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Additive Synthesis

To obtain oscillator-control envelopes for additive synthesis, the amplitude, frequency, and phase trajectories are estimated once per FFT hop by the STFT. It is customary in computer music to linearly interpolate the amplitude and frequency trajectories from one hop to the next. Call these signals $ \hat{A}_k(n)$ and $ \hat{F}_k(n)$, defined now for all $ n$ at the normal signal sampling rate. The phase is usually discarded at this stage and redefined as the integral of the instantaneous frequency when needed: $ \hat{\Theta }_k(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\hat{\Theta }_k(n-1) +
2\pi T \hat{F}_k(n)$. When phase must be matched in a given frame, the frequency can instead move quadratically across the frame to provide cubic polynomial phase interpolation [12], or a second linear breakpoint can be introduced somewhere in the frame for the frequency trajectory.

6. Apply any desired modification to the analysis data, such as time scaling, pitch transposition, formant modification, etc.

7. Use the (possibly modified) amplitude and frequency trajectories to control a summing oscillator bank:

$\displaystyle \hat{x}(n)$ $\displaystyle \mathrel{\stackrel{\mathrm{\Delta}}{=}}$ $\displaystyle \frac{1}{N}
\sum_{k=-N/2+1}^{N/2-1} \hat{A}_k(n)e^{j\hat{\Theta }_k(n)}$ (4)
  $\displaystyle =$ $\displaystyle \frac{2}{N}\sum_{k=0}^{N/2-1}\hat{A}_k(n)\cos(\hat{\Theta }_k(n))$ (5)

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Download parshl.pdf

``PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation'', by Julius O. Smith III and Xavier Serra, Proceedings of the International Computer Music Conference (ICMC-87, Tokyo), Computer Music Association, 1987.
Copyright © 2005-12-28 by Julius O. Smith III and Xavier Serra
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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