# solution on the time domain becomes “periodic” after the inverse fourier transform

I was trying to solve european option pricing problem using Conv method (introduced by Lord in 2008 https://pdfs.semanticscholar.org/0632/460bd50b2151f74ac40028df4cc60e73a884.pdf).

The final step of this algorithm is to transform the solution on the frequency domain to the time domain by inverse fast Fourier transform, which means that I can retrieve approximations for all the grid points.

However, I noticed that only the solutions in the center of the time domain (in this case, x=40, and I used log grid points) are accurate. Here is the plot: the orange line represents the approximation using FFT, the blue line represents the real solution.

Moreover, the approximation looks periodic. So I was wondering what kind of situation will inverse FFT give a 'periodic-like' solution? I did some research on this topic, and foundthat DFT assumes the input is periodic, which may cause "time domain aliasing" (https://www.dspguide.com/ch10/3.htm).

I also found that "spectral leakage" may be related to my problem. Can anyone give me some advice on this? Thank you very much.

P.S. By periodic, I mean the solutions on the two ends are very close to each other. I changed lots of parameters, and this always holds.

## 1 Answer

Multiplying a kernel and signal spectrum in Fourier domain lead to a circular convolution and not a linear convolution, so in order to it become linear convolution you must zero pad your signal and kernel before taking the Fourier transform (up to M+N-1 where M is the signal's length and N is the kernels's length). (if you compare the blue and orange signals, the beginning of orange signal has higher values because of of higher value at the end of blue signal and it has lower values in the end because of lower values in the beginning of the blue signal).

• Thank you sir. Do you mean zero pad the time-domain input? In my case, only the Fourier transform of the kernel is known analytically, while the original kernel is unknown. So I can't FFT the kernel, but I'm able to evaluate it on any points in the Fourier domain. How can I zero pad it in this case? Nov 12, 2019 at 20:48
• @Wen,for a N-point DFT the signal assumed to be periodic and its periodicity is N, you could assume an aperiodic signal is a periodic signal with a periodicity which is infinite, so by increasing the samples of your analytical FT you could alleviate this periodicity problem to some extent (zero padding in time domain lead to higher sampling in Fourier domain). Nov 14, 2019 at 23:18