Digital Signal Processing Reference
In-Depth Information
where
x(t)
is the input signal,
y(t)
is the output signal, and the constants
a
k
, b
k
,
and
n
are parameters of the system.
We now derive the transfer-function model for (7.47). From (7.29), recall the
differentiation property:
d
k
f(t)
dt
k
= s
k
F(s) - s
k- 1
f(0
+
) -
Á
- f
(k- 1)
(0
+
).
B
R
l
Initial conditions must be ignored when we derive transfer functions, because a sys-
tem with non-zero initial conditions is not linear. The transfer function shows the
relationship between the input signal and the output signal for a linear system. The
differentiation property is then
d
k
f(t)
dt
k
= s
k
F(s).
l
B
R
We use this property to take the transform of (7.47):
n
n
a
k
s
k
Y(s) =
a
b
k
s
k
X(s).
a
k= 0
k= 0
Expanding this equation gives
+
Á
+ a
1
s + a
0
]Y(s)
[a
n
s
n
+ a
n- 1
s
n- 1
+
Á
+ b
1
s + b
0
]X(s).
= [b
n
s
n
+ b
n- 1
s
n- 1
(7.48)
The system transfer function
H(s)
is defined as the ratio
Y(s)
/
X(s),
from
(7.48). Therefore, the transfer function for the model of (7.47) is given by
+
Á
+ b
1
s + b
0
b
n
s
n
+ b
n- 1
s
n- 1
Y(s)
X(s)
=
H(s) =
.
(7.49)
+
Á
+ a
1
s + a
0
a
n
s
n
+ a
n- 1
s
n- 1
For this case, the transfer function is a
rational function
(a ratio of polynomials).
Note that this transfer function is identical to that derived in Chapter 3; however,
the derivation in Chapter 3 applies only for a complex-exponential input signal. The
transfer function (7.49) applies for any input that has a Laplace transform and,
hence, is a generalization of that of Chapter 3. An example is now given.
LTI system response using Laplace transforms
EXAMPLE 7.12
Consider again the RL circuit of Figure 7.8, and let
R = 4Æ
and
L = 0.5H.
The loop equa-
tion for this circuit is given by
0.5
di(t)
dt
+ 4i(t) = v(t).
