convolution

Concept

Convolution

• slinding a kernel(window) over the signal
• take the weighted sum
• taking the weighted average of the signal over the window, giving a blurring effect

Linear convolution

• pad the rest to 0

kernel is “flipped”, weighted sum of overlapping, rest is assumed to be 0

Circular convolution

• wrap the signal instead of padding to 0
• analogous to periodic signals
  TODO

Convolution theorem

• convolution in one domain <-> multiplication in the other

more specifically, convolution in the time domain is equal to point-wise multiplication in the frequency domain

Application

Standard example

Polynomial coefficients

multiplication of polynomials <=> convolution of coefficients
hints that we can find the convolution using multiplication