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Complex demodulation of visual evoked responses
446
TECHNICAL
Electroencephalography and Clinical Neurophysiology~ 1973, 34 : 446 447 :~ Elsevier Scientific Publishing Company, A m s t e r d a m - Printed in The Netherlands
CONTRIBUTION
COMPLEX D E M O D U L A T I O N OF VISUAL EVOKED RESPONSES D . G . CHILDERS Department o f Electrical Engineering, University o f Florida, Gainesville, Fla. 32601 ( U . S . A . )
(Accepted for publication:September 12, 1972)
The technique of complex demodulation has been described sufficiently in the literature (Bingham et al. 1967; Walter and Adey 1968 ; Walter 1969 ; Childers and Pao 1972 ; and Levine et al. 1972), and has recently been shown to be equivalent to several other signal-processing procedures (Childers and Pao 1972). While Levine et al. (1972) have analyzed data none is shown; Walter and Adey (1968) show results for EEG data. We offer here succinct results of complex demodulation applied to a s u m m a t e d (averaged) visual evoked response (VER) from a h u m a n subject. For several years one aspect of research in our laboratory has been directed toward the decomposition of composite signals such as that of the E E G and VERs. The decomposition of the VER into its separate components or wavelets might provide us with insight into the possible electrogenesis of these components, i.e., we might speculate more quantitatively on the possible locations of the anatomically disparate neuronal populations which give rise to the separate VER wavelets. We have investigated inverse filtering (Childers et al. 1970), adaptive filtering and the complex cepstrum (Kemerait and Childers 1972 : Senmoto and Childers 1972) and complex demodulation (Childers and Pao 1972) : all have been successfully applied to VER data. This note is intended to demonstrate that initiation of the after-discharge (or ringing) often seen in VERs can be detected, that its envelope and frequency of oscillation can be estimated and that an estimate of the duration of the afterdischarge can also be obtained, all by complex demodulation. For the mathematical details and other results, the reader is referred to Childers and Pao 1972. The data to be analyzed here are of a VER monitored by scalp electrodes over the occiput and evoked by visual stimulation. The data were digitized with 128 samples/sec. The VER appears in Fig. 1, a with its modified periodogram (Cooley et al. 1970) in Fig. 1, b. The wavelet of interest is the string of 3M oscillations beginning in the vicinity of 0.2~0.3 sec latency with a period of approximately 0.1 scc in Fig. 1, a : this wavelet is presumably the major contribution to the peak in the periodogram at approximately 10.5 c/see in Fig. 1, b. The 10.5 c/see estimate was derived from an examination of the computer listing of the amplitudes of the discrete frequency components and selecting that frequency component with the largest amplitude in this vicinity. This frequency is a first
estimate of the frequency of oscillation. We can extract the envelope and phase history of this wavelet by either high-pass or low-pass filtering (the spacing between output samples was 1/128 sec) following demodulation (Childers and Pao 1972) : they appear in Fig. I, e and 1, d, respectively for both high-pass (solid) and low-pass (dashed) filtering. The phase has an abrupt change at 0.28 sec and the envelope contains two peaks, one starting at 0.28 sec, thus, we estimate that there is a wavelet in the alpha frequency range near 10.5 c/sec which arrives (or is initiated) at 0.28 sec. The duration of the wavelet can be estimated from the phase curve by measuring the interval over which the phase is zero slope. This is approximately 0.4 sec. This can be confirmed from an examination of the envelope curve (Childers and Pao 1972). The details of estimating the frequency of oscillation and the other parameters appear in Childers and Pao (1972) where it is shown that in effect the final estimate of the frequency of oscillation is determined by that frequency of complex demodulation which nulls the phase curve. The first estimate, however, can be obtained from an examination of the periodogram as mentioned above. These same data have been previously processed by inverse filtering, Childers et al. (1970), where we estimated that there was a damped sinusoid present of approximately 10 c/see. We designed an appropriate inverse filter which estimated the arrival time of the wavelet to be 0.2 sec as compared with 0.28 sec found above. Of course, we do not know which, if either, is more nearly correct. kevine et al. (1972) discuss the limitations of analyzing imperfect data and the uncertainty of estimating time and frequency parameters. A related discussion appear in Childers and Pao (1972) along with references to additional sources of information on the basic uncertainty relationship between signal (wavelet) time duration and bandwidth. From our simulated examples and other VER data it appears that complex demodulation is a technique of value for decomposing EEGs and evoked responses which can improve upon the results obtained via fast Fourier transform (FFT) techniques by improving upon the frequency resolution within a frequency band: further, the algorithm for complex demodulation expands the F F T technique by giving estimates of arrival or event initiation time, cessation time, and thus an estimate of the duration of the event being studied.
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Fig. I. Complex demodulation of a VER by high-pass and low-pass filtering, a : VER. b : Peri0dogram. c: Recovered envelope for the 10.5 c/sec transient, high-pass filtering (solid), low-pass filtering (dashed). d: Estimated phase for the 10.5 c/sec transient. high-pass filtering (solid), low-pass filtering (dashed). SUMMARY Complex demodulation is applied to decompose the after-discharge of a visual evoked response. The initiation and cessation times of the after-discharge are estimated along with the envelope and frequency of oscillation. RESUME D E M O D U L A T I O N C O M P L E X E DES REPONSES EVOQUEES VISUELLES La ddmodulation complexe est appliqu6e pour d6composer la post-d6charge d'une r6ponse 6voqu6e visuelle. Les temps d'initiation et de cessation de la post-d6charge sont 6valu6s en m6me temps que l'enveloppe et la fr6quence de l'oscillation. This work was supported in part by U.S. Public Health Service Grant EY 00581-06 from the National Eye Institute of the National Institutes of Health. REFERENCES BINGHAM, C., GODFREY, M. D. and TUKEY, J. W. Modern techniques of power spectrum estimation IEEE Trans. Audio Electroacoustics, 1967, A U-15: 56-66. CltlLDERS, D. G. and PAO, M-T. Complex demodulation for transient wavelet detection and extraction IEEE Trans'.
Auctlo Electroacoustics, 1972, AU-20: 295 308. CHILDERS, D. G., VARGA, R. S. and PERRY JR., N. W. Composite signal decomposition IEEE Trans. Audio Electroacoustics, 1970, A U-18 : 471-477. COOLEY, J. W., LEWIS, P. A. W. and WELCH, P. O. The application of the fast Fourier transform algorithm to the estimation of spectra and cross spectra In J. Fox (Ed.). Computer processing in communications. Polytechnic Press, Brooklyn, 1970:5 20. KEMERAIT, R. C. and CHILDERS, D. G. Signal detection and extraction by cepstrum techniques IEEE Trans. hl/b. Theor., 1972, IT-18:745 759. LEV1NE, D. A., ELASHOFF, R., GALLOWAY, E., PAYNE, D. and JONES, R. T. Evoked potential analysis by complex demodulation Eleetroenceph. clin. Neurophysiol., 1972, 32:513 520. SENMOTO, S. and CHILDERS, D. G. Adaptive decomposition of a composite signal of identical u n k n o w n wavelets in noise. IEEE Trans. Sys. Man Cyber., 1972, SMC-2: 59 -66. WALTER, D. O. The method of complex demodulation, Appendix I1. In D. O. WALTER and M. A. B. BRAZlER (Eds.), Advances in EEG analysis, Electroenceph. clhE Neurophysiol., 1969, Suppl. 27:51 58. WALTER, D. O. and ADEY, W. R. Is the brain linear'? In A. S. IBERALL and J. B. RENWIClt (Eds.), Teehnieal and biological problems o[ control. Instrument Soc. Amer., Pittsburgh, 1968:11 22.
TECHNICAL
Electroencephalography and Clinical Neurophysiology~ 1973, 34 : 446 447 :~ Elsevier Scientific Publishing Company, A m s t e r d a m - Printed in The Netherlands
CONTRIBUTION
COMPLEX D E M O D U L A T I O N OF VISUAL EVOKED RESPONSES D . G . CHILDERS Department o f Electrical Engineering, University o f Florida, Gainesville, Fla. 32601 ( U . S . A . )
(Accepted for publication:September 12, 1972)
The technique of complex demodulation has been described sufficiently in the literature (Bingham et al. 1967; Walter and Adey 1968 ; Walter 1969 ; Childers and Pao 1972 ; and Levine et al. 1972), and has recently been shown to be equivalent to several other signal-processing procedures (Childers and Pao 1972). While Levine et al. (1972) have analyzed data none is shown; Walter and Adey (1968) show results for EEG data. We offer here succinct results of complex demodulation applied to a s u m m a t e d (averaged) visual evoked response (VER) from a h u m a n subject. For several years one aspect of research in our laboratory has been directed toward the decomposition of composite signals such as that of the E E G and VERs. The decomposition of the VER into its separate components or wavelets might provide us with insight into the possible electrogenesis of these components, i.e., we might speculate more quantitatively on the possible locations of the anatomically disparate neuronal populations which give rise to the separate VER wavelets. We have investigated inverse filtering (Childers et al. 1970), adaptive filtering and the complex cepstrum (Kemerait and Childers 1972 : Senmoto and Childers 1972) and complex demodulation (Childers and Pao 1972) : all have been successfully applied to VER data. This note is intended to demonstrate that initiation of the after-discharge (or ringing) often seen in VERs can be detected, that its envelope and frequency of oscillation can be estimated and that an estimate of the duration of the afterdischarge can also be obtained, all by complex demodulation. For the mathematical details and other results, the reader is referred to Childers and Pao 1972. The data to be analyzed here are of a VER monitored by scalp electrodes over the occiput and evoked by visual stimulation. The data were digitized with 128 samples/sec. The VER appears in Fig. 1, a with its modified periodogram (Cooley et al. 1970) in Fig. 1, b. The wavelet of interest is the string of 3M oscillations beginning in the vicinity of 0.2~0.3 sec latency with a period of approximately 0.1 scc in Fig. 1, a : this wavelet is presumably the major contribution to the peak in the periodogram at approximately 10.5 c/see in Fig. 1, b. The 10.5 c/see estimate was derived from an examination of the computer listing of the amplitudes of the discrete frequency components and selecting that frequency component with the largest amplitude in this vicinity. This frequency is a first
estimate of the frequency of oscillation. We can extract the envelope and phase history of this wavelet by either high-pass or low-pass filtering (the spacing between output samples was 1/128 sec) following demodulation (Childers and Pao 1972) : they appear in Fig. I, e and 1, d, respectively for both high-pass (solid) and low-pass (dashed) filtering. The phase has an abrupt change at 0.28 sec and the envelope contains two peaks, one starting at 0.28 sec, thus, we estimate that there is a wavelet in the alpha frequency range near 10.5 c/sec which arrives (or is initiated) at 0.28 sec. The duration of the wavelet can be estimated from the phase curve by measuring the interval over which the phase is zero slope. This is approximately 0.4 sec. This can be confirmed from an examination of the envelope curve (Childers and Pao 1972). The details of estimating the frequency of oscillation and the other parameters appear in Childers and Pao (1972) where it is shown that in effect the final estimate of the frequency of oscillation is determined by that frequency of complex demodulation which nulls the phase curve. The first estimate, however, can be obtained from an examination of the periodogram as mentioned above. These same data have been previously processed by inverse filtering, Childers et al. (1970), where we estimated that there was a damped sinusoid present of approximately 10 c/see. We designed an appropriate inverse filter which estimated the arrival time of the wavelet to be 0.2 sec as compared with 0.28 sec found above. Of course, we do not know which, if either, is more nearly correct. kevine et al. (1972) discuss the limitations of analyzing imperfect data and the uncertainty of estimating time and frequency parameters. A related discussion appear in Childers and Pao (1972) along with references to additional sources of information on the basic uncertainty relationship between signal (wavelet) time duration and bandwidth. From our simulated examples and other VER data it appears that complex demodulation is a technique of value for decomposing EEGs and evoked responses which can improve upon the results obtained via fast Fourier transform (FFT) techniques by improving upon the frequency resolution within a frequency band: further, the algorithm for complex demodulation expands the F F T technique by giving estimates of arrival or event initiation time, cessation time, and thus an estimate of the duration of the event being studied.
447
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Fig. I. Complex demodulation of a VER by high-pass and low-pass filtering, a : VER. b : Peri0dogram. c: Recovered envelope for the 10.5 c/sec transient, high-pass filtering (solid), low-pass filtering (dashed). d: Estimated phase for the 10.5 c/sec transient. high-pass filtering (solid), low-pass filtering (dashed). SUMMARY Complex demodulation is applied to decompose the after-discharge of a visual evoked response. The initiation and cessation times of the after-discharge are estimated along with the envelope and frequency of oscillation. RESUME D E M O D U L A T I O N C O M P L E X E DES REPONSES EVOQUEES VISUELLES La ddmodulation complexe est appliqu6e pour d6composer la post-d6charge d'une r6ponse 6voqu6e visuelle. Les temps d'initiation et de cessation de la post-d6charge sont 6valu6s en m6me temps que l'enveloppe et la fr6quence de l'oscillation. This work was supported in part by U.S. Public Health Service Grant EY 00581-06 from the National Eye Institute of the National Institutes of Health. REFERENCES BINGHAM, C., GODFREY, M. D. and TUKEY, J. W. Modern techniques of power spectrum estimation IEEE Trans. Audio Electroacoustics, 1967, A U-15: 56-66. CltlLDERS, D. G. and PAO, M-T. Complex demodulation for transient wavelet detection and extraction IEEE Trans'.
Auctlo Electroacoustics, 1972, AU-20: 295 308. CHILDERS, D. G., VARGA, R. S. and PERRY JR., N. W. Composite signal decomposition IEEE Trans. Audio Electroacoustics, 1970, A U-18 : 471-477. COOLEY, J. W., LEWIS, P. A. W. and WELCH, P. O. The application of the fast Fourier transform algorithm to the estimation of spectra and cross spectra In J. Fox (Ed.). Computer processing in communications. Polytechnic Press, Brooklyn, 1970:5 20. KEMERAIT, R. C. and CHILDERS, D. G. Signal detection and extraction by cepstrum techniques IEEE Trans. hl/b. Theor., 1972, IT-18:745 759. LEV1NE, D. A., ELASHOFF, R., GALLOWAY, E., PAYNE, D. and JONES, R. T. Evoked potential analysis by complex demodulation Eleetroenceph. clin. Neurophysiol., 1972, 32:513 520. SENMOTO, S. and CHILDERS, D. G. Adaptive decomposition of a composite signal of identical u n k n o w n wavelets in noise. IEEE Trans. Sys. Man Cyber., 1972, SMC-2: 59 -66. WALTER, D. O. The method of complex demodulation, Appendix I1. In D. O. WALTER and M. A. B. BRAZlER (Eds.), Advances in EEG analysis, Electroenceph. clhE Neurophysiol., 1969, Suppl. 27:51 58. WALTER, D. O. and ADEY, W. R. Is the brain linear'? In A. S. IBERALL and J. B. RENWIClt (Eds.), Teehnieal and biological problems o[ control. Instrument Soc. Amer., Pittsburgh, 1968:11 22.