#### Document Type

Thesis - Open Access

#### Award Date

1988

#### Degree Name

Master of Science (MS)

#### Department

Electrical Engineering

#### First Advisor

Douglas B. Miron

#### Abstract

In 1942, Hartley proposed a new formulation of the Fourier integral identity by using cas (wt) as the transform kernel, where cas(wt) = cos(wt) + sin(wt), is an abbreviation for cosine and sine. It is called the Hartley transform and has many properties similar to those of the Fourier transform. The important distinctions are that the Hartley transform of a real-valued function is also real valued, and the inverse transformation is operation as the direct transformation. the same integral Furthermore, its evaluation does not involve complex functions. This is a potential advantage if the transform is to be explicitly computed. The discrete Fourier transform (DFT) has the same transform kernel as the Fourier transform. In 1983, Bracewell introduced the discrete Hartley transform (DHT) by using Hartley’s transform kernel, i.e. cas(wt). The DHT can apply to numerical spectral analysis and convolution. Unlike the DFT, no additional program is required for the inverse DHT as it is the same as the direct transformation. If the real and imaginary parts of the DFT are expressly required, then they are directly obtainable as the even and odd parts of the DHT. The power spectrum can also be obtained directly from the DHT without first calculating the real and imaginary parts of the DFT as in the usual way of calculating power spectra. In 1984, Bracewell worked out a fast algorithm for performing the DHT of a data sequence of H elements in a time proportional to Nlog2N. He proved that the fast Hartley transform (FHT) is as fast as or faster than the fast Fourier transform (FFT) and serves for all uses such as spectral analysis, digital signal processing, and convolution to which the FFT is at present applied. Since then, many discussions on the FHT [5-10] and its applications were aroused. The vector processor (VP) was initiated by Miron in It is an adjunct to an IBM personal computer in which a sequence of data is passed from the host's memory to a row of mathematical coprocessors, operated on simultaneously by each of a sequence of the coprocessor's instructions, and then the results are passed back to the main memory. It is a single-instruction, multiple-data stream (SIMD) computer system. The purpose of this device is to economically achieve maximum speed in the computer execution of a group of scientific calculations. Since the FHT is applied on a data sequence of N elements, where N is usually very large, it is a good candidate to be performed on the vector processor and theoretically, a great improvement on the computer execution time should be obtained. The Hartley transform, the DHT, and their properties will be presented in the subsequent chapters. The derivation of the DFT from the DHT will also be illustrated. Finally, a software system for computing the DHT on a VP will be discussed. and its performance will also be evaluated.

#### Library of Congress Subject Headings

Hartley transforms

Transformations (Mathematics)

Parallel processing (Electronic computers)

Parallel programming (Computer science)

#### Format

application/pdf

#### Number of Pages

91

#### Publisher

South Dakota State University

#### Recommended Citation

Lim, Boon Pock, "The Fast Hartley Transform on a Parallel Processor" (1988). *Electronic Theses and Dissertations*. 4534.

https://openprairie.sdstate.edu/etd/4534