Audio phase shift using Hartley Transform I want to shift the response phase of an audio signal (float array, length N, value range -1..1). I'm about using Fast Hartley Transform to transform it to the frequency-domain, process phase shift, then convert back to the time-domain and output the result.
I'm now can do the Hartley transform, and I also can calculate the phase of kth bin:

real(k) = (x(k) + x(-k))/2
imag(k) = (x(k) - x(-k))/2
phase(k) = atan2(imag(k), real(k))


I want to shift the phase(k) by an α degree. What should I do?
I think your notation is a bit fishy but I'll use it. Assuming that your $\text{phase}(k)$ calculation is correct, do a complex multiplication of $\text{real}(k) + i\ \text{imag}(k)$ by $e^{i\alpha} = \cos(alpha) + i\sin(alpha)$. That can be done using four real multiplications and a few add/sub. Then you need to reverse the process of how you obtained $\text{real}(k)$ and $\text{imag}(k)$:

$$x(k) = \text{real}(k) + \text{imag}(k)$$
$$x(-k) = \text{real}(k) - \text{imag}(k)$$

Finally retransform.