Discrete Fourier transform - Continuous and Discrete Time Signals and Systems - page 530

The discretized version of the DTFT X w ( Ω ) is referred to as the discrete
Fourier transform (DFT) and is generally represented as a function of the fre-
quency index r corresponding to DTFT frequency Ω r

= 2 r π/ M , for 0 ≤ r ≤
( M − 1). To derive the expression for the DFT, we substitute Ω = 2 r π/ M in
the following definition of the DTFT:

N − 1

x 2 [ k ]e − j k Ω ,

X 2 ( Ω ) =

(12.11)

k = 0

where we have assumed x 2 [ k ] to be a time-limited sequence of length N .
Equation (12.11) reduces as follows:

N − 1

x 2 [ k ]e − j(2 π kr / M ) , X 2 ( Ω r ) = (12.12)

k = 0

for 0 ≤ r ≤ ( M − 1). Equation (12.12) defines the DFT and can easily be imple-
mented on a digital device since it converts a discrete number N of input samples
in x 2 [ k ] to a discrete number M of DFT samples in X 2 ( Ω r ). To illustrate the
discrete nature of the DFT, the DFT X 2 ( Ω r ) is also denoted as X 2 [ r ]. The DFT
spectrum X 2 [ r ] is plotted in Fig. 12.1(r).

Let us now return to the original problem of determining the CTFT X ( ω )of
the original CT signal x ( t ) on a digital device. Given X 2 [ r ] = X 2 ( Ω r ), it is
straightforward to derive the CTFT X ( ω ) of the original CT signal x ( t )by
comparing the CTFT spectrum, shown in Fig. 12.1(b), with the DFT spectrum,
shown in Fig. 12.1(r). We note that one period of the DFT spectrum within the
range − ( M − 1) / 2 ≤ r ≤ ( M − 1) / 2 (assuming M to be odd) is a fairly good
approximation of the CTFT spectrum. This observation leads to the following
relationship:

N − 1

X ( ω r ) ≈ MT 1
N

X 2 [ r ] = MT 1
N

− j(2 π kr / M ) ,

x 2 [ k ]e (12.13)

k = 0

where the CT frequencies ω r =

Ω r / T 1

= 2 π r / ( M T 1 ) for − ( M − 1) / 2 ≤

r ≤ ( M − 1) / 2.
Although Fig. 12.1 illustrates the validity of Eq. (12.13) by showing that the
CTFT X ( ω ) and the DFT X 2 [ r ] are similar, there are slight variations in the two
spectra. These variations result from aliasing in Step 1 and loss of samples in
Step 2. If the CT signal x ( t ) is sampled at a sampling rate less than the Nyquist
limit, aliasing between adjacent replicas distorts the signal. A second distortion
is introduced when the sampled sequence x 1 [ k ] is multiplied by the rectangular
window w [ k ] to limit its length to N samples. Some samples of x 1 [ k ] are lost in
the process. To eliminate aliasing, the CT signal x ( t ) should be band-limited,
whereas elimination of the time-limited distortion requires x ( t ) to be of finite
length. These are contradictory requirements since a CT signal cannot be both
time-limited and band-limited at the same time. As a result, at least one of the