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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 $ 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

$\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 $ t$ by $ nT$ in Eq.(A.1):

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

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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