Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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2.5. POWER SPECTRUM 23 2.5 Power Spectrum The power spectrum density (PSD) estimation C2(ω) (or Pxx(ω)) is defined as: C2(ω) = ∞ τ1=−∞ c2(τ) exp(−i(ωτ)) (2.31) where c2(τ) is the autocorrelation. Equation 2.31 is also known as the Wiener- Khintchine theorem (Nikias and Mendel, 1993). At this point it is important to clarify that a non parametric estimator of the power spectrum can be computed using the Discrete Fourier Transform (DFT). In this case, the autocorrelation is estimated from the data and the DFT is used to evaluate an estimator of Pxx(ω) from equation 2.31 . Parametric methods for Power spectrum estimation, on the other hand, operate by defining the power spectrum of the parametric model. In the case of AR process the power spectrum is given by PAR(z) = σ 2 ε |1 − ( p j=1)ajz j | 2 (2.32) where σ 2 ε variance of the innovation sequence of the AR model. If z = exp(−iω) then PAR(z) becomes PAP (ω). In general, I first compute the AR coefficients using one of the methods described in the preceding sections, then the coefficients are plugged in the formula for the power spectrum. The latter is the so called AR- spectrum, one form of parametric spectral analysis, often used because of its ability to produce smooth spectra from short time series. Figures 2.4(a)-(d) I compare the classical spectral density for the AO-a data computed with nonparametric analysis and AR analysis using Yule-Walker, Burg and the LS methods.
2.6. SUMMARY 24 Relative PSD Relative PSD 25 10 (a) 0 0 50 100 125 Sampling Frequency (Hz) 1 0.5 (c) 0 0 50 100 125 Sampling Frequency (Hz) Relative PSD Relative PSD 1 0.5 (b) 0 0 50 100 125 Sampling Frequency (Hz) 1 0.5 (d) 0 0 50 100 Sampling Frequency (Hz) Figure 2.5: PSD Estimation of Arctic Oscillation data for standardized nonlinear sea-level pressures. (a) Nonparametric DFT based PSD estimation. (b) PSD AR estimation using Yule-Walker equations . (c) PSD AR estimation using Burg’s algorithm. (d) PSD AR estimation using the first order Volterra series which is equivalent to a linear prediction . 2.6 Summary In this chapter, I have covered some of the theoretical and practical aspects of linear prediction. Computation of AR coefficients with different AR models has been ex- plored using Yule-Walker equations based on autocorrelation sequences solved with the Levinson algorithm which ensures a fast inversion of the Toeplitz matrix. In addition, Burg’s method was used to directly estimate AR coefficients with back-
Page 1 and 2: Name of Author: Soner Bekleric Univ
Page 3 and 4: University of Alberta Faculty of Gr
Page 5 and 6: Abstract Linear filter theory has p
Page 7 and 8: Contents 1 Introduction 1 1.1 Appli
Page 9 and 10: List of Tables 3.1 Filter length an
Page 11 and 12: 4.4 (a) Prediction of Figure 4.2(b)
Page 13 and 14: List of symbols Symbol Name or desc
Page 15 and 16: List of abbreviations Abbreviation
Page 17 and 18: In applied seismology predictive de
Page 19 and 20: 1.1. APPLICATIONS 4 applied to vide
Page 21 and 22: 1.3. THESIS OUTLINE 6 • Chapter 4
Page 23 and 24: 2.2. LINEAR PROCESS 8 Finally, I pr
Page 25 and 26: 2.3. LINEAR PREDICTION 10 E(z) 1 X(
Page 27 and 28: 2.3. LINEAR PREDICTION 12 In my the
Page 29 and 30: 2.3. LINEAR PREDICTION 14 which can
Page 31 and 32: 2.3. LINEAR PREDICTION 16 ∂ρ fb
Page 33 and 34: 2.3. LINEAR PREDICTION 18 I have an
Page 35 and 36: 2.4. 1-D SYNTHETIC AND REAL DATA EX
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Page 41 and 42: Chapter 3 Nonlinear Prediction 3.1
Page 43 and 44: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 45 and 46: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 47 and 48: 3.2. NONLINEAR MODELING OF TIME SER
Page 61 and 62: 3.4. SUMMARY 46 3.4 Summary In this
Page 63 and 64: 4.1. LINEAR PREDICTION IN THE F −
Page 65 and 66: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 67 and 68: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 69 and 70: 4.3. NONLINEAR PREDICTION OF COMPLE
Page 73 and 74: RMSE 2 4.3. NONLINEAR PREDICTION OF
Page 77 and 78: 4.4. NOISE REMOVAL AND VOLTERRA SER
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4.6. SUMMARY 74 4.6 Summary In this
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5.1. INTRODUCTION 76 ples remains a
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5.3. SYNTHETIC DATA EXAMPLES 78 Not
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5.3. SYNTHETIC DATA EXAMPLES 80 Tim
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5.3. SYNTHETIC DATA EXAMPLES 82 Tim
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5.4. REAL DATA EXAMPLES 84 Time [s]
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5.5. SUMMARY 90 5.5 Summary In this
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series in this thesis has been trun
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6.1. FUTURE DIRECTIONS 94 Noise red
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BIBLIOGRAPHY 96 Canales, L. L. (198
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BIBLIOGRAPHY 98 Marple, S. L. (1980
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BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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