Assessing the Hartley transform

The fast algorithm for the (real) Hartly transform is discussed in relation to the established fast algorithm for the (complex) Fourier transform. The two transforms are compared by timing comparably written programs on a given machine, and the discipline of timing is discussed as an adjunct to complexity analysis. With real data, one Hartley transform program can economically replace such packages as a complex-valued unilateral Fourier transform combined with a real-valued unilateral inverse Fourier transform. The Hartley transform is favorable for fast convolution of real data sets. The utility of spectral analysis into Fourier series throughout physics suggested that the Hartley transform might have less physical significance, but the construction of Hartley diffraction planes in the microwave and optical laboratories, where electromagnetic phase is encoded as real-valued field amplitudes, has revealed interesting complementarity.>