Units

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

where 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 radians, which is 360 degrees (). Therefore, Hz is the same frequency as 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 ).

For example, a periodic signal with a period of seconds has a frequency of Hz, and a radian frequency of rad/s. The sampling rate, , is the reciprocal of the sampling period , i.e.,

Since the sampling period is in seconds, the sampling rate 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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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.