tags: dsp synthesis
Sine Generator
A sine generator is an [oscillator] that creates a [sine wave].
Analog
Given a frequency and amplitude, the general (analog, continuous) formula for a sine wave is
$$ s(t) = A sin (2 \pi ft + \phi) $$
- $f$: frequency in Hz
- $A$: amplitude in range [0, 1]
- $t$: time in seconds
- $\phi$: initial phase (for most cases, assume $\phi$ is 0)
Digital
A digital sine wave is given by the following formula:
$$ s[n] = A sin (2 \pi f n / f_s) $$
- $n$: the [sample] number
- $f_s$: [sample rate]
In most digital scenarios, such as calculators and computers, any sin() function is computed using the Taylor expansion of the sine function:
$$ sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} $$
However, this computation is generally expensive, especially in the context of [digital audio] where thousands of samples are computed a second.
- digital implementations usually resort to [wave tables], storing pre-computed values in memory