Can Two Distinct Functions Share Magnitude and Fourier Magnitude?

[StatusX]

I'm trying to uniquely determine a complex function given pairs of real valued functions derived from it. For example, if you have its real and imaginary parts, or phase and the magnitude, the function is uniquely determined from them.

But what if you have the magnitude of the function and the magnitude of its Fourier transform? ie, are there two distinct functions for which these two properties are identical? What are some other pairs of real valued functions related to the Fourier transform that I might use?

 

[reilly]

I think not. But, work out the problem with a complex function, like a complex Gaussian , or a truncated complex exponential.
Regards,
Reilly Atkinson

 

[StatusX]

Really? To summarize, I'm asking if there exist two distinct complex-valued functions f(t) and g(t) (t any real number), such that:

1) |f(t)|²=|g(t)|² for all t
2) |F(ω)|²=|G(ω)|² for all ω (where, eg, F(ω) is the Fourier transform of f(t).)

Now, clearly the two functions may differ by a constant phase and satisfy these two properties, so in that way this is false. But excluding this trivial case, are you still saying there are functions that differ by more than a constant phase and still satisfy the above two conditions? It seems like if there is any time variation in the phase seperating two functions of equal magnitude, this will add extra frequency components and so (2) will not be satisfied. I can't think of any counterexamples to this, but I'm not sure how to prove it.

EDIT:

I thought of another pair of functions that would satisfy this property. If |f(t)|² is even, but f(t) is not, then f(t) and f(-t) are two distinct functions which satisfy these properties, the first by assumption and the second since the Fourier transform of f(-t) is the complex conjugate of F(ω), and so has the same magnitude. Maybe I should rephrase the question to ask which functions have these two properties with resepct to a given function f(t). So far we have any function differing from f(t) by a constant phase, as well as the example I just mentioned if |f(t)|² is even, and functions differing from this one by a constant phase. Is this all?