Autoregressive Modeling of Temporal Envelopes

Autoregressive (AR) models are commonly obtained from the linear autocorrelation of a discrete-time signal to obtain an all-pole estimate of the signal's power spectrum. We are concerned with the dual, frequency-domain problem. We derive the relationship between the discrete-frequency linear autocorrelation of a spectrum and the temporal envelope of a signal. In particular, we focus on the real spectrum obtained by a type-I odd-length discrete cosine transform (DCT-Io) which leads to the all-pole envelope of the corresponding symmetric squared Hilbert temporal envelope. A compact linear algebra notation for the familiar concepts of AR modeling clearly reveals the dual symmetries between modeling in time and frequency domains. By using AR models in both domains in cascade, we can jointly estimate the temporal and spectral envelopes of a signal. We model the temporal envelope of the residual of regular AR modeling to efficiently capture signal structure in the most appropriate domain.