ECE 3250
DISCRETETIME CONVOLUTION
Fall 2008
We’ve been modeling discretetime signals as functions
x
:
Z
→
F
, where
F
denotes
R
or
C
. I introduced the notation
F
Z
to stand for the set of all
F
valued discretetime signals.
If you want, you can view the typical element
x
of
F
Z
as a twosided inﬁnite sequence of
numbers from
F
, i.e.
. . . , x
(

3)
, x
(

2)
, x
(

1)
, x
(0)
, x
(1)
, x
(2)
, . . .
x
stands for a signal and
x
(
n
) stands for the value of the signal
x
at time
n
.
Given two signals
x
1
and
x
2
in
F
Z
, the
convolution of
x
1
and
x
2
, if it exists, is the
signal
x
∈
F
Z
with speciﬁcation
(1)
x
(
n
) =
∞
X
k
=
∞
x
1
(
k
)
x
2
(
n

k
)
,
n
∈
Z
.
Some notations I’ll be using for the convolution of
x
1
and
x
2
are
x
1
*
x
2
and Conv(
x
1
, x
2
).
Alternative terminologies for the convolution of
x
1
and
x
2
are “the convolution of
x
1
with
x
2
” and “
x
1
convolved with
x
2
.”
The convolution of
x
1
and
x
2
exists if and only if the sum in (1) converges for every
n
∈
Z
. Note that the sum is a series that’s potentially “inﬁnite in both directions.”
We haven’t encountered such series before, so suppose that
{
a
n
}
is a twosided inﬁnite
sequence of numbers from
F
, i.e.,
. . . a

2
, a

1
, a
0
, a
1
, a
2
, . . .
We say that the doubly inﬁnite series
P
∞
n
=
∞
a
n
converges if and only if the two onesided
series
∞
X
n
=0
a
n
and

1
X
n
=
∞
a
n
=
∞
X
m
=1
a

m
both converge. In this case, the limit of the doubly inﬁnite series is the sum of the limits
of the two onesided series.
Let’s begin with an elementary observation about convolution. If
x
1
*
x
2
exists, then
the sum in (1) converges for every
n
∈
Z
. Change the index of summation as follows:
x
1
*
x
2
(
n
) =
∞
X
k
=
∞
x
1
(
k
)
x
2
(
n

k
) =
∞
X
m
=
∞
x
1
(
n

m
)
x
2
(
m
) =
∞
X
k
=
∞
x
1
(
n

k
)
x
2
(
k
)
.
Setting
m
=
n

k
yields the middle equality. To get the last, rename the “dummy
index of summation”
m
as
k
. The bottom line is that on the righthand side of equation
(1), it doesn’t matter where we put the
k
and where we put the (
n

k
) — the result
is the same. One could dignify this observation by saying something along the lines of,
“convolution, deﬁned by (1), is a commutative operation in the sense that if
x
1
*
x
2
exists,
then
x
1
*
x
2
=
x
2
*
x
1
.” That’s ﬁne, but it’s a bit unnecessary in my view.
A slightly less elementary observation about convolution is that it is an
associative
operation in the sense that if
x
1
*
(
x
2
*
x
3
) exists, then so does (
x
1
*
x
2
)
*
x
3
, and vice
versa, and both convolutions are the same. Proving this fact is an exercise in summation
manipulation. I’ll be a bit casual about interchanging orders of summation here. It turns
out that the interchanges are legal since all the sums converge. Assuming that
x
1
*
(
x
2
*
x
3
)
1