Diss / 10

As we can see from (15.295) and (15.296), the average values are equal to the probabilities of nonexceeding and exceeding the level M by the stochastic process realization x(t) at the instants t1 and t2

Taking into consideration (15.294) through (15.298), introducing new variables t = t1 − t2, and changing the order of integration, we can write

–

where Var{N*}/[NT]2 is the normalized variance of the average number of stochastic process spikes or the relative variance of the average number of stochastic process spikes.

As applied to the Gaussian and Rayleigh stochastic processes, we can present the two-dimensional probability distribution functions in the form (14.50) and (14.71). In the case of Gaussian stochastic process with zero mathematical expectation, that is, F(M0) = 1 − F(M0) at M0 = 0. Substituting (14.50) into (15.297) and (15.298), we obtain

where av and cv are defined analogously as in (14.56) and (14.58). In doing so, the values of the coefficients av are presented in Table 14.1 as a function of v and the normalized level z = Mσ −1.

Since

we can see from (15.302) that the normalized variance of the average number of stochastic process

spikes Var{ *}/[ – ] is symmetrical with respect to the level /σ = 0. Because of this, we can write

N NT 2 M

The normalized squared deviation of the average number of stochastic process spikes

Var{N } / [NT] as a function of the normalized level |z = Mσ −1| for realizations of stochastic process with fixed duration, the parameter p = αT is shown in Figure 15.16. As we can see from

FIGURE 15.16  Normalized squared deviation of the average number of stochastic process spikes as a function of the normalized level for Gaussian realizations of a stochastic process with fixed duration.

Figure 15.16, the normalized squared deviation Var{N } / [NT] of the average number of stochastic process spikes is increased with increasing in the absolute level value |z = Mσ −1| and decreasing the parameter p = αT. At p = αT ≥ 10 and |z = Mσ−1| = 0, we can write

Substituting (14.71) into (15.297) and (15.298), we obtain

where bv and dv are defined analogously as in (14.86) and (14.83). The values of coefficients are given in Table 14.3. The normalized squared deviation of the average number of stochastic process spikes

Var{N } / [NT] as a function of the normalized level M / 2σ2 at p1 = 2αT = 10 in the case of the exponential normalized correlation function given by (15.305) is shown in Figure 15.17. As we can see from Figure 15.17, the normalized squared deviation of the average number of stochastic

process spikes Var{N }[NT] increases with an increase in the deviation of the normalized level

M / 2σ2 with respect to the median of the probability distribution function M0 / 2σ2 = ln 2 at the fixed p1 = 2αT . This phenomenon is explained by a decrease in the number of spikes higher

or lower than the level M0 / 2σ2 = ln 2 .

15.6.2  Estimation of Average Spike Duration and Average Interval between Spikes

Considering the stochastic process realization presented in Figure 15.15a, we can see that with higher number of spikes N within the limits of the observation time interval [0, T] the estimate of

Var{N*}

8.0NT

6.0

4.0

2.0

M

√ 2σ

0.0

FIGURE 15.17  Normalized squared deviation of the average number of stochastic process spikes as a function of the normalized level for Rayleigh realizations of the stochastic process with fixed duration.

spike duration average τ* and the estimate of average of the interval θ* between the spikes can be presented as

for the given realization. The relationships given by (15.313) can be presented in the following form:

where ητ(t) and ηθ(t) are given by (15.282) and (15.283).

The device to measure the spike duration average τ* and the average of interval θ* between the spikes can be designed based on (15.313) and (15.314). As applied to the estimate of the spike duration average τ*, the flowchart of measurer is depicted in Figure 15.18. This measurer consists

Divider τ*

M

Counter

N*

FIGURE 15.18  Measurement of spike duration and the average of interval between the spikes.

As applied to the Gaussian stochastic process, if the condition T τcor is satisfied the normalized correlation functions τ(t) and θ(t) are given by (15.300) and (15.301) and the average number of stochastic process spikes at level M can be determined as

Taking into consideration (15.285) and substituting (15.319), we obtain

where αv is given by (14.56) and Table 14.1. The formula (15.322) is correct for measuring the estimate variance of the average of interval θ* between the spikes.

As applied to the normalized correlation function given by (15.305), the estimate variance of the average spike duration τ*

where p = αT. As we can see from (15.323), in the Gaussian stochastic process and the fixed duration of the observation interval [0, T] case, the estimate variance of the spike duration average τ* and the estimate variance of the average of interval θ* between the spikes are minimum at M/σ = 0.

As applied to the Rayleigh stochastic process, if the condition T τcor is satisfied the normalized correlation functions τ(t) and θ(t) are given by (15.310) and (15.311) and the average number of stochastic process spikes at level M can be determined as

Substituting τ(t) into (15.319) and taking into consideration (15.285) and (15.324), we obtain

It is easy to prove that (15.235) is true to define the estimate variance of the average of interval θ* between the spikes.

15.7  MEAN-SQUARE FREQUENCY ESTIMATE OF SPECTRAL DENSITY

The mean-square frequency f– given by (15.6) is –widely used as a parameter characterizing the spectral density of stochastic process. The value f defines a dispersion of component of the stochastic process spectral density relative to zero frequency. As applied to low-frequency stationary stochastic processes, the mean-square frequency f– characterizes the effective bandwidth–of spectral density. Taking into consideration (15.259), we can present the mean-square frequency f in the following form:

τ = 0

Here and further, we assume that the investigated stochastic process possesses zero mathematical expectation. As applied to the Gaussian stochastic process, the mean-square frequency f– is matched with the average number of stochastic process spikes per unit time at the zero level (15.321).

According to (15.326), it is worthwhile to consider for the investigated stationary stochastic process the following value

as the estimate of the mean-square frequency that tends to approach f– as T → ∞. The flowchart of measurer of the mean-square frequency of the stochastic process is shown in Figure 15.19. To define the characteristics of the mean-square frequency estimate we can use the following representation of numerator and denominator in (15.327) in the following form:

where

Varx and Varx· are the mathematical expectations of variances of the stochastic process and its derivative

Varx and Varx· are the random errors of definition of the earlier-mentioned variances within the limits of the finite observation time interval [0, T]

FIGURE 15.19  Measurement of the averaged squared frequency of stochastic process.