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Fourier Transformers

subspace oscillations
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!


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Comments

( 14 Transmissions — Comment )
cosmic_reverie
Apr. 8th, 2009 01:14 pm (UTC)
I could have used you when I was trying to figure out FTs in undergrad!

I wonder how many parents of kids who watch the Transformers movie would know what a Fourier transform is if their kid asked (or even know how to spell it to look it up).
mareserinitatis
Apr. 8th, 2009 03:44 pm (UTC)
I wondered the same thing myself. I think this is why a lot of kids who go into math and science are more likely to have parents who are also technical. They probably get more of intuition for it when they are young from their parents, rather than relying on people who probably don't have as much interest in math.

I actually was first introduced to the concept in a physics class, where the professor gave me a very lousy explanation. I decided to take matters into my own hands and went and talked to a math professo I knew, who explained all this (with substantially more detail, including the notion of orthogonality) in about all of 15 minutes. He was awesome. I think he's the reason I made it through physics. :-D
pammalamma
Apr. 8th, 2009 02:18 pm (UTC)
At NI, where I used to work, they made boards and software that did FFT, plus a lot of other data transformations. That's their thing. http://www.ni.com
mareserinitatis
Apr. 8th, 2009 03:45 pm (UTC)
You know, I thought you worked for TI. Or did you work there, too? I've had quite a bit of experience with NI's hardware and software. :-)
pammalamma
Apr. 8th, 2009 10:16 pm (UTC)
Nope, just NI. Worked there 6 years, in three different groups: GPIB, CVI, and DAQ. :o)
flyingflux.blogspot.com
Apr. 8th, 2009 04:57 pm (UTC)
"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."

Phase information is also needed to recreate the true original time domain signal.
mareserinitatis
Apr. 8th, 2009 05:08 pm (UTC)
True...but I was trying to keep it as simple as possible. :-) Even with what I had, he admitted he didn't fully understand.
flyingflux.blogspot.com
Apr. 8th, 2009 08:58 pm (UTC)
A big gap to go from algebra to Fourier transform.

Besides, tell him Fourier was a ninny (or whatever kids call each other these days), the real genius was Laplace. Don't forget to teach him sqrt(-1) while you're at it.
mareserinitatis
Apr. 8th, 2009 10:52 pm (UTC)
You have a lot of expectations, methinks. :-)
flyingflux.blogspot.com
Apr. 9th, 2009 03:14 am (UTC)
Shouldn't every child understand Maxwell's equations, play 3 instruments, and speak 4 languages by the time they enter high school? Uh oh...I think someone has low expectations. ;-) (just kidding, eh)
mareserinitatis
Apr. 11th, 2009 01:23 am (UTC)
All I can say is that I'm glad you weren't my parent. :-)
prof_brian
Apr. 10th, 2009 05:19 pm (UTC)
Yay, I love Fourier Transforms! The theoretical physics class I'm teaching next semester is rife with them :)
mareserinitatis
Apr. 11th, 2009 01:23 am (UTC)
You know, I don't remember using them as much in physics as I did in engineering. :-)
prof_brian
Apr. 11th, 2009 03:34 am (UTC)
You can even use them in paleontology! Damn useful things all around.

http://arxiv.org/abs/0807.4729
( 14 Transmissions — Comment )

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