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In this book, time $ t$ is always in physical units of seconds (s), while time $ n$ or $ m$ is in units of samples (counting numbers having no physical units). Time $ t$ is a continuous real variable, while discrete-time in samples is integer-valued. The physical time $ t$ corresponding to time $ n$ in samples is given by

$\displaystyle t = nT,

where $ T$ is the sampling interval in seconds.

For frequencies, we have two physical units: (1) cycles per second and (2) radians per second. The name for cycles per second is Hertz (Hz) (though in the past it was cps, which is more mnemonic). One cycle equals $ 2\pi$ radians, which is 360 degrees ($ \hbox{${}^{\circ}$}$). Therefore, $ f$ Hz is the same frequency as $ 2\pi f$ radians per second (rad/s). It is easy to confuse the two because both radians and cycles are pure numbers, so that both types of frequency are in physical units of inverse seconds (s $ \null^{-1}$).

For example, a periodic signal with a period of $ P$ (Greek ``tau'') seconds has a frequency of $ f = (1/P)$ Hz, and a radian frequency of $ \omega = 2\pi/P$ rad/s. The sampling rate, $ f_s$, is the reciprocal of the sampling period $ T$, i.e.,

$\displaystyle f_s = \frac{1}{T}

Since the sampling period $ T$ is in seconds, the sampling rate $ f_s=l/T$ is in Hz. It can be helpful, however, to think ``seconds per sample'' and ``samples per second,'' where ``samples'' is a dimensionless quantity (pure number) thrown in for clarity. The amplitude of a signal may be in any arbitrary units such as volts, sound pressure (SPL), and so on.

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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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