Journal of Signal and Information Processing, 2013, 4, 423-429 423 Published Online November 2013 (http://www.scirp.org/journal/jsip) http://dx.doi.org/10.4236/jsip.2013.44054 Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform Yunhai Zhong1, Xiaotong Wang1, Guanlei Xu2, Chengyong Shao1, Yue Ma1 1Navigtion Department of Dalian Naval Academy, Dalian, China; 2Ocean Department of Dalian Naval Academy, Dalian, China. Email: [email protected], [email protected], [email protected], [email protected], [email protected] Received September 24th, 2013; revised October 20th, 2013; accepted October 30th, 2013 Copyright © 2013 Yunhai Zhong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Uncertainty principle plays an important role in physics, mathematics, signal processing and et al. In this paper, based on the definition and properties of discrete linear canonical transform (DLCT), we introduced the discrete Hausdorff- Young inequality. Furthermore, the generalized discrete Shannon entropic uncertainty relation and discrete Rényi en- tropic uncertainty relation were explored. In addition, the condition of equality via Lagrange optimization was devel- oped, which 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 touch their lowest bounds. On one hand, these new uncer- tainty relations enrich the ensemble of uncertainty principles, and on the other hand, these derived bounds yield new understanding of discrete signals in new transform domain. Keywords: Discrete Linear Canonical Transform (DLCT); Uncertainty Principle; Rényi Entropy; Shannon Entropy 1. Introduction Shannon entropic uncertainty principle in LCT domain for discrete data, based on which we also derived the Uncertainty principle [1-20] plays an important role in conditions when these uncertainty relations have the physics, mathematics, signal processing and et al. Un- equalities via Lagrange optimization. The third contribu- certainty principle not only holds in continuous signals, tion is that we derived the Renyi entropic uncertainty but also in discrete signals [1,2]. Recently, with the de- principle in DLCT domain. As far as we know, there velopment of fractional Fourier transform (FRFT), con- have been no reported papers covering these generalized tinuous generalized uncertainty relations associated with discrete entropic uncertainty relations on LCT. FRFT have been carefully explored in some papers such as [3,4,16], which effectively enrich the ensemble of 2. Preliminaries FRFT. However, up till now there has been no reported article covering the discrete generalized uncertainty rela- 2.1. LCT and DLCT tions associated with discrete linear canonical transform Before discussing the uncertainty principle, we introduce (DLCT) that is the generalization of FRFT. From the some relevant preliminaries. Here we first briefly review viewpoint of engineering application, discrete data are the definition of LCT. For given analog signal widely used. Hence, there is great need to explore dis- f tLRLR 12 and ft 1 (where f t crete generalized uncertainty relations. DLCT is the dis- 2 2 crete version of LCT [5,6], which is applied in practical denotes the l2 norm of function f t ), its LCT [5,6] is engineering fields. In this article we will discuss the en- defined as tropic uncertainty relations [7,8] on LCT. FuAA F ft ftKuttA ,d In this paper, we made some contributions such as fol- lows. The first contribution is that we extend the tradi- idu22 iut iat tional Hausdorff-Young inequality to the DLCT domain 12i b e22bbb e e f t d t b 0, ad bc 1 with finite supports. It is shown that these bounds are 2 icdu 2 connected with lengths of the supports and LCT parame- dfdue0 b ters. The second contribution is that we derived the (1) Open Access JSIP 424 Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform 22 idu iut iat 1 N 2 22bb b Hx ˆˆAAln xn . where KutA ,12eee ib , n Z and i 1 n1 ab Clearly, if 1 as shown in [13], is the complex unit, A is the transform pa- cd H xHxˆAA ˆ . rameter defined as that in [5,6]. In addition, FABFft ft. 2.3. Discrete Hausdorff-Young Inequality on DLCT If A B1 , f tFuKut ,du, i.e., the in- A A1 Lemma 1: For any given discrete time series verse LCT reads: f tFuKut 1 ,d u. A A Xxxxx ,,,, Let 123 N N Xxxxx 123,,,, N x1,xx 2, 3, , xNC x 1,xx 2, 3, , xNCN with length N and X 1, UU is the DLCT 2 AB transform matrix associated with the transform parameter be a discrete time series with length N and X 1. 2 ab11 ab22 Assume its DLCT (discrete FLCT) A (B , respectively), then we can ˆ N cd cd X xxxˆ123,,,,ˆˆ x ˆN C under the transform pa- 11 22 rameter . obtain the generalized discrete Hausdorff-Young ine- Then the DLCT [5] can be written as quality 2 ian2 p2 N idk ikn 22bNbbN2 UX N ab ab2 p UX xˆ kibNx 1eeen ABqp12 21 l1 . (2) N with 1 p 2 and uknxnA ,,1, nkN. 11 l1 1. pq Also, we can rewrite the definition (2) as ˆ Proof: Let X AA UX, Xxxxx ,,,, where Uukn , . 123 N AANN Clearly, for DLCT we have the following property [5]: x1,xx 2, 3, , xNC N XUXˆ 1. AA2 In 2the following, we will assume that the transform be a discrete time series with length N and its DLCT ˆ N ˆ parameter b 0 . Note the main difference between the X CNxxxˆˆˆ123,,,, x ˆ C with X CC UX and the discrete and analog definitions is the length: one is finite ab transform parameter C . and discrete and the other one is infinite and continuous. cd ˆ Since X 1, XUXBC 1 from Parseval’s 2 2 2 2.2. Shannon Entropy and Rényi Entropy 1 N 2 2 For any discrete random variable x n 1, , N and theorem. Here Xx . Clearly, we can ob- n 2 n1 n its probability density function pxn , the Shannon entropy [9] and the Rényi Entropy [10] are defined as, tain the inequality [13]: respectively UX M X CC 1 N H xpxpnn ln xn, with MU . n1 C Here Uu sup l with CC 1 N Hxln px . l nnn1 1 UullCC ,1,, N. Hence, in this paper, we know that for any DLCT Hence, we have UXCC M X with MUCC . Xˆ xxxˆˆˆ,,,, x ˆ CN (with X 1 and 1 AN123 2 Then from Riesz’s theorem [11,12], we can obtain the XUXˆ 1), the Shannon entropy and the Ré- AA2 discrete Hausdorff-Young inequality [11,12] nyi Entropy2 [13] associated with DLCT are defined as, 2 p respectively UX Mp X CCq p 2 2 H xxnxˆˆˆ N ln n, AAAn1 with 1 p 2 and Open Access JSIP Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform 425 11 which are Shannon entropies. The equality in (3) holds 1. pq 1 1 iff xˆA and xˆB . N N Set UU , then UU UU [5], we A B C AB1 CAB1A B1 obtain Proof: From lemma 1, we have 2 p p2 1 UU X Mp X N p p AC1 p Nabab 2 p xˆ m B q 12 21 m1 B p1 1 . p with MUCA11 UU . N p AB B p1 xnˆA Let YUX , then X UY. In addition, from n1 B1 B the property of DLCT [5] we can have Take natural logarithm in both sides in above inequal- 1 ity, we can obtain MUU11A . AB B Nabab12 21 Tp 0 , Hence we can obtain from the above equations where 2 p p 21N p Tpln Nabab ln xˆ m UY Mp UY 12 21 m1 B 1 ABqpAB 2 pp with p p 1 N ln xnˆ p1 1 n1 A M . p AB1 Nabab12 21 Since 1 p 2 and X 1 and Parseval equality, 2 Since the value of X can be taken arbitrarily in C N , we know T 2 0. Note Tp 0 if 12 p . Y can also be taken arbitrarily in C N . Therefore, we Hence, Tp 0 if p 2 . Since can obtain the lemma. 11N p ˆ Clearly, this lemma is the discrete version of Haus- Tp22ln Nabab12 21 ln xmB pp m1 dorff-Young inequality. In the next sections, we will use p this lemma to prove the new uncertainty relations. N ln xmˆ xmˆ 1 m1 BB p 3. The Uncertainty Relations p N ˆ m1 xmB 3.1. Shannon Entropic Principle p , 1 N ˆ p1 2 ln n1 xnA Theorem 1: For any given discrete time series p Xxxxx 123,,,, N p N ˆˆp1 n1 xnAA ln xn x1,xx 2, 3, , xNC N 1 p pp1 N with length N and X 1, xˆˆx is the DLCT se- ˆ p1 2 AB n1 xnA ries associated with the transform parameter we can obtain the final result in theorem 1 by setting p = ab ab 11 22 2. A (B , respectively), NNAB cd11 cd22 Now consider when the equality holds. From theorem counts the non-zero elements of xˆA (xˆB , respectively), 1, that the equality holds in (3) implies that then we can obtain the generalized discrete Shannon en- H xHxˆA ˆB reaches its minimum bound, which tropic uncertainty relation means that Minimize H xHxˆA ˆB subject to xxˆ ˆ 1, i.e. AB22 HxˆAB n Hxˆ m (3) Minimize lnNabab , nm , 1, , N N 22 12 21 ln xnˆ xnˆ n1 AA where , 22 N 22 N ln xmˆˆ xm Hxˆ ln xnˆˆ xn m1 BB AAn1 A 22 and NNˆ ˆ subject to nn11xnAB xn 1.