Spectral estimation via selective harmonic amplification: a canonical decomposition for state-covariances

The value of the state-covariance consitutes a generalized interpolation constraint on the spectrum of the input signal. We explore this basic fact by considering singular spectra which are consistent with state-covariance estimates. Accordingly, we present a canonical decomposition for state covariances. The special case where the covariance matrices are Toeplitz reduces to the well-known Caratheodory-Fejer-Pisarenko (CFP) theorem which underlies modern subspace-based signal estimation techniques. As an application of the theory, we present two alternative methods for identifying spectral lines in a time-series (generalizations of MUSIC and ESPRIT respectively), we identify spectral envelopes consistent with covariance data (suitable generalizations of the Capon method of maximum likelihood), and we construct a canonical spectrum which is dual to the CFP decomposition and generates spectra made up of absorption lines instead (i.e., notches). Our method relies on selecting a suitably designed Input-to-State filter to enhance resolution over any preselected harmonic interval by amplifying the contribution of components in such an interval on the state covariance.