Digital Signal Processing Reference
In-Depth Information
As a check of this solution, we solve the difference iteratively for the first three unknown
values.
y[0] = 0;
y[1] = 0.6y[0] + 4 = 0.6(0) + 4 = 4;
y[2] = 0.6y[1] + 4 = 0.6(4) + 4 = 6.4;
y[3] = 0.6y[2] + 4 = 0.6(6.4) + 4 = 7.84.
We now find the first three values from our solution:
y[0] = 10[1 - 0.6
0
] = 0;
y[1] = 10[1 - 0.6
1
] = 4;
y[2] = 10[1 - 0.6
2
] = 6.4;
y[3] = 10[1 - 0.6
3
] = 7.84.
The solution checks for these values. The difference equation is evaluated by the following
MATLAB program:
n=[0:9];
y(1)=0;
for m=2:10;
y(m)=0.6*y(m-1)+4;
end
y
stem(n,y,'fill'),grid
What is the relationship of
m
in this program to
n
in
y
[
n
]?
The steady-state output
y
ss
[n]
can also be verified. From the solution,
y[n] = 10[1 - (0.6)
n
]u[n] Q lim
n:
q
y[n] = y
ss
[n] = 10.
For the difference equation, as
n : q, y
ss
[n - 1] : y
ss
[n],
and the difference equation is
given by
y
ss
[n] - 0.6y
ss
[n] = 0.4y
ss
[n] = 4 Q y
ss
[n] = 10.
Hence, the steady-state value also checks.
■
The natural-response part of the general solution of a linear difference equa-
tion with constant coefficients,
N
M
[eq(10.48)]
a
k
y[n - k] =
a
b
k
x[n - k],
a
k= 0
k= 0
is independent of the forcing function
x
[
n
] and is dependent only on the structure
of the system [the left side of (10.48)]; hence, the name
natural response
. It is
also called the
unforced response
, or the
zero-input response
. In the preceding
example,
y
c
[n] = C(0.6)
n
.
