FFTW 3.3.5: The Discrete Hartley Transform

If you are planning to use the DHT because you’ve heard that it is Chris@42: “faster” than the DFT (FFT), stop here. The DHT is not Chris@42: faster than the DFT. That story is an old but enduring misconception Chris@42: that was debunked in 1987. Chris@42:

The discrete Hartley transform (DHT) is an invertible linear transform Chris@42: closely related to the DFT. In the DFT, one multiplies each input by Chris@42: cos - i * sin (a complex exponential), whereas in the DHT each Chris@42: input is multiplied by simply cos + sin. Thus, the DHT Chris@42: transforms n real numbers to n real numbers, and has the Chris@42: convenient property of being its own inverse. In FFTW, a DHT (of any Chris@42: positive n) can be specified by an r2r kind of FFTW_DHT. Chris@42: Chris@42: Chris@42: Chris@42:

Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of Chris@42: size n followed by another DHT of the same size will result in Chris@42: the original array multiplied by n. Chris@42: Chris@42:

The DHT was originally proposed as a more efficient alternative to the Chris@42: DFT for real data, but it was subsequently shown that a specialized DFT Chris@42: (such as FFTW’s r2hc or r2c transforms) could be just as fast. In FFTW, Chris@42: the DHT is actually computed by post-processing an r2hc transform, so Chris@42: there is ordinarily no reason to prefer it from a performance Chris@42: perspective.⁵ Chris@42: However, we have heard rumors that the DHT might be the most appropriate Chris@42: transform in its own right for certain applications, and we would be Chris@42: very interested to hear from anyone who finds it useful. Chris@42:

If FFTW_DHT is specified for multiple dimensions of a Chris@42: multi-dimensional transform, FFTW computes the separable product of 1d Chris@42: DHTs along each dimension. Unfortunately, this is not quite the same Chris@42: thing as a true multi-dimensional DHT; you can compute the latter, if Chris@42: necessary, with at most rank-1 post-processing passes Chris@42: [see e.g. H. Hao and R. N. Bracewell, Proc. IEEE 75, 264–266 (1987)]. Chris@42:

For the precise mathematical definition of the DHT as used by FFTW, see Chris@42: What FFTW Really Computes. Chris@42:

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Footnotes

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We provide the DHT mainly as a byproduct of some Chris@42: internal algorithms. FFTW computes a real input/output DFT of Chris@42: prime size by re-expressing it as a DHT plus post/pre-processing Chris@42: and then using Rader’s prime-DFT algorithm adapted to the DHT.

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2.5.3 The Discrete Hartley Transform

Footnotes

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