We formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros (n poles and m zeros inside the unit disc and their conjugates). The approximation utilizes the Kullback-Leibler divergence as a distance measure. The stationarity condition for optimality requires that the approximant matches n+1 covariance moments of the given power spectrum and m cepstral moments of the corresponding logarithm, although the latter with possible slack. The solution coincides with one derived by Byrnes, Enqvist, and Lindquist who addressed directly the question of covariance and cepstral matching. Thus, the present paper provides an approximation theoretic justification of such a problem. Since the approximation requires only moments of spectral densities and of their logarithms, it can also be used for system identification.
Bibliographical noteFunding Information:
Manuscript received November 17, 2006; revised June 29, 2007 and August 29, 2007. Published August 27, 2008 (projected). Recommended by Associate Editor J.-F. Zhang. This research was supported by grants from AFOSR, NSF, VR, SSF, and the Göran Gustafsson Foundation.
- ARMA modeling
- Cepstral coefficients
- Convex optimization
- Covariance matching