Question

I need an 1D Convolution against 2 big arrays. I’m using this code in C# but it takes a loooong time to run.

I know, i know! FFT convolutions is very fast. But in this project i CAN’T use it.
It is a constraint of the project to not use FFT (please don’t ask why :/).

This is my code in C# (ported from matlab, by the way):

var result = new double[input.Length + filter.Length - 1];
for (var i = 0; i < input.Length; i++)
{
    for (var j = 0; j < filter.Length; j++)
    {
        result[i + j] += input[i] * filter[j];
    }
}

So, anyone knows any fast convolution algorithm widthout FFT?

Answer 1

Convolution is numerically the same as a polynomial multiplication with an extra wrap-around step. Therefore, all the polynomial and large integer multiplication algorithms can be used to perform convolution.

FFT is the only way to get the fast O(n log(n)) run-time. But you can still get sub-quadratic run-time using the divide-and-conquer approaches like Karatsuba’s algorithm.

Karatsuba’s algorithm is fairly easy to implement once you understand how it works. It runs in O(n^1.585), and will probably be faster than trying to super-optimize the classic O(n^2) approach.