Analysis and Design of Morphological Filter In the Fractional Domain and Under Alpha Stable Distribution Environment for Signal Processing Applications

Abstract

Non-stationary signal analysis is a widely researched field in signal processing since we come across various non-stationary kind of signals such as human speech, music, bio-medical signals etc., in our day-to-day lives. Within the dynamic domain of signal processing, non-stationary signals challenge conventional analysis with their ever-changing characteristics. With the inefficiency of Fourier transform (FT) in analyzing such time-varying signals, the concept of time-frequency (TF) tools was formulated, revolutionizing non-stationary signal analysis by providing time-dependent power spectrum and simultaneous localization in both time and frequency domains. Impressed by the realm of TF tools, this research is focused on improving the analysis of very important non-stationary signal existing in nature—an electrocardiogram (ECG) signal, which is quasiperiodic and known to encompass a non-stationary nature. Therefore, research work in the thesis is centered around improving the performance of ECG analyzing system with a primary focus on its noise analysis model, denoising and QRS complex detection. ECG is considered a health biomarker and its one cardiac cycle comprises P, QRS and T waves. These components contain important clinical information that can be interpreted in different ways such as any change in their morphological pattern is an indication of cardiac arrhythmia, wave interval timings of QRS complex provide information about heart rate variability (HRV) and other features extracted from these beats can help in identifying various heart disorders. However during signal acquisition, ECG signal is distorted by various types of noise such as powerline interference, baseline drift and muscular artifacts, which have spectral characteristics that coincide with those of ECG signals. The variability in the physiology of QRS complex influences the morphology of ECG signal, alongside its susceptibility to noise. Thus, noise removal is an essential part of ECG processing, along with accurately delineating the QRS complex in designing computer-aided diagnosis tools to assist physicians and doctors in providing suitable medical interventions. Furthermore, several research methods employing time and frequency domain approaches have been proposed in the literature, but non-stationary characteristics of ECG signals hinder their effectiveness. Since TF tools have the virtue of dealing with non-stationarity, various TF tool-based methods have been documented in the literature for ECG denoising and detecting QRS complexes. However, getting a high reconstruction rate and better retaining of significant information is still an open problem. The work in this thesis utilizes the benefits of fractional Fourier transform (FrFT) to expand conventional time-frequency (TF) tools to time-fractional frequency domain with fractional order parameter 𝑎, thus addressing the challenges and ambiguities of time-varying morphology of ECG signal faced by existing state-of-the-art methods. Moreover taking a step backward, susceptibility to various noises and artifacts for an ECG signal is very high. Yet, established denoising methods always employ the additive white Gaussian noise model to compensate for such undesirable additions and use this noise assumption to validate their efficacy for reliable performance. Although, justification for using this Gaussian distribution is backed by the Central Limit Theorem (CLT), defined for random variables with finite variance. But what if variables possess infinite variance and thus do not have a Gaussian nature? Therefore, an improvement window exists for analyzing the noise analysis model, which brings us to our next research aim of investigating the nature of noises that corrupt an ECG signal. This investigation study for the noise model is segregated into three parts. The first part employs statistical tools to prove that the noises corrupting ECG signals deviate from Gaussianity. Second part focuses on proving the resemblance of ECG noises with non-Gaussian 𝛼-stable distribution model. Finally, negative impact on the performance of conventional methods (employing the implicit assumption of noises following Gaussian distribution) for R-peak detection and classification is illustrated by considering their functioning in real-time scenarios modelled by 𝛼-stable distribution model. After establishing the non-Gaussian nature of ECG noises, further research aims to design methods based on alternative noise analysis model. The study delves into the domain of morphological signal processing to thoroughly examine design techniques for morphological filters (MFs). However, it is observed that less attention is paid to the design of structuring elements (SE) while working towards designing MF in the literature. Thus, a design of novel SE is put forward by merging the concept of cross-convolution with FrFT. In the context of preprocessing ECG signals, it is imperative to ensure that noise suppression techniques do not result in losing vital information. Hence, utilization of proposed SE in conjunction with proposed morphological operation (MO) enables adaptation to the non-stationary changes in ECG signals and ensures the suppression of noises along with facilitating the preservation of distinctive attributes of ECG signals. Furthermore, as a major contribution to research work, a fractional lower-order fractional Stockwell transform (FLO-FrST) based on fractional lower-order statistics (FLOS) is proposed, which aims to resolve the shortcomings of existing TF tools by providing better 3Rs’, namely resolution, reconstruction and robustness. This tool provides four degrees of freedom, leading to superior performance, particularly in terms of 3Rs’ compared to its counterpart TF tools. Additionally, study includes an electroencephalogram (EEG) as another biomedical signal to demonstrate the effectiveness of proposed FLO-FrST in classifying epileptic activity. The scope of presented work is broadened by including these non-stationary EEG signals for analysis. The performance of proposed FLO-FrST system is validated through the utilization of various convolutional neural network (CNN) models, showcasing its ability to excel in diverse modeling scenarios and providing good performance. In a broader sense, presented study has proposed solutions for non-stationary signals under non-Gaussian environment. This study has enabled a new perspective to adopt noise model analysis, ranging from impulsive to Gaussian noise environments. Additionally, proposed study amalgamates the concepts from multidisciplinary signal processing domains and comes up with robust tools and methods to analyse non-stationary signals in non-Gaussian environment. For instance, with an amalgamation of FrFT with ST, flexibility in terms of degrees of freedom has been added over traditional methods in processing ECG signals, putting together design methods of SE with cross-convolution window concept and FrFT has provided shape adaptability to classical morphological filtering, and by bringing together concepts of FrFT and ST with FLOS has provided fractional frequency domain support to existing fractional lower-order time-frequency (FLO-TF) approaches by improving over resolution, reconstruction and robustness for noise environments ranging from impulsive to Gaussian. Finally, this opens up new horizons and calls for more focus on research, opting for alternate noise analysis models to overcome performance shortcomings in conventional methods and exploring the applications of proposed tools in other applications of signal processing as well.