I'm trying to implement a Fortran program to compute the derivative of a function using the FFT. To begin with, just to test my installation of fftpack, I computed the Fourier transform of $\mathrm{sin}(x^{2})$, followed by the inverse transform of that result, which gave me back $\mathrm{sin}(x^{2})$ as expected.

So once I was satisfied that the FFT routines were working, I began trying to use it to find the derivative, via the formula $$\frac{\mathrm{d}f(x)}{\mathrm{d}x}=\mathcal{F}^{-1}[ik\hat{f}(k)],$$ where $f(x) = \mathrm{sin}(x^{2})$, for $x$ between $0$ and $2\pi$. There are $N=256$ points on the mesh, so $x_{n}=2\pi n/N.$ My simple problem is, I can't work out how to compute the values of $k$. I've tried some stuff, but it resulted in garbage. I'd really appreciate any help, thanks.


You perform FFT on $N=256$ points with equal distance $2\pi/N$ between them and as a result you get $256$ points in frequency domain. You want to know which frequency values these correspond to. You can consider them to be indexed as [-128,128) or [-128,127] (integers)

The leftmost point will be $-\frac{\pi}{\Delta x}$ where $\Delta x$ is the distance between your points before FFT. Rightmost point will not be $\frac{\pi}{\Delta x}$ since it will be non symmetric.

So, for k, you need 256 linearly spaced points starting from $-\pi/(2\pi/N)$ increasing with increments $2\pi /(N\Delta x)=1$

  • $\begingroup$ Thank you, my program now works! $\endgroup$ – O Smith Jun 4 '15 at 0:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.