Some years ago I became fascinated by uncertainty principles. I got to know them via signal processing and not via physics, although, from a mathematical point of view they are the same.
I recently supervised a Master’s thesis on this topic and the results clarified a few things for me which I used to find obscure and I’d like to illustrate this here on my blog. However, it takes some space to introduce notation and to explain what it’s all about and hence, I decided to write a short series of posts, I try to explain, what new insights I got from the thesis. Here comes the first post:
1. The Fourier transform and the windowed Fourier transform
Let’s start with an important tool from signal processing you all know: The Fourier transform. For the Fourier transform is
(I was tempted to say “whenever the integral is defined”. However, the details here are a little bit more involved, but I will not go into detail here; is defined for -functions, for functions and even for tempered distributions…) Roughly speaking, the Fourier transform decomposes a signal into its frequency components, which can be seen from the Fourier inversion formula:
i.e. the (complex) number says “how much the frequency (i.e. the function ) contributes to ”. In the context of signal processing one often speaks of the “time representation” and the “frequency representation” .
One drawback of the Fourier transform, when used to analyze signals, is its “global” nature in that the value depends on every value of , i.e. a change of in a small interval results a change of all of . A natural idea (which is usually attributed to Gabor) is, to introduce a window function which is supposed to be a bump function, centered at zero, then translate this function and “localize” by multiplying it with . The resulting transform
is called windowed Fourier transform, short-time Fourier transform or (in the case of ) Gabor transform.
In other words: The localization in time is precisely determined by the “locality” of , that is, how well is concentrated around zero. The better is concentrated around , the more “localized around ” is the information of , the windowed Fourier transform uses.
For the localization in frequency one obtains (by Plancherel’s formula and integral substitution) that
In other words: The localization in frequency is precisely determined by the “locality” of , that is, how well is concentrated around zero. The better is concentrated around , the more “localized around ” is the information of , the windowed Fourier transform uses.
Hence, it seems clear that a function is well suited as a window function, if it both well localized in time and frequency. If one measures the localization of a function around zero by its variance
then there is the fundamental lower bound on the product of the variance of a function and the variance of its Fourier transform, know under the name “Heisenberg uncertainty principle” (or, as I learned from Wikipedia, “Gabor limit”): For it holds that
Proof: A simple (not totally rigorous) proof goes like this: We use partial integration, the Cauchy-Schwarz inequality and the Plancherel formula:
Moreover, the inequality is sharp for the functions for . In this sense, these Gaussians are best suited for the windowed Fourier transform.
While this presentation was geared towards usability, there is a quite different approach to uncertainty principles related to integral transforms which uses the underlying group structure.
2. The group behind the windowed Fourier transform
The windowed Fourier transform (1) can also be seen as taking inner products of with the family of functions . This family is obtained from the single function by letting the so-called Weyl-Heisenberg group act on it:
Definition 1 The Weyl-Heisenberg group is the set endowed with the operation
The Weyl-Heisenberg group admits a representation of the space of unitary operators on , that is a map
It indeed the operators are unitary and it holds that
Moreover, the mapping is continuous for all , a property of the representation which is called strong continuity.
In this light, the windowed Fourier transform can be written as
Now there is a motivation for the uncertainty principle as follows: Associated to the Weyl-Heisenberg group there is the Weyl-Heisenberg algebra, a basis of which is given by the so called infinitesimal generators. These are, roughly speaking, the derivatives of the representation with respect to the group parameters, evaluated at the identity. In the Weyl-Heisenberg case:
(In the last case, my notation was not too good: Note that with and the derivative has to be taken with respect to .)
All these operators are skew adjoint on and hence the operators
are self adjoint.
For any two (possibly unbounded) operators on a Hilbert space there is a kind of abstract uncertainty principle (apparently sometimes known as Robertson’s uncertainty principle. It uses the commutator of two operators:
Theorem 2 For any two self adjoint operators and on a Hilbert space it holds that for any in the domain of definition of and any real numbers and it holds that
Proof: The proof simply consists of noting that
Now use Cauchy-Schwarz to obtain the result.
Now the three self-adjoint operators , and have three commutators but since is a multiple of the identity and hence commutes with the others. I.e. there is only one commutator relation:
Hence, using , and in Robertson’s uncertainty principle gives (in sloppy notation)
which is exactly the Heisenberg uncertainty principle.
Moreover, by (2), equality happens if fulfills the differential equation
the solution of which are exactly the functions .
Since the Heisenberg uncertainty principle is such an impressing thing with a broad range of implications (a colleague said that its interpretation, consequences and motivation somehow form a kind of “holy grail” in some communities), one may try to find other cool uncertainty principles be generalizing the approach of the previous sections to other transformations.
In the next post I am going to write about other group related integral transforms and its “uncertainty principles”.