A noob learns FFT

I have never done any time–frequency transformations. I understand the general idea, but I have never worked with the details. Today I had a reason to use it, but that means learning the basics first.

This article contains a bunch of fancy graphs, and we'll try to build an intuition from the basics without doing complicated maths ourselves.

Full article (10–19 minute read): A noob learns FFT

Flashcard of the week

A simpler question from Deming's Some Theory of Sampling:

If the average height of all n adults is x, and the average height of all m children is y, what is the population average?

The classical mistake is taking an average of averages, i.e. thinking the population average must be (x+y)/2. That's not true in general, because it lets the more populous group influence the result less than it should. (To see this, take an extreme case, like there being 100 adults and just two children. With the naïve approach, each child would have as much influence on the result as 50 adults! That is backwards: with so many adults compared to children, the population average should be basically the adult average and the children should have little-to-no effect.)

The next level of mistake is rejecting the premise by saying "you can't take an average of averages and get something sensible out of it". I see this a lot in discussions of production metrics in the software world.

Then there's the way to actually do it. We can do it because we know how many adults and children there are. This is how:

(nx + my)/(n+m)

This expression scales the averages up by their respective counts, and then divides by total count, and this gives the correct number for the population average. If there are 100 adults and only two children, the first term corresponding to adults will be much larger than that corresponding to children, as we'd expect.

What's neat about this is that when n and m are close to each other, they kind of cancel out. If n ~= k and m ~= k the expression almost reduces to

(kx + ky)/(k+k) = k(x+y)/(2 k) = (x+y)/2

I.e. when the groups are around the same size, the naïve method of taking averages of averages actually works! (The error of this approximation scales linearly with both the difference in size of the groups, and the difference between x and y.)

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