Abstract
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 its
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 (O2NMNT), 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-2i 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 DFT, 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