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

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Z 1 ;1 tx(t) d dt x(t) dt = 1 2 Z 1 ;1 t d dt x2 (t) dt = tx2 (t) 2 +1 ;1 ; 1 2 Z 1 ;1 x 2 (t) dt = ; 1 2 replacing in eq. 68 and then in eq. 67, the uncertainty principle is proved. A.3 Time-frequency resolution of the Fourier, Gabor and Wavelet Transform Fourier Transform is based in making a correlation between the original signal and complex sinusoidal functions (see eq. 3). Since the Fourier Transform of these functions is: x(t) =e i! 0t ! X(!) =2 (! ; ! 0 ) (69) then, the sinusoidal \mother functions" of the Fourier Transform are perfectly localized in the frequency domain. However, they are not localized in time. As stated in sec. x3.2, Gabor Transform consist in making a correlation between the signal and amplitude modulated complex sinusoids, by using a windowing function. Gabor (1946) suggested the use of a Gaussian function as window due to its simultaneous localization in time and frequency domains. In fact, the Fourier Transform of a Gaussian function is also a Gaussian: r x(t) =g (t) =e ;t2 ! X(!) = e ; !2 4 (70) Moreover, for a Gaussian function the inequality 68isanequality and ! t = 1 2 ,thus having Gaussian functions the best time-frequency resolution allowed by the uncertainty principle. In the case of the mother functions of the Gabor Transform, i.e. g e ;i!t where g is the Gaussian windowing function, the equality in eq. 68 also holds, thus having an optimal time-frequency resolution. The frequency resolution (bandwidth) of a normalized Gaussian function can be calculated by using eq. 63: 2 ! = 2 (71) and in the case of the functions of the Gabor Transform, g e ;i!t , by applying eq. 59 and eq. 63 we obtain for one of these mother functions (! = ! 0 ) the same bandwidth but localized in the frequency ! 0 , i.e.: 113
Figure 36: Time-frequency resolution of the Gabor Transform. = ! 0 2 ! = 2 (72) Then, in the case of Gabor Transform, the frequency resolution depends on the window wide, corresponding to small values of (wide windows see eq. 16) a better frequency resolution (but a worst time resolution see also discussion of the window sizein sec. x3.2). analyzing The area determined by the frequency window multiplied by the time window is called time-frequency window. Plotting of the time-frequency window is an easy way to visualize the time-frequency resolution: its wide represents the time resolution, its height represents the frequency resolution and its area represents the time-frequency resolution, this last one having alower bound determined by the uncertainty principle. In the case of the Gabor Transform, once the window is xed, the frequency resolution is the same for all the frequencies (and therefore the time resolution too see g. 36). Then, for low frequency signals, a wide window will be suitable and in the case of high frequencies, a narrow window will do the job. However, due to its xed window size, Gabor Transform is not suitable for analyzing signals with dierent ranges of frequencies. On the other hand, in the case of the Wavelet Transform, the size of the window is adapted, thus giving an optimal resolution for all frequencies (see g. 37). The Wavelet Transform consists in making a correlation between the original signal 114
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Quantitative analysis of EEG signal
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1. Berichterstatter: Prof. Dr.-Ing.
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an okzipitalen Positionen und (c) d
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To my family: Mama, Consuelo, Hugui
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Preface In this work, I will descri
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I would certainly like to remember
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Contents Zusammenfassung Preface Ac
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5.2.6 Calculating Lyapunov Exponent
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Summary Since the rsts recordings i
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1 Outline of Neurophysiology: Brain
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Figure 2: Normal 20 channels (10-20
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1.1.2 EEG in Epilepsy One important
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Figure 3: Scalp EEG of 18 channels
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Figure 4: Averaged responses upon N
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2 Fourier Transform 2.1 Introductio
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I xx (k) =jX(k)j 2 = X(k) X (k) (
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17 Figure 5: Topographical mapping
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1987) is that coherence gives the c
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3 Gabor Transform (Short Time Fouri
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where k g D k= R 1 ;1 jg D(t 0 )j 2
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and the band maximum peak frequency
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Figure 7: Intracraneal seizure reco
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Figure 10: Mean (soft line) and max
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EEG as the one with least amount of
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Figure 11: Scalp EEG of the right c
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3.5 Conclusion In this chapter I de
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Original signal FOURIER TRANSFORM G
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4.2.3 Multiresolution Analysis Cont
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Original signal -15 0 15 -1sec 0 1s
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4.2.5 Wavelet Packets In the approa
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marked decrease in computational ti
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ain activity during an epileptic se
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3 2 3 3 3 3.2 Hz 3.6 Hz 4.0 Hz 4.4
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Silva et al.,1973a,1973b Lopes da S
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sweep #1 sweep #2 sweep #3 sweep #4
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VEP non-target target F3 F4 F3 F4 F
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VEP non-target target F3 F4 F3 F4 F
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Electrode F3 F4 Cz P3 P4 O1 O2 Dela
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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
Page 86 and 87: the position vs. the linear momentu
Page 88 and 89: unknown topology it is necessary to
Page 90 and 91: the sum of the positive exponents (
Page 92 and 93: performed. On one hand, must be lar
Page 94 and 95: Duke (1992) and Elbert et al. (1994
Page 96 and 97: agation time. They applied this pro
Page 98 and 99: Figure 28: Histogram for the EEG da
Page 100 and 101: Figure 29: Attractors corresponding
Page 102 and 103: able for automatic detection proces
Page 104 and 105: ENTROPY Thermodynamics Signal analy
Page 106 and 107: 6.3 Application to visual event-rel
Page 108 and 109: VEP Non Target Target F3 -20 -10 0
Page 110 and 111: 1 VEP NON-TARGET TARGET 1 1 F3 F3 0
Page 112 and 113: 1 VEP NON-TARGET TARGET 1 1 F3 F3 0
Page 114 and 115: and the Lorenz equations (a model o
Page 116 and 117: P300 deection. Due to the relation
Page 118 and 119: 7 General Discussion In this chapte
Page 120 and 121: Wavelet analysis was also applied t
Page 122 and 123: aect the whole spectrum and for thi
Page 124 and 125: peaks. 7.2.3 Wavelet Transform vs.
Page 126 and 127: the ones having wide peaks (being t
Page 128 and 129: A Time-frequency resolution and the
Page 132 and 133: Figure 37: Time-frequency resolutio
Page 134 and 135: References Abarbanel HDI, Brown R,
Page 136 and 137: Berger, H. Uber das Elektrenkephalo
Page 138 and 139: Gastaut H and Broughton R. Epilepti
Page 140 and 141: Lopes da Silva FH, van Lierop THMT,
Page 142 and 143: Pradhan N, Sadasivan P, Chatterji S
Page 144 and 145: Serrano E, Ph.D. Thesis, Department
Page 146: Biographical sketch 21.03.1967 born
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