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where E(t0 + t) is the true mathematical expectation value of the investigated stochastic process at the instant t = t0. Thus, the mathematical expectation of estimate of the time-varying mathematical expectation of stochastic process in contrast to the stationary case is obtained by smoothing the estimate within the limits of time interval [t0 − 0.5T; t0 + 0.5T].

In general, as a result of considered averaging, there is the mathematical expectation bias that can be presented in the following form:

If the magnitude E(t) is described about the point t = t0 within the limits of time interval [t0 − 0.5T; t0 + 0.5T] by the series with odd powers in the following form

The variance of the mathematical expectation estimate of the investigated stochastic process is defined in the following form:

where R(t1, t2) is the correlation function of the investigated stochastic process ξ(t).

In practice, the nonstationary stochastic processes with time-varying mathematical expectation or variance or both of them simultaneously are widely used. In doing so, the mathematical expectation and variance vary slowly in comparison with variations of the investigated stochastic process. In other words, the mathematical expectation and variance of stochastic process are constant within the limits of the correlation interval. In this case, to define the variance of the time-varying mathematical expectation estimate we can assume that the centralized stochastic process ξ0(t) = ξ(t) − E(t) is the stationary stochastic process within the limits of the interval t0 ± 0.5T with the correlation function that can be presented in the following form:

Taking into consideration the given approximation, the variance of the mathematical expectation estimate given by (12.316) after transformation of the double integral by introducing new variables τ = t2 − t1, t2 = t and changing the order of integration takes the following form:

As we can see from (12.313) and (12.318), the dispersion of the mathematical expectation estimate is defined in the following form:

In principle, we can define the optimal integration time T, under which the dispersion will be minimum at the instant t0 minimizing the dispersion of estimate by the parameter T. However, we can present a solution to this problem in an acceptable analytical form by giving a specific function of E(t).

Evaluate how the mathematical expectation estimate varies when the mathematical expectation E(t) deviates from the linear function. At the same time, we assume that the estimated mathematical expectation E(t) possesses the first and second continuous derivatives with respect to the time t. Then, according to the Taylor formula, we can write

where 0 < ϑ < 1. Substituting (12.320) into (12.313), we obtain

Denoting M as the maximum value of the second derivative of the mathematical expectation E(t) with respect to the time t, we obtain the top bound of the mathematical expectation estimate bias by module

As a rule, the maximum magnitude of the second derivative of the mathematical expectation E(t) with respect to the time t can be evaluated based on an analysis of specific physical problems.

To estimate the optimal time of integration T minimizing the dispersion of estimate given by (12.319), we assume that the correlation interval of the investigated stochastic process is much less than the integration time, that is, τcor T. Then, the following written form is true:

Taking into consideration (12.322) and (12.323) and based on the condition of minimization of the estimate dispersion given by (12.319), we obtain the optimal estimation of the integration time:

As we can see from (12.324), the larger the integration time, the larger the correlation interval and the variance of the investigated stochastic process. The lesser the integration time, the larger the maximum absolute value of the second derivative of the mathematical expectation measured. This statement agrees well with the physical interpretation of measuring the time-varying mathematical expectation.

In some applications, the time-varying mathematical expectation can be approximated by the series given by (12.256). The values minimizing the function

can be considered as the estimates of the coefficients αi . This representation of the coefficients αi is possible only if the mathematical expectation E(t) and the functions φi(t) vary in time slowly in comparison with the variation of the first derivative of the function x0(t) with respect to the time; that is, the following condition must be satisfied:

of the mathematical expectation E(t) is much less than the effective bandwidth of energy spectrum of the stochastic component x0(t). Based on the condition of minimization the function ε2, that is,

where the determinant A of the system of linear equations given by (12.328) and the determinant Am obtained by changing the mth column cim of the determinant A by the column yi are determined based on (12.270) and (12.271).

mathematical expectation estimate is given by (12.183). In practice, we sometimes use the constant values of the factor, γ[N] that is,

In this case, the estimation given by (12.346) can be presented in the following form [19,20]:

As we can see from (12.350), the mathematical expectation estimate is unbiased. To define the variance of estimate, we assume that the samples xi are uncorrelated. Then, the variance of mathematical expectation estimate takes the following form:

As 1 − γ < 1 and N → ∞ or N 1, (12.351) can be simplified and takes the limiting form:

that is, the considered estimate is not consistent.

The ratio between the variance of the mathematical expectation estimate defined by (12.351) and the variance of the optimal mathematical expectation estimate given by (12.183) as a function of the number of uncorrelated samples N and various magnitudes of the factor γ is shown in Figure 12.17. As we can see from Figure 12.17, for each value N there is a definite magnitude γ, at which the ratio of variances reaches the maximum.

FIGURE 12.17  Ratio between the variance of mathematical expectation estimate given by (12.351) and the variance of optimal mathematical expectation estimate given by (12.183) as a function of the number of uncorrelated samples N and various values of the factor γv.

12.10  ESTIMATE OF MATHEMATICAL EXPECTATION WITH UNKNOWN PERIOD

In some applications, we can suppose that the time-varying mathematical expectation E(t) of the stochastic process ξ(t) is the periodic function

and the value of the period T0 is unknown. At the same time, the practical case, when the period T0 is much more than the correlation interval τcor of the observed stochastic process, is of interest for us. We employ the adaptive filtering methods widely used in practice under interference cancellation to measure the time-varying mathematical expectation estimate [21]. We consider the discrete sample x(ti) = xi = x(iTs) where the sampling period is Ts. The sample can be presented in the form discussed in (12.168).

The observed samples xi come in at the inputs of the main and reference (the adaptive filter) channels (see Figure 12.18). The delay τ = kTs, where k is integer, is chosen in such a way that the samples at the main and reference channels would be uncorrelated. There is a filter with varying parameters in the reference channel. The incoming samples xi and the process yi forming at the adaptive filter output are sent at the subtractor input. At the subtractor output, the following process

takes place. Taking into consideration that the samples are uncorrelated, the mathematical expectation of quadratic signal in the main and reference channels is defined as

The minimum of ε2 corresponds to the minimum of the second term in the right side (12.355). Thus, if the parameters of adaptive filter are changed before definition of the minimum ε2 , there is a need to use the signal yi at the adaptive filter output as the estimate Ei of time-varying mathematical expectation. As was shown in Ref. [21], under the given structure of interference and noise canceller, the value yi is the best estimate in the case of the quadratic loss function given by (11.25).

The adaptive filter with required impulse response is realized in the form of linear vector summing of signals with the weight coefficients Wj, where j = 0, 1,…, P − 1 are the numbers of

FIGURE 12.18  Adaptive filter flowcharts.

parallel channels, at that, the delay between neighboring channels is equal to the sampling period Ts. Tuning of the weight coefficients Wj is carried out in accordance with the recurrent Widrow–Hopf

μis the adaptation parameter characterizing the algorithmic convergence rate and the tuning accuracy of the weight coefficients

As was shown in Ref. [22], if the parameter μ satisfies the condition 0 < µ < λmax−1 , where λmax−1 is the largest eigenvalue [3] of the covariance matrix consisting of elements Cij = cicj  , then the algorithm

(12.356) is converged. In the physical sense the eigenvalues of covariance matrix characterize the power of input stochastic process and, under the other equal conditions, the larger λ, the more power of input stochastic process. It was proved that

where Wj are the elements of optimal vector of the weight coefficients satisfying the Wiener–Hopf equation that takes the following form:

j= 0

in the stationary mode. The block diagram of computation algorithm for the weight coefficients of adaptive filter is shown in Figure 12.19. To stop the adaptation process we can use the procedures discussed in Ref. [22]. The most widely used procedure applied to the considered problem is based on the following inequality:

where ν is the number given before.

The estimated mathematical expectation Ei is the periodical function and can be approximated by the Fourier series with finite number of terms

=1

where