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is the signal at the output of linear filter operating in stationary mode with impulse response given by

matched with the earlier-given orthogonal function φk(t). As we can see from (15.51), the mathematical expectation of estimate

is matched with the true value; in other words, the estimate of coefficients of expansion in series is unbiased.

The estimate variance of expansion in series coefficients can be presented in the following form:

As applied to the stationary Gaussian stochastic process, the stochastic process forming at the output of orthogonal filter will also be the stationary Gaussian stochastic process for the considered case. Because of this, we can write

Introducing new variables t2 − t1 = τ, t1 = t and changing the order of integration analogously as shown in Section 12.4, we obtain

yk(t)

h(t) = k(t)

FIGURE 15.4  One-channel measurer of coefficients αk.

Taking into consideration that

and the condition given by (14.123), we can define the “integral” variance of correlation function estimate

As we can see from (15.62), the variance of correlation function estimate increases with an increase in the number of terms of expansion in series v. Because of this, we must take into consideration choosing the number of terms under expansion in series.

The foregoing formulae allow us to present the flowchart of correlation function measurer based on the expansion of this correlation function in series and the estimate of coefficients of this expansion in series. Figure 15.4 represents the one-channel measurer of the current value of the coefficient αk . Operation of measurer is clear from Figure 15.4. One of the main elements of block diagram of the correlation function measurer is the generator of orthogonal signals or functions (the orthogonal filters with the impulse response given by [15.53]). If the pulse with short duration τp that is much shorter than the filter constant time and amplitude τ−p1 excites the input of the orthogonal filter, then a set of orthogonal functions φk(t) are observed at the output of orthogonal filters.

The flowchart of correlation function measurer is shown in Figure 15.5. Operation control of the correlation function measurer is carried out by the synchronizer that stimulates action on the generator of orthogonal signals and allows us to obtain the correlation function estimate with period that is much longer compared to the orthogonal signal duration.

The functions using the orthogonal Laguerre polynomials given by (14.72) and presented in the following form

The multistep filter based on RC elements, that is, α = 2(RC)−1, which has the transfer characteristic given by (15.66) accurate with the constant factor 2α−0.5 is shown in Figure 15.6. The phase inverter is assigned to generate two signals with equal amplitudes and shifted by phase on 90° and the amplifiers compensate attenuation in filters and ensure decoupling between them.

If the stationary stochastic process is differentiable ν times, the correlation function R(τ) of this process can be approximated by expansion in series about the point τ = 0:

The even 2ith derivatives of the correlation function at the point τ = 0 accurate within the coefficient (−1)i are the variances of ith derivatives of the stochastic process

Microprocessor

R*(τ)

FIGURE 15.7  Correlation function measurement.

correlation function estimate is formed at the calculator output. Define the statistical characteristics of the coefficient estimate αi . The mathematical expectation of the estimate αi has the following form:

We can see that

where

As applied to the Gaussian stochastic process, its derivative is Gaussian too. Because of this, we can write

Substituting (15.77) into (15.76) and then into (15.75), we obtain

If the observation time interval is much longer than the correlation interval of stochastic process and its derivatives, we can write

As applied to the conditions, for which (15.80) is appropriate, the derivatives of the correlation function can be written using the spectral density S(ω) of the stochastic process:

The variance of the estimate αi of expansion in series coefficients can be presented in the following form:

Let us define the deviation of correlation function estimate from the approximated value, namely,

Averaging ε(τ) by realizations of the investigated stochastic process, we can see that in the considered case ε(τ) = 0, which means the bias of correlation function estimate does not increase due to the finite observation time interval.

The variance of correlation function estimate can be presented in the following form:

Let the correlation function of the stochastic process be approximated by

which corresponds to the spectral density defined as

As we can see from (15.89), the variance of correlation function estimate increases with an increase in the number of terms under correlation function expansion in power series.

15.3  OPTIMAL ESTIMATION OF GAUSSIAN STOCHASTIC PROCESS CORRELATION FUNCTION PARAMETER

In some practical conditions, the correlation function of stochastic process can be measured accurately with some parameters defining a character of its behavior. In this case, a measurement of correlation function can be reduced to measurement or estimation of unknown parameters of

We suppose that the correlation intervals of the stochastic processes ξ(t) and ζ(t) are sufficiently small compared to the observation time interval [0, T]. In this case, we can use infinite limits of integration. Under this assumption, we have

Introducing new variables t2 − t1 = τ, t2 = t and changing the order of integration, we obtain

Introducing new variables τ = −τ′, t′ = t + τ = t − τ′ and taking into consideration that the correlation interval of stochastic process η(t) is shorter compared to the observation time interval [0, T], we can write

is the correlation function estimate of investigated input process consisting of additive mixture of the signal and noise.

Applying the foregoing statements to (15.94), we obtain

Substituting (15.102) and (15.104) into (15.93), we obtain

In the case, when the observation time interval [0, T] is much longer compared to the correlation interval of the investigated stochastic process, we can use the spectral representation of the function given by (15.100)

−∞

Then (15.104) takes the following form:

In (15.106) and (15.107), ϑn(ω), ϑx(ω; l), and Sx(ω; l) are the Fourier transforms of the corresponding functions ϑn(τ), ϑx(τ; l), and Rx(τ; l). Applying the Fourier transform to (15.95) and (15.96), we obtain

Flowchart of the optimal measurer is shown in Figure 15.8. This measurer operates in the following way. The correlation function Ry (τ) is defined based on the input realization of additive mixture of the signal and noise. This correlation function is integrated as the weight function with the signal ϑ(τ; l). The signals forming at the outputs of the integrator and generator of the function H(l) come

FIGURE 15.8  Optimal measurement of correlation function parameter.