Covariance-Free Nonhomogeneity STAP Detector in Compound Gaussian Clutter Based on Robust Statistics

Abstract

Space time adaptive processing (STAP) detects targets by computing adaptive weight vectors for each cell under test
using its covariance matrix, as estimated from surrounding secondary cells. In this context, the non-homogeneity detector (NHD)
excludes the anomalous secondary cells that adversely affect the detection performance. The existing robust NHDs require estimating
the covariance matrix of each secondary cell, which hinders their implementation in modern radars with large-dimensional
range cells. In this paper, we propose a new low-complexity NHD that is suitable for highly correlated clutter environments with both Gaussian and non-Gaussian heavy-tailed distributions. The proposed detector, which is based on the projection depth function from the field of robust statistics, features a nonparametric and covariance-free test statistic. As a result, its computational complexity is much lower than that of current NHDs, such as the widely used normalized adaptive matched filter (NAMF) detector, especially for large-dimensional range cells. In Monte Carlo simulations with different clutter distributions and radar system configurations, the proposed detector shows a comparable performance to that of NAMF. The low complexity and robust performance of the new detector make it particularly attractive for real time applications.

Full text

IET Radar, Sonar & Navigation
Research Article
Covariance-free non-homogeneity STAP
detector in compound Gaussian clutter based
on robust statistics
ISSN 1751-8784
Received on 27th April 2019
Revised 15th July 2019
Accepted on 9th August 2019
E-First on 8th November 2019
doi: 10.1049/iet-rsn.2019.0201
www.ietdl.org
Ahmed A. Abouelfadl1 , Ioannis Psaromiligkos1, Benoit Champagne1
1Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada
E-mail: ahmed.abouelfadl@mail.mcgill.ca
Abstract: Space-time adaptive processing (STAP) detects targets by computing adaptive weight vectors for each cell under test
using its covariance matrix, as estimated from surrounding secondary cells. In this study, the non-homogeneity detector (NHD)
excludes the anomalous secondary cells that adversely affect the detection performance. The existing robust NHDs require
estimating the covariance matrix of each secondary cell, which hinders their implementation in modern radars with large-
dimensional range cells. In this study, the authors propose a new low-complexity NHD that is suitable for highly correlated clutter
environments with both Gaussian and non-Gaussian heavy-tailed distributions. The proposed detector, which is based on the
projection depth function from the field of robust statistics, features a non-parametric and covariance-free test statistic. As a
result, its computational complexity is much lower than that of current NHDs, such as the widely used normalised adaptive
matched filter (NAMF) detector, especially for large-dimensional range cells. In Monte Carlo simulations with different clutter
distributions and radar system configurations, the proposed detector shows comparable performance to that of NAMF. The low
complexity and robust performance of the new detector make it particularly attractive for real-time applications.
1 Introduction
Modern radar systems can jointly estimate the range, speed, and
direction of a detected target. The range and speed can be measured
based on the time difference and Doppler frequency shift between
the transmitted signal and the target return received by the radar,
respectively. For angle measurement, phased array radars employ
antenna arrays and beamforming techniques [1]. More specifically,
following radio-frequency down-conversion, the range information
is obtained by sampling the received antenna signals within a
temporal window in the pulse repetition interval (PRI). To estimate
the target Doppler frequency, coherent pulse-Doppler radars
transmit a sequence of identical time-shifted pulses that together
define the coherent pulse interval (CPI). The use of an antenna
array makes it possible to measure phase differences between
different spatial receiving channels, which after sampling result in
a set of directions [2].
Upon reflection of the transmitted pulses by a target, the radar
antenna receives distorted versions of these pulses due to other
scatterers, clutter, and noise. A space-time adaptive processing
(STAP) detector discretely scans the range dimension and, for each
range bin, arranges the data along the angle and Doppler
dimensions into a vector, called a range cell. It then linearly
combines the spatio-temporal data in each range cell to form the
test statistics. To this end, it needs to compute a set of weight
vectors corresponding to the different spatio-temporal ‘look’
directions, which depend on the covariance matrix of the
background clutter and noise within the cell under test (CUT), also
called the primary cell [2]. However, this covariance matrix is not
known in practice and it is commonly estimated from the adjacent
range cells, known as the secondary or training cells in this context.
The estimation of the covariance matrix from the secondary
cells relies on the assumption that they are homogeneous, i.e.
independent and identically distributed (iid). In reality, the
homogeneity assumption is hardly met due to the presence of
discrete scatterers, in-band interferers, target-dependent jammers
[3, 4], or a combination thereof. In this case, the estimated
covariance matrix does not represent accurately the background
clutter and noise, and hence, the weight vectors computed from this
matrix lead to a performance degradation of the STAP detector. To
tackle this problem, the non-homogeneity detector (NHD) was
introduced to detect the anomalous secondary cells to be censored
from covariance matrix estimation [5].
Conceptually, a secondary cell is considered to be homogeneous
to its surrounding secondary cells if it shares with them the same
covariance matrix up to a scalar. Since the true covariance matrix
of a given secondary cell is unknown, the work in [5] used the
generalised inner product (GIP) test to examine the similarity
between this unknown covariance matrix and the test covariance
matrix estimated from the surrounding secondary cells. Later, the
normalised adaptive matched filter (NAMF) test was used as an
NHD with Gaussian and non-Gaussian clutter models in [6], where
the NAMF detector was shown to be the most robust NHD.
Recent research efforts on NHD have focused on improving the
performance of the aforementioned classical detectors or reducing
their complexity. For instance, Jiang and Wang [7] proposed a soft
NHD concept, wherein the covariance matrix of the CUT is
calculated using the weighted secondary cells assuming Gaussian
distributed clutter. In turn, the calculation of weight for each
secondary cell is formulated as a non-linear optimisation problem
based on the output of a modified version of the adaptive matched
filter (AMF). An iterative approximate maximum-likelihood (ML)
approach based on the GIP detector was developed in [8] for
estimating the subset of non-homogeneous cells. This approach
shows a comparable performance to the iterative original GIP test
using the ML covariance estimator for the Gaussian interference
[9]. In addition, a large body of research has been devoted to
reduce the dimensions of the STAP detection problem using
different transformation and rank reduction techniques as in [10],
which can also be applied to NHD [11]. However, these partially
adaptive detectors generally exhibit inferior performance compared
to their fully adaptive counterparts. Based on the GIP detector,
other NHD procedures for the special cases of spaceborne or side-
looking radars were introduced in [12, 13].
The above referenced covariance-based NHDs share the need to
estimate the covariance matrix and its inverse (known as the
precision matrix) for each secondary cell, which leads to a high
computational cost, especially for non-Gaussian clutter. Some
covariance estimators need a priori knowledge about the clutter
distribution [14], which is imperfect in most cases, while other
estimators need to solve non-convex optimisation problems with
high computational complexity [15]. To avoid such difficulties, a
IET Radar Sonar Navig., 2019, Vol. 13 Iss. 12, pp. 2107-2119
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2107

covariance-free NHD with a comparable performance to the GIP
was introduced in [16]. However, it is known that the GIP test is
not robust, especially in non-Gaussian clutter scenarios [6].
In this paper, we focus on the problem of detecting the non-
homogeneous cells within the secondary cells for correlated clutter
with Gaussian as well as non-Gaussian distributions. We introduce
a novel covariance-free, non-parametric NHD based on the
projection depth (PD) function, a well-known tool in the field of
robust statistics [17]. By exploiting the properties of this function,
the new detector does not require a priori knowledge about the
clutter distribution, and more importantly, evades estimating the
covariance or the precision matrices, which reduces the
computational burden significantly. Simulation results for different
clutter distributions show that the proposed detector can maintain
the robust performance of covariance-based, fully-adaptive
detectors such as the NAMF while inheriting the simple
computational structure of the PD function. Preliminary results of
this work were presented in an abridged conference paper [18].
This paper provides new results and extensions including: proofs of
key results that are central to the derivation of the new detector,
detailed performance analysis including Fisher-consistency under
different clutter models, guidelines on the choice of the proposed
NHD parameters with complexity analysis, and extended
simulation studies for different radar operational conditions and a
more challenging benchmark.
The rest of the paper is organised as follows: Section 2 provides
the background about STAP and the clutter signal model. The non-
homogeneity detection problem is presented in Section 3. In
Section 4, the proposed NHD is introduced and its approximate
equivalence to the NAMF detector is proven. The comparative
evaluation of the proposed and NAMF detectors through Monte
Carlo simulation is presented in Section 5. Section 6 concludes the
paper.
Notation: Matrices and column vectors are denoted in boldface
uppercase and lowercase letters, respectively; IK indicates a K×K
identity matrix; (⋅)T, (⋅)H, ⊗, ∥⋅∥2, and det( ⋅ ) denote the
transpose, the Hermitian transpose, the Kronecker product, the ℓ2-
norm, and determinant operations, respectively. E(⋅) is the
mathematical expectation; ℜ( ⋅ ) and ℑ( ⋅ ) denote the real and
imaginary parts of their complex-valued arguments, respectively.
2 Signal model
In this section, we briefly review the STAP concept for pulse-
Doppler radars and present a clutter model based on the spherically
invariant random process (SIRP) that can represent both Gaussian
and non-Gaussian clutters.
2.1 STAP model
Consider a pulsed Doppler radar using a uniform linear array
(ULA) of N antenna elements that are spaced d=λ/2 apart, where
λ is the wavelength at the radar's centre frequency. The radar
simultaneously transmits from each antenna element a sequence of
M coherent pulses with a PRI T, which define the so-called slow
time domain. The transmitted signal from each antenna element is
assumed to be narrowband, i.e. its bandwidth B satisfies
B≪c/Nd, where c is the speed of light [2].
Upon reflection by a moving point target with azimuth angle θt
from the boresight of the radar antenna array (planar geometry is
assumed), a target return will be received with a Doppler frequency
shift fd, normalised with respect to the pulse repetition frequency
1/T. Let Tu be the time delay corresponding to the radar maximum
unambiguous range, while the time delay corresponding to the
radar range resolution is 1/B. Hence, the total number of range
cells in the so-called fast time domain is
L= ⌊TuB⌋
(1)
where ⌊⋅⌋ denotes the floor function. Therefore, the STAP data
can be visualised as an L×N×M data ‘cube’ as shown in Fig. 1.
If the target's range corresponds to the kth range cell, its spatio-
temporal data is an N×M matrix that contains the received signal
from each antenna element and PRI within the target's range cell.
The M-dimensional temporal and N-dimensional spatial
steering vectors of the target are, respectively, given by [2]
b(fd) = 1 ej2π f d⋯ ej2π(M− 1) fdT
(2a)
a(θt) = 1 ej2πd
λsin(θt)⋯ ej2π(N− 1)d
λsin(θt)T
(2b)
The compound spatio-temporal steering vector s(fd,θt)∈ℂJ,
where J=MN , is defined as
s(fd,θt) = b(fd) ⊗ a(θt)
∥b(fd) ⊗ a(θt) ∥2
(3)
To simplify the presentation, s(fd,θt) will be denoted as s
hereinafter. The baseband signal r received from the target is
r=as
(4)
where a is an unknown deterministic complex amplitude (i.e.
Swerling case 0 [19]).
Depending on whether a target is present or not, the total
received signal vector z∈ ℂJ in a given range cell is expressed as
H1:z=r+c+n
(5a)
H0:z=c+n
(5b)
where H0 and H1 are the null and alternative hypotheses,
respectively, r is the target return as in (4), c is the clutter vector
and n is the noise vector; n and c are assumed to be statistically
independent. The noise vector n is drawn from a complex circular
symmetric Gaussian distribution CN(0,ςn
2IJ) with zero mean and
covariance matrix ςn
2IJ, where ςn
2 is the noise variance. The clutter
vector is modelled as [2]
c=∑
i= 0
Nc− 1
κiei
(6)
Fig. 1 Illustration of STAP data ‘cube’ and formation of range cells
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where Nc is the number of clutter patches, κi is the complex
amplitude of the ith patch and ei∈ ℂJ is the corresponding steering
vector, which admits the form of (3).
Let R=EzzH be the covariance matrix of the received signal
z in (5). For each CUT, STAP aims at forming the optimal
beamforming (or weight) vector in real-time to maximise the
received signal-to-interference-plus-noise ratio with respect to s.
Under the minimum variance distortionless response criterion, the
optimal weight vector takes the form [2]
w=gR−1s
(7)
where g is a complex scalar.
For the complex vector z=zR+jzI, where zR= ℜ(z) and
zI= ℑ(z), the covariance matrix is expressed as [20]
R=RzRzR+RzIzI+jRzRzI
T−RzRzI
(8)
where RzRzR=EzRzR
T, RzRzI=EzRzI
T, RzIzI=EzIzI
T, and z is a
proper complex signal, i.e. RzRzI= − RzRzI
T and RzRzR=RzIzI, which
is common in the radar context. Moreover, it is customary to
assume that the in-phase and quadrature components of z are
independent, i.e. RzRzI=0, where 0 is J×J zero matrix [21, 22].
Hence
R= 2RzRzR= 2RzIzI
(9)
In practice, the covariance matrix R is unknown and different
techniques are used to estimate it from the adjacent L− 1
secondary cells, assuming no guard cells. In the case of Gaussian
clutter, the ML estimator is the sample covariance matrix (SCM)
given below:
R
^
SCM =1
L− 1 ∑
l= 1
L− 1
zlzl
H
(10)
where zl denotes the total received signal in the lth secondary cell,
and the condition L− 1 ≥ 2J is needed to ensure robustness. If c
follows a non-Gaussian distribution, the SCM is neither a
consistent nor robust estimator and other estimators should be
used. More details on these estimators will be presented shortly.
2.2 SIRP clutter model
In this paper, we are concerned with coherent processing of the
received signal vector z in (5), where both the real and imaginary
parts (i.e. in-phase and quadrature components) of each vector
entry are considered. In this regard, it is essential to employ a
probabilistic model of the clutter vector c in (6) that takes into
account the joint statistics of the real and imaginary parts of all its
entries. In particular, for proper clutter modelling, both the spatio-
temporal correlation properties and probability density function
(PDF) of the clutter envelope should comply with experimental
data. Under the SIRP model, which meets these requirements [23],
the clutter vector is modelled as a product of two independent
components, i.e. a zero-mean complex Gaussian vector, known as
the speckle component, and a positive random variable, known as
the texture component and assumed to vary slowly across range
cells. Therefore, the clutter vector in (6) can be represented as
c=vy,
(11)
where y∈ ℂJ follows a complex Gaussian distribution CN(0,Σ)
with zero mean and covariance matrix Σ, and v is a positive
random variable. By choosing the proper PDF of the texture
component v in the SIRP model (11), denoted as fV(v) in the
sequel, we can obtain different non-Gaussian clutter distributions,
also known as the compound Gaussian distributions, while the
particular choice v= 1 (with probability one) yields the Gaussian
clutter model. Moreover, through a suitable choice of the
covariance matrix Σ of the Gaussian speckle vector y, the desired
spatio-temporal correlation properties can be fulfilled. The PDF of
c can be expressed as [6]
fc(c) = (2π)−Jdet(Σ)−1/ 2h2J(cHΣ−1c)
(12)
where the function h2J(x) is defined as
h2J(x)=∫0
∞
v−Jexp − x
v2fV(v)dv.
(13)
The covariance matrix of the SIRP vector c is given by
Rc=E(v2)Σ.
Other models for compound Gaussian clutter use zero memory
non-linear (ZMNL) transformations. These methods apply non-
linear transformations on sequences of coherent Gaussian samples
that result in the desired marginal PDF of the clutter envelope.
However, due to the non-linear transformations, the covariance
matrix of the resulting non-Gaussian clutter is related to that of the
original Gaussian samples in an intricate manner, which makes it
difficult to obtain the desired covariance matrix. Moreover, these
methods do not guarantee that the resulting covariance matrix is
non-negative definite [23]. On the contrary, the SIRP model in (12)
allows controlling both the envelope PDF and the covariance
matrix of the generated clutter.
One of the most common clutter distributions is the K-
distribution, which provides a good fit to the envelope of the data
acquired from different environments. The K-distribution of the
clutter envelope is given by [6]
f(r) = 2δ
Γ(α)
δr
2
α
Kα− 1(δr),
(14)
where α> 0 and δ> 0 are the shape and scale parameters,
respectively, Γ( ⋅ ) is the Gamma function, and Kα(⋅) is the
modified Bessel function of the second kind of order α. In order to
arrive at the K-distribution for the clutter envelope using the SIRP
model, the PDF of the texture component fV(v) should be selected
as [6]
fV(v) = 2δ
Γ(α)2αδv 2α− 1exp −δ2v2.
(15)
In this case, the second moment of v is given by E(v2)=2α/δ2.
3 Non-homogeneity detector
To calculate the adaptive weight vector w in (7) for a given CUT
within the available L range cells, one needs to estimate the
covariance matrix of this CUT from the adjacent L1=L− 1
secondary cells zl, where l∈ ℒ = {1,…, L1}, that together form
the secondary sample matrix Z=z1, …, zL1∈ ℂJ×L1. To censor
non-homogeneous secondary cells from the estimation, the NHD
decides if a secondary cell, say zk for k∈ ℒ, is non-homogeneous
with respect to the remaining L2=L1− 1 secondary cells zl for
l∈ ℒ − {k}, which together forms a matrix Zk (obtained from Z
by removing the column zk). The NHD is basically a STAP detector
that sequentially processes the L1 secondary cells with one of them,
zk, temporarily considered as the CUT (also termed secondary
CUT), while the remaining secondary cells Zk are used to estimate
the covariance matrix of zk.
A basic test employs the general inner product (GIP), which is
equivalent to the square of the Mahalanobis distance [5], i.e.,
ΛGIP = (zk−μ
^)HR
^−1(zk−μ
^) ≷
H1
H0
η1
(16)
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where μ
^∈ ℂJ× 1 is the sample mean of zk, R
^∈ ℂJ×J is its estimated
covariance matrix, and η1 is a threshold that is determined based on
the required probability of false alarm PF. In this test, H0 is the null
hypothesis that zk is homogeneous with respect to Zk, while H1 is
the alternative hypothesis. However, the GIP test is not robust in
non-Gaussian clutter environment as reported in [6], where a more
robust detector, namely the normalised adaptive matched filter
(NAMF), is proposed as
ΛNAMF =w
^Hzk
2
(w
^HR
^w
^)(zk
HR
^−1zk)
=sHR
^−1zk
2
(sHR
^−1s)(zk
HR
^−1zk)≷
H1
H0
η2
(17)
where w
^=gR
^−1s. For this detector, PF has been derived in [6]
assuming Gaussian clutter, but it is not tractable analytically for
non-Gaussian SIRP clutter. In the latter case, Monte Carlo
simulations are used to set the threshold.
While the SCM in (10) is the ML estimator in the case of
Gaussian clutter, the ML estimator of the covariance matrix in the
case of compound Gaussian clutter cannot generally be obtained in
analytical form. Tyler introduced a generalisation of the ML
estimator for elliptical distributions (which include Gaussian along
with other distributions) that can be expressed as the solution to the
non-linear equation [24]
R
^=J
L1∑
l= 1
l≠k
L1zlzl
H
zl
HR
^−1zl
(18)
However, besides the difficulties posed by solving (18) due to the
high computational cost, it needs a large number of secondary cells
L1 for estimator accuracy [6]. An approximation to the ML
estimator for the covariance matrix of SIRP clutter is given by [6]
R
^
SIRP =1
L1∑
l= 1
l≠k
L1
ζlzlzl
H,
(19)
where
ζl=h2J+2(zl
HR
^
SIRP
−1 zl)
h2J(zl
HR
^
SIRP
−1 zl)
(20)
where the function h2J(⋅) is defined in (13). The scalar ζl cannot be
expressed in a closed form, since both sides of (19) contain R
^
SIRP,
but it can be found by the iterative expectation-maximisation (EM)
algorithm [25]. However, the EM algorithm converges slowly,
especially for low values of α [6]. Moreover, the estimator in (19)
needs a priori knowledge of the clutter distribution. Another
approximation to the ML covariance estimator in case of non-
Gaussian clutter is the iterative normalised SCM (NSCM) that is
obtained through the following recursive formula [21]
R
^
NSCM
(t+ 1) =J
L1∑
l= 1
l≠k
L1ℜ(zl)ℜ(zl
T)
ℜ(zl
T)(R
^
NSCM
(t))−1ℜ(zl)
(21)
where t denotes the iteration index. The computation is initialised
with the estimator [21]
R
^
NSCM
(0) =J
L1∑
l= 1
l≠k
L1ℜ(zl)ℜ(zl
T)
ℜ(zl
T)ℜ(zl)
(22)
Although it is also based on iterative procedures, its rate of
convergence is faster than the EM-based algorithm mentioned
above for the solution of (19) and (20), and it has been reported to
converge after only four iterations [21]. Moreover, the NSCM
shows a detection performance that is very close to that of the EM-
based estimator [26]. Henceforth, whenever we use R
^ we mean
R
^
NSCM.
4 Proposed NHD
In this section, we first introduce the PD function and use it to
provide covariance-free interpretations of the GIP and NAMF test
statistics. We then introduce a covariance-free NHD that employs a
novel non-parametric (distribution-free) test statistic based on the
PD function and extend it to the case of correlated clutter.
4.1 Projection depth function
Let z∈ ℂJ be a random vector with joint cumulative distribution
function (CDF) F(z). A depth function is a random scalar
D(z,F) ∈ [0, 1], defined as a function of z and taking into account
the features of its distribution F. Ideally, the value of D(z,F)
provides an inverse measure of ‘distance’ from a central point
(such as the median or the mean of the distribution F), which can
used for the centre-outward ordering of observations of vector z
[27]. Based on this ordering, outliers can be detected when their
distance from the center is larger than a certain threshold. Hence,
the concepts of depth function and outliers are related. Specifically,
we can define a measure of outlyingness as the function [28]
O(z,F) = 1
D(z,F)− 1
(23)
Let μ(⋅) and σ(⋅) be univariate location and scale measures,
respectively. Then, the projection-based outlyingness of z is [28]
O(z,F) = sup
u∈ ℂJ, ∥ u∥ = 1
uHz−μ(Fu)
σ(Fu)
(24)
where Fu is the CDF of uHz. In practice, the sample version of (24)
is found by replacing Fu by its empirical version F
^
u.
The projection-based outlyingness has a higher breakdown
value in comparison to other types of outlyingness functions [28],
which motivates its use in this work. To understand the concept of
the breakdown value, consider the estimation of a scalar parameter
θ from n observations Xn= {x1, …, xn}, with Tθ(Xn) denoting the
resulting estimator. Let us assume that out of these observations, m
are replaced by arbitrary values (outliers), resulting in the
contaminated sample set Xn,m. The estimator TθXn,m is calculated
for the same parameter θ, but from the contaminated set Xn,m. The
finite sample breakdown value of the estimator Tθ(⋅) is the
smallest ratio of contamination m/n for which the distance between
Tθ(Xn) and Tθ(Xn,m) can become arbitrarily large for certain choices
of outliers [29].
The projection-based outlyingness function in (24) is a robust
alternative to the Mahalanobis distance and, hence, to the GIP. A
non-parametric GIP NHD detector was introduced in [16] for
Gaussian clutter using the outlyingness function in (24) that evades
the high computational burden of estimating the covariance matrix
and its inverse with increasing dimensions of the range cells. We
refer to it as the projection depth GIP (PD-GIP). However, the
generation of the projection vectors u to approximate the
supremum operation requires calculating the median of the
secondary cells in Zk for each zk; besides, the performance of the
original GIP detector in case of non-Gaussian clutter environment
is not robust [6]. Below, we propose a covariance-free detector
based on (24) that is approximately equivalent to the NAMF
detector in its robust performance, while at the same time sharing
the non-parametric character of (24) and its lower computational
complexity.
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4.2 Covariance-free reformulation of GIP and NAMF
We begin by stating a proposition about the equivalence of the
outlyingness function in (24) to the GIP in (16). This equivalence,
which was demonstrated in [30] for the case of real-valued data in
image processing applications, is extended here to complex-valued
radar observations, as needed to comply with the case of coherent
clutter model under consideration in this paper.
Proposition 1: Let Z=z1, …, zL1∈ ℂJ×L1 be a secondary
sample matrix. For any target steering vector s as in (3) and an
arbitrary secondary cell zk, where k∈ {1, …, L1}, is associated with
an estimated covariance matrix R
^ and mean vector μ
^, we have
sup
∥u∥ = 1
uHzk−μ
^(uHZk)
σ
^(uHZk)
2
=zk−μ
^HR
^−1 zk−μ
^
(25)
and
sup
∥u∥ = 1
uHs
σ
^(uHZk)
2
=sHR
^−1s
(26)
where μ
^(uHZk), σ
^(uHZk) are the sample mean and standard
deviation (SD) of uHZk, respectively, and Zk denotes the secondary
cells after excluding zk.
In (25) and (26), the supremum operation is taken over all unit-
norm vectors u∈ ℂJ. The proof of Proposition 1 is given in
Appendix 1. We note that the denominator of the test statistic of the
NAMF detector in (17) can be expressed using the outlyingness
function in (24) as shown in Appendix 1, specifically (48) and (50).
However, the numerator of (17), sHR
^−1zk, cannot be directly
expressed in terms of (24). To circumvent this difficulty, we
suggest replacing zk in the numerator of (17) by sHzks to obtain
sHR
^−1 sHzks
2=sHzk
2
sHR
^−1s
2
(27)
The following proposition states that, in case of a dominant target,
the expression in (27) is approximately equivalent to sHR
^−1zk
2,
which is the numerator of (17).
Proposition 2: Let s, zk, and R
^ be as defined in Proposition 1,
then sHzksHR
^−1s has the same target's signal component as
sHR
^−1zk.
The proof of this proposition is provided in Appendix 2. The
next proposition introduces a modified test statistic, which
approximates the original NAMF test statistic in (17) in terms of
the projection-based outlyingness in (24).
Proposition 3: Let s, zk, Zk, and R
^ be as defined in Proposition
1. Then
Λ′NAMF ≜sHR
^−1(sHzk)s2
(sHR
^−1s)(zk
HR
^−1zk)
=
sHzk
2
sup∥u∥ = 1
uHs
σ
^(uHZk)
2
sup∥u∥ = 1
uHzk
σ
^(uHZk)
2
(28)
The proof of this proposition is provided in Appendix 3. As
observed from (28), the test statistic ΛNAMF
′ is covariance-free.
Moreover, besides its approximate equivalence to the NAMF test
in (17) as shown in Proposition 2, it inherits the non-parametric
characteristic of the projection-based outlyingness.
4.3 Robust, covariance-free, and non-parametric NHD
Although the projection-based outlyingness in (24) does not dictate
a specific scale measure, the median absolute deviation (MAD) has
been widely used in robust statistics to detect outliers due to its
robustness with respect to heavy-tailed distributions and higher
breakdown value compared to the SD [31].
For the real-valued random sample data Xn=x1, …, xn with
order statistics x(1) ≤ ⋯ ≤ x(n), the sample median med(Xn) and
sample MAD mad(Xn) are calculated as [32]
med(Xn) = x((n+ 1)/2) nis odd
0.5(x(n/2) +x((n/2) + 1))nis even
(29)
and
mad(Xn) = med( xi− med(Xn) ), i= 1, …, n
(30)
respectively. The population MAD of the random variable X
MAD(X) is related to its population SD σ(X) as [33]
MAD(X)=kfσ(X)
(31)
where kf is a positive constant to achieve consistency and its value
depends on the population CDF of X. For the standard normal
distribution, kf≃ 0.6745 [31]. In the absence of outliers, the
sample versions mad(Xn) and σ
^(Xn) are related, approximately, by
the same constant kf even with a sample size as low as 10 [31].
The breakdown value of the MAD is 0.5 [33], which is the best
possible breakdown value, compared to a value of 0 for the SD
[29]. Therefore, the MAD is more robust than the SD, especially
for heavy-tailed clutter distributions as the K-distribution. Since
heavy-tailed distributions tend to have many outliers with very
high values, the MAD constitutes a better estimate for the scale
parameter than the SD and leads to a lower threshold for the same
false alarm rate and, consequently, a better detection.
By employing the mad uHZk as a robust scale measure instead
of σ
^(uHZk) in (28), we obtain
ΛNAMF
′≃
sHzk
2
sup∥u∥ = 1
kfuHs
mad(uHZk)
2
sup∥u∥ = 1
kfuHzk
mad(uHZk)
2
(32)
where for the proper complex signal vectors Zk, we have according
to (8)
mad(uHZk) = 2mad(ℜ(uHZk)) = 2mad(ℑ(uHZk))
(33)
Under the SIRP model considered in this paper, all the CDFs Fu of
the projections of a given SIRP vector are the same [22]; this
means that the value of kf, that is determined based on Fu, does not
depend on the projection vector u. To verify this independence for
the considered signal vector, a secondary sample matrix Z∈ ℂJ×L
of uncorrelated clutter vectors is simulated with the dimension of
the secondary cells fixed at J= 20, while L/J changes from 2 to
10. For each sample size L, a sample of kf from different 1000
projection vectors u is calculated using (31), but using the sample
MAD mad uHZk as defined in (30). The relative SD (RSD) of the
kf sample is
RSD(kf) = σ
^(kf)
μ
^(kf)
(34)
where σ
^(kf) and μ
^(kf) are the sample SD and mean of kf,
respectively, averaged over 104 trials. As Fig. 2 shows, the value of
kf exhibits a low variation for both of the considered distributions
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at all considered values of L. This also shows that (31) holds also
for mad uHZk even for low sample size, as shown in [31].
With the agreement of the presented simulation results with the
theoretical analysis in [22, 31], the constant kf can be taken out of
the supremum in (32). Therefore, the proposed test ΛPD based on
the NAMF and PD outlyingness is
ΛPD −NAMF ≜
sHzk
2
sup∥u∥ = 1
uHs
mad(uHZk)
2
sup∥u∥ = 1
uHzk
mad(uHZk)
2≷
H1
H0
η3
(35)
We call the detector based on the test statistic in (35) as the PD-
NAMF. Theoretically, implementing the supremum requires
calculating the projections of an infinite number of vectors that
cover the unit hypersphere in J-dimensional space. In practice, as
shown in [30, 34] for different applications, the supremum can be
approximated by taking the maximum magnitude of a finite
number Q of projections of zk or s on randomly generated vectors
over this hypersphere. As suggested in [35], each of these vectors
is obtained by first generating J independent complex Gaussian
variates ui∼CN(0,1), 1 ≤ i≤J with zero mean and unit
variance to form the vector u=u1, …, uJ, and then normalising u
with respect to ∥u∥2. As the steps above show, the generation
method used in this work is totally independent of the steering
vector s or the CUT zk, hence it is performed once and the obtained
vectors are stored to be used for all range cells and any steering
vector s. This off-line method of generation is different from that
used in [16]. In the latter, the projection vectors were recomputed
for each CUT from the secondary cells Z. The discussion on the
choice of Q is left for Section 5.
4.4 Correlated clutter
The correlation matrix of the clutter signal vector c in (6) is a
Kronecker product of the temporal (i.e. between pulses) and spatial
(i.e. between antenna elements) covariance matrices Ψt and Ψs,
respectively, i.e. [36]
Ψ=Ψt⊗Ψs
(36)
The spatial correlation of the clutter depends on the inter-element
spacing of the antenna array as in [37, 38]. It can be approximated
as
Ψs=ρsi−j, 1 ≤ i,j≤N
(37)
where ρs is the one-lag spatial correlation coefficient. Based on
experimental measurements for different clutter environments, e.g.
[39, 40], the temporal covariance matrix of the clutter can be
expressed similarly as
Ψt=ρti−j, 1≤i,j≤M
(38)
where ρt is the one-lag temporal correlation coefficient.
To achieve the constant false alarm rate (CFAR) property, a
crucial feature of a robust radar detector is the scale invariance of
its test statistic. However, the distribution of the sample version of
(24) depends on the scale parameter estimator if the latter is not
Fisher consistent, as shown in [28, Remark 3.4]. Since (24) is the
main building block of the PD-NAMF in (35), the dependency of
the detector's test statistic on the scale measure estimator degrades
its detection performance. This is more emphasised considering the
rapid variations of the clutter signal statistics in many real radar
scenarios. Therefore, we need to delve into the Fisher-consistency
of the mad(uHZk) as a scale measure in the case of correlated
clutter considered in this paper.
To define the Fisher consistency, let Tθ be an estimator for the
parameter θ such that Tθ=f(G
^), where G
^ is the empirical CDF
based on independent sample data and f is a continuous function in
the space of distribution functions. The estimator Tθ is said to be
Fisher-consistent if f(G)=θ, where G is the population CDF from
which the sample data is drawn [17]. From (31), we can outline
two necessary conditions for the Fisher consistency of the
mad(uHZk): (i) the convergence of the sample SD σ
^ to its
population version, and (ii) the convergence of empirical CDF F
^
u
to the population CDF Fu, since kf depends on Fu.
First, we begin the discussion with the Fisher consistency of σ
^
in the case of correlated data. Based on the results of [41], the
convergence rate of σ
^ to its population version σ in the case of
correlated data is lower compared to that of uncorrelated data. This
slowdown depends on the correlation coefficient ρ. More
specifically, for ρ= 0.75, the rate of convergence is 3.5 times
lower than that for the uncorrelated data [41]. Second, we consider
the convergence of the empirical CDF to the population CDF for
correlated data, which is addressed in [42, Theorem 1]. Let {xi}i= 1
n,
be random univariate samples that follow a joint normal
distribution with the correlation matrix Φ. If {xi}i= 1
n are not weakly
correlated, then E[G
^−G]2 does not tend to 0 as n→ ∞.
Based on the aforementioned discussions, σ
^ and F
^
u calculated
from correlated data samples do not converge to their
corresponding population versions, even if the data samples follow
the Gaussian distribution for which σ
^ is the ML estimator of the
SD. Therefore, given the strong correlation shown by the available
experimental data for different clutter environments [39, 40], the
mad uHZk is not a Fisher-consistent scale estimator in the
considered application. To handle this problem, we propose
decorrelating Z before applying (35). The decorrelated secondary
cells are
Zd=Ψ
^−1/ 2Z
(39)
where Ψ
^ is an estimate of the correlation matrix of Z. To keep the
non-parametric characteristic of the PD-NAMF, we use a non-
parametric correlation estimator.
There are two prevalent nonparametric rank correlation
coefficients, namely, the Kendall's and Spearman correlation
coefficients. Compared to the Spearman coefficient, the Kendall's
has a lower bias, shows better accuracy at lower number of
samples, and has a lower mean square error (MSE) for heavily
correlated data [43]. However, calculating the Spearman coefficient
has lower computational complexity than that of the Kendall
coefficient. Generally, the Kendall correlation matrix estimator of
the E-dimensional vector x calculated from the sample data
X∈ ℝE×D is given by [44]
ψ
^jk
K=2
D(D− 1) ∑
i<i′
sign xji −xji′sign xki −xki′
(40)
where ψ
^jk
K is the (j,k)th entry of Ψ
^K, 1 ≤ j,k≤E, 1≤i≤D, and
2≤i′ ≤ D. The Spearman correlation matrix estimator calculated
from the same sample data is [44]
ψ
^jk
S=∑i= 1
Do(xji) − D
¯o(xki) − D
¯
∑i= 1
Do(xji) − D
¯2∑i= 1
L1o(xki) − D
¯2
(41)
where D
¯=D+ 1/2 and o(xki) denotes the order of xki within
xk1, …, xkD.
The correlation estimators in (40) and (41) cannot be directly
applied to the complex-valued secondary cells assumed in this
paper. Based on (8), the correlation matrix Ψ
^∈ ℂMN ×M N is given
by
Ψ
^= 2Ψ
^
zRzR= 2Ψ
^
zIzI
(42)
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where Ψ
^
zRzR and Ψ
^
zIzI are the estimated autocorrelation of ℜ(z) and
ℑ(z), respectively. Equation (42) is applied to both Kendall and
Spearman correlation matrix estimators Ψ
^K and Ψ
^S, respectively.
Remark 1: It should be emphasised that the estimation of the
correlation matrix R
^−1 =pij is not the same as estimating the
covariance matrix R. The estimation of Ψ can be seen as a step to
estimate R; a step that should be followed by estimating the SDs of
the components of zk. To illustrate this, the correlation matrix
Ψ= [ψi j] is related to the covariance matrix R= [ri j] as
rij =σiσjψi j, 1 ≤ i,j≤M N
(43)
where σi, σj are the SDs of the ith and jth components of zk,
respectively. The step of estimating the SDs of zk’s components is
cumbersome for non-Gaussian clutter models in addition to the
need to calibrate the resulting covariance matrix by solving
multiple optimisation problems, as shown in [45]. Under the
proposed algorithm, it suffices to estimate Ψ; avoiding the
complexity of estimating R.
Remark 2: In the case of Gaussian distributed data, both the
Spearman and Kendall coefficients are related to the linear Pearson
correlation coefficient ψ
^jk
P by [44]
ψ
^jk
P= sin π
2ψ
^jk
K= 2sin π
6ψ
^jk
S.
(44)
Nonetheless, we do not use the transformed coefficients for two
reasons. The first is that they are derived for the Gaussian
distributed data, while we do not make any assumptions about the
distribution of the received signal vector. The second is that the
transformations in (44) do not guarantee the positive semi-
definiteness of the estimated matrices [46], in contrast to the
estimators in (40) and (41).
The flow of the PD-NAMF with both of Kendall and Spearman
decorrelation matrices is shown in Figs. 3 and 4, respectively.
5 Performance assessment
In this section, the performance of the PD-NAMF in (35) is
compared to that of the NAMF detector in (17) using Monte Carlo
simulations. To justify the robustness of the PD-NAMF, we
evaluate its performance with different clutter distributions and
signal configurations. Moreover, we study the different choices for
the algorithm parameters; specifically, the type of correlation
estimator used and the minimum required a number of projections.
Finally, we investigate the complexity and the execution time of
the PD-NAMF compared to the NAMF detector for different
design parameters.
5.1 Simulation parameters
The simulated radar signal has a fixed dimension J= 16 and L1 is
either 65 or 33 cells. The NHD is applied on a sequential basis on
each secondary cell where R
^ and mad(uHZk) are estimated from the
remaining L2= 64 or 32 cells. An interfering target is injected in a
secondary CUT, representing a non-homogeneous cell, with a
normalised Doppler frequency fd= 0.3 and azimuth angle
θt= 35°.
The Kendall correlation matrix Ψ
^K is estimated once from all
the L1 secondary cells Z including the secondary CUT zk, i.e. it is
not recalculated for each zk. However, the Spearman correlation
matrix Ψ
^S, due to its lower immunity to outliers, is calculated for
each secondary CUT zk from the remaining secondary cells. If the
secondary CUT is included in the calculations of Ψ
^S, a self-nulling
effect appears at the output of the detector, especially at low
number of secondary cells and/or high interfering target's power.
Regarding the clutter vectors, they are generated as proper
complex SIRP vectors with independent quadrature and in-phase
components. For the clutter's envelope distribution, we consider
two extreme cases: K-distributed clutter with α= 0.1, which
represents heavy-tailed spiky clutter, and Gaussian clutter. For the
K-distributed clutter, δ is allowed to be randomly and
independently changed from a range cell to another as indicated by
Michels et al. [14].
As for the value of δ, Melebari et al. [47] give the measured
values between (0, 1], which is the range of values considered in
most of the relevant work in the literature, where only the shape
parameter is considered to have an impact on the detection
performance [6]. However, Antipov [48] provides experimental
data showing that 1≤δ≤ 2. Therefore, we examine the
performance of both the NAMF and the PD-NAMF in K-
distributed clutter with the foregoing two cases of δ for each range
cell: δ∼U(0, 1], as a default case, and δ∼U[1, 2], where U
denotes the uniform distribution. The average clutter-to-noise ratio
(CNR) is assumed to be 20 dB. The one-lag spatial and temporal
correlation coefficients of the clutter in (37) and (38) are
ρs=ρt= 0.99 [38, 40], unless other values are specified.
The projection vectors are generated randomly over the J-
dimensional unit hypersphere as defined in Proposition 1. The
Fig. 2 RSD of kf at different number of samples (secondary cells) with
J= 20
Fig. 3 Algorithm 1a: Using Kendall
Fig. 4 Algorithm 1b: Using Spearman
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default number of projections is Q= 4J, however, we consider
other values for Q later in this section. For all detection
performance simulations, PF is set to 0.01. The probability of
detection PD is evaluated versus the input signal-to-noise (SNR) of
the interfering target using Monte Carlo simulation with 105 trials.
5.2 Results
5.2.1 Low-dimensional case J/L2= 0.25 : The STAP radar
signal is considered low-dimensional when J≤ 0.5L2. Fig. 5 shows
the detection performance of the PD-NAMF and the NAMF
detectors in K-distributed clutter with α= 0.1 and δ∼U(0, 1]. We
can observe that the detection performance of the PD-NAMF is
comparable to that of the NAMF detector with a maximum loss in
PD of 0.051 in the case of Ψ
^S. For SNR > − 7 dB, it is also
observed that using Ψ
^K provides a relative improvement over Ψ
^S,
narrowing the loss in PD relative to the NAMF detector to 0.036.
The difference in the performance between the NAMF detector and
the PD-NAMF is getting narrower for SNR values beyond 5 dB
and below−6 dB. While the observed slight advantage of the
NAMF over the PD-NAMF is not common among all the cases
studied in this paper, it also comes with the cost of much higher
complexity, as we show shortly. In Fig. 5, we can also point out
that the decorrelation is not only dictated by the theoretical need to
achieve the Fisher consistency of mad uHZk, but it also has a
significant effect on the performance of the PD-NAMF.
The performance in Gaussian distributed clutter is shown in
Fig. 6. Compared to the NAMF detector, the PD-NAMF has a
maximum loss in PD of 0.051 with both Ψ
^S and Ψ
^K. Moreover, the
performance of the PD-NAMF with Ψ
^S is similar to, or slightly
better than that with Ψ
^K.
To validate our claim of the robustness of the PD-NAMF using
the MAD, we replaced MAD by SD in (32) and we evaluated the
resulting detection performance in both Gaussian and K-distributed
clutters. The results are shown in Fig. 7. As we can observe, the
performance of the PD-NAMF using the SD is almost equivalent to
that using the MAD in the case of the Gaussian distribution for
both Spearman Ψ
^S and Kendall Ψ
^K decorrelation matrices. This is
attributed to the equivalence of the SD and MAD, up to a constant
kf, in the case of the Gaussian distribution as we indicate in (31),
which is based on [31]. In the case of the K-distribution, however,
the performance of the PD-NAMF with both Ψ
^S and Ψ
^K degrades
when SD is used in place of MAD. This is consistent with the
theoretical reasoning provided in Section 4.3. We can also notice
that when using SD, the PD-NAMF with Ψ
^K is more robust than
the one with Ψ
^K, which is ascribed to the higher robustness of the
former in the presence of outliers as shown in [49].
It is important to demonstrate the performance of the original
GIP and the PD-GIP detectors compared to the PD-NAMF in
simulation. Interestingly, to the best of our knowledge, the
detection performance of GIP in correlated compound Gaussian
clutter has not been investigated in the open literature.
Furthermore, the detection performance of the PD-GIP has not yet
been investigated. For a fair comparison between the PD-GIP and
the proposed PD-NAMF, we use a modified version of PD-GIP
that differs from the one originally proposed in [16] in the
following ways: the secondary cells are decorrelated using Ψ
^K, and
the projection vectors are generated in the same way as in the
proposed PD-NAMF. To make this point clear, we refer to this
detector as the ‘modified PD-GIP’. Fig. 8 shows the performance
of GIP, PD-GIP, modified PD-GIP, and PD-NAMF for both
Gaussian and K-distributed clutters. We first note from the figure
that the modifications made to the PD-GIP contribute to improve
its performance. Furthermore, both the GIP and the modified PD-
GIP show a performance degradation in the case of Gaussian
clutter of ∼5 and 7 dB compared to the proposed PD-NAMF,
respectively. However, this degradation is much greater in the case
of K-distributed clutter. These results are consistent with the false
alarm results for GIP presented in [6].
The effect of the scale parameter on the detection performance
of both detectors can be observed in Fig. 9. The PD-NAMF
performs approximately the same with both Ψ
^S and Ψ
^K. The
maximum detection loss by the PD-NAMF relative to NAMF is
Fig. 5 Detection performance in K-distributed clutter
α= 0.1, δ∼U(0, 1], J= 16, L2= 64
Fig. 6 Detection performance in Gaussian clutter J= 16, L2= 64
Fig. 7 Detection performance in Gaussian and K-distributed clutter for
PD-NAMF with MAD and SD J= 16, L2= 64
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0.034. In general, the detection performance of both the PD-NAMF
and NAMF detectors are considerably affected by the change in the
scale parameter of the clutter. In addition to the typical choice of
0.99 for both ρs and ρt, we consider the case of a low value
ρs=ρt= 0.2 for these coefficients. The results are also shown in
Fig. 9. The lower correlation coefficients of the clutter lead to a
degraded detection performance, which coincides with the results
in [50, Fig. 8].
5.2.2 Higher-dimensional case J/L2= 0.5 : The performance
of the PD-NAMF is investigated at a higher-dimensional case,
where L2= 32 and J= 16. As we notice in Fig. 10, in the presence
of K-distributed clutter, the PD-NAMF with Ψ
^K provides a relative
advantage over the NAMF at SNR<–6 dB, with a maximum
increase in PD of 0.06. Beyond this point, the maximum loss in
detection of the PD-NAMF with Ψ
^K relative to NAMF detector is
0.03. With Ψ
^S the PD-NAMF shows a maximum loss in PD of
0.061 relative to the NAMF for −10 dB ≤ SNR ≤ 0 dB. It is
noteworthy that the overall performance of both detectors is
relatively degraded by lowering L2/J as we observe by comparing
the performance of each detector in Fig. 5 with its counterpart in
Fig. 10.
As Fig. 11 depicts for the Gaussian clutter, PD of the PD-NAMF
with Ψ
^K is higher than that of the NAMF with a maximum
difference of 0.16. It is noteworthy that this improvement in PD
provided by the PD-NAMF is higher than any loss it shows relative
to the NAMF in the previous cases. When the PD-NAMF uses Ψ
^S,
it shows a maximum improvement of 0.014 over the NAMF.
Beyond the crossover point at SNR = −6 dB, the PD-NAMF with
Ψ
^S shows a comparable detection performance to the NAMF with
a maximum loss in PD of 0.041.
To summarise, using Ψ
^K with the PD-NAMF improves the
detection performance over that of the NAMF detector in case of
high-dimensional signals at all SNR values in Gaussian clutter and
at lower SNR values for K-distributed clutter. This is explained by
the robustness of the Kendall's coefficient in small sample
conditions as mentioned before.
5.2.3 Number of projections: Theoretically, the higher the
number of random projections Q, the more accurate (25) holds
[30]. However, in practice, the used number of projections should
be as small as possible for fast computations. Unfortunately, there
is no analytical method to determine the minimum number of
projections required for (25) to hold at a given approximation
level; consequently, simulations are used to determine this value as
in [30, 34]. While the simulations in these references are concerned
with the convergence of (25), the simulation in this paper is
concerned with maximising PD at a given level of false alarm.
Fig. 12 illustrates the power of a secondary CUT with
homogeneous interference (clutter and noise) or an interfering
target's signal power at the output of the PD-NAMF with Ψ
^K and
Ψ
^S, as a function of the ratio Q/J, averaged over 105 trials. The
powers of both interference and target signals of the PD-NAMF,
with both correlation estimators, are normalised with respect to
those of the NAMF detectors, respectively. The simulated clutter
Fig. 8 Detection performance in Gaussian and K-distributed clutter for
GIP, modified PD-GIP, and PD-NAMF J= 16, L2= 64
Fig. 9 Detection performance in K-distributed clutter (α= 0.1,
δ∼U[1, 2] or U(0, 1], J= 16, L2= 64, ρs=ρt= 0.99 or 0.2)
Fig. 10 Detection performance in K-distributed clutter
α= 0.1, δ∼U(0, 1], J= 16, L2= 32
Fig. 11 Detection performance in Gaussian distributed clutter
J= 16, L2= 32
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envelope follows the K-distribution and the radar signal is low-
dimensional J=L/4 . As shown in Fig. 12, using Ψ
^K results in a
lower interference power level at the output of the PD-NAMF than
Ψ
^S, which explains its superior performance relative to the latter.
The interference level of both of the correlation estimators
decreases as Q increases up to Q= 4J, beyond this point the
interference level is almost constant with Ψ
^S while it decreases
slightly using Ψ
^K. In general, it is also obvious that the interference
power at the output of the PD-NAMF for Ψ
^K and Ψ
^S is lower than
that at the output of the NAMF detector. This demonstrates the
validity of the proof in Appendix 1. Fig. 12 also reveals that Q has
a negligible effect on the target signal level at the output of the PD-
NAMF for both decorrelators with a relative higher target's signal
level for Ψ
^K than that of Ψ
^S.
The final choice of the minimum required Q is based on the
detection performance of the PD-NAMF as depicted in Fig. 13,
where Ψ
^S is used in the presence of K-distributed clutter
δ∼U(0, 1] and J=L2/4. We can see that the increase of Q
beyond 4J has a negligible impact on the detection performance
and for most values of SNR there is no difference in the
performance. When Q is reduced to 2J, PD decreases slightly with
a maximum loss of 0.022. The same is shown for Ψ
^K in Fig. 14,
but with a slight improvement with Q= 10J at lower SNR values
even over the NAMF. To investigate the dependence of Q on the
clutter distribution, we performed additional simulations for
different values of Q, but in the presence of Gaussian clutter. As
shown in Figs. 15 and 16, for both Ψ
^K and Ψ
^S, the performances of
the PD-NAMF in the Gaussian clutter for different values of Q
exhibit the same trend as in the K-distributed clutter shown in
Figs. 13 and 14. Therefore, we can conclude that using Q= 4J
projections is an appropriate rule of thumb that does not depend on
the clutter distribution.
Fig. 12 Interference and the interfering target powers at the output of the
PD-NAMF in K-distributed clutter α= 0.1, δ∼U(0, 1], J= 16, L2= 64
Fig. 13 Detection performance in K-distributed clutter with different Q
values (Ψ
^S, α= 0.1, δ∼U(0, 1], J= 16, L2= 64 )
Fig. 14 Detection performance in K-distributed clutter with different Q
values (Ψ
^K, α= 0.1, δ∼U(0, 1], J= 16, L2= 64)
Fig. 15 Detection performance in Gaussian distributed clutter with
different Q (Ψ
^K, J= 16, L2= 64)
Fig. 16 Detection performance in Gaussian distributed clutter with
different Q (Ψ
^S, J= 16, L2= 64)
2116 IET Radar Sonar Navig., 2019, Vol. 13 Iss. 12, pp. 2107-2119
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5.2.4 Complexity analysis: The complexity of the PD-NAMF
(with both Ψ
^K and Ψ
^S) compared to the NAMF is analysed in
terms of the required arithmetic operations and the run time of each
detector. Table 1 summarises the mathematical operations
performed by the PD-NAMF compared to the NAMF for each
secondary cell. The reported complexities are based on the
Gaussian–Jordan elimination, Schoolbook, and merge sort
algorithms for matrix inversion, matrix multiplication, and sorting,
respectively [51]. By 1/L1 we mean that Ψ
^K is calculated once for
all the L1 secondary cells and not for each cell in contrast to Ψ
^S,
which is estimated for each secondary cell from the remaining L2
cells.
The computation reduction is more obvious in Fig. 17, where
the run times of the NAMF detector and the PD-NAMF are
computed on the same platform dedicated only for this job. For
both versions of the PD-NAMF with Ψ
^S and Ψ
^K, the figure shows
their average run times normalised by the run time of NAMF with
L2= 4J secondary cells for different J. It is conspicuous that the
PD-NAMF, either with Ψ
^S or Ψ
^K, substantially reduces the NHD
run time depending on J. The larger the dimension of the cell J
(and consequently L2), the greater the reduction, which is of a great
importance for modern radar systems with large antenna arrays.
Remark 2: The complexity of the PD-NAMF can be reduced
further using parallel processing, given the independence of the
random projections from each other. Moreover, the use of parallel
programming on graphical processing units (GPUs) can reduce the
complexity of calculating the median, and consequently the MAD,
as in [52]. Furthermore, the calculation of Ψ
^K, the most
computationally demanding step, can be parallelised as well [53].
Nevertheless, the parallelisation is only possible partially in the
NAMF due to the iterative nature of the robust covariance
estimators.
6 Conclusion
In this paper, we introduced a novel covariance-free, non-
parametric NHD detector for correlated clutter environments with
Gaussian and non-Gaussian distributions. Based on the projection
depth function, the proposed PD-NAMF avoids the
computationally expensive estimation of the covariance matrix.
Interestingly, the larger the dimension of the radar signal vector,
the higher the computation reduction the PD-NAMF provides
relative to the NAMF. This advantage fosters the application of the
PD-NAMF in modern radars with large antenna arrays. Further,
this significant complexity reduction is not achieved at the expense
of degraded performance. That is, the detection performance of the
new detector is shown to be comparable to, and in some cases
better than, the full adaptive NAMF detector at different
dimensions and clutter distributions. With this robust performance
and the considerable reduction in computations, the PD-NAMF is
superior to its covariance-based counterparts in the literature for
real-time applications and it can be a more efficient replacement of
the computationally demanding GIP and NAMF detectors in
iterative NHD approaches. The feasible utilisation of parallel
processing and GPUs paves the way for more efficient
implementations of the PD-NAMF in the future.
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8 Appendix
8.1 Appendix 1: Proof of Proposition 1
Proof: Assuming that the covariance matrix R
^ is positive
definite, it is invertible and admits of a square-root R
^1/2, which is
also invertible. Hence, applying the Cauchy–Schwartz inequality
uHzk
2=uHR
^1
2R
^−1
2zk
2
≤ ∥ uHR
^1
2∥2∥R
^−1
2zk∥2
≤ (uHR
^u)(zk
HR−1zk)
(45)
Consequently, for any non-zero vector u∈ ℂJ, we have
uHzk
2
uHR
^u≤zk
HR
^−1zk
(46)
Hence
sup
∥u∥ = 1
uHzk
2
uHR
^u=zk
HR
^−1zk
(47)
Among the possible scale measures of the scalar random variable
uHzk, let us consider the sample variance σ
^2(uHZk)=uHR
^u. Then
according to (47)
sup
∥u∥ = 1 (uHzk
σ
^(uHZk))2=zk
HR
^−1zk
(48)
Based on (9), σ
^(uHZk) is given by
σ
^(uHZk) = 2σ
^(ℜ(uHZk)) = 2σ
^(ℑ(uHZk))
(49)
Repeating the same steps from (45)–(48), but with replacing zk by
s, a similar relation can be proven for the steering vector s, i.e.
sup
∥u∥ = 1
uHs
σ
^(uHZk)
2
=sHR
^−1s
(50)
For the centralised vector zk−μ
^, we can write
sup
∥u∥ = 1
uHzk−μ
^
σ
^(uHZk)
2
= (zk−μ
^)HR
^−1(zk−μ
^) ◻
(51)
8.2 Appendix 2: Proof of proposition 2
Proof: Considering the kth secondary cell zk=as+c+n,
where for convenience we let s= [s1, …, sJ]T, c= [c1, …, cJ]T, and
n= [n1, …, nJ]T, then we have
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sHzksHR
^−1s=a∑
i= 1
J
sisi
*+∑
i= 1
J
si
*ci+ni
.∑
i= 1
J
(si
*∑
j= 1
J
pijsj)
(52)
where R
^−1 =pij . From (3), ∑i= 1
Jsisi
*= 1, hence
sHzksHR
^−1s=a∑
i= 1
J
si
*∑
j= 1
J
pijsj
+∑
j= 1
J
sj
*∑
i= 1
J
si
*∑
k= 1
J
pikskcj+nj
(53)
sHR
^−1zk=a∑
i= 1
J
si
*∑
j= 1
J
pijsj+∑
i= 1
J
si
*∑
j= 1
J
pij cj+nj
(54)
From (53) and (54), we can observe that the two tests have the
same target signal component (i.e. the first term in each equation,
which is equivalent to a(sHR
^−1s)), but they differ in the interference
component. There is no analytical way to compare the interference
components in the two tests due to the different random weights of
each term, hence, we rely on simulation to compare them.
Nevertheless, as (53) considers only the interference component in
the spatio-temporal direction of the target, its average interference
power is anticipated to be lower than that of (54), which is
confirmed by simulation in Section 5. □
8.3 Appendix 3: Proof of proposition 3
Proof: Using the term in (27), we obtain the modified NAMF
test statistic
ΛNAMF
′=sHR
^−1(sHzk)s2
(sHR
^−1s)(zk
HR
^−1zk)
(55)
Based on (50), we have
(sHzk)sHR
^−1s
2= (sHzk) sup
∥u∥ = 1
uHs
σ
^(uHZk)
22
(56)
By substitution of (48), (50), and (56) into (55) and after simple
manipulations, ΛNAMF
′ reduces to
ΛNAMF
′=
sHzk
2
sup∥u∥ = 1
uHs
σ
^(uHZk)
2
sup∥u∥ = 1
uHzk
σ
^(uHZk)
2◻
(57)
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