Lower and upper bound for parametric Useful R-norm information measure

Transcription

1 Iteratoal Joral of Statstcs ad Aled Mathematcs 206; (3): 6-20 ISS: Maths 206; (3): Stats & Maths wwwmathsjoralcom eceved: Acceted: haesh Garg Satsh Kmar ower ad er bod for arametrc Usefl -orm formato measre haesh Garg ad Satsh Kmar Abstract A arametrc mea legth s defed as the qatty, Where 0, 0, 0, s a teger, Ths beg the sefl mea legth of code words weghted by tltes, ower ad Uer bods for are derved terms of sefl -orm formato measre AMS Sbject classfcato: 94A5, 94A7, 94A24, 265 Keywords: -orm Etroy, Usefl -orm Iformato, Utltes, Kraft eqalty, Holder s eqalty Corresodece: haesh Garg Itrodcto Cosder the followg model for a radom exermet S, S E ; P; U E E 2 Where, E,, E s a fte system of evets haeg wth resectve P, 2,, 0 robabltes,, ad credted wth tltes U, 2,,, 0,, 2,, S eote the model by, where, E, E2,, E S, 2,,, 2,, () We call () a Utlty Iformato Scheme (UIS) Bels ad Gas [3] roosed a measre of formato called sefl formato for ths scheme, gve by H U ; P log, (2) HU; P Where redces to Shao s [6] etroy whe the tlty asect of the scheme s gored e, whe for each throghot the aer, wll stad for otherwse stated ad logarthms are take to base less Gas ad Pcard [5] cosdered the roblem of ecodg the otcomes () by meas of a w, w, w refx code wth codewords 2,, havg legths 2 ad satsfyg Kraft s eqalty [4] ~ 6 ~

2 Iteratoal Joral of Statstcs ad Aled Mathematcs (3) Where s the sze of the code alhabet The sefl mea legth of code was defed as:, (4) HU; P ad the athors obtaed bods for t terms of Geeralzed codg theorems by cosderg dfferet geeralzed [7, 8,, 2, measres der codto (3) of qe decherablty were vestgated by several athors, see for stace the aers 3, 7] I ths aer, we stdy some codg theorems by cosderg a ew fcto deedg o the arameters, ad a tlty fcto Or motvato for stdyg ths ew fcto s that t geeralzes sefl -orm formato measre already exstg the lteratre sch as Boekee ad bbe [2] ad Satsh [,3] 2 Codg Theorems I ths secto, Satsh [] stded sefl -orm formato measre as: H U; P, (2) 0, 0, 0, 0,, 2,, where ad () If, the (2) becomes sefl -orm formato measre stded by Sgh ad ajeev [8] () Whe for each, e, whe the tlty asect s gored,, ad, the (2) redces to -orm etroy as cosdered by Boekee ad bbe [2] H P e, (22) () Whe, ad, the (2) redces to a measre of sefl formato de to Hooda ad Bhaker [] log H U; P e, (23) (v) Whe for each, e, whe the tlty asect s gored,,, ad, the measre (2) redces to Shao s etroy [6] H P e, log (24) (v) If, the (2) redces to -orm etroy for ower of robablty dstrbto H P e, (25) Frther cosder efto: The sefl mea legth wth resect to sefl -orm formato measre s defed as:, (26) Uder the codto, (27) Clearly the eqalty (27) s the geeralzato of Kraft s eqalty (3) A code satsfyg (27) wold be termed as a sefl ersoal robablty code (>2) s the sze of the code alhabet Whe, for each ad,, (27) redces to (3) () For for each ad, ad, becomes the otmal code legth defed by Shao [6] ~ 7 ~

3 Iteratoal Joral of Statstcs ad Aled Mathematcs () For for each ad, the (26) redced to cosdered by Boekee ad bbe [2] ad Satsh ad Ar [3] e, (28) () For for each, the (26) redces to mea code word legth corresodg to the etroy (25) e, (29) (v) For, the (26) becomes, (20) whch s a sefl -orm mea codeword legth H U ; P We establsh a reslt, that a sese, rovdes a characterzato of der the codto of qe decherablty,,,,2,,, Theorem 2 et satsfy the eqalty (27) The H( U; P), 0, 0 (2) Proof: By Holder s eqalty, we have q q x y x y, (22) where q ; ( 0), q0 or q( 0), 0; x, y 0 for each ( ), q Settg, ad, x y Pttg these vales (22) ad sg the eqalty (27), we get (23) ( ) (24) It mles ( ) (25) ow cosder two cases: Case : et 0 asg both sdes of (25) to the ower ( ) Sce ( ) 0 ( ) 2, we get (26) for 0, we get from (26) the eqalty (2) Case 2: et The roof follows o the same les It s clear that the eqalty (2) s tre f ad oly f whch mles that ~ 8 ~

4 Iteratoal Joral of Statstcs ad Aled Mathematcs log (27) Ths, t s always ossble to have a codeword satsfyg the reqremet log log, whch s eqvalet to (28) I the followg theorem, we gve a er bod for ( ; ) terms of HU P Theorem 22 By roerly choosg the legths, 2,, the code of Theorem 2, ca be made to satsy the followg eqalty: ( ) ( ) H( U; P) ( ) (29) Proof: From (28), t s clear that (220) We have aga the followg two ossbltes () et asg both sdes of (220) to the ower ( ), we have ( ) ( ) Mltlyg both sdes by ad the smmg over to we get ( ) (22) Obvosly (22) ca be wrtte as ( ) ( ) Sce 0 for, we get the eqalty (29) from (222) () If 0, the roof follows smlarly Bt the eqalty (222) s reversed (222) Theorem 23 For arbtrary, 0, 0,,,2,, ad for every codeword legths of Theorem 2, ca be made to satsy the followg eqalty: H( U; P) H( U; P) (223) Proof: Sose, log, 0 (224) Clearly ad satsfy the eqalty Holder s eqalty (22) Moreover, satsfes (27) Sose s the qe teger betwee ad, the obvosly, satsfes (27) Sce 0, 0, we have ( ) ( ) ( ) (225) ~ 9 ~

5 Iteratoal Joral of Statstcs ad Aled Mathematcs Sce, Hece (225) becomes ( ) ( ) Whch gves (223) 3 efereces Bhaker US, Hooda S Mea vale Characterzato of sefl formato measres, Tamkag J Math 993; 24: Boekee E, Vader bbe JCA The -orm Iformato Measre, Iformato ad Cotrol, 980; 45: Bels M, Gas S A Qaltatve-Qattatve Measre of Iformato Cyberetcs Systems, IEEE Tras Iformato Theory, IT, 968; 4: Feste A Fodato of Iformato Theory, McGraw Hll, ew York, Gas S, Pcard CF Bore Iferctre de la ogerr Utle de Certa Codes, C Acad Sc, Pars, 97; 273A: Grdal, Pessoa F O Usefl Iformato of Order, J Comb Iformato ad Syst Sc 977; 2: ogo G A oseless Codg Theorem for Sorces Havg Utltes, SIAM J Al Math 976; 30(4): Sgh P, Satsh Kmar, Parametrc -orm Iformato Measre, Pre ad Aled Mathematka Sceces, 2007; 65(- 2):4-6 9 Satsh Kmar Some more reslts o -orm formato measre Tamkag Joral of Mathematcs 2009; 40() Satsh Kmar Some more reslts o a geeralzed sefl -orm formato measre Tamkag Joral of Mathematcs 2009; 40(2)2-26 Satsh Kmar, Ar Chodhary Some More oseless Codg Theorem o Geeralzed -orm Etroy Joral of Mathematcs esearch 20; 3(): Satsh Kmar, Ar Chodhary A Codg Theorem Coected o -orm Etroy, Iteratoal Joral of Cotemorary Mathematcal Sceces, 20; 6(7): Satsh Kmar, Ar Chodhary -orm Shao-Gbbs Ieqalty Joral of Aled Sceces 20; (5): Satsh Kmar, Ar Chodhary Some Codg Theorems Based o Three Tyes of the Exoetal Form of Cost Fctos, Oe Systems ad Iformato yamcs, 202; 9(4):-4 5 Satsh Kmar, ajesh Kmar, Ar Chodhary Some more reslts o a geeralzed arametrc -orm formato measre of tye Alha Joral of Aled Scece ad Egg 204; 7(4): Shao CE A Mathematcal Theory of Commcato, Bell System Tech-J 948; 27( ): ShshaO Ieqaltes, Academc Press, ew York, Tteja K, Sgh P, ajeev Kmar Alcato of Holder s Ieqalty Iformato Theory, Iformato Sceces, 2003; 52:45-54 ~ 20 ~