0
\$\begingroup\$

A very naive question: How do we use Fourier transform for real world signals - for which you have the information only up to the present instant (and the present time keeps moving continuously)?

The Fourier integral is defined from -inf to +inf. The standard approach I see is we would assume/know a priori that the signal is zero beyond a certain time span of our interest.

enter image description here

  1. But for real world signals, lets say the audio signal from a speaker- If we need to do the Fourier analysis of the resultant signal - Iam not sure what should be the right approach?

For eg: if there is a single tone from speaker, the FT would be different based on whether I assume the tone continues or tone goes to zero. Of course both approaches would give me identical waveforms valid till time t'( Upon Fourier inverse).

enter image description here

  1. During a concert, if different musical instruments are played together, it seems our brain has the ability to distinguish the multiple sources/tones.

Looks like what matters for our brain is the instantaneous info of sound coming to our ears -without bothering about past and future info.

How do I link these 2 concepts?

\$\endgroup\$
1
  • \$\begingroup\$ Often the array of samples is multilpied with a "Hanning window" that reduces the significance of the values at the interval ends. \$\endgroup\$
    – Jens
    May 26, 2023 at 19:55

1 Answer 1

1
\$\begingroup\$

When we look at the mathematical proofs of Fourier transforms we see that it actually doesn't work for non-periodic signals. The signal in your diagram is an example of a a non-periodic signal, as it is not infinitely repeating.

That's not too much of a problem, we usually will apply a trick to solve this limitation by assuming that whatever time domain signal sample we have is exactly one single period of an infinitely repeating periodic signal.

EDIT: @Tim Williams pointed out the bit above isn't correct, and that FTs do work for periodic signals, but that when we do the Fast Fourier Transform we use a finite section of an assumed periodic wave.

This is an incredibly neat trick, as it not only makes the maths work, but it allows us to consider any signal, it doesn't have to actually be periodic anymore. It could be a 1 second duration of complete random noise, but the maths assumes that we actually have an infinite periodic signal constantly repeating, where our 1 second sample is a single period.

So we go from this:

enter image description here

To this:

enter image description here

Now if we do the FFT of this little 1 second snippet, it will give us an answer of the amplitudes and phases of each frequency in the snippet, where the maximum frequency is decided by our sample rate.

So in answer to point 1: the maths actually assumes the tone continues!

For point 2, you're correct that our brains are able to distinguish tones, I'm not a biologist/phycologist by any means, but my understanding is that we are actually doing some form of FFT-like process in our brains, and over time we learn the shapes/patterns in the result to be able to distinguish the sound of a flute vs violin etc (for example a violin being a very clean fundamental/overtones, where a flute may have a lot of higher frequencies due to all the wind rushing through).

What our brains are doing (and what basically any 'continuous' monitoring FFT plot) is doing, is actually taking a defined period of time (for us it was 1 second, for our brains it might be microseconds), doing an FFT of that 'chunk' then doing the next chunk etc etc. That way you get an FFT that is able to change over time to reflect not an 'instant' but a small time window that is very close to that instant.

\$\endgroup\$
5
  • 2
    \$\begingroup\$ This isn't quite correct: the Fourier Transform is a continuous map from infinite time ± to infinite frequency ± (and vice versa). The Fourier Series is the special case applied to periodic functions. (You can take an FT of a periodic function and get the same (infinite series) result, just with extra steps.) When we FFT, we transform a finite segment, which we assume repeats indefinitely. The harsh windowing of an FFT can be tempered with window functions, and using a fractional advance (sliding the window say 1/4 of the way along at a time). More generally, see: wavelet transforms. \$\endgroup\$ May 26, 2023 at 11:59
  • 1
    \$\begingroup\$ No, the Fourier transform can absolutely handle non-periodic functions. \$\endgroup\$
    – Hearth
    May 26, 2023 at 14:54
  • \$\begingroup\$ @TimWilliams thanks for the correction, I've added an edit. \$\endgroup\$
    – Entropy
    Jun 1, 2023 at 7:06
  • \$\begingroup\$ The edit doesn't really clarify it, since you say that the Fourier transform doesn't work for non-periodic waveforms, which is the thing it absolutely works for. It is periodic waveforms where it gets tricky since the integral on the Fourier transform won't converge to a finite value given an infinitely repeating input. \$\endgroup\$ Jun 1, 2023 at 13:39
  • \$\begingroup\$ Also the question asks about Fourier transforms, but most of the text talks about the FFT, which is an implementation of the DFT not the Fourier transform. \$\endgroup\$ Jun 1, 2023 at 13:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.