Sound Modelling
93
resynthesis
overlap and add
FFT-based synthesis
better to run the analysis backward in time, since in most cases a sharp attack
is followed by a stable release, and peak tracking is more effective when stable
states are reached gradually and suddenly released, rather than vice versa.
If we can rely on the assumption of harmonicity of the analyzed sounds, the
partial tracking algorithm can be "encouraged" by superposition of a harmonic
comb onto the spectral profile.
For a good separation, frequencies and phases must be determined accu-
rately, following the procedures described in section 4.1.5. Moreover, for the
purpose of smooth resynthesis, the amplitudes of partials should be interpolated
between frames, the most common choice being linear interpolation. Frequencies
and phases should be interpolated as well, but one should be careful to ensure
that the frequency track is always the derivative of the phase track. Since a
third-order polynomial is uniquely determined by four degrees of freedom, by
using a cubic interpolating polynomial one may impose the instantaneous phases
and frequencies between any couple of frames.
Resynthesis of the sinusoidal components
In the resynthesis stage, the sinusoidal components can be generated by any of
the methods described in section 5.2, namely the digital oscillator in wavetable
or recursive form, or the FFT-based technique. The latter will be more conve-
nient when the sound has many sinusoidal components.
The DTFT of a windowed sinusoidal signal is the transform of the window,
centered on the frequency of the sinusoid, and multiplied by a complex num-
ber whose magnitude and phase are the magnitude and phase of the sine wave.
A signal that is the weighted sum of sinusoids gives rise, in the frequency do-
main, to a weighted sum of window transforms centered around different central
frequencies.
If the window has a
A. sufficiently-high sidelobe attenuation,
we are allowed to consider only a restricted neighborhood of the window trans-
form peak. The sound resynthesis can be achieved by anti-transformation of a
series of STFT frames, and by the procedure of overlap and add applied to the
time-domain frames. The signal reconstruction is free of artifacts if
B. the shifted copies of the window overlap and add to give a constant.
If w is the window that fulfills property (A), and is the window that fulfills
property (B), we can use w for the analysis and multiply the sequence by /w
after the inverse transformation [35]. Using two windows gives good flexibility in
satisfying both the requirements (A) and (B). A particularly simple and effective
window that satisfies property (B) is the triangular window.
This FFT-based synthesis (or FFT
-1
synthesis) is convenient when the si-
nusoidal model gives many sine components, because its complexity is largely
due to the cost of FFT, which is independent on the number of components. It
is quite easy to introduce noise components with arbitrary frequency distribu-
tion just by adding complex numbers with the desired magnitude (and arbitrary
phase) in the frequency domain.