6124 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 66, NO. 23, DECEMBER 1, 2018
Persymmetric Adaptive Detection of Subspace
Signals: Algorithms and Performance Analysis
Jun Liu , Senior Member, IEEE, Weijian Liu , Member, IEEE, Yongchan Gao , Member, IEEE,
Shenghua Zhou
, Senior Member, IEEE, and Xiang-Gen Xia , Fellow, IEEE
Abstract—The problem of detecting a subspace signal in colored
Gaussian noise with unknown covariance matrix is investigated by
incorporating persymmetric structure of received data. The signal
of interest is described with a subspace model, namely, it belongs to
a subspace spanned by the columns of a known matrix, but with un-
known coordinates. We propose a persymmetric detector with two
tunable parameters, which includes many existing persymmetric
detectors as special cases. Approximate expressions for the prob-
abilities of false alarm and detection of the proposed detector are
derived, which are verified via Monte Carlo simulations. Numerical
results reveal that the exploitation of the persymmetric structure
leads to a significant gain in the detection performance, especially
in the case of limited training data. In addition, a further gain in
the detection performance can be obtained by optimally selecting
the tunable parameters.
Index Terms—Adaptive d etection, persymmetric detector, gen-
eralized likelihood ratio test, adaptive matched filter, adaptive sub-
space detection, subspace signal.
I. INTRODUCTION
A
DAPTIVE detection in colored noise with unknown co-
variance matrix is an area of longstanding interest [1]–
[10]. The noise here is in a broad sense, which may include
clutter, interference, and thermal noise. To handle the detection
problem with unknown noise covariance matrix, one usually
poses a standard assumption that a set of training data, free
of target signal, is available [11]. These training data are used
to estimate the unknown noise covariance matrix in the test
Manuscript received March 31, 2018; revised September 16, 2018; accepted
October 1, 2018. Date of current version October 22, 2018. The associate editor
coordinating the review of this manuscript and approving it for publication
was Prof. Jean-Yves Tourneret. This work is supported in part by the National
Natural Science Foundation of China under Contracts 61501351, 61871469,
and 61501505, in part by the National Natural Science Foundation of Shaanxi
Province under Grant 2018JM6051, in part by the Natural Science Foundation of
Hubei Province under Contract 2017CFB589, and in part by the Key Research
Program of the Frontier Sciences, CAS, under Grant QYZDY-SSW-JSC035.
(Corresponding author: Weijian Liu.)
J. Liu is with the Department of Electronic Engineering and Information
Science, University of Science and Technology of China, Hefei 230027, China
(e-mail:,jun_liu_math@hotmail.com; junliu@ustc.edu.cn).
W. Liu is with the Wuhan Electronic Information Institute, Wuhan 430019,
China (e-mail:,liuvjian@163.com).
Y. Gao and S. Zhou are with the National Laboratory of Radar Signal Pro-
cessing, Xidian University, Xi’an 710071, China (e-mail:,ycgao@xidian.edu.cn;
shzhou@mail.xidian.edu.cn).
X.-G. Xia is with the Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA (e-mail:,xxia@ee.udel.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2018.2875416
data, which leads to the concept of adaptive processing. It is
well known that the estimation error is non-negligible when
the number of training data is limited. It is proved in [12] that
the adaptive processing loss caused by this estimation error is
more than 3 dB when the number of training data is less than
twice the dimension of the received data.
Resorting to the training data, researchers have proposed
many adaptive detectors, such as Kelly’s generalized likelihood
ratio test (GLRT) [13], adaptive matched filter (AMF) [14],
adaptive coherence estimator (ACE) [15], and adaptive subspace
detector (ASD) [16]. The GLRT and AMF were developed for
the detection problem in a homogeneous environment where the
test and training data are independent and identically distributed
(IID). The ACE and ASD were designed for the detection prob-
lem in a partially homogeneous environment where the test and
training data share the same noise covariance matrix up to an
unknown scaling factor. Note that these detectors mentioned so
far do not exploit any prior knowledge on the structure of noise
covariance matrix. The number of training data required in these
detectors is no less than the data dimension to ensure with prob-
ability one the nonsingularity of the estimated noise covariance
matrix [17]. Unfortunately, this requirement may not be satisfied
in sample-starved environments. Some prior knowledge on the
structure of noise covariance matrix can be exploited to alleviate
the requirement on the number of training data.
In practice, the noise covariance matrix has an Hermitian per-
symmetric (also called centrohermitian) form, when the system
is equipped with a symmetrically spaced linear array [18, chap.
7] or symmetrically spaced pulse trains [19]. Hermitian persym-
metry has a property of doubly symmetry, i.e., Hermitian about
its principal diagonal and persymmetric about its cross diagonal.
Unless otherwise stated, “persymmetric” always denotes “Her-
mitian persymmetric” for brevity in the following. Note that the
persymmetry includes the Toeplitz structure as a special case
when a uniformly spaced array or pulse train is adopted.
The investigation on the persymmetric structure of t he noise
covariance matrix can be traced to Nitzberg’s paper [20], where
the maximum likelihood (ML) estimate of the persymmetric co-
variance matrix was obtained. De Maio reconsidered the ML es-
timation problem in the constraint where the covariance matrix
is a sum of a positive semidefinite Hermitian and persymmet-
ric term (accounting for the interference) and a matrix propor-
tional to the identity (accounting for the thermal noise) [21]. In
[22], the average output signal-to-interference-plus-noise ratio
(SINR) of a persymmetric sample matrix inversion beamformer
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