-
Can addition of noise improve distributed
detectionperformance?
Hao Chen, Pramod K. Varshney James H. MichelsDepartment of EECS
JHM Technologies
335 Link Hall, Syracuse University P.O. Box 4142Syracuse, NY
13244 Ithaca, NY 14852
hchen21{varshney}(syr.edu jmichels Oamericu.net
Steven M. KayECE Department
University of Rhode IslandKingston, RI 02881kay Qele.uri.edu
Abstract - Stochastic resonance (SR), a nonlinearphysical
phenomenon in which the performance ofsome nonlinear systems can be
enhanced by addingsuitable noise, has been observed and applied
inmany areas. However, it has not been shown whetheror not this
phenomenon plays a role in distributeddetection. It seems
counterintuitive that adding ad-ditional noise to the received
decisions at the fusioncenter can improve detection performance.
How-ever, in this paper, we demonstrate the existence ofthe SR
phenomenon in decision fusion by examples.An explanation for its
existence is provided.
Keywords: Stochastic resonance, distributed detection,decision
fusion.
1 Introduction
Stochastic Resonance (SR) is a nonlinear phenomenonfirst
reported and analyzed in [1] in terms of a nonlineardynamic effect
where the system performance can beenhanced by adding suitable
noise under certain condi-tions. Since then, the SR effect has been
observed andapplied in a wide range of applications [2]
includingaudio systems, neural networks, hyperspectral imag-ing,
neuroscience, medical imaging, and visual percep-tion. The classic
SR signature is the signal-to-noiseratio (SNR) gain of certain
nonlinear systems, i.e, theoutput SNR is significantly higher than
the input SNRwhen an appropriate amount of noise is added [3,
2].Some approaches have been proposed to tune the SRsystem by
maximizing SNR [4, 5, 6, 7]. SR was alsofound to enhance the mutual
information (MI) betweeninput and output signals [8, 9]. Although
it has beenshown that the capacity of a SR channel can not
exceedthe actual capacity at the input, Mitaim and Kosko [9]showed
that almost all noise probability density func-tions produce some
SR effect in threshold neurons anda new statistically robust
learning law was proposed
to find the optimal noise level. Compared to SNR, MIis more
directly correlated with the transferred inputsignal
information.
In signal detection theory, SR also plays a very im-portant role
in improving the signal detectability. In[10] and [11], improvement
of detection performanceof a weak sinusoidal signal is reported. To
detect aDC signal in a Gaussian mixture noise background,Kay [12]
showed that under certain conditions, perfor-mance of the sign
detector can be enhanced by addingsome white Gaussian noise. For a
more general twohypotheses detection problem, the underlying
mecha-nism of the stochastic resonance phenomenon is beingexplored
[13]. The signal detection optimization prob-lem involving the
determination of the stochastic res-onance probability density
function (pdf) for a fixeddetector was solved and reported in
[14].
Despite the progress achieved over the past twodecades, it has
not been shown whether this phenom-enon plays a role in distributed
detection. In this pa-per, we investigate the existence of the SR
effect in dis-tributed detection systems for the two hypotheses
de-tection problem. We restrict ourselves to binary localsensor
outputs, denoted by Uk, and assume conditionalindependence among
sensor observations. The perfor-mance degradation of detection
performance caused bytransmission errors between local sensor
outputs andthe fusion center is assessed. The relationship
betweenthe additive SR noise and system performance is ex-plored.
For the traditional two-stage approach usingthe Chair-Varshney
fusion rule [15], the role of additiveSR noise at both the decoding
stage and the decisionstage is discussed. We show that the SR
phenomenonexists under certain circumstances, for both cases.
The paper is organized as follows. In Section 2,the detection
framework using stochastic resonance isbriefly discussed. The
channel model used in this pa-per is discussed in Section 3. The
existence of SR effectis demonstrated by two constructive
distributed detec-tion examples in Section 4. Conclusions and
further
-
comments are given in Section 5.
2 Stochastic Resonance in De-tection
We briefly summarize the mathematical framework toanalyze the
stochastic resonance (SR) effect in binaryhypothesis testing
problems [13, 14]. Given a N dimen-sional data vector x e RN, we
have to decide betweentwo hypotheses H1 or Ho,
In general, two different fusion rules are applicable atthe
fusion center depending on the different definitionsof the output
x.
For the traditional two-stage approach, the outputof each
transmission channel Xk is the estimate of Uk.In other words, the
kth channel can be described asa binary channel with crossover
error probabilities akand Qk. The fusion rule -Y,, assuming perfect
connec-tions between the local sensors and the fusion center,is
given by
{ Ho: px(x; HO)Hl: px(x; Hi)
Po (x)P1 (x)
(1)
where po (x) and P1 (x) are the pdfs of x under Ho andH1,
respectively. In order to make a decision, a testwhich can be
completely characterized by a criticalfunction (decision function)
X where 0
-
1 kHo 0 0
XkUk
PI)k
Figure 1: Parallel fusion model
andX2k
p(xkUk 0) 2w7(l + 2r2)eE l JUakXki (akxk)2 (a )]
1 k
Figure 2: A two-layer transmission channel model fora
distributed detection system
Figure 3: Channel model for the signal detection prob-lem for
local sensor k
(10)H0
PC14k
0
where ak and Q(x) = 1- 2 dt isUkV12u2 fx V2w7akthe complementary
distribution function of the stan-dard Gaussian distribution.
Several decision fusion rules that require differentdegrees of a
priori knowledge have been proposed in[16] and [17]. We summarize
the test statistics for afew of them here.
1. Chair-Varshney Fusion Rule.
T3 = E 0log (1 pk)PAk I(Xk), (11)
where I(x) = 1,x>= 0 and I(x) 0,x < 0 is anindicator
function.
2. Equal Gain Combining (EGC) Fusion Statistic.
IKT4=KEXk, (12)
k=1
3. Likelihood Ratio Test Based on Channel Statistics(LRT-CS).
This test is based on the knowledge ofchannel statistics and local
detection performanceindices
K
5 = logk=1
(akxk)21 + 2-7akkeX 2 Q(-akXk)
(akxk)2(1- v2-7akXk 2 Q(akXk) J
(13)
It has been shown that although -3 is near optimalwhen the
channel SNR is high, it suffers significant per-formance loss at
low to moderate channel SNR. How-ever, as shown in the next
section, the detection perfor-mance of -y3 can be improved by
adding an independentSR noise.
4 Noise Enhanced Decision Fu-sion
In this section, we use two examples to demonstratethe possible
SR effect in decision fusion.
First, let us discuss the first decision fusion ap-proach where
an estimate of Uk is obtained before itis sent to the fusion
center. In this particular exam-ple, two sensors are involved in
the system. For sensor1, we assume that PD1 = 0.8, PFA1 = 0.1 and
thedetection performance for sensor 2 is PD2 = 0.95 andPFA2 = 0.05.
We further assume that channel one is aperfect channel while
channel 2 is a noisy channel withcrossover error probabilities a2 =
/32 Therefore,
3.
on the fusion center side, the detection performance ofthe
second sensor is actually equivalent to PD2 = 0.65and PFA2 = 0.35.
The detection performance of fusionrules -Yj and -Y2 is shown in
Fig. 4. Clearly, due to theperformance loss in the noisy channel,
-Yj is no longerthe optimum fusion rule and its detection
performanceis degraded. In order to improve the detection
per-formance of the fusion rule -Y,, we add some noise tothe
observed data x2 to obtain a new data sample Y2.Since x2 is a
discrete random variable, we use the noisybinary channel model with
crossover probabilities aSRand 13SR to generate the new noisy SR
data sampleY2. The fusion performance of -Yj using the new
datasamples Y2 is also plotted in Fig. 4. When aSR = 0and 13SR =
0.5, compared to the original -Y,, a higherPD for this SR modified
fusion system is observed forPFA C [0.07, 0.35]. A similar effect
is also observed forthe parameter setting with aSR = 0.5 and 13SR =
0.Furthermore, it can be shown that performance en-hancement for
the shaded region in Fig. 4 is possibleby adding suitable SR
noise.
In the next example, in order to examine the pos-sible SR effect
in decision fusion in a wireless sensornetwork, we choose the
number of sensors K = 8,PDk = 0.5 and PFAk = 0.05 for each sensor.
The SRnoise n here is chosen to be a DC value A, i.e., instead
H11
Xk
1
PDk
I . I
-
0.2 0.4 0.6 0.8
FA
Figure 4: Detection performance comparison of differ-ent fusion
rules and SR noise
35 T I
30~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
,,,,,,,,.....I30 - '- -
25
.0a)
C:t 0
15
10
5
d wa-10
Figure 5:A =-0.2.
-5 0 5 10 15SNR(dB)
Deflection coefficient for dif
LRT-CSEGC
S3SR 73
0,-05 0
A05
Figure 6: Deflection coefficient for SR enhanced -3 de-cision
fusion for different channel SNR and A.
tion of SNR. When SNR is very high, Ao 0 which isconsistent with
the conclusion drawn in [16, 17] wherethe asymptotic optimum of -3
is proved; i.e., SR noisewill not improve performance in very high
SNR.
Fig. 7 gives the ROC curves corresponding to dif-ferent fusion
statistics at channel SNR of 5dB. Clearly,the SR modified -3 fusion
rule provides a better detec-tion performance than both EGC and the
original -3rule.
20 25 To explain this SR enhanced detection phenomenon,we first
obtain the relationship between the Rayleighfading channel model
and the binary channel model.
ferent statistics. Corresponding to the transmission channel
model il-lustrated in Fig.2, we have, for '3,
of using the original observed data Xk to perform deci-sion
fusion using -3, new SR modified data Yk = X+Ais used. Due to the
computational complexity of thisdetection problem, the detection
performance evalua-tion is obtained by intensive Monte Carlo
simulations.Fig.5 gives the deflection measures [18] defined as
D (a) = [E(' 1Ho) -E('y H1 )]Var('y Ho) (14)
of different fusion rules where the SR noise A =-0.2.As we can
see from Fig.5, for most values of channelSNR, by adding a
stochastic resonance noise n = A =-0.2 to the observed data x, the
deflection coefficientis improved. One very interesting observation
is thatwhen the channel SNR is between 10dB and 20dB, thedeflection
coefficient of SR modified -3 is even higherthan that of LRT-CS.
However, this does not implythat the detection performance of SR
modified -3 isbetter than LRT-CS since these test statistics are
notGaussian distributed.
For a fixed channel SNR and SR enhanced -3 deci-sion fusion, the
relationship between different values Aand deflection coefficient D
is shown in Fig. 6. WhenA starts becoming negative, first the
deflection coef-ficient D increases and then after attaining its
peak,it decreases when A decreases further. The optimumvalue of A,
Ao, that maximizes D is an increasing func-
k = p(I(yk) = Uk = 0), (15)
and
3k= P(I(Yk) = Uk = 1) = P-p(I(Yk) = IUk = 1)(16)
From (9) and (10), after some calculation, we have
P(I(Yk) Uk 1) p((Xk + A)> 0Uk 1)P(Xk > -A 1)+00
2(7k2 2 2k I akt2wQ(-a t) dt
2 1 + 272) L
A Q(Aak)A2Q( )+ xexp( A2 2) (17)and ~-1+2or
and
P(I(Yk) = 1 1Uk = 0) =
-Q(_A ) _ Q(-Aak) exp(-2+
A2
1 + 2o72 ) (18)
From (17) and (18), it can be shown that akmonotonically
decreases and /3k monotonically in-
creases when A decreases. An illustration of such re-lationship
is shown in Fig. 8 for the case of channelSNR 5dB. Also, from (6)
and (7), it can be shownthat for any fixed channel SNR and the
probability of
0.9
0.8
35
30
0.6
a.0.5
0.4
0.3
0.2
0.1
0
17aSR1 0' PSR1 = 0.5aSR1 0.5' PSR1 =
SNR = 5dBSNR = 10dBSNR = 25dB
C 25.5
ci) 20-0C)
. 150
o- 10
-
equivalence between this fading channel and the binarychannel
model for the widely used two-stage Chair-Varshney fusion rule. We
further demonstrated theexistence of the SR phenomenon in this
fusion problemby adding a discrete DC value to the observed
signalon the fusion center side. A significanlt improvement
ofdetection performance is reported when suitable noise
---LRT-CS is selected.--EGC
y3 SR AcknowledgmentThis work was supported by AFOSR under
contract
1o-4 io-3 102 101i 100 FA9550-05-C-0139.pFA
Figure 7: ROC curves for various fusion statistics.
SNR= 5dB,A = 0.2.
0.71
0.6-
ci)
L-Q
-0
0
ci)
0
C)
0.1
0.5 0
A
0.5
Figure 8: The equivalent channel crossover error prob-
abilities as a function of A, SNR= 5dB
false alarm PFA, the probability of detection PD given
by the SR modified fusion rule -i3 is determined by thecrossover
error probabilities ak and 13k, k= 1, 2,..- , Kwhich are functions
of A. Therefore, there exists a suit-
able A which yields the best detection performance,i.e.,
maximizes the PD for a given PFA. When A= 0,
Cvk3k =1 1 When SNR is very high,
-*k 0 and ak,/13k -*> 0, the channel Ck becomes anear perfect
channel. As a result, -}3 becomes a near
optimum fusion rule.
5 Concluding Ruemarks
In this paper, we have investigated the detection per-
formance of distributed detection and fusion systems
in the presence of non-ideal transmission channels. For
fusion of decisions transmitted over channels that can
be modeled as a binary channel, we showed that the
detection performance of some decision fusion systems
can be improved by randomly changing the received bi-
nary signal, i.e., by adding stochastic resonance noise.
For the problem of fusion of decisions transmitted
though a Rayleigh fading channel, we established the
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