Kullback-Leibler approximation of spectral density functions

We introduce a Kullback-Leibler (1968) -type distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density Ψ by one that is consistent with prescribed second-order statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose spectral density is sought. In this context, we show (i) that there is a unique spectral density Φ which minimizes this Kullback-Leibler distance, (ii) that this optimal approximate is of the form Ψ/Q where the "correction term" Q is a rational spectral density function, and (iii) that the coefficients of Q can be obtained numerically by solving a suitable convex optimization problem. In the special case where Ψ = 1, the convex functional becomes quadratic and the solution is then specified by linear equations.