I am confused about the definition of 1D convolution.
Given $ a = [-\frac{1}{2}, \frac{1}{2}] $ and $b = [1, 1, 1, -1, -1, -1] $, what will be the result of the convolution $( a * b )$?
From my understanding, the length of $ a * b $ should be $ 2 + 6 - 1 = 7 $. I tried converting the vectors into polynomials for easier calculation:
$ a = -\frac{1}{2}x + \frac{1}{2} $
$ b = x^5 + x^4 + x^3 - x^2 - x - 1 $
Multiplying them gives:
$ -\frac{1}{2}x^6 + x^3 - \frac{1}{2} $
So, $ a * b = [-\frac{1}{2}, 0, 0, 1, 0, 0, -\frac{1}{2}] $
However, I am not sure if this is correct because I tried it in ChatGPT, and it gave a completely different output:
ChatGpt Convolution Computation
In this context, the kernel is $ a$.
Could anyone help explain this kernel's purpose and clarify the convolution's correct result?
