1
A Neural Solution for Signal Detection In
Non-Gaussian Noise
D. G. Khairnar, S. N. Merchant, U. B. Desai
SPANN Laboratory
Department of Electrical Engineering
Indian Institute of Technology, Bombay,
Mumbai-400 076, India
phone: +(9122) 25720651, email: dgk,merchant,ubdesai @ee.iitb.ac.in
Abstract— In this paper, we suggest a neural network signal detector using radial basis function network for detecting a known signal
in presence of Gaussian and non-Gaussian noise. We employ this RBF
Neural detector to detect the presence or absence of a known signal
corrupted by different Gaussian and non-Gaussian noise components.
In case of non-Gaussian noise, computer simulation results show
that RBF network signal detector has significant improvement in
performance characteristics. Detection capability is better than to
those obtained with multilayer perceptrons and optimum matched
filter detector.
Index Terms— Radial basis function neural network, nonGaussian noise, signal detection.
radial basis function network and we employ this neural
detector to detect the presence or absence of a known signal
corrupted by Gaussian and non-Gaussian noise components.
For many non-Gaussian noise distributions such as double
exponential, Contaminated Gaussian, Cauchy noise etc. We
found that RBF network signal detector performance is very
close to that of MF and BP detector for Gaussian noise.
While, we observed that in non-Gaussian noise environments
the RBF network signal detector show better performance
characteristics and good detection capability compared to
neural detector using BP.
I. I NTRODUCTION
In radar, sonar and communication applications, ideal signals are usually contaminated with non-Gaussian noise. Detection of known signals from noisy observations is an important
area of statistical signal processing with direct applications in
communications fields. Optimum linear detectors, under the
assumption of additive Gaussian noise are suggested in [1]. A
class of locally optimum detectors are used in [2] under the
assumptions of vanishingly small signal strength, large sample
size and independent observation. Recently, neural networks
have been extensively studied and suggested for applications in
many areas of signal processing. Signal detection using neural
network is a recent trend [3] - [6]. In [3] Watterson generalizes
an optimum multilayer perceptron neural receiver for signal
detection. To improve performance of the matched filter in
the presence of impulsive noise, Lippmann and Beckman [4]
employed a neural network as a preprocessor to reduce the
influence of impulsive noise components. Michalopoulou it et
al [5] trained a multilayer neural network to identify one of
orthogonal signals embedded in additive Gaussian noise.
They showed that, for
, operating characteristics of the
neural detector were quite close to those obtained by using
the optimum matched filter detector. Gandhi and Ramamurti
[6], [7] has shown that the neural detector trained using BP
algorithm gives near optimum performance. The performance
of the neural detector using BP algorithm is better than the
Matched Filter (MF) detector, used for detection of Gaussian
and non-Gaussian noise. In our previous work [10], we suggest
the signal detector for two non-Gaussian cases such as Double
exponential and Contaminated Gaussian. In this work , we
explore it further and propose a neural network detector using
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II. P RELIMINARIES
Probability of detection
and the probability of false
alarm
are the two commonly used measures to assess
performance of a signal detector [1] That is,
is defined as
the probability of choosing
given that
is true, and
is defined as the probability of choosing
given that
is
true.
(1)
and
(2)
Consider a data vector
as an input to the detector in Figure 1. Using the additive
observational model, we have
(3)
for the hypothesis that the target signal is present (denoted by
) and
(4)
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for the hypothesis that the signal is absent (denoted by
),
where
is the target signal
vector and
is the noise
vector. The likelihood ratio is defined by
(5)
and
are the jointly condiwhere
tional probability density functions of
under
and
,
respectively. Denoting the decision threshold by , we choose
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as,
Fig. 1.
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where
is a set of basis functions.
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The
layer. When using RBF the basis is
Block diagram of a signal detector.
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(7)
where
with as unknown centers
to be determined. is a symmetric positive definite weighting
matrix of size
.
represents a multivariate Gaussian
distribution with mean vector and covariance matrix . By
using above equations we redefine
as
(8)
We determine the set of weights
and
the set of vectors of centers such that the cost functional,
Fig. 2.
Signal detector based on the RBF.
(9)
(the output of the detector is 1) if
; otherwise,
we choose
(the output of the detector is 0) [2].
With
as the marginal (symmetric) probability density
function (pdf) of
, here we consider the
following pdf’s:
1) Gaussian pdf with
.
2) Double exponential pdf with
.
3) Contaminated Gaussian pdf with
4) Cauchy pdf with
.
where
is a new set of basis
functions. The first term on the right hand side of the
equation may be expressed as the squared Euclidean norm
, where
and
[8].
and
and
..
.
..
.
..
.
..
.
.
and
where
is the nominal variance,
is contaminated
variance,
is the degree of contamination, and
. These non-Gaussian pdfs are
commonly used to model impulsive noises.
III. A RBF N EURAL S IGNAL D ETECTOR
The structure of the signal detector based on an RBF
network is shown in Figure 2 [9]. This neural network signal
detector consists of three layers. The input layer has number
of neurons with a linear function. One hidden layer of
neurons with nonlinear transfer functions such as the Gaussian
function. The output layer has only one neuron whose inputoutput relationship should be such that it approximates the
two possible states. The two bias nodes are included as part
of the network. A real-valued input
to a neuron of the
, where
hidden or output layer produces neural output
. The Gaussian function
that we choose
here is
. The RBF neural
network detector test statistic
may now be expressed
The first step in the learning procedure is to define the
instantaneous value of the the cost function.
(10)
where
is the size of the training sample used to do the
learning, and is the error signal defined by
(11)
We assume
. is to be minimized
with respect to the parameters
, , and
. The cost
function is convex with respect to the linear parameters
, but non convex with respect to the centers
and matrix
. The search for the optimum values of
and
may get stuck at a local minimum in parameter space. The
different learning-parameters assigned updated values to
,
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, and
. RBF networks with supervised learning were
able to exceed substantially the performance of multilayer
perceptrons [8]. After updating at the end of an epoch, the
training is continued for the next epoch and it continues until
the maximum error among all K training patterns is reduced
to a prespecified level.
IV. E XPERIMENTAL R ESULTS AND P ERFORMANCE
E VALUATIONS
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Neural weights are obtained by training the network at 10and
. During simulation,
dB SNR using
the threshold
is set to
, and the bias weight
value that gives a
value in the range
. For
each
value that gives a
value in the above range, the
corresponding
value are also simulated. These
values
are plotted against the corresponding
values to obtain the
receiver operating characteristics. Of course, for a given
value, larger
value implies a better signal detection at that
. The 10-dB-SNR-trained neural network is tested in the
5-dB and 10-dB SNR environment. This latter experiment is
carried out to study the neural detector’s sensitivity to the
training SNR. To achieve 5-dB SNR environment, we keep
at
and sufficiently increase the noise variance
.
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A. Performance in Gaussian Noise (Constant Signal, 10 dB
Performance characteristics of neural detectors using RBF,
MLP and MF detectors are presented in Figure 3 for Gaussian
noise. The RBF and MLP neural detectors are trained using
the constant signal and ramp signal with SNR = 10 dB. And
then both neural detectors and match filter detector are tested
with 10-dB SNR inputs.
Fig. 4.
Performance in Gaussian Noise (Ramp Signal, 10 dB).
(a) Ramp Signal, 10 dB.
Fig. 5.
(b) Ramp Signal, 5 dB.
Performance comparison in double exponential noise.
C. Testing of Signal Detector in Non-Gaussian Noise
In this work, we consider the classical problem of detecting
known signals in non-Gaussian noise. Performance characteristics of RBF and MLP neural detectors are presented at small
false alarm probabilities (in the range
to
) that are of
typical practical interest.
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D. Performance in Double Exponential Noise (Ramp Signal)
Fig. 3.
Performance in Gaussian Noise (Constant Signal,10 dB).
Here we, illustrates performance comparisons of the LR,
MF, LO, and neural detectors using RBF and MLP for
ramp signal embedded in additive double exponential noise.
Figure 5-a and 5-b show the comparison for a 10-dB-SNRtrained neural detectors operating in the 10-dB and 5-dB SNR
environment.
In this testing, the signal detector using RBF network
continues to provide performance improvement, compare to
MLP neural, MF and LO signal detectors.
B. Performance in Gaussian Noise (Ramp Signal,10 dB)
E. Performance in Contaminated Gaussian Noise (Ramp Signal)
For gaussian noise, the receiver operating characteristics
of neural detectors as well as matched filter detectors are
presented in Figure 4. In this case, RBF and MLP neural
detectors are trained using the ramp signal with SNR = 10 dB.
All detectors are then tested with 10-dB SNR inputs. In both
Constant and Ramp Signal cases, the RBF and MLP neural
detectors performance is very close to that of the MF detector.
The same experiment is repeated for the ramp signal
embedded in contaminated Gaussian noise with parameters
,
and
. Figure 6-a and 6-b show the
comparison for a 10-dB-SNR-trained neural detectors operated
in the 10-dB-SNR and 5-dB-SNR environment respectively.
In all cases, we see that both MF and LO detectors perform
similarly and that the neural detector using RBF network
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4
(a) 10 dB SNR.
Fig. 6.
(b) 5 dB SNR.
Performance comparison in contaminated Gaussian noise.
clearly provides the best detection performance compare to
MLP neural detector.
Fig. 8.
Performance in Cauchy Noise (Triangular Signal).
F. Performance in Cauchy Noise (Constant and Triangular
Signal)
In this case, we are not consider SNR as the random variable
is not finite in Cauchy noise. Here, we consider the signal
energy
and
of the Cauchy pdf. Performance
of detector are illustrated in Figure 7 and 8. We observe that
the neural detector using RBF outperforms compare to other
detectors. But for relatively high
values its performance
decreases compare to the matched filter and locally optimum
detectors.
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þEúmÿ ý
(a) NN Trained at 0 dB SNR. (b) NN Trained at 10 dB SNR.
(c) NN Trained at 15 dB SNR.
Fig. 9.
Fig. 7.
Performance in Cauchy Noise (Constant Signal).
Performance comparison in double exponential noise.
function of SNR. The neural detector using RBF network
clearly yields superior performance characteristics in all three
cases.
V. C ONCLUSION
G. Detection Performance as a Function of SNR (Ramp
Signal)
Here we try to study the behavior of MF, LO and RBF,
MLP neural detector’s for fixed and varying values.
The noise variance is set to unity during training and testing.
Here, we consider the case of contaminated Gaussian noise
distribution with and , as before.
, The neural detectors are trained using the ramp signal at 0, 10
and 15-dB SNR. During testing, we adjust the bias weight in
both the neural detector’s to ensure that the neural detector’s
. We set to unity and vary
operation at
for SNR values between 0-15 dB. These probability of
detection values are plotted in Figure 9-a,9-b and 9-c as a
›úÝý ´û þ úœý ´ûwü þ ú
ú ý ýgýjÿ
In this paper radial basis function network is proposed for
known signal detection in non-Gaussian noise. Neural detector
using radial basis function network show better performance
characteristics for many non-Gaussian noise distributions such
as double exponential, contaminated Gaussian and Cauchy
noise. We observed that in non-Gaussian noise environments
the RBF neural network signal detector show good detection
capability compared to neural detector using multilayer perceptron (BP) and conventional signal detectors.
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