Additive Synthesis - Definitions

Definitions

See also: Fourier series and Fourier analysis

Harmonic additive synthesis is closely related to the concept of a Fourier series which is a way of expressing a periodic function as the sum of sinusoidal functions with frequencies equal to integer multiples of a common fundamental frequency. These sinusoids are called harmonics, overtones, or generally, partials. In general, a Fourier series contains an infinite number of sinusoidal components, with no upper limit to the frequency of the sinusoidal functions and includes a DC component (one with frequency of 0 Hz). Frequencies outside of the human audible range can be omitted in additive synthesis. As a result only a finite number of sinusoidal terms with frequencies that lie within the audible range are modeled in additive synthesis.

A waveform or function is said to be periodic if

for all and for some period .

The Fourier series of a periodic function is mathematically expressed as:

\begin{align} y(t) &= \frac{a_0}{2} + \sum_{k=1}^{\infty} \left \\ &= \frac{a_0}{2} + \sum_{k=1}^{\infty} r_k \cos\left(2 \pi k f_0 t + \phi_k \right) \\ \end{align}

where

is the fundamental frequency of the waveform and is equal to the reciprocal of the period,
is the amplitude of the th harmonic,
is the phase offset of the th harmonic. atan2( ) is the four-quadrant arctangent function,

Being inaudible, the DC component, and all components with frequencies higher than some finite limit, are omitted in the following expressions of additive synthesis.

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