Additive Synthesis (Early Sinusoidal Modeling)

Additive synthesis is evidently the first technique widely used for analysis and synthesis of audio in computer music [232,233,184,186,187]. It was inspired directly by Fourier theory [264,23,36,150] (which followed Daniel Bernoulli's insights (§G.1)) which states that any sound $ s(t)$ can be expressed mathematically as a sum of sinusoids. The `term ``additive synthesis'' refers to sound being formed by adding together many sinusoidal components modulated by relatively slowly varying amplitude and frequency envelopes:

$\displaystyle y(t)= \sum\limits_{i=1}^{N} A_i(t)\sin[\theta_i(t)]$ (11.17)

$\displaystyle A_i(t)$ $\displaystyle =$ $\displaystyle \hbox{amplitude of $i$th partial over time $t$}$  
$\displaystyle \theta_i(t)$ $\displaystyle =$ $\displaystyle \int_0^t \omega_i(t)dt + \phi_i(0)$  
$\displaystyle \omega_i(t)$ $\displaystyle =$ $\displaystyle d\theta_i(t)/dt = \hbox{radian frequency of $i$th partial vs.\ time}$  
$\displaystyle \phi_i(0)$ $\displaystyle =$ $\displaystyle \hbox{phase offset of $i$th partial at time $0$}
\protect$ (11.18)

and all quantities are real. Thus, each sinusoid may have an independently time-varying amplitude and/or phase, in general. The amplitude and frequency envelopes are determined from some kind of short-time Fourier analysis as discussed in Chapters 8 and 9) [62,187,184,186].

An additive-synthesis oscillator-bank is shown in Fig.10.7, as it is often drawn in computer music [235,234]. Each sinusoidal oscillator [166] accepts an amplitude envelope $ A_i(t)$ (e.g., piecewise linear, or piecewise exponential) and a frequency envelope $ f_i(t)$ , also typically piecewise linear or exponential. Also shown in Fig.10.7 is a filtered noise input, as used in S+N modeling (§10.4.3).

% latex2html id marker 27324\psfrag{A1} []{ \Large$ A_1(t)$\ }\psfrag{A2} []{ \Large$ A_2(t)$\ }\psfrag{A3} []{ \Large$ A_3(t)$\ }\psfrag{A4} []{ \Large$ A_4(t)$\ }\psfrag{f1} []{ \Large$ f_1(t)$\ }\psfrag{f2} []{ \Large$ f_2(t)$\ }\psfrag{f3} []{ \Large$ f_3(t)$\ }\psfrag{f4} []{ \Large$ f_4(t)$\ }\psfrag{s} [b]{ {\Huge$\sum$} }\psfrag{noise} [][b]{{\Large$\stackrel{\hbox{[new] noise}}{u(t)}$}}\psfrag{Filter} [][c]{{\Large$\stackrel{\hbox{filter}}{h_t(\tau)}$}}\psfrag{out} []{
{\large$ y(t)= \sum\limits_{i=1}^{4}
A_i(t)\sin\left[\int_0^t\omega_i(t)dt +\phi_i(0)\right]
+ (h_t \ast u)(t) $\ = sine-sum + filtered-noise}}\begin{figure}[htbp]
\caption{Sinusoidal oscillator bank and filtered
noise for sines+noise spectral modeling synthesis.}

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Additive Synthesis Analysis
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Spectral Envelope Examples