Quantitative analysis of EEG signals: Time-frequency methods and ...

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3 Gabor Transform (Short Time Fourier Transform) 3.1 Introduction Since the Fourier Transform is based in comparing the signal with complex sinusoids that extend through the whole time domain, its main disadvantage is the lack of information about the time evolution of the frequencies. Then, if an alteration occurs at some time boundary, the whole Fourier spectrum will be aected, thus also needing the requirement of stationarity. In many occasions, signals have time varying features that can not be resolved with the Fourier Transform. A traditional example is a chirp signal (i.e. a signal characterized by awell dened but steadily rising frequency). In the case of a chirp, Fourier analysis can not dene the instantaneous frequency because it integrates over the whole time, thus giving a broad frequency spectrum. This is partially resolved by using the Gabor Transform, also called Short-Time Fourier Transform. With this approach, Fourier Transform is applied to time-evolving windows of a few seconds of data smoothed with an appropriate function (Cohen, 1995 Chui, 1992 Qian and Chen, 1996). Then, the evolution of the frequencies can be followed and the stationarity requirement is partially satised by considering the signals to be stationary in the order of the window length (Lopes da Silva, 1993a). In other words, the procedure consists in breaking the signal in small time segments and then in applying a Fourier Transform to the successive segments. One application of Gabor Transform is to the analysis of Tonic-Clonic (Grand Mal) seizures (see sec. x1.1.2). As we will see in this chapter, these seizures have interesting time evolving frequency characteristics that can not be resolved with methods suchasthe visual inspection of the EEG or the Fourier Transform (Quian Quiroga et al., 1997b Blanco et al., 1998b). Moreover, since it is dicult to extract any information from the traditional time-frequency graphic representation of the Gabor Transform, called spectrogram, I will introduce three parameters, the relative intensity ratio and the mean and maximum band frequencies, dened from the Gabor Transform, in order to obtain quantitative measures of the dynamics of epileptic seizures. After a brief theoretical outline, I will describe two dierent type of applications. First, the application to scalp recorded tonic-clonic seizures (Quian Quiroga et al., 1997b) and second, the application to intracranially recorded tonic-clonic seizures (Blanco et al., 1995b,1996a). 21
3.2 Theoretical background The Gabor transform of a signal x(t) is dened as follows: G D (f t) = Z 1 ;1 x(t 0 ) g D(t 0 ; t) e ;i2ft0 dt 0 (14) where denotes complex conjugation. Note that G D (f t) is the same as the Fourier Transform (eq. 2) but with the introduction of the window g D(t 0 ; t) of wide D and center in t. Furthermore, in analogy with the Fourier Transform, the Gabor Transform can be considered as an inner product between the signal and the complex sinusoidal functions e ;i2ft0 modulated by the window g D (compare with eq. 3), i.e. G D (f t) = (15) The window function g D is introduced in order to localize the Fourier Transform of the signal at time t. For this reason, the window function mustbepeaked around t and falling o rapidly, thus emphasizing the signal in the rst case and suppressing it for distant times. Several window functions can be used for achieving this, among them, Gabor (1946) proposed the use of a Gaussian function: g (t) = 1=4 e ; 2 t2 (16) Since the Fourier Transform of a Gaussian is again a Gaussian, this function allows a simultaneous localization in time and frequency (see appendix xA). It should be noted that the Gaussian is described in function of a positive constant and the window function in eq. 14 was dened in function of the window length D. This is because Gaussian functions do not have compact support (i.e. they are not 0 outside a certain boundary). However, they approach asymptotically to 0 with a rate determined by the parameter . In consequence, the Gaussian function can be truncated to have a length D, by assuming that in the borders their values are nearly 0 due to the use of a reasonable decay . Then, in the case of Gaussian windows the Gabor Transform can be explicitly dened as a function of the parameter G D (f t) ! G D(f t) (17) Let us return now to the general case of any arbitrary window function and consider the inverse transformation. Taking the inverse Fourier Transform of eq. 14 and after some algebraic manipulation we obtain: x(t) = 1 k g D k Z 1 Z 1 ;1 ;1 G D (f t 0 ) g D (t ; t 0 ) e i2ft df dt 0 (18) 22
Page 1 and 2: Quantitative analysis of EEG signal
Page 3 and 4: 1. Berichterstatter: Prof. Dr.-Ing.
Page 5 and 6: an okzipitalen Positionen und (c) d
Page 8 and 9: To my family: Mama, Consuelo, Hugui
Page 10 and 11: Preface In this work, I will descri
Page 12 and 13: I would certainly like to remember
Page 14 and 15: Contents Zusammenfassung Preface Ac
Page 16 and 17: 5.2.6 Calculating Lyapunov Exponent
Page 18 and 19: Summary Since the rsts recordings i
Page 20 and 21: 1 Outline of Neurophysiology: Brain
Page 22 and 23: Figure 2: Normal 20 channels (10-20
Page 24 and 25: 1.1.2 EEG in Epilepsy One important
Page 26 and 27: Figure 3: Scalp EEG of 18 channels
Page 28 and 29: Figure 4: Averaged responses upon N
Page 30 and 31: 2 Fourier Transform 2.1 Introductio
Page 32 and 33: I xx (k) =jX(k)j 2 = X(k) X (k) (
Page 34 and 35: 17 Figure 5: Topographical mapping
Page 36 and 37: 1987) is that coherence gives the c
Page 40 and 41: where k g D k= R 1 ;1 jg D(t 0 )j 2
Page 42 and 43: and the band maximum peak frequency
Page 44 and 45: Figure 7: Intracraneal seizure reco
Page 46 and 47: Figure 10: Mean (soft line) and max
Page 48 and 49: EEG as the one with least amount of
Page 50 and 51: Figure 11: Scalp EEG of the right c
Page 52 and 53: 3.5 Conclusion In this chapter I de
Page 54 and 55: Original signal FOURIER TRANSFORM G
Page 56 and 57: 4.2.3 Multiresolution Analysis Cont
Page 58 and 59: Original signal -15 0 15 -1sec 0 1s
Page 60 and 61: 4.2.5 Wavelet Packets In the approa
Page 62 and 63: marked decrease in computational ti
Page 64 and 65: ain activity during an epileptic se
Page 66 and 67: 3 2 3 3 3 3.2 Hz 3.6 Hz 4.0 Hz 4.4
Page 68 and 69: Silva et al.,1973a,1973b Lopes da S
Page 70 and 71: sweep #1 sweep #2 sweep #3 sweep #4
Page 72 and 73: VEP non-target target F3 F4 F3 F4 F
Page 74 and 75: VEP non-target target F3 F4 F3 F4 F
Page 76 and 77: Electrode F3 F4 Cz P3 P4 O1 O2 Dela
Page 78 and 79: In conclusion, our results point to
Page 80 and 81: 63 Figure 22: Gamma responses for a
Page 82 and 83: Figure 24: T-test comparison of the
Page 84 and 85: wavelet coecients allowed an easy d
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unknown topology it is necessary to
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the sum of the positive exponents (
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performed. On one hand, must be lar
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Duke (1992) and Elbert et al. (1994
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agation time. They applied this pro
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Figure 28: Histogram for the EEG da
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Figure 29: Attractors corresponding
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able for automatic detection proces
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ENTROPY Thermodynamics Signal analy
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6.3 Application to visual event-rel
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VEP Non Target Target F3 -20 -10 0
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1 VEP NON-TARGET TARGET 1 1 F3 F3 0
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1 VEP NON-TARGET TARGET 1 1 F3 F3 0
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and the Lorenz equations (a model o
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P300 deection. Due to the relation
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7 General Discussion In this chapte
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Wavelet analysis was also applied t
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aect the whole spectrum and for thi
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peaks. 7.2.3 Wavelet Transform vs.
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the ones having wide peaks (being t
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A Time-frequency resolution and the
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Z 1 ;1 tx(t) d dt x(t) dt = 1 2 Z 1
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Figure 37: Time-frequency resolutio
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References Abarbanel HDI, Brown R,
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Berger, H. Uber das Elektrenkephalo
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Gastaut H and Broughton R. Epilepti
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Lopes da Silva FH, van Lierop THMT,
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Pradhan N, Sadasivan P, Chatterji S
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Serrano E, Ph.D. Thesis, Department
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Biographical sketch 21.03.1967 born
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