Abstract

The discrete Fourier transform (DFT) and discrete Hartley transform (DHT) play a crucial role in one- and multi-dimensional digital signal processing applications. Traditionally, the main concern in the design of fast Fourier transform (FFT) and fast Hartley transform (FHT) algorithms has been the reduction of the arithmetic complexity. However, with the recent advances in the digital technology and the present demands of such transforms in low-power high-performance real-time applications, a more comprehensive treatment of the computational and structural complexities must be considered in the design of the algorithms. The objective of this thesis is to design one- and multi-dimensional FFT and FHT algorithms that address the problem of reducing the number of arithmetic operations, data transfers, address generations, and twiddle factor evaluations or accesses to the lookup table, while possessing features such as simplicity, regularity, modularity, easy indexing scheme, and butterfly-style and in-place computations that are highly desirable characteristics for software or hardware implementations of the algorithms. To achieve these objectives, radix-based algorithms are proposed by introducing new decomposition strategies, efficient index mappings, and by an appropriate use of the Kronecker product. A general decomposition method, which is based on the radix-2 approach, valid for any dimension and applicable to both the DHT and DFT, and which significantly reduces the complexity of the FHT algorithms, is proposed. This method enables us to develop multidimensional FHT and FFT algorithms. A new approach for computing the DFT and DHT using a unified structure is proposed by establishing a close relationship, valid for any dimension, between the radix-2 based FHT and FFT algorithms. An efficient method, based on the radix-2 approach, for pruning output samples of a 1-D or 2-D DFT is proposed by grouping in its 1-D or 2-D FFT algorithm all the stages that involve unnecessary operations into a single stage and by introducing a new recursive technique for the computations required in the resulting stage. A technique is presented to improve the performance of the radix-4, radix-8 and radix-16 FFT algorithms in terms of the number of twiddle factor evaluations or accesses to the lookup table without any increase in the computational or structural complexities of the algorithms. In order to take advantage of the lowest structural complexity provided by the radix-2 approach and reduced computational complexity offered by the radix-4 approach, a technique suitable for combining these two approaches is introduced in order to develop efficient 3-D FFT and FHT algorithms. A radix-2/8 approach for reducing the complexity in the computation of the 1-D DFT and DHT of lengths N = q {604} 2 m is proposed by appropriately mixing the radix-2 and radix-8 index maps. This approach is extended to 2-D and 3-D DFTs. It is shown that the proposed radix-2/8 approach is superior to all the other existing radix-based approaches in providing low-complexity 1-D, 2-D and 3-D FFT, and 1-D FHT algorithms.