Digital Signal Processing Reference
In-Depth Information
Find the state transition matrix by two different procedures.
(b)
Draw a simulation diagram for the system, and describe how the system can be
realized physically.
13.15.
Consider the system described by the state equations
00
10
1
1
B
R
B
R
x
[n + 1] =
x
[n] +
u[n];
y[n] = [0
1]
x
[n].
(a)
Find the state transition matrix.
(b)
Verify the results of part (a), using a different procedure.
(c)
Find the initial-condition response for
(d)
Verify the calculation of the state vector
x
[
n
] in part (c), by substitution in the
equation
(e)
Calculate the system unit step response, with using iteration.
(f)
Calculate the system unit step response, with using (13.32).
(g)
Verify the results of parts (e) and (f), using the system transfer function and the
z
-transform.
(h)
Verify the results in part (g), using MATLAB.
13.16.
Consider the system described by the state equations
2]
T
.
x
(0) = [1
x
[n + 1] =
Ax
[n].
x
(0) =
0
,
x
(0) =
0
,
x[n + 1] = 0.95x[n] + u[n];
y[n] = 3x[n].
(a)
Find the state transition matrix.
(b)
Find the initial-condition response for
(c)
Verify the calculation of the state
x
[
n
] in part (b), by substitution in the equation
x(0) = 1.
x[n + 1] = Ax[n].
(d)
Calculate the system unit step response, with using (13.32).
(e)
Verify the results of part (d), using the system-transfer function and the
z
-transform.
(f)
Verify the results in part (e), using MATLAB.
13.17.
Consider the system of Problem 13.15, given by
x(0) = 0,
00
10
1
1
B
R
B
R
x
[n + 1] =
x
[n] +
u[n];
y[n] = [0
1]
x
[n].
(a)
Find the transfer function for this system.
(b)
Use a similarity transformation to find a different state model for this system.
(c)
Use MATLAB to verify the results of parts (a) and (b).
(d)
Calculate the transfer function of part (b). This function should equal that of part (a).
(e)
Verify the results in part (d), using MATLAB.
(f)
You have just verified Property 4, (13.64), of similarity transformations. Verify the
other three properties in (13.61), (13.62), and (13.63).
