Journal of Signal and Information Processing, 2013, 4, 120-124 doi:10.4236/jsip.2013.43B021 Published Online August 2013 (http://www.scirp.org/journal/jsip)
Discrete Entropic Uncertainty Relations Associated with FRFT*
Guanlei Xu1, Xiaotong Wang2, Lijia Zhou1, Limin Shao1, Xiaogang Xu2
1Ocean department of Dalian Naval Academy, Dalian, China, 116018; 2Navgation department of Dalian Naval Academy, Dalian, China, 116018. Email: [email protected]
Received March, 2013.
ABSTRACT Based on the definition and properties of discrete fractional Fourier transform (DFRFT), we introduced the discrete Hausdorff-Young inequality. Furthermore, the discrete Shannon entropic uncertainty relation and discrete Rényi en- tropic uncertainty relation were explored. Also, the condition of equality via Lagrange optimization was developed, as shows that if the two conjugate variables have constant amplitudes that are the inverse of the square root of numbers of non-zero elements, then the uncertainty relations reach their lowest bounds. In addition, the resolution analysis via the uncertainty is discussed as well.
Keywords: Discrete Fractional Fourier Transform (DFRFT); Uncertainty Principle; Rényi Entropy; Shannon Entropy
1. Introduction zation. The third contribution is that we derived the Renyi entropic uncertainty principle in FRFT domain for Uncertainty principle not only holds in analog signals, discrete case. The final contribution is that we discussed but also in discrete signals [1,2]. Recently, with the de- the resolution in multiple FRFT domains as a succession velopment of fractional Fourier transform (FRFT), ana- of above derivative, including new proofs. In a word, log generalized uncertainty relations associated with there have been no reported papers covering these gener- FRFT have been carefully explored in some papers such alized discrete entropic uncertainty relations on FRFT. as [3,4,16], which effectively enrich the ensemble of FRFT. However, up till now there has been no reported 2. Preliminaries article covering the discrete generalized uncertainty rela- tions associated with discrete fractional Fourier trans- 2.1. FRFT and DFRFT form (DFRFT). From the viewpoint of engineering ap- First, confirm that you have the correct template for your plication, discrete data are widely used and seem to be paper size. This template has been tailored for output on more profitable than the analog ones. Hence, there is the custom paper size (21 cm * 28.5 cm). enough need to explore discrete generalized uncertainty Before discussing the uncertainty principle, we intro- relations. DFRFT is the discrete version of FRFT [5,6], duce some relevant preliminaries. Here we first briefly which is applied in practical engineering fields. In this review the definition of FRFT. For given analog signal article we will discuss the entropic uncertainty relations 1 2 RLRLtf )()()( and tf 1)( , its FRFT [5,6] is [7,8] associated with DFRFT. In this article, we made 2 defined as some contributions such as follows. The first contribu- tion is that we extend the traditional Hausdorff-Young F ()u F ft () ftK () (,) utdt inequality to the FRFT domain with finite supports. It is iu22cot iut it cot shown that these bounds are connected with lengths of (1ieeeftdt cot ) / 222sin () n the supports and FRFT parameters. The second contribu- tion is that we derived the Shannon entropic uncertainty ft() 2 n , ft()(21) n principle in FRFT domain for discrete case, based on which we also derived the conditions when these uncer- tainty relations have the equalities via Lagrange optimi- (1) *This work was fully supported by the NSFC (61002052) and partly where n Z and i is the complex unit, is the trans- supported by the NSFC (61250006). form parameter defined as that in [5,6]. In addition,
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tfFtfFF )()( . If , tftfFF )()( , and the transform parameter . ˆ i.e., the inverse FRFT reads: tf )( F ),()( dutuKu . Since X 1, XUX 1 from Parseval’s 2 2 2 1 N 2 N theorem. Here X x 2 . Clearly, we can ob- Let 321 ,,,, xxxxX N ),2(),1( xxx ( )(,),3 CNx n1 n 2 be a discrete time series with cardinality N and X 1. 2 tain XMXU with UM . ˆ N 1 Assume its DFRFT X ˆ , ˆ , xxx ˆ321 ,, ˆN Cx under the transform parameter . Here U lu )(sup with ,,1,)( NlluU . r r Then the DFRFT [5] can be written as l 2 2 N ik cot ikn in cot Hence, we have XMXU with UM . 1 xˆ()ki (1 cot)/Neeexn2 NNsin 2 2 () l1 Then from Riesz’s theorem [11,12], we can obtain the N discrete Hausdorff-Young inequality 2 p uknx (,) (n), ,1 kn N . (2) 11 l1 p XMXU with p 21 and 1. q p Also, we can rewrite the definition (2) as qp Set UU , then UUUU , we obtain Xˆ XU , 2p p XMXUU with UUUM . where nkuU ),( NN . q p Clearly, for DFRFT we have the following proper- Let XUY , then YUX . In addition, from the ties[5]: property of DFRFT we can have UUXUU X XU XU , 2k 1 UUM . Hence we obtain UU X UX X , 2k N )sin( ˆ UX X 1 . 2 2 2p 1 p YUMYU with M . In the follows, we will assume that the transform pa- q p N )sin( rameter 20 and . The max difference Since the value of X can be taken arbitrarily in C N , between the discrete and analog definitions is the support: Y can also be taken arbitrarily in C N . Therefore, we one is finite and discrete and the other one is infinite and can obtain the following lemma. analog. Lemma 1: For any given discrete time series )(,),3(),2(),1(,,,, CNxxxxxxxxX N 2.2. Shannon Entropy and Rényi Entropy 321 N with cardinality N and X 1 , U ( U ) is the For any discrete random variable x ( ,,1 Nn ) and 2 n DFRFT transform matrix associated with the transform its probability density function p(x ) , the Shannon en- n parameter ( , respectively), then we can obtain the tropy [9] and the Rényi Entropy [10] is defined as, re- generalized discrete Hausdorff-Young inequality spectively p2 N NXU )sin( 2 p XU xpxpxH )(ln)( , q n n1 n n p 11 1 N with p 21 and 1. xH n xp n )(ln . 1 n1 qp Hence, in this article, we know that for any DFRFT Clearly, this is the discrete version of Hausdorff- ˆ N ˆ , ˆ , ˆ321 ,, ˆN CxxxxX , the Shannon entropy and Young inequality. In the next sections, we will use this the Rényi Entropy associated with DFRFT is defined as, lemma to prove the new uncertainty relations. respectively 2 2 N ˆˆ ln)( ˆ nxnxxH )( , 3. The Uncertainty Relations n1
1 N 2 3.1. Shannon Entropic Principle xH ˆ ln ˆ nx )( . n1 1 Theorem 1: For any given discrete time series Clearly, if ,1 xHxH ˆˆ . N 321 N NxxxxxxxxX )(,),3(),2(),1(,,,, C with cardinality N and X 1, xˆ ( xˆ ) is the DFRFT 2 2.3. Discrete Hausdorff-Young Inequality series associated with the transform parameter ( ,
Associated with DFRFT respectively), N (N ) counts the non-zero elements
N of xˆ ( xˆ , respectively), then we can obtain the gener- Let 321 N )(,),3(),2(),1(,,,, CNxxxxxxxxX be a discrete time series with cardinality N and its alized discrete Shannon entropic uncertainty relation ˆ N ˆ ˆˆ DFRFT ˆ , ˆ , ˆ321 ,, ˆ N CxxxxX with XUX NxHxH )sin(ln , (3)
Copyright © 2013 SciRes. JSIP 122 Discrete Entropic Uncertainty Relations Associated with FRFT
where 22 22 LxnxnxmxNNlnˆˆ ( ) ( ) ln ˆ ( ) ˆ (m ) N 2 2 nm11 xH ˆ ln ˆ )( ˆ nxnx )( , n1 NN2 2 12xnˆ() 1xnˆ () 1. 2 2 nn11 xH ˆ N ln ˆ )( ˆ mxmx )( , m1 In order to simplify the computation, we set 2 2 ˆ )( pnx and ˆ )( pnx . Hence we have which are Shannon entropies. The equality in (3) holds n n 1 1 L pn 1 01ln , iff xˆ and xˆ . pn N N L Proof: From lemma 1, we have p 01ln , p n 2 p2 1 n N p p N )sin( 2 p ˆ mx )( N m1 p 1 , n1 n p1 1 . p N N p p 1. ˆ nx )( p1 n1 n n1 Solving the above equations, we finally obtain Take natural logarithm in both sides in above inequal- 1 1 ˆ nx )( , ˆ nx )( . From the definition of ity, we can obtain N N pT 0)( , Shannon entropy, we know that if ˆ ln NxH and where ˆ ln NxH , the equality in (3) holds. In addition, we p 21N p Tp() ln N sin( ) ln xˆ () m also have N NN )sin( . From the proof we m1 2 pp 1 1 know that ˆ nx )( and ˆ nx )( means p p 1 N N N ˆ p1 ln n1 xn ( ) . p that ˆ )( ˆ nxandmx )( can be complex, and only if their Since p 21 and X 1 ,we know T 0)2( . 2 amplitudes are constants, the equality will hold. Now we Note T p 0)( if p 21 . Hence, T ' p 0)( if can obtain the following corollary out of above analysis. p 2 . Since Corollary 1: For any given discrete time series ~ N 11N p NxxxxxxxxX )(,),3(),2(),1(,,,, C with Tp'( ) ln N sin( ) lnxmˆ ( ) 321 N ~ 22 m1 pp cardinality N and X 1 , xˆ ( xˆ ) is the DFRFT se- 2 p ries associated with the transform parameter ( , re- N ln()xmˆ xmˆ () m1 1 spectively), N (N ) counts the non-zero elements N p 1 p ˆ ˆ ˆ ˆ xm () of x ( x , respectively), if nx )( and m1 N p 1 N lnxnˆ ( ) p1 1 2 n1 ˆ p nx )( , then we have N p N xnˆˆ()p1 ln xn () n1 NNN )sin( 1 p , pp(1) N and ˆ p1 n1 xn () ˆˆ NxHxH )sin(ln . we can obtain the final result in theorem 1. Now consider when the equality holds. From theorem 3.2. Rényi Entropic Principle 1, that the equality holds in (3) implies that Theorem 2: For any given discrete time series xHxH ˆˆ reaches its minimum bound, which NxxxxxxxxX )(,),3(),2(),1(,,,, CN with means that 321 N cardinality N and X 1, xˆ ( xˆ ) is the DFRFT se- 2 Minimize xHxH ˆˆ subject to xx ˆˆ 1 , 2 2 ries associated with the transform parameter ( , re- i.e. spectively), N (N ) counts the non-zero elements of Minimize xˆ ( xˆ , respectively), then we can obtain the generalized 2 2 N ˆ 2 ˆ 2 N ˆ ˆ discrete Renyi entropic uncertainty relation ln )( nxnx )( ln )( mxmx )( , n 1 m 1 ˆˆ NxHxH )sin(ln N 2 N 2 subject to ˆ )( ˆ nxnx 1)( . 1 11 n1 n1 with 1 and 2 . (4) To solve this problem let us consider the following 2 Lagrangian where
Copyright © 2013 SciRes. JSIP Discrete Entropic Uncertainty Relations Associated with FRFT 123
~ 1 N 2 NxxxxxxxxX )(,),3(),2(),1(,,,, with xH ˆ ln ˆ mx )( , 321 N ~ m1 ˆ ˆ 1 cardinality N and X 1 , x ( x ) is the DFRFT se- 2 1 N 2 ries associated with the transform parameter ( , re- xH ˆ ln ˆ nx )( , n1 spectively), N (N ) counts the non-zero elements of 1 xˆ ( xˆ , respectively), then we can obtain the generalized which are Rényi entropies. discrete entropic uncertainty relation Proof: In lemma 1, set q 2 and p 2 , we have H N ))sin(ln( , (6) 1 11 1 1 and 2 . Then from lemma 1, we 2 2 Proof: For variable xˆ , we have for 0 p 1 obtain N p p 1 1 1 2 ˆˆ nxx )( . N 2 2 N 2 p ˆ 2 ˆ n1 mx )( N )sin( nx )( . m1 n1
Set xˆ max ˆ nx )( . Therefore, we have Take the square of the above inequality, we have Nn 1 1 1 1 1 N 2 N 2 q q N ˆ ˆ ˆ )( Nmx )sin( ˆ nx )( . d nx )( nx )( m1 n1 ˆ log x 1 1 log 1 , (7) dq q n1 q q xˆ 1 xˆ 1 Take the power of both sides in above ine- q q 1 d 2 ˆ 1 quality, we obtain 2 log x 1 1 dq q 2 N 2 1 1 N 1 ˆ ˆ 2 mx )( N )sin( nx )( , m1 n1 111 1 NNqqqˆ q 1 1 xnˆ() xnˆ () xnˆ () xn () 2 log log . 1 N 1 1111 ˆ nn11qqqq N )sin( nx )( t xxxxˆ ˆ ˆ ˆ n1 1111 qqqq i.e., 1 1. (5) 2 N ˆ mx )( 1 Taking into account of the non-negative second de- m1 rivative, therefore we know that log xˆ 1 is convex. Take natural logarithm in both sides of (5), we can q obtain Therefore, we obtain dd 11NN2 2 Hxˆˆlog x11 lim log xˆ lnxnˆˆ ( ) lnxm ( ) q nm11 dqqq1 dq q 11 2 lnN sin( ) . log N sin( ) . (8) ~ Since ˆˆ Xxx 1 and Clearly, as 1 and 1 , the Renyi entropy 2 2 2 reduces to Shannon entropy, thus the Renyi entropic un- 1 xˆ xˆ and the Riesz-Thorin certainty relation in (4) reduces to the Shannon entropic N sin( ) 1 uncertainty relation (3). Hence the proof of equality in theorem [4], one has theorem 2 is trivial according to the proof of theorem 1. ˆ x 1 1 Note that although Shannon entropic uncertainty rela- 1q q 1 N )sin( 2 ,1 xq ˆ 0 tion can be obtained by Rényi entropic uncertainty rela- xˆ 1 2 tion, we still discuss them separately in the sake of inte- q grality. By applying negative logarithm to both sides of above inequality, we obtain the following relation 4. Resolution Analysis in Time-Frequency logxxˆ log ˆ 1 Domain 1 q 1q In many cases [13-15], we often discuss the data concen- 1 1 qNlog sin( ) , q 1 . tration in both time domain and frequency domain, 2 2 therefore, we will adopt a new measure on entropy: ~ Taking into account of ˆˆ Xxx 1, then 2 2 p ˆ )1( xHpxpHH ˆ , 2 where xˆ ( xˆ ) is the DFRFT series associated with the both sides of above inequality are equal to zero for 1 transform parameter ( , respectively) for time series q . Hence, (7) and (8) imply ~ 2 xX x21 ,, x3 ,, xN x ),1( Nxxx )(,),3(),2( , xH ˆ 1 1 1 and H xˆ are the Shannon entropies for the DFRFT ˆ ˆ NxHxH )sin(ln . series xˆ ( xˆ ). 2 2 2 Theorem 3: For any given discrete time series Thus the proof is completed. Note that this proof can
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