Spectral Envelope Extraction

There are many definitions of spectral envelope. Piecewise-linear (or polynomial spline) spectral envelopes (applied to the spectral magnitude of an STFT frame), have been used successfully in sines+noise modeling of audio signals (introduced in §10.4). Here we will consider spectral envelopes defined by the following two methods for computing them:

1. cepstral windowing to lowpass-filter the log-magnitude spectrum (a ``nonparametric method'')

2. using linear prediction (a ``parametric method'') to capture spectral shape in the amplitude-response of an all-pole filter in a source-filter decomposition of the signal (where the source signal is defined to be spectrally flat)

In the following, $X_m(\omega_k)$ denotes the $m$ th spectral frame of the STFT (§7.1), and $Y_m(\omega_k)$ denotes the spectral envelope of $X_m(\omega_k)$ .