Fourier Transform and Hilber transform, properties

1. Mar 5, 2015

kidsasd987

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?

2. Mar 5, 2015

mathman

Fourier transform is 1-1. The inverse transform will give you the same function back (except on a set of measure 0).

3. Mar 10, 2015

micromass

With what domain and codomain?

4. Mar 10, 2015

mathman

Reals and reals.

5. Mar 10, 2015

micromass

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