Center for New Music and Audio Technologies

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Controlling the Oscillator

An important requirement of digital additive synthesis is the accurate and independent mapping of the control parameters, amplitude, frequency, and phase into sinusoids as described earlier in equation 1.1:

$\displaystyle y\left(t\right)=\sum_{i=0}^{N} A_i\left(t\right)\cos\left(2\pi f_i \left(t\right)t+\theta_i\left(t\right)\right).$

The amplitude $ A_i$ represents the maximum displacement of the varying quantity from its average value. The frequency $ f_i$ gives the mumber of peaks per second in a sine wave whereas the reciprocal of $ f_i$, the period $ T_i$, is the time between amplitude peaks. Finally, the phase $ \theta_i$, describes the displacement in time from its origin [15]. The required resolutions for frequency $ f_i$ and phase $ \phi_i$ can be estimated from the just noticeable difference (JND) curve derived for the human ear [16]. The curve shows that the human ear is capable for noticing pitch differences of $ 1$ percent at $ 50$Hz which corresponds to a resolution of $ 0.5$ Hz. With a sampling rate of 48kHz the resolution needs to be 16 bit for frequency $ f_i$ and phase $ \phi_i$. Amplitude resolution is dictated largely by what is considered acceptible in audio terms, and the resources that are available. With this in mind, 16 bits was chosen for the amplitude parameter.

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Norbert Lindlbauer
2000-01-19