Digital Signal Processing Reference
In-Depth Information
Appendix B
Convolution and Correlation - Some Case
Studies
B.1 Convolution
Generally, we define a system (like a digital communication system) as a function
or mapping which maps the input sequence (sampled signal) to output sequence.
The mapping rule may be defined clearly or not. Say, the output or response
y
(
n
)of
asystem
0.5
x
2
(
n
). If that
is the case, then we can predict the output for any query input, accurately. But in
practice, the situation is not so easy. The mapping
τ
with input or excitation
x
(
n
) is clearly defined as
y
(
n
)
=
is unknown in most of the cases
and we have to consider the system as a black box as shown in the Fig.
B.1
.
τ
Fig. B.1
Digital system with
unknown transfer function
System as
a
Black-box
with transfer
function
x
(
n
)
δ
(
n
)
y
(
n
)
h
(
n
)
τ
Now, if one asks for the predicted output for any input, there must also be some
procedure taking the help of the black-box. With this objective let us try to define
any given sequence
x
(
n
)asEq.(
B.1
).
x
(
n
)
=
x
(
−∞
)
δ
(
n
+∞
)
+
...............
+
x
(
−
1)
δ
(
n
+
1)
+
x
(0)
δ
(
n
)
+
x
(1)
δ
(
n
−
1)
+
...
x
(
∞
)
δ
(
n
−∞
)
(B.1)
∞
⇒
x
(
n
)
=
x
(
k
)
δ
(
n
−
k
)
k
=−∞
This is true for any sequence of infinite length, for finite length sequence the limits
of summation of Eq. (
B.1
) will be finite. We know that
δ
(
n
) is the unit impulse
defined as
0;
n
=
0
δ
=
(
n
)
(B.2)
1;
n
=
0
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