Journal of Medical Systems, Vol. 24, No. 4, 2000

Application of Periodogram and AR Spectral Analysis toEEG Signals

M. Akin1 and M. Kemal Kiymik2,3

In this study, in order to analyze the EEG signal, the conventional and modernspectral methods were investigated. Interpretation and performance of these methodswere detected for clinical applications. For this purpose EEG data obtained fromdifferent persons were processed by PC computer using periodogram and AR modelalgorithms. Periodogram and AR modeling approaches were compared for theirresolution and interpretation performance. It was determined that the AR approachis better for the use in clinical and research areas, because of the clear spectra thatare obtained by it.

KEY WORDS: periodogram; spectral analysis; EEG.

INTRODUCTION

EEG signals involve a great deal of information about the function of thebrain. For the use of this information in medical diagnoses, several studies havebeen carried out. However, although the function of EEG equipment has diminishedin imaging pathological syndromes in brain through the use of brain tomography andmagnetic resonance monitoring equipment, EEG equipment keeps its importance inroutine clinical diagnoses because of being economical and especially because itidentifies the epileptic discharges. Among several analysis methods, especially spec-tral analysis methods are important because the frequencies and characteristics ofbrain waveform change depending on the brain function affected from disordersand physiological state. There are many applications of FFT, which are present. FFTis the base of conventional spectral analysis methods. However, several problemsresulting from the property of this method are known. For instance, these problemsincrease in the low frequency samples which are not present in the original signal,and necessitates the use of windowing for decreasing the error rate. To reach a

1Department of Electrical Engineering of Dicle University, Diyarbakir/Turkey.2Department of Electric and Electronic Engineering of Kahramanmaras S. University, 46100Kahramanmaras/Turkey.

3To whom correspondence should be addressed.

247

0148-5598/00/0800-0247$18.00/0 2000 Plenum Publishing Corporation

248 Akin and Kiymik

definite decision the specialist must observe frequency contents of the signal prop-erly and clearly. Also, this depends on the quality of the spectrum signals.

In this study, conventional and modern spectral analysis methods are compared,and the method by which the highest rightness and neatness rate are obtained andthe doctors can use in diagnostic studies is determined. To that end, real EEG signals,and those which were taken from many patients by using a personal computer, andsoftware were evaluated. The spectrums of signals were obtained by applying theperiodogram and AR modeling to one simulated and two real EEG signals that wereobtained from patients at Medicine Faculty Dicle University. The most appropriatemethod, which is used in routine diagnosis and studies, was determined by detectingthese spectra.1, 2 In previous studies original EEG signals were sampled at 200 Hzand spectral analysis methods were applied by Isaksson, et al.4 In this study ARand ARMA modeling were used. It is claimed that using the spectral parameteranalysis variables, (f � peak frequency, � � band width, and P � power) insteadof activity coefficients such as �, �, �, and � in parametric analysis has moreadvantage. When AR and ARMA modeling were compared, it was observed thatthe results were closely related. Only from the viewpoint of the operation process,AR modeling was determined to have more advantages. The studies on this subjectare given in references.3–5,8,9

METHOD

The Analysis of EEG Signals

The most widespread analysis method applied to signals is spectral analysis,which is used to detect the EEG signals with visual methods and identify the disease.For the application of these analysis methods, EEG signals at that time domainare sampled at an appropriate frequency. Sampled signals are grouped as framesthat contain evident sample numbers. The most widespread lengths of frames are64, 128, and 256. Then, the power spectral density p(f) is found for each windowby applying conventional and modern spectral analysis methods such as FFT, AR,MA, and ARMA, respectively. A three-dimensional spectrum graphic shows thevarying spectrum of EEG signal according to time by the function of this spectrumbeing arranged on time axis. The best appropriate length of the used frame dependson the steadiness of signal and sampling frequency. In this study, frame length istaken as 64, because sampling frequency is low and the number of samples are fewas well. FFT and conventional spectral analysis methods like FFT give sufficientresults when the length of frame time is long. But it is reported in literature thatmodern spectral methods like AR, MA, and ARMA modeling give better resultsthan FFT, especially where the number of samples are fewer.6 In the AR method,the modeling degree is identified according to different criteria. In this study, AIC(Akaike Information Criteria) was taken as the base. However, as a result ofscanning the references, modeling degree p � 10 was taken because the determinedmodeling degree was lower.

Application of Periodogram 249

PERIODOGRAM METHOD

The periodogram method used for determining the power density of the fre-quency components in a signal is based on Fourier conversion. Fourier series canrepresent a periodic signal and again a waveform can be obtained by using Fouriercoefficients. For obtaining the power spectrum of an EEG signal with periodogrammethod, EEG signals were divided into frames 64, 128, 256, at the power of 2. Toincrease the resolution, zero-padding process was performed, because the resolutionof FFT is inverse at the time when the sampled data were presented.

The signal is windowed with a proper window since it’s impossible to workwith an infinite lengthened signal practically. Several windows have been developedwith studies carried out, and the best appropriate windows for the used data, andinformation that were necessary to obtain have been suggested. When the biomedi-cal signals are examined, especially better appropriate windows are rectangular andHanning windows on the spectral analysis of this signal.7

The Discrete Fourier Transformation of a discrete time X[n] signal is definedas follows.

X �m� � �N�1

n�0x[n] exp ��j2�mn

N � m � 0, 1, 2, 3, .................. N � 1 (1)

To find of the power spectrum of this signal, the following equation is used.

P�

per( fm) �1

N�t�X[m]2 �t �

1fs

(2)

Fig. 1. Epileptic signal (EEG 1) taken from a 1.5-year-old patient.

250 Akin and Kiymik

Fig. 2. Comparation of periodogram and AR spectrum of the EEG1 signal.

Where, N, �t, and fs show estimated value, sample numbers in signal, samplingperiod, and sampling frequency, respectively.

AR MODELING METHOD

In this study, the AR parametric modeling method was used. The parametricalspectral analysis method may be summarized with a sequence of parameter aspower spectrum, peak frequency, and bandwidth or power contents. These methodsare also called parametrical analysis methods. FFT processes are applied to eitherwindowed data or windowed value estimations in unparametrical methods. Whileperforming the operations, it causes the values that remain out of the window tobe zero. Such a condition is not probable in parametric modeling.

The properties of the signal should be taken into consideration for an appro-priate modeling of the signal. For instance, AR modeling is proper for a signalcontaining sudden peaks in frequency spectrum. Inversely, Moving Average (MA)is used for signals that have no sharp peaks. But the Autoregressive Moving Average

Application of Periodogram 251

Fig. 3. Representation of 3D EEG 1 signal. (a) Periodogram spectrum. (b) AR spectrum.

252 Akin and Kiymik

Fig. 4. Normal signal (EEG 2) taken from a 29-year-old healthyperson.

can be used for the two signals. AR and ARMA are preferred in EEG signalsbecause their structure consists of peaks discrete frequency intervals. In this study,The application of AR modeling methods application was carried out, because ARhas more advantages than ARMA in view of process numbers.

In the AR modeling method, the amplitude of a signal at a given period isobtained by summing up the different amplitudes of previous samples, and addingthe estimation error. AR process from the order p is defined with the following defi-nition.

x(n) � � �p

m�1amx(n � m) � e(n) (3)

Where x(n), am, and e(n) define the sequence of samples, modeling coefficients,and error term which is white noise, respectively. Mode 1 contains p � 2 parameter,which is estimated from data. These are coefficients, expected values of samples,and variance of white noise. The estimation problem of these parameters coversthe formation and resolution of the set of equations, which can be easily calculated.AR spectral estimation of data’s estimation from the order p was given with thefollowing equation.8

P(f) ��p2�t

�1 � �p

m�1apme�j2� fm�t�2

(4)

Where ap0 � 1. Thus, for estimation of spectral power density only the existenceof p number apm parameters, and �2

p parameters, which are the variances of whitenoise, are sufficient. AR coefficients, which identify the amplitude rates, can be

Application of Periodogram 253

Fig. 5. Comparation of periodogram and AR spectrum of the EEG2 signal.

calculated by using several methods. Levinsion–Durbin and Burg algorithms areamong these methods. In the Levinsion–Durbin algorithm, AR coefficients arefound out by solving the Yule–Walker equations. Eigen relationships are usedwhen this operation is carried out. The Levinson method is useful for the real timeapplications. However, in the Burg method, AR coefficients are found out withadvanced reverse errors, which depend on samples taken from the signal. The orderof the model, namely, the filter, is identified with the number of AR coefficients.5

The initial values of the algorithm calculate the parameters of an AR process ofan order p

a11 � �Rxx (1)/Rxx (0) (5)

� 21 � 1 � a2

11Rxx (0) (6)

is chosen as above. Then, for k � 2,3,4,..........

254 Akin and Kiymik

Fig. 6. Representation of 3D EEG 2 signal. (a) Periodogram spectrum. (b) AR spectrum.

Application of Periodogram 255

akk � �

Rxx(k) � �k�1

m�1ak�1,mRxx(k � m)

� 2k�1

(7)

aki � ak�1,i � akkak�1,k�i i � 1,2,3,....... k � 1 (8)

� 2k � 1 � a2

kk� 2k�1 (9)

Where Rxx(k) is the estimation of the eigen relationship function of the process.When the processes are finished

ai � api i � 1,2,3,........ p (10)

�2 � � 2p (11)

the real AR model parameters are determined by means of a expression above.7

To choose the order of modeling is very important in the AR method. Definitepeaks do not exist when the chosen degree is low. That is, the contents of thesignal’s frequency is not defined. When the order of model is very high, misleadingand wrong peaks occur and spectra degenerate.3 In literature, several criteria aresuggested to define the modeling order.6 AIC, offered by AKAIKE (Akaike Infor-mation Criteria), FPE (Final Prediction Error), and CAT function (Auto RegressiveCriteria) are among these criteria.6 Use of these critera varies according to signaland aim. The use of the Akaike information criterion has been suggested for thesimulated data.5, 6 Also due to the fact that AIC criterion is advised to be used todefine the model order when the model degree to be used is lower than �N, inthis study, AIC criteria was taken as the base, and the order of model (p) waschosen between 10 and 15.

EXPERIMENTAL RESULTS AND DISCUSSION

The signals used in this study are real EEG signals and are recorded in Neurol-ogy Laboratory D.U. Records are made a long with a data-collecting unit developedin the previous study.1 The sampling frequency of the signal recorded during 6 secis 50 Hz. The periodogram and AR spectrum of these real EEG signals wereobtained by using Turbo-C software. Frame length was chosen as 64 when thespectra are obtained, because the sampling frequency was low. First, periodogramand AR spectra obtained for each frame were drawn from the same frequency axisand the comparison of the spectra was simplified. Then three-dimensional drawingexisting with time, frequency, and power density for seeing the time-varying periodo-gram and AR spectra collectively make the use of treatment possible.

Figure 1 is the changing of an epileptic EEG signal taken from a 1.5-year-oldpatient with respect to time. The variations of the periodogram spectra are muchmore in all the frames when the AR and periodogram of the signal are comparedin Fig. 2. The excessiveness of number of depictive peaks is seen although thereare definite peaks in the same frequencies. In the first frame of the AR spectrum,peaks are observed in the neighborhood of 4 Hz, 9.5 Hz, and 14 Hz, and also in

256 Akin and Kiymik

the second frame a top between 0 and 5 Hz is seen. In the third frame AR spectrumpeaks are observed in the 2 Hz and 14 Hz. Namely, the variation of periodogramspectrum is excessive, and it only has a more definite peak between 11 and 12 Hzthan the others. In Fig. 3 (a) the periodogram spectrum variation of the EEG signalwith respect to time is seen and also in Fig. 3 (b) the time-varying of AR spectrumis observed. When the two figures are compared, explaining the AR spectrum,which has more definite peaks, and defining the activities in the signal are easy.However, the explaining of the periodogram spectrum is more difficult because ithas more depictive peaks.

In Fig. 4 the EEG signals taken from a 29-year-old healthy person are seen.When the comparison given in Fig. 5 is detected, the existence of depictive peaksin the periodogram spectrum is observed. In the first two frames of the AR spectrumthere are definite peaks at 0 and 10 Hz. However, in the third, peaks are at 0.5,11.5, and 16 Hz. Although periodogram spectrum has peaks at the same points,because of the depictive peaks being closure, an absolute decision shouldn’t begiven. Also in Fig. 6, the three-dimensional graphics of the frames of both of thetwo spectra placed to the time axis is observed.

As a result when the Periodogram and AR modeling approach are comparedaccording to their resolution and interpretation performance, it is determined thatthe AR approach is better for the use in clinical and research areas, because of theclear spectra, which are obtained by it.

REFERENCES

1. Akin, M. Investigation of excited brain potential with spectral analysis methods. PhD. Thesis,University of Erciyes, Turkey, 1995.

2. Inouye, T., Analysis rapidly changing EEG’s before generalized spike and wave complexes. Electro-encephalography and Clinical Neurophsiology, 75:204–210, 1990.

3. Birch, G.E., and Lawrence, P.D., Application of prewhitening to AR spectral estimation of EEG.IEEE Transaction on Biomedical Eng., BME-35/8, 640–646, 1988.

4. Isaksson, A., Wennberg, A., and Zetterberg, L.H., Computer analysis of EEG signals with 4parametric models. Proceedings of the IEEE 69/4:451–461, 1981.

5. Kay, S.M., and Marple, S.L., Spectrum analysis—A modern perspective. Proceedings of the IEEE69/11:1380–1419, 1981.

6. Burshtein, D., and Weinstein, E., Some relations between the various criteria for AR modelorderdetermination. IEEE Transaction on Ac. Speech and Signal Processing ASSP-34/4:1017–1019, 1985.

7. Proakis, J., and Manolakis, D., Digital Signal Processing Principles, Algorithms, and Applications.Prentice Hall, NJ, 1996.

8. Guler, N.F., Kiymik, M.K., and Guler, I., Comparison of FFT and AR-based sonogram outputs of20 MHz pulsed Doppler Data in real Time. J. Computers Biol. Med. 25:383–391, 1995.

9. Roessgen, M., Zoubir, M., and Boashash, B., Seizure detection of newborn EEG using a modelapproach. IEEE Transactions on Biomedical Eng. IEEE 45:673–685, 1998.