Diss / 10

FIGURE 12.10  Relative increase in the variance of equidistributed estimate of mathematical expectation as a function of the normalized correlation function between samples.

as a function of values of the normalized correlation function between the samples ψ for various numbers of samples. Naturally, if the relative increase in the variance is low, then the magnitude of the normalized correlation function between samples will be low as well. Similar to the mathematical expectation estimate by the continuous realization of stochastic process, the presence of maxima is explained by the fact that in the case of small numbers of samples N and sufficiently large magnitude ψ, the variance of the optimal estimate decreases rapidly in comparison with the variance of the equidistributed estimate of mathematical expectation. As we can see from Figure 12.10, the magnitude of the normalized correlation function between samples is less than ψ = 0.5, then the optimal and equidistributed estimates coincide practically.

As applied to the normalized correlation function (12.146), the normalized variance of estimate is defined as

Var(E*)

σ2

where, as before, ψ = exp{−αΔ}. At ϖ = 0, we obtain the formula (12.195). In the case of large numbers of samples, the formula (12.198) is simplified

As we can see from (12.199), in the case of stochastic process with the correlation function given by (12.146) the equidistributed estimate may possess the estimate variance that is less than the variance of estimate by the same number of the uncorrelated samples. Actually, if the samples are taken over the interval

then the minimal magnitude of the normalized variance of estimate can be presented in the following form:

Otherwise, if the interval between samples is chosen such that

then the maximum value of the normalized variance of estimate takes the following form:

Thus, for some types of correlation functions, the variance of the equidistributed estimate of mathematical expectation by the correlated samples can be lesser than the variance of estimate by the same numbers of uncorrelated samples.

If the interval between samples is taken without paying attention to the conditions discussed previously, then the value Δϖ = φ can be considered as the random variable with the uniform distribution within the limits of the interval [0, 2π]. Averaging (12.199) with respect to the random variable φ uniformly distributed within the limits of the interval [0, 2π], we obtain the variance of the mathematical expectation estimate of stochastic process by N uncorrelated samples

Of definite interest for the definition of the mathematical expectation estimate is the method to measure the stochastic process parameters by additional signals [13,14]. In this case, the realization x(ti) = xi of the observed stochastic process ξ(ti) = ξi is compared with the realization v(ti) = vi of the additional stochastic process ζ(ti) = ζi. A distinct feature of this measurement method is that the values xi of the observed stochastic process realization must be with the high probability within the limits of the interval of possible values of the additional stochastic process. Usually, it is assumed that the values of the additional stochastic process are independent from each other and from the values of the observed stochastic process.

If we know a priori that the observed stochastic sequence v(ti) = vi is a positive value, then we can use the following pdf:

as the nonlinear transformation η = g(ε). In doing so, the following condition must be satisfied:

As we can see from (12.220), this condition is analogous to the condition given by (12.206).

The conditional mathematical expectation of random variable ηi at ξi = x takes the following form:

In doing so, the unconditional mathematical expectation of the random variable ηi is defined in the following form:

For this reason, if the mathematical expectation estimate of the random sequence ξi is defined by (12.211) it will be unbiased at the first approximation.

The variance of the mathematical expectation estimate, as we discussed previously, is given by (12.213). In doing so, the conditional second moment of the random variable ηi is determined analogously as shown in (12.221):

Taking into consideration (12.223) and (12.224) under definition of the variance of the mathematical expectation estimate, we have

FIGURE 12.12  Staircase characteristic of quantization.

f (z)

1

z

FIGURE 12.13  Rectangular function.

where Tk = Σiτi is the total time when the observed realization is within the limits of the interval (k ± 0.5) during its observation within the limits of the interval [0, T]. In doing so, lim Tk is the probability that the stochastic process is within the limits of the interval (k ± 0.5) . T →∞ T

The mathematical expectation of the realization y(t) forming at the output of inertialess element (transducer) with the transform characteristic given by (12.228) when the realization x(t) of the stochastic process ξ(t) excites the input of inertialess element is equal to the mathematical expectation of the mathematical expectation estimate given by (12.229) and is defined as

where p(x) is the one-dimensional pdf of the observed stochastic process. In general, the mathematical expectation of estimate E  differs from the true value E0; that is, as a result of quantization we obtain the bias of the mathematical expectation of the mathematical expectation estimate defined as

To determine the variance of the mathematical expectation estimate according to (12.116), there is a need to define the correlation function of process forming at the transducer output, that is,

where p2(x1, x2; τ) is the two-dimensional pdf of the observed stochastic process.

Since the mathematical expectation and the correlation function of process forming at the transducer output depend on the observed stochastic process pdf, the characteristics of the mathematical expectation estimate of stochastic process quantized by amplitude depend both on the correlation function and on the pdf of observed stochastic process.

Apply the obtained results to the Gaussian stochastic process with the one-dimensional pdf given by (12.153) and the two-dimensional pdf takes the following form:

where c is the integer; −0.5 ≤ d ≤ 0.5. The value d is equal to the deviation of the mathematical expectation from the middle of quantization interval (step). Further, we will take into consideration that for the considered staircase characteristic of transducer (transformer) the following relation is true:

where w = 0, ±1, ±2,… is the integer. Substituting (12.153) into (12.230) and taking into consideration (12.234) and (12.235), we can define the conditional mathematical expectation in the following form:

k= −∞

where

is the ratio between the quantization step and root-mean-square deviation of stochastic process; Q(x) is the Gaussian Q function given by (12.68).

The conditional bias of the mathematical expectation estimate can be presented in the following form:

k= −∞