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

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Original signal FOURIER TRANSFORM GABOR TRANSFORM WAVELET TRANSFORM Figure 13: Frequency and time-frequency methods. Fourier Transform is obtained by correlating the original signal with complex sinusoids of dierent frequencies (upper part). In Gabor Transform, the signal is correlated with modulated sinusoidal functions that slides upon the time axis, thus giving a time-frequency representation (middle part). Wavelets give an alternative time-scale representation but due to their varying window size, a better resolution for each scale is achieved (bottom part). Furthermore, the function to be correlated with the original signal can be chosen depending on the application (e.g. in the graph quadratic B-Splines wavelets are shown). 37
4.2 Theoretical Background 4.2.1 Continuous Wavelet Transform A wavelet family ab is a set of elemental functions generated by dilations and translations of a unique admissible mother wavelet (t): ab(t) = jaj ;1=2 t ; b where a b 2 R, a 6= 0 are the scale and translation parameters respectively. a (27) As a increases the wavelet becomes more narrow and by varying b, the mother wavelet is displaced in time. Thus, the wavelet family gives a unique pattern and its replicas at dierent scales and with variable localization in time. The continuous wavelet transform of a signal X(t) 2 L 2 (R) (nite energy signals) is dened as the correlation between the signal and the wavelet functions ab , i.e. (W X)(a b) = jaj ;1=2 Z 1 ;1 X(t) ( t ; b a ) dt = < X(t) ab > (28) where denotes complex conjugation. Then, the dierent correlations < X(t) ab > indicates how precisely the wavelet function locally ts the signal at every scale a. Since the correlation is made with dierent scales of a single function, instead of with complex sinusoids characterized by their frequencies, wavelets give a time-scale representation. 4.2.2 Dyadic Wavelet Transform The continuous Wavelet Transform maps a signal of one independent variable t onto a function of two independent variables a b. This procedure is redundant and not ecient for algorithm implementations. In consequence, it is more practical to dene the Wavelet Transform only at discrete scales a and discrete times b. One way toachieve this, is by choosing the discrete set of parameters fa j = 2 j b jk = 2 j kg, with j k 2 Z. Replacing in eq. 27 we obtain the discrete wavelet family jk(t) = 2 ;j=2 ( 2 ;j t ; k ) j k 2 Z (29) that forms a basis of the Hilbert space L 2 (R), and whose correlation with the signal is called Dyadic Wavelet Transform. 38
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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 38 and 39: 3 Gabor Transform (Short Time Fouri
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 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
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
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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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