1D convolution for neural networks, part 4: Convolution equation

Key Takeaways

so of course math is great at finding more efficient ways to write things like that so we can use the summation sign and we can introduce this index K K is going to be our counter that counts along the length of our kernel and so it'll go for a - P 2 plus P and that will always cover the length of our kernel so to write this more concisely for the first element of our result Y sub 0 we take and increment K starting at minus P all the way to plus P and at each one we multiply X sub minus K times W sub K so the very first element would be X sub - minus P or X sub P times W sub - B so you can go through element by element in the equations we just wrote out longhand and see how these line up the one thing that is different about these is if you keep going you'll see here by the time K is equal to P we're multiplying X sub minus P times W sub P X isn't defined at minus P it's only defined at 0 through M minus 1 so we have to add the footnote any time we try to access an X index that's outside that range we just assume X equals 0 so really X extends to plus and minus infinity it's at 0 for most of that but between indexes 0 and M minus 1 that's when it's interesting and we can repeat that then for the next position of the result Y sub 1 and Y sub 2 etc all the way up to the last position Y sub M and if we want to we can take and instead of explicitly counting through all the positions of our signal we can use the index J and just say hey for any position J between 0 and M minus 1 we can use this expression so Y sub J is equal to the sum of X sub J minus K times W sub K and in each of the summations K goes from minus B to P so this is the beautifully short way to write all of this that we wrote out longhand the first time so math is beautiful exhibit 556