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+1 for the Brigham reference. It's the best explanation I've ever read.

@andand: thanks, yes, excellent book, even though it's quite old now: the applications chapters are good too, and still relevant.

The FFT is just an efficient implementation of the DFT. The results should be identical for both, but in general the FFT will be much faster. Make sure you understand how the DFT works first, since it is much simpler and much easier to grasp.

When you understand the DFT then move on to the FFT. Note that although the general principle is the same, there are many different implementations and variations of the FFT, e.g. decimation-in-time v decimation-in-frequency, radix 2 v other radices and mixed radix, complex-to-complex v real-to-complex, etc.

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If you seek a plain English explanation of DFT and a bit of FFT as well, instead of academic goggledeegoo, then you must read this: http://www.dspdimension.com/admin/dft-a-pied/

Shame there's no explanation of the actual FFT algorithm. Great link anyway, thanks!

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(If you want to learn more about Fourier analysis, I recommend the book Fourier Analysis and Its Applications by Gerald B. Folland)

Yes, the FFT is merely an efficient DFT algorithm. Understanding the FFT itself might take some time unless you've already studied complex numbers and the continuous Fourier transform; but it is basically a base change to a base derived from periodic functions.

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