Chapter 6
LINEAR CONVOLUTION
Copyright
c 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the
students attending the undergraduate DSP course EE113 in the Electrical Engineering Department
at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A.
H. Sayed, Electrical Engineering Department, UCLA, CA 90095, [email protected]
The response of an
LTI system with impulse response sequence
h
(
n
) to any input
sequence
x
(
n
) is given by the convolution sum
y
(
n
) =
x
(
n
)
? h
(
n
) =
∞
X
k
=
∞
x
(
k
)
h
(
n

k
)]
In this chapter we study more closely such convolution sums and derive several of their
properties.
We shall also provide some physical interpretations for the properties of the
convolution sum.
Properties of the Convolution Sum
1.
Commutativity
. It holds that
x
(
k
)
? h
(
k
) =
h
(
k
)
? x
(
k
)
That is,
∞
X
k
=
∞
x
(
k
)
h
(
n

k
) =
∞
X
k
=
∞
h
(
k
)
x
(
n

k
)
Proof:
Introduce the new variable
j
=
n

k
. Then
∞
X
k
=
∞
x
(
k
)
h
(
n

k
) =
∞
X
j
=
∞
x
(
n

j
)
h
(
j
)
.
Rename
j
as
k
again to obtain
∞
X
j
=
∞
x
(
n

j
)
h
(
j
) =
∞
X
k
=
∞
h
(
k
)
x
(
n

k
)
.
♦
Physical interpretation.
Think of
x
(
n
) and
h
(
n
) as the impulseresponse sequences of
two
LTI systems. Consider further the series cascades:
δ
(
n
)
x
(
n
)
h
(
n
)
y
1
(
n
)
49
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50
Linear Convolution
Chapter 6
δ
(
n
)
h
(
n
)
x
(
n
)
y
2
(
n
)
The impulse response of the first cascade is
y
1
(
n
) =
x
(
n
)
? h
(
n
) (as established
in Chapter 5).
On the other hand, the impulse response of the second cascade is
y
2
(
n
) =
h
(
n
)
? x
(
n
).
By the commutativity property, both output sequences must
agree,
y
1
(
n
) =
y
2
(
n
).
Therefore, the commutativity property tells us that we can
always switch the order of
LTI systems in a series cascade.
2.
Distributivity
. The following relations hold:
x
(
n
)
?
[
h
1
(
n
) +
h
2
(
n
)] =
x
(
n
)
? h
1
(
n
) +
x
(
n
)
? h
2
(
n
)
and
[
x
1
(
n
) +
x
2
(
n
)]
? h
(
n
) =
x
1
(
n
)
? h
(
n
) +
x
2
(
n
)
? h
(
n
)
Algebraic proof:
We prove the first relation. An identical argument applies to the second
one. Using the definition of convolution we write
x
(
n
)
?
[
h
1
(
n
) +
h
2
(
n
)] =
∞
X
k
=
∞
x
(
k
)[
h
1
(
n

k
) +
h
2
(
n

k
)]
=
∞
X
k
=
∞
x
(
k
)
h
1
(
n

k
) +
∞
X
k
=
∞
x
(
k
)
h
2
(
n

k
)
=
x
(
n
)
? h
1
(
n
)
+
x
(
n
)
? h
2
(
n
)
.
♦
Physical interpretation.
Consider two LTI systems with impulse responses
h
1
(
n
) and
h
2
(
n
) and assume they are connected in parallel, as shown in Fig. 6.1.
x
(
n
)
h
1
(
n
)
h
2
(
n
)
y
(
n
)
Figure 6.1.
Parallel connection of two LTI systems.
Let
x
(
n
) be the input sequence to this cascade connection. We know from Chapter 5
that the impulse response sequence of the parallel connection is
h
1
(
n
)+
h
2
(
n
). There
fore, the output of the system will be
x
(
n
)
?
[
h
1
(
n
) +
h
2
(
n
)]. On the other hand, the
output of
h
1
(
n
) is
x
(
n
)
? h
1
(
n
) and the output of
h
2
(
n
) is
x
(
n
)
? h
2
(
n
). Hence,
x
(
n
)
?
[
h
1
(
n
) +
h
2
(
n
)] =
x
(
n
)
? h
1
(
n
) +
x
(
n
)
? h
2
(
n
)
.
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 Spring '08
 Walker
 Electrical Engineering, Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Commutativity

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