Diss / 10

Taking into consideration the approximation given by (13.116), the variance of the variance estimate presented in (13.114) can be presented in the following form:

As applied to the Gaussian stochastic process at the given assumptions, we can write

and the variance of the variance estimate at the instant t0 takes the following form:

Consider the characteristics of the estimate of the time-varying stochastic process variance for the case, when the investigated stochastic process variance can be presented in the series expansion form

i=1

where

βi are some unknown numbers ψi(t) are the given functions of time

With increase in the number of terms in series given by (13.120), the approximation errors can decrease to an infinitesimal value. Similar to (12.258), we can conclude that the coefficients βi are defined by the system of linear equations

based on the condition of minimum of quadratic error of approximation (13.120). Thus, the problem with definition of the stochastic process variance estimate by a single realization observed within the limits of the interval [0, T] can be reduced to estimation of the coefficients βi in the series given by (13.120). In doing so, the bias and the variance of the variance estimate of the investigated stochastic process caused by errors occurring while measuring the coefficients βi take the following form:

i =1, j =1

In this case, the series coefficient estimations are defined by a simple integration of the quadratic realization of stochastic process with the corresponding weight function ψm(t):

In doing so, the flowchart presented in Figure 13.6 is essentially simplified because there is no need to generate the coefficients cij and to solve the system of linear equations. The number of delay block is decreased too.

Determine the bias and mutual correlation functions between the estimates β*m and β*q . Taking into consideration (13.127) we can rewrite (13.133) in the following form:

where Aim are the algebraic complement of the determinant Am ≡ Bm (12.276) in which cip and gi are given by (13.132) and (13.138).

Since ζ(t) = 0, then gi = 0 and, consequently, the estimations β*m of the coefficients in series

given by (13.120) are unbiased. The correlation functions R(β*m,β*q ) and the variance Var (β*m ) are defined by formulas that are analogous to (12.333) and (12.334). For these formulas, we have

We can show that by applying the correlation function R(t1, t2) to the Gaussian stochastic process we have

If the functions ψi(t) are the orthonormalized functions the correlation function of the estimations β*m and β*q of coefficients is defined by analogy with (12.338):

At that time, the formulas for the current and averaged variance of the time-varying variance estimate can be presented as in (12.339) and (12.240)

If in addition to the condition of orthonormalization of the functions ψi(t), we assume that the frequency band of time-varying variance is much lower than the effective spectrum bandwidth of the investigated stochastic process ξ(t), the correlation function of the centralized stochastic process ζ(t) can be presented in the following form:

If the functions ψi(t) are the orthonormalized functions satisfying the Fredholm equation of the second type

In doing so, the variance of estimations β*m of the coefficients and the current and averaged variations in the variance estimate can be presented in the following form:

investigated and measured stochastic process ζ(t) and receiver noise ς(t) are distributed uniformly within the limits of amplifier bandwidth.

The realization x(t) of stochastic process ξ(t) at the squarer input can be presented in the follow-

accurate within the constant coefficient characterizing the average value of amplifier coefficient, where υ(t) is the realization of random variations of the receiver amplifier coefficient β(t). Since the amplifier coefficient is a positive characteristic, the pdf of the process 1 + β(t) must be approximated by the positive function, too. In practice, measurement of weak signals is carried out, as a rule, under the condition of small value of the variance Varβ of variations of the amplifier coefficient compared to its mathematical expectation that is equal to unit in our case. In other words, the condition Varβ 1 must be satisfied.

Taking into consideration the foregoing statements and to simplify analysis, we assume that all stochastic processes ζ(t), ς(t), and β(t) are the stationary Gaussian stochastic processes with zero mathematical expectation and correlation function defined as

We assume that the measured stochastic process and receiver noise are the narrow-band stochastic processes. The stochastic process β(t) characterizing the random variations of the amplifier coefficient is the low-frequency stochastic process. In addition, we consider situations when the stochastic processes ζ(t), ς(t), and β(t) are mutually independent.

Realization of stochastic process at the ideal integrator output can be presented in the following form:

The variance estimate of the investigated stochastic process after cancellation of amplifier noise takes the following form:

To define the sensitivity of compensation procedure we need to determine the mathematical expectation and the variance of estimate z(t). We can obtain that

After cancellation of variance of the amplifier noise (1 + Varβ)Varn, the mathematical expectation of the output signal with accuracy within the coefficient 1 + Varβ corresponds to the true value

of ­variance of the observed stochastic process. Thus, in the case of random variations of the amplification coefficient, the variance possesses the following bias:

Determine the variance of the variance estimate that limits the sensitivity of compensation procedure to measure a variance of the investigated stochastic process. Given that the considered stochastic processes are stationary, we can change the integration limits in (13.156) from 0 to T. In this case, the variance of the variance estimate takes the following form:

where

is the ratio between the amplifier noise and the noise of investigated stochastic process within the limits of amplifier bandwidth. Double integral in (13.160) can be transformed into a single integral by introducing new variables and changing the order of integration. Taking into consideration the condition Varβ 1 and neglecting the integrals with double frequency 2ω0, we obtain

The time interval of observation corresponding to the sensitivity threshold is determined as

As applied to the compensation procedure of measurement, we obtain

As an example, the stochastic processes with exponential normalized correlation functions can be considered:

r(τ) = exp{−α | τ |};

(13.165)rβ (τ) = exp{−γ | τ |},

where α and γ are the characteristics of effective spectrum bandwidth of their corresponding stochastic processes. As a result, we have

The obtained general and particular formulas for the variation in the variance estimate of stochastic process are essentially simplified in practice since the time interval of observation is much more than the correlation interval of stochastic processes ζ(t) and ς(t). In other words, the inequalities αT 1 and α γ are satisfied. In this case, (13.162) and (13.166) take the following form, correspondingly

When the random variations of the amplifier coefficient are absent (Varβ = 0) the variance of the variance estimate can be presented in the following form:

Consequently, when there are random variations of the amplification coefficient, there is an increase in the variance of the variance estimate of the investigated stochastic process on the value

As we can see from (13.170), with an increase in average time (the parameter γT) the additional random errors decrease correspondingly, and in the limiting case at γT 1 we have

Let T0 be the time required to measure the variance of investigated stochastic process with the given root-mean-square deviation if random variations of the amplification coefficient are absent. Then, Tβ is the time required to measure the variance of investigated stochastic process with the given root-mean-square deviation if random variations of the amplification coefficient are present and is given by

FIGURE 13.8  Relative increase of the variance of the variance estimate as a function of the parameter γT = 1 at 8(α/γ)Varβ = 1.

The relative increase in the variance of variance estimate

as a function of the parameter γT when 8(α/γ)Varβ = 1 is shown in Figure 13.8. Since Varβ 1, this case corresponds to the condition (α/γ) 1; that is, the spectrum bandwidth of the investigated stochastic processes is much more than the spectrum bandwidth of variations in amplification coefficient. Formula (13.173) is simplified for two limiting cases γT 1 and γT 1. At γT 1; in this case, the correlation interval of the amplification coefficient is greater than the time interval of observation of stochastic process, that is,

As we can see from Figure 13.8, at definite conditions the random variations in the amplification coefficient increase essentially the variance of the variance estimate of stochastic process; that is, the radiometer sensitivity is decreased.

Formula (13.169) allows us to obtain a value of the time interval of observation corresponding to the sensitivity threshold at Varβ = 0. This time can be defined as

As we can see from (13.176), the time interval of observation essentially increases with an increase in the variance of amplifier noise. The amplifier intrinsic noise can be presented in the form of product between the independent stationary Gaussian stochastic processes, namely, the narrow-band