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
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 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 (Greek ``tau'') 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.,