# fft formula convention

I try to code up an algorithm from a scientific paper where they solve a differential equation using Fourier transforms. In the paper, they use the FT convention $$\rho(x_i) = \sum_{k=-m}^m r_k e^{i2\pi kx_i}$$, where $$x_i$$ is the ith data point.

Numpy uses the convention described in the fft routine documentation.

Both of these should be completely equivalent (the negative $$k$$ in one convention correspond to the highest $$k$$ in the other convention).

Now, in the paper they frequently multiply $$r_k$$ with a function that depends on this $$k$$, i.e. $$r_k f(k)$$. This should be the same as if I wrote $$A_k\cdot f(k)$$ in Numpy's convention. And the $$A_k$$ are simply given by A[k] if A=numpy.fft(ρ, N). Is this correct?

I was confused by the statement

and A[n/2+1:] contains the negative-frequency terms

But this simply refers back to the other convention and should not change anything in the way I calculate $$A_k\cdot f(k)$$, shouldn't it?

• Hi! check your formula please. It uses $\rho$ on both sides ? – Fat32 Nov 25 '19 at 1:00
• Sorry, the $\rho_k^*$ is supposed to be the fourier amplitude. I should not have used the $*$ symbol. So left is $\rho$ in real space, right is in $k$ space. – SchroedingersLion Nov 26 '19 at 21:37
• please edit the question to reflect the change... – Fat32 Nov 26 '19 at 21:39
• better like this – SchroedingersLion Nov 26 '19 at 21:49
• sorry but still not better. what's $x_i$ and whay cannot you use another letter for $\rho$ in one of the sides ??? – Fat32 Nov 26 '19 at 22:01