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AFRL-HE-WP-TP-2005-0016

AIR FORCE RESEARCH LABORATORY

Enhancement of Stochastic Resonance by Tuning

System Parameters and Adding Noise

Simultaneously

Xingxing Wu

Zhong-Ping Jiang

Polytechnic University

Brooklyn, NY 11201

Daniel Repperger

Human Effectiveness Directorate

Warfighter Interface Division

Wright-Patterson AFB OH 45433-7022

November 2005

20070103048

Human Effectiveness Directorate

Warfighter Interface Division

Collaborative Interfaces Branch

Wright-Patterson AFB OH 45433

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Enhancement of Stochastic Resoncance by Tuning

System Parameters and Adding Noise Simultaneously

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* Xingxing Wu, * Zhong-Ping Jiang, ** Daniel Repperger

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14. ABSTRACT

The stochastic resonance effect can be realized by tuning system parameters or by adding noise. This paper investigates the possibility to

enhance the stochastic resonance effect by tuning system parameters and adding noise simultaneously. First, we use some examples to

demonstrate the situation where only the system parameters or noise can be adjusted for maximizing the stochastic resonance effect. Then,

it is shown using standard optimization theory that the normalized power normal <C > of the bistable double-well system with a periodic

input signal can reach a larger maximal value by tuning the system parameter and adding noise simultaneously. Finally, for the purpose of

practical implementation, searching for the optimal system parameter and noise intensity is realized by an on-line fast-converging

optimization algorithm.

15. SUBJECT TERMS

Stochastic resonance, signal processing, optimization

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Enhancement of Stochastic Resonance by Tuning System

Parameters and Adding Noise Simultaneously

Xingxing Wu, Zhong-Ping Jiang, and Daniel W. Repperger

Abstract- The stochastic resonance effect can be realized

by tuning system parameters or by adding noise. This paper

investigates the possibility to enhance the stochastic

resonance effect by tuning system parameters and adding

noise simultaneously. First, we use some examples to

demonstrate the situation where only the system parameters

or noise can be adjusted for maximizing the stochastic

resonance

effect. Then, it is shown using standard

optimization theory that the normalized power normal < C, >

of the bistable double-well system with aperiodic input signal

can reach a larger maximal value by tuning the system

parameter and adding noise simultaneously. Finally, for the

purpose of practical implementation, searching for the

optimal system parameter and noise intensity Is realised by

an on-line fast-converging optimization algorithm,

and

information [11], are used instead. Over the years,

stochastic resonance has been applied in wide-range of

areas, such as physics, chemistry, biomedical sciences, and

engineering [2-3]. One of the important applications of

stochastic resonance is in signal processing. As a nonlinear

signal processor, it has been used for signal detection [12-

s13, signal transmission [14-15] and signal estimation [162.

In order to realize the stochastic resonance so as to make

the chosen quantifier, e.g. the output signal-to-noise ratio

information-based measures,

such as mutual

(SNR), reach its maximal value, certain conditions must be

satisfied. The traditional way is to adjust the noise intensity

by adding optimal amount of noise. Recently, tuning system

parameters have been demonstrated to be a better method to

realize stochastic resonance, especially when the initial

input noise level already exceeds the resonance region [17-

19]. The output SNR will reach a higher maximal value by

tuning system parameters than by adjusting noise intensity

[18]. Among this research, either the noise intensity is

adjusted or the system parameters are tuned in order to

maximize the chosen measure, but not both. This paper will

investigate the possibility to further increase the maximum

by tuning the system parameters and by adding noise

simultaneously. This will in turn improve the system

performance when used as the nonlinear signal processor

for signal

detection, signal

estimation.

The rest of this paper is organized as follows. In Section

2, we demonstrate the cases when only system parameters

or noise intensity can be adjusted. Section 3 will prove the

possibility to further increase the maximal value of

normalized power norm [8] of the aperiodic stochastic

resonance in bistable double-well system by adjusting

system parameters and noise intensity, based on the

conventional first-order necessary condition and second-

order sufficient condition in optimization theory. Section 4

will provide an on-line fast-convergent optimization

algorithm to search the optimal system parameters and

noise intensity. Section 5 is devoted to verify, via computer

simulations, the improvement

comparing the maximal normalized power norm obtained

by tuning system parameters, by adding noise and by both.

Finally, Section 6 closes the paper with brief concluding

remarks.

Index Terms- stochastic resonance, signal processing, and

optimization.

I. INTRODUCTION

Stochastic resonance (SR) is the phenomenon that the

noise can be used to enhance rather than hinder the

system performance. The noise can excite the richness of

the nonlinearities and provides improved dynamics which

better enables the system to increase signal-to-noise ratio

(SNR) or mutual information. The concept of stochastic

resonance was first proposed by Benzi in 1981, addressing

the problem of the periodically recurrent ice ages [1]. Over

the last two decades, stochastic resonance has been

continuously attracting considerable attention. It is a

ubiquitous and conspicuous phenomenon. Many nonlinear

systems have demonstrated the stochastic resonance effects,

such as discrete systems [4], dynamic systems [2], static

systems [5], coupled systems [6] and random systems [7].

The signal can be periodic [2], aperiodic [8], subthreshold

[2] or suprathreshold [9]. In order to quantify the stochastic

resonance phenomena and reveal the synchronization

between signals and noise, different measures have been

adopted. For the periodic signals, the most commonly used

quantifier is signal-to-noise ratio (SNR) [2]. For aperiodic

signals, cross-correlation measures [10], power norm [8]

transmission or signal

of maximal <c, > by

This work has been partially supported by the Polytechnic CATT Center

sponsored by New York State, NSF grants ECS-009317 and OISE-0408925.

X. Wu and Z. P. Jiang are with the Control and Telecommunications

Research Lab, Electrical and Computer Engineering Department, Polytechnic

University, Brooklyn, NY 11201 USA, (email: xwu03(,utopia.poly.edu,

ziiangacontrol.poly.edu).

D. W. Repperger is with the Air Force Research Laboratory, AFRL/HECP,

Wright-Patterson

AFB, OH

Daniel.Reoer=,c rtwyfb.af.milD.

45433 USA. (email:

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II. STOCHASTIC RESONANCE VIA TUNING SYSTEM

PARAMETERS OR ADDING NOISE

In some stochastic resonance systems, the chosen

measures can be described as a function of both the system

parameters and the noise. This, however, does not ensure

that both system parameters and noise can be adjusted at the

same time to maximize or minimize these quantifiers. In

[20], the output signal-to-noise ratio is affected by the ratio

of noise variance and system parameter (threshold), rather

than by the parameter value and noise individually. The

paper [17] shows that the noise intensity cannot be adjusted

and must be fixed at the initial value (no noise will be

added) in order to minimize the bit error rate (BER) for the

aperiodic stochastic resonance (ASR). In this case, only

system parameters tuning is meaningful. In what follows we

will demonstrate this is also true for the periodic stochastic

resonance in double-well bistable dynamic systems.

The double-well bistable system is described by the

equation [21]:

t=t/T,,

y==x T..-,

- =X,1T.,/D,

A=A4IY

T

. = r / T,

(

For the rescaled y-system, its noise density becomes unit

and the output SNR is:

22

SNR = 'r2 A X,

exp(-

/-- 4)-(

(1/,',)

1/0

Now, the optimization parameters are A and x,,.

Obviously, A should take the maximal value in order to

maximize SNR. From (4), this means that the noise

intensity should be fixed to its initial value D0 which is also

the minimal noise intensity, assuming the signal amplitude

A is not changeable. Again, only system parameter Xb can

be adjusted.

In the next section, we will examine some interesting

situations where both the system parameters and noise

intensity can be adjusted simultaneously to improve the

stochastic resonance effect.

r Xt) = xt)W-

+sO)+r7Q),

(1) III. STOCHASTIC RESONANCE ENHANCEMENT

In [22], the aperiodic stochastic resonance (ASR) was

demonstrated in the bistable-well system. The cross-

Xb

where system parameters Ta > 0, Xb > 0. The periodic input

is s(t) = A cos(2

noise with zero mean average and autocorrelation of

< rq(t)(0) >= 2DQ(t).

For small and slow input signal, the output signal-to-

noise ratio is given by [21]:

SNRA= X/•

e/4 Xp( 1

(DO/rý)

x

Dir

Assuming parameter T', is fixed, the output SNR is a

function of both system parameter Xb and noise intensity D.

One constraint on the noise intensity is that it should not be

less than the initial value Do. There are also constraints on

the parameter Xb. For example, X, > -. ffAI2 for the

subthreshold system [17].

To prove our above claim, assume the SNR is maximized

at the optimal values (X ,,D'). Obviously, there will be a

local maximizer for this optimization problem if both

system parameter Xb and noise intensity D can be adjusted

at the same time. According to the first-order necessary

condition for a local maximizer, we have VSNR(Xb,D) = 0,

and thus:

X' = 8D,X' = 4D

The derived solution (Xb = 0 and D = 0) does not meet

the constraint requirements.

constrained optimization problem has no local maximizer.

This means either system parameter Xb or noise D, but not

both, can be adjusted to maximize the output SNR.

In order to determine which one will take the extremum

and which one is not adjustable, we can rescale the

variables as:

/ T,). 17(t) is an additive Gaussian white correlation measures (power norm Co

power norm CI) were adopted for characterizing the ASR

and normalized

behaviour:

Co =max{St)RQt+ )}

C=.

[S'()]"'{iR(t)-

where S(t) is the zero-mean aperiodic input signal, R(t) is

the mean transition rate of the system.

The symmetric bistable-well system with a fluctuating

barrier is given by [22]:

= -

+(t),

dt

ax

Co ,

(6)

'

(2)

SN ,2

'

2x( t

(7)

where U(x) =-[A-SQ)]x

Usually, A is a constant. Here A will be taken as a system

parmt. A

is a

ussan

parameter. ý(t) is Gaussian white noise with zero mean

average and autocorrelation of <•(t)(s) >= 2DS(t -s). The

angular brackets denote an ensemble average.

In general, the power norm does not have an explicit

expression. For the specific case where the signal amplitude

is small compared with the barrier height, i.e., S(Q)2 << A2,

and S(t) is a Gaussian-distributed signal, <Co> and <C1>

are given as [22]:

<C.>- QAexp[-e + Aý7(S)/12]S (t),

+x

24

is the potential function.

wite ne

th zem

(3)

In other words, this (8)

< C. >(

(exp

's'(t)] - I + o(D)Q-'exp[2 -6? S()]}'2

wherea(D)= K <Q-) >, < RQ) >= Qexp[-O+A2S'(t)/2]

Q= kA/-F2r, 0 = A 2/4D,A = A /2D

Page 5

We will choose <C1> as the objective function to be

maximized. The theoretic expression of<C1> can predict its

real shape, even when the noise intensity is outside the

range of its validity [22]. So, we can form the following

constrained optimization problem:

max < C, >,

subject to S(t)' << A', D > D,

The optimization parameters are the system parameter A

and the noise intensity D. From the expressions of Q, E

and A, we notice that E is a function of Q and A:

Proof- First, it is shown that the first-order necessary

condition has at least one solution:

From the first-order necessary condition:

a 'Ond

a<C,>=0.

ae aQ

We get:

caAQ=l,

(2-2s'A2)exp[s'A']-2

+ k, (2Q-' -caA - 2dAQ-') exp[caQA + A] 0

(15)

(10)

,

(16)

f-2.,•a_ (11) Letting Q-'= ca A, we have

(2-22s2A2)exp[s2A2]-2

E) = 2

-cQA'

(11

(17)

where c=,.-2[/12k is a constant, Q and A are functions of A

and D.

The <C1> can be expressed in term of Q and A:

< >As,

(exp[A's2]_-+k

+cak, (A - 2dA') exp[l + dA2] = 0

Letf(A)denote the left hand of equation (17).

Obviously, f(0) = 0 and f(+co) = _oo. Also:

- = (-4s2A - 4s4A') exp[s2A ]

(12)

1Q-'exp[cAQ+dA2]}) '

whes

2

+kca(l+2s'A2_s A,)exp[l+dA']

,

(18)

where s=[s'(t)]"' " d=S'(t)/2-S'()=--.

Therefore we will be interested in maximizing <C1> at

some nonzero optimal values Q' and AY.

From the simulation, we find that there is a unique local

maximizer for the unconstrained optimization problem, i.e.

(10) without the constraints. Unfortunately, the local

maximizer (Q"

andSA)

cannot meet the constraint

requirements for some input signal. For example, when

s=0.01, we will have A7=142.791, Q'=0.0032, A*=O.014.

It cannot meet the requirement of small signal (s(«), << A').

So we introduce two additional parameters (a>O, b>0)

X2

X4itst b ei

If A -- 0+, we get af > 0* We can conclude that there is

TA

at least one A > 0 satisfying eq. (17).

Now, we need to prove that the solution is unique for

fixed parameters a and b.

Letf,(A)=(2-2sWA')exp[sWA']. The function f,(A) will

decrease monotonically to -

f,(0) = 2. We denote the rest part of the LHS of equation

(17) as function f2 (A) = cak1A(1 + s2A' )exp[l -s

will first increases from zero, and then decreases to zero.

From these special characteristics of f.((A) and f2(A), it

follows readily that equation (17) can only have one positive

- as A -- c,

starting from

2A' / 2]. It

U(x) = -[A - S(t,)](13)

We can then get:

W e an

solution.

2a 4b

(14)

Proposition 2: The parameter a can be used to continuously

adjust the values of Q' and A9 satisfying the first-order

necessary condition of the unconstrained optimization

problem to ensure A* and D* will meet the constraint

requirements.

Proof. From (17) and definitions of f,(A) and f,(A), we

notice that parameter a only affects f, (A), but not f,(A)

w i h i

e r a i g f n t o

which is a decreasing function of A. The increase of

parameter a will increase the value off,(A) for the same

A. This will in turn increase AS. Also, A' will approach

zero when parameter a approaches zero. This means that

AN can be changed continuously by adjusting parameter a.

(xp[s2AI]_

I+kQ,

exp[caaQ+dA'])1,'

where: Q=k kA/V, re O=bA'/4a2D,A=bA/2a2D,O=caAQ.

From (14), one sees that the bigger the parameter a, the

smaller the <C,> will be. Here parameter a will be taken as

a supporting parameter used to adjust the local maximizer

( Q *, A 7 ) t o e a l A * a n d D * t o m e e t t h e c o n s t r a i n t s .

Parameter b. is also taken as a supporting parameter which

will be used to match parameter a to keep the potential

function in good shape and make the optimal noise intensity

D* reasonable. For example, we can let b/ (2a2) be a proper

constant.

f A

h

n r a e o

Proposition 1:

parameters (Q,,A*) satisfying the first-order necessary

condition of this unconstrained optimization problem.

There exists one and only one pair of

Proposition 3: There is one and only one local maximizer

for the unconstrained optimization problem with small

input (s<<1) and properly chosen parameters a and b.