Sinusoids

The term sinusoid is used here to designate a waveform of the type

$$A\cos(2\pi ft + \phi) = A \cos(\omega t + \phi).$$

Thus, a sinusoid is defined as a cosine at amplitude $A$, frequency $f$, and phase $\phi$. (See [83] for a fuller development and discussion.) We could also have defined a sinusoid based on the sine function as $A\sin(2\pi ft + \phi)$, in which case $\phi$ would be different by $90\degrees$ ($\pi/2$ radians). A sinusoid's phase is typically denoted by either $\phi$ or $\phi$, and is in radian units. We may call

$$\theta(t) \isdef \omega t + \phi$$

the instantaneous phase, as distinguished from the phase offset $\phi$. The instantaneous frequency of a sinusoid is defined as the derivative of the instantaneous phase with respect to time (see [83] for more):

$$f(t) \isdef \frac{d}{dt} \theta(t) = \frac{d}{dt} \left[\omega t + \phi\right] = \omega$$

A discrete-time sinusoid is simply obtained from a continuous-time sinusoid by replacing $t$ by $nT$:

$$A\cos(2\pi f nT + \phi) = A \cos(\omega n T + \phi).$$