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Crossposted at Electrical Engineering SE


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 $-\infty$ to $+ \infty$. 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.

Forward Fourier Transform of a signal:

$$F \left( \omega \right) = \int_{- \infty}^{\infty} f \left( t \right) e^{-j \omega t} dt \tag{1} \label{FourierTranform} $$

Inverse Fourier Transform of a signal:

$$f \left( t \right) = \int_{- \infty}^{\infty} F \left( \omega \right) e^{j \omega t} d \omega \tag{2} \label{InverseFourierTranform} $$

  1. But for real-world signals, let's say the audio signal from a speaker - if we need to do the Fourier analysis of the resultant signal - I am not sure what should be the right approach.

For eg: if there is a single tone from a speaker, the FT would be different based on whether I assume the tone continues or goes to zero. Of course, both approaches would give me identical waveforms valid till time $t^{\prime}$ (upon Fourier inverse).

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  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?

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    $\begingroup$ This is pretty much what STFT is doing and this is why it is used quite extensively in audio. STFT finds use in Audio, Sound engineering, Speech Processing, Natural Language Processing, Audio Machine Learning and many more. I suggest you have a look at that if haven't yet. $\endgroup$
    – ZaellixA
    Jun 2, 2023 at 10:06
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    $\begingroup$ Looks like what matters for our brain is the instantaneous info of sound coming to our ears - without bothering about past and future info.”. Are you sure about that? $\endgroup$
    – Jdip
    Jun 2, 2023 at 14:57
  • $\begingroup$ The "updating FT" is a convolution (stack of). $\endgroup$ Jun 2, 2023 at 20:49
  • $\begingroup$ @Jdip: For eg: a person just entering a concert hall in the middle of the session and a person hearing it for long, experience the sound effect. $\endgroup$ Jun 6, 2023 at 9:14
  • $\begingroup$ That’s not really what “instantaneous” means though. $\endgroup$
    – Jdip
    Jun 6, 2023 at 11:06

2 Answers 2

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If the signal is not stationary, and you want to know the change of the spectrum component with time, you may use the short-time Fourier transform (STFT), or the more advanced one, the wavelet transform (WT). For example, for a chirp signal (i.e., a swept-frequency signal whos frequency varies as time passes), with WT (or STFT), you can identify what frequency component is at different brief time intervals, while you cannot do so with the traditional FT. WT has many advantages over STFT and is more advanced.

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The most common method for time frequency analysis is the Short Term Fourier Transform (or STFT for short).

Basically the input signal is chopped up into small chunks and then you perform a Fourier transform on each chunk. So you get a representation that depends BOTH on time and frequency.

That's actually a pretty good model to describe Human auditory perception. You can follow a melody, i.e. the pitch is changing over time.

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

Yes and no. It's certainly NOT instantaneous but the time window is relatively short. The time constants of the auditory system for spectral perception are in the range of about 10ms - 100ms (depending on what exactly we are listening for).

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