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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