Fourier Transform: Nonperiodic vs Periodic Signals

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Discussion Overview

The discussion revolves around the Fourier transform, specifically contrasting nonperiodic and periodic signals. Participants explore the conditions under which the Fourier transform can be applied, the implications of periodicity, and the nature of certain signals like decaying exponentials.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that while the Fourier transform is defined for nonperiodic deterministic signals, it raises the question of why nonperiodicity is necessary, given that periodic signals like sine functions can also have Fourier transforms.
  • Another participant explains that the Fourier transform can be viewed as a continuous form of the Fourier series, where periodic signals lead to discrete frequency components, while nonperiodic signals result in a continuous spectrum.
  • A participant questions whether it is absolutely necessary for a signal to be nonperiodic in order to compute its Fourier transform.
  • One contributor suggests that the term nonperiodic broadens the applicability of the Fourier transform to any integrable functions, regardless of periodicity.
  • Another participant discusses the derivation of the Fourier transform from the Fourier series, emphasizing that it involves repeating finite-length signals and raises concerns about how decaying exponentials, which are not finite in length, fit into this framework.
  • There is speculation about whether the Fourier transform for decaying exponentials might stem from a different derivation that has yet to be discovered.
  • One participant introduces the idea of "multiple infinities," questioning how to conceptualize the Fourier transform of signals that do not fit neatly into the finite-length framework.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of nonperiodicity for the Fourier transform and the implications of periodicity on the nature of the frequency spectrum. The discussion remains unresolved regarding the treatment of decaying exponentials and their Fourier transforms.

Contextual Notes

Participants highlight limitations in current literature, suggesting that existing explanations may not adequately address the nuances of Fourier transforms for various signal types.

RaduAndrei
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In a book the Fourier transform is defined like this. Let g(t) be a nonperiodic deterministic signal... and then the integrals are presented.

So, I understand that the signal must be deterministic and not random. But why it has to be nonperiodic (aperiodic).
The sin function is periodic and we can calculate its Fourier transform.

Is it because a nonperiodic signal is absolutely integrable?

And with the sin function. Yes, I can calculate. But deltas appear.

This is the answer?
 
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Fourier transform is a continuous form of Fourier series. In computing Fourier series, the signal in time domain must be periodic, i.e.it has finite period, and that you find that the spectrum contains combs separated by a fixed value which is reciprocal to the signal's period. So, the longer the period, the closer the frequency combs are to its neighbors. When the signal is not periodic, we can suppose that its period is infinitely long, therefore the corresponding frequency combs is separated by infinitesimal distance, which leads to a continuous spectrum.
 
Ok.

But in computing the Fourier transform of a signal, that signal must be absolutely necessary nonperiodic?
 
The use of the term non-periodic generalize the applicability of Fourier transformation to any integrable functions, be it periodic or non-periodic.
 
Aa, ok. Thanks for the answer.

I think this is a problem with today's books. They are not written in a more Euclidean way.
 
Also. When deriving the Fourier transform from the Fourier series, we have a finite-length signal and repeat it multiple times over the time axis. And then expand it into a Fourier series. And then calculations. And then we get the Fourier transform.

So the Fourier transform is for finite-length signals.

The fact that we can calculate Fourier transforms for periodic signals or signals like the unit step is because we involve deltas functions there.

But what about the decaying exponential? Its Fourier transform does not involve deltas and it is not of finite length.
How can this decaying exponential be viewed as a finite-length signal that gets repeated multiple times over the time axis. Its period is infinite.

So my question. How does one attach a Fourier transform to such decaying exponential? It could be the fact that for such signals we actually have other derivation of the Fourier transform but we haven't found it yet?
We derived the Fourier transform for finite-length signals and with it we just calculated the Fourier transform for decaying exponential?
Or one can think of having multiple infinities into one infinite. So we have that notion that some infinities are bigger than others?
 

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