Digital Signal Processing Reference
In-Depth Information
y
ss
(t) = C
0y
+
q
k= 1
2 ƒC
ky
ƒ cos(kv
0
t + u
ky
),
(4.42)
where, from (4.40) and (4.41),
C
ky
= ƒC
ky
ƒ ∠
u
ky
= H(jkv
0
)C
kx
.
(4.43)
This equation is identical to the one given in (4.39), since both equations yield
An example is now presented to illustrate these relationships.
C
ky
.
LTI system response for a square-wave input
EXAMPLE 4.7
Suppose that for the LTI system of Figure 4.16, the impulse response and the transfer func-
tion are given by
1
s + 1
.
h(t) = e
-t
u(t) 3 H(s) =
For example, the interested reader can show that the circuit of Figure 4.17 has this transfer
function, with Suppose that the input signal
x
(
t
) is the square
wave of Figure 4.18, where
t
is in seconds. Since the fundamental period is
x(t) = v
i
(t)
and
y(t) = v
o
(t).
T
0
= 2p,
the fun-
damental frequency is
v
0
= 2p/T
0
= 1 rad/s and kv
0
= k.
From Table 4.3, the Fourier series
of
x
(
t
) is given by
q
k=-
q
kZ 0
q
k=-
q
k odd
4
pk
e
-jp/2
e
jkt
.
C
kx
e
jkv
0
t
= 2 +
x(t) = C
0x
+
1 H
v
i
(
t
)
v
o
(
t
)
1
Figure 4.17
RL
circuit.
x
(
t
)
4
Figure 4.18
Input signal for Example 4.7.
0
2
3
t
