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If I have a sound recording with a sinusodial frequency with a constant amplitude and I do an FFT on the time domain of this recording to get the frequency back:

If I were to ignore the imaginary part and just watched the amplitude (real part) would there be anything different about it? Would it also be constant?

I'm trying to do an FFT and I notice that the frequency spike oscillates up and down as if it were a sine wave, I have ignored the imaginary part on my FFT.

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The real part isn't the amplitude.

The imaginary part is the sine wave part of the FFT. Thus exactly periodic- in-aperture sine waves (with a phase of zero at the window start) won't show up on the real part, and the FFT real part will see a zero signal. Only sinusoids with a periodic cosine wave (with a phase of zero at the window start) component will show up at all.

The "beating" you are seeing is your signal alternating between being in and out of phase compared to a cosine basis vector.

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