Digital Signal Processing Reference
In-Depth Information
We now develop the solution procedure. Note that
y
c
[n] = Cz
n
,
y
c
[n - 1] = Cz
n- 1
= Cz
-1
z
n
,
y
c
[n - 2] = Cz
n- 2
= Cz
-2
z
n
,
(10.51)
o
y
c
[n - N] = Cz
n-N
= Cz
-N
z
n
.
Substitution of these terms into (10.50) yields
+
Á
+ a
N- 1
z
-(N- 1)
( a
0
+ a
1
z
-1
+ a
N
z
-N
)Cz
n
+
Á
+ a
N- 1
z + a
N
)Cz
-N
z
n
= (a
0
z
N
+ a
1
z
N- 1
= 0.
(10.52)
y
c
[n] = Cz
n
We assume that our solution
is nontrivial
(C Z 0);
then, from
(10.52),
+
Á
+ a
N- 1
z + a
N
= 0.
a
0
z
N
+ a
1
z
N- 1
(10.53)
This equation is called the
characteristic equation
, or the
auxiliary equation
, for the
difference equation (10.48). Note that the characteristic equation is a polynomial in
z
set to zero. The polynomial may be factored as
+
Á
+ a
N- 1
z + a
N
= a
0
(z - z
1
)(z - z
2
)
Á
(z - z
N
) = 0.
a
0
z
N
+ a
1
z
N- 1
(10.54)
Hence,
N
values of
z
, denoted as satisfy this equation. For the
case of no repeated roots, the solution of the homogeneous equation (10.50) may be
expressed as
z
i
, i = 1, 2, Á , N,
+
Á
+ C
N
z
n
,
y
c
[n] = C
1
z
n
+ C
2
z
n
(10.55)
since the equation is linear. This solution is called the
natural response
(
comple-
mentary function
) of the difference equation (10.48) and contains the
N
unknown
coefficients These coefficients are evaluated in a later step of the
solution procedure. An example is now given.
C
1
, C
2
, Á , C
N
.
Complementary response for an LTI discrete system
EXAMPLE 10.8
As an example, we consider the first-order difference equation given earlier in the section:
y[n] - 0.6y[n - 1] = x[n].
From (10.53), the characteristic equation is
a
0
z + a
1
= z - 0.6 = 0 Q z = 0.6
;
thus, the natural response is
y
c
[n] = C(0.6)
n
,
where
C
is yet to be determined.
■
