Digital Signal Processing Reference
In-Depth Information
Step 2: Obtaining the ideal filter impulse response
h
d
(
n
)
The impulse response can be obtained by the inverse DTFT, from
Chapter 2
,
as follows:
π
1
2
()
∫
j
ωω
j
n
hn
()
=
H e
e d
ω
(5.8)
d
d
π
−
π
Considering the low-pass example given in
Figure 5.4
,
the impulse
response, obtained from Equation 5.8 is:
ω
π
sin(
ω
n
)
,
c
c
hn
()
=
−∞ ≤
n
≤ ∞
(5.9)
d
ω
n
c
A rough sketch of the impulse response, given in Equation 5.9, is shown
in Figure 5.5. On observing the impulse response in Figure 5.5, there are two
fundamental problems:
•
The impulse response
h
d
(
n
) exists on both positive and negative sides
of the time axis; hence, the system is
not causal
.
•
The impulse response
h
d
(
n
) exists to infinite extent on both sides of
the time axis; hence, the system is
not
finite
.
h(n)
1
-
∞
∞
------ -------------
n
-4 -3 -2 -1 0
1 2 3 4
FIGURE 5.5
Impulse response of ideal low-pass filter.
