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EEL 3135: Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: Convolution
Lesson Number: 10 (Section 53)
Background:
In Chapter 5 the concept of convolution was introduced in the context of an FIR filter. A finite
impulse response (FIR) filter, as the name implies, has an impulse response that persists for
only of a finite number of sample values.
The impulse response of an
N
th
order FIR is given by
(remember the order controversy):
{
}
1
1
0
,...,
,
]
[

=
N
h
h
h
k
h
1.
The coefficients
h
i
are called
tapweights
. Observe that if
x
[
k
]=
δ
Κ
[
k
], the output
y
[
k
] is first
h
[0]
δ
Κ
[k]=
h
[0], followed by
h
[1]
δ
Κ
[k1]=
h
[1], and so forth.
The last nonzero output would be
h
[
N
1]
δ
Κ
[N1] ]=
h
[
N
1].
The
discretetime convolution sum
shown in Equation 2, defines how
an input timeseries
x
[
k
] is modified and manipulated by a
N
th
order FIR to produce an output
timeseries
y
[
k
].
[
]
∑

=

=
1
0
y[k]
N
m
m
m
k
x
h
2.
where
h
m
is the
m
th
tap weight coefficient. Assuming that the input
x
[
k
] is infinitely long, Equation
2 states that:
...
...
x[N]
h
..
.
+
x[2]
h
x[1]
h
y[N]
1]

x[N
h
...
+
x[1]
h
x[0]
h
1]

y[N
...
...
x[2]
h
x[1]
h
x[0]
h
y[2]
x[1]
h
x[0]
h
y[1]
x[0]
h
y[0]
0
2

N
1

N
0
2

N
1

N
0
1
2
0
1
0
=
+
+
=
+
+
=
=
+
+
=
+
=
=
3.
Suppose that the input timeseries consists of
L
nonzero samples {x[0], x[1], … , x[L1]}, which
is presented to an N
th
order FIR. Then Equation 2 yields:
1]
x[L
h
2]
L
y[
1]
x[L
h
2]
x[L
h
3]
L
y[
...
....
x[k]
h
..
.
+
x[1]
h
x[0]
h
y[k]
...
...
x[2]
h
x[1]
h
x[0]
h
y[2]
x[1]
h
x[0]
h
y[1]
x[0]
h
y[0]
1

N
2

N
1

N
0
1

k
k
0
1
2
0
1
0

=

+

+

=

+
=
+
+
=
=
+
+
=
+
=
=
N
N
4.
1
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View Full DocumentEEL 3135: Signals and Systems
Dr. Fred J. Taylor, Professor
Example
Suppose an FIR filter has an impulse response h={1, 2, 3, 2,1} and input x={1, 2, 3, 4, 5}.
Then
the filter operation can be studied as follows:
n=0
y[0]=1x[0]+2x[1]+3x[2]+2x[3]+1x[4]= 1*1+2*0+3*0+2*0+1*0=1
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