Biomedical Engineering Reference
In-Depth Information
exact equations to calculate the coefficients for a Butterworth or a critically
damped filter are as follows:
(
tan(
π f
c
/f
s
))
C
ω
c
=
(2.19)
where
C
is the correction factor for number of passes required, to be explained
shortly. For a single pass filter
C
=
1.
=
√
2
ω
c
for a Butterworth filter,
or, 2
ω
c
for a critically damped filter
K
K
2
K
2
=
ω
c
,
a
0
=
K
2
)
,
a
1
=
2
a
0
,
a
2
=
a
0
(
1
+
K
1
+
2
a
0
K
2
K
3
=
,
b
1
=−
2
a
0
+
K
3
b
2
=
1
−
2
a
0
−
K
3
,
or
b
2
=
1
−
a
0
−
a
1
−
a
2
−
b
1
For example, a Butterworth-type low-pass filter of second order is to be
designed to cutoff at 6 Hz using film data taken at 60 Hz (60 frames per
second). As seen in Equation (2.19) the only thing that is required to deter-
mine these coefficients is the ratio of sampling frequency to cutoff frequency.
In this case it is 10. The design of such a filter would yield the following
coefficients:
a
0
=
0
.
067455,
a
1
=
0
.
13491,
a
2
=
0
.
067455,
b
1
=
1
.
14298,
b
2
=−
0
.
41280
Note that the algebraic sum of all the coefficients equals 1.0000. This gives
a response of unity over the passband. Note that the same filter coefficients
could be used in many different applications, as long as the ratio
f
s
/f
c
is the
same. For example, an EMG signal sampled at 2000 Hz with cutoff desired
at 400 Hz would have the same coefficients as one employed for movie film
coordinates where the film rate was 30 Hz and cutoff was 6 Hz. The number
of passes, C, in Equation (2.17) are important when filtering kinematic data
in order to eliminate the phase shift of the filtered data. This aspect of digital
filtering of kinematic data will be detailed later in Section 3.4.4.2.
2.2.4.5 Fourier Reconstitution of Original Signal.
Figure 2.18 is pre-
sented to illustrate a Fourier reconstitution of the vertical trajectory of the
heel of an adult walking his or her natural cadence. A total of nine harmonics
is represented here because the addition of higher harmonics did not improve
the curve of the original data. As can be seen, the harmonic reconstitution
is visibly different from the original, sufficiently so as to cause reasonable
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