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2.5.3 The Discrete Hartley Transform

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

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

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

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

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

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