Sunday, January 12, 2020

fourier transform - When can we write Heisenberg uncertainty Principle as a equality?


We know that Heisenberg uncertainty Principle states that ΔfΔt14π.


But (in many case for Morlet wavelet) I have seen that they changed the inequality to an equality. Now my question is when are we allowed to change the inequality to an equality: ΔfΔt=14π

why =



Answer



It is important to define the time and frequency widths Δt and Δω of a signal before discussing any special forms of the uncertainty principle. There is no unique definition of these quantities. With appropriate definitions it can be shown that only the Gaussian signal satisfies the uncertainty principle with equality.



Consider a signal f(t) with Fourier transform F(ω) satisfying


f2(t)dt=1(unit energy)t|f(t)|2dt=0(centered around t=0)ω|F(ω)|2dω=0(centered around ω=0)


None of these conditions is actually a restriction. They can all be satisfied (for signals with finite energy) by appropriate scaling, translation and modulation.


If we now define time and frequency widths as follows


Δ2t=t2|f(t)|2dtΔ2ω=ω2|F(ω)|2dω


then the uncertainty principle states that


Δ2tΔ2ωπ2


(if f(t) vanishes faster than 1/t for t±)


where the inequality is satisfied with equality for the Gaussian signal


f(t)=απeαt2



The equation numbers above correspond to the proof below which is from Wavelets and Subband Coding by Vetterli and Kovacevic (p.80):


enter image description here


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