Diss / 10

and applying the limiting process to (13.5) from discrete observation to continuous one at T = const ( → 0, N → ∞), we obtain

Determining the mathematical expectation of variance estimate, we can see that it is unbiased. The variance of the variance estimate can be presented in the following form:

Since

we can see from (13.16) that the variance of the optimal variance estimate of Gaussian stochastic process approaches symmetrical to zero for any value of the observation interval [0, T].

Analogous statement appears for a set of problems in statistical theory concerning optimal signal processing in high noise conditions. In particular, given the accurate measurement of the stochastic process variance, it is possible to detect weak signals in powerful noise within the limits of a short observation interval [0, T]. In line with this fact, in [1] it was assumed that for the purpose of resolving detection problems, including problems related to zero errors, we should reject the accurate knowledge of the correlation function of the observed stochastic processes or we need to reject the accurate measurement of realizations of the input stochastic process. Evidently, in practice these two factors work. However, depending on which errors are predominant in the analysis of errors, limitations arise due to insufficient knowledge about the correlation function or due to inaccurate measurement of realizations of the investigated stochastic process.

It is possible that errors caused by inaccurate measurement of the investigated stochastic process have a characteristic of the additional “white” noise with Gaussian pdf. In other words, we think that the additive stochastic process

comes in at the input of measurer instead of the realization x(t) of stochastic process ξ(t), where n(t) is the realization of additional “white” noise with the correlation function defined as Rn(τ) = 0.5 0δ(τ) and 0 is the one-sided power spectral density of the “white” noise.

To define the characteristics of the optimal variance estimate in signal processing in noise we use the results discussed in Section 15.3 for the case of the optimal estimate of arbitrary parameter of the correlation function R(τ, l) of the Gaussian stochastic process combined with other Gaussian stochastic processes with the known correlation function. As applied to a large observation interval compared to the correlation interval of the stochastic process and to the variance estimate (l ≡ σ2), the variance estimate, in the first approximation, is unbiased. The variance of optimal variance estimate of the correlation function parameter given by (15.152) can be simplified and presented in the following form:

where S(ω) is the spectral power density of the stochastic process ξ(t) with unit variance.

To investigate the stochastic process possessing the power spectral density given by (12.19), the variance of the variance estimate in the first approximation is determined in the following form:

as the noise power within the limits of effective spectrum band of investigated stochastic process, and according to (12.284) feff = 0.25α, and

as the ratio of the noise power to the variance of investigated stochastic process, in other words, the noise-to-signal ratio. As a result, we obtain the relative variation in the variance estimate of the stochastic process with an exponential normalized correlation function:

where p = αT = T τcor−1 , as mentioned earlier, is the ratio of the observation time to correlation interval of stochastic process. In practice, the measurement errors of instantaneous values xi = x(ti) can be infinitely small. At the same time, the measurement error of the normalized correlation function depends, in principle, on measurement conditions and, first of all, the observation time of stochastic process. Note that in practice, as a rule, the normalized correlation function is defined in the course of the joint estimate of the variance and normalized correlation function.

102

101

FIGURE 13.1  Variance of the variance estimate versus the true value of the correlation coefficient.

as σ2 and is independent of the true value ρ0 of the correlation coefficient. The random error under definition of the variance of variance estimate

as a function of the true value ρ0 of the correlation coefficient is shown in Figure 13.1. The value κ shows how much the variance of variance estimate increases with the increase in the error under definition of the correlation coefficient ρ. As we can see from (13.35) and Figure 13.1, with increase in the absolute value of the correlation coefficient, the errors in variance estimate also rise rapidly. In doing so, if the module of the correlation coefficient between samples tends to approach unit, then independent of the small variance Varε of the definition of the correlation coefficient the variance of the variance estimate of stochastic process increases infinitely. Compared to ρ0 = 0, at ρ0 = 0.95, the value κ increases 30 times and at ρ0 = 0.99 and ρ0 = 0.999, 150 and 1500 times, respectively. For this reason, at sufficiently high values of the module of correlation coefficient between samples we need to take into consideration the variance of definition of the correlation coefficient ρ. Qualitatively, this result is correct in the case of the optimal variance estimate of the Gaussian stochastic process under multidimensional sampling.

Although in applications we define the correlation function with errors, in particular cases, when we carry out a simulation, we are able to satisfy the simulation conditions, in which the normalized correlation function is known with high accuracy. This circumstance allows us to verify experimentally the correctness of the definition of optimal variance estimate of Gaussian stochastic process by sampling with high values given by the module of correlation coefficient between samples.

Experimental investigations of the optimal (13.5) and nonoptimal (13.24) variance estimations by two samples with various values of the correlation coefficient between them were carried out. To simulate various values of the correlation coefficient between samples, the following pair of values was formed:

L

xi = L1 ∑yi − p , (13.37)

p=1

L

xi − k = L1 ∑yi − k − p , k = 0,1,…, L. (13.38)

p=1

As we can see from (13.37) and (13.38), the samples xi and xi−k are the sums of independent samples yp obtained from the stationary Gaussian stochastic process with zero mathematical expectation.

1.0

FIGURE 13.2  Normalized correlation function of stationary Gaussian stochastic process.

The samples xi and xi−k are distributed according to the Gaussian pdf and possess zero mathematical expectations. The correlation function between the newly formed samples can be presented in the following form:

To obtain the samples xi and xi−k subjected to the Gaussian pdf we use the stationary Gaussian stochastic process with the normalized correlation function presented in Figure 13.2. As we can

see from Figure 13.2, the samples of stochastic process yp with the sampling period exceeding 0.2 ms are uncorrelated practically. In the course of test, to ensure a statistical independence of the samples yp we use the sampling rate equal to 512 Hz, which is equivalent to 2 ms between samples. Experimental test with the sample size equal to 106 shows that the correlation coefficient between neighboring readings of the sample yp is not higher than 2.5 × 10−3.

Statistical characteristics of estimations given by (13.5) and (13.24) at N = 2 are computed by the microprocessor system. The samples xi and xi− k are obtained as a result of summing 100 independent readings of yi−p and yi−k−p, respectively, which guarantees a variation of the correlation coefficient from 0 to 1 at the step 0.01 when the value k changes from 100 to 0. The statistical characteristics of estimates of the mathematical expectation  Varj  and the variance Var{Varj} at j = 1 corresponding to the optimal estimate given by (13.5) and at j = 2 corresponding to the estimate given by (13.24) are determined by 3 × 105 pair samples xi and xi−k. In the course of this test, the mathematical expectation and variance have been determined based on 106 samples of xi. In doing so, the mathematical expectation has been considered as zero, not high 1.5 × 10−3, and the variance has been determined as σ2 = 2.785.

Experimental statistical characteristics of variance estimate obtained with sufficiently high accuracy are matched with theoretical values. The maximal relative errors of definition of the mathematical expectation estimate and the variance estimate do not exceed 1% and 2.5%, correspondingly. Table 13.1 presents the experimental data of the mathematical expectations and variations in the variance estimate at various values of the correlation coefficient ρ0 between samples. Also, the theoretical data of the variance of the variance estimate based on the algorithm given by (13.24) and determined in accordance with the formula

are presented for comparison in Table 13.1. Theoretical value of the variance of the optimal variance estimate is equal to Var{VarE} = 7.756 according to (13.5). Thus, the experimental data prove the previously mentioned theoretical definition of estimates and their characteristics, at least at N = 2.

TABLE 13.1

Experimental Data of Definition of the Mathematical

Expectation and Variance of the Variance Estimate

13.2  STOCHASTIC PROCESS VARIANCE ESTIMATE UNDER AVERAGING IN TIME

is considered as the variance estimate, where x(t) is the realization of observed stochastic process; E is the mathematical expectation; h(t) is the weight function with the optimal form defined based on the condition of unbiasedness of the variance estimate