Fourier Transform and Hilber transform, properties

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Discussion Overview

The discussion revolves around the properties of the Fourier transform and the Hilbert transform, focusing on their mappings and the nature of the signals they produce. Participants explore theoretical aspects and implications of these transforms in the context of signal representation.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant asserts that the Fourier transform expresses a function in the time domain as a function in the frequency domain, suggesting that they represent the same signal but questions the 1-to-1 mapping due to the integration over all time.
  • Another participant claims that the Fourier transform is indeed 1-to-1, noting that the inverse transform returns the original function except on a set of measure zero.
  • A further inquiry is made regarding the domain and codomain of the Fourier transform, with a focus on whether it applies to real numbers.
  • Participants discuss the Hilbert transform, with one suggesting it conserves the structure of the original function while producing a signal that is considered physically different.
  • There is a challenge regarding the nature of the Fourier transform's output, questioning whether the Fourier transform of a real number results in a real number.

Areas of Agreement / Disagreement

Participants express disagreement regarding the 1-to-1 nature of the Fourier transform and its implications. Some assert it is 1-to-1, while others question this characterization based on the integration process. The discussion remains unresolved regarding the nature of the outputs of the Fourier and Hilbert transforms.

Contextual Notes

Participants do not fully clarify the assumptions regarding the domains and codomains of the transforms, nor do they resolve the implications of the measure zero exception in the context of the Fourier transform.

kidsasd987
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Textbook says, Fourier transform expresses a function in time domain as a function in frequency domain. Basically, Fourier transform gives two different expressions in terms of t domain and f domain but they represent the same signal.

It also says Hilbert transform is a different type of transform because it gives a new signal not equal to the previous one.

However, I do not think Fourier transform does 1 to 1 mapping since we have to integrate t for -infinity to +infinity. (ex, t=1 converts to f=constant.) I understand F.T shifts the domain, but strictly speaking, I don't think they have exactly the same structure (function-wise).

In contrast, Hilber transform does 1 to 1 mapping as it shifts the original function by 90 degree. In this case function structure is conserved.I wonder why we say Hilbert transform is a totally different signal from the original one. is it because they are physically different?
 
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Fourier transform is 1-1. The inverse transform will give you the same function back (except on a set of measure 0).
 
mathman said:
Fourier transform is 1-1.

With what domain and codomain?
 
micromass said:
With what domain and codomain?
Reals and reals.
 
mathman said:
Reals and reals.

So the Fourier transform of a real number is a real number?
 

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