7 ms·

Fourier Transform existed, but I never had intuitive understanding until now

61 points by GianIsAlive 3y ago

32gbsd 3y ago
Hmmmm i am still missing something. Are you analysing in real time or testing chunks of the wave? It would seem like chunks
- scantis 3y ago
  Real time with DFT is a bit sketchy. Just view it as a matrix vector multiplication. The vector is the sampled signal at discrete time steps of equal spacing, the matrix is the DFT of fixed size. You need to fill your vector before the multiplication. This time needs to pass. You might shift it by just one value in time to create a sliding DFT, but nonetheless you will need to sample your time vector first. The length of each chunk is fixed by the size of the DFT. Let's say it is 1 millisecond long, each sample is 1 microsecond spaced, so it has a length of 1000. You may then create a spectrogram that captures 1 millisecond and then advances by 1 microsecond every step and this will be as close as real time as you can get. The initial delay of your analysis vector, then advancing by one sample, plus all the time you need for the calculations included cause a delay. Although this is sometimes called real time DFT. Since you may observe the change in the signal basically in real time, but after some delay. It is always done in chunks. Hope this helps.
  - 32gbsd 3y ago
    Its helps thanks.
- 112233 3y ago
  short loops made from single chunks, actually. since frequencies are in a limited band, signal must be infinite and repeating. That is the reason analysis usually multiplies signal by window function before FFT, to "fade out" discontinuity at the loop point
userbinator 3y ago
Multiplication is used because when a frequency is present and in phase (meaning peaks are aligned), it would result in more 'positive curves' sitting above the x-axis than negative curves below. This is also the principle behind DSSS modulation ( https://en.wikipedia.org/wiki/Direct-sequence_spread_spectrum https://en.wikipedia.org/wiki/Direct-sequence_spread_spectru... ), which is used in numerous communications systems.
- GianIsAlive 3y ago
  Thank you for sharing!
voidhorse 3y ago
https://www.dspguide.com/ https://www.dspguide.com/ is a great, general introduction to digital signal processing and includes coverage of the Fourier Transform. I'd recommend checking it out for anyone struggling to grok DSP concepts. This article seems to have the right intent, but I felt the quality of the writing would need to improve for it to become truly useful for learners.
- GianIsAlive 3y ago
  Thank you. I read a lot of papers drive equations from Euler's formula, which helped with result, but I didn't have any intuitive understanding, but when I watched geometric breakdown of it, it suddenly made sense. It's just my attempt to share my excitement. I do agree my writing could definitely be improved.
- shaklee3 3y ago
  I think the writing was poor, but also there are some things that are just wrong. Like most of the time people sample at 10x Nyquist. That would be a huge waste of processing given that it doesn't give you any more information about the signal than just Nyquist.
  - GianIsAlive 3y ago
    I wouldn't say over sampling is wrong. Inefficiency and inaccuracy has different consequences in my opinion. Maybe I'm missing something idk.
  - mitthrowaway2 3y ago
    Almost by definition, it does give you more information about the signal than sampling at the Nyquist frequency though? If you have any noise at all, oversampling will be very helpful to improve the signal to noise ratio.
    - shaklee3 3y ago
      for band limited signals the snr is not increased by oversampling
tempodox 3y ago
For me this produced more confusion than it explained anything.
- GianIsAlive 3y ago
  Sorry it didn't hit the mark for you. It's my attempt to explain it with geometry, but I clearly need to improve my writing.
- totetsu 3y ago
  Yes, This reads more like someones notes to self, than an article written with a general audience in mind.
deleted 3y ago
[deleted]
rubatuga 3y ago
For those wanting a visual understanding to Fourier transforms, I enjoyed 3blue1brown's video [0] which is a bit more of a continuous approach. [0]: https://www.youtube.com/watch?v=spUNpyF58BY https://www.youtube.com/watch?v=spUNpyF58BY
- GianIsAlive 3y ago
  I like his video too! Thanks for sharing.
mejutoco 3y ago
For anyone wanted an intuitive understanding of Fourier transforms I recommend "Who Is Fourier?: A Mathematical Adventure". It seems a bit childish, but it actually explains everything perfectly. https://www.goodreads.com/en/book/show/706622 https://www.goodreads.com/en/book/show/706622

61 points by GianIsAlive 3y ago

Hmmmm i am still missing something. Are you analysing in real time or testing chunks of the wave? It would seem like chunks

scantis 3y ago
Real time with DFT is a bit sketchy. Just view it as a matrix vector multiplication. The vector is the sampled signal at discrete time steps of equal spacing, the matrix is the DFT of fixed size. You need to fill your vector before the multiplication. This time needs to pass. You might shift it by just one value in time to create a sliding DFT, but nonetheless you will need to sample your time vector first. The length of each chunk is fixed by the size of the DFT. Let's say it is 1 millisecond long, each sample is 1 microsecond spaced, so it has a length of 1000. You may then create a spectrogram that captures 1 millisecond and then advances by 1 microsecond every step and this will be as close as real time as you can get. The initial delay of your analysis vector, then advancing by one sample, plus all the time you need for the calculations included cause a delay. Although this is sometimes called real time DFT. Since you may observe the change in the signal basically in real time, but after some delay. It is always done in chunks. Hope this helps.
- 32gbsd 3y ago
  Its helps thanks.
112233 3y ago
short loops made from single chunks, actually. since frequencies are in a limited band, signal must be infinite and repeating. That is the reason analysis usually multiplies signal by window function before FFT, to "fade out" discontinuity at the loop point

userbinator 3y ago

Multiplication is used because when a frequency is present and in phase (meaning peaks are aligned), it would result in more 'positive curves' sitting above the x-axis than negative curves below. This is also the principle behind DSSS modulation ( https://en.wikipedia.org/wiki/Direct-sequence_spread_spectrum https://en.wikipedia.org/wiki/Direct-sequence_spread_spectru... ), which is used in numerous communications systems.

GianIsAlive 3y ago
Thank you for sharing!

voidhorse 3y ago

https://www.dspguide.com/ https://www.dspguide.com/ is a great, general introduction to digital signal processing and includes coverage of the Fourier Transform. I'd recommend checking it out for anyone struggling to grok DSP concepts. This article seems to have the right intent, but I felt the quality of the writing would need to improve for it to become truly useful for learners.

GianIsAlive 3y ago
Thank you. I read a lot of papers drive equations from Euler's formula, which helped with result, but I didn't have any intuitive understanding, but when I watched geometric breakdown of it, it suddenly made sense. It's just my attempt to share my excitement. I do agree my writing could definitely be improved.
shaklee3 3y ago
I think the writing was poor, but also there are some things that are just wrong. Like most of the time people sample at 10x Nyquist. That would be a huge waste of processing given that it doesn't give you any more information about the signal than just Nyquist.
- GianIsAlive 3y ago
  I wouldn't say over sampling is wrong. Inefficiency and inaccuracy has different consequences in my opinion. Maybe I'm missing something idk.
- mitthrowaway2 3y ago
  Almost by definition, it does give you more information about the signal than sampling at the Nyquist frequency though? If you have any noise at all, oversampling will be very helpful to improve the signal to noise ratio.
  - shaklee3 3y ago
    for band limited signals the snr is not increased by oversampling

tempodox 3y ago

For me this produced more confusion than it explained anything.

GianIsAlive 3y ago
Sorry it didn't hit the mark for you. It's my attempt to explain it with geometry, but I clearly need to improve my writing.
totetsu 3y ago
Yes, This reads more like someones notes to self, than an article written with a general audience in mind.

deleted 3y ago

[deleted]

rubatuga 3y ago

For those wanting a visual understanding to Fourier transforms, I enjoyed 3blue1brown's video [0] which is a bit more of a continuous approach. [0]: https://www.youtube.com/watch?v=spUNpyF58BY https://www.youtube.com/watch?v=spUNpyF58BY

GianIsAlive 3y ago
I like his video too! Thanks for sharing.

mejutoco 3y ago

For anyone wanted an intuitive understanding of Fourier transforms I recommend "Who Is Fourier?: A Mathematical Adventure". It seems a bit childish, but it actually explains everything perfectly. https://www.goodreads.com/en/book/show/706622 https://www.goodreads.com/en/book/show/706622