Biomedical Engineering Reference
In-Depth Information
√
2
ω
c
for a Butterworth filter
or, 2
ω
c
for a critically damped filter
K
=
K
2
(
1
+
K
1
+
K
2
)
,
ω
c
,
K
2
=
a
0
=
a
1
=
2
a
0
,
a
2
=
a
0
2
a
0
K
2
K
3
=
,
b
1
=−
2
a
0
+
K
3
b
2
=
1
−
2
a
0
−
K
3
,
r
b
2
=
1
−
a
0
−
a
1
−
a
2
−
b
1
As well as attenuating the signal, there is a phase shift of the output signal
relative to the input. For this second-order filter there is a 90
◦
phase lag at
the cutoff frequency. This will cause a second form distortion, called
phase
distortion
, to the higher harmonics within the bandpass region. Even more
phase distortion will occur to those harmonics above
f
c
, but these components
are mainly noise, and they are being severely attenuated. This phase distortion
may be more serious than the amplitude distortion that occurs to the signal
in the transition region. To cancel out this phase lag, the once-filtered data
was filtered again, but this time in the reverse direction of time (Winter
et al., 1974). This introduces an equal and opposite phase lead so that the
net phase shift is zero. Also, the cutoff of the filter will be twice as sharp
as that for single filtering. In effect, by this second filtering in the reverse
direction, we have created a fourth-order zero-phase-shift filter, which yields
a filtered signal that is back in phase with the raw data but with most of the
noise removed.
In Figure 3.18 we see the frequency response of a second-order
Butter-worth filter normalized with respect to the cutoff frequency. Super-
imposed on this curve is the response of the fourth-order zero-phase-shift
filter. Thus, the new cutoff frequency is lower than that of the original
single-pass filter; in this case, it is about 80% of the original. The correction
factor for each additional pass of a Butterworth filter is
C
=
(
2
1
/n
1
)
0
.
25
,
−
where
n
is
the
number
of
passes.
Thus,
for
a
dual
pass,
C
=
0
.
802.
=
(
2
1
/
2
n
1
)
0
.
5
; thus, for a dual pass,
For a critically damped filter,
C
−
C
0
.
435. This correction factor is applied to Equation (3.8) and results
in the cutoff frequency for the original single-pass filter being set higher,
so that after the second pass the desired cutoff frequency is achieved. The
major difference between these two filters is a compromise in the response
in the time domain. Butterworth filters have a slight overshoot in response
to step- or impulse-type inputs, but they have a much shorter rise time.
Critically damped filters have no overshoot but suffer from a slower rise
time. Because impulsive-type inputs are rarely seen in human movement
data, the Butterworth filter is preferred.
The application of one of these filters in smoothing raw coordinate data
can now be seen by examining the data that yielded the harmonic plot in
Figure 3.16. The horizontal acceleration of this toe marker, as calculated by
=
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