Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 83

Consider now the problem of sampling the DTFT in the frequency domain. It is
assumed that the frequency sampling is uniform, and that the spacing between
samples is F s . Now it is critical that this frequency domain sampling does not lead
to information loss. That is, it is necessary for the time domain signal to still be
perfectly recoverable, despite this sampling.
It is now required to determine the maximum frequency sampling interval
which ensures no loss of information. Assume initially that this rate is F s = f s /
N. This choice implies that the number of samples in the frequency domain is the
same as the number of samples in the time domain. Then the resulting DFT is
defined by:

X s ð k Þ¼ X

N 1

x s ð n Þ e j2pkn = N :

n ¼ 0

Now consider the inverse Fourier Transform of X s (k). By using an analysis very
similar to that done in Sect. 2.3.2.1 it is possible to show that the inverse is given
by:

X

X s ð k Þ e j2pkn = N () X s ð k Þ¼ X

1 N 1

x p ð n Þ¼ 1
N

x p ð n Þ e j2pkn = N

k ¼1 n ¼ 0

Now it is important to realize that both X s (k) and e -j2p kn/N are periodic, and
because of this periodicity, x p (t) is also periodic, with period N. This indicates that
just as sampling in the time domain causes periodicity in the frequency domain, so
sampling in the frequency domain causes periodicity in the time domain. Fur-
thermore, with a frequency sampling interval of F s = f s /N the periodicity is seen
to be N, which is just adequate to prevent the time domain images from ''running
into each other''. That is, the frequency sampling interval of F s = f s /N is just
enough to prevent time-domain aliasing. If the sampling interval were any greater
aliasing in the time domain would occur.
In summary then, sampling in the time and frequency domains causes repetition
in both domains. The DFT is normally obtained from the DTFT by sampling at a
rate of F s = f s /N, because this is just enough to avoid time-domain aliasing. With
this sample rate the number of samples in both the time and frequency domains is
N. Although the time and frequency domains both repeat doubly discredited
systems, it is common to only display one image. With this in mind the usual
definitions for the DFT and inverse DFT (IDFT) are:

N 1

X

X ð k Þ e j2pkn = N () X ð k Þ¼ X

N 1

x ð n Þ¼ 1
N

x ð n Þ e j2pkn = N : ð 2 : 15 Þ

k ¼ 0 n ¼ 0

Note that

IDFT f DFT ½ x ð n Þg¼ 1; ð 2 : 16 Þ

that is, the DFT and IDFT are reversible transforms.