Fast and parallel algorithms for Fourier-like signal transforms and convolutions

Neng-Chung Hu, Purdue University

Abstract

This thesis deals with a new approaches to the fast computation of discrete Fourier-like signal transforms. The fast algorithms are considered in terms of 2 measures. The first one is the reduction of the number of arithmetic operations such as additions and multiplications. The second one is massive parallelism in terms of high degree of pipelining and multiprocessing. Two types of algorithms are discussed to achieve high performance in terms of these measures. Better performance than demonstrated by previous algorithms is seeked in terms of minimum number of operations, minimum number of pipelining stages, minimized communication costs, and high degree of parallelism. The first type of algorithm involves the fast computation of the real discrete Fourier transform (RDFT) as a basic building block. They achieve the minimum number of operations as well as minimum number of pipelining stages, thereby being most attractive for pipelined implementations in real-time applications. Important features of the algorithms are the use of only real arithmetic, the same basic block of computation for both forward and inverse transforms, replacement of the complex butterfly with Givens' plane rotation as the basic unit of computation, and optimal computation of all discrete trigonometric transforms such as the discrete cosine (DCT) and the discrete sine (DST) transform with the same basic algorithm, and thereby the same VLSI or electro-optical designs. The second type of algorithm is based upon the factorization of the transform matrix into 2 stages. The first stage is preprocessing, with only simple matrix elements such as $\pm$1 and 0. The second state is postprocessing, where independent, small-size circular convolutions are performed. The two-stage algorithms are especially suitable for parallel implementations, since the preprocessing stage consists of only simple operations, and the postprocessing stage consists of circular convolutions, both of which can be implemented in architectures such as systolic and wavefront arrays. The applications which are studied with these methods are the fast computation of linear and circular, one-dimensional and multidimensional convolutions. By neglecting postprocessing stage, the preprocessing stage has the potential to drastically decrease the time of computation and cost of implementation in shape recognition, signal recognition and image recognition. It may actually perform better than the DFT in terms of classification accuracy.

Degree

Ph.D.

Advisors

Ersoy, Purdue University.

Subject Area

Electrical engineering

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