#### 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
, where
is random and
uniformly distributed between
and
. 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.

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Sines+Noise Summary

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Sines+Noise Analysis