Additive Synthesis Overview

Additive synthesis is a technique in which a signal is reconstructed from a summation of sinusoids and possibly other components. Each sinusoid has a time varying amplitude and phase:

$$y(t)= \sum\limits_{i=1}^{N} A_i(t)\sin[\theta_i(t)]$$

where

$$A_i(t) = \hbox{Amplitude of $i$th partial over time $t$}$$

$$\theta_i(t) = \int_0^t \omega_i(t)dt + \phi_i(t)$$

$$\omega_i(t) = d\theta_i(t)/dt = \hbox{Radian frequency of $i$th partial vs.\ time}$$

$$\phi_i(t) = \hbox{Phase offset of $i$th partial at time $t$} \protect$$ (8.3)

and all quantities are real.

As mentioned previously, the sinusoidal signal model is efficient for tonal signals, such as voiced speech, steady-state wind instrument tones, plucked/struck strings, etc. It is inefficient for noise-like signals, such as unvoiced speech, and the ``chiff'' portion of flute/organ tones. It is also inefficient for attacks, (sharp time-domain transients) such as percussive note onsets.

An additive-synthesis oscillator-bank is shown in Fig.7.1, as it is often drawn in computer music [218,217]. Each sinusoidal oscillator [246] accepts an amplitude envelope $A_i(t)$ (typically piecewise linear) and a frequency envelope $f_i(t)$, also typically provided as a piecewise linear function (in computer music). Also shown in Fig.7.1 is a filtered noise input, used in sines plus noise spectral modeling, to be discussed in §7.4.