A DFT is a "Discrete Fourier Transform". An FFT is a "Fast Fourier Transform". An FFT is a DFT, but is much faster for calculations. The whole point of the FFT is speed in calculating a DFT.

Why Do This?

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The reason to learn about the DFT and FFT is in order to get a frequency spectrum of a wave or to understand better what frequencies it is composed of. This might allow you to better identify, for example, a sound wave that you have sampled than could be done with the time wave, which is useful for speech recognition. Or, maybe you want to add or subtract frequencies and recreate the original wave with these modifications using an inverse Fourier Transform. Doing this with light waves you could, for example, remove dirty spots or noise from an image, or find recurring patterns in an image. You may have other ideas as to what you can do with the frequency components of a wave. The sky is the limit!

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