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

$\displaystyle 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

$\displaystyle \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):

$\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$:

$\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, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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