[HN Gopher] Fourier Transforms
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Fourier Transforms
Author : o4c
Score : 87 points
Date : 2025-11-05 17:24 UTC (11 days ago)
(HTM) web link (www.continuummechanics.org)
(TXT) w3m dump (www.continuummechanics.org)
| thinkzilla wrote:
| I generally agree with the point of the article ("Fourier
| transform is not magical").
|
| However saying it is "just" curve fitting with sinusoids fails to
| mention that, among an infinite number of basis functions, there
| are some with useful properties, and sinusoids are one such: they
| are eigenvectors of shift-invariant linear systems (and hence are
| also eigenvectors of derivative operators).
| physicsguy wrote:
| All quite good examples but I would say that these are quite well
| known. It's also missing that there are mitigation strategies for
| some - for e.g. in vibration analysis it's typical to look at the
| Hann windowed data to remove the effect of partial cycles, and
| it's common to overlap samples too. Similarly there are other
| tools like the Cepstrum which help you identify periodic peaks in
| the spectral data.
| CamperBob2 wrote:
| Fourier tutorials are a dime a dozen, so it would likely have
| been a better idea to link to his excellent wavelet tutorial at
| https://www.continuummechanics.org/wavelets.html . Good
| explanations of that concept are a lot harder to come by.
| seam_carver wrote:
| I made a video about a cool application of the Discrete Fourier
| Transform regarding color eink Kaleido 3 and manga:
|
| https://youtu.be/Dw2HTJCGMhw?si=Qhgtz5i75v8LwTyi
|
| Learning about Fourier is really interesting in image processing,
| I'm glad I found a good textbook explaining it.
| lyelibi wrote:
| It's not just curve fitting because basis functions have
| characteristics which make them desirable for the kind of
| decomposition one is trying to find. We typically assume in
| factor analysis that factors are gaussian random variables
| without clear and repeating patterns. Fourrier transforms force
| us to think in similar terms but accounting for specific dynamics
| factor (I. E. Basis functions) should capture.
|
| Also how do we construct those orthogonal basis functions for any
| downstream task is an interesting research question!
| nickpsecurity wrote:
| "Few people appreciate statistics. But at least they seem OK with
| this and don't go off starting religious wars over the subject."
|
| Frequentist vs Bayesian get debated constantly. I liked this
| video about the difference:
|
| https://youtu.be/9TDjifpGj-k?si=BpjlTCWIFMu506VL
| behnamoh wrote:
| This gets brought up quite often here but something people don't
| talk about is _why_ Fourier needed to do this. Historical context
| is really fun! In the late 1800s, Fourier wanted to
| mathematically describe how heat diffuses through solids, aiming
| to predict how temperature changes over time, such as in a heated
| metal rod. Observing that temperature variations evolve smoothly,
| he drew inspiration from the vibrating string problem studied by
| Euler and D'Alembert, where any complex motion could be expressed
| as a sum of simple sine waves. Fourier hypothesized that heat
| distribution might follow a similar principle; that any initial
| temperature pattern could be decomposed into basic sinusoidal
| modes, each evolving independently as heat diffused.
| johnp314 wrote:
| Minor correction, Fourier made his breakthroughs in the early
| 1800's. He worked under the reign of Napoleon and continued in
| the decade thereafter.
| cycomanic wrote:
| I really don't get the point the article is making. I think the
| whole point about curve fitting is really a distraction, they
| could have simply stated that the FFT has periodic boundary
| conditions, so if you take the FFT of something that only extends
| a finite time of your sampling window, you will see your delta
| functions in the frequency domain spaced by the inverse of the
| length of your sampling window, i.e. the FFT "sees" your finite
| window as a pulse train. That's well known and a fundamental
| aspect of Fourier transforms.
|
| But then the statements about the discontinuous "vibrations".
| E.g. in the case of the 1 Hz cycle over half the window the
| author states that:
|
| > Yet the FFT of this data is also very complex. Again, there are
| many harmonics with energy. They indicate that the signal
| contains vibrations at 0.5Hz, 1.0Hz, 1.5Hz, etc. But the time
| signal clearly shows that the 'vibration' was only at 1Hz, and
| only for the first second.
|
| The implication that there is a vibration only at 1Hz is plain
| wrong. To have a vibration abruptly stop, you need many
| frequencies (in general the shorter a feature in the time domain
| the more frequency components you need in the frequency domain).
| If we compare for example a sine wave with a square wave at the
| same frequency, the square wave will have many more frequency
| components in the Fourier domain (it's a sinc envelope of delta
| functions spaced at the frequency of the wave in fact). That's
| essentially what is done in the example the sine wave is
| multiplied by a square wave with half the frequency (similar
| things apply to the other examples). Saying only the fundamental
| frequency matters is just wrong.
|
| This is also not just a "feature of the fitting to sines", it's
| fundamental and has real world implications. The reason why we
| e.g. see ringing on an oscilloscope trace of a square wave input
| is because the underlying analog system has a finite bandwidth,
| so we "cut-off"/attenuate higher frequency components, so the
| square wave does not have enough of those higher frequencies
| (which are irrelevant according to the author) to represent the
| full square wave.
| krackers wrote:
| > FFT has periodic boundary conditions
|
| FFT is simply an algorithm to efficiently compute the DFT. The
| fact that the article makes no mention of DFT vs fourier series
| vs DTFT is going to end up creating more confusion that it
| solves. For some reason introductory tutorials always start
| with the DFT (usually mistakenly using FFT and DFT
| interchangeably), even though to me the continuous fourier
| transform is far easier to conceptually understand. Going from
| continuous fourier transform to the DTFT, is just applying the
| FT to a dirac-combed (sampled) function. Then from DTFT to DFT
| you introduce periodic boundary condition. Fourier series is
| just applying FT to a function that happens to already be
| periodic, resulting in a finite set of discrete frequencies.
|
| There is a connection between fourier series and DFT in that if
| the fourier series is computed for the periodic resummation of
| a signal, and then the DFT is computed for the original signal
| (which implicitly involves applying a periodic boundary
| condition), the DFT is just the periodic resummation of the
| fourier series.
|
| I spent ages meditating on this image
| https://en.wikipedia.org/wiki/Discrete_Fourier_transform?#/m...
| before everything finally clicked, it's a shame that
| introductions never once mention DTFT
| cycomanic wrote:
| Completely agree the transition between continuous and
| discrete domains is often glossed over and people use DFT,
| FT, DTFT and FFT almost interchangeably (I certainly have
| been guilty of that myself, and you are correct the FFT and
| DFT are equivalent for this discussion).
|
| An interesting fact (somewhat related to your mentioning of
| the DTFT) is that one can consider the DFT as a filter with a
| sinc transfer function. That's essentially how you can
| understand the spectrum of an OFDM signal. You perform a
| block based FFT on your input bit/symbol stream, so you have
| waves at different carriers. However, because the stream is
| timevarying you essentially get sinc shaped spectra spaced at
| the symbol rate (excluding cycling prefixes etc.). So your
| OFDM spectrum is composed of many sincs spaced at fb, which
| is very squarish which is one of the reasons why OFDM is so
| advantageous.
| a-dub wrote:
| seems to be missing some stuff. first, the notion that most real-
| valued functions can be decomposed to an infinite sum of
| orthogonal basis functions of which fourier bases are one. this
| is the key intuition that builds up the notion of linear
| decomposition and then from which the practical realities of
| computing finite dfts on sampled data arise. second, the talk of
| transients absent the use of stfts and spectrograms seems really
| weird to me. if you want to look at transients in nonstationary
| data, the stft and spectrogram visualization are critical.
| computing one big dft and looking at energy at dc to detect drift
| seems weird to me.
|
| maybe this is the way mechanical engineers look at it, but
| leaving out stfts and spectrograms seems super weird to me.
| cbondurant wrote:
| This feels like a very indirect way of saying "yes the fourier
| transform of a signal is a breakdown of its component
| frequencies, but depending on the kind of signal you are trying
| to characterize for it might not be what you actually need."
|
| Its not that unintuitive to imagine that if all of your signals
| are pulses, something like the wavelet transform might do a
| better job at giving you meaningful insights into a signal than
| the fourier transform might.
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