21–30 of 33 posts
I have an article explaining step by step how to implement code for the discrete version of the Fourier transform: https://www.nayuki.io/page/how-to-implement-the-discrete-fou...
He touches on it - but I’d love to see an intuitive explanation of why the response of each frequency to the input function is linearly independent. i.e the fact that Fourier transform of the sum is equal to the sum of the Fourier transforms. This is “why it works” - it’s what makes the frequency space an orthonormal basis - but it’s never been intuitively obvious to me. Otherwise, there would be more than one way of…
This is also why peaks on an fft are gaussian (finite window), and get sharper as the fft window is increased.
* for cosine, technically there is a peak at -ƒ too. this is because a real cosine signal is ambiguous whether it is "moving forward or backwards in time". Hence it has a peak at +/-ƒ. A complex exponetial (helix through time) has chirality due to the real and imag components, so it has a single peak at ƒ. And if you take a +ƒ (lefthanded) and -ƒ helix (righthanded) and add them, the complex part cancels out, leaving only a real "up and down" wave.
How do you animate something like this?
He touches on it - but I’d love to see an intuitive explanation of why the response of each frequency to the input function is linearly independent. i.e the fact that Fourier transform of the sum is equal to the sum of the Fourier transforms. This is “why it works” - it’s what makes the frequency space an orthonormal basis - but it’s never been intuitively obvious to me. Otherwise, there would be more than one way of…
> what would be useful is to give an example of a set of functions which are not linearly independent.
See [1,2] for example, which (I believe) has applications in compressed sensing and dictionary learning.
Fourier analysis is also approachable from the discrete setting of finite vectors instead of functions, where the fourier analysis is just an orthogonal (orthonomal when sanely defined) linear function, i.e. it acts by matrix multiplication and is represented as that matrix.
This appropriately extended to the continous setting leads to the fourier transform on functions, and also gives intuition why the fourier transform uses integrals.
I would like a general term for frequency space of a signal, without the use of the word `frequency` . This is because `frequency` is also used when describing histograms in general image processing, and is in general an overloaded term.
Any established words or phrases in the corpus? any tips?
In fMRI data, we refer to frequency space of volumetric image data as K-Space. I would like a general term for frequency space of a signal, without the use of the word `frequency` . This is because `frequency` is also used when describing histograms in general image processing, and is in general an overloaded term. Any established words or phrases in the corpus? any tips?
I think it's much easier and more direct to visualize the time-domain as superposition of helical components and the transform as an exploration of what happens when you twist the "cylinder" with varying "intensities". You avoid the vague center-of-mass spike depicted here and start from the get-go with the terms of the transform.
That doesn't sound very clear at all to me.
> You avoid the vague center-of-mass spike depicted here and start from the get-go with the terms of the transform.
The center-of-mass spike is the result of summing across all the different complex points/vectors, this is stated very clearly by the FT formula (sum (int_^) of points on a circle (exp(t)) amplified by signal strength (f(t))). Seems very explicit to me.
He touches on it - but I’d love to see an intuitive explanation of why the response of each frequency to the input function is linearly independent. i.e the fact that Fourier transform of the sum is equal to the sum of the Fourier transforms. This is “why it works” - it’s what makes the frequency space an orthonormal basis - but it’s never been intuitively obvious to me. Otherwise, there would be more than one way of…
The orthogonality is essentially follows from (1) integer frequency complex sinusoids have an average value of zero over [0,2π], and (2) if you multiply two distinct integer frequency complex sinusoids, you get another integer frequency complex sinusoid. I'm not sure that this is any more intuitive. > what would be useful is to give an example of a set of functions which are not linearly independent. See [1,2] for ex…
i think these kinds of explanations are hilariously pointless. and i don't mean to disparage because you're just trying to answer op's question but all you've done is restated the proof in english - i.e. of course it follows from that because what you've just said is the inner product of basis functions is 0. well yes of course that's definition of orthogonal.