fixed off by one error for FFTmagnitude()

[paradigmn]

The output off a Real-FFT is N/2+1 due to the dc offset. Hence the last frequency bin was discarded.

[swharden]

Hi @paradigmn, thank you for spotting this and creating a PR! It would be excellent if we could point to a reference to support that this is accurate. In reviewing The Fundamentals of FFT-Based Signal Analysis and Measurement (by NI) they say:

"...to convert from a two-sided spectrum to a single-sided spectrum, discard the second
half of the array and multiply every point except for DC by two."

This implies the output length should be N/2, with the first element being the DC component... yet this PR suggests length should be N/2 + 1 🤔

Do you have an authoritative reference we can point to to confirm the correct behavior? Absent this I might hop over to some of the other popular libraries (like numpy for matplotlib) and see what they tend to do...

[paradigmn]

I was rewriting some Numpy Python code, when I notices this issue. They explicitly mention the buffer size in their documentation: https://numpy.org/doc/stable/reference/generated/numpy.fft.rfft.html. Would that be sufficient?

[swharden]

I was rewriting some Numpy Python code, when I notices this issue. They explicitly mention the buffer size in their documentation: https://numpy.org/doc/stable/reference/generated/numpy.fft.rfft.html. Would that be sufficient?

Yeah, great reference! They say length should be N/2 + 1, and regarding that last value:

If N is even, A[-1] contains the term representing both positive and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely real.

I'll add some tests to this PR and merge it in shortly. Thanks for your help here @paradigmn!

[paradigmn]

no problem, you are welcome :)