Biomedical Engineering Reference
In-Depth Information
Note that if we know the input magnitudes, we can determine the output Fourier coeffi-
cients by multiplying the transfer function magnitude by the input Fourier coefficients:
B
m
¼
H
j j
A
m
. The phase angle of the transfer function describes the phase relationship
between the input and output for the
m-th
frequency component
∠
H
m
¼
y
m
f
m
:
ð
11
:
31
Þ
.
Equations (11.30) and (11.31) are the two critical pieces of information that are necessary
to fully describe a linear system. If these two properties of the transformation are known,
it is possible to determine the output for any arbitrary input.
If we know the input phase, the output phase is determined as y
m
¼
∠
H
m
þ
f
m
11.6.2 Time Domain Representation of Linear Systems
The relationship between the input and output of a linear system can be described by
studying its behavior in the time domain (Figure 11.15b). The
impulse response
function,
h
), is a mathematical description of the linear system that fully characterizes its behavior.
As we will see subsequently, the impulse response of a linear system is directly related to
the system transfer function as outlined for the periodic signal. If one knows
(
t
h
(
t
), one can
readily compute the output,
y
(
t
), to any arbitrary input,
x
(
t
), using the
convolution integral
1
y
ð
t
Þ¼
h
ð
t
Þ
*
x
ð
t
Þ¼
h
ð
t
Þ
x
ð
t
t
Þ
d
t
:
ð
11
:
32
Þ
1
The symbol * is shorthand for the convolution between the input and the system impulse
response. Integration is performed with respect to the dummy integration variable t. For
the discrete case, the output of a discrete linear system is determined with the convolution
sum
1
y
ð
k
Þ¼
h
ð
k
Þ
x
ð
k
Þ¼
h
ð
m
Þ
x
ð
k
m
Þ
ð
11
:
33
Þ
*
m
¼1
where
) is the impulse response of the discrete system. A detailed treatment of the
convolution integral is found in many signal processing textbooks and is beyond the scope
of this text. As shown in a subsequent section, a simpler treatment of the input-output
relationship of a linear system is obtained by analyzing it in the “frequency-domain.”
h
(
m
EXAMPLE PROBLEM 11.18
A cytoplasmic current injection
i
ð
t
Þ¼
u
ð
t
Þ
to a cell membrane produces an intracellular
change in the membrane voltage,
v
(
t
). The membrane of a cell is modeled as a linear system with
h
ð
t
Þ¼
A
e
t
=t
impulse response
is a constant in units V/s/A and t is the cell
membrane time constant (units: seconds). Find the cell membrane voltage output in closed form.
Simulate in MATLAB, for A
u
ð
t
Þ
,where
A
¼
100 and
t ¼
0.01 sec. Compare the closed form solution to the
simulated results.
Continued
