Guanlei Xu,Xiaogang Xu,Xiaotong Wang

Abstract:Uncertainty principle plays an important role in multiple fields such as physics,mathematics,signal processing,etc.The linear canonical transform (LCT) has been used widely in optics and information processing and so on.In this paper,a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced.These newly deduced uncertainty relations not only introduce new physical interpretation in signal processing,but also build the relations between the uncertainty lower bounds and the LCT transform parameters a,b,c and d for the first time,which give us the new ideas for the analysis and potential applications.In addition,these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts.Furthermore,some numeric examples are given to demonstrate the efficiency of these newly deduced uncertainty inequalities.

Keywords:linear canonical transform (LCT);Fisher information;uncertainty principle

1 Introduction

Uncertainty principle is a key principle in physics,mathematics,signal processing,etc.[1?23].Especially,in conventional time-frequency domains,the product of time spread and frequency spread can not be small unlimitedly.On the other hand,if the resolution in one domain increases,then the resolution in another domain necessarily decreases,and vice verse.At the same time,in mathematics,physics and information theory,the Fisher information is the current manner of measuring the amount of information.Based on the Fisher information,some new derived uncertainty relations including the Cramér-Rao inequality have been discussed recently(such as [9?14]).Up till now,in the traditional time-frequency domain,the uncertainty relations have been addressed fully [3?8].

As the generalization of the traditional Fourier transform (FT),Fresnel transform and the fractional Fourier transform (FrFT),the linear canonical transform (LCT) has more freedoms and has been applied in various fields such as physical optics and information science and so on[24?32],which will result in new properties and more applications in signal processing and so on.Therefore,LCT will have more advantages over the traditional counterparts in multiple fields.For more details,see [24,26,27].On the other hand,this paper is also an improvement of [33,34]for more parameters with new physical interpretation.

We must note that there have been lots of related papers involving generalized uncertainty principles.For example,the entropy based uncertainty principles have been addressed in [12,35]to show the relations between the uncertainty bounds and LCT parameters.The generalized Heisenberg uncertainty principles have been addressed in [15?25]in LCT domains and so on,which have been demonstrated the new physical senses on signal resolution analysis.Also,there have been some papers involving the uncertainty relations on Fisher information in traditional time-frequency domain and FRFT domain.However,there have been no any report addressing the uncertainty relations on Fisher information in LCT domain,which will be shown in this paper.

This paper is organized as follows.Section 2 reviews the traditional uncertainty relations,the LCT along with some properties.Then the new generalized Cramér-Rao inequality is proven and explained in the new physical sense as well.In addition,the newly proposed uncertainty relations on Fisher information are discussed.In Section 3,some experiments are given to show the efficiency of the newly derived inequality relations.In Section 4,the results and discussion are given.Finally,our last section will concludes this paper.

2 Proposed Method

2.1 The LCT

For a given functionf(t)∈L1(R)∩L2(R) (we assume‖f(t)‖2=1 andf(t) is absolutely integrable in this paper) and its traditional Fourier transformF(u),then we have

More details of the properties can be found in [24,26,27].

2.2 The Generalized Cramér-Rao Inequalities on LCT

We know that the traditional Cramér-Rao inequality only involves one single transform domain (either the time domain or the frequency domain) [9?14].Differently,in this section we will deduce the generalized Cramér-Rao inequality involving both the time domain and the frequency domain (including the generalized frequency domain such as the LCT transform domain) at the same time.

The proof is finished.f(t) defined in above,its LCT’s variance and LCT ’s gradient integral must satisfy certain equality relation,i.e.,their square sum cannot be less than some low bound.The biggest novelty here is that this uncertainty inequality builds the relation between the four parameters (a,b,c,d)

Note that Theorem 2 is the generalized Cramér-Rao uncertainty inequality,which only involves the single transform domain (in one single LCT domain).In other words,Theorem 2 is a new uncertainty relation involving only the single transform domain.For the given function and the lower bound,which is sharper and tighter than the traditional counterparts because the new bound is more refined with more parameters.

2.3 Fisher Information Based Generalized Uncertainty Relation on LCT

Like the Shannon entropy and Renyi entropy,the Fisher information is another significant information measure and plays an important role in signal processing.In the same manner,the uncertainty relation based on the Fisher information has the important effect on information processing and signal processing.In this section,we will deduce the Fisher information based generalized uncertainty inequalities in the LCT domain and explain the corresponding physical significance.

Lemma 1 is another new version of the generalized uncertainty relation on the Fisher information.This lemma builds the new relation between the variance,the Fisher information on the LCT of the functionf(t) and the LCT transform parameterbfor the first time.

Similarly,since there are more parameters in these new bounds than their traditional counterparts,the bounds will be more related with the transform with tighter and sharper values.

3 Experiments

which means that Lemma 1 holds true.

Furthermore,we also have tried other real functionf(t) with different forms,and the same conclusion has been obtained as well.Taking into account of the limited length of this paper,we only show the above examples.In other words,these examples prove that the newly deduced Cramér-Rao inequalities and the uncertainty inequalities based on the Fisher information hold true,which will give us new potential values in information processing and signal processing and so on.

4 Results and Discussion

As shown in Section 3,these experiments prove that the newly deduced Cramér-Rao inequalities and the uncertainty inequalities based on the Fisher information hold true,which will give us new potential values in information processing and signal processing and so on.

On one hand,these novel uncertainty inequalities are the extension of the traditional Cramér-Rao inequality and the uncertainty relation on Fisher information.At the same time,the generalized Cramér-Rao inequalities’ bounds are much sharper and tighter than the traditional counterparts.

On the other hand,the generalized Cramér-Rao uncertainty inequalities build the relation between the Cramér-Rao bounds and the LCT transform parameters.In addition,the generalized Cramér-Rao inequalities build the relation between the LCT’s variance and LCT’s gradient integral in only one single transform domain.Also,compared with the traditional counterparts,the newly deduced uncertainty relations on Fisher information have the sharper and tighter bounds,which show the potential advantage for the parameter estimation in the LCT domain.

5 Conclusion

In this paper,a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced.These newly deduced uncertainty relations make full use of the properties of LCT with more freedoms that will yield more potential values.That is to say,these new uncertainty inequalities employ every parameter of (a,b,c,d).In addition,in our paper some numeric experiments are given to show the efficiency of these newly deduced inequalities.Our future work will include the relations on the complex functions and other more complicated case such as the Hilbert transform on LCT[34?36]including some equality conditions of these theorems and so on.