S+N Synthesis

A sines+noise synthesis diagram is shown in Fig.10.12. The spectral-peak amplitude and frequency trajectories are possibly modified (time-scaling, frequency scaling, virtual formants, etc.) and then rendered into the time domain by additive synthesis. This is termed the deterministic part of the synthesized signal.

The stochastic part is synthesized by applying the residual-spectrum-envelope (a time-varying FIR filter) to white noise, again after possible modifications to the envelope.

To synthesize a frame of filtered white noise, one can simply impart a random phase to the spectral envelope, i.e., multiply it by $\exp[j\phi(\omega_k)]$ , where $\phi(\omega_k)$ is random and uniformly distributed between $-\pi$ and $\pi$ . In the time domain, the synthesized white noise will be approximately Gaussian due to the central limit theorem (§D.9.1). Because the filter (spectral envelope) is changing from frame to frame through time, it is important to use at least 50% overlap and non-rectangular windowing in the time domain. The window can be implemented directly in the frequency domain by convolving its transform with the complex white-noise spectrum (§3.3.5), leaving only overlap-add to be carried out in the time domain. If the window side-lobes can be fully neglected, it suffices to use only main lobe in such a convolution [239].

In Fig.10.12, the deterministic and stochastic components are summed after transforming to the time domain, and this is the typical choice when an explicit oscillator bank is used for the additive synthesis. When the IFFT method is used for sinusoid synthesis [239,94,139], the sum can occur in the frequency domain, so that only one inverse FFT is required.