Convoluted
January 28th, 2004
So today I was trying to understand convolution. I'm looking at this picture of a box with an arrow entering it and an arrow leaving it. Inside the box is a function called h[n], next to the arrow entering the box is a function called x[n], and next to the arrow leaving the box is the function y[n]. x[n] is the input function and y[n] is the output function. h[n] is the devil and it tempts the pure and innocent x[n] into becoming the dirty, dirty, evil y[n].
Right, so I understand that much. But they call it convolution for a reason, right? What I want to know is how h[n] manages to tempt happy-go-lucky x[n] into being this wildly transformed nether-function called y[n]. Because it can't be as simple as just sticking the values of x[n] into h[n] and getting out y[n] for every value. Cuz when you convolve something you mess it up good.
Here's convolution. You take x[n] and h[n] and you shove 'em together in a special way. You flip one of them. Doesn't matter which. Then you stick it on top of the other. You follow? Can you see? Already it's getting dirty and perverted! So after you've forced them together against their consent, each point gets multiplied with the point on top of it (You shouldn't be surprised, people multiply when you put them on top of each other too). Then you sum all of the results of each point and that's your value for y[n] when n = 0. Isn't that great? Then you shift the flipped function to the right one point and do it all again: multiply each point, sum all the points, bing! The result is the value of y[n] for n = 1. If you shift the flipped function to the left and do the stuff, you get the value of y[n] for n = -1.
Of course, since they're functions you don't need to keep doing that crap again and again until infinity. You express it in the form of a summed function and use some of those crazy summation properties that I can never remember and BAM! you've got the function for y[n].
So that's all well and good, but why?! Why does it have to be so random and senseless?! Why the flipping and the multiplying and the summing and the shifting and the devil's work and the sexual innuendos and the naughty ideas?! Why?! *sob* *sob*
