Biomedical Engineering Reference
In-Depth Information
Input
Output
x
ð
t
Þ!
y
ð
t
Þ
1
1
x
ð
t
Þ!
y
ð
t
Þ
,
2
2
we can easily determine the output of a linear system to any arbitrary combination of these
inputs. More generally, a linear superposition and scaling of the input signals produces a
linear superposition and scaling of the output signals
Input
Output
ð
11
:
27
Þ
x
ð
t
Þþ
k
x
ð
t
Þ!
k
y
ð
t
Þþ
k
y
ð
t
Þ
k
1
1
2
2
1
1
2
2
where
k
2
are arbitrary amplitude scaling constants. These constants scale the input
amplitudes by making them larger (
k
1
and
1). This produces a comparable
change in the net outputs, which are likewise scaled by the same constants.
k
>
1) or smaller (
k
<
EXAMPLE PROBLEM 11.16
The following information is given for a linear system
Input
Output
x
1
ð
t
Þ¼
cos
ð
t
Þ
!
y
1
ð
t
Þ¼
cos
ð
t
þ p=
2
Þ
x
2
ð
t
Þ¼
cos
ð
t
Þþ
sin
ð
2
t
Þ!
y
2
ð
t
Þ¼
cos
ð
t
þ
p
=
2
Þþ
5 sin
ð
2
t
Þ
x
3
ð
t
Þ¼
cos
ð
3
t
Þ
!
y
3
ð
t
Þ¼
2 cos
ð
3
t
Þ
Find the output if the input is:
x
ð
t
Þ¼
3 sin
ð
2
t
Þþ
1
=
2 cos
ð
3
t
Þ
Solution
The input is represented as a superposition of
x
1
,
x
2
, and
x
3
x
3
ð
t
Þ
Applying the superposition and scaling properties produces an output
x
ð
t
Þ¼
3
ð
x
2
ð
t
Þ
x
1
ð
t
Þ
Þ þ
1
=
2
x
3
ð
t
Þ¼
3
x
2
ð
t
Þ
3
x
1
ð
t
Þþ
1
=
2
y
ð
t
Þ¼
3
y
2
ð
t
Þ
3
y
1
ð
t
Þþ
1
=
2
y
3
ð
t
Þ¼
3 cos
ð
ð
t
þ p=
2
Þþ
5 sin
ð
2
t
Þ
Þ
3 cos
ð
ð
t
þ p=
2
Þ
Þ þ
1
=
2 2 cos
ð
ð
3
t
Þ
Þ
¼
15
sin
ð
2
t
Þþ
cos
ð
3
t
Þ
EXAMPLE PROBLEM 11.17
Consider the system given by the expression
y
ð
t
Þ¼
fx
ð
t
fg¼
A
x
ð
t
Þþ
B
:
Determine if this is a linear system.
Solution
To solve this problem, consider a superposition of two separate inputs,
x
1
(
t
) and
x
2
(
t
), that
independently produce outputs
y
1
(
t
) and
y
2
(
t
). Apply the input
x
ð
t
Þ¼
k
1
x
1
ð
t
Þþ
k
2
x
2
ð
t
Þ
. If the
system is linear, the output obeys
Continued