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The rule of 1 s for padding kinematic data prior to digital filtering: Influence of sampling and filter cutoff frequencies
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Journal of Electromyography and Kinesiology 19 (2009) 875–881 www.elsevier.com/locate/jelekin
The rule of 1 s for padding kinematic data prior to digital filtering: Influence of sampling and filter cutoff frequencies Samuel J. Howarth, Jack P. Callaghan * Faculty of Applied Health Sciences, Department of Kinesiology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 3 February 2008; received in revised form 15 March 2008; accepted 20 March 2008
Abstract The influence of signal sampling frequency and the low-pass digital filter cutoff frequency on the minimum number of padding points when applied to kinematic data are factors often absent in data processing descriptions. This investigation determined a relationship between the number of padding points and the ratio of filter cutoff to signal sampling frequencies (fc/fs). Two kinematic recordings were used which represented signals with high and low deterministic variation magnitudes at the signals’ beginning. Signal sampling rates (40– 128 Hz) were generated at intervals of 1 Hz. Filter cutoff frequency was iterated from 2 to 10 Hz at 0.5 Hz intervals. Data extrapolation was performed using three different techniques (first order polynomial, third order polynomial, and data reflection). A maximum of 2 s of padding points were added to the beginning of each test signal which was then dual-pass filtered using a second order Butterworth filter. For each successive increase in the number of padding points, the filtered test signal was compared to a criterion signal and the root mean square difference (RMSD) over the first second was calculated. The number of padding points required to attain a constant RMSD was recorded as the minimum number of padding points needed for that ratio of filter cutoff to sampling frequency. As fc/fs increased, the number of padding points decreased non-linearly. More padding points were required for the signal with higher deterministic variation at the beginning than the signal with lower deterministic variation. Additional padding points (beyond the determined minimum) did not further reduce the RMSD. The largest temporal extrapolation determined by the algorithm to produce a stable RMSD was 1 s. It is suggested that a minimum of 1 s of extraneous data be used when using a low-pass recursive digital filter to remove noise from kinematic data. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Data extrapolation; Reflection; Polynomial; Butterworth; Kinematic
1. Introduction Digital filtering of biomechanical data is often necessary for removing noise from the collected data. High frequency noise removal is particularly important and common when differentiating marker kinematics to obtain velocities and accelerations since high frequency components are amplified by the differentiation process (Pezzack et al., 1977). Underdamped and recursive filters, such as the Butterworth digital filter exhibit amplitude distortion in response *
Corresponding author. Tel.: +1 519 888 4567x37080; fax: +1 519 746 6776/885 0470. E-mail address: [email protected] (J.P. Callaghan). 1050-6411/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jelekin.2008.03.010
to a step input before converging on the true value of the input (Robertson and Dowling, 2003). Due to the initial period of amplitude distortion that occurs when using underdamped and recursive digital filters, it is important to either collect extra data beyond the events of interest or add points to the beginning and end of data records to reduce amplitude distortion caused by the filtering process. The current investigation seeks to determine characteristics of the digital filter, and the data being filtered that will influence the minimum number of padding points required to add to the beginning and end of data records to reduce endpoint amplitude distortion after digital filtering. Previous investigations have sought to determine the minimum number of padding points to reduce endpoint
S.J. Howarth, J.P. Callaghan / Journal of Electromyography and Kinesiology 19 (2009) 875–881
discrepancies with kinematic data as well as seeking the optimal method for producing padding points which reduce the endpoint discrepancies. Smith (1989), using a pre-existing data set (Pezzack et al., 1977), showed that there was no improvement in endpoint distortion of a single kinematic signal beyond the addition of 10 extrapolated data points using three different techniques. The same investigation also demonstrated that the extrapolation technique of data reflection was a superior method for reducing endpoint discrepancy when compared to linear extrapolation or endpoint duplication. Typical extrapolation techniques reported in the literature are data reflection (Smith, 1989), polynomial extrapolation (Vint and Hinrichs, 1996; Giakis et al., 1998), and autoregressive models (D’Amico and Ferrigno, 1990). While these investigations attempted to provide a criterion for the number of padding points as well as evaluating the merits of different extrapolation techniques when filtering kinematic data with a recursive digital filter, the influence that altering filter characteristics such as cutoff frequency and signal sampling frequency would have on their objective criteria and evaluations have not been examined. The coefficients that mediate the amplitude response for a digital Butterworth filter are dependent upon the ratio of the sampling frequency to the cutoff frequency and the number of times that the data is passed through the filter (Robertson and Dowling, 2003). Therefore, it is likely that the number of padding points required for adequately reducing endpoint amplitude distortion is a function of the selected signal sampling and filter cutoff frequencies. The purpose of this study was to determine a criterion that explicitly states the minimum number of padding points required for minimizing endpoint error with a given set of signal sampling frequencies and a concurrent set of filter cutoff frequencies. A secondary purpose of this investigation was to evaluate three different extrapolation techniques (reflection, first order polynomial and third order polynomial) over a range of sampling frequencies and filter cutoff frequencies. The hypothesis was that the number of padding points required to minimize endpoint discrepancy will increase as the ratio between the filter’s cutoff frequency and the signal’s sampling frequency decreases. 2. Methods 2.1. Data characteristics For this investigation, the authors used two kinematic patterns from individual infrared emitting diodes (IRED) which were collected for a separate investigation that compared low back loading and vertebral joint rotational stiffness during pushups that were performed while the hands were interfaced with both stable and labile support surfaces (Beach et al., in press). The first kinematic signal was taken from the vertical component of a single marker attached to a fin which was secured to the T12-L1 vertebral joint while the second kinematic signal was obtained from a marker placed on the third metacarpal of the right hand. Both signals were sampled at a rate of 64 Hz during the original
1200
Displacement (mm)
876
1000 800 600 400 200 0 0
5
10
15
20
25
30
Time (sec) Fig. 1. The kinematic signals (spine marker, ; hand marker, ) used for this investigation. The thick black section denotes the temporal duration over which the test signal was extracted.
investigation (Fig. 1). During the original experiment, the pushup rate was controlled with a metronome such that a full pushup cycle was completed in 2 s. The fin marker was selected for this investigation since it had a larger excursion in the vertical direction while the hand marker represented a kinematic signal that demonstrated minimal variation. Thus, the signal obtained from the fin marker represented a signal with larger deterministic fluctuations than the signal obtained from the hand marker. 2.2. Algorithm description The general algorithm used for determining the minimum number of padding points to achieve a constant discrepancy between a test signal and a criterion signal is shown in Fig. 2. The raw kinematic signals (originally sampled at 64 Hz) were resampled (Eq. (1)) from 40 to 128 Hz at intervals of 1 Hz: y j ¼ ð1 rÞxl þ rxlþ1 ; j ¼ 1; . . . ; Tfo 8j k > < j ffoi þ 1 ; fi > fo l¼ j k > : j ffi ; fi 6 fo o ( fi j fo þ 1 l; fi > fo r¼ j ffoi ; fi 6 fo
ð1Þ
This is where; x is the original signal, yj is the resampled signal at point j, fi and fo are the initial signal sampling frequency and the resampling frequency, respectively, the b c operator denotes rounding down to the nearest integer and T is the total signal sampling duration. The cutoff frequency of a dual-pass, second order digital Butterworth filter was varied from 2 to 10 Hz at 0.5 Hz intervals. All possible combinations of signal, sampling frequency and filter cutoff frequency resulted in a set of 1513 filtered signals. A window of data between 4.65 and 10.94 s was removed from the resampled raw signal (Fig. 1). For the purpose of this investigation, the entire kinematic record will be called the raw signal while the kinematic record of the removed section will be called the test signal. Initially, the entire raw signal was re-sampled and filtered with all of the same combinations. These steps generated the criterion signal used in this investigation. The raw signal is composed of a deterministic and a non-deterministic component. Thus, the raw signal was filtered and served as an estimate of the kinematics after any unwanted high frequency noise was removed.
S.J. Howarth, J.P. Callaghan / Journal of Electromyography and Kinesiology 19 (2009) 875–881
877
Raw signal Re-sample
Remove section
Set filter cutoff
Set filter cutoff
Filter
Pad signal
Remove section
Filter
Increase number Increase filter of padding points cutoff until 2*fs
Remove padding points
RMSD over 1 sec of filtered partitioned signal, R(n) Compute R (n), R (n) Find the first n* such that R (n*) = R (n*) = 0
Fig. 2. Algorithm layout for determining the minimum number of padding points.
The identical temporal window to the test signal was removed from the raw signal after processing for comparison. Padding points were added to the beginning of the test signal using each of three extrapolation techniques (reflection, first order polynomial, third order polynomial) that are described in the following section. The number of padding points (n) that were added to each test signal ranged from 1 to 2fs (fs = sampling frequency) so that a maximum of 2 s of data were extrapolated at the beginning of each test signal. The raw signal and the padded test signal were filtered using a dual-pass second order digital Butterworth filter using the range of cutoff frequencies that was previously described. The padding points were removed from the filtered test signal and the identical temporal duration of kinematic data (between 4.65 and 10.94 s) was then removed from the criterion signal. The root mean square difference (RMSD) between the subsection of the criterion and the test signal was computed, over the first second (i.e. from 4.65 to 5.65 s) of the filtered data, to obtain a measure of the discrepancy induced by the extrapolation techniques. The preceding steps were repeated by successively adding padding points which permitted description of the RMSD as a function of the number of padding points (R(n)). The first and second derivatives of R(n) were computed with respect to the number of padding points for each successive padding point using finite difference equations (Winter, 1990). The criteria for determining the minimum number of padding points (n*) to be added prior to digital filtering was the number of padding points at which the first and second derivatives of R(n) were both zero (i.e. constant RMSD). One limitation of using finite difference equations for determining derivatives is that a minimum of three padding points must be added prior to determining the first and second derivatives. Thus, the lower bound on the minimum number of padding points was three. 2.3. Extrapolation techniques Padding points were added to the beginning of the unfiltered test signals using one of three previously published methods. The
first is a reflection method whereby the data near the endpoints are reflected and added prior to the endpoint: zniþ1 ¼ 2y 0 y i ;
i ¼ 1; . . . ; n
ð2Þ
Here y0 is the first point in the unfiltered test signal prior to padding. The first n points of the padded test signal (z) are the reflected padding points while the remaining points of z are identical to the data of y. The number of padding points to be added is given by n. The second and third methods were polynomial extrapolations (first and third order, respectively) of the data at the beginning of the test signal. The coefficients of the third order polynomial were determined by a least squares estimate over the first 0.2 s of each test signal while the first order polynomial was fit with a similar least squares approach using the first two points of the test signal prior to padding. Data extrapolation was performed according to the following equation which is a generalization of the polynomial extrapolation for a polynomial of any order: 2 p i i i zniþ1 ¼ a0 þ a1 þ a2 þ þ ap ; fs fs fs i ¼ 1; . . . ; n
ð3Þ
The padded test signal is given by z, and aj (j = 0, . . ., p) are the coefficients from the least squares estimates. Since the least squares estimate yields a polynomial as a function of the time i at which the sample is taken, it is necessary to compute the fs extrapolated points also as function of time. The value p denotes the order of the polynomial. In the case of a first order polynomial, p = 1, while in the third order polynomial, p = 3.
3. Results The RMSD between each criterion and the corresponding filtered test signal exhibited an oscillatory response that eventually achieved a constant RMSD as the number of padding points was increased (Fig. 3).
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Fig. 3. The root mean square difference (RMSD) as a function of the number of padding points for a signal that is sampled at a frequency of 64 Hz and is filtered using a dual-pass second order digital Butterworth ), third order polynomial filter for the first order polynomial ( ) and reflection ( ) extrapolation techniques. (
The minimum number of padding points was defined as the number of padding points required to achieve a constant RMSD between the criterion and filtered test signals. The constant RMSD was achieved, for each combination of sampling and cutoff frequencies, prior to the maximum number of padding points which was defined by our algorithm as twice the sampling frequency of the resampled kinematic signal. The RMSD at the minimum number of padding points was lowest for the first order polynomial extrapolation technique (Fig. 4). It should be noted that discrepancies between the filtered criterion and filtered test signals were unavoidable regardless of the padding method or the number of padding points that were used (Fig. 4). A non-linear decreasing trend occurred between the minimum required number of padding points and the ratio between the cutoff frequency and sampling frequency. As the ratio between the cutoff frequency and sampling frequency increased, the minimum number of padding points decreased. The minimum number of padding points required to attain a constant RMSD for the signal with larger deterministic variation was fit using a least squares estimate of a power law relationship (Table 1). This relationship quantifies the minimum number of padding points for a given ratio between the cutoff frequency and sampling frequency. Additional padding points beyond the calculated minimum value will not reduce, nor increase the magnitude of the RMSD between the criterion and filtered test signals. The kinematic signal with less deterministic variation required fewer padding points than the kinematic signal with more deterministic variation for each of the three extrapolation techniques (Fig. 5). The first order polynomial and reflection extrapolation methods required fewer padding points to attain a constant and lower RMSD than the third order polynomial method (Fig. 5). The maximum extrapolated time (number of padding points divided by resampling frequency) across the three extrapolation techniques and each of the 1513 test signals was 1 s (Fig. 6).
Fig. 4. The root mean square difference (RSMD) between the criterion and test signals when the number of padding points is equal to the determined minimum number of padding points as a function of the ratio between the cutoff and sampling frequencies (fc/fs) for the first order polynomial (a), third order polynomial (b) and reflection (c) extrapolation techniques. Light grey squares ( ) represent the RMSD at the minimum number of padding points for the signal with less deterministic variation while the dark grey diamonds ( ) represent the RMSD at the minimum number of padding points for the signal with more deterministic variation.
4. Discussion The primary goal of this investigation was to determine how filter cutoff frequency, as well as signal sampling frequency, and the deterministic component of the signal affects the number of padding points that should be added
S.J. Howarth, J.P. Callaghan / Journal of Electromyography and Kinesiology 19 (2009) 875–881
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Table 1 Coefficients from the least squares curve fit relating the minimum number of padding points (n*) to the ratio between the cutoff frequency and the sample frequency (fc/fs) for the kinematic signal with larger deterministic variation Padding method
a
b
First order polynomial Third order polynomial Reflection
2.538 1.706 2.584
0.748 0.985 0.804
R2
0.937 0.873 0.912 fc The general equation used for the least squares curve fit was n fs ¼ b a ffcs .
prior to digital filtering of kinematic data. As a measure of safety, the minimum number of padding points that should be added prior to signal filtering is equal to the signal’s sampling rate (i.e. 1 s of data). The same criterion should be followed if using extraneous data as the padding points. The minimum number of padding points can vary greatly depending on the ratio between the cutoff frequency and the sampling frequency. Moreover, the signal with smaller deterministic variation at the starting point required fewer padding points than the signal with larger deterministic variations near the starting point. An estimate of the true kinematics was chosen for this investigation as a criterion to determine discrepancies induced by the extrapolation techniques prior to digital filtering. Estimations of the true kinematics are necessary in biomechanics research since the investigators do not have direct knowledge of the exact kinematics. An example of a situation where the exact kinematics could be determined would be tracking a marker on a rigid oscillating pendulum. Under ideal conditions (frictionless pin joint for the pendulum, zero air resistance, known mass distribution of the pendulum), the exact kinematics of a marker located at a distance from the rigid pendulum’s axis of rotation could be determined using Newtonian mechanics. Previous investigations have estimated the discrepancies, from directly measured accelerations using an accelerometer, induced by different data extrapolation techniques on calculated accelerations from displacement data (Giakis et al., 1998). Although the accelerations can be directly measured, there will be a non-deterministic component in the accelerometer signal implying that the accelerations derived from the accelerometer are also an approximation of the true accelerations. Smith (1989) used the original accelerations calculated by Pezzack et al. (1977) as the criterion for determining the minimum number of padding points using three extrapolation techniques with a fixed sample rate and filter cutoff frequency. Vint and Hinrichs (1996) also used a criterion signal for estimating errors induced in calculated accelerations from filtered data, but these authors do not explicitly state what they used as their criterion. For the current investigation, the entire kinematic signal was filtered using the same range and increments of filter cutoff frequencies that were applied to the test signals. The temporally matched section to the test signals was removed from the entire, filtered kinematic record
Fig. 5. The minimum number of padding points as a function of the ratio between the cutoff and sampling frequencies (fc/fs) for the first order polynomial (a), third order polynomial (b) and reflection (c) extrapolation techniques. Light grey squares ( ) represent the minimum number of padding points for the signal with less deterministic variation while the dark grey diamonds ( ) represent the minimum number of padding points for the signal with more deterministic variation. The black line represents the least squares estimate of a power law relationship for the relationship between the minimum number of padding points and fc/fs.
and used as the criterion for comparison with the filtered test signals. The criterion signal represented an estimation of the true underlying kinematics. One further limitation of this study was that padding points were only added to the beginning of the test signals while a dual-pass Butterworth filter was implemented. Thus, the second pass of the filter would not include preced-
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Fig. 6. The maximum extrapolated time calculated as the minimum number of padding points divided by the resampling frequency for each of the three extrapolation techniques.
ing extrapolated data points. However, the results from this study demonstrated that the RMSD between the criterion and test signals did not change following a minimum of 1 s after the beginning of the data record. The data record considered for the test signal was 6.29 s (between 4.65 and 10.94 s) with the RMSD being calculated for the first second of the data (4.65–5.65 s) which means that the last second of the test signal served as the padding points for the reverse pass of the Butterworth filter. When implementing padding points on experimental data both the beginning and end must be padded with 1 s of data. The RMSD (for a single combination of sampling and cutoff frequency) expressed as a function of the number of padding points had a similar oscillatory response followed by a constant error as the previously reported response of the mean residuals for the linear extrapolation, digitized points and reflection extrapolation techniques (Smith, 1989). The results from the current investigation demonstrate that endpoint error following digital filtering is unavoidable regardless of the extrapolation technique or the number of extrapolation points that are used. This is consistent with the findings from previous investigations that quantified endpoint errors, in accelerations derived from differentiated displacement data, induced by the digital filtering process (Smith, 1989; Vint and Hinrichs, 1996; Giakis et al., 1998). Smith (1989) considered a single sampling frequency of 100 Hz (Pezzack et al., 1977) and filter cutoff frequency of 4 Hz and demonstrated that endpoint error did not improve beyond addition of 10 padding points for the reflection and first order polynomial techniques. This is comparable to the findings in this study of 10 padding points (reflection) and 7 padding points (first order polynomial) for the signal with smaller deterministic variation with a sample rate of 100 Hz and a filter cutoff frequency of 4 Hz. The current investigation extends Smith (1989) original work by showing that the minimum required number of padding points to attain a consistent endpoint error will vary as a non-linearly decreasing function of the ratio between the filter cutoff frequency and the sampling frequency. The relationship between the minimum required
number of padding points and the ratio of the filter cutoff frequency to the sampling frequency was also dependent upon the extrapolation technique as well as the deterministic variation at the endpoints. The derived functions from the least squares curve fits relating the minimum number of padding points to the ratio between the filter cutoff frequency and sampling frequency create an estimate of a lower boundary for the number of padding points that are required to achieve a constant error between the test and criterion kinematic signals used for this investigation. It is imperative to point out that the reported curve fitting coefficients are specific to the data that is considered in this investigation, and will likely change depending on the signal being analyzed should the current algorithm be repeated with a different set of data. Analysis of the RMSD at the determined minimum number of padding points showed that the first order polynomial extrapolation method had smaller discrepancies than the reflection and third order polynomial extrapolation techniques. This result is consistent with the findings of Vint and Hinrichs (1996) who demonstrated that the endpoint error, caused by digital Butterworth filtering, for the first order polynomial extrapolation was smaller than the endpoint error for the reflection technique. However, Giakis et al. (1998) showed that extrapolation using a least squares estimate of a third order polynomial induced smaller error in computed angular accelerations from the data set described by Pezzack et al. (1977) than the linear extrapolation technique. The errors computed by Giakis et al. (1998) were based on discrepancies between calculated angular accelerations and measured angular accelerations as reported by Pezzack et al. (1977). Thus, the discrepancies calculated in this investigation, for kinematic displacement data, should not be directly compared with those of Giakis et al. (1998). Similar to the result that the minimum number of padding points is a function of the signal characteristics such as the slope near the endpoints, it is possible that the error induced by a particular extrapolation technique is also a function of the same signal characteristics. This investigation demonstrated with two different kinematic signals (high and low deterministic variation) that the variation of the kinematic recording at the endpoints will influence the minimum number of padding points required to attain a constant discrepancy between the criterion and test signals. Conceivably, the minimum number of padding points will continually vary as a function of the kinematic signal’s deterministic variation. As a general criterion, the authors suggest a minimum duration of 1 s of data be extrapolated for all kinematic recordings prior to digital filtering. The most suitable alternative to data extrapolation is to collect an extra second of superfluous data at the beginning and end of the desired sample period that can be removed following digital filtering. This suggested benchmark is based on the result that 1 s was the maximum extrapolated time, for all 1513 combinations of sampling and filter cutoff frequencies, across each of the three extrapolation techniques. Furthermore, additional
S.J. Howarth, J.P. Callaghan / Journal of Electromyography and Kinesiology 19 (2009) 875–881
padding points beyond the determined minimum did not increase nor decrease the discrepancy between the filtered test and criterion signals which implies that overestimation of the minimum number of padding points is not detrimental to error determination. Future studies should focus on the differences between underdamped and critically damped digital filters as well as the effects that filter order has on the minimum number of required padding points. In conclusion, this study demonstrates that signal sampling and filter cutoff frequencies interact to influence the minimum number of padding points that must be added prior to digital filtering of kinematic data. Discrepancies from the true signal, following digital filtering are unavoidable regardless of the number of padding points that are added to the signal. In the absence of a thorough analysis to determine the minimum number of padding points, it is suggested that at least 1 s of data be extrapolated onto the beginning and end of data trials as a standard prior to dual-pass digital filtering. Acknowledgements The authors are thankful for the financial support offered by the Natural Sciences and Engineering Research Council of Canada (NSERC). Dr. Jack Callaghan is supported by a Canada Research Chair (CRC) in Spine Biomechanics and Injury Prevention. References Beach TAC, Howarth SJ, Callaghan JP. Muscular contribution to lowback loading and stiffness during standard and suspended push-ups. Hum Movement Sci, in press. D’Amico M, Ferrigno G. Technique for the evaluation of derivatives from noisy biomechanical displacement data using a model-based bandwidth-selection procedure. Med Biol Eng Comput 1990;28(5):407–15. Giakis G, Baltzopoulos V, Bartlett RM. Improved extrapolation techniques in recursive digital filtering: a comparison of least squares and prediction. J Biomech 1998;31(1):87–91.
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Pezzack JC, Norman RW, Winter DA. An assessment of derivative determining techniques used for motion analysis. J Biomech 1977;10(5–6):377–82. Robertson DGE, Dowling JJ. Design and responses of Butterworth and critically damped digital filters. J Electromyogr Kinesiol 2003;13(6):569–73. Smith G. Padding point extrapolation techniques for the digital Butterworth filter. J Biomech 1989;22(8-9):967–71. Vint PF, Hinrichs RN. Endpoint error in smoothing and differentiating raw kinematic data: an evaluation of four popular methods. J Biomech 1996;29(12):1637–42. Winter DA. Biomechanics and motor control of human movement. 2nd ed. Toronto: John Wiley and Sons; 1990.
Samuel J. Howarth received a B.Sc. in Kinesiology with a Joint Honours in Mathematics (2004) as well as an M.Sc. in Biomechanics (2006) from the University of Waterloo. He is currently a Ph.D. candidate in the Department of Kinesiology at the University of Waterloo. His research investigates how passive tissue damage in the lumbar spine under acute as well as repetitive shear loading is influenced by modulating factors such as posture, compressive load and shear load rate. Jack P. Callaghan received his Ph.D. in Kinesiology from the Faculty of Applied Health Sciences at the University of Waterloo in 1999. From 1998 to 2003, he was a faculty member in the Department of Human Biology at the University of Guelph. In 2003, he was awarded a Canada Research Chair in Spine Biomechanics and Injury Prevention and returned to the Kinesiology Department at the University of Waterloo. He has also received an Ontario Distinguished Researcher Award, and a Canada Foundation for Innovation infrastructure grant. He is a project leader in the AUTO21 Network of Centres of Excellence and an NSERC, CIHR and WSIB funded researcher. He is an author on 70 peer reviewed journal articles, has presented over 140 papers at conferences and supervised 20 graduate students. His main research interest is injury mechanisms from exposure to cumulative loading exposure.
Journal of Electromyography and Kinesiology 19 (2009) 875–881 www.elsevier.com/locate/jelekin
The rule of 1 s for padding kinematic data prior to digital filtering: Influence of sampling and filter cutoff frequencies Samuel J. Howarth, Jack P. Callaghan * Faculty of Applied Health Sciences, Department of Kinesiology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 3 February 2008; received in revised form 15 March 2008; accepted 20 March 2008
Abstract The influence of signal sampling frequency and the low-pass digital filter cutoff frequency on the minimum number of padding points when applied to kinematic data are factors often absent in data processing descriptions. This investigation determined a relationship between the number of padding points and the ratio of filter cutoff to signal sampling frequencies (fc/fs). Two kinematic recordings were used which represented signals with high and low deterministic variation magnitudes at the signals’ beginning. Signal sampling rates (40– 128 Hz) were generated at intervals of 1 Hz. Filter cutoff frequency was iterated from 2 to 10 Hz at 0.5 Hz intervals. Data extrapolation was performed using three different techniques (first order polynomial, third order polynomial, and data reflection). A maximum of 2 s of padding points were added to the beginning of each test signal which was then dual-pass filtered using a second order Butterworth filter. For each successive increase in the number of padding points, the filtered test signal was compared to a criterion signal and the root mean square difference (RMSD) over the first second was calculated. The number of padding points required to attain a constant RMSD was recorded as the minimum number of padding points needed for that ratio of filter cutoff to sampling frequency. As fc/fs increased, the number of padding points decreased non-linearly. More padding points were required for the signal with higher deterministic variation at the beginning than the signal with lower deterministic variation. Additional padding points (beyond the determined minimum) did not further reduce the RMSD. The largest temporal extrapolation determined by the algorithm to produce a stable RMSD was 1 s. It is suggested that a minimum of 1 s of extraneous data be used when using a low-pass recursive digital filter to remove noise from kinematic data. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Data extrapolation; Reflection; Polynomial; Butterworth; Kinematic
1. Introduction Digital filtering of biomechanical data is often necessary for removing noise from the collected data. High frequency noise removal is particularly important and common when differentiating marker kinematics to obtain velocities and accelerations since high frequency components are amplified by the differentiation process (Pezzack et al., 1977). Underdamped and recursive filters, such as the Butterworth digital filter exhibit amplitude distortion in response *
Corresponding author. Tel.: +1 519 888 4567x37080; fax: +1 519 746 6776/885 0470. E-mail address: [email protected] (J.P. Callaghan). 1050-6411/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jelekin.2008.03.010
to a step input before converging on the true value of the input (Robertson and Dowling, 2003). Due to the initial period of amplitude distortion that occurs when using underdamped and recursive digital filters, it is important to either collect extra data beyond the events of interest or add points to the beginning and end of data records to reduce amplitude distortion caused by the filtering process. The current investigation seeks to determine characteristics of the digital filter, and the data being filtered that will influence the minimum number of padding points required to add to the beginning and end of data records to reduce endpoint amplitude distortion after digital filtering. Previous investigations have sought to determine the minimum number of padding points to reduce endpoint
S.J. Howarth, J.P. Callaghan / Journal of Electromyography and Kinesiology 19 (2009) 875–881
discrepancies with kinematic data as well as seeking the optimal method for producing padding points which reduce the endpoint discrepancies. Smith (1989), using a pre-existing data set (Pezzack et al., 1977), showed that there was no improvement in endpoint distortion of a single kinematic signal beyond the addition of 10 extrapolated data points using three different techniques. The same investigation also demonstrated that the extrapolation technique of data reflection was a superior method for reducing endpoint discrepancy when compared to linear extrapolation or endpoint duplication. Typical extrapolation techniques reported in the literature are data reflection (Smith, 1989), polynomial extrapolation (Vint and Hinrichs, 1996; Giakis et al., 1998), and autoregressive models (D’Amico and Ferrigno, 1990). While these investigations attempted to provide a criterion for the number of padding points as well as evaluating the merits of different extrapolation techniques when filtering kinematic data with a recursive digital filter, the influence that altering filter characteristics such as cutoff frequency and signal sampling frequency would have on their objective criteria and evaluations have not been examined. The coefficients that mediate the amplitude response for a digital Butterworth filter are dependent upon the ratio of the sampling frequency to the cutoff frequency and the number of times that the data is passed through the filter (Robertson and Dowling, 2003). Therefore, it is likely that the number of padding points required for adequately reducing endpoint amplitude distortion is a function of the selected signal sampling and filter cutoff frequencies. The purpose of this study was to determine a criterion that explicitly states the minimum number of padding points required for minimizing endpoint error with a given set of signal sampling frequencies and a concurrent set of filter cutoff frequencies. A secondary purpose of this investigation was to evaluate three different extrapolation techniques (reflection, first order polynomial and third order polynomial) over a range of sampling frequencies and filter cutoff frequencies. The hypothesis was that the number of padding points required to minimize endpoint discrepancy will increase as the ratio between the filter’s cutoff frequency and the signal’s sampling frequency decreases. 2. Methods 2.1. Data characteristics For this investigation, the authors used two kinematic patterns from individual infrared emitting diodes (IRED) which were collected for a separate investigation that compared low back loading and vertebral joint rotational stiffness during pushups that were performed while the hands were interfaced with both stable and labile support surfaces (Beach et al., in press). The first kinematic signal was taken from the vertical component of a single marker attached to a fin which was secured to the T12-L1 vertebral joint while the second kinematic signal was obtained from a marker placed on the third metacarpal of the right hand. Both signals were sampled at a rate of 64 Hz during the original
1200
Displacement (mm)
876
1000 800 600 400 200 0 0
5
10
15
20
25
30
Time (sec) Fig. 1. The kinematic signals (spine marker, ; hand marker, ) used for this investigation. The thick black section denotes the temporal duration over which the test signal was extracted.
investigation (Fig. 1). During the original experiment, the pushup rate was controlled with a metronome such that a full pushup cycle was completed in 2 s. The fin marker was selected for this investigation since it had a larger excursion in the vertical direction while the hand marker represented a kinematic signal that demonstrated minimal variation. Thus, the signal obtained from the fin marker represented a signal with larger deterministic fluctuations than the signal obtained from the hand marker. 2.2. Algorithm description The general algorithm used for determining the minimum number of padding points to achieve a constant discrepancy between a test signal and a criterion signal is shown in Fig. 2. The raw kinematic signals (originally sampled at 64 Hz) were resampled (Eq. (1)) from 40 to 128 Hz at intervals of 1 Hz: y j ¼ ð1 rÞxl þ rxlþ1 ; j ¼ 1; . . . ; Tfo 8j k > < j ffoi þ 1 ; fi > fo l¼ j k > : j ffi ; fi 6 fo o ( fi j fo þ 1 l; fi > fo r¼ j ffoi ; fi 6 fo
ð1Þ
This is where; x is the original signal, yj is the resampled signal at point j, fi and fo are the initial signal sampling frequency and the resampling frequency, respectively, the b c operator denotes rounding down to the nearest integer and T is the total signal sampling duration. The cutoff frequency of a dual-pass, second order digital Butterworth filter was varied from 2 to 10 Hz at 0.5 Hz intervals. All possible combinations of signal, sampling frequency and filter cutoff frequency resulted in a set of 1513 filtered signals. A window of data between 4.65 and 10.94 s was removed from the resampled raw signal (Fig. 1). For the purpose of this investigation, the entire kinematic record will be called the raw signal while the kinematic record of the removed section will be called the test signal. Initially, the entire raw signal was re-sampled and filtered with all of the same combinations. These steps generated the criterion signal used in this investigation. The raw signal is composed of a deterministic and a non-deterministic component. Thus, the raw signal was filtered and served as an estimate of the kinematics after any unwanted high frequency noise was removed.
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877
Raw signal Re-sample
Remove section
Set filter cutoff
Set filter cutoff
Filter
Pad signal
Remove section
Filter
Increase number Increase filter of padding points cutoff until 2*fs
Remove padding points
RMSD over 1 sec of filtered partitioned signal, R(n) Compute R (n), R (n) Find the first n* such that R (n*) = R (n*) = 0
Fig. 2. Algorithm layout for determining the minimum number of padding points.
The identical temporal window to the test signal was removed from the raw signal after processing for comparison. Padding points were added to the beginning of the test signal using each of three extrapolation techniques (reflection, first order polynomial, third order polynomial) that are described in the following section. The number of padding points (n) that were added to each test signal ranged from 1 to 2fs (fs = sampling frequency) so that a maximum of 2 s of data were extrapolated at the beginning of each test signal. The raw signal and the padded test signal were filtered using a dual-pass second order digital Butterworth filter using the range of cutoff frequencies that was previously described. The padding points were removed from the filtered test signal and the identical temporal duration of kinematic data (between 4.65 and 10.94 s) was then removed from the criterion signal. The root mean square difference (RMSD) between the subsection of the criterion and the test signal was computed, over the first second (i.e. from 4.65 to 5.65 s) of the filtered data, to obtain a measure of the discrepancy induced by the extrapolation techniques. The preceding steps were repeated by successively adding padding points which permitted description of the RMSD as a function of the number of padding points (R(n)). The first and second derivatives of R(n) were computed with respect to the number of padding points for each successive padding point using finite difference equations (Winter, 1990). The criteria for determining the minimum number of padding points (n*) to be added prior to digital filtering was the number of padding points at which the first and second derivatives of R(n) were both zero (i.e. constant RMSD). One limitation of using finite difference equations for determining derivatives is that a minimum of three padding points must be added prior to determining the first and second derivatives. Thus, the lower bound on the minimum number of padding points was three. 2.3. Extrapolation techniques Padding points were added to the beginning of the unfiltered test signals using one of three previously published methods. The
first is a reflection method whereby the data near the endpoints are reflected and added prior to the endpoint: zniþ1 ¼ 2y 0 y i ;
i ¼ 1; . . . ; n
ð2Þ
Here y0 is the first point in the unfiltered test signal prior to padding. The first n points of the padded test signal (z) are the reflected padding points while the remaining points of z are identical to the data of y. The number of padding points to be added is given by n. The second and third methods were polynomial extrapolations (first and third order, respectively) of the data at the beginning of the test signal. The coefficients of the third order polynomial were determined by a least squares estimate over the first 0.2 s of each test signal while the first order polynomial was fit with a similar least squares approach using the first two points of the test signal prior to padding. Data extrapolation was performed according to the following equation which is a generalization of the polynomial extrapolation for a polynomial of any order: 2 p i i i zniþ1 ¼ a0 þ a1 þ a2 þ þ ap ; fs fs fs i ¼ 1; . . . ; n
ð3Þ
The padded test signal is given by z, and aj (j = 0, . . ., p) are the coefficients from the least squares estimates. Since the least squares estimate yields a polynomial as a function of the time i at which the sample is taken, it is necessary to compute the fs extrapolated points also as function of time. The value p denotes the order of the polynomial. In the case of a first order polynomial, p = 1, while in the third order polynomial, p = 3.
3. Results The RMSD between each criterion and the corresponding filtered test signal exhibited an oscillatory response that eventually achieved a constant RMSD as the number of padding points was increased (Fig. 3).
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Fig. 3. The root mean square difference (RMSD) as a function of the number of padding points for a signal that is sampled at a frequency of 64 Hz and is filtered using a dual-pass second order digital Butterworth ), third order polynomial filter for the first order polynomial ( ) and reflection ( ) extrapolation techniques. (
The minimum number of padding points was defined as the number of padding points required to achieve a constant RMSD between the criterion and filtered test signals. The constant RMSD was achieved, for each combination of sampling and cutoff frequencies, prior to the maximum number of padding points which was defined by our algorithm as twice the sampling frequency of the resampled kinematic signal. The RMSD at the minimum number of padding points was lowest for the first order polynomial extrapolation technique (Fig. 4). It should be noted that discrepancies between the filtered criterion and filtered test signals were unavoidable regardless of the padding method or the number of padding points that were used (Fig. 4). A non-linear decreasing trend occurred between the minimum required number of padding points and the ratio between the cutoff frequency and sampling frequency. As the ratio between the cutoff frequency and sampling frequency increased, the minimum number of padding points decreased. The minimum number of padding points required to attain a constant RMSD for the signal with larger deterministic variation was fit using a least squares estimate of a power law relationship (Table 1). This relationship quantifies the minimum number of padding points for a given ratio between the cutoff frequency and sampling frequency. Additional padding points beyond the calculated minimum value will not reduce, nor increase the magnitude of the RMSD between the criterion and filtered test signals. The kinematic signal with less deterministic variation required fewer padding points than the kinematic signal with more deterministic variation for each of the three extrapolation techniques (Fig. 5). The first order polynomial and reflection extrapolation methods required fewer padding points to attain a constant and lower RMSD than the third order polynomial method (Fig. 5). The maximum extrapolated time (number of padding points divided by resampling frequency) across the three extrapolation techniques and each of the 1513 test signals was 1 s (Fig. 6).
Fig. 4. The root mean square difference (RSMD) between the criterion and test signals when the number of padding points is equal to the determined minimum number of padding points as a function of the ratio between the cutoff and sampling frequencies (fc/fs) for the first order polynomial (a), third order polynomial (b) and reflection (c) extrapolation techniques. Light grey squares ( ) represent the RMSD at the minimum number of padding points for the signal with less deterministic variation while the dark grey diamonds ( ) represent the RMSD at the minimum number of padding points for the signal with more deterministic variation.
4. Discussion The primary goal of this investigation was to determine how filter cutoff frequency, as well as signal sampling frequency, and the deterministic component of the signal affects the number of padding points that should be added
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Table 1 Coefficients from the least squares curve fit relating the minimum number of padding points (n*) to the ratio between the cutoff frequency and the sample frequency (fc/fs) for the kinematic signal with larger deterministic variation Padding method
a
b
First order polynomial Third order polynomial Reflection
2.538 1.706 2.584
0.748 0.985 0.804
R2
0.937 0.873 0.912 fc The general equation used for the least squares curve fit was n fs ¼ b a ffcs .
prior to digital filtering of kinematic data. As a measure of safety, the minimum number of padding points that should be added prior to signal filtering is equal to the signal’s sampling rate (i.e. 1 s of data). The same criterion should be followed if using extraneous data as the padding points. The minimum number of padding points can vary greatly depending on the ratio between the cutoff frequency and the sampling frequency. Moreover, the signal with smaller deterministic variation at the starting point required fewer padding points than the signal with larger deterministic variations near the starting point. An estimate of the true kinematics was chosen for this investigation as a criterion to determine discrepancies induced by the extrapolation techniques prior to digital filtering. Estimations of the true kinematics are necessary in biomechanics research since the investigators do not have direct knowledge of the exact kinematics. An example of a situation where the exact kinematics could be determined would be tracking a marker on a rigid oscillating pendulum. Under ideal conditions (frictionless pin joint for the pendulum, zero air resistance, known mass distribution of the pendulum), the exact kinematics of a marker located at a distance from the rigid pendulum’s axis of rotation could be determined using Newtonian mechanics. Previous investigations have estimated the discrepancies, from directly measured accelerations using an accelerometer, induced by different data extrapolation techniques on calculated accelerations from displacement data (Giakis et al., 1998). Although the accelerations can be directly measured, there will be a non-deterministic component in the accelerometer signal implying that the accelerations derived from the accelerometer are also an approximation of the true accelerations. Smith (1989) used the original accelerations calculated by Pezzack et al. (1977) as the criterion for determining the minimum number of padding points using three extrapolation techniques with a fixed sample rate and filter cutoff frequency. Vint and Hinrichs (1996) also used a criterion signal for estimating errors induced in calculated accelerations from filtered data, but these authors do not explicitly state what they used as their criterion. For the current investigation, the entire kinematic signal was filtered using the same range and increments of filter cutoff frequencies that were applied to the test signals. The temporally matched section to the test signals was removed from the entire, filtered kinematic record
Fig. 5. The minimum number of padding points as a function of the ratio between the cutoff and sampling frequencies (fc/fs) for the first order polynomial (a), third order polynomial (b) and reflection (c) extrapolation techniques. Light grey squares ( ) represent the minimum number of padding points for the signal with less deterministic variation while the dark grey diamonds ( ) represent the minimum number of padding points for the signal with more deterministic variation. The black line represents the least squares estimate of a power law relationship for the relationship between the minimum number of padding points and fc/fs.
and used as the criterion for comparison with the filtered test signals. The criterion signal represented an estimation of the true underlying kinematics. One further limitation of this study was that padding points were only added to the beginning of the test signals while a dual-pass Butterworth filter was implemented. Thus, the second pass of the filter would not include preced-
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Fig. 6. The maximum extrapolated time calculated as the minimum number of padding points divided by the resampling frequency for each of the three extrapolation techniques.
ing extrapolated data points. However, the results from this study demonstrated that the RMSD between the criterion and test signals did not change following a minimum of 1 s after the beginning of the data record. The data record considered for the test signal was 6.29 s (between 4.65 and 10.94 s) with the RMSD being calculated for the first second of the data (4.65–5.65 s) which means that the last second of the test signal served as the padding points for the reverse pass of the Butterworth filter. When implementing padding points on experimental data both the beginning and end must be padded with 1 s of data. The RMSD (for a single combination of sampling and cutoff frequency) expressed as a function of the number of padding points had a similar oscillatory response followed by a constant error as the previously reported response of the mean residuals for the linear extrapolation, digitized points and reflection extrapolation techniques (Smith, 1989). The results from the current investigation demonstrate that endpoint error following digital filtering is unavoidable regardless of the extrapolation technique or the number of extrapolation points that are used. This is consistent with the findings from previous investigations that quantified endpoint errors, in accelerations derived from differentiated displacement data, induced by the digital filtering process (Smith, 1989; Vint and Hinrichs, 1996; Giakis et al., 1998). Smith (1989) considered a single sampling frequency of 100 Hz (Pezzack et al., 1977) and filter cutoff frequency of 4 Hz and demonstrated that endpoint error did not improve beyond addition of 10 padding points for the reflection and first order polynomial techniques. This is comparable to the findings in this study of 10 padding points (reflection) and 7 padding points (first order polynomial) for the signal with smaller deterministic variation with a sample rate of 100 Hz and a filter cutoff frequency of 4 Hz. The current investigation extends Smith (1989) original work by showing that the minimum required number of padding points to attain a consistent endpoint error will vary as a non-linearly decreasing function of the ratio between the filter cutoff frequency and the sampling frequency. The relationship between the minimum required
number of padding points and the ratio of the filter cutoff frequency to the sampling frequency was also dependent upon the extrapolation technique as well as the deterministic variation at the endpoints. The derived functions from the least squares curve fits relating the minimum number of padding points to the ratio between the filter cutoff frequency and sampling frequency create an estimate of a lower boundary for the number of padding points that are required to achieve a constant error between the test and criterion kinematic signals used for this investigation. It is imperative to point out that the reported curve fitting coefficients are specific to the data that is considered in this investigation, and will likely change depending on the signal being analyzed should the current algorithm be repeated with a different set of data. Analysis of the RMSD at the determined minimum number of padding points showed that the first order polynomial extrapolation method had smaller discrepancies than the reflection and third order polynomial extrapolation techniques. This result is consistent with the findings of Vint and Hinrichs (1996) who demonstrated that the endpoint error, caused by digital Butterworth filtering, for the first order polynomial extrapolation was smaller than the endpoint error for the reflection technique. However, Giakis et al. (1998) showed that extrapolation using a least squares estimate of a third order polynomial induced smaller error in computed angular accelerations from the data set described by Pezzack et al. (1977) than the linear extrapolation technique. The errors computed by Giakis et al. (1998) were based on discrepancies between calculated angular accelerations and measured angular accelerations as reported by Pezzack et al. (1977). Thus, the discrepancies calculated in this investigation, for kinematic displacement data, should not be directly compared with those of Giakis et al. (1998). Similar to the result that the minimum number of padding points is a function of the signal characteristics such as the slope near the endpoints, it is possible that the error induced by a particular extrapolation technique is also a function of the same signal characteristics. This investigation demonstrated with two different kinematic signals (high and low deterministic variation) that the variation of the kinematic recording at the endpoints will influence the minimum number of padding points required to attain a constant discrepancy between the criterion and test signals. Conceivably, the minimum number of padding points will continually vary as a function of the kinematic signal’s deterministic variation. As a general criterion, the authors suggest a minimum duration of 1 s of data be extrapolated for all kinematic recordings prior to digital filtering. The most suitable alternative to data extrapolation is to collect an extra second of superfluous data at the beginning and end of the desired sample period that can be removed following digital filtering. This suggested benchmark is based on the result that 1 s was the maximum extrapolated time, for all 1513 combinations of sampling and filter cutoff frequencies, across each of the three extrapolation techniques. Furthermore, additional
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padding points beyond the determined minimum did not increase nor decrease the discrepancy between the filtered test and criterion signals which implies that overestimation of the minimum number of padding points is not detrimental to error determination. Future studies should focus on the differences between underdamped and critically damped digital filters as well as the effects that filter order has on the minimum number of required padding points. In conclusion, this study demonstrates that signal sampling and filter cutoff frequencies interact to influence the minimum number of padding points that must be added prior to digital filtering of kinematic data. Discrepancies from the true signal, following digital filtering are unavoidable regardless of the number of padding points that are added to the signal. In the absence of a thorough analysis to determine the minimum number of padding points, it is suggested that at least 1 s of data be extrapolated onto the beginning and end of data trials as a standard prior to dual-pass digital filtering. Acknowledgements The authors are thankful for the financial support offered by the Natural Sciences and Engineering Research Council of Canada (NSERC). Dr. Jack Callaghan is supported by a Canada Research Chair (CRC) in Spine Biomechanics and Injury Prevention. References Beach TAC, Howarth SJ, Callaghan JP. Muscular contribution to lowback loading and stiffness during standard and suspended push-ups. Hum Movement Sci, in press. D’Amico M, Ferrigno G. Technique for the evaluation of derivatives from noisy biomechanical displacement data using a model-based bandwidth-selection procedure. Med Biol Eng Comput 1990;28(5):407–15. Giakis G, Baltzopoulos V, Bartlett RM. Improved extrapolation techniques in recursive digital filtering: a comparison of least squares and prediction. J Biomech 1998;31(1):87–91.
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Pezzack JC, Norman RW, Winter DA. An assessment of derivative determining techniques used for motion analysis. J Biomech 1977;10(5–6):377–82. Robertson DGE, Dowling JJ. Design and responses of Butterworth and critically damped digital filters. J Electromyogr Kinesiol 2003;13(6):569–73. Smith G. Padding point extrapolation techniques for the digital Butterworth filter. J Biomech 1989;22(8-9):967–71. Vint PF, Hinrichs RN. Endpoint error in smoothing and differentiating raw kinematic data: an evaluation of four popular methods. J Biomech 1996;29(12):1637–42. Winter DA. Biomechanics and motor control of human movement. 2nd ed. Toronto: John Wiley and Sons; 1990.
Samuel J. Howarth received a B.Sc. in Kinesiology with a Joint Honours in Mathematics (2004) as well as an M.Sc. in Biomechanics (2006) from the University of Waterloo. He is currently a Ph.D. candidate in the Department of Kinesiology at the University of Waterloo. His research investigates how passive tissue damage in the lumbar spine under acute as well as repetitive shear loading is influenced by modulating factors such as posture, compressive load and shear load rate. Jack P. Callaghan received his Ph.D. in Kinesiology from the Faculty of Applied Health Sciences at the University of Waterloo in 1999. From 1998 to 2003, he was a faculty member in the Department of Human Biology at the University of Guelph. In 2003, he was awarded a Canada Research Chair in Spine Biomechanics and Injury Prevention and returned to the Kinesiology Department at the University of Waterloo. He has also received an Ontario Distinguished Researcher Award, and a Canada Foundation for Innovation infrastructure grant. He is a project leader in the AUTO21 Network of Centres of Excellence and an NSERC, CIHR and WSIB funded researcher. He is an author on 70 peer reviewed journal articles, has presented over 140 papers at conferences and supervised 20 graduate students. His main research interest is injury mechanisms from exposure to cumulative loading exposure.