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In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a path algebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n2n!) time. We also discuss several future directions for this research.

A discrete Fourier transform, or DFT, is an isomorphism from a group algebra to a direct sum of matrix algebras. An algorithm that efficiently applies a DFT is called a fast Fourier transform, or FFT. The concept of a DFT will be introduced and examined from both a general and algebraic perspective. We will then present and analyze a specific FFT for the symmetric group.

This project estimates interwell connectivity, a characteristic that is crucial to determine reservoir continuity while developing a waterflooding project. It tests the combination of Fourier transforms (FT’s) of the flow rate data and analytical solutions from analog electrical circuits to infer the inverse diffusivity coefficient (IDC). I solved the transmission line equation analytically for 0D, 1D, and 2D resistance/capacitance (RC) network models and used those solutions to compare with the flow rate FT’s to determine the diffusivity parameters. I used the analogy between the electrical response of RC networks and the fluid response of permeable reservoirs on the basis of the similarities in the governing equations. I conclude that the analogy works accurately in simple reservoirs, where the assumptions of an analytical solution are met, i.e. single-phase fluid and a homogeneous system. For two-phase liquid cases, I determined that the analogy remains applicable because we still could produce accurate interwell connectivity information. When I investigated cases with dissolved-gas production around the wellbore, however, the analogy broke down and the results were not as good as the liquid systems.

Yes / Two fractional Fourier transforms are used to define bi-fractional displacement operators, which interpolate between displacement operators and parity operators. They are used to define bi-fractional coherent states. They are also used to define the bi-fractional Wigner function, which is a two-parameter family of functions that interpolates between the Wigner function and the Weyl function. Links to the extended phase space formalism are also discussed.

A Discrete Fourier Transform (DFT) changes the basis of a group algebra from the standard basis to a Fourier basis. An efficient application of a DFT is called a Fast Fourier Transform (FFT). This research pertains to a particular type of FFT called Decimation in Frequency (DIF). An efficient DIF has been established for commutative algebra; however, a successful analogue for non-commutative algebra has not been derived. However, we currently have a promising DIF algorithm for CSn called Orrison-DIF (ODIF). In this paper, I will formally introduce the ODIF and establish a bound on the operation count of the algorithm.

Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N(^2). The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages, random sampling has been adopted in measurement Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N"^. The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% , of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages, random sampling has been adopted in measurement instruments where computing a spectrum is either minimal or not required. Such applications in instrumentation are easily found in the literature. In this thesis, two applications in digital signal processing are introduced. (5) To suggest an inverse transformation for random sampling so as to complete a two-way process and to broaden its scope of application. Apart from the above, a case study of realizing in a transputer network the prime factor algorithm with regular sampling is given in Chapter 2 and a rough estimation of the signal-to-noise ratio for a spectrum obtained from random sampling is found in Chapter 3. Although random sampling is alias-free, problems in computational complexity and noise prevent it from being adopted widely in engineering applications. In the conclusions, the criteria for adopting random sampling are put forward and the directions for its development are discussed.

The FIRST program (Far Infrared Spectroscopy in the Troposphere) was created as an Instrument Incubator Program (IIP) by NASA Langley to demonstrate improved technology readiness levels (TRLs) for two technologies needed in the design of new imaging Fourier transform spectrometers (IFTS). The IIP IFTS was developed at the Space Dynamics Laboratory and flown to an altitude of 103,000 feet on an instrumented NASA balloon payload. The sensor collected approximately 15,000 interferograms during its 6-hour flight. Fourier transforms (FFT) produced acceptable results except for noise equivalent temperature differences (NETD) that were five times higher than goal and inconclusive transforms at seven strong absorption features. An alternate transform technique, maximum likelihood estimation (MLE), was implemented to improve spectral estimations at the absorptions and to improve the NETD for the sensor. Iterative expectation-maximization (EM) algorithms provide numerical solutions for the MLE. Four combinatorial forms of the EM algorithm were developed. Forms of the EM algorithm were developed to optimize amplitude estimations as a function of assumed noise distributions. 'Direct' and 'indirect' EM forms were developed to process the asymmetrical interferograms recorded by the FIRST sensor. The direct method extends the standard even (cosine) EM algorithm to simultaneously transform both the sine and cosine components of the interferogram. The indirect method, uses Fourier and inverse Fourier transforms as pre-processors to convert the measured asymmetrical interferograms to even (cosine) interferograms. Using the indirect Gaussian EM form improved the measured NETD by approximately twenty percent between 100 and 700 wavenumbers. For wavenumbers less than 100 or greater than 700, the improvement increased to a factor of at least two out to 1500 wavenumbers. The indirect Gaussian produced inconclusive results in the areas of high absorption because of large bias errors introduced by the FFT/IFFT pre-processing. The indirect method was found to be inadequate for estimating spectra at the deep absorptions. The direct EM method, on the other hand, has the potential to produce improved amplitude estimations at the absorptions since there are no inherent biases in the algorithm's initial conditions at a cost in computer resources and execution times that are four times those needed for the indirect method.

Published Article / The reproductive status of animals is of utmost importance to the modern farmer. Decisions concerning the management of the flock are influenced by the knowledge of the percentage of animals that are pregnant at any specific time. The aim of the project was to gain knowledge for the development of an instrument that is affordable and with which a farmer can do pregnancy determination himself/herself, thereby enabling him/her to make the correct management decisions. Experimental data were obtained from pregnant Dorper ewes with the aid of a portable Doppler instrument. Using real data as input, simulations of Wavelet and Short Time Fourier Transforms (STF) were done in MathCAD. In the simulations known levels of noise were added to the Doppler signals. Satisfactory results were obtained from the simulations of Wavelet Transforms. In the simulation of the Wavelet Transforms, signals with a SNR of -6.5 dB were successfully identified. It can thus be concluded that Wavelet Transforms can be used successfully for the detection of the fetal heartbeat in noisy ultrasonic Doppler signals.

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / In many spacecraft receiver applications, the Fast Fourier Transform (FFT) provides a powerful tool for measuring the amplitude and frequency of an unmodulated RF signal. By increasing the FFT acquisition time, tiny signals can be coaxed from the noise and their frequency measured by determining which frequency bin the signal energy appears. The greater the acquisition time, the narrower the bin bandwidth and the more accurate the frequency measurement. In modern satellite operations it is often desirable for the receiver to measure the frequency of a carrier which is modulated with BPSK data. The presence of the BPSK data limits the FFT acquisition time since the signal may switch polarities a number of times while the FFT samples are being acquired. This polarity switching spreads the signal energy into multiple frequency bins making frequency measurement difficult or impossible. The Bit Templating Technique, used for the first time in the CMC Electronics Cincinnati TDRSS / BPSK Spacecraft Receiver, collects the modulated waveform energy back into a signal bin so that accurate amplitude and frequency information can be calculated.