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in at the input of summator. Thus, the receiver output signal is formed. The decision device issues the value of the parameter lm, under which the output signal takes the maximum value.

If the correlation function of the investigated Gaussian stochastic process has several unknown parameters l = {l1, l2,…, lμ}, then the likelihood ratio functional can be found by (15.93) changing the scalar parameter l on the vector parameter l and the function H(l) is defined by its derivatives

The function ϑx(t1, t2; l) is the solution of integral equation that is analogous to (15.95).

To obtain the estimate of maximum likelihood ratio of the correlation function parameter the optimal measurer (receiver) should define an absolute maximum lm of logarithm of the likelihood ratio functional

To define the characteristics of estimate of the maximum likelihood ratio lm introduce the signal

Prove that if the noise component is absent in (15.116), that is, n(l) = 0, the logarithm of likelihood ratio functional reaches its maximum at l = l0, that is, when the estimated parameter takes the true value. Define the first and second derivatives of the signal function given by (15.114) at the point l = l0:

544 Signal Processing in Radar Systems

Now, let us define the second derivative of the signal component at the point l = l0:

is satisfied forever. For this purpose, we define the averaged quadratic first derivative of the likelihood ratio functional logarithm at the point l = l0

In addition, as follows from the definition, the mathematical expectation of the noise function n(l) is zero. Taking into consideration the introduced notations, we can write the logarithm of the likelihood ratio functional in the following form:

where ε = 1/ SNR. Taking into consideration (15.133), the likelihood ratio equation for the estimate­ of correlation function parameter of Gaussian stochastic process can be presented in the following form:

Usually, under the measurement of stochastic process parameters, SNR is high and, consequently, ε 1. Then, by analogy with Ref. [5], the approximated solution of likelihood ratio equation can be searched in the form of expansion in power series

To define the approximations l1, l2, l3 and grouping the terms with small value ε of the same power, we obtain

Since the system of functions 1, x, x2,… is linearly independent, the equality given by (15.136) is satisfied for any ε if and only if all coefficients of terms with power equal to ε are equal to zero. Zero approximation is matched with the true value of the parameter l0 since S(l) reaches its absolute maximum at

Equating to zero the coefficients at ε, ε2, and ε3, we obtain equations to define l1, l2, and l3. We can write solutions of these equations in the following form:

Taking into consideration the first three approximations l1, l2, and l3, the conditional bias and variance of maximum likelihood ratio take the following form:

Var{lm |l0} = ε2[ l12 − l1 2 ]+ 2ε3[ l1l2 − l1 l2 ]

Averaging is carried out by all possible realizations of the total stochastic process η(t) at the fixed value of estimated parameter l0. The relative error of estimate bias and variance that can be defined as the ratio of the first term with small order to the first term of expansion takes the order ε2.

We are limited by consideration of the first approximation. In doing so, the random error of a single measurement can be presented in the following form:

For the first approximation, the estimate of arbitrary parameter of correlation function will be unbiased, since n(l) = 0. Taking into consideration (15.128) and (15.129), the variance of estimate can be presented in the following form:

If the observation time interval is much longer than the correlation interval of the investigated stochastic process η(t), a flowchart of optimal measurer is significantly simplified. In this case, the logarithm of the likelihood ratio functional is given by (15.111), where the signal function can be described in the following form:

q2 is the ratio of the investigated stochastic process variance to the white noise power within the limits of effective bandwidth of the signal; p is the ratio between the observation time interval of the investigated stochastic process and its correlation interval. Using these notations, we can write

where α0 is the true value of estimated correlation function parameter. Substituting (15.158) and (15.159) into (15.151) and (15.152), we obtain

When q2 1 and 4σ2T −01  1, in this case (15.160) and (15.161) are correct, then β ≈ 1 and (15.160) and (15.161) take a simple form:

to power noise q2 within the limits of effective spectral bandwidth are presented in Figures 15.9 and 15.10.

Consider the second example. For this purpose, we analyze the correlation function of the nar-

FIGURE 15.9  Relative estimate bias shift as a function of the ratio between the variance of the investigated stochastic process and power noise.

pD(αm|α0) 1/2

2α20

2.0

1.6

1.2

0.8

0.4

q2

0.0

FIGURE 15.10  Relative root-mean-square deviation of estimate as a function of the ratio between the variance of the investigated stochastic process and power noise.

is satisfied. We estimate the parameter ν. In the narrowband stochastic process case, the parameter ν plays a role of the central spectral density frequency. We assume that the stochastic process with the correlation function given by (15.166) is investigated in the white noise with the correlation function

where δ(τ) is the Dirac delta function and the observation time interval [0, T] is much longer than the correlation stochastic process interval.

In accordance with (15.111), the logarithm of likelihood ratio functional can be presented in the following form: