Kullback-Leibler approximation of spectral density functions

We introduce a Kullback-Leibler 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 particular, 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.