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.(adsbygoogle = window.adsbygoogle || []).push({});

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 and Hilber transform, properties

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