Biomedical Engineering Reference
In-Depth Information
y
Lin
ð
t
Þ¼
k
1
y
1
ð
t
Þþ
k
2
y
2
ð
t
Þ¼
k
1
A
x
1
ð
t
Þþ
B
ð
Þ þ
k
2
A
x
2
ð
t
Þþ
B
ð
Þ
¼
Ak
1
x
1
ð
t
Þþ
k
2
x
2
ð
t
Þ
ð
Þ þ
k
1
þ
k
2
ð
Þ
B
The true system output, however, is determined as
y
ð
t
Þ¼
fx
ð
t
fg¼
fk
1
x
1
ð
t
Þþ
k
2
x
2
ð
t
f g ¼
A
ð
k
1
x
1
ð
t
Þþ
k
2
x
2
ð
t
ÞÞ þ
B
We need to compare our expected linear system output,
y
lin
(
t
), with the true system output,
y
(
t
). Note that
y
ð
t
Þ 6¼
y
Lin
ð
t
Þ
, and therefore the system is not linear.
The superposition principle takes special meaning when applied to periodic signals.
Because periodic signals are expressed as a sum of cosine or complex exponential functions
with the Fourier series, their output must also be expressed as a sum of cosine or exponen-
tial functions. Thus, if a linear system is stimulated with a periodic signal, its output is also
a periodic signal with identical harmonic frequencies. The output,
y
(
t
), of a linear system to
a periodic input,
x
(
t
), is related by
Input
Output
þ1
þ1
x
ð
t
Þ¼
A
0
f
m
Þ)
x
ð
t
Þ¼
B
0
ð
11
:
28
Þ
2
þ
A
m
cos
ð
m
o
0
t
þ
2
þ
B
m
cos
ð
m
o
0
t
þ
y
m
Þ
m
¼1
m
¼1
meaning the input and output contain cosines with identical frequencies,
o
0
, and are
expressed by equations with similar form. A similar form of this expression is also obtained
for the exponential Fourier series:
m
Input
Output
þ1
þ1
ð
11
:
29
Þ
c
m
e
jm
o
0
t
b
m
e
jm
o
0
t
x
ð
t
Þ¼
)
y
ð
t
Þ¼
m
¼1
m
¼1
where the input and output coefficients,
c
m
and
b
m
, are explicitly related to
A
m
and
B
m
via
Eq. (11.5b).
FromEqs. (11.28) and (11.29) the input and output of a linear system to a periodic input dif-
fer in two distinct ways. First the amplitudes of each cosine are selectively scaled by different
constants,
for the output. These constants are uniquely determined by
the linear system properties. Similarly, the phases angle of the input components, f
m
A
m
for the input and
B
m
, are dif-
ferent from the output components, y
m
, meaning that the input and output components are
shifted in time in relationship to each other. As for the amplitudes, the phase difference
between the input and output is a function of the linear system. Thus, if we know the mathe-
matical relationship of how the input components are amplitude scaled and phase shifted
between the input and output, we can fully describe the linear system. This relationship is
described by the system
H
m
. The transfer function fully describes how the
linear systemmanipulates the amplitude and phases of the input to produce a specific output.
This transformation is described by two separate components: the
transfer function
,
magnitude
and the
phase
.
The magnitude of
H
m
is given by the ratio of the output to the input for the
m-th
component
j ¼
B
m
A
m
j
H
m
ð
11
:
30
Þ
