Fast Fourier transformation (FFT) is a powerful analytical tool with wide-ranging applications in many fields. The standard FFT algorithms inherently assume that the length of the input and output sequence are equal. In practice, it is not always an accurate assumption. In certain case only some of the inputs to the transformation function are non-zero but lot of other are zero. In this thesis, a novel architecture of a 1024-point FFT, which adopts the transform decomposition (TD) algorithm, is presented to further reduce the complexity when the non-zero input data are consecutive. To implement this FFT processor, fixed point simulation is a conducted by using MATLB. The hardware implementation is realized by using the Verilog Hardware Description Language (HDL) which is taped out in TSMC0.18 Cell-Based Library for system verification.

We present a new method to stabilize the three-term AVO inversion using Curvelet and Wavelet transforms. Curvelets are basis functions that effectively represent otherwise smooth objects having discontinuities along smooth curves. The applied formalism explores them to make the most of the continuity along reflectors in seismic images. Combined with Wavelets, Curvelets are used to denoise the data by penalizing high frequencies and small contributions in the AVO-cube. This approach is based on the idea that rapid amplitude changes along the ray-parameter axis are most likely due to noise. The AVO-inverse problem is linearized, formulated and solved for all (x, z) at once. Using densities and velocities of the Marmousi model to define the fluctuations in the elastic properties, the performance of the proposed method is studied and compared with the smoothing along the ray-parameter direction only. We show that our method better approximates the true data after the denoising step, especially when noise level increases.

The wavelet transform technique has been frequently used in time-frequency analysis as a relatively new concept. Compared to the traditional technique Short-time Fourier Transform (STFT), which is theoretically based on the Fourier transform, the wavelet transform has its advantage on better locality in time and frequency domain, but not significant as the solutions in spectrum. Wavelet transform has dynamic ‘window functions’ to represent time-frequency positions of raw signals, and can get better resolutions in time-frequency analysis. In this report, we shall first briefly introduce fuzzy sets and related concepts. And then we will evaluate their similarities and differences by not only the theoretic comparisons between STFT and wavelet transform, but also the process of the de-nosing to a noisy recorded signal.

Yes / A new wavelet transform (WT) is introduced based on the fractional properties of the traditional Fourier transform. The new wavelet follows from the fractional Fourier order which uniquely identifies the representation of an input function in a fractional domain. It exploits the combined advantages of WT and fractional Fourier transform (FrFT). The transform permits the identification of a transformed function based on the fractional rotation in time-frequency plane. The fractional rotation is then used to identify individual fractional daughter wavelets. This study is, for convenience, limited to one-dimension. Approach for discussing two or more dimensions is shown.

The High Resolution Electrocardiogram (HRECG) is used to detect Ventricular Late Potentials (VLPs) in post-myocardial infarction patients. VLPs are low-amplitude, high-frequency signals that are usually found within the terminal part of the QRS complex. The aim of this research was to develop possible alternative methods and improve existing methods of detecting VLP activity. There are two main topics in this work: applications of the Continuous Wavelet Transform (CWT) and the Discrete Wavelet Transform (DWT) to the HRECG. For the CWT application, a Fractionation Factor (FF) method proposed by previous work was further investigated and improved by combining the CWT and DWT for distinction between patients with and those without VLPs. A Differential Fractionation Factor was proposed as an alternative approach to the FF with better results. Observation in the time-scale plot showed a difference in the energy distribution. A 2-dimensional Fractionation Factor was proposed to quantify this difference. A new concept of local intermittency was investigated to exhibit energy nonuniformity and then a Local Intermittency Factor was developed to quantify the degree of nonuniformity. The energy computed with the CWT was also used for patient distinction. Patients with VLPs may be also characterised by a slow rate of energy decay. The CWT can reveal a difference in ECG irregularity between the patients. A new approach of approximate entropy was implemented to quantify this irregularity. For the DWT application, the DWT can reveal irregularity of VLP activity and it was quantified by the approximate entropy to identify patients with VLPs. The wavelet entropy was utilised as an alternative method to the FF. The energy computed with the DWT was used for patient classification. The potentially promising results of both the CWT and DWT applications were obtained from the methods of computing the energy and approximate entropy

published_or_final_version / Physics / Doctoral / Doctor of Philosophy

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