Digital Signal Processing Reference
In-Depth Information
3.7
First in this section, we consider further the linearity property for systems. Then
the response of continuous-time LTI systems to a certain class of input signals is
derived.
Consider the continuous-time LTI system depicted in Figure 3.20. This system is
denoted by
x(t) : y(t).
(3.60)
For an LTI system, (3.60) can be expressed as the convolution integral
q
y(t) =
L
x(t - t)h(t)dt.
(3.61)
-
q
The functions and are all real for models of physical systems.
Suppose now that we consider two real inputs
H(s),
y(t),
h(t)
x
i
(t),
i = 1, 2.
Then, in (3.60),
x
i
(t) : y
i
(t),
i = 1, 2.
(3.62)
Thus, are real, from (3.61). Because the system of (3.60) is linear, the
principle of superposition applies, and it follows that
y
i
(t), i = 1, 2,
a
1
x
1
(t) + a
2
x
2
(t) : a
1
y
1
(t) + a
2
y
2
(t).
(3.63)
No restrictions exist on the constants and in (3.63); hence, these constants
may be chosen to be complex. For this development, we choose the constants to be
a
1
a
2
a
1
= 1,
a
2
= j =
2
-1.
With this choice, the superposition property of (3.63) becomes
x
1
(t) + jx
2
(t) : y
1
(t) + jy
2
(t).
(3.64)
This result may be stated as follows: For a linear system model with a complex input
signal, the real part of the input produces the real part of the output, and the imagi-
nary part of the input produces the imaginary part of the output.
x
(
t
)
y
(
t
)
h
(
t
)
Figure 3.20
LTI system.
