IEEE Aerospace and Electronic Systems Magazine - June 2020 - 70

Cognitive Radar Target Detection and Tracking With Multifunctional Reconﬁgurable Antennas
positioned at pn will be
y'n ðtÞ ¼

M
X

2p T

sm ðtÞgðum ; fm ; 'n ÞeÀj  pn um þ nðtÞ

(2)

m¼1

where u m is the look vector in the direction of the mth
target defined as, u m ¼ ½ sin um cos fm ; sin um sin fm ;
cos um T ,  is the wavelength of the signal and nðtÞ is the
zero mean and circularly symmetric complex Gaussian
noise with variance s 2 . The vector of sample outputs of
N antennas at time instance t will be ym ðtÞ ¼ ½y'1 ðtÞ;
y'2 ðtÞ; . . . ; y'N ðtÞT and it can be expressed as
y m ðtÞ ¼ Am sðtÞ þ nðtÞ;

t 2 f1; 2; :::; T g

(3)

where sðtÞ ¼ ½s1 ðtÞ; s2 ðtÞ; :::; sM ðtÞT , nðtÞ ¼ ½n1 ðtÞ; n2 ðtÞ;
:::; nN ðtÞT and the antenna array mode vector m ¼
½'1 ; '2 ; :::; 'N . Am is N Â M matrix with entries
Am ðn; mÞ ¼

½glun

cos g m þ

gl?n

sin g m e

jhm

e

T
Àj2p
 p n um

(4)

where g'un and g'fn are the complex antenna patterns for
mode 'n of antenna n. The mth column of Am for the signal DoA and polarization parameters ðum ; fm ; g m ; hm Þ is
denoted as a m
m . Further details about signal models for
diverse arrays can be found in [38] and [42]-[44].

PROCESSOR

(5)

where Fðx
xk Þ is the angle of the DoA of the target with
respect to radar array origin and w k is the estimation error.
We assume that the estimator is unbiased and model the
estimation error as Gaussian with zero mean and variance
70

fðzzk jx
xk ; m k Þ ¼

expfÀjjzzk À Fðx
xk Þjj2 =2s 2z ðm
mk Þg
:
2
mk Þ
2ps z ðm

(6)

Denoting the set of DoA observations and the array
modes up to time k as Z k  fzz1 ; z2 ; ::::; z k g and M  fm
m1 ;
m 2 ; ::::; mk g respectively, the predicted density f À ðx
xk Þ ¼
fðx
xk jZ
Z kÀ1 ; M Þ and the posterior density f þ ðx
xk Þ ¼
fðx
xk jZ
Z k ; M Þ can be calculated using
xk Þ ¼
f À ðx

The radar array observes the raw data y m ðtÞ and processes
it to estimate the DoA parameters ðum ; fm Þ efficiently. A
large number of DoA estimation algorithms have been
proposed in the literature [45]-[47]. A more detailed overview of these techniques can be found in [48] and [49].
Although these techniques are applied on arrays of identical antenna elements with fixed radiation patterns, techniques for diverse arrays, where each element of the array
has fixed, but different radiation and polarization patterns
have also been studied in literature for both DoA and
polarization estimation [42]-[44]. A multiple signal classification (MUSIC) [46] based DoA and polarization estimation technique that is suitable for MRA arrays is
utilized using the approaches detailed in [42]-[44]. Since
the mode selection for MRA creates diverse arrays where
each antenna element might have a different radiation pattern, techniques in [42]-[44] are very suitable for the
MRAA scenarios. MUSIC generates a DoA estimate of
the target; hence the measurement model for the MRAAradar can be stated as
z k ¼ Fðx
xk Þ þ w k

of s 2z ðmk Þ. In (5), estimated DoA can be either azimuth,
elevation, or both depending on the application. Through
simulations and previous results presented in [38], we
observe that the MUSIC based estimator is unbiased and
the variance of the DoA estimates depends on m , the
modes of the MRAs in the array. This dependence creates
an opportunity to select the antenna modes m cognitively
to minimize the variance of the DoA estimate. However,
the variance of the DoA estimate does not have a closed
form expression and is difficult to calculate. Instead of
minimizing s 2z ðm
mk Þ, we minimized the CRLB of zk since
the CRLB is easier to work with and it is a lower bound
on the estimation variance. This is achieved as detailed in
the Controller part of our. Following the measurement
model in (5), zk $ NðFðx
xk Þ; s 2z ðm
mk ÞÞ and the measurement likelihood function can be stated as

Z

xk Þ ¼ R
f þ ðx

qðx
xk jx
xkÀ1 Þf þ ðx
xkÀ1 Þdx
xkÀ1
fðzzk jx
xk ; M Þf À ðxk Þ
xk ; M Þf À ðx
xk Þdx
xk
fðzzk jx

(7)

(8)

x0 Þ ¼ qðx
x0 Þ.
with an initialization step of f þ ðx

CONTROLLER
The objective of the controller is to find the next set of
MRA array modes mk for the kth time step using the current state and all the previous measurements up to z k to
optimize the state estimator. The cognitive radar will take
an "action" based on the decision by the controller and
obtain the next set of measurements zk by using the
designed array mode vector m k . This control action can
be done by selecting the array mode mk that minimizes the
CRLB of the DoA estimate at the predicted posterior
mean c
xk À ¼ E½f À ðx
xk Þ as shown in the following:
m k ¼ arg min trðF
F À1
xk À Þ; mÞ:
m ðFðc
m

(9)

The MRAA signal model is a Gaussian process and for the
case when mode state vector is the same for the coherent
processing interval of T snapshots, the Fisher information
matrix F m is given in [50] as

IEEE A&E SYSTEMS MAGAZINE

JUNE 2020
IEEE Aerospace and Electronic Systems Magazine - June 2020

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