What you describe is the Hartley transform.
In your case it is the fast discrete Hartley transform.
It is an involuntory form of the FFT.
It is a very interesting find indeed.
Edit: fyi it retains all properties of the FFT of course.
It is an involuntory form of the FFT.
It is a very interesting find indeed.
Edit: fyi it retains all properties of the FFT of course.
the inverse transformation, which allows one to recover the xn from the Hk, is simply the DHT of Hk multiplied by 1/N. That is, the DHT is its own inverse (involutory), up to an overall scale factor.
The DHT can be used to compute the DFT, and vice versa. For real inputs xn, the DFT output Xk has a real part (Hk + HN−k)/2 and an imaginary part (HN−k − Hk)/2. Conversely, the DHT is equivalent to computing the DFT of xn multiplied by 1 + i, then taking the real part of the result.
It's so cool to discover something like this in the wild, then find out it has a formal name (DHT) and that people have already done the hard work of figuring out all the useful properties and relations.
I wish I had a way of contacting you (mostly to thank you in person, but also to ask what you use DHT for / how you learned about it), but unfortunately your HN profile is empty. Feel free to DM me on twitter sometime to say hello (https://twitter.com/theshawwn). Otherwise, cheers for brilliantly pointing out the exact answer I was hoping for!