Four contributions to the multitaper method of applied spectrum estimation are presented. These are a generalization of the multitaper method of spectrum estimation to time-series possessing irregularly spaced samples, a robust spectrum estimate suitable for cyclostationary, or quasi cyclostationary time-series, an improvement over the standard, multitaper spectrum estimates using quadratic inverse theory, and finally a method of scan-free spectrum estimation using a rotational shear-interferometer. Each of these topics forms a chapter in this thesis. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-02-05 18:01:45.187

Robust nonparametric spectral estimation includes generating an accurate estimate of the Power Spectral Density (PSD) for a given set of data while trying to minimize the bias due to data outliers. Robust nonparametric spectral estimation is applied in the domain of electrical communications and digital signal processing when a PSD estimate of the electromagnetic spectrum is desired (often for the goal of signal detection), and when the spectrum is also contaminated by Impulsive Noise (IN). Power Line Communication (PLC) is an example of a communication environment where IN is a concern because power lines were not designed with the intent to transmit communication signals. There are many different noise models used to statistically model different types of IN, but one popular model that has been used for PLC and various other applications is called the Middleton Class A model, and this model is extensively used in this thesis. The performances of two different nonparametric spectral estimation methods are analyzed in IN: the Welch method and the multitaper method. These estimators work well under the common assumption that the receiver noise is characterized by Additive White Gaussian Noise (AWGN). However, the performance degrades for both of these estimators when they are used for signal detection in IN environments. In this thesis basic robust estimation theory is used to modify the Welch and multitaper methods in order to increase their robustness, and it is shown that the signal detection capabilities in IN is improved when using the modified robust estimators. / Master of Science / One application of blind spectral estimation is blind signal detection. Unlike a car radio, where the radio is specifically designed to receive AM and PM radio waves, sometimes it is useful for a radio to be able to detect the presence of transmitted signals whose characteristics are not known ahead of time. Cognitive radio is one application where this capability is useful. Often signal detection is inhibited by Additive White Gaussian Noise (AWGN). This is analogous to trying to hear a friend speak (signal detection) in a room full of people talking (background AWGN). However, some noise environments are more impulsive in nature. Using the previous analogy, the background noise could be loud banging caused by machinery; the noise will not be as constant as the chatter of the crowd, but it will be much louder. When power lines are used as a medium for electromagnetic communication (instead of just sending power), it is called Power Line Communication (PLC), and PLC is a good example of a system where the noise environment is impulsive. In this thesis, methods used for blind spectral estimation are modified to work reliably (or robustly) for impulsive noise environments.

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Signals recorded from the brain often show rhythmic patterns at different frequencies, which are tightly coupled to the external stimuli as well as the internal state of the subject. These signals also have transient structures related to spiking or sudden onset of a stimulus, which have a duration not exceeding tens of milliseconds. Further, brain signals are highly non-stationary because both behavioral state and external stimuli can change over a short time scale. It is therefore essential to study brain signals using techniques that can represent both rhythmic and transient components of the signal. In Chapter 2, we describe a multi-scale decomposition technique based on an over-complete dictionary called matching pursuit (MP), and show that it is able to capture both sharp stimulus-onset transient and sustained gamma rhythm in local field potential recorded from the primary visual cortex. Gamma rhythm (30 to 80 Hz), often associated with high-level cortical functions, has been proposed to provide a temporal reference frame (“clock”) for spiking activity, for which it should have least center frequency variation and consistent phase for extended durations. However, recent studies have proposed that gamma occurs in short bursts and it cannot act as a reference. In Chapter 3, we propose another gamma duration estimator based on matching pursuit (MP) algorithm, which is tested with synthetic brain signals and found to be estimating the gamma duration efficiently. Applying this algorithm to real data from awake monkeys, we show that the median gamma duration is more than 330 ms, which could be long enough to support some cortical computations.