So is the Fourier transform, when scaled by 1/sqrt(N) (instead of scaling one direction by 1/N). Bracewell described the transform that way in his books. And Bracewell introduced the Discrete Hartley Transform, so...I suppose he was inclined that way. [edit: Oops—sorry, thought the reference was to scaling symmetry, not running the same transform twice.]resynthesis wrote:...and it being its own inverse.
I messed with the FHT after working with Bracewell (we were experts on a patent case—what a nice and brilliant man, made a big impression on me). I think that Byte article was pretty fresh at the time, and he pointed it out to me. The code in it wasn't great though, as I recall, as far as missing a lot of obvious optimizations. Anyway, even after I improved it, it was obvious that the FHT could not be made faster than an equivalently optimized FFT. That's why we don't hear much of it.I seem to remember that Hartley transforms were at one time computationally more efficient than Fourier transforms but that might have changed (I'm trying to remember this from a Byte article in about 1990). These days it's certainly easier to get an optimized FFT library than FHT.