355
Electroencephalography and Clinical Neurophysiology, 1979, 4 6 : 3 5 5 - - 3 5 6 © Elsevier/North-Holland Scientific Publishers, Ltd.
Technical contribution THE APPLICATION OF LOW-PASS LINEAR FILTERS TO EVOKED POTENTIAL DATA: FILTERING WITHOUT PHASE DISTORTION D.G. WASTELL 1
Department of Psychology, University of Durham, Durham (England) (Accepted for publication: October 9, 1978)
Analogue filtering in order to eliminate high frequency noise from evoked potential (EP) data yields much 'cleaner' wave forms but, depending upon the bandpass of the filter (Desmedt et al. 1974), introduces phase errors, distorting the final wave form. Dawson and Doddington (1973) indicate that the solution to this problem is to use filters with zero phase shift, and they provide details of a family of suitable interference filters. This contribution describes a simple linear digital filter (Chatfield 1975) with this same desirable property, and illustrates the application of the technique to the filtering of the wave form of the average evoked potential. Essentially, the linear filtering of a wave form amounts to the 'smoothing out' of local fluctuations by the computation of a moving average. If x t represents the unfiltered value of the average EP at time t, then the filtered value, Yt, is given by: q
Yt =
~
akXt--k
k=--q
~k=±3 ~4k=±2 15
hk=
~k=±l 20
~-~ k = 0 0 otherwise
(1)
q
and H(co) =
~
h k e- i c~ k
k=--q
= (2 cos 3oj + 12 cos 2C0 + 30 cos Co + 20)/64
(2)
H(co) is generally a complex function of the form G(69) ei~(c°) where G(co) and ~b(co) are the gain and phase functions respectively. However, the transfer function of a symmetrical weighting function such as equation (1) happens to be real (equation 2) and so the phase is given by:
~(~) o =
where (ak~ are a set of weights and ~ a k = 1 In order to plot the gain and phase characteristics of such a filter, its transfer function, H(09), which is the Fourier transform of the weighting function, hk, is first computed. The weights employed here were generated by the expansion of the binomial expression (1/2 + 1]2) 2q with q = 3; and Yt, hk and H(co) were thus given by: Yt = (xt--3 d. 6 x t _ 2 + 15Xt_l + 20xt + 15Xt+l + 6xt+ 2 + xt+3)]64
1 Present address and address for reprint requests: MRC Applied Psychology Unit, 15 Chaucer Road, Cambridge CB2 2EF, England.
i.e., no phase distortion is introduced by the filter. The gain function for the h k of equation (1) is thus simply given by the expression for H(co) of equation (2) and is plotted in Fig. 1A. For a discrete process, such as the present average EP, the m a x i m u m frequency that can be resolved is 0.5 cycles, i.e. ~z radians, per unit time; and thus the abscissa of the gain diagram is limited to the frequency range 0 to ~'. For a sampling interval of At this m a x i m u m frequency, the so-called Nyquist frequency, is given by 1/2zXt. For the sampling rate of 5 msec per point used in the computation of the average EP of Fig. 1C, this means that the filtereffectively eliminates frequencies above 50 c/sec (?r/2), increasingly attenuates those above 15 c/sec and largely preserves those below 15 c/sec, i.e. the frequency range of the late components of the EP. Fig. 1B shows the gain functions of the filters based on the weights generated by the expansion of
356
D.G. W A S T E L L
FR ['~.
A
. ""',,, ... ,//"
B
÷--
]O0m$"
When the linear filter of Fig. 1A is applied to the unfiltered visual EP of Fig. 1C the 'clean' wave form of Fig. 1D is produced. The high f r e q u e n c y noise contaminating the unfiltered record, complicating comp o n e n t identification and m e a s u r e m e n t , is effectively abolished by the filter w i t h o u t significantly affecting the latencies and amplitudes of the EP c o m p o n e n t s . Linear filtering, using sets of weights based on the binomial coefficients associated with the expansion of (1/2 + 1/2) 2q, manifestly affords a simple and satisfactory t e c h n i q u e for s m o o t h i n g EP records witho u t distorting them.
Summary A linear filter, whose weights are based on the binomial coefficients, is described. It is shown that the phase distortion i n t r o d u c e d by the filter is zero and the application of the filter to s m o o t h i n g EP records is illustrated.
Rdsum~ Fig. 1. Gain functions of the filters based on the weights generated by the expansion of (1/2 + 1/2) 2q with q = 3 (A), 2, 4 and 7 (B). The gain f u n c t i o n of the simple m o v i n g average (SMA) described in the t e x t is also shown in B. C shows an unfiltered vertex visual EP o b t a i n e d in response to 100 L E D flashes. The time c o n s t a n t of the Grass 7P58 preamplifier was 0.1 sec and the high f r e q u e n c y c u t - o f f on the Grass 7 D A F driver amplifier was set at 500 Hz. The sampling rate was 5 msec per point. The filtered version o f C is presented in D.
Application de filtres phase-bas d l'enregistrement de potentiels dvoquds: filtrage sans distorsion de phase On d~crit un filtre lin~aire, d o n t les caract~ristiques de pond~ration sont 6tablies fi partir des coefficients du bin6me. On m o n t r e que la distorsion de phase ainsi introduite par le filtre est nulle. Par ailleurs, on applique le filtre au lissage d'enregistrements de potentiels ~voqu~s. This research was supported by a SRC studentship.
References (1/2 + 1/2) 2q with q -- 2, 4, 7, and it can be seen that the expansion is generating a family of curves of similar shape with the desirable characteristic of positive gain t h r o u g h o u t . In contrast, the gain f u n c t i o n o f the simple m o v i n g average given by:
k - - - - l , o, 1 hk
[ 0 otherwise
is also p l o t t e d and can be seen to go negative at high frequencies, which effectively a m o u n t s to the introd u c t i o n of a 180 ° phase shift.
Chatfield, C. The Analysis of T i m e Series: T h e o r y and Practice. C h a p m a n and Hall, L o n d o n , 1975. Dawson, W.W. and Doddington, H.W. Phase distortion of biological signals: e x t r a c t i o n of signal f r o m noise w i t h o u t phase error. Electroenceph. clin. Neurophysiol., 1973, 34: 207--211. Desmedt, J.E., Brunko, E., Debecker, J. and Carmeliet, J. The system handpass required to avoid distortion of early c o m p o n e n t s when averaging s o m a t o s e n s o r y evoked potentials. Electroenceph. clin. Neurophysiol., 1974, 37: 407--410. "
Electroencephalography and Clinical Neurophysiology, 1979, 4 6 : 3 5 5 - - 3 5 6 © Elsevier/North-Holland Scientific Publishers, Ltd.
Technical contribution THE APPLICATION OF LOW-PASS LINEAR FILTERS TO EVOKED POTENTIAL DATA: FILTERING WITHOUT PHASE DISTORTION D.G. WASTELL 1
Department of Psychology, University of Durham, Durham (England) (Accepted for publication: October 9, 1978)
Analogue filtering in order to eliminate high frequency noise from evoked potential (EP) data yields much 'cleaner' wave forms but, depending upon the bandpass of the filter (Desmedt et al. 1974), introduces phase errors, distorting the final wave form. Dawson and Doddington (1973) indicate that the solution to this problem is to use filters with zero phase shift, and they provide details of a family of suitable interference filters. This contribution describes a simple linear digital filter (Chatfield 1975) with this same desirable property, and illustrates the application of the technique to the filtering of the wave form of the average evoked potential. Essentially, the linear filtering of a wave form amounts to the 'smoothing out' of local fluctuations by the computation of a moving average. If x t represents the unfiltered value of the average EP at time t, then the filtered value, Yt, is given by: q
Yt =
~
akXt--k
k=--q
~k=±3 ~4k=±2 15
hk=
~k=±l 20
~-~ k = 0 0 otherwise
(1)
q
and H(co) =
~
h k e- i c~ k
k=--q
= (2 cos 3oj + 12 cos 2C0 + 30 cos Co + 20)/64
(2)
H(co) is generally a complex function of the form G(69) ei~(c°) where G(co) and ~b(co) are the gain and phase functions respectively. However, the transfer function of a symmetrical weighting function such as equation (1) happens to be real (equation 2) and so the phase is given by:
~(~) o =
where (ak~ are a set of weights and ~ a k = 1 In order to plot the gain and phase characteristics of such a filter, its transfer function, H(09), which is the Fourier transform of the weighting function, hk, is first computed. The weights employed here were generated by the expansion of the binomial expression (1/2 + 1]2) 2q with q = 3; and Yt, hk and H(co) were thus given by: Yt = (xt--3 d. 6 x t _ 2 + 15Xt_l + 20xt + 15Xt+l + 6xt+ 2 + xt+3)]64
1 Present address and address for reprint requests: MRC Applied Psychology Unit, 15 Chaucer Road, Cambridge CB2 2EF, England.
i.e., no phase distortion is introduced by the filter. The gain function for the h k of equation (1) is thus simply given by the expression for H(co) of equation (2) and is plotted in Fig. 1A. For a discrete process, such as the present average EP, the m a x i m u m frequency that can be resolved is 0.5 cycles, i.e. ~z radians, per unit time; and thus the abscissa of the gain diagram is limited to the frequency range 0 to ~'. For a sampling interval of At this m a x i m u m frequency, the so-called Nyquist frequency, is given by 1/2zXt. For the sampling rate of 5 msec per point used in the computation of the average EP of Fig. 1C, this means that the filtereffectively eliminates frequencies above 50 c/sec (?r/2), increasingly attenuates those above 15 c/sec and largely preserves those below 15 c/sec, i.e. the frequency range of the late components of the EP. Fig. 1B shows the gain functions of the filters based on the weights generated by the expansion of
356
D.G. W A S T E L L
FR ['~.
A
. ""',,, ... ,//"
B
÷--
]O0m$"
When the linear filter of Fig. 1A is applied to the unfiltered visual EP of Fig. 1C the 'clean' wave form of Fig. 1D is produced. The high f r e q u e n c y noise contaminating the unfiltered record, complicating comp o n e n t identification and m e a s u r e m e n t , is effectively abolished by the filter w i t h o u t significantly affecting the latencies and amplitudes of the EP c o m p o n e n t s . Linear filtering, using sets of weights based on the binomial coefficients associated with the expansion of (1/2 + 1/2) 2q, manifestly affords a simple and satisfactory t e c h n i q u e for s m o o t h i n g EP records witho u t distorting them.
Summary A linear filter, whose weights are based on the binomial coefficients, is described. It is shown that the phase distortion i n t r o d u c e d by the filter is zero and the application of the filter to s m o o t h i n g EP records is illustrated.
Rdsum~ Fig. 1. Gain functions of the filters based on the weights generated by the expansion of (1/2 + 1/2) 2q with q = 3 (A), 2, 4 and 7 (B). The gain f u n c t i o n of the simple m o v i n g average (SMA) described in the t e x t is also shown in B. C shows an unfiltered vertex visual EP o b t a i n e d in response to 100 L E D flashes. The time c o n s t a n t of the Grass 7P58 preamplifier was 0.1 sec and the high f r e q u e n c y c u t - o f f on the Grass 7 D A F driver amplifier was set at 500 Hz. The sampling rate was 5 msec per point. The filtered version o f C is presented in D.
Application de filtres phase-bas d l'enregistrement de potentiels dvoquds: filtrage sans distorsion de phase On d~crit un filtre lin~aire, d o n t les caract~ristiques de pond~ration sont 6tablies fi partir des coefficients du bin6me. On m o n t r e que la distorsion de phase ainsi introduite par le filtre est nulle. Par ailleurs, on applique le filtre au lissage d'enregistrements de potentiels ~voqu~s. This research was supported by a SRC studentship.
References (1/2 + 1/2) 2q with q -- 2, 4, 7, and it can be seen that the expansion is generating a family of curves of similar shape with the desirable characteristic of positive gain t h r o u g h o u t . In contrast, the gain f u n c t i o n o f the simple m o v i n g average given by:
k - - - - l , o, 1 hk
[ 0 otherwise
is also p l o t t e d and can be seen to go negative at high frequencies, which effectively a m o u n t s to the introd u c t i o n of a 180 ° phase shift.
Chatfield, C. The Analysis of T i m e Series: T h e o r y and Practice. C h a p m a n and Hall, L o n d o n , 1975. Dawson, W.W. and Doddington, H.W. Phase distortion of biological signals: e x t r a c t i o n of signal f r o m noise w i t h o u t phase error. Electroenceph. clin. Neurophysiol., 1973, 34: 207--211. Desmedt, J.E., Brunko, E., Debecker, J. and Carmeliet, J. The system handpass required to avoid distortion of early c o m p o n e n t s when averaging s o m a t o s e n s o r y evoked potentials. Electroenceph. clin. Neurophysiol., 1974, 37: 407--410. "
