Signal Processing in Radar Systems

13 Estimation of Stochastic Process Variance

13.1 Optimal Variance Estimate of Gaussian Stochastic Process

Let the stationary Gaussian stochastic process ξ(t) with the correlation function

R(t1,t2)=σ2ℛ(t1,t2)⁢(13.1)

be observed at N equidistant discrete time instants t_i, i = 1, 2,…, N in such a way that

ti+1−tj=Δ=const.⁢(13.2)

Then, at the measurer input we have a set of samples x_i = x(t_i). Furthermore, we assume that the mathematical expectation of observed stochastic process is zero. Then, the conditional N-dimensional pdf of Gaussian stochastic process can be presented in the following form:

p(x1,x2,...,xN|σ2)=1(2πσ2)N/2det‖ ℛij ‖exp{ −12σ2∑​i=1,j=1NxixjCij }⁢(13.3)

where

• det ‖ℛ_ij‖ is the determinant of matrix consisting of elements ...