B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

3.9 Numerical Computation of Fourier Transform: The DFT 121
Equation (3.102) reveals the interesting fact that g(m+N o ) = g m . This means that the
sequence gk is also periodic with a period of No samples (representing the time duration
No T s = To seconds). Moreover, G q
is also periodic with a period of No samples, representing
a frequency interval Nofo = (To/T s )(To) = I/T s = ls hertz. But I/T s is the number of samples
of g(t) per second. Thus, I/T s = ls is the sampling frequency (in hertz) of g(t). This means
that G q
is No-periodic, repeating every ls Hz. Let us summarize the results derived so far. We
have proved the discrete Fourier transform (DPT) pair
N o- 1
G q
= L gke - f q rl o k
k=O
No-1
gk = _!__ L G q
J krl oq
No q=O
(3.103a)
(3.I03b)
where
G q
= G(qfo)
2n
2nlo = - To
2n
2nfs = -
Ts
(3.104)
To ls 2n
No = - = - S1o = 2nloT s = -
T s Jo No
Both the sequences gk and G q
are periodic with a period of No samples. This results in gk
repeating with period To seconds and G q
repeating with periodfs = I/T s rad/s, orfs = I/T s Hz
(the sampling frequency). The sampling interval of gk is T s seconds and the sampling interval
of G q
islo = I/To Hz. This is shown in Fig. 3.39c and d. For convenience, we have used the
frequency variable l (in hertz) rather than w (in radians per second).
We have assumed g(t) to be time-limited to r seconds. This makes G(f) non-bandlimited.
* Hence, the periodic repetition of the spectra G q
, as shown in Fig. 3.39d, will cause
overlapping of spectral components, resulting in error. The nature of this error, known as
aliasing error, is explained in more detail in Chapter 6. The spectrum G q
repeats every ls
Hz. The aliasing error is reduced by increasing fs, the repetition frequency (see Fig. 3.39d).
To summarize, the computation of G q
using DPT has aliasing error when g(t) is time-limited.
This error can be made as small as desired by increasing the sampling frequency ls = I /T s ( or
reducing the sampling interval T s ). The aliasing error is the direct result of the nonfulfillment
of the requirement T s --+ 0 in Eq. (3.95).
When g(t) is not time-limited, we need to truncate it to make it time-limited. This will
cause further error in G q
. This error can be reduced as much as desired by appropriately
increasing the truncating interval To .t
In computing the inverse Fourier transform [by using the inverse DPT in Eq. (3.I03b)],
we have similar problems. If G(f) is band-limited, g(t) is not time-limited, and the periodic
repetition of samples gk will overlap (aliasing in the time domain). We can reduce the aliasing
error by increasing To, the period of g k ( in seconds). This is equi valent to reducing the frequency
* We can show that a signal cannot be simultaneously time-limited and band-limited. If it is one, it cannot be the
other, and vice versa. 3
t The DFf relationships represent a transform in their own right, and they are exact. If, however, we identify gk and
Gq as the samples of a signal g (t) and its Fourier transform G(f) , respectively, then the DFT relationships are
approximations because of the aliasing and truncating effects.