Russian Journal of Electrochemistry, Vol. 36, No. 11, 2000, pp. 1163-1166. Translated from Elektrokhimiya, Vol. 36, No. 11, 2000, pp. 1315-1318. Original Russian Text Copyright �9 2000 by Grafov.

Theory for a Wavelet Analysis of Electrochemical Noise in the Laplace Space with Use of Two Operational Frequencies1

B. M. Grafov Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 117071 Russia

Received October 8, 1999

Abstract--The image of a random process in the Laplace space may be viewed as resulting from use of a one- way continuous wavelet transform with an exponential as the basic function, i.e. as resulting from the applica- tion of the Laplace wavelet. If the Laplace-wavelet variance of an electrochemical noise allows one to determine the Laplace transform of a time-correlation function, i.e. a factual operational spectral density of the noise, then the covariance of two Laplace waveletes of an electrochemical noise, each of which corresponds to its own operational frequency, allows one to verify a local consistency of the initial experimental noise data. The Laplace waveletes are applied rather widely. In fact, any stationary random process and stationary random time sequence can be described with operational noise spectra.

INTRODUCTION

Fluctuations and random processes play a very important role in the electrochemistry. This refers to the theory of absolute rates of reactions [1-3], the noise diagnostics of electrochemical systems [4-9], the study of a turbulent electrochemical noise [10], and the mea- surements of diurnal variations in the Earth's microseismic fields by the hydroelectrochemical impedance method [ 11 ].

The method most commonly used for studying the noise processes is the measurement of spectral densi- ties of the noise (the noise energy spectra) in the Fou- rier space [12]. However, since Grossman and Morlet [13] had used the wavelet transform [14, 15] for analyz- ing the seismic prospecting data, it became clear that in some cases the wavelet transform was a more powerful means than the Fourier transform. The one-way wave- let transforms of the Haare type [14] are most suitable for the noise analysis. The Laplace transform may be classified with the one-side wavelet transforms as well [16]. In [17, 18], for example, the Laplace transform was used as a wavelet transform. This opened a straight route to measuring the noise spectra in a Laplace space without involving the stage of measuring a correlation or intercorrelation function. By analogy with the termi- nology accepted in the theory of the operational elec- trochemical impedance [19, 20], spectral densities of the electrochemical noise in a Laplace space may be designated as the operational noise spectra densities (the operational noise spectra).

The aim of this work is a further development of a mathematical method of the continuous Laplace wave- letes, which would allow one to use two independent Laplace variables instead of one when analyzing the

I Dedicated to the ninetieth anniversary of Ya.M. Kolotyrkin's birth.

electrochemical noise. In doing so, one could use two independent operational frequencies simultaneously. Note that the possibilities of discrete wavelet trans- forms in an electrochemical noise analysis have been recently demonstrated in [21].

We emphasize that the results of this study may come useful for analyzing any random processes, not only those of an electrochemical nature. In particular, they can be used in the noise diagnostics of materials and interfaces [7, 8, 22-25] and in econometric studies [26, 27].

DESCRIPTION OF NOISE IN A LAPLACE SPACE

According to the standard definition [28], to within

a factor of four, the spectral density ( y2 ) of a random process y(t) is a real part of the Fourier image K(jo~) of the time autocorrelation function k(t):

(Y~ = 4ReK(jto), (1)

K(jo3) = ~dtexp(-jo)t)k(t) , (2)

o

k(t) = (y(O)y(t)). (3)

Here, t is the time, j = ~ - i , and o~ is the angular fre- quency of the analysis. The angle brackets (...) desig- nate the operation of averaging over an ensemble of realization of the random process y(t). The latter is pre- sumably stationary, with the finite intersection prop- erty. The factor "4" is of a historical origin. It corre- sponds to an analog measurement of a spectral density with the narrow-band filters. Formally, the product jm can be replaced with the Laplace variable p. As a result, we obtained a description of the noise in a Laplace

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1 ! 64 GRAFOV

space in the form of the Laplace transform of the auto- correlation function [29]:

c , o

K(p) = Idtexp(-pt)k( t ) . o

(4)

In [17], we managed to show that the Laplace image K(p) of the autocorrelation function k(t) can be deter- mined directly via the variance of the Laplace image Y(p) of the initial random process according to the expression

K(p) = p(Y(p)Y(p)) , (5)

where

Y(p) = Idtexp(-pt )y( t ) , 0

(6)

and the angle brackets (...) designate the averaging over the ensemble of realizations of the random process y(t).

The Laplace variable p can also be a complex num- ber with a positive real part [16]. Therefore, expression (5) with a complex Laplace variable opens quite a def- inite way to measuring both the real and the imaginary parts of an operational spectral density of the electro- chemical noise. Note that the Laplace variable in the form of a complex quantity plays an important role in the electrochemical impedance theory as well [30].

A WAVELET ANALYSIS INVOLVING TWO OPERATIONAL FREQUENCIES

Expression (6) may be viewed as a wavelet trans- form [14, 15] with an exponential as the basic function. In econometric studies [27], the exponential smoothing is a standard procedure in the determination of the mean of a random time series. However, expressions of the type (5) are not used in econometric studies. Expression (5) affords an interesting generalization performed by introducing the correlator (Y(p)Y(q)) involving two independent Laplace variables p and q:

(Y(P)Y(q))

(i = I e x - I I 2 e x - 2 2 �9

0

(7)

Expression for covariance (7) is symmetrical about operational frequencies p and q. Bearing this in mind, it is convenient to present expression (7) in the form of the sum of two integrals

(Y(p)Y(q)) = 1, (p, q) + 12(p, q), (8)

where

t l (p ,q)

(i i ' = dt jexp(-pt l )Y(h) dt2exp(-qt2)y(t2), (9)

r

I2(P, q)

(i ,J " = dtlexp(-ptl)Y(h dt2exp(-qt2)y(t2 �9 0

For the symmetry reasons,

12(p,q) = l~(q, p). (11)

Let us introduce new integration variable, 13 = t 2 - tl, instead of h in (9). After a number of operations, we obtain

l l (p ,q)

= Idt lexp(-pt , ) Id t2exp(-qt2)k( t2- t , )

0 ,, (12)

= Idt, exp(-pt ,)exp(-qt ,)Idt3exp(-qt3)k(t3) o o

= K(q)l(p + q).

With allowance made for (11), then

12(p, q) = K(p)l(p + q). (13)

Therefore,

(Y(p)Y(q)) = [ g ( p ) + g ( q ) ] l ( p + q ) . (14)

Multiplying (I 4) by the sum of operational frequencies (p + q), we obtain the sought generalization of expres- sion (5)

g ( p ) + g ( q ) = (p+q) (Y(p)Y(q) ) . (15)

DISCUSSION

If we set the second Laplace variable q in (I 5) equal to the first Laplace variable p, (15) reduces to formula (5) obtained earlier in [17], just as expected. Now, let the second operational frequency q coincide with the complex conjunction of the first operational frequency p, i.e.

From (15), then

K(p)+ K(p*)

q = p*. (16)

= ( p + p * ) ( r ( p ) Y ( p * ) ) . (17)

As follows directly from (5),

K(p) + K(p*) = p(Y(p)Y(p) )

+p*(Y(p*)Y(p*)) . (18)

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THEORY FOR A WAVELET ANALYSIS 1165

Thus, the real part of an operational spectral noise density can be measured by two different means, spe- cifically, by using either (17) or (18). Since the mea- surement result should not depend on the measurement technique, the following relationship needs must be ful- filled:

(p + p*) (Y(p)Y(p*) ) = p (Y(p )Y(p ) ) (19)

+p*(Y(p*)Y(p*) ) .

This relationship can be considered as an analogue, although very vague, of the Kramers-Kronig relations [31, 32] and may be generalized on the basis of (5) and (15):

(p + q)(Y(p)Y(q)) = p (Y(p )Y(p ) ) (20)

+ q (Y(q) Y(q)).

The Kramers-Kronig relations serve for checking the consistency of experimental electrochemical- impedance data [33-35]. Similarly, (20) can be used for checking the consistency of experimental electrochem- ical-noise data. The essential advantage of (20) is its localization. In (20), the integration with respect to the operational frequency is absent.

Final formulas (15) and (20) are invariant relative to the time inversion in the expression for the Laplace wavelet of the random process under study. This means that, when analyzing a noise spectra in a Laplace space, along with the Laplace wavelet in the form of (6), one can also use Laplace waveletes in the following form:

0

Y(p) = f dtexp(pt)y(t), (21)

i.e. with use of the past time.

CONCLUSIONS

(1) An integral Laplace transform can be viewed as a one-side continuous wavelet transform with the basic function in the form of an exponential.

(2) By analogy with an operational electrochemical impedance, the spectral noise densities in a Laplace space can be called the operational spectra densities (operational noise spectra).

(3) Use of the covariance of one-way Laplace wave- letes with two different operational frequencies opens additional possibilities for directly measuring opera- tional noise spectra without preliminary measurement of the time correlation function.

(4) The method of Laplace waveletes with two oper- ational frequencies makes it possible to verify the con- sistency of the initial noise data obtained experimen- tally. The verification has a local character and involve no integration with respect to the operational fre- quency.

(5) The application of Laplace waveletes is rather wide. Actually, any stationary random process and time series can be described with operational noise spectra.

ACKNOWLEDGMENTS

The author is grateful to Profs. A.M. Kuznetsov, S.E Timashev, and L.N. Nekrasov for helpful discus- sions of the problem of a wavelet analysis of random processes.

This work was supported by the Russian Foundation for Basic Research, project no. 99-03-32 086a.

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