Hartley transform

Abstract

The Hartley transform is an integral transformation that maps a real-valued temporal or special function into a real-valued frequency via the kernel, cas (vx) cos (vx)Þ sin (vx). This novel symmetrical formulation of the traditional Fourier transform (FT), 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 components, respectively. This is in contrast to the complex exponential, exp(jvx), used in classical Fourier analysis. For periodic power signals, various mathematical forms of the familiar Fourier series (FS) 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 complex-valued fast Fourier transform (FFT).