ch04 - Olejniczak K.J “ The Hartley Transforms.” The...

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Unformatted text preview: Olejniczak, K.J . “ The Hartley Transforms .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 4 The Hartley Transform 4.1 Introduction 4.2 Historical Background 4.3 Fundamentals of the Hartley Transform The Relationship Between the Hartley and the Sine and Cosine Transforms • The Relationship Between the Hartley and Fourier Transforms • The Relationship Between the Hartley and Hilbert Transforms • The Relationship Between the Hartley and Laplace Transforms • The Relationship Between the Hartley and Real Fourier Transforms • The Relationship Between the Hartley and the Complex and Real Mellin Transforms 4.4 Elementary Properties of the Hartley Transform 4.5 The Hartley Transform in Multiple Dimensions 4.6 Systems Analysis Using a Hartley Series Representation of a Temporal or Spacial Function Transfer Function Methodology and the Hartley Series • The Hartley Series Applied to Electric Power Quality Assessment 4.7 Application of the Hartley Transform via the Fast Hartley Transform Convolution in the Time and Transform Domains • An Illustrative Example • Solution Method for Transient or Aperiodic Excitations 4.8 Table of Hartley Transforms Appendix 4.1 Introduction The Hartley transform is an integral transformation that maps a real-valued temporal or spacial function into a real-valued frequency function via the kernel, cas( ν x ) ≡ cos( ν x ) + sin( ν x ). This novel symmetrical formulation of the traditional Fourier transform, attributed to Ralph Vinton Lyon Hartley in 1942, 1 leads to a parallelism that exists between the function of the original variable and that of its transform. Furthermore, the Hartley transform permits a function to be decomposed into two independent sets of sinusoidal components; these sets are represented in terms of positive and negative frequency compo- nents, respectively. This is in contrast to the complex exponential, exp( j ω x ), used in classical Fourier analysis. For periodic power signals, various mathematical forms of the familiar Fourier series come to mind. For aperiodic energy and power signals of either finite or infinite duration, the Fourier integral can be used. In either case, signal and systems analysis and design in the frequency domain using the Hartley transform may be deserving of increased awareness due necessarily to the existence of a fast algorithm that can substantially lessen the computational burden when compared to the classical com- plex-valued Fast Fourier Transform (FFT). Kraig J. Olejniczak University of Arkansas © 2000 by CRC Press LLC Throughout the remainder of this chapter, it is assumed that the function to be transformed is real valued. In most engineering applications of practical interest, this is indeed the case. However, in the case where complex-valued functions are of interest, they may be analyzed using the novel complex Hartley transform formulation presented in Reference 10.Hartley transform formulation presented in Reference 10....
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This note was uploaded on 12/03/2009 for the course EEE transforn taught by Professor Profcenk during the Spring '09 term at Dogus University.

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ch04 - Olejniczak K.J “ The Hartley Transforms.” The...

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