# How would you compute Fourier transform of a real world signal where the signal keeps getting updated (not a static one)?

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). 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?

• 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. Jun 2 at 10:06
• 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?
– Jdip
Jun 2 at 14:57
• The "updating FT" is a convolution (stack of). Jun 2 at 20:49
• @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. Jun 6 at 9:14
• That’s not really what “instantaneous” means though.
– Jdip
Jun 6 at 11:06