Heisenberg Uncertainty Principle for n-Dimensional Linear Canonical Transforms

Yonggang Li,Chuan Zhang,Huafei Sun

Abstract:The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain,in this manuscript,an improved Heisenberg uncertainty principle is obtained in linear canonical transforms domain.

Keywords:Heisenberg uncertainty principle;linear canonical transforms;Pitt inequality

1 Introduction

In 1927,the German physicist Heisenberg proposed the uncertainty principle also called Heisenberg-Weyl inequality.Now the Heisenberg uncertainty principle has been applied to quantum mechanics and signal processing.It means that the position and the momentum of a particle can not be determined simultaneously in a quantum mechanical systems [2].Equivalently,Heisenberg uncertainty principle can be seen the temporal resolution and the frequency resolution of any signal which can not be infinitely improved simultaneously in signal processing community [3].

In fact,the temporal resolution of a unit signalis defined as [4]

and the frequency resolution off(x) is defined as

The so-called Heisenberg-Weyl inequality can be obtained by using Cauchy inequality.In 1995,a sharp form of Pitt inequality is obtained by Beckner [5],and then logarithmic uncertainty principle

can be gotten for any unitf(x) belonged to Schwartz space [6].

Heisenberg uncertainty principle and logarithmic uncertainty principle are associated with Fourier transform (FT) and there are some general results such as logarithmic uncertainty principle in linear canonical transforms (LCT) domain [7].LCT,firstly introduced by Collins in 1970,is a generalization of FT and can be used in differential equations,optical systems analysis[8].With the rapid development of fractional Fourier transform,LCT has been paid more and more attention in applied mathematics and signal processing community such as the filtering theory [9],the frame theory [10],the sampling theory [11]and the discrete algorithms [12].Furthermore,LCT can be used to radar signal processing,communication signal processing,the image encryption,denoising and so on.

Since LCT is very useful,it is worth to study Heisenberg uncertainty principle in LCT domain.Hence,in this manuscript,we will improve Heisenberg uncertainty principle in LCT domain andn-dimensional Heisenberg uncertainty principle also be obtained.The manuscript is organized as follows.Section 2 introduces some general definitions and gives some results.In Section 3,we calculaten-dimensional Heisenberg uncertainty principle in LCT domain.

2 Preliminaries

Before we proceed,some important definitions and results related to the LCT and Heisenberg uncertainty principle are reviewed in this section.

2.1 Linear Canonical Transforms

We frequently use the one-dimensional LCT in signal processing [1]as

where the parameters satisfyad −bc=1.While the inverse transform of the one-dimensional LCT (ILCT) is given by the following formula

Generally,for each 2n×2nsymplectic mat-MTis conjugate transpose ofM.n-dimensional LCT [13]is defined as

where

For more detailed definitions and properties of the LCT,one can refer to [1].

2.2 Heisenberg Uncertainty Principle for Complex Signal

In this subsection,we recall Heisenberg uncertainty principle for complex signal.Supposing a unit signalf(x)=ρ(x)eiφ(x),we have

Using this result,Heisenberg uncertainty principle in LCT domain and some other uncertainty principle can be improved [14].

2.3 The Sharp Pitt Inequality

Since logarithmic uncertainty principle is obtained from the sharp Pitt inequality,in this subsection,the next lemma is needed [5].

Lemma 1Letf(x) be a rapidly decreasing function [6]and 0 ≤α≤1,then

where Γ(x) is the Gamma function.In fact,Lemma 1 can be obtained from Stein-Weiss fractional integral inequality [5].

3 Main Results

In this section,letf(x) be a unit signal with 2 ≤α,then Rassias obtained the following inequality by using the Hölder inequality and the Plancherel-Parseval-Rayleigh identity [15]

Lemma 2Supposingf(x) is a rapidly decreasing function and 2 ≤α≤3,we have

ProofLetg(x)=xf(x),then,we have

By using Lemma 1,we get

Then,by using Cauchy inequality,we have

From Moyal identical equations,we obtain

This has finished the proof.

Associated with LCT,the following theorem can be obtained.

Theorem 1Letf(x) be a rapidly decreasing function and 2 ≤α≤3,then we have

ProofLetg(x)=,then,we have

By using Lemma 2,we get

In fact,Theorem 1 is just Heisenberg uncertainty principle in LCT domain whenα=2.Next,the following theorem can be obtained associated with the definition ofn-dimensional LCT.

Theorem 2Letf(x) be a rapidly decreasing function and 2 ≤α≤3,then we have

In fact,Theorem 2 is a generalization of Theorem 5 in [16].In particularly,Heisenberg uncertainty principle is just Theorem 2 whenα=2 andn=1.

4 Conclusion

In this manuscript,we obtain a Heisenberg uncertainty principle in LCT domain by using sharp Pitt inequality and this result improves the theorem of Rassias.In fact,Theorem 2 can be obtained only using the integral transform,sharp Pitt inequality and Cauchy.In the following work,we find Pitt inequality holds only for 0 ≤α≤1,hence,it is worth to discuss Pitt inequality for 1 ≤α≤2.