Le problème de la détection d'un signal aléatoire noyé dans un bruit additif non gaussien modélisé par un processus sphériquement invariant est adressé. Une structure asymptotiquement optimale pour la détection d'un signal gaussien est synthétisée. Les performances de cette structure de détection sont obtenues par des simulations de Monte Carlo. De plus, des comparaisons sont effectuées avec le détecteur localement optimal et le détecteur optimal dans un bruit gaussien corrélé

TANDA, Mario

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SEIZIÈME COLLOQUE GRETSI — 15-19 SEPTEMBRE 1997 — GRENOBLE 1261
Random signal detection in correlated
non-Gaussian noise
Mario Tanda
Università di Napoli Federico II, Dipartimento di Ingegneria Elettronica,
Via Claudio 21, I-80125 Napoli, Italy
RÉSUMÉ
Le problème de la détection d’un signal aléatoire noyé dans un
bruit additif non gaussien modélisé par un processus sphériquement
invariant est adressé. Une structure asymptotiquement optimale pour
la détection d’un signal gaussien est synthétisée. Les performances
de cette structure de détection sont obtenues par des simulations
de Monte Carlo. De plus, des comparaisons sont effectuées avec le
détecteur localement optimal et le détecteur optimal dans un bruit
gaussien corrélé.
ABSTRACT
The problem of detecting the presence of a random signal embedded
in additive correlated non-Gaussian noise modeled as a spherically
invariant random process is addressed. An asymptotically optimum
structure for the detection of a Gaussian signal is synthesized. The
performance of the detection structure is assessed via Monte Carlo
computer simulations. Moreover, a comparison with the locally
optimum detector and with the fully optimum detector for correlated
Gaussian noise is made.
1 Introduction
The synthesis and the performance analysis of detection al-
gorithms optimized against non-Gaussian noise are generally
very difficult. Therefore, to obtain detection structures of easy
implementation, some simplifying assumptions about the sig-
nal of interest (SOI) and the noise have usually been made. For
example, under the weak-signal assumption, locally optimum
(LO) (that is, optimum for a vanishingly small signal) detec-
tors have been synthesized, which, in the case of noise sam-
ples modeled as independent and identically distributed (IID)
random variables (RVs), are slight modifications of the detec-
tion structures optimized against Gaussian noise. Moreover,
since in weak-signal conditions relatively large sample sizes
are needed to obtain a reasonable value of the detection proba-
bility, asymptotically optimum (AO) detectors have also been
synthesized, whose operating characteristics approach opti-
mality as the sample size increases and the signal amplitude
vanishes (see [3] and references therein).
To study the detection of signals embedded in non-
Gaussian noise without resorting to the assumption (often vi-
olated in practice) of IID noise samples, the noise is modeled
here as a spherically invariant random process (SIRP). The ra-
tionale for adopting the SIRP model is twofold: (i) it is ade-
quate in many application fields, such as underwater and seis-
mology; (ii) it is a generalization of the Gaussian model and
hence retains many of its properties.
As regards the SOI, in this paper it is modeled as a
random process. The random signal model is appropriate
in situations where the signal propagates through turbulent
media or along multiple paths. For example, in underwater
sound detection the random dispersion due to turbulence and
inhomogeneities in propagation media does not allow one
to adopt a deterministic or quasi-deterministic model for the
SOI. Moreover, regardless of the channel characteristics, the
random signal model is also adequate when the understanding
of the signal generating mechanism is insufficient.
Recently, the locally optimum array detector for a random
signal embedded in spherically invariant (SI) noise has been
synthesized [2]. It has been shown that in weak-signal condi-
tions, the LO detector assures a significant performance im-
provement over the LO detector designed for the Gaussian
noise environment. However, for moderate and high values
of the signal-to-noise ratio, the LO detector presents a per-
formance collapse, and then other detection structures need to
be considered. In particular, when only one sensor is utilized,
the LO performs very poorly and worse than the conventional
detector.
In this paper, to propose a detection structure that over-
comes the drawbacks of the LO detector when single chan-
nel detection is considered, the AO detector for a random sig-
nal embedded in additive SI noise is synthesized. At first, an
asymptotic sufficient statistic is derived for an arbitrary ran-
dom signal; then, the AO detector for the case of a Gaussian
SOI is synthesized. The obtained detection structure does not
depend on the noise univariate probability density function
(PDF). However, it requires the knowledge, but for a scale fac-
tor, of the noise correlation matrix. The performance of the
synthesized detection structure is assessed via Monte Carlo
computer simulations, by assuming the generalized Cauchy
model for the univariate PDF of the noise. Moreover, a per-
formance comparison with the locally optimum detector syn-
thesized in [2] and the fully optimum detector designed for
a Gaussian signal embedded in additive correlated Gaussian
noise is made.
1262
2 Synthesis of the detector
The problem of detecting a random signal in additive noise can
be represented by the hypothesis test
H0 : x D z;
H1 : x D v C z;
(1)
where x and z denote N-dimensional column vectors whose
components are samples of the received signal and the noise.
The N components of the vector v are samples of the random
SOI. The vector z is assumed to be an SI random vector, that
is,
z D a ng;
where a is a nonnegative RV independent of ng , which is
a zero-mean Gaussian vector characterized by a covariance
matrix  2g K, with 
2
g the variance of each component of ng .
The signal vector v has zero mean and covariance matrix
K
v
and, moreover, is statistically independent of z.
The closure property of the SI vectors under deterministic
linear transformations assures that the detector synthesized on
the basis of the hypothesis test
H0 : y D w;
H1 : y D sC w;
(2)
retains the optimality properties of the detector synthesized
starting from (1). In (2), y D CÄ1 x and s D CÄ1 v, where
C is an NN invertible lower triangular matrix obtained by
the Cholesky decomposition K D C CT of the matrix K.
The vector w D CÄ1 z, due to the closure property of the SI
vectors, is still an SI vector with the same modulating RV a as
z, zero mean, and covariance matrix
K
w
D E[a2] 2g I;
where E[] denotes statistical expectation and I is the NN
identity matrix.
The optimum test for H0 vs H1, in the sense of maximizing
the detection probability for a fixed false-alarm rate (Neyman-
Pearson criterion), compares the likelihood ratio
3. y/ D
p
y
. y j H1/
p
y
. y j H0/
to some threshold chosen to achieve the desired false-alarm
probability.
Taking into account the adopted noise model, the likelihood
ratio can be written as
3. y/ D
Z C1
0
p
y
. y j H1; a/pa.a/daZ C1
0
p
y
. y j H0; a/pa.a/da
;
where p
a
.a/ is the PDF of the modulating random variable.
Note that the conditional joint PDF p
y
. y j H0; a/ is a
product of Gaussian PDFs:
p
y
. y j H0; a/ D
NY
iD1
1
ag
p
2
exp
 
Ä y
2
i
2 2g a
2
!
Moreover, the conditional joint PDF p
y
. y j H1; a/ can be
written as
p
y
. y j H1; a/ D
Z
RN
p
s
. s/p
y
. y j H1; a; s/d s
D
Z
RN
p
s
. s/
NY
iD1
1
ag
p
2
exp[Ä .yi Ä si/
2
2 2g a
2 ]d s
where p
y
. y j H1; a; s/ is the conditional joint PDF of the N
components of y, which is just the N-dimensional Gaussian
PDF p
w
. yÄ s j a/ of the noise vector w conditioned to a,
and p
s
. s/ is the joint PDF of the N components of the signal
vector s.
Owing to the implementation difficulties of the fully opti-
mum detector, in this paper, an asymptotically optimum struc-
ture of easier implementation is proposed. Such a detector can
be synthesized following an approach similar to that consid-
ered in [1] to obtain a detection structure for weak radar sig-
nals with unknown parameters embedded in SI noise.
At first, the likelihood ratio can be written as
3. y/ D
Z 1
0
aÄN e
Ä q
2a2 W . y; a/ p
a
.a/ da
Z 1
0
aÄN e
Ä q
2a2 p
a
.a/da
; (3)
where
q 1
NX
iD1
y2i
 2g
D 1
 2g
jj y jj2
and
W . y; a/1
Z
RN
e
Ä 1
2 2g a
2 jj s jj2 C
1
 2g a
2 y
T s
p
s
. s/d s;
where T denotes matrix transposition. Then, by substituting
 D q
Na2
in both the integrals in (3), it results that
3. y/ D
Z 1
0
9N ./ 
Ä 12 W

y;
r
q
N

p
a
r
q
N

dZ 1
0
9N ./ 
Ä 12 p
a
r
q
N

d
;
where the function
9N ./ D
.N=2/N=2
0.N=2/
N=2Ä1eÄ
N
2 
has been introduced. Finally, since
lim
N!1
9N ./ D .Ä 1/;
1263
it results that the likelihood ratio can be asymptotically ex-
pressed as
3. y/ D W

y;
r
q
N

D
Z
RN
e
Ä N
2 2g q
jj s jj2 C N
 2g q
yT s
p
s
. s/d s:
Since the obtained AO detection structure depends on the
joint PDF of the N components of the signal vector, to go
further one needs to specify such a PDF. In the sequel, the
SOI is modeled as a zero-mean Gaussian process, so that s
is a Gaussian vector with zero mean and covariance matrix
K s. Thus, after simple manipulations, the AO detector for a
Gaussian signal embedded in SI noise can be expressed as
T AO. y/ D yT
24 I C  K s1
N jj y jj2
!Ä135Ä1y 11
N jj y jj2
Ä ln
(
det
"
I C K s1
N jj y jj2
#)
;
where det[] denotes the determinant of the matrix in the
brackets.
For comparison purposes, let us report the LO detector
statistic for the problem at hand, which was derived in [2]:
T L O. y/ D yT K s y

G2.q/Ä 2 PG.q/
 2g
Ätr K sG.q/; (4)
where trfg denotes the trace of the matrix in the brackets,
PG.q/ is the derivative of G.q/ with respect to the quadratic
form q and the nonlinearity with memory
G.q/ 1
Z C1
0
aÄNÄ2 exp.Ä q
2a2
/pa.a/daZ C1
0
aÄN exp.Ä q
2a2
/pa.a/da
(5)
has been introduced.
Note that, the AO detection structure does not depend on
the PDF of the modulating RV and, consequently, on the
univariate PDF of the noise. Moreover, the knowledge, but for
a scale factor, of the noise correlation matrix is required. On
the other hand, the synthesis of the LO detector requires the
knowledge of the univariate PDF and the correlation matrix
of the noise. As regard the a priori knowledge on the signal
to be detected, the synthesis of the AO detector requires the
knowledge of the joint PDF of the N components of the
signal vector, whereas the implementation of the LO detector
requires only the knowledge of the covariance matrix of the
signal vector.
Finally, let us report the well known optimum (for any value
of N) detection statistic for a Gaussian signal embedded in
Gaussian noise:
T OG. y/ D yT
24 I C  K s
 2g
!Ä135Ä1 y:
3 Performance assessment
The present section is aimed at assessing the performance of
the AO detector synthesized in the previous section. Moreover,
the performance is compared with that of the LO detector for
a random signal embedded in SI noise and with that of the
OG detector, that is, the fully optimum detector for correlated
Gaussian noise.
Since the analytical evaluation of the conditional PDFs of
the decision variable under both hypotheses is an intractable
problem, the performance (detection probability P.d/ for a
fixed false-alarm rate P. f a/) has been carried out by Monte
Carlo simulations.
In the simulations, the generalized Cauchy model has been
considered for the univariate PDF of the noise RVs:
pz.x/ D
.2 2g /
 0. C 12 /p
 0./.2 2g  C x2/C
1
2
; (6)
where 0./ is the gamma function and  is a positive shape
parameter. Note that as  approaches infinity, the resulting
PDF approaches the Gaussian density. Moreover, the variance
of z is given by  2z D  2g =. Ä 1/ for  > 1 and is not finite
for any value of  belonging to the interval .0; 1], which has
not been considered in the simulation experiments. This model
is quite flexible encompassing a wide class of PDFs and,
moreover, adequately models the univariate statistics of certain
data arising from noise sources. For example, it can suitably
model the extremely-low frequency atmospheric noise caused
by lightning strokes.
The PDF of the nonnegative modulating RV a, which
generates the SI noise vector z with univariate PDF given by
(6), has the following expression [4]:
pa.a/ D
2
a2C10./
exp.Ä=a2/ .a > 0/
Therefore, the nonlinearities with memory G.q/ and PG.q/ of
the LO detector (see (4) and (5)) are expressed by
G.q/ D N C 2
2 C q ;
PG.q/ D Ä N C 2
.2 C q/2 :
As regards the correlation of the noise samples, in the
simulations a noise correlation matrix K z whose .im/th
element is
Kz.i Ä m/ D

 Ä 1 
2
g 
jiÄmj; j  j< 1
has been considered.
The components of the SOI vector v have been modeled as
statistically independent Gaussian RVs with zero means and
unit variances and, hence, K s D CÄ1. CÄ1/T .
Under the previous assumptions on noise and SOI, the
signal-to-noise ratio at the input of the whitening filter CÄ1,
say SNR, is given by
SN R D  Ä 1
 2g
:
1264
Figures 1 and 2 present the detection probability P.d/
of all considered detectors as a function of SNR for two
different values of the false-alarm rate (P. f a/ D 10Ä2 in
Fig.1 and P. f a/ D 10Ä3 in Fig.2). In both figures, N=100
samples, a correlation coefficient  D 0:9 and two different
values ( D 1:2 and  D 10) of the noise shape parameter
have been considered. The number of simulation runs per
datum has been assumed to be 100=P. f a/. The results show
that the AO detector outperforms the OG detector on the
whole range of values of SNR. Moreover, although the LO
detector outperforms both AO and OG detectors in very
weak-signal conditions, in such a SNR range the performance
is largely unsatisfactory. Thus, the AO detector overcomes
the performance limits exhibited by the LO detector and
outperforms the OG detector.
References
[1] E. Conte, M. Lops, and G. Ricci, “Distribution-free radar
detection in compound-Gaussian clutter,” in Proceedings
of the International Conference “RADAR ’92”, Brighton,
Great Britain, Oct. 1992, 98-101.
[2] L. Izzo and M. Tanda, “Array detection of random signals
in spherically invariant noise,” J. Acoust. Soc. Am. 94,
2682-2690 (1993).
[3] S.A. Kassam, Signal Detection in Non-Gaussian Noise.
Springer, New York, 1988.
[4] K. Yao, “A representation theorem and its applications to
spherically invariant random processes,” IEEE Trans. Inf.
Theory IT-19, 600-608 (1973).


Random signal detection in correlated non-Gaussian noise

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