Analysis of linear prediction, coding, and spectral estimation from subbands

Linear prediction schemes make a prediction xˆ_i of a data sample x_i using p previous samples. It has been shown by Woods and O´Neil (1986) as well as Pearlman (1991) that as the order of prediction p→∞, there is no gain to be obtained by coding subband samples. This paper deals with the less well understood theory of finite-order prediction and optimal coding from subbands which are generated by ideal (brickwall) filtering of a stationary Gaussian source. We first prove that pth-order prediction from subbands is superior to pth-order prediction in the fullband, when p is finite. This fact adduces that optimal vector p-tuple coding in the subbands is shown to offer quantifiable gains over optimal fullband p-tuple coding, again when p is finite. The properties of subband spectra are analyzed using the spectral flatness measure. These results are used to prove that subband DPCM provides a coding gain over fullband DPCM, for finite orders of prediction. In addition, the proofs provide means of quantifying the subband advantages in linear prediction, optimal coding, and DPCM coding in the form of gain formulas. Subband decomposition of a source is shown to result in a whitening of the composite subband spectrum. This implies that, for any stationary source, a pth-order prediction error filter (PEF) can be found that is better than the pth-order PEF obtained by solving the Yule-Walker equations resulting from the fullband data. We demonstrate the existence of such a “super-optimal” PEF and provide algorithmic approaches to obtaining this PEF. The equivalence of linear prediction and AR spectral estimation is then exploited to show theoretically, and with simulations, that AR spectral estimation from subbands offers a gain over fullband AR spectral estimation