Units

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

$$t = nT,$$

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.,

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

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.