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 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.