# Fourier Transforms For Dummies

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Could someone recommend a resource where I could get a thorough, but gentle explanation of Fourier Transforms? For whatever reason I am quite intimidated by this topic. Hopefully like most things it will turn out to be much simpler than I imagine it to be.

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The best two sources are wikipedia and mathworld. To understand Fourier transforms from the mathematical perspective, you need to know up through integral calculus and maybe some linear algebra.

But the basic idea is that you can express any function f(x) as an infinite sum of sine and cosine waves of varying frequencies.
The Fourier transform takes f(x) and produces a new function, F(k), where k is a spatial frequency. F(k) is related to the amplitude of a sine wave at spatial frequency k.

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http://ocw.mit.edu/OcwWeb/Mathematics/18-310Fall-2004/LectureNotes/index.htm

See chapter 23 on FT and FFT.

I don't know if it's good or not.

I'm interested in more links to resources also.

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May I had, its invaluable that you *really* get imaginary numbers too, before you get into fourier transforms.

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I don't know what is your math background, but if you know some linear algebra and single variable calculus, then to get started with Fourier series and integral functions I'd recommend Boundary Value Problems, 5ed by David Powers. If you really want to dive into Fourier Transforms first (and you know enough math), then Mathematical Methods for Physicists by George Arfken is really good, and it covers almost all the math you'd ever want to know unless you plan to be a mathematician.

Hope it helps.

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Do you really need to understand the mathematics behind the transform, or do you just want to feel comfortable with the operation? There is a lot of foundational knowledge and mathematical intuition to be established before the ingredients for a FT can be understood, and even then, you have to get your head around some very abstract complex analysis before the integral itself seems to make any sense. After an extensive education in mathematics, I can see and make sense of the operation in terms of its parts, but I don't think anyone can truly grok the transform for what it truly is: a quasi-two-dimensional contour integral through infinity.

It would be much more worthwhile for you to get the hang of orthogonal bases in the context of linear algebra. Once you're comfortable with the concept of arbitrary objects in a vector-space being representable in terms of the basis vectors of that space, you should recognise the special case of Fourier series. From here it's not much of a leap to envisage the Fourier transform (informaly, an infinitely fine Fourier series transformation) for what it does: represent an arbitrary analytic function in the infinite-dimensional basis of harmonic waves.

With that intimidating mouthful out of the way, you'll probably get the most satisfaction by jumping straight in and ignoring as much of the theory as you can get away with. Once you know how to use it, understanding it will be that much simpler. I wouldn't be surprised if many of the world's experts on signal processing are clueless about the analysis behind their most powerful tool.

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Well said man, well said!

I have seen enough math to understand a little of just what you said. The FT was never really discussed in any of my classes, but I realize that I have the background to really understand things.

Thanks!

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Quote:
 Original post by TheAdmiral[...]With that intimidating mouthful out of the way, you'll probably get the most satisfaction by jumping straight in and ignoring as much of the theory as you can get away with. Once you know how to use it, understanding it will be that much simpler. I wouldn't be surprised if many of the world's experts on signal processing are clueless about the analysis behind their most powerful tool.Admiral
Couldn't have put it better myself. What I've heard from my fellow colleagues in engineering involved with signals, they are forced to learn FFT, of course, with all of the details and then they go forward with their lives and prefer wavelets as they are, I've heard, easier to handle. [smile]