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stochastic process χ(t) and the low-frequency stochastic process θ(t) with zero mathematical expectations. In doing so, we assume that the stochastic process θ(t) possesses the unit variance, that is,θ2(t) = 1, and the correlation functions can be presented in the following form:

Moreover, we assume that αT 1, α η, α γ. In this case, the variation in the variance estimate or the variance of the signal at the ideal integrator output can be presented in the following form:

(13.179)

Comparing (13.179) with (13.168), we see that owing to the low-frequency fluctuations of amplifier intrinsic noise the variance of the variance estimate increases as a result of the value yielded by the second term of (13.179).

13.5.2  Method of Comparison

This method is based on a comparison of constant signal components formed at the output of twochannel amplifier by the investigated stochastic process and the generator signal coming in at the amplifier input. Moreover, the generator signal is calibrated by power or variance. The wideband Gaussian stochastic process with known power spectral density or the deterministic harmonic signal can be employed as the calibrated generator signal. Flowchart of stochastic process variance measurer employing a comparison of signals at the low-pass filter output is presented in Figure 13.9.

Assuming that the receiver channels are identical and independent, the signal at the squarer inputs can be presented in the following form:

FIGURE 13.9  Flowchart of the stochastic process variance measurer.

where

υ1(t) and υ2(t) are the realizations of Gaussian stationary stochastic processes describing the random variations β1(t) and β2(t) relative to the amplification coefficient value at the first and second channels of signal amplifier

n1(t) and n2(t) are the realizations of signal amplifier noise at the first and second channels s(t) and s0(t) are the investigated stochastic process and the reference signal, respectively

Let the reference signal be the stationary Gaussian stochastic process with the variance Vars0. The signal at the subtractor output takes the following form:

The variance estimate is determined in the following form:

Define the mathematical expectation and the variance of the variance estimate under assumptions made earlier

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

As we can see from (13.184), the bias of variance estimate can be presented in the following form:

If the variance of the reference stochastic process is controlled then the variance measurement process can be reduced to zero signal aspects at the output indicator. This procedure is called the method with zero instant. Naturally, the variance estimate is unbiased.

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

0

As applied to the exponential normalized correlation function given by (13.165), we can transform (13.191) into the following form:

Comparing (13.193) with (13.189) at Varβ = 0, we can see that in the case of weal signals, that is, the signal-to-noise ratio is small or q 1, with the use of the harmonic signal the variance of the variance estimate is twice as less as when the wideband stochastic process is used. Otherwise, at the high signal- to-noise ratio, q > 1, under the use of the harmonic signal the variance of the variance estimate can be higher compared to the case when we use the stochastic process. This phenomenon is explained by the presence of uncompensated components of the output signal formed by the terms of high order under expansion of the amplifier noise in series. In addition, the considered procedure possesses the errors caused by the nonidentity of channels and presence of statistical dependence between amplifier channels.

13.5.3  Correlation Method of Variance Measurement

While using the correlation method to measure the stochastic process variance, the stochastic process comes in at the inputs of two channels of amplifier. The intrinsic amplifier channel noise samples are independent of each other. Because of this, their mutual correlation functions are zero and the mutual correlation function of the investigated stochastic process is not equal to zero and the coincidence instants are equal to the variance of the investigated stochastic process. A flowchart of the variance measurement made using the correlation method is presented in Figure 13.10.

Let the independent channels of amplifier operate at the same frequency. Then the stochastic processes forming at the amplifier outputs can come in at the mixer inputs directly. We assume, as before, that the integrator is the ideal, that is, h(t) = T−1. The signal at the integrator output defines the variance estimate

FIGURE 13.10  Correlation method of variance measurement.

are the realizations of stochastic processes at the outputs of the first and second channels, respectively. The mathematical expectation of estimate can be presented in the following form:

that is, the variance estimate is unbiased under the correlation method of variance measurement and when the identical channels are independent.

The variance of the output signal or the variance of the variance estimate of the input stochastic process in the case when the input stochastic process is Gaussian takes the following form:

where Rβ1(τ) and Rβ2(τ) are the correlation functions of random components of the amplification coefficients of the first and second channels of amplifier.

We can simplify (13.197) taking into consideration that the correlation interval of random components of amplification coefficients is longer compared to the correlation interval of the intrinsic noise n1(t) and n2(t) and the investigated stochastic process ς(t). At the same time, we can introduce the function Rβ(0) = Varβ instead of the correlation function Rβ(τ) in (13.197). Then, taking into consideration the condition accepted before, that is, Varβ < 1 or Varβ2 1, identity of channels, and (13.153) through (13.155), we obtain

Formula (13.198) allows us to define the time interval of observation T corresponding to the sensitivity threshold at the condition given by (13.188). Let the normalized correlation functions r(τ) and rβ(τ) be defined by the exponents given by (13.165) and substituting into (13.198), we obtain

(13.199)

When the random variations of the amplification coefficients are absent and the condition T τcor is satisfied, the variance of the variance estimate can be presented in the following form:

The factor 2 before the integral is not important, in principle, but it allows us, as we can see later, to obtain the unbiased variance estimate and the variance of the variance estimate that are convenient to compare with the variance of the variance estimates obtained by other procedures and methods.

As we can see from (13.211), when the random variations of the amplification coefficients are absent Varβ = 0, the variance estimate of stochastic process using the modulation method of variance measurement is unbiased as is the case with the compensation method. To determine the variance of the variance estimate we need to present the function u(t) by the Fourier series expansion

is the switching function frequency.

Ignoring oscillating terms and taking into consideration that Varβ 1 and the correlation interval of random component of the amplification coefficients is much more the correlation ­interval of the investigated stochastic process, the variance of the variance estimate, can be presented in the following form:

(13.214)

Taking into consideration (13.153) through (13.155), (13.214) can be written in the following form:

In practice, the correlation interval of the investigated stochastic process is much less than the modulation period T0 and, consequently, the time interval T of observation, since in real applications, the effective spectrum bandwidth of the investigated stochastic process is more than 104–105 Hz and at the same time, the modulation frequency fmod = Ω × (2π)−1 is several hundred hertz. In this case, we can assume that the functions cos[(2k − 1)Ωτ] are not variable functions within the limits of the correlation interval, and we can think that this function can be approximated by unit, that is, cos[(2k − 1)Ωτ] ≈ 1. This statement is true for the components of variance of the variance estimate caused by stochastic character of variation of the processes s(t) and n(t). Taking into consideration that

we obtain

The approximation cos[(2k − 1)Ωτ] ≈ 1 is true within the limits of the correlation interval of investigated stochastic process owing to fast convergence of the series (13.216) to its limit. If, in addition to the aforementioned condition, the correlation interval of the amplification coefficients is much less than the time interval of observation, that is, τβ 1 but τβ > T0, then