perform fourier-type integration using DFT misunderstanding

Question

I need to perform the following integral ($i^2 = -1$) (with a fixed value of $m$):

$ \int_{\phi=0}^{\phi=2\pi} \psi(\phi) e^{-im \phi} \hspace{0.7mm} d\phi $

for all values of $m$ in $\{-N_{\theta}, -N_{\theta}+1, ..., 0, 1, ..., N_{\theta}-1, N_{\theta} \}$, that would be $2N_{\theta} + 1$ naive calculations of the above integral (plug $m$ into the formula and use Riemann sums, for example), each time with a different value of $m$. $N_{\theta}$ is an integer $> 0$.

I recognize the above as a Fourier Transform from $\phi$ to $m$.

How could I use the DFT (i.e. scipy's FFT implementation) to help me perform the integral above?

[ I have the function $\psi$ evaluated at equidistant points $\phi_j$ for $j \in \{0, 1, ..., 2N_{\theta} \}$ in memory. ]

My problem is that reading the scipy's documentation, it outputs a result of the form: $ I(m_x) = \sum_{j=0}^{j=M-1} \psi(\phi_j) \hspace{0.8mm} e^{-i \frac{2\pi x}{M} j} $, with $x \in \{ 0, 1, ..., M-1 \}$, thus I cannot get results for $m_x = - 20$ or $m_x = -15$ or any other negative value of $m_x$, because $x$ is $\geq 0$ in the scipy's implementation. I define $m_x = \frac{2\pi x}{M\Delta_{\phi}}$

Am I missing something obvious?

Thank you!

Answer 1

Am I missing something obvious?

Yes. The FFT implements the DFT (Discrete Fourier Transform) which is periodic in both time and frequency. So we simply have $X(-k) = X(N-k)$ or more generally $X(-k) = X(mN-k), m \in \mathbb{Z}$, where $N$ is the FFT length.

Whether the approximation will be any good or not depends a lot on how exactly the continuous function is being sampled.