CHAPTER 7 Channel Capacity of N Identical Cascaded Channel with Utility

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1 CHAPER 7 Chae Caacty of Idetca Cascaded Chae wth Utty 7. Itroducto Cascaded chaes are very ofte used ractce. hey become ecessary because some waves e mcrowave s ad the eectromagetc wave do ot foow the curvature of the earth, because the shrg suffered by the sga becomes rohbtve whe the dstace becomes arge. I such codtos the desger forced to brea u the whoe chae to a cascade of chaes. It s aso caed as termedate chae. he erformace of chaes cascade wth themseves has bee dscussed by varous authors. Some of them cosder the case of bary symmetrc chae cascaded wth tsef tmes refer to Ash965,Cover ad homas99,mceece977. Gaager968 shows that for og cascades of fxed chaes, the caacty teds to zero at east as fast as a certa exoetay decreasg fucto. A eary aer of Sverma955 comares BSC s ad Z chaes of the same caacty ad exames whch woud have hgher caacty whe cascaded wth tsef.e.a cascade of egth 2. Maa 988 cosders such cascades for the cass of very osy chaes, ad aso exames the effect of a verter aced betwee the chaes. Smo97 gves uder certa assumtos a formua for the caacty of X chae that s cascaded wth tsef tmes. he caacty s exressed terms of egevaues ad egevectors of the dvdua trasto matrx. I the vew of formato theory, the arge umber of mcrowave s recety but ehaces the dscusso of chaes cascade.desgers ow that 3

2 cascaded chaes t s mortat to use moduato system exhbtg ose reducg roertes such as Frequecy ModuatoF.M. ad Puse Code ModuatoP.C.M.. here s a ot of dfferece betwee the robem of trasmttg formato through a sge chae ad that of trasmttg formato through a cascade of chaes. I the frst case, the trasmtter has a the formato to be trasmtted; whereas, the secod case, excet for the frst trasmtter the formato whch s avaabe to each trasmtter to be recse formato about what was trasmtted by the frst trasmtter s o more the form of a symbo but rather the form of a set of a osteror robabtes. We shoud therefore exect to fd that maer whch the termedate chae w be oerate ad that s very mortat for the erformace of cascaded chaes. Caes ca be cascaded dfferet ways deedg o codto of trasmtter ad recever. I the reset chater we dscuss roertes of cascaded chaes secto7.2. I secto 7.3 we gve the cassfcato of cascaded chaes I secto 7.4 we defe mutua formato wth utty ad rove that t s cocave fucto. I secto 7.5 we study chae caacty of detca cascaded chaes wth utty. 7.2 Basc Proertes of Cascades Let C, C2, C be a st of dscrete memoryess chaes DMC s. here are! ways to form a ew DMC by cascadg the orders. Itay we ca th of each C dfferet C as a stochastc matrx, so that the stochastc matrx of the ew DMC w be equa to a matrx roduct of dvdua DMC s. Sce matrx mutcato s ot commutatve, we exect that the order whch the chaes are cascaded w effect the caacty of overa cascade. Our frst observato s that sce the determat of the chaes are arraged, t suffces to aayze the behavour of the caacty fucto for chaes of costat determat. 32

3 Let we cosder the Bary chaes,.e. 2 X 2 stochastc matrx. If s a bary DMC havg trasto matrx the t w be coveet to th of as the coum vector,.e. the ot the ut square whose coordates are the trasto error robabtes for the chae. If deotes the chae α,β, the Sverma955 defe the caacty bts/chae as foows: H H H H og 2 2, f, c, otherwse, for, ad otherwse. Here H x x og x xog x s the bary etroy fucto measured bts. he uque otmzg ut dstrbuto, - s gve by Sverma955. H H, 2, for Cascaded Pars of Chaes If C ad C2 are the bary chaes wth trasto error robabtes a,b ad α,β resectvey, we et C C2 deote the vector corresodg to the stochastc matrx obtaed from the matrx roduct CC 2. Sce the matrx mutcato s assocatve, the so s, whch oerates o vectors. Sce the determat of C s -a-b, for the C we defe m C a b. he chae vectors C ad C2ca be gve as C C2 C2 m C2 C ow observe that the dstace from the vector C to BSC e s a b 2 ; f we et r C a b, the ths dstace s r C 2. he caacty of cascade C s maxmzed 33

4 mmzed f r C s maxmzed mmzed. We ow dscuss some sme roertes of r. From the defto of r t s easy to show that r C C2 r C2 m C2 r C ext, suose C s a chae vector wth r C r C 2. he f m C2,.e. the matrx C2 has a ostve determat, the r C C2 r C2 m C2 r C r C2 m C2 r C r C C Smary, f r C2 r C2, ad f m C2 m C2, the r C C2 r C2 m C2 r C r C2 m C2 r C r C C he age that the chae vector C maes wth the BSC e s C arcta a b a b. By sme cacuato t ca be verfed that r C C2 r C2 C C C By combg these resuts, we ca characterze the otma cascade uder certa codtos Cascades of Chaes Let we cosder the ocatos of the! ots corresodg to each ossbe cascade of the chaes s C, C, C. Sce the determat of the chae matrx of such a cascade s deedet of the order of the C a of these ots e o a e the ut square havg so -. From the covexty ad symmetry of the caacty fucto, the cascade havg hghest caacty w be the oe that es the farthest from the BSC e. Hece we eed oy cosder the sequece whch maxmze r for the overa cascade. 34

5 Defto 7.2.: A cascade C C2... C s sad to be r max r m otma f o other cascade of the C roduces a chae wth arger smaer r. Hece f rmax rm the the sequece achevg r max yeds the cascade of hghest caacty, whe f rmax hghest caacty. rm the the sequece achevg r m yeds the cascade of 7.3 Cassfcato of Cascaded Chaes Cascadg of the chaes ca be cassfed four categores as dscussed beow: 7.3. Equvaet Chaes It s ofte coveet to cosder the cascade of chaes as a ut, that s, cosderg the cascade oy the term of ts ut ad outut. hs ut w be ca the cascaded chae. Aso, the equvaet chaes s the chae whch has statstca roertes detca to those of the cascade, at east as far as ts ut-outut reatos are cocered. he statstca roertes of the equvaet chae deed very much o the assumed oerato of the termedate chaes. he chage the oerato of the termedate stato roduces very drastc chages the erformace of the equvaet chae. A the roertes of equvaet chaes ca be defe by ts trasto robabty matrx; the trasto robabty matrx of the equvaet chae s equa to the roduct of the trasto robabty matrces of each chae of the cascade ad the factor matrces s detca to the order of the chaes the cascade. he chae caacty of the equvaet chae s aways smaer or equa to the smaest chae caacty of the cascaded chaes. Whe the trastorobabty matrces of a chaes are o-sguar, the equa sg hods oy f ot a but oe of them s oseess chae Dscrete Chaes Cosder a cascade of dscrete chaes. Sce each of these chaes must trasmt the same message, therefore we may assume that they have a commo ahabet of m symbos. I each chae, arorate sgas are assocated to each 35

6 symbo. We assume that a artcuar chae, a the sgas have the same durato ad aso the termedate chae oerates as a maxmum a-osteror robabty detector. Uder these codtos, addto to the roagato tme, a deay at east equa to w occure the th chae because the recever must have receved the comete sga before beg abe to comute the a- osteror robabtes. For each chae, o the bass of the ose statstcs ad the decodg rocedure, t s ossbe to obta the trasto robabtes, that s, the robabty that a artcuar symbo, say x, beg trasmtted ad some other symbo say y, w be receved. Let ths robabty, for the th chae, be rereseted by x. y Cascade of Reeaters he termedate chae oerato dscussed revous case may occur addto deay each chae but certa cases ths cumuatve deay becomes udesrabe. It s therefore of terest to cosder a case where ths deay reduce to a mmum. I artcuar we wsh to cosder here the case where the sgas are retrasmtted exacty as they are receved. Let us assume that a chaes are bad mted ad have the same badwdth w. hus the sgas are cometey defed by a sequece of equdstat same tae at rate of 2w sames er secod. For the smcty, et us assume that the termedate chaes are reeaters that are retrasmttg the sga exacty as they receved. hus order to obta the ut-outut statstcs of the cascade, we eed oy to cosder the sga as oe same at a tme. Let the same x of the frst trasmtter wth ts corresodg robabty x. he same x w trasmt dow the frst chae ad w be receved as y by the frst termedate stato, y 2 by the secod termedate stato, ad fay as y by the ast recever. Each chae s rereseted by a codtoa robabty desty; for the th chae y gves the robabty dstrbuto of the sames y y. It w be cometey defe by the trasto robabty desty y. x 36

7 7.3.4 Cascadg of Idetca Chaes Qut ofte ractce, oe s cocered wth trasmttg data over a chae that s comosed of a cascade of detca sub chaes, e.g. the reeated teehoe e. Cacuato of the chae caacty va the stadard techque of fst fdg the overa chae trasto matrx becomes extremey tedous as the umber of sub chaes become extremey arge tycay =3 for og hau teehoe. A ateratve smfed aroach to fd the chae caacty of chae s by combg a Ege vaue techque wth a chae caacty. It has bee exa a stadard text o Iformato theory due to Robert Ash965 aog wth the codto that sub chaes shoud be dscrete, memory ess, ad have a o-sguar trasto matrx. 7.4 Mutua Iformato wth Utty Let X x, x2,..., x be a set of ut ahabet of source X ad Y y, y,..., y be the set of outut ahabet of th destato Y. Let 2 x,,2,..., ad y,, 2,..., be the robabty dstrbuto fucto defed o X ad Y resectvey. Let A deote the trasto matrx of the th sub chae. As A s assumed o- sguar, t must be square. Sce umber of etters to be trasmtted,.e. A s, therefore the ut ad outut state coum vector for the th sub chae are Y ad Y, resectvey, ad are reated by Y A. Y It s aso true that Y X, where X deotes the robabty dstrbuto vector for the source X. For detca sub chaes, the source dstrbuto s reated to the outut state by X [ A ]. Y, here the th subscrt o A s gored because we are assumg that a sub chaes are detca. By Kere 966 f B deote A -, the X ca be exressed as 37

8 m X a v where λ ad v are resectvey, the th ege vaue ad corresodg ege vector of the matrx B. he coeffcets a are determed from the set of equatos Y a v Sce the egevaues of A are aways ess tha or equa to magtude, the egevaues of B, say ;,2,3,..., ad satsfy A,, 2,3,..., m Aso, s aways oe of the egevaues, ad a reeated ege vaue w occur f ad oy f the matrx A s reducbe. he egevectors of B are detca to those of A. hus, the egevectors corresodg to the o ut egevaues have comoets that sum to zero. he eemets of the ege vectors corresodg to the ege vaues are a ostve or zero sce they must corresod to robabtes. ow, the mutua formato betwee source X ad destato Y ca be gve as I X, Y y og y H. X, where y are the eemet of Y. If H s the coum vector wth th comoet H Y x y x og y x, where y x s the th eemet of the matrx A. Substtutg , we obta I X, Y y og y a H v Let u be the utty corresodg to the ut robabtes x. Here we are assumg that the utty for a the sub chaes w be same for the cascaded chaes. he the usefu mutua formato for detca cascaded chaes s defed as 38

9 I X, Y ; U H Y ; U a r, where r H. v ad H Y x ; U H s the trasose of the coum vector H ad u. y x og y x x, y. u heorem 7.4. he average usefu mutua formato rocessed by detca cascaded chae s covex fucto of the ut robabtes. Proof. Let b, b 2,..., b are o-egatve umbers such that a.. Let us defe ut dstrbuto P x b P x, 7.4. the we sha rove that the average usefu mutua formato corresodg to P x satsfes I X, Y ; U b I X, Y ; U, 7.4. where I X, Y ; U s the average usefu mutua formato corresodg to ut dstrbuto PK x. r Let I I X, Y ; U ak I K X, Y ; U,, the, I H Y U a r H Y U b a r ; ; 39

10 4 m m r a b y u y y u b r a y u y y u,,,,. og.. og. Sce y x s the sum of the chae robabtes,, 2,3,... y x, therefore, we have,. og ; b u y x y x H Y x U y x u or ; ; H Y x U b H Y x U It mes that. H b H ad hece r b r Equato together wth reduced to ;. ; r I H Y U a H Y U It mes that

11 I u. y og y b. u. y og y u. y u. y ow sce P b P therefore ca be wrtte as = b u y og y b u y og y y. u b y u y og y y u y og y I. m y u y By the geerazed Shao s equaty, we have u y og y u y og y wth equaty oy f y y for each, t mes u. y og y u. y og y Hece together wth gves I, sce y u s aways ostve.hus the roof of heorem7.4. cometes., 4

12 7.5 Chae Caacty of Cascaded Chaes Chae caacty, as deveoed by Shao948 s a ey cocet formato theory. It s a mortat measure of chae-erformace but t etas a free choce of ut dstrbuto. I ractce, however, severa ureated factors for exame, tme, eergy ad moey costrats, may restrct ths choce ad chae erformace has the to be cosdered uder these restrctos. A geera method for determg the chae caacty of cascade detca dscrete memory ess o-sguar chaes was studed by Smo97. We dscuss the same techque ad exted ts acatos the case whe source ahabet has uttes addto to robabtes. From 4.7.8, we have I X, Y ; U H Y ; U a r 7.5. So the chae caacty of detca cascaded chaes s gve by C max I X, Y [ Y ] heorem 7.5. Let A be a stochastc matrx of th sub-chae, whch s o-sguar ad square. Aso q be the eemet of A - for, 2,...,, the chae caacty of detca cascaded chaes wth uttes s gve by rovded a C U H Y ; U q r x u. y costat. Proof. As we ow usefu mutua formato for detca cascaded s gve by I X, Y ; U H Y ; U a r

13 he equato may be regarded as beg defed for a o-egatve rea vaues of robabty x aog wth corresodg ostve uttes u. We have to maxmze subect to the codto x. For ths we assume that the souto does ot vove ay x ad ay the method of Lagrage s muter. I X, Y ; U x Dfferetatg wth resect to x ad equate to zero. I X, Y ; U x As we ow H Y ; U u y og y m u y It mes H Y ; U u y og y, where u y. Partay dfferetatg both sdes of wth resect to x, we have H Y ; ; U H Y y U x y x u og y. y x he equato together wth reduces to a.. r u [ og y ]. y x. x, 43

14 It mes that a.. r u y [ og y ]. y x , Sce, y x, therefore ca be wrtte as [ u { og y } ] y x ar x Sce A s a o-sguar matrx, therefore, ts verse exsts. Muty both sdes by We have A A u y [ og y ] q a r 7.5., Mutyg both sdes of 7.5. by y ad summg over a H Y ; U q r 7.5. x By the revous theorem s the sum of a covex ad ear fucto ad s therefore covex o the set of o-egatve umbers whose sum s uty. It mes that for gve we ca foud a absoute maxmum of the fucto over the doma x ad formato rocessed. x. hus the souto yed a absoute maxmum for the If we muty both sdes of by x ad summg over, we have u y. y u y. y og y a r, 44

15 or H Y ; U a r. or H Y ; U a r. Hece C U max I X, Y ; U From 7.5. ad together, we get a C U H Y ; U q r, x whch s the chae caacty of detca cascaded chaes. Hece the roof of heorem 7.5. cometes Cocuso he characterstc dfferece betwee the robem of commucato through chaes cascade ad that of commucato through a sge chae s that, the atter case, the trasmtter ossesses the comete owedge of what t shoud trasmt. I ths chater we have studed the dfferet tye of cascaded chaes. We have aso defed the chae caacty of detca cascaded chae ad roved two theorems o t. he geera robem of determg the otma orderg of arbtrary chose bary chaes remas oe. However, we have troduced a cass of formato theoretc robem that deas wth chae erformace cascade ad estmates the chae caacty wth utty. 45

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