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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). 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$ 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=1/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) included 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, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University