An interesting q . The FT ( if  the function is satisfying the Diriclhlet condition ) condition is unique , and reversible so I expect your claim to be correct. Gibbs phenomena may be another aspect to watch out i.e hopefully no jump discontinuities in the time function.

Cheers

If the Fourier Transforms are identical (both real and imaginary parts) then the functions are identical. Both for continuous and discrete functions.

No. You cannot say that. It is absolutely wrong. There can be two totally different functions with identical Fourier Transform as the attached figure shows:

@ Rajesh,

Thank you for the example. From the plots, it is not difficult to see that the "energy content" of the two signals is the same in time domain and so you can expect that in the FFT magnitude plot as well. Not a surprise!

But, you have not compared the imaginary parts of the FT of the two signals. Please provide the phase plot. If the phase plot also looks identical, then your claim can be acceptable, otherwise not.

FT is complex in nature, so to establish the equality (or, unequality) of the two functions, you must compare both their real parts and imaginary parts of FT.

Where is the phase spectrum plot?

I am afraid you are wrong, Rajesh Sanghvi. What you plotted is not the Fourier Transform, but the magnitude of the FT.

Yes, you can draw the conclusion.

Thanks! Xinbo. Never thought about that simple proof. That was kind of cute!

If two or more functions are equal, almost everywhere, except on a set of points with zero Lebesgue measure, then the FT of these functions is same. Thus, we may obtain many functions which differs on a set of points and their FT are same. Moreover, reverse of this is also true.

Some pointers:

1. FT integrates over time interval..[0 to infinity] so in a sense time location is lost.

2. Wavelets show approximately location and time with Heisenberg uncertainty principle as a limit.

3. Also FT exists only for some functions meeting Dirichlet's conditions .

4 If 3 is met we do get back the function with Inverse FT

Conclusion : Cannot conclude with certainty that functions are identical with equal FT but they maybe good approximations.

Cheers

Please look at the book (Linear Time-Invariant Systems)

by

Schetzen

from

Northeastern Universiy

in

Boston, MA

I agree with Pushpendra Singh Sir, FT may same but reverse may true but not all the cases.

Or course, not. This is a common fact of the degeneracy for FT: two different functions may yield the same FT image.

Dear;

for more information please check the following documents:

book : Linear Time-Invariant Systems

Regards

No, There are many different functions may yield the same FT image.