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

128 ANALYSIS AND TRANSMISSION OF SIGNALS
Filtering
We generally think of filtering in terms of a hardware-oriented solution (namely, building
a circuit with RLC components and operational amplifiers). However, filtering also has a
software-oriented solution [ a computer algorithm that yields the filtered output y(t) for a given
input g(t)] . This can be conveniently accomplished by using the DFT. If g(t) is the signal to
be filtered, then G q
, the DFT of gk, is found. The spectrum G q
is then shaped (filtered) as
desired by multiplying G q
by H q
, where H q
are the samples of the filter transfer function H (f)
[H q
= H(qjo )]. Finally, we take the inverse DFT or (IDFT) of G q
H q
to obtain the filtered
output Yk [yk = Ts y(kT)]. This procedure is demonstrated in the following example.
COMPUTER EXAMPLE C3.3
The signal g(t) in Fig. 3.42a is passed through an ideal low-pass filter of transfer function H (f), shown
in Fig. 3.42b. Use DFT to find the filter output.
Figure 3.42
Filtering g(t)
through H(J).
8 g(t ) (a)
1
H (f)
(b)
-0.5
0.5 t -+
-2
2 f(HZ) -+
J
-40 -32
[ J
- 24 - 16 - 8 -4
l
4 8
1
16 q -+ 24
r
32 40
(c)
- 10 -8
-6 -4 -2
"'
· ·· /\
Jli'
0
2
4 6
f(Hz) -+
8 10
/ .._I\
(d)
-16
-··•·
8 ···-
........
/1 111
• i ! ! !
f: : ! : !
0
-
1. . ...
·-&.·' · 8
16
.- .•. ,t . t'
. 24 -.f 31
k -+
We have already found the 32-point DFT of g(t) (see Fig. 3.41d). Next we multiply G q
by
H q
, To compute H q
, we remember that in computing the 32-point DFT of g(t), we have
used fo = 0.25. Because G q
is 32-periodic, H q
must also be 32-periodic with samples
separated by 0.25 Hz. This means that H q
must be repeated every 8 Hz or lfor rad/s (see
Fig. 3.42c). This gives the 32 samples of H q
over O S f S 8 as follows:
H q
= {
0.5
0 S q S 7 and 25 S q S 31
9 S q S 23
q = 8, 24