# turn circular convolution into linear convolution by zero padding: A special case

We know that, multiplying a kernel and signal spectrum in Fourier domain will lead to a circular convolution and not a linear convolution, so in order to it become linear convolution we 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 kernel's length). However, I have already known the analytical solution to the Fourier transform of the kernel, while I don't know the analytical form of the kernel. I know that I can inverse discrete Fourier transform $$\mathcal{F} \{kernel\}$$, then sample and do the zero-padding, but I will lose the advantage of the "analytical form" in this case. Is there any good way for me to turn the circular convolution into linear convolution?

• "in Fourier domain": you mean in discrete Fourier domain; I know this reads like nitpicking, but it's usually really important to be sure what you're talking about here; the confusion between Fourier Transform of a continuous function and Discrete Fourier Transform is the Nr. 1 reason for "my" students having problem when we confront them with OFDM in the fourth semester. – Marcus Müller Nov 13 '19 at 20:08
• So, do you know the analytical form of the continous-time Fourier Transform, or of the DFT? That makes all the difference here! – Marcus Müller Nov 13 '19 at 20:09
• @MarcusMüller Thanks for the clarification. I know the analytical form of the continous-time Fourier Transform – Wen Nov 13 '19 at 20:38
• Could you please give an example of the Fourier transform of the continuous-time kernel? Is the continuous-time kernel band-limited? – Olli Niemitalo Nov 14 '19 at 8:25
• @OlliNiemitalo 1 I was trying to solve an European option pricing problem using Conv method (pdfs.semanticscholar.org/8e38/…). The kernel is a probability density function(PDF) and the Fourier transform of it is called characteristic function, and for a certain class of PDF, its characteristic function is known analytically. You can read page 5 of that paper. – Wen Nov 14 '19 at 14:22