Digital Signal Processing Reference
In-Depth Information
0
1 −
|
k
|
|
k
|
> 3
≤ 3
0.67
0.67
0.6
0.33
0.33
0.36
k
k
−5
−4
−3
−2
−1
012345
−5
−4
−3
−2
−1
012345
(a)
(b)
FIR filters
Fig. 14.3. (a) FIR filter; (b) IIR
filter.
1
−
k
3
k
≤
3
Triangular sequence
h
[
k
]
=
(
N
=
5);
0
elsewhere
shifted impulse sequence
h
[
k
]
=
0
.
1
δ
[
k
−
2]
+ δ
[
k
]
+
0
.
2
δ
[
k
−
2]
(
N
=
5);
5
0
.
4
k
δ
[
k
−
m
]
exponentially decaying triangular sequence
h
[
k
]
=
m
=−
5
(
N
=
11);
10 000
1
m
+
1
δ
[
k
−
m
]
N
=
10 001)
.
decaying impulses
h
[
k
]
=
m
=
0
IIR filters
h
[
k
]
=
0
.
6
k
u
[
k
]
N
=∞
);
Causal decaying exponential
h
[
k
]
=
0
.
5
k
sin(0
.
2
π
k
)
u
[
k
]
N
=∞
)
.
causal decaying sinusoidal
Other examples of IIR filters include non-causal ideal filters as shown in
Table 14.1.
Figure 14.3(a) plots the triangular sequence with length
N
=
5 as an example
of the FIR filter. Likewise, Fig. 14.3(b) plots the causal decaying exponential
sequence with infinite length as an example of the IIR filter. An important
consequence of a finite-length impulse response
h
[
k
] is observed during the
determination of the output response of an FIR filter resulting from a finite-
length input sequence. Since the output response is obtained by the convolution
of the impulse response and the input sequence, the output of an FIR filter is
finite in length if the input sequence itself is finite in length. On the other hand,
an IIR filter produces an output response that is always infinite in length.
A second consequence of the finite length of the FIR filters is observed in the
stability characteristics of such filters. Recall that an LTID system with impulse
response function
h
[
k
] is BIBO stable if
∞
h
[
k
]
< ∞.
k
=−∞
Search WWH ::
Custom Search