Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Dysfunction in the central nervous system of the neonate is often first identified through seizures. The diffculty in detecting clinical seizures, which involves the observation of physical manifestations characteristic to newborn seizure, has placed greater emphasis on the detection of newborn electroencephalographic (EEG) seizure. The high incidence of newborn seizure has resulted in considerable mortality and morbidity rates in the neonate. Accurate and rapid diagnosis of neonatal seizure is essential for proper treatment and therapy. This has impelled researchers to investigate possible methods for the automatic detection of newborn EEG seizure. This thesis is focused on the development of algorithms for the automatic detection of newborn EEG seizure using adaptive time-frequency signal processing. The assessment of newborn EEG seizure detection algorithms requires large datasets of nonseizure and seizure EEG which are not always readily available and often hard to acquire. This has led to the proposition of realistic models of newborn EEG which can be used to create large datasets for the evaluation and comparison of newborn EEG seizure detection algorithms. In this thesis, we develop two simulation methods which produce synthetic newborn EEG background and seizure. The simulation methods use nonlinear and time-frequency signal processing techniques to allow for the demonstrated nonlinear and nonstationary characteristics of the newborn EEG. Atomic decomposition techniques incorporating redundant time-frequency dictionaries are exciting new signal processing methods which deliver adaptive signal representations or approximations. In this thesis we have investigated two prominent atomic decomposition techniques, matching pursuit and basis pursuit, for their possible use in an automatic seizure detection algorithm. In our investigation, it was shown that matching pursuit generally provided the sparsest (i.e. most compact) approximation for various real and synthetic signals over a wide range of signal approximation levels. For this reason, we chose MP as our preferred atomic decomposition technique for this thesis. A new measure, referred to as structural complexity, which quantifes the level or degree of correlation between signal structures and the decomposition dictionary was proposed. Using the change in structural complexity, a generic method of detecting changes in signal structure was proposed. This detection methodology was then applied to the newborn EEG for the detection of state transition (i.e. nonseizure to seizure state) in the EEG signal. To optimize the seizure detection process, we developed a time-frequency dictionary that is coherent with the newborn EEG seizure state based on the time-frequency analysis of the newborn EEG seizure. It was shown that using the new coherent time-frequency dictionary and the change in structural complexity, we can detect the transition from nonseizure to seizure states in synthetic and real newborn EEG. Repetitive spiking in the EEG is a classic feature of newborn EEG seizure. Therefore, the automatic detection of spikes can be fundamental in the detection of newborn EEG seizure. The capacity of two adaptive time-frequency signal processing techniques to detect spikes was investigated. It was shown that a relationship between the EEG epoch length and the number of repetitive spikes governs the ability of both matching pursuit and adaptive spectrogram in detecting repetitive spikes. However, it was demonstrated that the law was less restrictive forth eadaptive spectrogram and it was shown to outperform matching pursuit in detecting repetitive spikes. The method of adapting the window length associated with the adaptive spectrogram used in this thesis was the maximum correlation criterion. It was observed that for the time instants where signal spikes occurred, the optimal window lengths selected by the maximum correlation criterion were small. Therefore, spike detection directly from the adaptive window optimization method was demonstrated and also shown to outperform matching pursuit. An automatic newborn EEG seizure detection algorithm was proposed based on the detection of repetitive spikes using the adaptive window optimization method. The algorithm shows excellent performance with real EEG data. A comparison of the proposed algorithm with four well documented newborn EEG seizure detection algorithms is provided. The results of the comparison show that the proposed algorithm has significantly better performance than the existing algorithms (i.e. Our proposed algorithm achieved a good detection rate (GDR) of 94% and false detection rate (FDR) of 2.3% compared with the leading algorithm which only produced a GDR of 62% and FDR of 16%). In summary, the novel contribution of this thesis to the fields of time-frequency signal processing and biomedical engineering is the successful development and application of sophisticated algorithms based on adaptive time-frequency signal processing techniques to the solution of automatic newborn EEG seizure detection.

The quadratic class of time-frequency distributions (TFDs) forms a set of tools which allow to effectively extract important information from a nonstationary signal. To determine which TFD best represents the given signal, it is a common practice to visually compare different TFDs' time-frequency plots, and select as best the TFD with the most appealing plot. This visual comparison is not only subjective, but also difficult and unreliable especially when signal components are closely-spaced in the time-frequency plane. To objectively compare TFDs, a quantitative performance measure should be used. Several measures of concentration/complexity have been proposed in the literature. However, those measures by being derived with certain theoretical assumptions about TFDs are generally not suitable for the TFD selection problem encountered in practical applications. The non-existence of practically-valuable measures for TFDs' resolution comparison, and hence the non-existence of methodologies for the signal optimal TFD selection, has significantly limited the use of time-frequency tools in practice. In this thesis, by extending and complementing the concept of spectral resolution to the case of nonstationary signals, and by redefining the set of TFDs' properties desirable for practical applications, we define an objective measure to quantify the quality of TFDs. This local measure of TFDs' resolution performance combines all important signal time-varying parameters, along with TFDs' characteristics that influence their resolution. Methodologies for automatically selecting a TFD which best suits a given signal, including real-life signals, are also developed. The optimisation of the resolution performances of TFDs, by modifying their kernel filter parameters to enhance the TFDs' resolution capabilities, is an important prerequisite in satisfying any additional application-specific requirements by the TFDs. The resolution performance measure and the accompanying TFDs' comparison criteria allow to improve procedures for designing high-resolution quadratic TFDs for practical time-frequency analysis. The separable kernel TFDs, designed in this way, are shown to best resolve closely-spaced components for various classes of synthetic and real-life signals that we have analysed.