Digital Signal Processing Reference
In-Depth Information
the impulse response of the inverse of system (ii) is given by
h
i
[
k
]
=
∞
(
−
1)
m
δ
[
k
−
m
]
= δ
[
k
]
− δ
[
k
−
1]
+ δ
[
k
−
2]
− δ
[
k
−
3]
m
=
0
We can show indirectly that
h
i
[
k
] is indeed the impulse response of the inverse
of system (ii) by proving that
h
[
k
]
∗
h
i
[
k
]
= δ
[
k
]:
h
[
k
]
∗
h
i
[
k
]
=
(
δ
[
k
]
+ δ
[
k
−
1])
∗
h
i
[
k
]
=
h
i
[
k
]
+
h
i
[
k
−
1]
=
(
δ
[
k
]
− δ
[
k
−
1]
+ δ
[
k
−
2]
− δ
[
k
−
3]
)
+
(
δ
[
k
−
1]
− δ
[
k
−
2]
+ δ
[
k
−
3]
− δ
[
k
−
4]
)
= δ
[
k
]
.
10.9 Experiments w
ith M
ATLAB
M
ATLAB
provides several functions (also referred to as M-files) for processing
DT signals and LTID systems. In this section, we will focus on the M
ATLAB
implementations of the difference equations with known ancillary conditions,
convolution of two DT signals, and deconvolution.
10.9.1 Difference equations
Consider the following linear, constant-coefficient difference equation:
y
[
k
+
n
]
+
a
n
−
1
y
[
k
+
n
−
1]
++
a
0
y
[
k
]
=
b
m
x
[
k
+
m
]
+
b
m
−
1
x
[
k
+
m
−
1]
++
b
0
x
[
k
]
,
(10.31)
which models the relationship between the input sequence
x
[
k
] and the output
response
y
[
k
] of an LTID system. The ancillary conditions
y
[
−
1],
y
[
−
2],...,
y
[
−
n
] are also specified.
To solve the difference equation, M
ATLAB
provides a built-in function
filter
with the syntax
>> [y] = filter(B,A,X,Zi);
In terms of the difference equation, Eq. (10.31), the input variables
B
and
A
are
defined as follows:
A
=
[1
,
a
n
−
1
,...,
a
0
] and
B
=
[
b
m
,
b
m
−
1
,...,
b
0
]
,
while
X
is the vector containing the values of the input sequence and
Zi
denotes
the initial conditions of the delays used to implement the difference equation.
The initial conditions used by the
filter
function are not the past values of
the output
y
[
k
] but a modified version of these values. The initial conditions
used by M
ATLAB
can be obtained by using another built-in function,
filtic
.
The calling syntax for the
filtic
function is as follows:
>> [Zi] = filtic(B,A,yinitial);
Search WWH ::
Custom Search