Biomedical Engineering Reference
In-Depth Information
In the time domain, the ideal high-pass filter impulse response is obtained as
ð
t
Þ
W
c
p
h
HP
ð
t
Þ¼
d
ð
t
Þ
h
LP
ð
t
Þ¼
d
sinc
ð
W
c
t
Þ
ð
11
:
40
Þ
A schematic depiction of
the ideal high-pass filter transfer function is shown in
Figure 11.17b.
The final class of filter we will consider is the band-pass filter. The prototypical band-pass
filter is somewhat more complex than the low-pass and high-pass filters, since it requires the
definition of a lower and upper cutoff frequency,
W
2
. Figure 11.17c illustrates the
magnitude response of the ideal band-pass filter transfer function and impulse response.
Only signals between the two cutoff frequencies are allowed to pass through to the output.
All other signals are rejected. The transfer function of the ideal band-pass filter is given by
W
1
and
0
1
W
<
jj<
W
o
2
,
1
ð
11
:
41
Þ
H
HP
ð
o
Þ¼
otherwise
In the frequency domain, the ideal band-pass filter can be obtained by combining a high-
pass filter with cutoff
2
. The band-pass filter transfer
function can therefore be expressed as the product of transfer functions for a low-pass and
high-pass filter:
W
1
and a low-pass filter with cutoff
W
H
BP
ð
o
Þ¼
H
HP
ð
o
Þ
H
LP
ð
o
Þ:
ð
11
:
42
Þ
In the time domain, the band-pass filter impulse response is obtained by the inverse Fourier
transform of the filter transfer function
h
BP
ð
t
Þ¼
h
HP
ð
t
Þ
*
h
LP
ð
t
Þ:
ð
11
:
43
Þ
This is done by applying the convolution theorem (Eqs. (11.14) and (11.15))toEq. (11.42).
EXAMPLE PROBLEM 11.20
Consider the cell membrane cytoplasmic current injection for Example Problem 11.18. Find the
cell's output voltage in the Fourier domain. Also, compute and plot in MATLAB the transfer func-
tion magnitude of the cell membrane. What type of filter is this?
Solution
The Fourier transform of the step current input is
ð
1
1
0
¼
e
o
t
dt
¼
e
o
t
j
1
i
ð
t
Þ
e
j
o
t
dt
¼
I
ðoÞ¼
1
o
j
o
0
The transfer function is determined as the Fourier transform of the impulse response
ð
1
0
A
e
t
=t
A
j
o þ
A
e
t
=t
e
j
o
t
dt
¼
H
ð
o
Þ¼
FT
h
ð
t
fg¼
u
ð
t
Þ
dt
¼
1
=t
Continued
