Digital Signal Processing Reference
In-Depth Information
5.2
Filtering in the Time Domain
[
]
[
]
The summation in (5.3) is called the
convolution
of sequences
x
n
and
h
n
l
g
r
,
y
i
d
.
,
©
,
L
s
and is denoted by the operator “
∗
” so that (5.3) can be shorthanded to
y
[
n
]=
x
[
n
]
∗
h
[
n
]
This is the general expression for a filtering operation in the discrete-time
domain. To indicate a specific value of the convolution at a given time index
n
0
, we may use the notation
y
[
]=(
)[
]
n
0
x
∗
h
n
0
5.2.1
The Convolution Operator
Clearly, for the convolution of two sequences to exist, the sum in (5.3) must
be finite and this is always the case if both sequences are absolutely summa-
ble. As in the case of the DTFT, absolute summability is just a sufficient
condition and the sum (5.3) can be well defined in certain other cases as
well.
Basic Properties.
The convolution operator is easily shown to be linear
and time-invariant (which is rather intuitive seeing as it describes the be-
havior of an LTI system):
]
∗
α
·
]
=
α
·
x
[
n
y
[
n
]+
β
·
w
[
n
x
[
n
]
∗
y
[
n
]+
β
·
x
[
n
]
∗
w
[
n
]
(5.4)
w
[
n
]=
x
[
n
]
∗
y
[
n
]
⇐⇒
x
[
n
]
∗
y
[
n
−
k
]=
w
[
n
−
k
]
(5.5)
The convolution is also commutative:
x
[
n
]
∗
y
[
n
]=
y
[
n
]
∗
x
[
n
]
(5.6)
which is easily shown via a change of variable in (5.3). Finally, in the case of
square summable sequences, it can be shown that the convolution is asso-
ciative:
x
]
∗
]
∗
h
]
[
n
]
∗
h
[
n
w
[
n
]=
x
[
n
[
n
]
∗
w
[
n
(5.7)
This last property describes the effect of connecting two filters
and
in
cascade and it states that the resulting effect is that of a single filter whose
impulse response is the convolution of the two original impulse responses.
As a corollary, because of the commutative property, the order of the two
filters in the cascade is completely irrelevant. More generally, a sequence of
filtering operations can be performed in any order.
