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Introduction to Digital Filters
    Background Fundamentals
       Signal Representation and Notation
          Sinusoids

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Sinusoids

The term sinusoid means a waveform of the type

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

(A.1)

Thus, a sinusoid may be defined as a cosine at amplitude

, frequency

, and phase $\phi$ . (See [84] for a fuller development and discussion.) A sinusoid's phase $\phi$ is in radian units. We may call

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

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):

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

A discrete-time sinusoid is simply obtained from a continuous-time sinusoid by replacing

in Eq. $\,$ (A.1):

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

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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