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sure, that’s the standard way of teaching and is useful background, but you don’t need calculus or the continuous Fourier transform to come up with or understa
by rrss 5y ago
sure, that’s the standard way of teaching and is useful background, but you don’t need calculus or the continuous Fourier transform to come up with or understand the DFT. Once you know what a basis is for a finite dimensional vector space, the DFT is a straightforward change of basis to a particularly useful set of basis vectors which are easy to write down.
The manuals you cite cover several different transforms (not just the DFT) as well as the complexities of the FFT.
- hasmanean 5y agoOr more generally, it’s the correlation of your signal with a set of sine waves of different frequencies. It doesn’t have to be a sine wave. You could correlate your signal to a fart sound (or a set of them). It’s a valid transform but it would not be invertible—given a vector of scalar values showing how much your signal resembled each type of fart, you could not reconstruct the original signal. However you can do that with the FT. It’s invertible and you can recover the signal from the spectrum. So in some sense it doesn’t lose information. Energy in the signal is equal to energy in the Fourier spectrum.