Cherish (mareserinitatis) wrote,
  • Mood: nerdy

Fourier Transformers

Tonight, I was talking with my son about some math, and I mentioned basis functions. This is not a clear concept for people who don't have at least a little advanced math, so I talked specifically about Fourier Series.

My son's eyes got big.

"Oh! Are those the same as Fourier transfers?"

"You mean tranforms?"

"Yeah!"

"Where did you hear about those?!"

"The Transformers movie."

You may remember (as I vaguely do) the part where they haul in a bunch of bright people to analyze some sort of signal, and Maggie mentions Fourier Transforms.

Thus, I had to explain what Fourier Transforms are to a thirteen year old who is doing basic algebra.



First, we need a signal:



Let's use the example of the triangle wave. Fourier's idea was that you could make a triangle wave (or any other function) by taking sine (and cosine) waves with different wavelengths and amplitudes and add them together.



Looking at the plot above, you can see that the wavelength (i.e., the distance from one wavecrest to the next) is given by λ. The height of the wave is y. Although the horizontal axis is said to be distance, it could just as easily be time. We can imagine standing at a fixed point, watching the height of the wave as it moves past us and calling out, every second or so, the amplitude of the wave. Thus, we could also say that the wavelength could also be the period, or the amount of time that it takes between wavecrests to move past us.

Fourier's idea is officially called a Fourier Series. Although for the series to equal the function or signal exactly, you could technically need an infinite number of wavelengths. However, you can generally see the idea with just a few waves of just a few wavelengths (or harmonics):



If we take those different waves and examine them individually, we could also note that they have a frequency. That is, the number of wavecrests that pass by us in a given amount of time is the same for a single wave no matter when we count the wavecrests. However, no two waves will have the same number of wavecrests pass by in a given time. This means that all these waves have a unique frequency. (The frequency can also be calculated by dividing the number one by the period.)

A Fourier Transform takes a signal and looks at the waves and then shows us the frequencies of all the waves. If we only have a single sine wave, like above, we will have a frequency that is zero everywhere except for the frequency of that sine wave. More complicated signals will be made up of several of these different frequencies and thus will have several peaks. The idea is that you could move back and forth between the period of the wave and the frequency by using the Fourier Transform. If you only have the frequency information, for example, you can use that to figure out which waves you need. Add them together, and you have your original signal back.

Back to Transformers, Maggie was looking at the signal sent by the Decepticons and trying to look at its frequency content. Of course, real signal processing would have required a lot more than knowing the frequencies involved, but it sounds pretty cool to talk about "Fourier Transforms". I mean, heck, I just devoted a whole post to it!
Tags: fourier series, fourier transforms, math, movies, transformers
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