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  ftKuttA ,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 ˆˆAAln  xn . where KutA ,12eee ib , n  Z and i 1  n1  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  B1 , f tFuKut ,du, i.e., the in-  A  A1    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 x1,xx  2,   3,  , xNC  

x 1,xx 2, 3, , xNCN 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 p2 N idk ikn 22bNbbN2 UX N ab  ab2 p UX xˆ kibNx 1eeen ABqp12 21 l1 . (2) N with 1 p  2 and uknxnA ,,1, nkN. 11 l1   1. pq Also, we can rewrite the definition (2) as ˆ Proof: Let X AA UX, Xxxxx  ,,,,  where Uukn , . 123 N AANN Clearly, for DLCT we have the following property [5]: x1,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 CNxxxˆˆˆ123,,,, 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 n1 n  its probability density function pxn , 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 . n1 C  Here Uu sup l with CC   1 N  Hxln px . l  nnn1  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  AN123  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, AAAn1   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 AB1 CAB1A B1 obtain Proof: From lemma 1, we have

2 p p2 1 UU X Mp X N p p AC1  p Nabab 2 p xˆ m B q 12 21 m1 B  p1  1 . p with MUCA11 UU . N p AB B p1  xnˆA  Let YUX , then X  UY. In addition, from n1 B1 B  the property of DLCT [5] we can have Take natural logarithm in both sides in above inequal- 1 ity, we can obtain MUU11A . AB B  Nabab12 21 Tp   0 , Hence we can obtain from the above equations where

2 p p  21N p Tpln Nabab ln xˆ m UY Mp UY  12 21 m1 B  1  ABqpAB 2 pp with p p 1 N  ln xnˆ p1 1 n1 A  M  . p  AB1 Nabab12 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- Tp22ln Nabab12 21 ln  xmB  pp m1  dorff-Young inequality. In the next sections, we will use p this lemma to prove the new uncertainty relations. N ln xmˆ  xmˆ  1 m1 BB  p 3. The Uncertainty Relations p N ˆ m1 xmB 

3.1. Shannon Entropic Principle p , 1 N ˆ p1  2 ln n1 xnA  Theorem 1: For any given discrete time series p 

Xxxxx 123,,,, N p N  ˆˆp1 n1 xnAA ln  xn x1,xx  2,   3, , xNC   N 1   p pp1 N with length N and X  1, xˆˆx is the DLCT se- ˆ p1 2 AB n1 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 HxˆAB n Hxˆ  m (3) Minimize lnNabab , nm ,  1, , N N 22 12 21  ln xnˆ  xnˆ  n1  AA  where , 22 N 22 N  ln xmˆˆ xm Hxˆ ln xnˆˆ  xn m1 BB  AAn1   A    22 and NNˆ ˆ subject to nn11xnAB  xn  1. 22 Hxˆ N ln xˆ  m  xˆ  m , To solve this problem let us consider the following BBBm1   Lagrangian

Open Access JSIP 426 Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform

N 22 1 Lxnxnln ˆ   ˆ  xnˆ  n1  AA A  N A 22 N ln xmˆ xmˆ . and m1  BB   1 xnˆ  , NN22 B  xnˆˆ11 xn  NB 12nn11AB     then we have NN Nabab and In order to simplify the computation, we set AB 12 21 2 2 xˆ np A and xˆ np B . Hence we have An B  n H xHxˆAB ˆ ln  Nabab 12 21 .

L A A lnpn  1 1  0 , pn 3.2. Rényi Entropic Principle L Theorem 2: For any given discrete time series lnpB  1   0 , pB n 2 n Xxxxx  123,,,, N 

N p A  1 , N n1 n x1,xx  2,   3,  , xNC   N B with length N and X  1, xˆ xˆ is the DLCT series n1 pn  1 . 2 AB ab11 Solving the above equations, we finally obtain associated with the transform parameter A    1 1 cd11 xnˆ  , xnˆ  . From the definition of A B ab N A NB 22 ( B    , respectively), NNAB counts the non- cd22 Shannon entropy, we know that if H xˆAA ln N and  

H xˆB  ln NB , then zero elements of xˆA (xˆB , respectively), then we can obtain the generalized discrete Renyi entropic uncer- H xHxˆ ˆ ln Nabab. AB  12 21  tainty relation

In addition, we also can obtain NNAB N ab12  ab 21 . HxHxˆˆAB ln  Nabab 12 21  1 (4) From the above proof, we know that xnˆA  111 N with 1 and  2 A 2  1 and xnˆB  imply that xˆA m and xˆB n where NB 1 N 2 can be complex values, and only if their amplitudes are Hx ˆAAln  xmˆ  , 1  m1  constants, the equality will hold. Now we can obtain the following corollary out of above analysis. 1 N 2 Hx ˆˆBBln  xn , Corollary 1: For any given discrete time series 1  n1   Xxxxx 123,,,, N which are Rényi entropies. Proof: In lemma 1, set q  2 and p  2 , we have N x1,xx  2,   3, , xNC   1 11    1 and   2 . Then from lemma 1, we ob- with length N and X  1, xˆˆx is the DLCT series 2  2 AB ab btain associated with the transform parameter A  11   1 cd11 2 N xmˆ  2 ab m1 A  22 . ( B  , respectively), NNAB counts the non-   1 1 cd22 N 2 Nabab 2  xˆ n 2 12 21 n1 B   zero elements of xˆA (xˆB , respectively), if

Take the square of the above inequality, we have

11 1 NN22 xmˆ   Nabab   xnˆ  . mn11AB  12 21     Take the power of both sides in above inequallity, we obtain 1

Open Access JSIP Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform 427

1  1  N 2  1 Hx ln x (7)  xmˆA     k  m1  1  k  , 1 12 when   1 , discrete Rényi entropy tend to discrete Nabab  N xˆ n 1 12 21 n1 B   Shannon entropy. In order to obtain the discrete spectrum, the sampling i.e., 1 must be done. For two continuous functions’ DLCT 12N  1 Nabab  xˆ n FA u and FB v with the transform parameter 12 21 n1 B    1. (5) ab11 ab22 1 A  (B  , respectively), we set the N 2  1     xmˆ cd11 cd22 m1 A   sampling periods T1 and T2 and assume that they sat- Take the natural logarithm on both sides of (5), we can isfy the Shannon sampling theorem [16]. Set obtain kT1 2  uFu 1 du 11NN22  kAkT  lnxnˆ  ln xmˆ 1 nm11BA     (8) 11 lT1 2 2 lBvFv  dv lT 2 lnNabab12 21 . Therefore

Clearly, as   1 and   1 , the Renyi entropy  22kT1 reduces to Shannon entropy, thus the Renyi entropic un- F uudd 1 Fu u (9)  AA  kT    certainty relation in (4) reduces to the Shannon entropic k  1  uncertainty relation (3). Hence the proof of equality in 22lT1 F vvdd 2 Fv v (10) theorem 2 is trivial according to the proof of theorem 1.  BB  lT    2 Note that although Shannon entropic uncertainty rela- l  tion can be obtained by Rényi entropic uncertainty rela- Since when   1 , f  xx   is a convex function, tion, we still discuss them separately in the sake of inte- and when   1 , g  yy   is a concave function, we grality. have the following inequalities   11kT111122kT 3.3. Another Shannon Entropic Principle F uudd Fu u kTAAkT   11 via Sampling TT11 (11) The discrete Shannon entropy can be defined as    11lT112222lT F vvdd Fvv Es kk sln   s (6) lT BB lT 22 k  TT22 (12) where k x is the density function of variable s . Discrete Rényi entropy can be defined as follows: Therefore

   22kT11111 kT 2 Fu dd uFu uT Fu d u  T1   u .  AA kT   11 kT  Ak 11 kk T1 k 

  22lT11221 lT 2 F vdd v Fv vT Fv d v  T1   v ,  BB lT   22 lT  Bl ll22  v l  i.e.,   2   Fu d uT 1   u (13)   A 1  k k 

  2   F vvTd  1   v. (14)   B 2  l l Therefore 1 12  11 11  1    ab ab T11 u T v (15) 12 21 1kl    2   ab12 ab 21 kl    

Open Access JSIP 428 Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform

 Take the power of on the both sides of above equation and use the relation between  and  , we have 1

1 1   21  1 TT12   l v π l    1 (16) 1 1 1 1  1  21    1 ab ab  1  u 12 21   k  ab12 ab 21 π k  Take logarithm on both sides of above equation

 11ln π ln  π ab12 ab 21 lnlkvu ln ln  (17) 11lk  2121  TT12 That is,

BAln π ln  π ab12 ab 21 HH  ln  (18) 2121TT12  If

abcd111, , , 1 cos ,sin , sin ,cos  and

abcd222, , , 2 cos ,sin , sin ,cos  , then we have

12  1  ln π ln π cos sin cos  sin  HH  ln T1  v  2  l  2121TT  l  when  2nnππ2  Z and  2llπ Z, we number of non-zero elements, the equality holds in the have the traditional case uncertainty relation. In addition, the product of the two numbers of non-zero elements is equal to Nabab , ln π ln π 12 21 HHT  ln 12 T. (19) i.e., NN N ab12  ab 21. On one hand, these new 2121 uncertainty relations enrich the ensemble of uncertainty Specially, when  1,  1 , have principles, and on the other hand, these derived bounds yield new understanding of discrete signals in new 2TT transform domain. EEln 2π 1ln12  abcd111,,, 1 a 2 , bc 2 , 2 , d 2  ab12 ab 21 5. Acknowledgements (20) where, This work was fully supported by the NSFCs (61002052,  2260975016, 61250006). kT1111kT   EF udlnuF udu, AA kT kT A  k  11 REFERENCES  22 lT1122lT  EF vdlnvF vdv. [1] R. Ishii and K. Furukawa, “The Uncertainty Principle in BB lT lT B  l22Discrete Signals,” IEEE Transactions on Circuits and Systems, Vol. 33, No. 10, 1986, pp. 1032-1034. 4. Conclusion [2] L. C. Calvez and P. Vilbe, “On the Uncertainty Principle in Discrete Signals,” IEEE Transactions on Circuits and In this article, we extended the entropic uncertainty rela- Systems-II: Analog and Digital Signal Processing, Vol. tions in DLCT domains. We first introduced the general- 39, No. 6, 1992, pp. 394-395. ized discrete Hausdorff-Young inequality. Based on this http://dx.doi.org/10.1109/82.145299 inequality, we derived the discrete Shannon entropic un- [3] S. Shinde and M. G. Vikram, “An Uncertainty Principle certainty relation and discrete Rényi entropic uncertainty for Real Signals in the Fractional Fourier Transform Do- relation. Interestingly, when the variable’s amplitude is main,” IEEE Transactions of Signal Processing, Vol. 49, equal to the constant, i.e. the inverse of the square root of No. 11, 2001, pp. 2545-2548.

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