Digital Signal Processing Reference
In-Depth Information
(ii) Assuming that the impulse response of the inverse system is
h
i
(
t
), the
stability condition is expressed as
h
(
t
)
∗
h
i
(
t
)
=
[
δ
(
t
)
− δ
(
t
−
2)]
∗
h
i
(
t
)
= δ
(
t
)
.
By applying the convolution property, Eq. (3.41), the above expression simpli-
fies to
h
i
(
t
)
−
h
i
(
t
−
2)
= δ
(
t
)
or
h
i
(
t
)
= δ
(
t
)
+
h
i
(
t
−
2)
.
The above expression can be solved iteratively. For example,
h
i
(
t
−
2) is given
by
h
i
(
t
−
2)
= δ
(
t
−
2)
+
h
i
(
t
−
4)
.
Substituting the value of
h
i
(
t
−
2) in the earlier expression gives
h
i
(
t
)
= δ
(
t
)
+ δ
(
t
−
2)
+
h
i
(
t
−
4)
,
leading to the iterative expression
∞
h
i
(
t
)
=
δ
(
t
−
2
m
)
.
m
=
0
To verify that
h
i
(
t
) is indeed the impulse response of the inverse system, we
convolve
h
(
t
) with
h
i
(
t
). The resulting expression is as follows:
∞
h
(
t
)
∗
h
i
(
t
)
=
[
δ
(
t
)
− δ
(
t
−
2)]
∗
δ
(
t
−
2
m
)
,
m
=
0
which simplifies to
∞
∞
h
(
t
)
∗
h
i
(
t
)
= δ
(
t
)
∗
δ
(
t
−
2
m
)
+ δ
(
t
−
2)
∗
δ
(
t
−
2
m
)
m
=
0
m
=
0
or
∞
∞
h
(
t
)
∗
h
i
(
t
)
=
δ
(
t
−
2
m
)
+
δ
(
t
−
2
−
2
m
)
= δ
(
t
)
.
m
=
0
m
=
0
Therefore,
h
i
(
t
) is indeed the impulse response of the inverse system.
3.8 Experiments w
ith M
ATLAB
In this chapter, we have so far presented two approaches to calculate the output
response of an LTIC system: the differential equation method and the convolu-
tion method. Both methods can be implemented using M
ATLAB
. However, the
convolution method is more convenient for M
ATLAB
implementation in the
discrete-time domain and this will be presented in Chapter 8. In this section,
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