## An ffmpeg and SDL Tutorial

The function we are using is a weighted mean using a geometric series as its weights. A weighted mean is defined like this:

w_0*x_0 + w_1*x_1 + ... + w_n*x_n
---------------------------------
w_0 + w_1 + ... + w_n

If you substitute 1 in for each `w_n`, you get your normal everyday arithmetic mean (a.k.a. an *average*).
Our function is basically a repetition of:

total = d_x + c*avg;

However, you can also look at it like this:
total = c^n*d_0 + c^(n-1)*d_1 + ... + c*d_(n-1) + d_n

in which case, this is just the top part of a weighted mean with `c^n, c^(n-1)...` as the weights. That means the bottom half is `c^n+c^(n-1)...`, which, as you may have guessed, is a simple geometric sum which works out to `1/(1-c)` as n approaches infinity.
So, by approximation, the weighted mean of our sequence of diffs is simply:

total
------- = total * (1-c)
1
-----
(1-c)

So when we get the final total and want to know the average, we just multiply it by 1-c and get the answer! There is probably a name for this way of taking the mean of a sequence, but I'm pretty ignorant and I don't know it. If you know it, please email me.

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Code examples are based off of FFplay, Copyright (c) 2003 Fabrice Bellard, and a tutorial by Martin Bohme.