Databases Reference
In-Depth Information
Scaling:
Given a linear system
L
with input
f
(
t
)
and output
g
(
t
)
, if we multiply the
input with a scalar
α
, then the output will be multiplied by the same scalar:
L
[
α
f
(
t
)
]=
α
L
[
f
(
t
)
]=
α
g
(
t
)
The two properties together are referred to as
superposition
.
12.6.1 Time Invariance
Of specific interest to us are linear systems that are
time invariant
. A time-invariant system
has the property that the shape of the response of the system does not depend on the time at
which the input was applied. If the response of a linear system
L
to an input
f
(
t
)
is
g
(
t
)
,
L
[
f
(
t
)
]=
g
(
t
)
and we delay the input by some interval
t
0
, then if
L
is a time-invariant system, the output will
be
g
(
t
)
delayed by the same amount:
L
[
f
(
t
−
t
0
)
]=
g
(
t
−
t
0
)
(25)
12.6.2 Transfer Function
Linear time-invariant systems have a very interesting (and useful) response when the input is a
sinusoid. If the input to a linear system is a sinusoid of a certain frequency
ω
0
, then the output
is also a sinusoid of the same frequency that has been scaled and delayed; that is,
L
[
cos
(ω
0
t
)
]=
α
cos
(ω
0
(
t
−
t
d
))
or in terms of the complex exponential
e
j
ω
0
t
e
j
ω
0
(
t
−
t
d
)
L
[
]=
α
Thus, given a linear system, we can characterize its response to sinusoids of a particular
frequency by a pair of parameters, the gain
and the delay
t
d
. In general, we use the phase
φ
=
ω
0
t
d
in place of the delay. The parameters
α
will generally be a function of
the frequency, so in order to characterize the system for all frequencies, we will need a pair
of functions
α
and
φ
. As the Fourier transform allows us to express the signal as
coefficients of sinusoids, given an input
α(ω)
and
φ(ω)
f
(
t
)
, all we need to do is, for each frequency
ω
,
e
j
φ(ω)
, where
multiply the Fourier transform of
f
(
t
)
with some
α(ω)
α(ω)
and
φ(ω)
are the
gain and phase terms of the linear system for that particular frequency.
This pair of functions
α(ω)
and
φ(ω)
constitute the
transfer function
of the linear time-
invariant system
H
(ω)
:
e
j
φ(ω)
H
(ω)
= |
H
(ω)
|
where
|
H
(ω)
| =
α(ω)
.
