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Discrete Entropic Uncertainty Relations Associated with FRFT* 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, Copyright © 2013 SciRes. JSIP Discrete Entropic Uncertainty Relations Associated with FRFT 121 F F f()() t F f t . If , F F f()() t f t , and the transform parameter . ˆ i.e., the inverse FRFT reads: f() t F ()(,)u K u t du . 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 X x1,,,, x 2 x 3 xN x(1), x (2), x(3), ,x ( N ) C n1 n 2 be a discrete time series with cardinality N and X 1. 2 tain UXMX with MU . ˆ N 1 Assume its DFRFT X xˆ1, xˆ 2, xˆ 3 ,, xˆN C under the transform parameter . Here U supu ( l ) with U u( l ) , l 1, , N . r r Then the DFRFT [5] can be written as l 2 2 N ik cot ikn in cot Hence, we have UXMX with MU . 1 nxˆ()ki 1( toc x/)N2 NNsin e 2 2 () e e l1 Then from Riesz’s theorem [11,12], we can obtain the N discrete Hausdorff-Young inequality 2 p uknx (,) (n), 1 n , k N . (2) 1 1 l1 UXMX p with 1p 2 and 1. q p Also, we can rewrite the definition (2) as p q Set UU , then UUUU , we obtain Xˆ UX , 2p UUXMX p with MUUU . where U u (,) k n NN . q p Clearly, for DFRFT we have the following proper- Let YUX , then XUY . In addition, from the ties[5]: property of DFRFT we can have UUXUU X UX UX , 2k 1 MUU . Hence we obtain UU X XU X , 2k N sin( ) ˆ XU X 1 . 2 2 2p 1 UYMUY p with M . In the follows, we will assume that the transform pa- q p Nsin( ) rameter 0 2 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 Xxxx , , , , x xxx (1), (2), (3), , xNC ( ) N 2.2. Shannon Entropy and Rényi Entropy 1 2 3 N with cardinality N and X 1 , U ( U ) is the For any discrete random variable x ( n 1, , N ) 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 UXN sin( ) 2 p UX H x p( x ) ln p ( x ) , q n n1 n n p 1 1 1 N with 1 p 2 and 1. H xn lnp ( xn ) . 1 n1 p q Hence, in this article, we know that for any DFRFT Clearly, this is the discrete version of Hausdorff- ˆ N X xˆ1, xˆ 2, xˆ 3 ,, xˆN C , 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 H xˆ N x ˆ ( n ) ln xˆ () n , 3. The Uncertainty Relations n1 1 N 2 3.1. Shannon Entropic Principle H xˆ ln xˆ () n . n1 1 Theorem 1: For any given discrete time series Clearly, if 1, H xˆ Hx ˆ . N X x1, x 2 , x 3 , , xN x (1), x (2), x (3), , x ( N ) 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 Xxxx 1, 2 , 3 , , xN xxx (1), (2), (3), , xNC ( ) be a discrete time series with cardinality N and its alized discrete Shannon entropic uncertainty relation ˆ N ˆ ˆ ˆ DFRFT X xˆ1, xˆ 2, xˆ 3 ,, xˆ N C with XUX H x H x ln N sin( ) , (3) Copyright © 2013 SciRes. JSIP 122 Discrete Entropic Uncertainty Relations Associated with FRFT where 22 22 x mNNln nˆˆ ( ) ( ) x ln ˆ ( ) x ˆ (m ) Lxn N 2 2 nm11 H xˆ ln xˆ () n xˆ () n , n1 NN2 2 12xnˆ() 1xnˆ () 1. 2 2 nn11 H xˆ N ln xˆ () m xˆ () m , m1 In order to simplify the computation, we set 2 2 xˆ () n p and xˆ () n p . Hence we have which are Shannon entropies. The equality in (3) holds n n 1 1 L lnpn 1 1 0 , iff xˆ and xˆ . pn N N L Proof: From lemma 1, we have lnp 1 0 , p n 2 p2 1 n N p p N sin( ) 2 p xˆ () m N m1 p 1 , n1 n p1 1 . p N N p p 1. xˆ () n p1 n1 n n1 Solving the above equations, we finally obtain Take natural logarithm in both sides in above inequal- 1 1 xˆ () n , xˆ () n . From the definition of ity, we can obtain N N T( p ) 0 , Shannon entropy, we know that if H xˆ ln N and where H xˆ ln N , the equality in (3) holds. In addition, we p 21N p Tp() nl N (nis ) nl xˆ () m also have N NN sin( ) . From the proof we m1 2 pp 1 1 know that xˆ () n and xˆ () n means p p 1 N N N ˆ p1 ln n1 xn ( ) . p that xˆ () m and xˆ () n can be complex, and only if their Since 1p 2 and X 1 ,we know T (2) 0 . 2 amplitudes are constants, the equality will hold. Now we Note T (p ) 0 if 1p 2 . Hence, T '(p ) 0 if can obtain the following corollary out of above analysis.
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