STRUCTURE OF STATISTICAL RELATIONS BETWEEN A GAUSSIAN RANDOM PERTURBATION

AND THE OUTPUT COORDINATE OF A STOCHASTIC SYSTEM

A. N. Malakhov and O. V. Muzychuk UDC 538.56:519.25

We obtain a generalization of the Furutsu-Novikov formula to joint cumulants which describe higher-order statistical relations between a Gaussian process and its functionals. We study the character of the statistical relations between a delta- correlated random perturbation and the output coordinate of an n-th order dynami-

cal system. It is shown, in particular, that in the case of an additive noise, there are only correlation relations, and higher-order cumulants are equal to zero.

In the determination of the stochastic characteristics of dynamical systems which are acted upon by random forces of an arbitrary physical nature, one encounters the problem of

decoupling of the statistical averages which contain the random perturbation and functions of the output coordinates (the state vectors of the system). To obtain one-time character-

istics of the state vector x, it is sufficient to establish a procedure for the decoupling of averages of the form < a ( i ) Z t ( x ) > , where ~(t) is a random force with given stochastic char- acteristics, and Z(x) is a function of the output coordinate. If a more detailed descrip-

tion is needed, for example, the determination is required of higher-order statistical rela- tions described by the joint cumulants*

<a (q) . . . . , a (t~), x ( t t ) . . . . . x ( G ) > ,

or, in general,

<~ ( t , ) , . . . , ~ (t3, z , (x), ..., z , (x)>, (1)

where Zi are functions of X(t~) (functionals of the random perturbation ~(t)), s, n = i, 2, .... In the case of Gaussian process e(t), these cumulants can be found, in principle, by using the Furutsu--Novikov formula [i, 2] and its generalizations obtained below. If, at the same time, the process can be assumed delta correlated, in a number of cases one can

obtain concrete results.

I. Furutsu-Novikov Formula for the Joint Cumulants

We obtain the generalization of the Furutsu-lqovikov formula to the case of joint cumu- lants of the form (i). Let us suppose that ~(t) is a Gaussian random process with zero average value and a given correlation function

<~ ( t , ) ~ ( t2) > = B~( t~ , t2) ,

and Zi(x) are some functions of the state vector of the dynamical system. The joint corre- lations will be decoupled by using the Furutsu--Novikov formula

<~x (t) Z, (x)) = f B,, (t, '=) ~ZtB= (x____~)(,:) d'=, (2)

where the variational derivative is given by

~Z, (x) = OZ~ ~x~ (t) ~ (~) Ox~ ~ (~)

*Here and below, we denote the joint cumulants by the cumulant brackets introduced in [3] :

< Y, ..., Y, Z ..., Z> --- < y,ls] Z,Inl>.

Gor'kii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 26, No. 12, pp. 1546-1551, December, 1983. Original article submitted

November 2, 1982.

1128 0033-8442/83/2612,1128507.50 �9 1984 Plenum Publishing Corporation

(we understand summation over repeating indices). For brevity, we introduce the operator

Equation (2) then takes the form

v S F~ = d~B~( t , ~) ~ (*)

V <~ZI> ~ <F~Z~>. (2a)

Using (2a) we have , f o r t he j o i n t moment o f t he random f o r c e and t he f u n c t i o n s o f the s t a t e v e c t o r o f o r d e r n + 1,

V

<~ Zl...Zn>---=<Fa (Z1.. ,Z, ,)>, (3)

V It should be noted that the action of the operator F~ is similar to the action of the usual differential operator:

V V

F~ (Zt ,.. Zn) : n{Zi ... Zi--t (F=Zi) Zi+i ... Z~} i. (4)

Here, n {...}~ is the symmetrization bracket with respect to the index i, which contains n terms. Using (4), Eq. (3) can be written in the form

, V V

: = F~Z>, <~ fT~> <F~Zn> n<f~-~ (3a)

a s suming t h a t n d e n o t e s a m u l t i p l e i ndex .

We shall show that a similar formula holds for a joint cumulant

V V

<% Z,I.1> = n <Z,ln-~J F~ Z> ~ <F~ (Z,I~))>.

It is known [3] that for a joint moment of a collection of any random variables {F, Z} , we have the cumulant expansion

m

<yz~> = <r, Z,Im]> + ~ C~ <Zk> <r, Z,I~-kl>. (6) k = l

We shall assume that Eq. (5) is valid for the joint cumulants of the form <~,Z,[~]> for m~n--l. Using (3a), (6), and this assumption, we transform the joint cumulant as

of order n + i, namely

(5)

n V

<~, Z,tnl> = <aZn> - - ~ C] <Zk> <~, Z,P'-*I> = n <Z "-1 F , Z > - - k = l

n V

- >] C~ (n - - k) <Zk> <Z,tn--~--~l F~Z> = k = l

V n V

= C . _ , <Zk> <Z,I"- ' -~JF~ Z>). n (<Z,,-1 F~Z> - y , k

k = l

We n o t e , by a c o m p a r i s o n w i t h Eq. (6 ) , t h a t t he e x p r e s s i o n in t he b r a c k e t s i s a cumulan t v

o f t he form <Z,[n-qFc, Z>. Thus, r e l a t i o n (5) i s p r o v e d f o r an a r b i t r a r y n.

S ince n can be a m u l t i p l e i n d e x , i t f o l l o w s f rom (5) t h a t

V v <a, y,ls-l] Z,In]) = (s - - 1) <Z, ln] y,[s-UlF~ y> -r n ( y, ls-l] Z)n - l l F~ Z>, (6a)

where Y and Z are arbitrary functionals of the process ~(t). We put here Y - ~. The re- V

sults of operation of the operator F~ on the process ~(t) is a deterministic quantity

V F, ~. (t,) = [ B, (t, ,) ~ (t, - ~) ct, = ~ , (t, q) ,

and, therefore, the first cumulant brackets is (6a) vanishes.* Therefore, we obtain the relat ion

*A basic property of cumulant bracket in that they vanish if at least one argument is sta- tistically independent of the rest, or it is deterministic.

1129

v v <~,[s] Z,tn]> = n <=,[s-u Z,In-ll F.Z> -- <~ [.-U F~ (Z,[n])>. (7)

In essence, this is a recurrence relation for lowering the order of the cumulant bracket. A multiple application of (7) gives the operator formula

v~ <~,[sl Z,[nI> = <F~ (Z,[nb>. (8 )

v v Here, Ks denotes s-fold application of the operator F~ inside the cumulant brackts [a

single application is shown in Eqs. (5) and (7)]. The indices in (8) can be understood as multiple indices. In particular, for s = i, (8) goes into formula (5), and for n = i, it gives' the useful relation*

<~,1;I Z> := </.s Z> ---- d%B. (t, ~,) ... d~sB. (t, ~s) ~ 5 ~ (~,). . . ~ ( % ) / t

Finally, for s = n = i, it gives the Furutsu--Novikov formula.

We note that from (8) one can, by "removing commas," obtain a similar formula for the mixed moments, as it was done above [see (3a) and (5)]. The decoupling of the moments has a considerably more complex form, namely

s

<~,z) = <~z> + <~> ~ cY(2k - ~)~ <>~-'~ z>, k=!

where s' = s/2 for even s, and s' = (s- i)/2 for odd s.

Formula (8) and its particular cases allow a limiting transition to a delta-correlated process ~(t); it is known that this idealization is important for the analytical solution of a wide circle of physical problems [4]. At the same time, this limiting transition is not permissible for the decoupling of joint moments since <@z> then diverges to infinity.

2. Statistical Relationship between the Random Perturbation and the Output Coordinate

We consider a stochastic system of order N which is described by an ordinary differen- tial equation of the form

dxJd t - - / k (x) + ~ (t) g~ (x), x O) = xo,

where x is the state vector of the system, Its, gh is a deterministic function, and a(t) Gaussian delta-correlated process with

<~(1)> = 0, B~(t,, &) ~- 2D6 ( t , - - & ) .

Using the above relations, we shall attempt to answer the question about the statistical

relationship between the random force and the output coordinate x in identical moments of time. Let us suppose that Z(x) is a deterministic function of the vector x (t). Then

(9)

i s a

Consequently,

v ; F , Z = 2 D d- ,a ( t - - , ) OZ axe(t) Dg~(x) OZ. Oxk ~ (9 Ox~. "

where, for brevity, we denoted

V F$ Z = (Dg (x) O/Ox)" Z, (10)

0 0 = gk, ( x ) ONe, "'" g k , ( x ) o Lr1~ ~ s = 1, 2,

(here and below, we understand summation over repeating indices).

For a joint correlation we find, using (2) and (i0),

<~ (t) z (x)> = D <gk (x) 0Z (x)IOx~> (11)

*Hence follows the adequacy of the functional [4] and cumulant [5, 6] approaches to the analysis of stochastic systems.

1130

This formula is widely used to obtain closed equations for the one-time characteristics of the output coordinate [4].

For the joint cumulant of order n + 1 we obtain

<~, Z 1 . . . . . Zn> -~- D ~ {<Z 1 . . . . . Z I - I~ gjOZi/Oxy, Zi+l . . . . . Z . } i ( 1 2 )

(here and below, the values of all variables are taken in moment of time t), {-..}~ is the symmetrization bracket with respect to the index i. If in (12) we put Zi(x)=xi, we obtain an expression for the joint cumulants of the set {x,~l of order n + I:

<~,Xl , .... Xn> = D N {(X 1 . . . . . Xi-1, g i ( x ) , Xi+l . . . . . Xn>} I. (13)

In particular, for a one-dimensional system

<~, ~ [n]> = ~ n <X,[ n - l l g ( x ) > . (13a)

The corresponding expressions for the joint moments have an analogous form; it is sufficient to "remove commas."

To find the higher statistical relations between x and ~, we shall present an operator formula which follows from (8) and (i0):

<%I~J Zt , ..., Zn> = O s <(g ( x ) O / O x ) ,~ ( Z , . . . . . Zn)>. ( 1 4 )

In practice, however, it is more convenient to use the following "recurrence" formula

<=,N Z, . . . . . Z~> = D <~,Is lj (g (x) a;ax) ( z , , .... z~)> = Dn {<%t~-~J Zl . . . . , Z~_I, gj a z i~x j , zi+l, .... z ,>} i . ( 1 5 )

By putting here Zi(x)='~-Xi, we Obtain a formula for the lowering of the order (with respect to a) of the joint cumulants of the set {x,=}:

< a , [ ' l x , , . . . , xn> = D n {<%[s-i1 x , . . . . , x j - l , g~ (x) , x t+l . . . . . xn>}t, s = 2, 3, ... ( 1 5 a )

It is seen from the above expressions that the structure of the statistical relations of the set {x,=l is determined only by the form of the functions g~(x) in Eqs. (9), and is inde- pendent of the functions fk (the form of function [r~(x) determines the probability distribu- tion of the state vector x([)). We now consider two important particular cases:

a) System with a "Simple" Parametric Interaction. We put in Eqs. (9) gh(x)=bkjx~, where bkj are deterministic functions of time or constants. Using (13) we then obtain

<a, Xt . . . . . Xn> ---= D,i{bhj <xt ..... xi-l, xj, x~+t . . . . . xn>}i . ( 1 6 )

Thus, here the joint cumulants of order n + i can be expressed in terms of a linear combina- tion of the cumulants of the state vector of order n. It is easily seen that cumulants of order n + s can be expressed in an analogous fashion. In particular, for a one-dimen- sional stochastic system described by the equation

dx/dt = [(x) + b ~ (t)x,

we find*

<%[sl x , lnl> - - (Dbn)S <x,[,l>. ( 1 7 )

b) System with Additive Random Interaction. We put in Eqs. (9) g ~ ( x ) : bk----const. Then, the cumulant brackets which contain gk as an argument vanish, and we obtain the relation

<~ ,x~> = <~x,~> = braD, <~,I~lxk . . . . . . xk,> ----- 0, s n > l . ( 1 8 )

Thus, an additive Gaussian noise is related to the output coordinate only by the correlation (Gaussian) relations, and the higher-order statistical relations which are described by higher joint cumulants are absent (for linear systems, this is well known; see, for example, [7]). Of course, the probability distribution of the vector x(t) is not normal, if at least one of the functions ~(x} is nonlinear. The absence of higher-order relations between the

*The cumulants of the input coordinate can be found if it is possible to solve the Fokker-- Planck equation for the probability density W(x;t). In the one-dimensional case, this has the form

OW O [ O ] dt --Ox f (x) W - D b 2 x ~ x x W .

1131

random force and the input coordinate resembles the Mil]ionshchikov hypothesis [8]. It was shown above that it is fully justified only for joint statistical relations of the above sets.

LITERATURE CITED

i. K. Furutsu, J. Res. NBS, D-67, No. 3, 303 (1963). 2. E.A. Novik~v, Zh. Eksp. Teor. Fiz., 47, 1919 (1964). 3. A.N. Malakhov, Cummulant Analysis of Non-Gaussian Random Processes and of Their Trans-

formations [in Russian], Sov. Radio, Moscow (1978). 4. V.I. Klyatskin, Stochastic Equations and Waves in Randomly Inhomogeneous Media [in

Russian], Nauka, Moscow (1980). 5. A.N. Malakhov, O. V. Muzychuk, and I. E. Pozumentsov, Izv. Vyssh. Uchebn. Zaved.,

Radiofiz., 21, 1279 (1978).

6. O.V. Muzychuk, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 22, 1246 (1979).

7. S.A. Akhmanov, Yu. E. D'yakov, and A. S. Chirkin, Introduction to Statistical RadiO- physics [in Russian], Nauka, Moscow (1981).

8. M.D. Mil!ionshchikov, Dokl. Akad. Nauk SSSR, 32, 611 (1941).

i/f NOISE AND AGING OF FREQUENCY SHIFT IN OSCILLATORS WITH SURFACE

ACOUSTIC WAVE RESONATORS

M. A. Krevskii and A. V. Yakimov UDC 621.391.822.4

Oscillators using surface acoustic wave resonators with gold and aluminum elec- trodes are considered. Two different types of aging mechanisms are studied, which determine the form and level of the flicker frequency fluctuation spectrum and drift with aging in the mean value of the latter. The results obtained agree well with available experimental data.

Surface acoustic wave (SAW) devices are used to generate highly stable signals in the hf range. Such devices undergo an aging process which manifests itself by irreversible changes in the frequency of o~scillators using the devices [1-3]. Moreover, the oscilla- tion frequency shows flicker fluctuations [3, 4] with a I/f type spectrum, which accord- ing to [5, 6] can be Considered a random process accompanying changes in device character- istics with age. The validity of this approach was demonstrated in [7] by analyzing corre- sponding changes in the resistance of point contacts.

Using the flicker fluctuation physical model, in the present study we will examine the possibility of establishing the relationship between frequency flicker fluctuation characteristics and the aging change (mean statistical) in frequency of an oscillator using

a SAW resonator.

i. We will assume that relative variations in frequency 6f(t) can be represented as a Poisson superposition of rectangular pulses with amplitude (6f)i and random duration e. The minimum possible pulse duration is to. For g > to the pulse distribution changes as ~-2. The mean frequency of pulse appearance is equal to ~.

In this case, according to [6], drift of the mean statistical frequency value and the frequency fluctuation spectrum are defined by the following expressions:

<6f(t)>=Aln(t/to), t>~to; (1) <6/z>F= B/F, F<< (2ato)-i. (2)

Gor'kii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 26, No. 12, pp. 1552-1556, December, 1983. Original article submitted November 30, 1982; revision submitted June 28, 1983.

1132 0033-8442/83/2612-1132507.50 �9 1984 Plenum Publishing Corporation