An application of generalized Tsalli s-havrda-charvat entropy in coding theory through a generalization of Kraft inequality

Transcription

1 Internatonal Journal of Statstcs and Aled Mathematcs 206; (4): 0-05 ISS: Maths 206; (4): Stats & Maths wwwmathsjournalcom Receved: Acceted: Maharsh Markendeshwar Unversty, Mullana, Ambala, Haryana, Inda An alcaton of generalzed Tsall s-havrda-charvat entroy n codng theory through a generalzaton of Kraft nequalty Abstract A arametrc mean length s defned as the quantty, where 0, 0, u 0, s an nteger, Ths beng the useful mean length of code words weghted by utltes, u Lower and Uer bounds for are derved n terms of useful Tsall s-havrda-charvat nformaton measure for ower robablty dstrbuton Keywords: Tsall s Entroy, Useful Tsall s entroy, Utltes, Kraft nequalty, Holder s nequalty AMS Subject classfcaton: 94A5, 94A7, 94A24, 265 Introducton Consder the followng model for a random exerment S, S E ; P; U, where E E, E2,, E s a fnte system of events haenng wth resectve robabltes P, 2,,, 0, and credted wth utltes U u, u2,, u, u 0,, 2,, enote the model by S, where, E, E2,, E S, 2,, u, u2,, u () We call () a Utlty Informaton Scheme (UIS) Bels and Guasu [2] roosed a measure of nformaton called useful nformaton for ths scheme, gven by H U ; P u log ( ), (2) Corresondence: Maharsh Markendeshwar Unversty, Mullana, Ambala, Haryana, Inda where HU; P reduces to Shannon s [5] entroy when the utlty asect of the scheme s gnored e, when u for each Throughout the aer, wll stand for unless otherwse stated and logarthms are taken to base Guasu and Pcard [4] consdered the roblem of encodng the outcomes n () by means of a refx code wth codewords w, w, 2, w havng lengths n, n, 2, n and satsfyng Kraft s nequalty [3] ~ ~

2 Internatonal Journal of Statstcs and Aled Mathematcs n (3) Where s the sze of the code alhabet The useful mean length u L u of code was defned as: n L u, (4) u and the authors obtaned bounds for t n terms of HU; P Generalzed codng theorems by consderng dfferent generalzed measures under condton (3) of unque decherablty were nvestgated by several authors, see for nstance the aers [4, 8-0, 3] In ths aer, we study some codng theorems by consderng a new functon deendng on the arameters, and a utlty functon Our motvaton for studyng ths new functon s that t generalzes useful nformaton measure already exstng n the lterature such Tsall s entroy [7], Havrda-Charvat [6] etc 2 Codng Theorems In ths secton, we defne a new nformaton measure as : ; H U P, (2) 0, 0, u 0, 0,, 2,, and where () If, Then (2) becomes a useful nformaton measure e, ; H U P (22) () When u for each, e, when the utlty asect s gnored,, and, then (2) reduces to Tsall s-havrda- Charvat entroy e, HP (23) () When, and, then (2) reduces to a measure of useful nformaton due to Hooda and Bhaker [] log ( ) e, HU; P (24) (v) When u for each, then (2) reduced to Satsh and Arun [3] entroy e, ; H U P (25) (v) When u for each, e, When the utlty asect s gnored,,, and, the measure (2) reduces to Shannon s entroy [5] H P (26) e, log ( ) Further consder, efnton: The useful mean length wth resect to useful R-norm nformaton measure s defned as :, (27) n under the condton, u (28) Clearly the nequalty (28) s the generalzaton of Kraft s nequalty (3) A code satsfyng (28) would be termed as a useful ersonal robablty code (>2) s the sze of the code alhabet When, u for each and,, (28) reduces to (3) ~ 2 ~

3 Internatonal Journal of Statstcs and Aled Mathematcs () For u for each and, and, becomes the otmal code length defned by Shannon [5] () For u for each and, then (27) becomes a new mean code word length corresondng to the Tsall s entroy n e, L (29) () If, then (27) becomes a new mean codewords length corresondng to the entroy (22) n e, (v) If u, then (27) becomes a mean codewords length corresondng to the entroy (25) e, L We establsh a result, that n a sense, rovdes a characterzaton of H U; P under the condton of unque decherablty Theorem 2 Let u,, n,,2,,, satsfy the nequalty (28) Then H ( U; P), 0, 0 (20) Proof: By Holder s nequalty, we have q q x y xy, (2) q ; ( 0), q 0 or q( 0), 0; x, y 0 for each where ( ) Settng,, q, and n, x y, (22) Puttng these values n (2) and usng the nequalty (28), we get n ( ) (23) It mles n ( ) (24) ow consder two cases: Case : Let 0 Rasng both sdes of (24) to the ower ( ), we get n ( ) Snce, ( ) 0for 0, we get from (25) the nequalty (20) (25) Case 2: Let The roof follows on the same lnes It s clear that the equalty n (20) s true f and only f n whch mles that log (26) n ~ 3 ~

4 Internatonal Journal of Statstcs and Aled Mathematcs Thus, t s always ossble to have a codeword satsfyng the requrement log n log, whch s equvalent to n (27) In the followng theorem, we gve an er bound for n terms of H ( U; P) Theorem 22 By roerly choosng the lengths n, n2,, n n the code of Theorem 2, can be made to satsy the followng nequalty: ( ) ( ; ) ( L ) u H U P (28) Proof: From (27), t s clear that n (29) We have agan the followng two ossbltes () Let n ( ) ( ) Rasng both sdes of (29) to the ower ( ) Multlyng both sdes by and then summng over we get n ( ) ( ), we have (220) Obvously (220) can be wrtten as n ( ) ( ) (22) Snce 0 for, we get the nequalty (28) from (22) () If 0, the roof follows smlarly But the nequalty (22) s reversed Theorem 23 For arbtrary, 0, 0, and for every codeword lengths n,,2,, of Theorem 2, L u can be made to satsy the followng nequalty: H ( U ; P) H ( U ; P) Proof: Sose, log, 0 n (222) (223) Clearly n and n satsfy the equalty n Holder s nequalty (2) Moreover, nteger between n and n, then obvously, n satsfes (28) Snce 0, 0, we have n ( ) n ( ) n ( ) n ( ) ~ 4 ~ n satsfes (28) Sose n s the unque (224) Snce,

5 Internatonal Journal of Statstcs and Aled Mathematcs Hence (224) becomes n ( ) Whch gves (222) 3 References Bhaker US, Hooda S Mean value Characterzaton of useful nformaton measures, Tamkang J Math 993; 24: Bels M, Guasu S A Qualtatve-Quanttatve Measure of Informaton n Cybernetcs Systems, IEEE Trans Informaton Theory, 968; IT-4: Fensten A Foundaton of Informaton Theory, McGraw Hll, ew York Guasu S, Pcard CF Borne Infercutre de la Longueur Utle de Certan Codes, CR Acad Sc, Pars, 97; 273A: Gurdal, Pessoa F On Useful Informaton of Order, J Comb Informaton and Syst Sc 977; 2: Havrda JF, Charvat F Qualfcaton Method of Classfcaton Process the concet of Structural α-entroy, Kybernetka, 967; 3:30-35, Kumar S Some more results on R-orm nformaton measure, Tamkang Journal of Mathematcs, 2009; 40(): Kumar S Some more results on a generalzed useful R-orm nformaton measure, Tamkang Journal of Mathematcs, 2009; 40(2): Kumar S, Choudhary A Some More oseless Codng Theorem on Generalzed R-orm Entroy, Journal of Mathematcs Research 20; 3(): Kumar S, Choudhary A Codng Theorem Connected on R-orm Entroy, Internatonal Journal of Contemorary Mathematcal Scences 20; 6(7): Kumar S, Choudhary A Some Codng Theorems Based on Three Tyes of the Exonental Form of Cost Functons, Oen Systems and Informaton ynamcs, 202; 9(4):-4 2 Kumar S, Kumar R, Choudhary A Some more results on a generalzed arametrc R-norm nformaton measure of tye Alha Journal of Aled Scence and Engg 204; 7(4): Kumar S, Choudhary A Some codng theorems on generalzed Havrda-Charvat and Tsall s entroy, Tamkang journal of mathematcs, 202; 43(3): Longo G A oseless Codng Theorem for Sources Havng Utltes, SIAM J Al Math, 976; 30(4): Shannon CE A Mathematcal Theory of Communcaton, Bell System Tech-J 948; 27: , Shsha O Inequaltes, Academc Press, ew York Tsall s C Possble generalzaton of Boltzmann Gbbs statstcs J Stat Phys 988; 52: ~ 5 ~