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

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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University