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Title: Development of efficient algorithms for fast computation of discrete transforms
Author: Hamood , Monir Taha
Awarding Body: University of Newcastle Upon Tyne
Current Institution: University of Newcastle upon Tyne
Date of Award: 2012
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Transforms are widely used in diverse applications of science, engineering and technology. In particular, the field of digital signal processing has grown rapidly and achieved their performance through the development of fast algorithms for computing discrete transforms. This thesis focuses on the computation, generalisation and applications of four commonly used transforms, namely the new Mersenne number transform (NMNT), the discrete Fourier transform (DFT), the discreet Hartley transform (DHT), and the Walsh-Hadamard transform (WHT). As a result, several new algorithms are developed and two number theoretic transforms (NTTs) are introduced. The NMNT has been proved to be an important transform as it is utilised for error- free calculation of convolution and correlation with long transform lengths. In this thesis, new algorithms for the fast calculation of the NMNT based on the Rader- Brenner approach are developed, where the transform's structure is enhanced and the lowest multiplicative complexity among all existing NMNT algorithms is achieved. Two new NTTs defined modulo the Mersenne primes, named odd NMNT (ONMNT) and odd-squared NMNT (02NMNT), are introduced for incorporation into generalised NMNT (GNMNT) transforms which are categorised by type, with detailed instructions regarding their derivations. Development of their radix-2 and split radix algorithms, along with an example of the calculation of different types of convolutions, shows that these new transforms are suitable for wide range of applications. In order to take advantage of the simplest structural complexity provided by the radix-2 algorithm and the reduced computational complexity offered by a higher radix algorithm, a technique suitable for combining these two algorithms has been proposed, producing new fast Fourier transform (FFT) algorithms known as radix-Z' FFTs. In this thesis, a general decomposition method for developing these algorithms is introduced based on the decimation in time approach, applicable to one and multidimensional FFTs. The DHT has been proposed as an efficient alternative to the DFT for real data applications, because it is a real-to-real transform and possesses similar properties as the DFT. Based on the relationship between the DHT and complex-valued OFT, a unified 'FFT to FHT transition approach' is presented, providing a translation of FFT algorithms into their fast Hartley transform (FHT) counterparts. Using this approach, many new FHT algorithms in one and multidimensional cases are obtained. Finally, the combination of the WHT with the DFT has been proved to be a good candidate for improving the performances of orthogonal frequency division multiplexing (OFDM) systems. Therefore, part of this thesis deals with the Walsh- Hadamard-Fourier transform (WFT) that combines these transforms into a single orthogonal transform. Development of fast WFT (FWFT) algorithms has shown a significant reduction in the number of arithmetic operations and computer run times.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available