Digital Signal Processing Reference
In-Depth Information
x
(
t
)
y
(
t
)
x
(
t
)
y
(
t
)
h
(
t
)
H
(
s
)
h
(
t
)
e
st
dt
H
(
s
)
Figure 3.23
LTI system.
TABLE 3.1
Input-Output Functions for an LTI System
q
h(t)e
-st
dt
H(s) =
L
-
q
Xe
s
1
t
:XH(s
1
)e
s
1
t
;
X = ƒXƒ e
jf
ƒXƒ cos (v
1
t + f) : ƒXƒ ƒH(jv
1
) ƒ cos [v
1
t + f +
l
H(jv
1
)]
q
h(t)e
-st
dt.
H(s) =
L
(3.77)
-
q
This equation is the desired result. Table 3.1 summarizes the results developed in
this section.
We can express these developments in system notation:
e
st
:H(s)e
st
.
(3.78)
We see that a complex exponential input signal produces a complex exponential
output signal.
It is more common in practice to describe an LTI system by the transfer func-
tion rather than by the impulse response However, we can represent LTI
systems with either of the block diagrams given in Figure 3.23, with
H(s)
h(t).
H(s)
and
h(t)
related by (3.77).
Those readers familiar with the bilateral Laplace transform will recognize
in (3.77) as the Laplace transform of Furthermore, with in
(3.77) is the Fourier transform of We see then that both the Laplace transform
(covered in Chapter 7) and the Fourier transform (covered in Chapter 5) appear
naturally in the study of LTI systems.
We considered the response of LTI systems to complex-exponential inputs in
this section, which led us to the concept of transfer functions. Using the transfer
function approach, we can easily find the system response to inputs that are con-
stant, real exponential, and sinusoidal. As a final point, the relationship between the
transfer function of a system and its impulse response was derived.
H(s)
h(t).
s = jv, H(jv)
h(t).
3.8
BLOCK DIAGRAMS
Figure 3.23 gives two block-diagram representations of a system. In this section, we
consider a third block-diagram representation. The purpose of this block diagram is
to give an
internal structure
to systems, in addition to the usual input-output description
