GENERALIZATION OF CAMPBELL S THEOREM TO NONSTATIONARY NOISE. Leon Cohen

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1 GENERALIZATION OF CAMPBELL S THEOREM TO NONSTATIONARY NOISE Leon Cohen City University of New York, Hunter College, 695 Park Ave, New York NY 10056, USA ABSTRACT Campbell s theorem is a fundamental result in noise theory is applied in many fields of science engineering. It gives a simple but very powerful expression for the mean stard deviation of a stationary rom pulse train. We generalize Campbell s theorem to the non-stationary case where the rom process is space time dependent. We also generalize it to a pulse train of waves, acoustic electromagnetic, where the intensity is defined as the absolute square of the pulse train. Index Terms nonstationary noise, Campbell s theorem, rom pulse train, reverberation, time-frequency 1. INTRODUCTION One of the most fundamental results in noise theory is Campbell s theorem, a result that was given by Campbell [1, ] in 1909 soon after the basic concepts of noise were developed [3]. Campbell s theorem applies to a rom process that is a pulse train of the following form He showed that covariance between two pulse trains is given by (s s (q q = N f(tg(tdt (5 We point out that a fruitful view point is to think of a pulse train as consisting of delta functions, N δ(t t n, letting the sum go through a linear filter whose impulse response function is f(t. In deriving the above results a number of assumptions are made, the most important is that the series are stationary. In this paper we consider a non-stationary model where we have spatial time dependence. We explicitly derive the mean stard deviations analogous to Campbell s theorem show that in the limit of infinite time we recover Campbell s theorem.. NON-STATIONARY PULSE TRAIN Consider the following model of a non-stationary pulse train as illustrated in the figure. At t = 0, N particles with width, s = f(t t n (1 where f(t is a deterministic function of time, t n are rom arrival times, N is the number of constituents. Campbell s result is that the mean stard deviation of the process are given by s = N σ = s s = N f(tdt ( f (tdt (3 This is a simple but very powerful result that is easy to apply, is used in almost all fields has found many applications. Furthermore, sometime later, Rowl [4] considered two series where in addition to Eq. (1 we have the series q = g(t t n (4 Work supported by the Office of Naval Research. as represented by the f(t, are generated at rom positions x n chosen from a uniform distribution ranging from a to 0. The particles move to the right with a constant velocity, c. At position x time t we observe the rom process given by V (x, t = f(x ct x n (6 We define the ensemble average as the average over the initial positions x n, V (x, t = N P (x n f(x ct x n dx n (7

2 where P (x n is the probability distribution of x n. For the case of a uniform distribution { 1/a a < xn < 0 P (x n = (8 0 otherwise therefore 0 V (x, t = N n dx n = a af( x N a (9 where we have made the substitution z = x ct x n changed the limits accordingly. Now, suppose we take N a large in such a manner that N ρ for N a (10 a where ρ is the constant density. Hence, we have that (11 This is the average at position x time t for N a. If we further take t which can be thought of as the steady state limit then we have (1 which is the first part of Campbell s theorem Eq. (. In Eq. (9 (11 the result is position dependent but as time goes to infinity we obtain a stationary process the resulting average, Eq. (1, is independent of position. To calculate the stard deviation we first consider V (x, t = k=1 f(x ct x n f(x ct x k (13 following the same procedure as above we obtain V (x, t = N a ( (14 Taking the limit as in Eq. (10 noting that N/ 0 we have V (x, t = ρ ρ ( (15 The stard deviation σ = V (x, t V (x, t is σ = N a ( N a ( (16 = N a N ( (17 If we take the limit as per Eq. (10 we obtain σ = ρ if we further take t then we have σ = ρ (18 (19 which is the second part of Campbell s theorem, Eq. (3. We consider Eq. (9 Eq. (11 as a nonstationary position time dependent Campbell s theorem for the mean; Eq. (17 (18 to be a generalization of the stard deviation. 3. NON-UNIFORM DENSITY In the above considerations we have taken the initial density to be uniform as given by Eq. (8. For a non uniform initial distribution given by a probability density P (x n, but still constant velocity motion, we have V (x, t = N f(x ct x n P (x n dx n (0 V (x, t = N f (x ct x n P (x n dx n ( f(x ct x n P (x n dx n (1 The stard deviation is given by σ = N f (x ct x n P (x n dx n ( N f(x ct x n P (x n dx n ( It is possible for the second term to be significant depending on P (x n f(x. 4. GENERAL MOTION In the above, we have taken motion to be constant velocity; however, there are situations where non-constant velocity is appropriate. Suppose the motion is governed by x = x n η(t (3 To take that into account, replace ct by a time function that describes the motion. We have

3 V (x, t = N f(x η(t x n P (x n dx n (4 further V (x, t = N f (x η(t x n P (x n dx n ( f(x η(t x n P (x n dx n (5 The stard deviation is then σ = N f (x η(t x n P (x n dx n ( N f(x η(t x n P (x n dx n (6 For the case of uniform density, as per Eq. (8 taking the same limits as in Section we have for the mean stard deviation that x η(t (7 σ = ρ (8 x η(t 5. TWO-TIME AUTOCORRELATION FUNCTION We now calculate the two-time correlation function at position x, R(t 1, t ; x = V (x, t 1 V (x, t (9 For the constant velocity case we have V (x, t V (x, t 1 = f(x ct x n f(x ct 1 x k (30 k=1 following the same procedure as in the previous sections one obtains V (x, t V (x, t 1 = N a ( a a f(zf(z c(t t 1 dz ( 1a 1 Taking the constant density limit one obtains (31 V (x, t V (x, t 1 = ρ f(zf(z c(t t 1 dz ( ( ρ (3 1 Letting we have t t 1 = τ (33 V (x, t V (x, t 1 = ρ f(zf(z cτdz ( ( ρ 1 (34 As expected, this is not a function τ, hence the process is not stationary. However, if we take the limit of large times but keeping (t t 1 a constant, then V (x, t V (x, t 1 = ρ t, t 1 (35 f(zf(z cτdz ( ρ t, t 1 ; t t 1 = τ we now see that the process becomes stationary. 6. TIME-VARYING SPECTRUM (36 Another fruitful approach is to calculate the time varying spectrum. Here we use the Wigner spectrum relate it to the autocorrelation function. The time-frequency Wigner distribution at position x, is defined by W (t, ω, x= 1 V (x, t 1 π τ V (x, t 1 τe iτω dτ (37 To deal with a nonstationary process [6 10] one takes the ensemble average of Eq. (37 W (t, ω; x= 1 V (x, t 1 π τ V (x, t 1 τ e iτω dτ (38 where W (t, ω; xis called the Wigner spectrum which can be thought of as the instantaneous spectrum at position x time t. The Wigner spectrum can be written as W (t, ω, x = 1 π inversely we have, R(t τ/, t τ/; x = R(t 1, t = R(tτ/, t τ/; xe iτω dτ (39 W (t, ω, xe iτω dω (40 ( t1 t W, ω; x e i(t t 1 ω dω (41

4 Consider now the Wigner distribution for the pulse train given by Eq. (1. Substituting Eq. (1 into Eq. (9 we have where W (t, ω; x= W nn = 1 π m=1 W nm (t, ω; x (4 = NW nn (t, ω; x W nm (43 f (x ct 1 τ x n f(x ct 1 τ x np (x n e iτω dτdx n (44 W nm = 1 f (x ct 1 π τ x n f(x ct 1 τ x mp (x n P (x m e iτω dτdx n dx m (45 Consider W nn where we consider the uniform distribution, Eq. (8. We obtain W nn = 1 π f (z 1 τ (46 f(z 1 τ e iτω dτdz (47 The inner integral is simply the Wigner spectrum of the function f(z. Therefore we have Now consider W nn = 1 a W nm = 1 1 π W (z, ωdz (48 f (z 1 τf(z 1 τe iτω dτdzdz (49 = 1 ( z z W, ω e i(z zω dzdz (50 If we substitute these results into Eq. (43 apply the limit as in the previous sections we obtain W (t, ω; x= ρ W (z, ωdz ρ W ( z z, ω e i(z zω dzdz (51 7. WAVE TRAINS In the above considerations we have taken the intensity to be given by Eq. (1 but if f(t are waves then the intensity the intensity squared are given, respectively, by I(x, t = f(x ct x n (5 I (x, t = f(x ct x n 4 (53 Furthermore, f(t can be complex. In this paper we consider the real case. One can show that [14] [ f I = N (N 1 f ] (54 I = (N (N 3 f 4 6(N f f 4 f f 3 3 f N f 4 ] (55 Consider first I. We can use the results obtained above to immediately write, as per Eq. (17, that I = N a ( (56 If we take the limit as given by Eq. (10 we have ( I = ρ ρ (57 further, the long time limit is I = ρ ( ρ (58 The calculation for I is more involved we just give the final answer, I ( 4 = ρ 4 ( ( 6ρ 3 ( ( 4 ρ ( 3 ( ρ f 4 (zdz f 3 (zdz (59

5 giving ( σ = ρ f 4 (zdz ( ρ ( 4ρ 3 ( ( ( 8. CONCLUSION f 3 (zdz (60 We derived an extension of Campbell s theorem to nonstationay situations. As with Campbell s theorem, there are a number of sensitive mathematical issues regarding convergence that will be addressed in a future paper. One of the applications of the model we have given is to the problem of reverberation which is often inherently nonstationary [11 14]. This is the case if the scatterers, which are spatially distributed, are excited by a pulse therefore the returns at a particular spatial point is a rom process where the function f(t is proportional to the return from each of the scatterers if indeed all the scatterers are the same. If the scatterers are different then the stochastic process has to be replaced by [11 14] V (x, t = f n (x ct x n (61 where f n reflects the scattering of each scatterer. This presents additional very interesting issues which are currently being investigated. REFERENCES [1] N. R. Campbell, The study of discontinuous phenomena, Proc. Cambr. Phil. Soc., 15, , [] N. R. Campbell, Discontinuities in light emission, Proc. Cambr. Phil. Soc., 15, , [3] L. Cohen, The History of Noise, IEEE Signal Processing Magazine,, 0-45, 005. [4] E. N. Rowl, The theory of the mean square variation of a function formed by adding known functions with rom phases, applications to the theories of the shot effect of light, Math. Proc. Cambridge Phil. Soc., 3, ,1936. [5] E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Review, vol. 40, pp , 193. [6] W. D. Mark, Spectral Analysis of the Convolution Filtering of Non-Stationary Stochastic Processes, Jour. Sound Vibration, 11, pp , [7] W. Martin, Time-Frequency Analysis of Rom Signals, Proc. ICASSP 8, pp , [8] W. Martin P. Flrin, Wigner Ville spectral analysis of nonstationary processes, IEEE Trans. Acoust. Speech, Signal Process., 33, (1985. [9] L. Galleani L. Cohen, The phase space of nonstationary noise, Journal of Modern Optics, vol. 51, pp , 004. [10] L. Cohen, Time-Frequency Analysis, Prentice-Hall, [11] P. Faure, Theoretical Model of Reverberation Noise, J. Acoust. Soc. Am., Vol. 36, 59-66, [1] V. V. Ol shveskii, Characteristics of Sea Reverberation, Consultants Bureau, [13] D. Middleton, A statistical theory of reverberation similar first-order scattered fields, Parts I II, IEEE Transactions on Information Theory, vol. IT-13, , 1967; A statistical theory of reverberation similar first-order scattered fields, Parts III IV, IEEE Transactions on Information Theory, vol. IT-18, , 197. [14] L. Cohen A. Ahmad, The scintillation index for reverberation noise, Proc. SPIE 8391, 83910F, 01.