Sound Modelling
95
transients
sines + noise + transients
SNT
DCT
Discrete Cosine Transform
Musical
Transformations
Musical
Transformations
Additive
Synthesis
Subtractive
Synthesis
Frequency
Magnitude
Intensity
Coefficients
Noise
Magnitude
Frequency
Spectral Shape
Stochastic
Deterministic (sinusoidal)
Sound
Control
Part
Part
Figure 3: Framework for performing music transformations.
5.1.2
Sines + Noise + Transients
The fundamental assumption behind the sinusoids + noise model is that sound
signals can are composed of slowly-varying sinusoids and quasi-stationary broad-
band noises. This view is quite schematic, as it neglects the most interesting part
of sound events: transients. Sound modifications would be much more easily
achieved if transients could be taken apart and treated separately. For instance,
in most musical instruments extending the duration of a note does not have any
effect on the quality of the attack, which should be maintained unaltered in a
time-stretching task.
For these reasons, a new sines + noise + transients (SNT) framework for
sound analysis was established [108]. The key idea of practical transient extrac-
tion comes from the observation that, as sinusoidal signals in the time domain
are mapped to well-localized spikes in the frequency domain, by duality short
pulses in the time domain would correspond to sine-like curves in the frequency
domain. Therefore, the sinusoidal model can be applied in the frequency domain
to represent these sinusoidal components. The scheme of the SNT decomposition
is represented in figure 4.
Sinusoidal
Modelling
DCT
Sinusoidal
Modelling
DCT
-1
Noise
Modelling
Sines
Sound
e
Transient Detector
Transients
e
Noise
1
2
Figure 4: Decomposition of a sound into sines + noise + transients.
The DCT block in figure 4 represents the operation of Discrete Cosine Trans-
form, defined as
C(k) =
N -1
n=0
x(n) cos
(2n + 1)k
2N
.
(5)
The DCT has the property that an impulse is transformed into a cosine, and a
cluster of impulses becomes a superposition of cosines. Therefore, in the trans-
formed domain it makes sense to use the sinusoidal model and to extract a
second residue that is given by transient components.