FFT

[willtebbutt]

Lots of FFT-related operations to support:

• - fft
• - ifft
• - bfft
• - plan_fft + associated operators
• - plan_ifft + associated operators
• - plan_bfft + associated operators
• - rfft
• - irfft
• - brfft
• - plan_rfft + associated operators
• - plan_irfft + associated operators
• - plan_rbfft + associated operators
• - dct
• - idct
• - plan_dct + associated operators
• - plan_idct + associated operators
• - fftshift (might not be differentiable)
• - ifftshift (might not be differentiable)
• - filt
• - deconv
• - conv
• - conv2
• - xcorr
• - r2r (not sure what this is yet)
• - plan_r2r (not sure what this is yet)

Note that a large subset of the above accept / produce complex-valued inputs / outputs. It might therefore be necessary to have extended the RMAD package to handle complex-valued quantities.

[ararslan]

Note that these functions are no longer in Base: the FFT functionality is now in FFTW and the DSP functionality in DSP. So extending these would require including them as dependencies, which I don't think we want to do.

[iamed2]

Would it? Or would it just be sufficient to include https://github.com/JuliaMath/AbstractFFTs.jl?

[ararslan]

Oh, I had forgotten about AbstractFFTs. Yeah, we could use that. Doesn't cover the DSP stuff but it would suffice for FFTs.

[willtebbutt]

I think I probably put the cart before the horse with this issue -- Nabla doesn't really have support for Complex numbers, so we would need to think carefully about this (if we want to do it). I'm half tempted to say that we should just close this issue because it's quite unlikely that we'll ever actually want to implement support for Complex numbers.