Digital Signal Processing Reference
In-Depth Information
13.2.
(
a)
Find a state model for the
a-filter
described by the equation
y[n + 1] - (1 - a)y[n] = ax[n + 1],
where
x
[
n
] is the input signal. (
Hint:
Draw a simulation diagram first.)
(b)
Verify the results of part (a) by (i) finding the transfer function from the describing
equation and (ii) finding the transfer function from the state equations.
13.3.
The simulation diagram for the filter is given in Figure P13.3. This filter is sec-
ond order and is used in radar-signal processing. The input
u
[
n
] is the unfiltered target-
position data, the output
y
[
n
] is the filtered position data, and the output is an
estimate of the target velocity. The parameter
T
is the sample period. The parameters
and
a - b
v[n]
a
b
are constants and depend on the design specifications for the filter.
(a)
Write the state equations for the filter, with the state variables equal to the outputs
of the delays and the system outputs equal to
y
[
n
] and
(b)
Let
v[n].
b = 0
in part (a). Show that the resulting equations are equivalent to those of
the
a-filter
of Problem 13.2.
y
[
n
]
u
[
n
]
x
1
[
n
]
D
1
1
v
[
n
]
T
x
2
[
n
]
/
T
D
Figure P13.3
13.4. (a)
Draw a simulation diagram for the system described by the difference equation
2y[n + 1] - 1.8y[n] = 3u[n + 1].
(b)
Write the state equations for the simulation diagram of part (a).
(c)
Give the system transfer equation for the simulation diagram of part (a).
(d)
Use the results in part (b), and MATLAB, to verify part (c).
(e)
Repeat parts (a) through (d) for the difference equation
y[n + 2] - 1.5y[n + 1] + 0.9y[n] = 2u[n].
(f)
Repeat parts (a) through (d) for the difference equation
y[n + 3] - 2.9y[n + 2] + 3.4y[n + 1] - 0.72y[n] = 2u[n].
