Additive Synthesis

Search JOS Website, CCRMA Website, or Web

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation

Introduction and Overview

The Phase Vocoder

Overlap-Add Synthesis

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

This paper describes a system for audio analysis, modification, and synthesis, based on the Short Time Fourier Transform (STFT). The system is intended both as a tool for sound manipulation, and as a means to reinforce people's intuitions regarding the relationships between timbre and the harmonic structure of music and other audio signals, as conveyed via their spectrograms. This is done by creating a 3D spectrogram which shows a sound's harmonic structure in great detail as it is sampled. — Click for http://cnmat.cnmat.berkeley.edu/~alan/MS-html/MSv2.html#RTFToC1

The Fast Fourier Transform (FFT) is a fast algorithm for computing the Discrete Fourier Transform (DFT) — Click for http://faculty.prairiestate.edu/skifowit/fft/

Click for http://ccrma.stanford.edu/~jos/parshl/

Search JOS Website, CCRMA Website, or Web

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation

Introduction and Overview

The Phase Vocoder

Overlap-Add Synthesis

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 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)

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
[Automatic-links disclaimer]