value of CMGLC increases as tr(R)
increases. Specifically, tr(R) will be large in the presence of strong interference, resulting
in a very high weighting value on I in RMGLC. Consequently, when such a RMGLC is used
in adaptive beamformers, their adaptive capability is greatly reduced because CMGLC is
large. To avoid the said problem of MGLC, an additional constraint can be enforced in the
MGLC formulations in (5-22) and (5-24):
0S+2 < 7Amin, (5-34)
where Amin is the smallest non-zero eigenvalue of R, and 7 is a scaling factor. Noting
the fact that 0s+2 should decrease as N/M increases, we may choose 7 to be inversely
proportional to N/M. In our simulations, we choose 7 = 103M/N. The constraint in
(5-34) is usually inactive when tr(R) is small.
5.4 Using R for Adaptive Beamforming
Assume that the true covariance matrix R of the array output has the following form:
R oaoao + kaka* + Q, (5-35)
k= 1
0 50 -50 0
Angle (degree) Angle (degree)
synthesized for N = 10, L = 256 and P = 1 by using CA under (a) PAR = 1,
accurate a priori knowledge can significantly improve the performance of the adaptive
beamformers, especially in the presence of strong interference.
Appendix: A Property of Bo in (5 16)
In this appendix, we show that 0 < Bo < 1. Note that 0o in (5-16) satisfies:
A0o = b,
where A and b are given in (5-14) and (5-12), respectively.
associated with (5-50) is [Alb], where
I|Ro 12 tr(RR) tr(R-)
[Alb] tr(R-R) |R112+p tr(R*)
tr(R*) tr(R*) ||I||2
tr(RoR)
D = 2 R o0 R,
||IRo 1
The augmented matrix
tr(R*Ro)
tr(R*)
tr (Ro)
D2 = Ro I.
I I Ro 12
Then making use of the Gaussian elimination, it follows that (5-51) is equivalent to
I|Ro 02 tr(RR) tr(R*) tr(R*Ro)
0 ||Dl|2 +p tr(DDD2) I||D |2
0 tr(D*D2) I|D2 2 tr (DD2)
and consequently it is also equivalent to
IIRo012
tr(R|R)
rDtr2 (DD2)
IIDll1 +p DII
tr(DTD2)
tr(RO)
|ID2 12
IDl 112 tr2(DD2)
tr (D ID2
tr (DTD2)
sine(azimuth)
Figure 1-1. Angle-Doppler interference image for a single range bin.
S Antennas
P pulses
Th-rl
w*y
Figure 1-2. A general block diagram for a space-time processor.
signal-to-clutter-and-noise ratio (SCNR) of the beamformer is given by [4]:
w = R-a, (1-1)
range bin of current interest, and a E CMxI denotes the steering vector of the target.
In standard STAP, in order to estimate the clutter-and-noise covariance matrix R, the
training (or secondary) data, {y(n)}j 1, associated with the range bins close to the range
the optimization by omitting (6-17). In the Appendix we show how to modify the
algorithm of [86, 87] in the case where only (6-16) is enforced. (In all numerical examples
presented in the following section, (6-17) will be enforced.)
6.4 Numerical Case Studies
signal synthesis in several diversity-based transmit beampattern design applications.
6.4.1 Beampattern Matching Design
We first review briefly the beampattern matching design (more details can be found in
[72, 76]). We then present a number of relevant numerical examples.
The power of the probing signal at a generic focal point with coordinates 0 can be
shown to be (see, e.g., [61, 69, 72]):
P() a*(O)Ra(O), (6-19)
where R is as defined before,
a(0) --'.I -L- () c -"' "- ) -N (e ) (6 20)
and where fo is the carrier frequency of the transmitted signal, and 7-,(0) is the time
needed by the signal emitted via the nth transmit antenna to arrive at the focal point;
unless otherwise stated, 0 will be a one-dimensional angle variable (expressed in degrees).
The design problem under discussion consists of choosing R, under a uniform elemental
power constraint,
R N, = n =l, N, (6-21)
where C is the given total transmitted power, to achieve the following goals:
(a) Control the spatial power at a number of given locations by matching (or approximat-
ing) a (scaled version of a) desired transmit beampattern.
(a) Minimize the sidelobe level in a prescribed region.
(b) Achieve a predetermined 3 dB main-beam width.
Assume that the main-beam is directed toward 00 and the prescribed 3-dB angles are
01 and 02 (i.e., the 3-dB mainbeam width is 02 01, with 01 < 00 < 02). Let ts denote the
sidelobe region of interest and {pg} a grid covering it. Then the design problem of interest
in this section can be mathematically formulated as follows:
min -t
t,R
s.t. a*(0o)Ra(0o) a*(p,)Ra(p,) > t, Vp, E Q,
a*(01)Ra(01) 0.5a*(0o)Ra(0o)
R>0
0.8 ( 5) R & 1.2 ) ,
n= ,N, "R" = C. (6-25)
n-l
Note that the relaxed elemental power constraint in (6-25), while still quite practical,
offers more flexibility than the strict elemental power constraint in (6-21). Note also that
the total transmit power is the same for both (6-25) and (6-21), viz. C. In the examples
below, we set C 1.
As shown in [76], this minimum sidelobe beampattern design problem is a semi-
definite program (SDP) that can be efficiently solved in polynomial time. Once the
optimal R has been determined, we can again use CA to synthesize the waveform matrix
Consider first an example where the main-beam is directed toward 00 = 0 with a
3-dB width equal to 200 (01 = -100 and 02 = 100). The sidelobe region is chosen to
be [-900, -200] U [200, 90], which allows for some transition between the main-
beam and sidelobe region. The same mesh grid size of 0.10 is used here. Figure 6-6 shows
-100 -
-140 I I I
-10 -5 0 5 10
Figure 7-4. Correlation levels versus p for the CA synthesized waveforms with N = 10,
L 256 and P 10 (R / I).
10S. o o ----....
0 -----'"
5- o -.-..
o ,'
0 oo ---SCB
-.MGLC2 w/A=0
MGLC
-10 MGLC2
-15 o Equal weight
-20 I I I o
10- -' o a
1 o o o
5 ,- 0 -
0 0o, ---SCB
-5 .- -.MGLC w/A=0
-15- o Equal weight
-20 I I opt
SINR versus the snapshot number N when ao = a = oa = 20 dB, and the a
prior covariance matrix is given by (a) Ro = Rol (see (5-42)), (b) Ro = R02
(see (5-43)), (c) Ro = R03 (see (5-44)) and (d) Ro = R4 (see (5-45)).
>- -. --- -0 ----0
o 0o O
So ---- -
oo00 -
i' -..MGLC2 w/A=
SM GLC
SMGLC2
o Equal weight
> I > -I I : .__I.-.. .1- -.__,--'-1
.-' o"" o o _o__0
0 o o -
0 *---
-- -SCB
.--.MGLC2 w/A=0
/MGLC
1'--
target detection performance of the radar in heterogeneous clutter environments; however,
inaccurate environmental knowledge, which is possible in practice due to environmental
changes or outdated information, can significantly degrade the radar's performance by
wasting adaptive array's DOF on false environmental clutter. Therefore, one of the
important problems in KA-STAP is to determine the degree of accuracy of the a prior
knowledge and how much emphasis one should place on it. To address this problem, in
this paper, we propose a maximum likelihood (\l I) based KA-STAP. The ML approach
is compared with the convex combination (CC) approach recently proposed in [48] using
numerically simulated data.
problem of interest. Section 4.3 discusses the methods for determination of the optimal
weight on the prior knowledge for KA-STAP. Section 4.4 presents numerical results
obtained by using simulated radar data and Section 4.5 provides the conclusions.
4.2 Problem Formulation
In KA-STAP, the prior knowledge usually consists of some information about the
clutter-and-noise covariance matrix R. Therefore, we assume that an initial guess of R,
let us -i Ro, is available. Ro can be obtained either by a previous scanning of the same
region or by a map-based study (see, e.g., [8, 10, 11, 12, 23]). This a priori Ro can then
be incorporated into STAP to improve its clutter suppression capability. As in [8], we
consider a "convex combination" of R and Ro, which corresponds to the direct use of prior
knowledge, as follows:
R caRo + (1 a)fR; a E(0, 1), (4-3)
where a is the weighting factor on the a priori knowledge, and the constraint on a is
imposed to guarantee that R > 0. This is also known as the "colored loadiin approach
[23]. However, in [23], the "(o !. ,I [ and "diagonal" loading levels were chosen manually.
Similarly, the authors of [15] considered equal weighting of R and R as a direct use of
prior knowledge: R = 0.5Ro + 0.5R (we refer to this as the "equal-,., i,!i1 approach
obtain an optimal estimate of R so as to minimize the MSE of R:
MSE= E{|R- RI2}.
We denote the optimal value of a obtained by the CC approach as acc.
Combining (4-3) and (4-11) and using the fact that R is an unbiased estimate of R
E{ |a(Ro- R) + (R- R) |2}
const + 2E{ Ro RI|2} 2aRe {tr [E{(R-
const + a2E{lRo RI|2} 2aE{|IR R||2},
R)(R -Ro)*}]
where tr denotes the trace, and Re stands for the real part. The minimization of (4-12)
with respect to a gives:
E {IIR- R1|2} E{IIR R112
a'" ceE = {-- R R2} .-
cc E{||IR-Ro||2} E{||IR-R||2}+ IIR-R 2
p E{i|R-R R2}.
(4 14)
To estimate co from the available data, we need an estimate of p. Note that
P Ef,
r, II ,
rm= N y y() (n),
rnm = E{y(n)y*(n)},
E{I|R- RI|2}
aO"""""'I
Figure 6-11. Temperature distribution for N = 50 and L = 128. (a): CA synthesized
constant modulus signals, and (b): CA synthesized signals with optimal R).
...~~~~..,
6-3 Differences between the beampatterns obtained from optimal R and the CA
synthesized waveforms under (a) PAR = 1, (b) PAR < 1.1, and (c) PAR < 2.. 100
6-4 MSE of the difference between R and R as a function of sample number L. 100
6-5 Beampattern matching design with each desired beam width of 200 and under
the uniform elemental power constraint. ................ ..... 101
6-6 Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and under the relaxed elemental power constraint. ........... ..102
to 200 and a -40 dB null at -300, under the relaxed elemental power constraint. 104
to 200 and a null from -550 to -450, under the relaxed elemental power con-
straint .. . . . . . . . . .. 105
6-9 Breast model .................. .................. .. 106
6-10 PAR values for CA synthesized waveforms with optimal R. . . .... 106
6-11 Temperature distribution for N = 50 and L = 128. .............. ..107
7-1 Beampattern matching design. The probing signals are synthesized for N = 10,
L = 256 and P = 1 by using CA. ............... ...... 119
7-2 Correlation levels versus p for the CA synthesized waveforms with N = 10, L
256 and P = 1 (R / I) ................ ........... .. 120
L = 256 and P = 10 by using CA ................ .... .. .. 121
256 and P = 10 (R / I) ................ .......... 122
512 and P = 1 (R / I) ................ .......... 124
L = 512 and P = 10 by using CA ............... ...... 125
7-8 Correlation levels versus p for the CA synthesized waveforms with N = 10, L =
7-9 Correlation levels versus p for the CA synthesized orthogonal waveforms with
N = 10, L = 256 and P = 10 (R I). .................. .... 126
3 --A
2.5- E-i-B
2 -.
1 ---------------------
-0.5 .
2 .5 -,
--B
1.5 -. ...... -
1 -__-__-..-- ------- -----=
when ao2 = a = o = 20 dB, and the a priori covariance matrix is given by (a)
Ro = Rol (see (5-42)), (b) Ro = Ro2 (see (5-43)), (c) Ro = Ro3 (see (5-44))
and (d) Ro = Ro4 (see (5-45)).
...,
[1] V. C. vannicola and J. A. Mineo, "Applications of knowledge-based systems to
surveillance," In Proceedings of the 1988 IEEE National Radar Conference, April
[2] V. C. vannicola, L. K. Slaski, and G. J. Genello, "Knowledge-based resource alloca-
tion for multifunction radars," In Proceedings of the 1993 SPIE Conference on Signal
and Data Processing of Small Targets, vol. 1954, April 1993.
[3] S. Haykin, "Radar vision," In Proceedings of the Second International Specialist
Seminar on the Design and Application of Parallel Digital Processors, April 1991.
[4] L. E. Brennan and I. S. Reed, "Theory of adaptive radar," IEEE Transactions on
Aerospace and Electronic S.,-1. mn vol. AES-9, pp. 237-252, March 1973.
[5] J. Ward, "Space-time adaptive processing for airborne radar," Technical Report 1015,
MIT Lincoln Laboratory, December 1994.
[6] R. Klemm, Principles of Space-Time Adaptive Processing. London, U.K.: IEE Press,
[7] J. R. Guerci, Space-Time Adaptive Processing for Radar. Norwood, MA: Artech
House, 2003.
[8] J. R. Guerci and E. J. Baranoski, "Knowledge-aided adaptive radar at DARPA,"
IEEE S.:g,.,l Processing 1.1.i; .:,.,. pp. 41-50, January 2006.
[9] I. S. Reed, J. D. Mallett, and L. E. Brennan, "Rapid convergence rate in adaptive
arrays," IEEE Trans. on Aerospace and Electronic Sl' 1"' vol. AES-10, pp. 853-863,
[10] D. Page and G. Owirka, "Knowledge-aided STAP processing for ground moving
target indication radar using multilook data," EURASIP Journal on Applied S.:,i.rl
Processing, vol. 2006, Article ID 7,!- ;: 2006.
[11] W. L. Melvin and G. A. Showman, "An approach to knowledge-aided covariance
estimation," IEEE Transactions on Aerospace and Electronic Sl 1 ii- vol. 42,
[12] C. T. Capraro, G. T. Capraro, I. Bradaric, D. D. Weiner, M. C. Wicks, and W. J.
Baldygo, I1l!,1. i i i i ig digital terrain data in knowledge-aided space-time adaptive
pro'. -in:. IEEE Transactions on Aerospace and Electronic S;qii. vol. 42,
[13] J. S. Bergin, C. M. Teixeira, P. M. Techau, and J. R. Guerci, Inpin clutter
mitigation performance using knowledge-aided space-time adaptive processing," IEEE
Transactions on Aerospace and Electronic S;ql. nm vol. 42, pp. 997-1009, July 2006.
as Ro for MGLC1 and MGLC2, respectively. (Note that the averaged estimate of A/ at is
shown in Figures 5-1(c) and 5-2(c).) We note that by enforcing R > 0, the so-obtained
AMGLC1) BMGLCI and CMGLCI can be negative, as shown in Figure 5-1. For Figures 5-1(a)
and 5-2(a), where the prior knowledge is accurately known, we see clearly that ambiguities
exist as N -> oc. Recall that R converges to R N -- o. For MGLC1, A = 1 B
and = 1 B for any B will result in R =R as N -> oc. MGLC2 is, however, more
restrictive. For MGLC2, A = 1 B and C = 1 B for any 0 < B < 1 will result in
R = R as N -- o. For all other cases of Figures 5-1 and 5-2, there is no ambiguity and
we observe that A and C approach 0 and B approaches 1 as N -- o, as desired. We
also observe from Figures 5-1(d) and 5-2(d) that A is close to 0 when inaccurate a priori
covariance matrix R04 is used as Ro. That is to ~-v, little or no emphasis is placed on the
a prior knowledge when it is unreliable.
Figures 5-3(a) 5-3(d) show the averaged array output SINR (in dB) versus the
snapshot number N when the a priori covariance matrix is given by Ro = Roi, i
1, 4, respectively. Note that we consider SINR because it is an important criterion in
many array processing applications, such as communications. As we can see from Figure
5-3(a), with accurate a priori knowledge, i.e., with Ro = Rol, MGLC2 significantly
outperforms MGLC2 with A = 0. MGLC1, however, is inferior to MGLC2 and will not be
considered further. Observe that with accurate knowledge of the interference, i.e., with
Ro = R02, both MGLCi and MGLC2 outperform MGLC2 with A = 0, as shown in Figure
5-3(b). Observe also that with only the DOA information on the first interference, i.e.,
with Ro = R03, slight improvements over MGLC2 with A = 0 can still be achieved by
using MGLC1 or MGLC2. It is interesting to note that when the a priori knowledge is
inaccurate, i.e., when R04 is used as Ro, the performance of MGLC2 is similar to that of
MGLC2 with A = 0, as shown in Figure 5-3(d). This is an appealing feature, indicating
that MGLC2 is robust to the error in the a priori knowledge. For comparison purposes,
we also consider equal weighting of Ro, R and I for the four cases. For example, one can
denote the mth columns of R and R, respectively, with ym(n) being the mth element of
y(n). It has been shown in [48] that E{ Ir, r, ||-} can be estimated as
N2 Iy(n)y(n) r. | (4 18)
It follows from (4-15) and (4-18) that
1M N
m= 1 n= 1
M N- t
m=l n=l
N N y(n) L', (n) 2 I 1 (4 19)
S N2 ||y(u)||4 [R[2. (4-20)
Using (4-20) and the unbiased estimate ||R Ro 2 for E{ R Ro 12} in (4-13) yields the
following estimate of acc:
ace = (4-21)
||R R0o 2
To ensure that the estimate of acc in (4-21) is less than 1, as required, we recommend
using min(1, &cc) in lieu of (4-21).
We refer the reader to [48] for the detailed derivation of CC. Note that the CC ap-
proach is fully automatic and has a simple closed form solution which is computationally
efficient.
4.3.2 Maximum Likelihood
In this subsection, we introduce an ML approach to determine an optimal value
of a by assuming that a in (4-3) is constant for all range bins (for the sake of reduced
computational complexity). We denote the optimal value of a obtained by the ML
approach as &ML.
Since Ro, R, and hence R in (4-3) as well as R vary with the range bin index i, in
this subsection, we will reinstate the dependence on the range bin index of these variables
for presentation's clarity. Specifically, let R(i) = aRo(i) + (1 a)R(i) be the estimate of
operation. Given a covariance matrix as the a priori knowledge, the problem becomes
that of determining a signal waveform matrix X whose covariance matrix is equal or
close to the given covariance matrix, and which also satisfies some practically motivated
constraints (such as constant-modulus or low peak-to-average-power ratio constraints). We
propose a computationally efficient cyclic optimization algorithm for the synthesis of such
an X. Moreover, in many radar imaging applications, the length (or sample support) of
the transmitted waveforms can be rather large, hence the reflected waveforms from near
and far ranges can overlap significantly, as shown in Figure 1-4. This in turn requires good
correlation property of the transmit waveforms. Therefore we also consider synthesizing an
X that has good auto- and cross-correlation properties.
Return pulse Return pulse
from near range from scene center
Return pulse
from far range
Fast time
Figure 1-4. Reflected pulses from different range bins can overlap significantly.
and consider the following MSE minimization problem for MGLC with the constraint
R > 0 enforced:
[0 o] TA-8
R(0) > 0. (5-18)
The above formulation is equivalent to
min MSE R(0)] s.t. R(0) > 0,
and it is a Semidefinite Program (SDP) that can be efficiently solved in polynomial time
using public domain software [52, 53].
For MCC, we only need to add the following additional constraint
to the MGLC formulation in (5-18). We use ul to denote a vector of l's of length 1. The
resulting problem is still a SDP.
In practice, 0o must be replaced by its estimate 0o, where
Oo A-1b,
with A and b being the estimates of A and b, respectively. Then (5-18) becomes:
s,0
0R >0o
R(o) > 0.
[ 0I
0-0o >0
u o = 1, u3 = [1 1 1]T,
o.2I ll N it 1 0.2l
200 400 600 800 1000 200 400 600 800 1000
0.4- 0.4-
Figure 3-2. The ratio in (3-41) as a function of range bin index, for four different values of
the perturbation level applied to R to obtain Ro: (a) of 0, (b) o = 0.01,
(c) 1a 0.1, (d) a? =0.5.
6.4.3 Waveform Diversity-Based Ultrasound Hyperthermia
In this final example, we consider an application of the waveform diversity-based
transmit beampattern design to the treatment of breast cancer via ultrasound hyperther-
mia. Of all women diagnosed with breast cancer, 21 have locally advanced disease and
even with ., i-iressive treatments, the risk of distant metastases remains high. Thermal
therapy provides a good treatment option for this type of cancer: the breast tumor is
heated, and the resulting heat distribution sensitizes tumor tissues to the anti-cancer
effects of ionizing radiation or chemotherapy [90, 91, 92]. Thermal therapy can also help
achieve targeted drug delivery.
A challenge in the local hyperthermia treatment of breast cancer is heating the
malignant tumors to a temperature above 43C for about thirty to sixty minutes, while
maintaining a normal temperature level in the surrounding healthy breast tissue region.
Ultrasound arrays have been recently used for hyperthermia treatment because they can
provide satisfactory penetration depths in the human tissue. Note that the elemental
power of an ultrasound array must be limited to avoid burning healthy tissue. As a result,
a large aperture array is needed to deliver sufficient energy for heating the tumor without
harming the healthy tissue. However, due to the short wavelength of the ultrasound, the
focal spots generated by a large ultrasound array are relatively small and therefore hun-
dreds of focal spots are required for complete tumor coverage, which results in excessively
long treatment times.
We have shown recently that flexible transmit beampattern design schemes can
provide a sufficiently large focal spot under a uniform elemental power constraint, which
can lead to more effective breast cancer therapies [84]. In the cited reference, the goal of
the transmit beampattern design was to focus the acoustic power onto the entire tumor
region while minimizing the peak power level in the surrounding L. i11 !: breast tissue
region, under a uniform elemental power constraint. The beampattern design problem is
In the literature, there are also several review papers on KA-STAP [8, 17, 18]. [8]
gives an overview of the KASSPER program at DARPA. The benefits of KA adpative
radar, the direct and indirect use of the a priori knowledge, the prewhitenning approach
for determining the degree of accuracy of the a priori knowledge (we will discuss this
in detail in ('!, lpter 4), and the look-ahead radar scheduling approach for real-time
KA-STAP are discussed.
In [18], and also in [17], the authors first discuss individually how to deal with
heterogeneous clutter training data selection and how to mitigate antenna array effects
such as mutual coupling and channel mismatch, with the aid of a priori knowledge. They
also introduce the use of hybrid processing for heterogeneous range bins (the hybrid
algorithm is a cascade of the direct data domain (D3) algorithm and the joint domain
localization (JDL) training [31, 32, 33].) Finally, these separate STAP issues are combined
into a preliminary, but comprehensive and practical KA-STAP approach.
2.3 Contributions of Our Work
The direct use of the a priori knowledge usually consists of constructing the a priori
clutter covariance matrix Ro from the available environmental knowledge [13, 11, 10, 15].
However, one of the important problems in KA-STAP is to determine the degree of
accuracy of the a priori knowledge and the optimal emphasis that should be placed on it.
That is, to determine the optimal weight on Ro. To our knowledge, there are no efficient
algorithms in the literature for solving this problem. As we mentioned before, in the
colored loading approach [23], the colored and diagonal loading levels are chosen manually.
And the pli, vI!i, iii, approach proposed in [8] -,--, --1; to optimize the weight on Ro
so as to maximally whiten the observed interference data. However, as we will show in
C'!i lpter 4, this approach may not work properly. In this dissertation, we propose several
efficient algorithms to determine the optimal weight on the a priori knowledge. In our
approaches, the a priori knowledge is used in both direct and indirect v--,v to enhance the
An interesting current research topic is the optimal synthesis of the transmitted
waveforms. For MIMO radar with widely separated antennas, waveform designs without
any practical constraint (such as the constant-modulus constraint) have been considered
in [73]. For MIMO systems with colocated antennas, on the other hand, the recently
proposed techniques for transmit beampattern design or for enhanced target parameter
estimation and imaging have focused on the optimization of the covariance matrix R of
the waveforms [61, 65, 67, 72, 75, 76, 79, 81]. For example, in a waveform diversity-based
ultrasound system, R can be designed to achieve a beampattern that is suitable for the
hyperthermia treatment of breast cancer [84]. Now, instead of designing R, as in the
cited references, we might think of designing directly the probing signals by optimizing
a given performance measure with respect to the matrix X of the signal waveforms.
However, compared with optimizing the same performance measure with respect to the
covariance matrix R of the transmitted waveforms, optimizing directly with respect
to X is a more complicated problem. This is so because X has more unknowns than
R and the dependence of various performance measures on X is more intricate than
the dependence on R (as R is a quadratic function of X). In effect, there are several
recent methods, as mentioned above, that can be used to efficiently compute an optimal
covariance matrix R, with respect to several performance metrics; yet the same cannot be
said about determining an optimal signal waveform matrix X, which is the ultimate .I..il of
the designing exercise. Furthermore, in some cases, the desired covariance matrix is given
(e.g., a scaled identity matrix), and therefore there is no optimization with respect to R
involved (directly or indirectly).
In this paper, we consider the synthesis of the signal waveform matrix X for diversity-
based flexible transmit beampattern design. With R obtained in a previous (optimization)
stage, our problem is to determine a signal waveform matrix X whose covariance matrix is
equal or close to R, and which also satisfies some practically motivated constraints (such
as constant-modulus or low peak-to-average-power ratio (PAR) constraints). We present a
Figure 3-1 shows the GLC estimates of ao and 0o obtained for different range bins
and for four values of ao (see (3-39)). As expected, )o increases and do decreases as ao
increases. The smaller variability of ao and )0 observed in the figure for the larger values
of ao can be explained by the fact that these estimates approach 0 and, respectively, 1 as
f increases. Figure 3-2 dip-~'i the estimation error ratio
min1 Ilt R12 (3
min.2 IH2R- R112
for the same cases as in Figure 3-1. Note that we chose to express the estimation error
ratio as in (3-41) because the AMF detector in (3-40) is invariant to the scaling of the
estimated covariance matrix. As can be seen from the figure, the optimally scaled R can
be a significantly more accurate estimate of R than the optimally scaled R when ao takes
on small values (i.e., Ro is not too far from R) -see Figures 3-2(a) and 3-2(b); however,
interestingly enough, the scaled R is more accurate than the scaled R even for a relatively
large perturbation level -see Figure 3-2(d), corresponding to af = 0.5, where (3-41)
fluctuates around 0.9.
Finally, Figure 3-3 shows the ROC curves (we remind the reader that target detection
is our main goal here) corresponding to the "ideal detector" (i.e., (3-40) with R = R), the
"initial detector" (i.e., (3-40) with R = Ro), the sample matrix inversion (SMI) detector
(i.e., (3-40) with R equal to the sample covariance matrix), and the GLC detector. The
SMI detector (with N = 375) is first used as initial detector to eliminate range bins
with strong targets from the secondary data, and then the sample covariance matrix is
computed, for each range bin, from the so-obtained secondary data corresponding to the
375 range bins that are nearest to the one of interest. The GLC detector outperforms
significantly the SMI detector and the initial detector for the larger values of ao. Observe
also that, as at decreases, the performance of the GLC detector approaches that of the
ideal detector, as one would desire -note that this is done in an entirely data-adaptive
manner, with the GLC detector using no direct information about the accuracy of Ro.
which has the form required by the algorithm of [86, 87]. Therefore, we can solve (6-34)
using the said algorithm and then compute s = cx with c given by (6-32).
The alternative discussed in Sec. 6.3 is to constrain I||s|2 = I|z|2 (which is the case
when we choose = R,, in (6-17)). The use of this constraint is logical if we want
to match R,, exactly (for strict transmission power control, for example). However, if
matching R, exactly is not a necessary condition, then a smaller matching error between
s and z is obtained using (6-32) and (6-34).
densities and temperature distributions inside the breast model when the synthesized
waveforms are transmitted via the acoustic transducers.
Figure 6-10 shows the actual PAR values of the CA -;;,,i/,. -.. ./ waveforms with
optimal R. Figure 6-11 shows the temperature distributions within the breast model, with
Figure 6-11(a) corresponding to the CA -;;,'/,1, -..,1 ; ,.ri. [.., ,, with PAR = 1 and Figure
6-11(b) to the CA -;u,,'/l,. ..: ,1 i.forms with optimal R. As shown in Figures 6-11(a)
and 6-11(b), by transmitting either of the synthesized diversity-based waveforms, the
entire tumor region is heated to a temperature equal to or greater than 43C, while the
temperature of the surrounding normal tissues is below 40 C. In contrast with this, when
a phased-array is used for transmission and the d. 1 i- ,il-sum technique is employ, .1 to
ensure that the energy is focused on the tumor center, the temperature distribution is far
from satisfactory (see [84]).
We have considered the problem of waveform synthesis for diversity-based flexible
transmit beampattern designs. Optimization of a performance metric dir './/l; with respect
to the signal matrix can lead to an intractable problem even under a relatively simple
low PAR constraint. For this reason, we proposed the following strategy: first optimize
the performance metric of interest with respect to the signal covariance matrix R; and
then synthesize a signal waveform matrix that, under the low PAR constraint, realizes
(at least approximately) the optimal covariance matrix derived in the first step. We have
presented a cyclic optimization algorithm for the synthesis of a signal waveform matrix
to (approximately) realize a given covariance matrix R under the constant-modulus
constraint or the low PAR constraint. The output of the cyclic algorithm can be used to
obtain either a waveform matrix whose covariance matrix is exactly equal to R but whose
PAR is slightly larger than the imposed value, or a waveform matrix with the imposed
PAR but whose covariance matrix may differ slightly from R -the type of application
will dictate which one of these two kinds of waveforms will be more useful. A number of
fast maximum likelihood (FML) method [22], reiterative censoring (RC), adaptive power
residue (APR) metric, concurrent block processing (CBP), two-weight method (TWM)
and adaptive coherence estimate (ACE) metric [28]. The a priori clutter covariance
matrix Ro obtained by KACE is combined with the sample covariance matrix R as
Ro + R to get a new estimate of the covariance matrix, which belongs to the direct use of
the a priori knowledge.
2.2 Indirect Use
Now, we consider the indirect use of the a priori knowledge in KA-STAP. A KA
spectral domain (or range Doppler domain) approach to estimate the clutter-and-noise
covariance matrix is discussed in [14]. In this paper, the a priori knowledge is used
indirectly for training data selection. The knowledge sources are first used to model the
observed clutter scene (i.e., to form RCS map versus range and Doppler bins). This a
prior knowledge of the clutter scene is then used to identify the homogeneous scattering
regions. The clutter power is then estimated from the collected radar data of these
homogeneous segments and the clutter-and-noise covariance matrix is generated from the
so-obtained clutter power file. The effects of inaccurate a priori knowledge, such as IC'\
channel calibration error, etc., on the proposed approach are also considered in this paper.
In [29], a KA radar detector is discussed, which is composed of three components: the
map based selector to eliminate range bins containing strong stationary scatterers from
the training data, which corresponds to the indirect use of the a priori knowledge; a data-
adaptive selection algorithm for removing dynamic outliers or targets; and an adaptive
detector for target detection. The performance of the proposed algorithm is evaluated
based on both simulated and measured data.
In [30], the authors give a detailed description on the numerical implementation of the
digital terrain data in airborne KA-STAP. The registration and corrections of the data are
discussed. Numerical examples show that the performance of STAP can be improved by
properly exploiting the a priori knowledge of the terrain probed by the radar.
To make use of a priori knowledge and at the same time to ensure that the estimate
of R is positive semidefinite, we consider a modified general linear combination (\ GLC)
of the a priori covariance matrix Ro, the sample covariance matrix R, and the identity
matrix I to get an enhanced estimate, let us call it R, of the true array covariance matrix
R = ARo + BR + CI. (5-3)
The combination weights A, B and C in (5-3) should be chosen to guarantee that R > 0.
We also consider the following modified convex combination (\ICC) of the three terms
(i.e., Ro, R and I):
R = ARo + BR + CI; A + B + C = 1. (5-4)
Again, constraints should be enforced to ensure that R obtained by (5-4) satisfies R > 0.
We refer to the use of (5-3) and (5-4) (with optimized A, B and C, see below) to obtain
an estimate of R as the MGLC and MCC approaches, respectively. With B normalized to
1, (5-3) is also known as the "colored loadiin approach proposed in [23], with the colored
loading level given by A and the diagonal loading level by C. However, in [23], the colored
and diagonal loading levels were chosen only manually.
The first goal of this paper is to obtain optimal estimates of the weighting factors A,
B and C that minimize the mean-squared error (\ Sl' ) of R:
MSE= E{|R- RI|2}, (5-5)
for both (5-3) and (5-4); the second goal of this paper is then to use R (with optimized
A, B and C) as a new estimate of R, in lieu of R, in SCB to improve the array output
SINR.
L 256 and P 1 (R / I).
ON USING A PRIORI KNOWLEDGE IN STAP
In standard STAP, the clutter-and-noise covariance matrix, let us call it R, is
estimated from secondary data (presumed to be target free), let us J- {y(n)} 1, by
means of the well-known formula (see, e.g., [5, 6, 7, 8, 16]):
R =y(n)y*(n), (3-1)
where (.)* denotes the conjugate transpose. However, frequently the dimension of R
(denoted by M in what follows) is larger than, or at best comparable with, N. The result
is that R is, more often than not, a poor estimate of R (particularly so when M > N, see
Section 3.3 of this note for such a case).
To estimate R more accurately, we can try to make use of prior knowledge on the
terrain probed by the radar, acquired either from a previous scanning or from a map-based
study. In knowledge-aided STAP (KA-STAP), an initial guess of R, let us ,- Ro, is
obtained in this way (see, e.g., [8, 16]). The problem is then to "(I 1iidoii, R and Ro into
an estimate of R, preferably much more accurate than both R and Ro.
In this note, we will consider a "linear combination" of R and Ro (see, e.g., [34, 11,
13]):
R = Ro + 3R; a> 0 and 3 > 0. (3-2)
Because a and 3 are constrained to be positive, and as typically Ro > 0 (positive definite),
we have that R > 0, which is a desirable feature. We will also consider the following
"convex combination":
R cRo + (1 a)R; a (0,1). (3-3)
The constraint in (3-3) on a is imposed, once again, to guarantee that R > 0. In general,
there is no obvious reason why 3 in (3-2) should be constrained to equal 1 a, like in
(3-3); however, the convex combination, (3-3), is a more parsimonious description of R
where I I denotes the determinant of a matrix. Our problem is that of finding the optimal
&ML that maximizes the log-likelihood function in (4-24). A closed-form solution to this
problem is not likely to exist, but the optimal &ML can be found by a search method (see,
e.g., [49]). With the optimal &ML determined by the ML approach, R(i) in (4-23) can be
readily obtained and used for target detection.
Note that the main computational part of ML for KA-STAP is due to the I | and
(*)-1 operations for R(i) E CMxM in (4-24), which are required for each i and each
candidate a. We present in the Appendix a fast algorithm that can be used to reduce the
computational complexity significantly for KA-STAP.
In ir ini, practical applications, when the computational resources and the available
secondary samples are limited, one often considers a more practical reduced-dimension
(RD) KA-STAP. One common approach to reduce the dimensionality is to apply a trans-
form matrix, T E CMXD, to the original snapshot of ROI [5]. Hence, after transformation,
the RD space becomes a D-dimensional subspace. Since D is usually much smaller than
the total number of DOFs, i.e., D
after the transformation. In the numerical examples, we consider a multi-bin element-
space post-Doppler STAP to reduce the dimension. This RD technique preserves the
number of spatial DOFs (equals to S), while reduces the number of the temporal DOFs
from P to Q (Q < P). Consequently, the dimensionality (or the DOFs) of the RD space is
given by D = QS. Focusing on the adaptive processing for a specific Doppler bin, v the
pth Doppler bin, we can write the transform matrix T as:
T = Fp Is, (4-25)
where Fp is a P x Q matrix whose columns are discrete Fourier transform filters at Q
.ilIi i.:ent Doppler frequencies with the pth Doppler bin as the central frequency bin, and
0 denotes the Kronecker matrix product. The resulting snapshot for the ith range bin,
One way to obtain A and b is to replace p and R in A and b by p and R, where p is
an estimate of p, which can be obtained as [51]:
p 2 jE y()4 R2. (5-23)
We refer to [51] for a detailed derivation of p.
As -i-.-i- -i. I in [34] (and also in [54, 36]), to estimate 0 and R consistently, we prefer
to use an alternative estimation scheme to obtain A and b as follows. Because
p + ||R||2 E{R RR 2} + RI2 E {|RI2},
we can estimate p + I||R|2 by I||R|2, which is an unbiased estimate that is smaller than
the previously s-l_-_. -1. 1 p + I|R||2; we can estimate l|R||2 by I|R||2 p, again, a smaller
estimate. We also replace R by R in tr(R) and tr(R*Ro). We refer to the resulting
SDP problem in (5-22) for MGLC as MGLC1, and similarly, (5-22) with the additional
constraint (5-20) for MCC as MCC1.
Alternatively, we can enforce in (5-22) A > 0, B > 0, and C > 0. Then the constraint
R(0) > 0 is trivially satisfied and (5-22) becomes a quadratic program (QP):
min ( o A( o
s.t. i > 0, i 1,2,3. (5-24)
We denote this formulation for MGLC as MGLC2. Similarly, adding (5-20) as an addi-
tional constraint to (5-24) yields a QP problem for MCC, which we denote as MCC2.
5.3.2 Extensions
In the previous subsection, we considered a linear combination of three terms to get
an enhanced estimate of R. The previous approach can be easily extended to a linear
combination of an arbitrary number of positive semidefinite matrices.
Let {R }')I denote S a priori covariance matrices representing knowledge about S
dominant sources or strong interference. We consider the following linear combination of
LITERATURE REVIEW FOR KA-STAP
In this chapter, we briefly review the KA-STAP literature. DARPA's recent
KASSPER program aims at exploiting environmental knowledge to enhance the de-
tection performance of STAP. The a priori knowledge can be used either in a direct way
or in an indirect way.
2.1 Direct Use
In this section, we first consider the direct use of the a priori knowledge in KA-
STAP. In [23], the authors present a framework for incorporating the low-,I' .:1/,/ a prior
knowledge directly in space-time adaptive beamformer. The linearly-constrained minimum
variance (LC'\ V) space-time beamformer proposed in this paper includes an additional
a prior knowledge-based constraint to force nulls at the clutter locations. The authors
showed that the knowledge-aided constraint resulted in "(o .1. i. loading of the adaptive
covariance matrix estimate, which can be expressed as
RCL = R + Ro + /3ld, (2-1)
where RCL is the colored loading covariance matrix and f3 and 3d, respectively, are the
colored and diagonal loading levels. However, in this colored loading approach, the colored
and diagonal loading levels are chosen manually. A reduced degree-of-freedom (DOF)
colored loading approach is considered in [24]. In [13], also by the same authors, applica-
tion of the colored loading approach to both full DOF and reduced DOF beamformer is
considered and the performance is demonstrated by using I,.:il,-I/7, 1. :/; a priori knowledge
radar data.
In [25], the authors consider a combination of the synthetic aperture radar (SAR)
processing and the ground moving target indication (GMTI) processing. SAR is used to
detect stationary targets with long CPIs, whereas GMTI is used to detect moving-targets
with a short CPI. In [25], discrete scatterers are identified by applying a (low) threshold
XUMIN ZHU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
wD was a slightly perturbed version of the following M x 1 "nominal" steering vector:
a(w), D) = a(w) d-(Un), (3-36)
where M = PS = 352, 0 denotes the Kronecker matrix product,
a( ) 1 ejw ... j(s-1)O (3-37)
a(wD) C JWD ... j(P ) -) (3-38)
with (.)T denoting the transpose. The spatial frequency w, has a fixed value corre-
sponding to steering towards 750 relative to the array normal, but the Doppler fre-
quency wD is only known to belong to the following set of 64 possible Doppler values
{0, 27/64,. 1267/64}. The number of range bins considered is equal to 1,000; of these
784 range bins contain no target, whereas the rest of the 216 bins contain at least one
target (at certain values of wD). Furthermore, the clutter-and-noise covariance matrix, R,
is specified for each range bin (R varies quite a bit with the range bin index, but we omit
its dependence on the bin index for notational simplicity).
The final goal of this exercise is to detect the range and Doppler bins comprising
targets. This, of course, has to be done without using the detailed knowledge of the
scenario that was mentioned above. We will only use the noisy space-time snapshots
corresponding to the 1,000 range bins, the nominal steering vectors, and a perturbed
version of R (that mimics the uncertainty in the a priori knowledge that alv--,i exists in
practice). The said perturbed clutter-and-noise matrix, Ro, is generated in the following
way:
Ro = R (tt*, (3-39)
where ( denotes the Hadamard matrix product, and t is a vector of i.i.d. Gaussian ran-
dom variables with mean 1 and variance of. Note that the perturbed matrices generated
in this way remain positive definite (we have also tried two other forms of perturbation of
Alternatively, we can use the unbiased estimate I|R Ro012 for E{IIR Rol2} in (3 7),
and the resulting estimate of ao:
ao" (3-20)
|R- Ro0l2
Unlike (3-19), the estimate of ao in (3-20) is not guaranteed to be less than 1, as required;
thus we recommend using min(1, do") in lieu of (3-20).
3.2.2 General Linear Combination (GLC)
For (3-2), we have that:
E{ ||R-R|2}I E{|jRo (1 -3)R+3 (R R)I|2}
lRo (1 )RI2 + f 2E{lR- R112}
2 11Ro112 2a(1 3)Re [tr(R*R)] + (1 3)211R 2
+/2E{i|R R||2}, (3-21)
where Re (the real part) can be omitted because tr(R*R) > 0. The unconstrained
minimization of (3-21) with respect to a, for fixed 3, gives:
(1 /3o) tr(R R)
ao (322)
where so is the minimizer of the function that is obtained by inserting (3-22) (with 9o
replaced by 3) in (3-21), viz.:
(3 1)2 [IIRII211Roll2 tr2(R;R)]
( t)2 [R o 2 2E{IR R1122}. (3 23)
IIR oll2
The unconstrained minimization of (3-23) with respect to f3 yields:
A0o (3-24)
P+7
where p has been defined previously in (3-15) and
IIRII2R112- t2(RR)
S= (3-25)
JIRo 112
{R }) 1, Rand I:
R = A(8)R ) + BR + CI, (5-25)
where {A(")}s are the weights applied to the a priori covariance matrices {R1 }s1.
Constraints again need to be imposed to ensure that R > 0. We observe that (5-3) is a
special case of (5-25) with S = 1.
Matrix A in (5-14) for this extended case becomes:
A = tr(R I s) ... il tr(S) tr(R)*) (5-26)
tr(R)*R) .. tr(Rf)) IRl+p tr(R*)
tr(R *) ... tr(R 5s)*) tr(R*) ||I||2
Similarly, b in (5-12) now has the form:
b [tr(R*R')) tr(R*Rs) |R||2 tr(R*) (5-27)
We redefine 0 as
0 [A(') .. A(s) B C]T. (5 28)
With A and b given in (5-26) and (5-27), the corresponding A and b can be
obtained by using the same scheme as before. Consequently, an estimate of 0 in (5-28)
can be obtained by solving (5-22) for extended MGLC1, and (5-24) for extended MGLC2
(with the constraints being replaced by 0i > 0, i = 1, 2, ... S, S + 1,S + 2). By solving
(5-22) and (5-24) with the following additional constraint:
u+20 (5-29)
we get extended MCC1 and MCC2, respectively.
0MGLCi AMGLCi SLCi MLCCi CMGLCi Oi 1t, 2, (5-30)
An important advantage of the above algorithm is that Step 2 has a closed-form solution.
This solution can be derived in a number of v--,v (see, e.g., [88, 89]). A simple derivation
of it runs as follows. For given X, we have that
X VUR/ 2 const 2Re tr [ R1/2X*U] (6-4)
LR1/2X* UEU* (6-5)
denote the singular value decomposition (SVD) of VR1/2X*, where U is N x N, E is
N x N, and U is L x N. Then
Re {tr [LR1/2X*U] }
Re {tr [Lu*UUE1 (6-6)
= Re { [U*UU] } ,,n. (6-7)
(u*uu) (u*u*u) *uu*
< U*U
I, (6-8)
it follows that
Re2 uu _}< L[u*UUI
< [(u*uu) (u*u*u)1]
< 1, (6-9)
This can be obtained either by previous radar probings or by a map-based study. In this
dissertation, assuming that Ro is full rank, we consider two shrinkage methods, called
the general linear combination (GLC) and the convex combination (CC), as well as a
maximum likelihood-based approach to obtain an enhanced estimate of true clutter-and-
noise covariance matrix R based on the a priori clutter covariance matrix Ro and the
sample covariance matrix R. Given the new estimate, it then can be used in the STAP
beamformer for better target detection.
1.2 Knowledge-Aided Adaptive Beamforming
We also consider KA signal processing algorithms in a general adaptive beamforming
application. It is well-known that in array pro'. --iir given the true array covariance
matrix R and the steering vector for the signal of interest (SOI), the standard Capon
beamformer (SCB) [19] can be used to maximize the array output signal-to-interference-
plus-noise ratio (SINR) adaptively. However, when the available snapshot number is
comparable with or even smaller than the sensor number, the sample covariance matrix
R is a poor estimate of the true covariance matrix R. To estimate R more accurately,
we can make use of prior environmental knowledge, which again is manifested as knowing
an a prior covariance matrix Ro. In practice, Ro usually represents prior knowledge on
dominant sources or interference. Unlike in some literature, for example, see [20, 21, 22],
where the noise power level is treated as prior knowledge, we now consider a more
practical case where the noise power level is assumed to be unknown, and thus cannot
be included into Ro, hence Ro is often rank deficient. In the dissertation, we consider
both modified general linear combinations (I GLC) and modified convex combinations
(\!CC) of the a priori covariance matrix Ro, the sample covariance matrix R, and an
identity matrix I to get an enhanced estimate of R, denoted as R. MGLC and MCC,
respectively, are the modifications of the GLC and CC methods. Both MGLC and MCC
synthesized for N = 10 and L = 256 by using CA under (a) PAR = 1
(resulting in 6 = -29.7891 dB), (b) PAR < 1.1 (resulting in 6 = -41.7237 dB),
and (c) PAR < 2 (resulting in 6 = -119.5251 dB).
Index of Transmit Antenna
Figure 6-2. PAR values for CA synthesized waveforms with optimal R and for colorized
Hadamard code.
-e-CA (PAR = 1): Optimal R
- -CA (PAR< 1.1): Optimal R
CA (PAR< 2): Optimal R
-Colored Hadamard
tit I
The concept of KA radar was first proposed by Vannicola in [1, 2] and Haykin in [3].
In KA radar, some a priori knowledge such as radar parameters and/or information about
the environment is available. This a priori knowledge can then be incorporated into signal
processing algorithms to improve the performance of the radar. In this dissertation, we
study several KA signal processing applications: we first consider KA space-time adpative
processing (KA-STAP) algorithms in airborne radar for wide area surveillance; then we
discuss KA adaptive beamforming for array applications where knowledge on dominant
sources or interference is given; finally, we introduce KA waveform synthesis for waveform
diversity-based systems (such as MIMO radar) where the a priori knowledge is manifested
as knowing a covariance matrix of the waveforms.
1.1 Knowledge-Aided Space-Time Adaptive Processing
STAP is widely used in ground moving target indication (GMTI) radar to detect
moving targets in the presence of severe interference, such as clutter, jamming and noise
[4, 5, 6]. Figure 1-1 shows a typical interference environment seen by the airborne radar
for a single range bin. The jamming is located at one angle and spread over all Doppler
frequencies. And the clutter lies on the diagonal "clutter ri~, (For a side-looking
airborne radar and small crab angle, it is known that the clutter Doppler frequency
depends linearly on the sinusoidal value of the azimuth angle.) In GMTI radar, the
velocity difference between moving targets and ground clutter is exploited (by using the
Doppler effect) for target detection.
A general block diagram for a space-time adaptive processor is shown in Figure 1-2.
Assume that the system has S receive antennas and P pulses. A space-time processor can
be viewed as an adaptive linear combiner or filter that combines all the snapshots from
range bin of interest. It is well-known that the optimal weight vector, w E CMX1, with
M = SP being the degrees of freedom (DOFs) of STAP, used to maximize the output
SINR values can be obtained by using (5-38) with the wo in (5-38) replaced by the
MGLC and MCC weight vectors.
In this section, we present several numerical examples comparing the performance of
SCB and MGLC. The performance is also compared with that obtained by setting A's to
zero in MGLC, resulting in a diagonal loading approach with the diagonal loading level
computed fully automatically. The performance of MCC was inferior to that of MGLC in
all of our examples and hence only the MGLC results are presented hereafter. We consider
a uniform linear array (ULA) with M = 10 sensors and half-wavelength spacing between
.,.i i:ent elements. Assume a spatially white Gaussian noise whose covariance matrix Q
is given by 10I. We assume that the direction-of-arrival (DOA) of the SOI relative to
the array normal is 0o = 0 and that there are K = 2 interference whose DOAs are
01 -40, 02 = 200. The power of the SOI is 20 dB, i.e., a2 20 dB. The powers of the
two interference, i.e., a, and aoj, will be specified later on. Also, we assume knowledge of
the steering vector ao of the SOI. In our simulations, averaged SINR values obtained from
200 Monte-Carlo trials are given. Note that unlike SCB, MGLC allows N to be less than
the number of sensors M. We use N = 4, 6, 8 for the N < M case.
Obviously, the performance of MGLC depends on the degree of accuracy of the a
prior knowledge. To investigate the effects of the accuracy of Ro on the performance of
MGLC, we consider the following six cases (we use Roi to denote the a priori covariance
matrix for the ith case, i = 1, 6):
(i). Accurate a priori knowledge, i.e.,
Rol R -Q aaoa + aka( (5-42)
(ii). Accurate a priori knowledge of the interference, i.e.,
R02 Jkakak. (5-43)
solution X to (6-3) obtained via CA as the transmitted signal waveform matrix, then X
will satisfy the PAR constraints, but will realize R only approximately. We refer to the
so-synthesized waveforms as the CA -;,',.'l,. -.., waveforms with PAR < p (abbreviated as
PAR < p).
In the following examples, the transmit array is assumed to be a uniform linear array
(ULA) comprising N = 10 sensors with half-wavelength inter-element spacing. The sample
number L is set equal to 256. The uniform elemental power constraint with C = 1 is used
for the design of R. For Q, we choose a mesh grid size of 0.1. Finally, the CA algorithm
is initialized using the U described in Step 0.
In the first example, the desired beampattern has one wide main-beam centered
at 0 with a width of 600. The weighting factor i,-, in (6-22) is set to 1 and ", is set to
0. Figures 6-1(a), 6-1(b), and 6-1(c) show the beampatterns using the CA synthesized
waveforms under the constraints of PAR = 1 (constant-modulus), PAR < 1.1, and
PAR < 2, respectively. For comparison purposes, we also show the desired beampattern
0(0) scaled by the optimal value of a. Note that the beampattern obtained using the
CA synthesized waveforms is close to the desired one even under the constant-modulus
constraint.
We also note from Figure 6-1 that the beampatterns obtained using the CA -;;,ill,.-
sized waveforms with optimal R are slightly different from those obtained using the CA
-,.Ii, ., .: I, ,,:, fi.,n' with PAR < p. Let
R X*X (6-23)
be the sample covariance matrix corresponding to the CA -;;,',/ /, -..,/ n, forms with PAR
< p. Let
6 R- R (6-24)
denote the norm of the difference between R and R. Then we have 6 = -29.7891 dB,
-41.7237 dB, and -119.5251 dB for Figures 6-1(a), 6-1(b), and 6-1(c), respectively. As
- 2 0 ... . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
1.-10
-20 -
Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and under the relaxed elemental power constraint. The probing signals
are synthesized for N = 10 and L = 256 by using CA under (a) PAR = 1 and
(b) PAR < 1.1.
.- . . . . . .
.- . . . . .
cyclic optimization algorithm for the synthesis of such an X. We also investigate how the
synthesized waveforms and the corresponding transmit beampattern design depend on the
enforced practical constraints. Several numerical examples are provided to demonstrate
the effectiveness of the proposed methodology.
Notation. Vectors are denoted by boldface lowercase letters and matrices by
boldface uppercase letters. The nth component of a vector x is written as x(n). The nth
diagonal element of a matrix R is written as R,,,. A Hermitian square root of a matrix
R is denoted as R1/2. We use ()T to denote the transpose, and (.)* for the conjugate
transpose. The Frobenius norm is denoted as || ||. The real part of a complex-valued
vector or matrix is denoted as Re(-).
6.2 Formulation of the Signal Synthesis Problem
Let the columns of X e CLxN be the transmitted waveforms, where N is the number
of the transmitters, and L denotes the number of samples in each waveform. Let
be the (sample) covariance matrix of the transmitted waveforms. We assume that L > N
(typically L > N). Note that X has 2NL real-valued unknowns, which is usually a much
larger number than the number of unknowns in R, viz. N2
The class of (unconstrained) signal waveform matrices X that realize a given covari-
ance matrix R is given by
X* R1/2U*, (62)
where U* is an arbitrary semi-unitary N x L matrix (U*U = I). Besides realizing (at least
approximately) R, the signal waveform matrix must also satisfy a number of practical
constraints. Let C denote the set of signal matrices X that satisfy these constraints. Then
a possible mathematical formulation of the problem of .;,,.l,,. .. ..,
matrix X is as follows:
min X- LUR12 (63)
* /
C o m b i n a t i o n o f f{ n ( t ) } 1 2 P h
N(T) MIMO Receive Array N ( Receive Phased-Array
MIMO Transmit Array Transmit Phased-Array
Figure 1-3. (a) MIMO radar. (b) Phased array.
Given R obtained from MGLC or MCC, it can be used instead of R in standard Capon
beamformer for KA adaptive beamforming.
1.3 Knowledge-Aided Waveform Synthesis
In this dissertation, we study KA waveform synthesis for MIMO radar as well. The
a prior knowledge in this case is manifested as knowing a covariance matrix of the
waveforms. This can be obtained in a previous optimization stage (for example, in the
stage of transmit beampattern design as we will discuss later) or simply pre-specified.
Transmit beampattern design is a critically important task in many fields including
defense and homeland security as well as biomedical applications. Unlike a standard
phased-array, which transmits scaled versions of a single waveform, a MIMO radar
system transmits multiple different waveforms that can be chosen at will (see Figure
1-3). Flexible transmit beampattern designs for MIMO radar can then be achieved by
properly choosing how the transmit waveforms are correlated with one another. Recently
proposed techniques for waveform diversity-based transmit beampattern design have
focused on the optimization of the covariance matrix of the waveforms, as optimizing a
performance metric directly with respect to the waveform matrix is a more complicated
Xumin Zhu received the B.Eng. degree in electrical engineering and information
science from University of Science and Technology of C('!i Hefei, C'ii i, in 2004. She
is expected to receive her Ph.D degree with the Department of Electrical and Computer
Engineering, University of Florida, Gainesville in 2008. Her research interests include
statistical and array processing, space-time adaptive processing, and multiple-input-
multiple-output radar.
for the ith range bin as
tr [o(i)a(u, D (4 31)
tr [R(i)]
where wD0 is the Doppler frequency bin in which the test targets are inserted. Hence the
average SCNR over I = 1000 range bins can be calculated as
1 tr [1o(i)a(, wD0)a* ( ) (4 32)
I 1 tr [R(i)]
In our numerical illustrations, we choose the training sample number as N = 50,
which is much less than the full DOFs, M = 352, of the simulated radar system. Hence
R(i) is severely rank deficient. We consider a multi-bin post-Doppler STAP using three
.,.i ,i:ent adaptive Doppler bins (i.e., Q = 3), which reduces the dimensionality of the
system to D = 33, and the resulting sample covariance matrix now has full rank (N > D).
Figure 4-1 di- p-, the log-likelihood function in (4-24) as a function of a for four
values of of. The optimal weighting factors obtained by the ML approach for the four
different perturbation levels are listed in Table 4-1. As expected, &ML decreases as of
increases. Observe that aML = 0.8 when of = 0, and &ML takes on small values at high
perturbation levels, as desired. Similarly to the CC approach, the ML approach is also
fully automatic.
Table 4-1. ML estimates of a for different values of of.
of 0 0.01 0.1 0.5
&ML 0.80 0.35 0.08 0.014
Figure 4-2 shows the CC estimate &cc as a function of the range bin index, and the
ML estimate &ML, for the four values of of It is clear that &cc decreases as af increases.
The average values of &cc's over the 1000 range bins, denoted as acc, for the different
perturbation levels are given in Table 4-2.
Table 4-2. Average values of the CC estimates of a for different values of of.
acc 0.343 0.077 0.004 0.0002
An important advantage of the above algorithm is that Step 2 has a closed-form
solution. A simple derivation of it runs as follows. For given X (X can be constructed
from X using (7-10)), we have that
X L- vUR12 2 const- 2Re tr [LR1/2X*U} (7-13)
Y = -LR1/2 (7 14)
where Re(.) denotes the real part of a complex-valued vector or matrix. Let
v/LR1/2X* UUJ* (7-15)
denote the singular value decomposition (SVD) of R1/2X*, where U is NP x NP, E is
NP x NP, and U is (P + L 1) x NP. Then the solution to the minimization problem in
Step 2 is given by [88, 89]:
U = UU*. (716)
The solution to the problem in Step 1 naturally depends on the constraint set C.
Note that the constraints in (7-1) and (7-4) are imposed on X in a column-wise manner.
Consequently, the solution to Step 1 is obtained by dealing with the columns of X in a one
by one fashion. Under the constant-modulus constraint in (7-1), Step 1 of the algorithm
also has a closed-form solution. The generic problem in such a case is:
min m e P 2
Smin const 2Re -zi ej }
ma ( P (717)
= max cos arg YZ, -p (7-17)
\(p=I