Apple Accelerate vDSP fft vs DFT and scaling factors

Question

I am an experienced programmer but I don't have a lot of experience implementing DSP routines.

I've been banging my head against this for weeks if not months. My question is two fold, concerning Apple's Accelerate framework:

1)

In the vDSP.h header there are comments to the effect of: please use vDSP_DFT_XXX instead of the (i guess) older versions vDSP_fft_XXX. However there are zero examples of this outside of Apple's https://developer.apple.com/library/prerelease/mac/samplecode/vDSPExamples/Listings/DemonstrateDFT_c.html#//apple_ref/doc/uid/DTS10004300-DemonstrateDFT_c-DontLinkElementID_6. Maybe it's just that the DFT functions are newer? If so, fine and dandy.

2)

Scaling factors. I can read the documentation (https://developer.apple.com/library/mac/documentation/Performance/Conceptual/vDSP_Programming_Guide/UsingFourierTransforms/UsingFourierTransforms.html#//apple_ref/doc/uid/TP40005147-CH202-16195), it says in the case of an FFT on a real input, like audio that I am working with, the resulting value of each of the Fourier coefficients is 2x the actual, mathematical value.

And yet, in every example, including Apple's own, the scaling factor used for the resulting vsmul() function looks like it is 1/2*N instead of 1/2 as I would expect.

Further, there is no documentation about the scaling factors for the vDSP_DFT_XXX routines, but I assume that they just wrap the older ones?

Any insight into either of these questions would be greatly appreciated! Hopefully I'm just missing something basic about the way that FFT's are implemented in this framework (or in general).

Answer 1

There are at least 3 different FFT scaling options that produce "mathematical" results, and there is no single standard scaling. Energy preserving (see Parseval's theorem) FFT libraries need to be scaled by on the order of 1/N for input magnitude results, since a longer signal of the same magnitude will have proportionally more energy. vDSP uses an energy preserving forward FFT.

Answer 2

On part 1), two answers in the Apple developer forum say "The DFT interfaces are newer and a little bit easier to use correctly, and support some lengths that the older FFT interfaces cannot." and "plus there are new DFT routines that support working directly with interleaved data"