Improved empirical wavelet transform for non-stationary signal analysis using fourier- bessel series expansion

Abstract

This dissertation presents an improved empirical wavelet transform (EWT) for the time-frequency (TF) representation of non-stationary signals. Though the EWT method has been shown its e ectiveness in some applications, it becomes di cult to analyze some non-stationary signals due to its improper segmentation in the frequency domain. Spectral analysis using the Fourier transform is a powerful tech- nique for stationary time series where the characteristics of the signal do not change with time. For non-stationary time series like modulated signals, the spectral content changes with time and hence time-averaged amplitude spectrum may be found using Fourier transform. There are serval TF domain based methods available for analy- sis of non-stationary signals namely short-time Fourier transform (STFT), wavelet transform(WT), Wigner-Ville distribution (WVD), and Hilbert-Huang transform (HHT). The features can be extracted in TF domain for the classi cation of non- stationary signals. There are other methods namely, tunable-Q wavelet transform (TWQT), empirical mode decomposition (EMD), and variational mode decomposi- tion (VMD). The conventional wavelet based method rely on pre- xed basis func- tions to analyze the signals, hence are considered to be rigid or non-adaptive. How- ever, the non-adaptive methods nd di culty in analyzing physical signals due to the existence of closely spaced frequency components. In this dissertation existing EWT has been enhanced using Fourier-Bessel series expansion (FBSE) in order to obtain improved TF representation of non-stationary signals. The FBSE uses Bessel functions as bases, which are non-stationary in nature. This makes FBSE suitable for analysis of signal with time-varying param- eters. There are certain advantages of the FBSE spectrum representation. It has been observed that FBSE spectrum has compact representation as compared to conventional Fourier representation. Secondly, FBSE spectrum avoids windowing for spectral representation. We have used the FBSE method for the spectral repre- sentation of the analyzed multi-component signals with good frequency resolution. The scale-space based boundary detection method has been applied for the accu- rate estimation of boundary frequencies in the FBSE based spectrum of the signal. After that, wavelet based lter banks have been generated in order to decompose non-stationary multi-component signals into narrow-band components. Finally, the normalized Hilbert transform has been applied to the estimation of amplitude en- velope and instantaneous frequency functions from the narrow-band components and the TF representation of the analyzed non-stationary signal is obtained. We have applied our proposed method for the TF representation of multi-component synthetic signals and real electroencephalogram (EEG) signals. The proposed method has provided better TF representation as compared to existing EWT method and HHTThis dissertation presents an improved empirical wavelet transform (EWT) for the time-frequency (TF) representation of non-stationary signals. Though the EWT method has been shown its e ectiveness in some applications, it becomes di cult to analyze some non-stationary signals due to its improper segmentation in the frequency domain. Spectral analysis using the Fourier transform is a powerful tech- nique for stationary time series where the characteristics of the signal do not change with time. For non-stationary time series like modulated signals, the spectral content changes with time and hence time-averaged amplitude spectrum may be found using Fourier transform. There are serval TF domain based methods available for analy- sis of non-stationary signals namely short-time Fourier transform (STFT), wavelet transform(WT), Wigner-Ville distribution (WVD), and Hilbert-Huang transform (HHT). The features can be extracted in TF domain for the classi cation of non- stationary signals. There are other methods namely, tunable-Q wavelet transform (TWQT), empirical mode decomposition (EMD), and variational mode decomposi- tion (VMD). The conventional wavelet based method rely on pre- xed basis func- tions to analyze the signals, hence are considered to be rigid or non-adaptive. How- ever, the non-adaptive methods nd di culty in analyzing physical signals due to the existence of closely spaced frequency components. In this dissertation existing EWT has been enhanced using Fourier-Bessel series expansion (FBSE) in order to obtain improved TF representation of non-stationary signals. The FBSE uses Bessel functions as bases, which are non-stationary in nature. This makes FBSE suitable for analysis of signal with time-varying param- eters. There are certain advantages of the FBSE spectrum representation. It has been observed that FBSE spectrum has compact representation as compared to conventional Fourier representation. Secondly, FBSE spectrum avoids windowing for spectral representation. We have used the FBSE method for the spectral repre- sentation of the analyzed multi-component signals with good frequency resolution. The scale-space based boundary detection method has been applied for the accu- rate estimation of boundary frequencies in the FBSE based spectrum of the signal. After that, wavelet based lter banks have been generated in order to decompose non-stationary multi-component signals into narrow-band components. Finally, the normalized Hilbert transform has been applied to the estimation of amplitude en- velope and instantaneous frequency functions from the narrow-band components and the TF representation of the analyzed non-stationary signal is obtained. We have applied our proposed method for the TF representation of multi-component synthetic signals and real electroencephalogram (EEG) signals. The proposed method has provided better TF representation as compared to existing EWT method and HHT method, especially when analyzed signal possesses closely spaced frequency components and of short time duration. method, especially when analyzed signal possesses closely spaced frequency components and of short time duration.