TECHNICAL NOTES
JOURNAL OF APPLIED BIOMECHANICS, 1994,10, 374-381
Q 1994 by Human Kinetics Publishers, Inc.
Construction of a High-Pass Digital Filter
From a Low-Pass Digital Filter
Stephen D. Murphy and D. Gordon E. Robertson
To remove low-frequency noise from data such as DC-bias from electromyograms (EMGs) or drift from force transducers, a high-pass filter was constructed from a low-pass filter of known characteristics. A summary of the
necessary steps required to transform the low-pass digital were developed.
Contaminated EMG and force platform data were used to test the filter. The
high-pass filter successfully removed the low-frequency noise from the EMG
signals. The high-pass filter was then cascaded with the low-pass filter to
produce a band-pass filter to enable simultaneous high- and low-frequency
noise reduction.
The smoothing of displacement signals prior to time differentiation is
essential to remove high-frequency artifacts. Higher frequencies become amplified during the differentiation process (see Winter, 1990). To remove highfrequency noise, low-pass digital filters are frequently used. A similar situation
occurs when low-frequency noise is present in a signal that needs to be integrated.
Even a relatively small bias error, upon integration, can become the dominant
component of the signal. The solution is a high-pass filter, which removes the
low-frequency interference and leaves the higher frequencies unaffected.
Digital high-pass filters are rarely used in biomechanics. These filters could
be valuable when low-frequency noise, such as amplifier drift, occurs during
electromyography (EMG), accelerometry, or dynamometry. The goal of this
research was to develop a high-pass digital filter using the same software already
developed for performing low-pass digital filtering, since most biomechanics
researchers already possess such technology.
Pezzack, Norman, and Winter (1977) showed that the process of digitally
filtering cinefilm coordinate data to remove high-frequency noise and then
applying finite differences was an appropriate solution for obtaining segmental
kinematics. More detail on the construction of the low-pass filter used by Pezzack
et al. can be found in Rader and Gold (1967) and Winter (1990).
Barr and Chan (1986) constructed low-pass, high-pass, band-pass, and
band-stop digital filters based on a transformation from an analog low-pass filter.
Barr and Chan's approach did not take advantage of the current knowledge that
exists regarding the construction of low-pass digital filters to build high-pass
digital filters. A high-pass digital filter can be constructed from a low-pass digital
Stephen D. Murphy and D. Gordon E. Robertson are with the School of Human
Kinetics, University of Ottawa, Ottawa, ON Canada K I N 6N5.
High-Pass Digital Filter
375
filter by means of a transformation. A brief presentation of this derivation, as
well as applications of the high-pass digital filter follows. For a more detailed
description of the theory behind this approach, consult the text by Oppenheim
and Schafer (1989), who developed the principles upon which this transformation
is made, but who did not provide a specific example with validation of its effects.
Theory
The general Nth order digital filter is a recursive equation of the form (Hamming,
1977)
N
N
Therefore, a second order (N = 2) digital filter is written
Yn
= aoxn + alxn-l+ W n - 2 + b,y,-t
+ b2yn-2.
Let the known low-pass digital filter have the following transfer function:
To get the corresponding high-pass digital filter let (Oppenheim & Schafer, 1989)
where
and q,= angular velocity of low-pass filter (radls), wP= angular velocity of highpass filter (radls), and T = sampling interval (seconds). Expanding and regrouping
the constants into the same form as (Equation 1) yields the high-pass filter
with the following coefficients:
Murphy and Robertson
376
Before the coefficients for the low-pass filter can be calculated, a suitable
low-pass cutoff frequency must be chosen. Since the high-pass cutoff frequency
is given and the low-pass cutoff frequency can be chosen arbitrarily based on
a, set a = 0 and solve for a&,. Choosing a = 0 simplifies the calculations of the
high-pass filter coefficients (see Equation 3) and Equation 2 implies that
q Q T+ Q,~T= a. Solving for wP gives qp= (IT
- @,T)/T. Since researchers
usually deal with frequencies rather than angular velocities, recall that
mQ= 27tfhp-Perform a similar substitution for ol, and simplify to get
To summarize, use the following steps to build a high-pass digital filter:
1. After selecting the desired cutoff frequency for the high-pass filter, use
Equation 4 to calculate the associated low-pass filter's cutoff frequency,
.hQ.
2. Next, compute the low-pass filter's coefficients (Ai and Bj) using the usual
equations. We recommend a zero-lag, Butterworth, critically damped filter
as presented by Winter (1990) (for suitable equations, see below).
3. Then by letting a = 0 in Equation 3, the new coefficients, a, and bi,for
the high-pass digital filter will be
4. Finally, filter the data using the same equation as for the low-pass filter
but using the coefficients, ai and bj,instead of Ai and Bi.
As an example, the following steps were used to construct a high-pass
filter from the critically damped, Butterworth, low-pass zero-lag filter as described
by Winter (1990). First, assume that the researcher has sampled the data at a
frequency, f,. Using Equation 4, the associated low-pass cutoff frequency is
Note that the cutoff frequency,j,, must be less than one half the sampling rate,
f,; otherwise, an aliasing error will occur due to violation of the sampling theorem.
Second, determine the cutoff frequency of the associated low-pass filter
(see Winter, 1990): o,= tan [n(f;&)]. Third, correct for the number of filter
passes of the critically damped, zero-lag filter (Robertson, Barden, & Dowling,
1992). The corrected cutoff is
where n is the number of filter passes. In the case of a zero-lag filter where the
data are passed (cascaded) through the filter twice (once in the forward direction
and once in the backwards direction) n will be 2.
Fourth, compute the low-pass filter coefficients. Let Kl = 203,* and
Then, the low-pass coefficients are as follows:
K1=
High-Pass Digital Filter
Fifth, from Equation 5 above, the high-pass coefficients are a, = A,, a, =
-Al, a2 = A,, b, = -B,, and b2 = B2. Finally, filter the data by running them
through the filter first in the forwards direction and then in the reverse direction.
This will produce a zero-lag filter with a roll-off of fourth order (i.e., -24 db/
+ blyi-I + boi-2,
octave). The equation of the filter is yi = a,gi+ alxi-~+ a 2 4 - ~
where, 4 are the raw data and yi are the filtered data.
Methods
Four applications of the high-pass digital filter demonstrate its usefulness in
removing low-frequency noise. In the first two applications, a high-pass filter
was constructed and applied to two 6-second samples of EMG from a series
of flexor digitorurn longus muscle contractions. ~ a c hsignal was collected at
approximately 2100 Hz. The first EMG had a DC-bias of 0.13 mV added while
the second sample had a drift of 0.42 mV/s added.
The third application was a vertical force history that was collected from
a piezoelectric force platform (Kistler Model 9865B) with the "time constant"
option set (Bioware Version 2.0, Kistler Corp.) to produce drift. This is similar
to using the "long" time constant on the older Kistler Model 9261A force
platforms. The subject stood on the platform and performed a series of deepknee bends.
In the fourth application, a digitally constructed signal containing 0.1, 1.O,
and 10.0 Hz sine waves, all with unity amplitude, was constructed. Low-pass
and high-pass filters were designed in the same manner as above. The multifrequency signal was both high-pass and low-pass filtered and then was filtered
with both the high-pass and low-pass filters providing a band-pass filter.
Results and Discussion
Figure 1 shows the frequency response of the high-pass filter. Notice that this
filter is very flat in the pass band. This is a property of Butterworth filters, which
are designed to be optimally flat in the pass band.
In Figure 2, a raw EMG signal with a DC-offset is displayed. Below it in
the same figure is the signal after applying a high-pass filter with a cutoff
frequency of 1.0 Hz. The filter successfully removed the bias from the signal
without adversely influencing the amplitudes of the high-frequency components.
Note that the closely dotted line in the middle of this and the following figures
are the zero voltage levels. Both curves have the same scaling factor, which is
displayed to the right beside each curve. The time-axis scaling is displayed in
the bottom right-hand side of each figure. Figure 3 shows a similar EMG that
had "drift" present. The lower curve was filtered at 1.0 Hz, which completely
removed the drift but left the EMG amplitudes unaffected. These results demon-
u o s w q o ~pue A y d ~ n ~
High-Pass Digital Filter
3 79
Figure 3 - Effects of a high-pass filter on an EMG signal with "drift."
strate the usefulness of a high-pass filter for removing unwanted low-frequency
noise from EMG signals.
In Figure 4 a vertical force history with drift due to using a piezoelectric
force platform (Kistler) with a low time constant is shown. After 12 seconds,
the apparent body weight of the subject dropped from 717 N to 633 N. After
running the data through a high-pass filter with a 0.5 Hz cutoff the body weight
line was reduced to zero. This was expected because a high-pass filter acts like
ac-coupling and so the dc-offset produced by the subject's body weight was
removed from the signal, as shown by the middle curve in Figure 4. The lower
curve shows the force signal after the body weight of 717 N was added back.
This signal very closely matches the original signal, except that the drift due to
the low time constant has been removed. Notice that there is a slight artifact
present in the first second of the filtered signal. This is due to the "warm-up"
that is needed before the filter functions correctly. These effects can be reduced
by having lead-in data to allow the filter time to settle.
Figure 5 shows the results of both high-pass and low-pass filtering a signal
containing 0.1, 1.0, and 10.0 Hz sine waves. The top curve is the raw data, the
second curve shows the data high-pass filtered with a 0.5 Hz cutoff, and the
third curve shows the effects of low-pass filtering at 5.0 Hz. The bottom curve
demonstrates the effects of cascading the data through a 0.5 Hz high-pass filter
followed by a 5.0 Hz low-pass filter. Examining the second and bottom curves
shows that the low frequency 0.1 Hz signal is no longer present. Its signal
amplitude has been reduced to 3% of its original amplitude. The higher frequency
10 Hz signal has also been reduced (to 12%) in the third and bottom curves, but
is still visible in the filtered signals. Lowering the low-pass filter's cutoff would
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High-Pass Digital Filter
381
reduce the 10 Hz signal more, but the amplitude of the 1.0 Hz signal will also
be reduced. Again, notice that these filters d o require time to reach their specified
operating characteristics. In the case of zero-lag filters, extra leading and trailing
data are required (see Smith, 1989).
References
Barr, R.E., & Chan, E.K.Y. (1986). Design and implementation of digital filters for
biomedical signal processing. Journal of Electrophysiological Techniques, 13,7393.
Hamming, R.W. (1977). Digitalfilters. Englewood Cliffs, NJ: Prentice Hall.
Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood
Cliffs, NJ: Prentice Hall.
Pezzack, J.C., Norman, R.W., & Winter, D.A. (1977). An assessment of derivative determining techniques used for motion analysis. Journal of Biomechanics, 10, 377382.
Rader, C.M., & Gold, B. (1967). Digital filtering design techniques in the frequency
domain. Proceedings of lEEE, 55, 149-171.
Robertson, D.G.E., Barden, J., & Dowling, J. (1992). In Proceedings of the Second
North American Conference on Biomechanics. Kingston, ON: Canadian Society
for Biomechanics, Queen's University.
Smith, G. (1989). Padding point extrapolation techniques for the butterworth digital filter.
Journal of Biomechanics, 22, 967-972.
Winter, D.A. (1990). Biomechanics and motor control of human movement. Toronto, ON:
Wiley.
Acknowledgment
For technical assistance, we wish to thank Dr. Tyseer Aboulnasr of the Department
of Electrical Engineering, University of Ottawa.