Let
, denote an -sample complex sequence,
i.e.,
. Then the spectrum of is defined by the
Discrete Fourier Transform (DFT):

The inverse DFT (IDFT) is defined by

In this chapter, we will omit mention of an explicit sampling interval
, as is most typical in the digital signal processing
literature. It is often said that the sampling frequency is .
In this case, a radian frequency
. is in
units of ``radians per sample.'' Elsewhere in this book,
usually means ``radians per second.'' (Of course, there's no
difference when the sampling rate is really .) Another term we use
in connection with the convention is normalized
frequency: All normalized radian frequencies lie in the range
, and all normalized frequencies in Hz lie in the range
.^{7.1} Note that physical units
of seconds and Herz can be reintroduced by the substitution