Audio Signal Processing and Coding

9.2 THE SINUSOIDAL MODEL

This section describes the sinusoidal model that forms the basis for the parametric audio coding and the extended hybrid model given in the latter portions of this chapter. In particular, standard methodologies are presented for sinusoidal analysis, tracking, interpolation, and synthesis. The classical sinusoidal model comprises an analysis-synthesis framework ([McAu86] [Serr90] [Quat02]) that represents a signal, s(n), as the sum of a collection of K sinusoids (“partials”) with time-varying frequencies, phases, and amplitudes, i.e.,

where A_k represents the amplitude, ω_k(n) represents the instantaneous frequency, and ϕ_k(n) represents the instantaneous phase of the k-th sinusoid. It is assumed that the amplitude, frequency, and phase functions evolve on a time scale substantially longer than a signal period. Analysis for this model amounts to estimating the amplitudes, phases, and frequencies of the constituent partials. Although this estimation is typically accomplished by peak picking in the short-time Fourier domain [McAu86] [Span91] [Serr90], analysis-by-synthesis estimation techniques that minimize explicitly a mean square error in terms of the sinusoidal parameters have also been proposed [Geor87] [Geor90] [Geor92]. Sinusoidal analysis-by-synthesis has also been presented within the more generalized framework of matching pursuits using overcomplete ...