# Expected Behavior of FFT

I am debugging an FFT. My FFT uses complex numbers in polar form, $(r,\theta)$.

I am trying to establish that it works, with a unit test. My real values are all correct, after a polynomial multiplication...

However, I pad out my polynomials to a power of two with complex zeros, and when the multiplication is complete, I have random angle values in the polar form of the complex numbers.

For instance, I multiply the following polynomials, listed in standard form by their coefficients:

$$A = \lbrace1,2,3\rbrace\\ B = \lbrace4,5,6\rbrace\\$$

And I am expecting the answer to be: $$C = \lbrace (4,0),(13,0),(28,0),(27,0),(18,0),(0,0),(0,0),(0,0)\rbrace$$

However, my FFT returns this exactly (ignoring floating point errors), except the last three terms are:

$$C = \lbrace \ldots,(0,4.518),(0,0.4769),(0,2.2877)\rbrace$$

My question is whether these angle values in the extra terms are expected, or whether this is a sign of something going wrong in my FFT code?

(this is tricky for me, because it seems like everything is calculating correctly; and shouldn't it not work at all in a d&c if something is wrong?)

• @evil for some reason, it is claiming I am trying to create a new tag. Not sure whether I should delete the question or not.... – Chris May 25 '17 at 2:33
• You are right, something is odd. The system suggests fft but then tells it is new one. Anyway, the core of your question is interesting one. I wouldn't just take a word for it. – Evil May 25 '17 at 2:42
• @evil ok, thanks man--will wait and see. – Chris May 25 '17 at 2:44

In polar coordinates, $(0,x) = (0,y)$ for any $x$ and $y$, so there's no issue. You've simply used a representation that has multiple representations of the complex number $0 + 0i$.