Sinusoids

The term sinusoid means a waveform of the type

$$A\cos(2\pi ft + \phi) = A \cos(\omega t + \phi). \protect$$ (A.1)

Thus, a sinusoid may be defined as a cosine at amplitude $A$ , frequency $f$ , and phase $\phi$ . (See [84] for a fuller development and discussion.) A sinusoid's phase $\phi$ is in radian units. We may call

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

the instantaneous phase, as distinguished from the phase offset $\phi$ . Thus, the ``phase'' of a sinusoid typically refers to its phase offset. The instantaneous frequency of a sinusoid is defined as the derivative of the instantaneous phase with respect to time (see [84] 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 $nT$ in Eq.(A.1):

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