Discrete Hartley transform

A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing and related fields. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no intrinsic involvement of complex numbers. Just as the DFT is the discrete analogue of the continuous Fourier transform, the DHT is the discrete analogue of the continuous Hartley transform, introduced by R. V. L. Hartley in 1942.

Because there are fast algorithms for the DHT analogous to the fast Fourier transform (FFT), the DHT was originally proposed by R. N. Bracewell in 1983 as a more efficient computational tool in the common case where the data are purely real. It was subsequently argued, however, that specialized FFT algorithms for real inputs or outputs can ordinarily be found with slightly fewer operations than any corresponding algorithm for the DHT (see below).

Formally, the discrete Hartley transform is a linear, invertible function H : Rⁿ-> Rⁿ (where R denotes the set of real numbers). The N real numbers x₀, ...., x_N-1 are transformed into the N real numbers H₀, ..., H_N-1 according to the formula

The combination ${\displaystyle \cos(z)+\sin(z)\!}$ ${\displaystyle ={\sqrt {2}}\cos(z-{\frac {\pi }{4}})}$ is sometimes denoted ${\displaystyle \mathrm {cas} (z)\!}$, and should be contrasted with the ${\displaystyle e^{-iz}=\cos(z)-i\sin(z)\!}$ that appears in the DFT definition (where i is the imaginary unit).

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