Convolution Theorem using the Discrete Hartley Transform (DHT)

Hello forum,

I'm trying to see if the convolution theorem for the DHT works as it should. I'm using the formulations on wiki in the "properties" section of two pages:

Discrete Hartley transform - Wikipedia, the free encyclopedia

and

Hartley transform - Wikipedia, the free encyclopedia

I'd need a worked example with real numbers so that I can match them with the inputs/outputs of my own Matlab code which I just can't get to work so far.

I'm basically computing the DHT of two short "real number" sequences and attempting to combine them according to the wiki formula to produce the DHT of
their circular convolution.

I can't seem to get a match despite any interpretation I put on each formula, e.g. what X(-f) or Y(-f) mean rather than X(f) or Y(f).

Has anyone done this and has a worked example with some simple number sequences?

Hey Johnny33.

Do you want to prove the convolution symbolically? If so, then there are a couple of ways to do this.

You could do it by starting from the original definition or you could use the relationship to the Fourier Transform since convolution already exists for that transform (i.e. use that as a gateway to proving the Hartley one).

Here is the relationship to the Fourier:

Hartley transform - Wikipedia, the free encyclopedia

Hi Chiro,

I wanted to do it numerically - I always like to see actual results. They're normally quick to generate using Matlab or Octave. Using the Fourier link you supplied its easy to prove the DHT convolution theorem - simply add the real and imaginary parts of the FFT together to get the same result as the DHT. Thanks for that. I'm close to getting a correct convolution result just using the DHT now - I'll keep you posted.