The uncertainty principle, as first formulated by Heisenberg, states that the product of the standard deviations of two conjugated variables, cannot be less than some minimum. This statement has been generalized to a precise mathematical theorem, in the frame of wavelet theory.
1 THE UNCERTAINTY THEOREM
2 THE CONDITION
A real function of the real variable will be said to satisfy the
condition if , and the derivative are all in .
If is its Fourier transform, is the transform of . All the following functions belong to :
and are the Fourier transforms of and .
and being in , we may define their finite norms:
By Parseval’s identity, they are related:
We are now able to define the probability distributions and for the ”random” variables and :
Since the integrals of and are 1, they are proper probability distributions. The mean value of is defined the usual way:
Note that ’s mean value is always 0 because is a real function. Finally, we have the standard deviations for the uncertainty theorem:
4 PROOF OF THE THEOREM
Equality occurs if, and only if, one of the functions is proportional to the other. For the two functions and , we have therefore:
The integral at the right hand side can be integrated by parts. Using the definition of :
But and are related by a factor, so:
Applying Parseval’s identity to the second integral of the left hand side, we get:
We have used the fact that the Fourier transform of if .
Now, dividing both sides by the norms, and simplifying by the factor, we get exactly the uncertainty theorem:
5 THE GAUSSIAN FUNCTION
The Cauchy-Schwarz inequality becomes an equality if, and only if, one of the functions is proportional to the other. In our case, this condition is expressed by where is a constant. This differential equation is readily solved: . must be in so that must be negative. Defining , we get the gaussian function in its traditional form:
The constant has been omitted because it cancels anyway in the probability distributions. The standard deviations are easily computed from their definitions:
Their product is , independent of . There is no other function with this property.
Roberto Celi Time-Frequency visualization of helicopter noise
Despite its frightening title, this paper is mostly theoretical and it is the only place where I saw the uncertainty theorem clearly stated.
Robi Polikar The wavelet tutorial
|Date of creation||2013-03-22 18:50:37|
|Last modified on||2013-03-22 18:50:37|
|Last modified by||dh2718 (16929)|