Fourier Transforms by Looking at it

[Mindscrape]

Some people are able to do Fourier transforms without doing a single integral (i.e. just looking at a function). After thinking about it for a while I discovered that convolution is really helpful. For example, because two square waves are convolved to make a triangle wave, then the Fourier transform will be the Fourier transform of the square multiplied by the Fourier transform of the square. I am sure there are other methods though, does anybody know of anything better or any good tutorials of approximating the Fourier transform of functions?

 

[fourier jr]

i've heard of a former prof at my old university who could do stuff like that. people always seemed to just think he was some sort of calculating freak because when talking to someone calculating a Fourier transform never seemed to slow him down. if there's a trick to it i think it would be cool to know.

 

[hypermorphism]

It may just be experience. Like the way most of us can do integrals, without having to go through pen/paper or even acknowledge intermediary steps.

 

[Hurkyl]

How many Fourier transforms have you done in your lifetime? A couple dozen? How many Fourier transforms do you think he has done in his lifetime?

In some sense, it's like ordinary arithmetic. If you don't even know your addition tables, it's hard to add things. When you learn your addition tables, you can add things a lot faster, and sometimes in your head. And if you do lots of addition (but not mindlessly), or go looking for them, you can pick up tricks that can let you add faster.

 

[fourier jr]

hmm working at it... that's a good trick

 

[Hurkyl]

That reminds me of a Simpsons episode.

Bart Simpson was once compelled to find a way to distract himself from a disturbing scene, and the only option was to repeatedly read off the names of the planets off of a nearby poster.

He later got an A on an astronomy test. He remarked that the answers were stuck in his brain; it was a whole new kind of cheating!

 

[StatusX]

Another useful trick is when you have more than one copy of a single shape. For example, consider two gaussians side by side. You can obtain this shape by convolving a single gaussian with two delta functions centered at, say, -T and +T. The Fourier transform of these is just e^iwT+e^-iwT=2cos(wT), and so the transform of two gaussians is just the transform of a single gaussian modulated by a cosine function. This readily generalizes to more than two copies of the shape (and you can even take the infinite limit to recover the transform of a periodic signal), or copies with different amplitudes.