Additive Synthesis

Additive synthesis or “sinusoidal modeling” involves the summation of multiple sinusoidal signals to approximate the spectral content of complex sounds. While this approach may at first appear quite simple, the synthesis of time-varying signals using additive techniques requires the control and careful manipulation of many sinusoidal parameters that continually vary over time.

Additive synthesis can be represented mathematically as:

$$y(t) = \sum_{n=1}^{N} A_{n}(t) \sin \left[\int_{0}^{t}\omega_{n}(t)dt + \phi_{n}(0)\right]$$

where $N$ is the number of sinusoidal components, $A_{n}$ and $\omega_{n}$ are the time-varying amplitude and frequency of the $n$th sinusoidal component, respectively, and $\phi_{n}$ is the initial phase of the $n$th sinusoidal component.