Digital Signal Processing Reference
In-Depth Information
x
()
=
[1 1 1 1]
h
()
=
[1 2 3]
Solution
The linear convolution is given by:
∞
∑
yn
()
=
xkhn k
()(
−
)
k
=−∞
Sliding tape method:
This method can be done by hand calculation, if the
number of points in both the sequences is quite small.
The procedure is as
1
follows:
•
Write the sequences
x
(
m
),
h
(
m
)
,
and
h
(
-m
) as shown below. The
sequence
h
(
-m
) is obtained by mirroring the sequence
h
(
m
) about the
m
= 0 axis. Then the dot product of the vectors
x
(
m
) and
h
(
-m
) gives
the convolution output
y
(
0
). Similarly, the next term in the table
below,
h
(1
-m
), is obtained by shifting
h
(
-m
) by
one
step to the right.
The dot product of the vectors
x
(
m
) and
h
(1
-m
) gives the convolution
output
y
(1). The process is continued until the output
y
(
n
) remains
at zero.
x
(
m
)
=
[000111 ]
h
(
m
)
=
[000123 ]
h
(0
-m
)
=
[032100 ]
; y
(0)
=
1
h
(1
-m
)
=
[003210 ];
y
(1)
=
3
h
(2-
m
)
=
[000321 ];
y
(2)
=
6
h
(3
-m
)
=
[000032 ];
y
(3)
=
6
h
(4
-m
)
=
[000003 ];
y
(4)
=
5
h
(5
-m
)
=
[000000 ];
y
(5)
=
3
Any more shift in the sequence
h
(
m
) will result in a zero output.
Hence, the output vector is:
y
(
n
)
=
[1 3 6 6 5 3].
Note that the length of the output vector y(n) = [length of
x
(
n
)]+
[length of
h
(
n
)] -1:
Length of
y
(
n
) = 4 +3 -1 = 6.
This is a general law of discrete-time linear convolution.
