Hartley transform
In mathematics, the Hartley transform is an integral transform closely related to the Fourier transform, but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by R. V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse.
The discrete version of the transform, the Discrete Hartley transform, was introduced by R. N. Bracewell in 1983.
The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform, with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase (Villasenor, 1994). However, optical Hartley transforms do not seem to have seen widespread use.
The Hartley transform of a function f(t) is defined by:
where can in applications be an angular frequency and
is the cosine-and-sine or Hartley kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).
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