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I think the best semi-intuitive, non rigorous explanation I've seen of the Fourier transform is still one that first explained signal correlation in the time do

I think the best semi-intuitive, non rigorous explanation I've seen of the Fourier transform is still one that first explained signal correlation in the time domain and then described the transform as basically performing correlation on the signal for all the possible sines at different frequencies. Essentially you're just testing for the presence of individual sine waves (of different frequencies) within the signal. Unfortunately I can't remember where I saw this explanation. It might've been in the Scientist's and Engineers Guide to DSP book.

ajross 4y ago
> best semi-intuitive, non rigorous explanation [...] signal correlation in the time domain [...] basically performing correlation on the signal for all the possible sines at different frequencies That's only going to make sense for someone who understands your jargon usage of "correlate", and who groks that integrals of sine curves are orthogonal under addition. That's precisely the hard part the linked explanation is trying to explain. If you've already gotten that far then the DFT is just calculator math.
- voidhorse 4y ago
  True! Though tbh, I did not even know that property of integrals, since I've studied this stuff only in the discrete realm. At the totally basic level, another thing that I think can really trip people up and that a lot of texts are not clear about is that the transform is about moving between representations and that we're not actually transforming anything. Not enough maths texts take the time to explain how the representation connects up with the objects we wish to study and how there are multiple possible ways to represent them, and in some cases, like Fourier, useful ways to "translate" between different representations.
tonmoy 4y ago
To me the sentence in the OP is the same as the sentence you posted (your sentence is easier to parse since I already know the meaning of correlate)
teldau 4y ago
I agree, FFT is simply a correlation of a bunch of sine waveforms. The extra tidbit - the sine waveforms are orthogonal to each other (their cross correlation is zero), so the transform can be inverted. (in math terms: they form a basis set. I'm mostly referring the STFT here, which is the 'common' FFT use. not infinite FFT).
wankle 4y ago
That's the best definition I've seen either in the OP post or any comment I've read on the page, it's exactly how I think of it.

ajross 4y ago

> best semi-intuitive, non rigorous explanation [...] signal correlation in the time domain [...] basically performing correlation on the signal for all the possible sines at different frequencies That's only going to make sense for someone who understands your jargon usage of "correlate", and who groks that integrals of sine curves are orthogonal under addition. That's precisely the hard part the linked explanation is trying to explain. If you've already gotten that far then the DFT is just calculator math.

voidhorse 4y ago
True! Though tbh, I did not even know that property of integrals, since I've studied this stuff only in the discrete realm. At the totally basic level, another thing that I think can really trip people up and that a lot of texts are not clear about is that the transform is about moving between representations and that we're not actually transforming anything. Not enough maths texts take the time to explain how the representation connects up with the objects we wish to study and how there are multiple possible ways to represent them, and in some cases, like Fourier, useful ways to "translate" between different representations.

tonmoy 4y ago

To me the sentence in the OP is the same as the sentence you posted (your sentence is easier to parse since I already know the meaning of correlate)

teldau 4y ago

I agree, FFT is simply a correlation of a bunch of sine waveforms. The extra tidbit - the sine waveforms are orthogonal to each other (their cross correlation is zero), so the transform can be inverted. (in math terms: they form a basis set. I'm mostly referring the STFT here, which is the 'common' FFT use. not infinite FFT).

wankle 4y ago

That's the best definition I've seen either in the OP post or any comment I've read on the page, it's exactly how I think of it.