Sine Wave Signal

An audio signal, y(t), composed of exactly one sine wave, can be completely described by the parameters $t, A, f$ and $\phi$, $$ y(t) = A \sin(2 \pi f t + \phi) $$ where $t$ represents time in seconds, $A$ is the wave’s amplitude (unit-less), $f$ is its frequency in Hz, and $\phi$ is its phase offset in radians (i.e., where in the cycle the wave is at $t=0$). If $t \ne 0$, then the sine wave appears shifted in time by $\frac{\phi}{2 \pi f}$, where negative values mean “delay” and positive “advance” it.

Fourier Series

Our old pal Fourier told us that any sound can be represented as an infinite summation of sine waves each with their own amplitudes, frequencies, and phase offsets. This means that any sound we hear can be represented as many, many tuples of $t, A, f, \phi$.