Unexpected result with custom FFT implementation

Question

To fully understand the Fourier Transform I have been working on a rather standard implementation of the FFT algorithm. I tried generating a 440 Hz sin wave and passing it through the algorithm, however, when plotting the real part of the result, I get a spiky mess, whereas I was expecting two spikes (as from my understanding it's mirrored, since the second half is just a representation of the first's complex conjugate).

Disappointed, I tried passing the sin wave through a third party library, but achieved the same result, so I suspect I am misunderstanding what the FFT actually outputs.

Not sure how useful it is, but here is the GitHub project https://github.com/kradicati/fourier-transform

The two most important classes are: https://github.com/kradicati/fourier-transform/blob/master/src/test/kotlin/me/krb/transform/FastFourierTransformTest.kt (where the actual plotting is done) https://github.com/kradicati/fourier-transform/blob/master/src/main/kotlin/me/krb/transform/FastFourierTransform.kt (the FFT implementation)

Answer 1

Looking at the y-axis (vertical scale) on your plot I see the values are really small. I am assuming that your input signal is of magnitude order one. This suggests that all you are seeing is roundoff error.

This is to be expected for a sine signal if you are only plotting the real part of the FFT. For a sine signal the Fourier transform is pure imaginary.

What you should probably be plotting is the absolute value of the FFT to see the combined magnitude of the real and imaginary part. You may also want to plot both the real and imaginary part of the result on the dame figure to be able to see the relationship between the two components.

From Wikipedia:

$$\mathcal {F}[\sin(ax)] = \frac{\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)}{2i}$$

where $\xi$ is the frequency, $\delta()$ is the Dirac delta function, and $i$ is the imaginary unit.