Spectral Analysis for Nonstationary Audio

Abstract

A new approach for the analysis of nonstationary signals is proposed, with a focus on audio applications. Following earlier contributions, nonstationarity is modeled via stationaritybreaking operators acting on Gaussian stationary random signals. The focus is on time warping and amplitude modulation, and an approximate maximum-likelihood approach based on suitable approximations in the wavelet transform domain is developed. This paper provides theoretical analysis of the approximations, and introduces JEFAS, a corresponding estimation algorithm. The latter is tested and validated on synthetic as well as real audio signal.

Time-Scale Synthesis for Locally Stationary Signals

“…When all subband signals W s in (1) are stationary, the resulting signal y is stationary. We are interested here in a specific situation where non-stationarity induces a time-dependent shift on the scale axis, as studied in [14,12,10]. It was shown there that such a model can account for signals obtained by time warping stationary signals, namely signals of the form…”

“…• The vectors w n are decorrelated, zero-mean, circular complex Gaussian vectors: w n ∼ CN c (0, C n ) • The corresponding covariance matrices C n are translates of a fixed function f as shown in [10], namely…”

“…In JEFAS-S, we use the expression (5), for which an initial estimate of S has to be provided. When successful, JEFAS [10] provides such an estimate. Otherwise, a rough estimate can be obtained from the Welch periodogram of the input signal y.…”

“…Spectrum estimate update. The spectrum update from the current estimate of W is performed in two steps: first correct for the translation by θ n , to obtain an approximately stationary subband transform, then average over time to obtain a wavelet based spectral estimate as in [10].…”

“…The signal duration is 83.1 seconds, sampled at F s = 10 Hz (N = 832 samples). Because of the fast instantaneous frequency variations, the wavelet transform of y (not shown here) contains interference patterns, the model in [10] is not adequate and JEFAS does not converge. We initialized JEFAS-S to θ = 0, and a constant function for S .…”