A Deterministic Model for Channel Capacity with Utility
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- Hilary Phillips
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1 CAPTER 6 A Detestc Model fo Chel Cct wth tlt 6. todcto Chel cct s tl oeto ssocted wth elble cocto d defed s the hghest te t whch foto c be set ove the chel wth btl sll obblt of eo. Chel codg theoes d the coveses ove tht the cct C c be defed s fol tht deeds o the chels obbltsc desctos s gve below: C 6.. Sho 948 oved tht 6.. s the cct of eo less chels whee s the chel t s the chel ott d s the vege tl foto betwee d. The eqto 6.. eteded to the ltg eesso C l s 6..2 f t ests the chel cct of the cet chel wth eo s eql to the of vege tl foto whee t s the seqece of legth wth coesodg ott seqece. Dobsh 964 oved tht the chel cct gve b eqto 6..2 s fo foto stble chels whle fo foto stble chels the fol 6..2 does ot covese to lt. Eles of foto stble chels clde the stto egl decoosble chels the stto otct chels d veges eo less chels. Now the qesto ses whethe thee est colete geel fol fo chel cct whch does ot eqe ssto sch s eoless foto stblt sttot cslt etc.ved d 994 hs defed geel fol fo chel cct s follows: 3
2 C s deote s t ocess the whee 2 fo of seqece of fte- desol dstbto d 2... s the coesodg ott seqece of fte desol dstbtos d s the foto te o tl foto betwee W W P d. ee s dced b v the chel : A B whch s bt seqece of desol codtol ott dstbtos fo A to B whee A d B e the t d ott lhbets esectvel. Chel cct gve b 6.. ossesses the followg oetes: The chel cct s o egtve.e. C becse C. C log sce C log. C log sce C log. s cotos d cocve fcto of. ths chte we dscss the dscete eoless chels d the clssfcto secto 6.2. secto 6.3 we std dffeet tes of chel cct. secto 6.4 we dscss chel cct wth tlt. secto 6.5 we defe chel cct of dscete eoless chels wth tlt d ove two theoes o t. 4
3 6.2 Clssfcto of Dscete Meoless Chels A tssso chel c be secfed tes of the set of ts vlble t the t tel the set of ott vlble t ott tel d fo ech t the obblt ese o the ott evets codtol o tht t. ee we dscss the Dscete Meoless Chel D.M.C.. Ths s chel fo whch the t d ott e ech seqece of lettes fo fte lhbet d fo whch the ott lette t gve te deeds sttcll ol o the coesodg t lette. Cosde Dscete Meoless Chel DMC wth t lhbets ott lhbets d chel t[ A ] A f s do vble tg o the vles wth the obbltes esectvel the the ott lso becoes do vble. The ot dstbto of the t d dstbto of s gve b DMC c be clssfed dffeet ws s follows: 6.2. Lossless Chel f fo ll t dstbtos. othe wods lossless chel s chctezed b the fct tht the t s deteed b the ott d hece o tsto eos c occ. b Detestc Chel f o fo ll..e. f s detee b o eqvletl fo ll t dstbto A ele of detestc chel 5
4 s oe whose t s the dett of lg cd ced fo od 52-cd c d whose ott s the st of the cd. f cd s ced t do so tht ll vles of d hece of e eqll lel the the foto ocessed s log 4. c Setc Chel f ech ow of the chel t cots the se set of bes d ech col of cots the se set of bes q q2... q. Fo ele the t d eesets setc chels. The ows of chel t e detcl ecet fo etto d sll fo cols. t s edte coseqece of the defto of setc chel tht fo sch chel s deedet of the t dstbto d deeds ol o the chel obbltes. t be oted tht f the obbltes ssocted wth the ott lhbet e ece log Theefoe
5 log β β β β Fge 6.2.: B setc chel Fo t dstbto. The bove fge shows the ele of B Setc Chel. 6.3 Clssfcto of Chel Cct Chel cct gve b Sho eqto 6.. c be clssfed s follows: 6.3. ε Cct of Chel The ε cct of chel s the cct stsfg < ε < s the se of the tes the bts e chel sbol tht e stotcll chevble b seqece of chel bloc codes ech hvg bloc eo obblt ε. Fo the geel fte-lhbet chel odel Ved d 994 obted foto theoetc fol fo ε-cct whch s vld fo ll bt t ost cotble vles of ε. Defto 6.3. Let s Ved- chel whee s fte set of chel t lhbet s set of fte ott lhbet d s codtol obblt. The fo ech d ech ostve tege N. Let N be the lgest ostve tege N sch tht thee est bloc code fo the chel of sze N wth bloc eo obblt ε. The ε- cct of the chel s defed s follows: 7
6 C l t s log N 6.3. The ε cct fcto of the chel s the fcto defed o whch s ech ε ϵ the ε-cct. The cct fcto s o-decesg d theefoe t hs t ost cotble dscottes. The cl eslts bot ε-cct e gve below: b Fo cet chel t s ow tht the ε-cct cocdes wth chel cct fo eve < ε <. Sch chel cct s sd to osses stog cct. Fo ele dscete chels fte stte decoosble chels d dtve ose chels. Fo ddtve ose fte- lhbet chels wth stto oegodc do ose Pthsth 964 estblshed othe defto fo ε-cct whch s vld t eve cott ot of the ε-cct fcto. Actll Pthsth ε-cct c be obted fo Ved- 994 cct bt t s sle sce Pthsth s chels e secl cse of Ved Vble Rte Chel Cct the vble te chel codg the ecode dst ts te d/o othe esoces sch s owe d bdwdth to the ctl chel codto o stte. Whe the chel stte s ow t tstte the vege of the cctes chevble fo the dvdl chel sttes s ott fdetl lt. ee we dscss soe codg stteges b whch we obt vble te chel cct. Fed Bloc legth Chel Cct ths cse the be of tstted foto bts d the be of obseved chel sbols bloc legth e esecfed. Let s seqece of t lhbets d s seqece of ott lhbets d let f d g e ecode d decode esectvel sch tht 8
7 f g : {} : {} whose vege bloc eo obblt d te stsf l l f R esectvel. The we obt the oto of covetol Sho chel cct. b Vble -to Fed Chel Cct Whe the be of obseved chel sbols bloc legth s esecfed bt the be of elbl ecoveed foto bts deeds o chel codtos. Let. log 2 The fo ecode-decode c be defed s : f : : g : 2... wth essge. Let deote the chel esose to the t : f The be of cosectvel ecoveed bts L s defed s the lgest tege sch tht : g : : g L whee deote bts 2 g d s the do vble tht deeds o the ecode decode essge d chel elzto. A llstto of L s gve the followg fge
8 ENCODER CANNEL DECODER Fge 6.3.: Ele of L N the bove fg = = whle the ecoveed bt L =6. f R bts/chel s vble-to-fed chevble te d f thee ests seqece of code f : g : whose eected be of ecoveed bts stsfes R l f E L whee the eectto s ove the chel do tsfoto d f bts the vble-to-fed chevble te s the chel cct. c Fed to-vble Chel Cct f the be of tstted foto bts s esecfed the the be of chel obsevtos eqed ecoveg the deeds o chel codtos. Ths s the set of te less fot codes. Let decode d ecode e defed s follows: : g : d 2
9 : f : Fo defe the be N of chel sbols eqed to ecove the tstted bts s the sllest tege sch tht :... g f R bts/chel se s fed-to-vble chevble te d f thee s seqece : : of f g sch tht R l f EN E N = l s whee the eectto s ove the chel do tsfoto d f bts. The fed to-vble chevble te s the fed to-vble cct. 6.4 Chel Cct wth tlt cocto sste foto s tstted d ocessed vew of gol wth egd to whch essge st be effcet. The obectve of the sste s flflet of the gol d tht es thee est logcl bloc whch s to be ble to dscte the qlt of vos sgls ccodg the gve cteo. These cteos fo qlttve dffeetto of sgls e bsed o the elevce o the sgfcce o tlt of the foto whch s beg tstted b the. Let... be do vble. Let d = 2 be 2 tlt d obblt dstbtos esectvel whee > fo eve d. Bels d Gs968 todce the followg qtttve- qlttve ese of foto: = log 6.4. whee > =2 s clled tlt o otce ssocted to ll ossble evets of. The ese 6.4. hs lso bee clled sefl b Logo972. t c hve ve sll s well s ve lge vle deedg o. Ths the ge of the 2
10 ese soetes s geble whle stdg ts lcto. Ths ese does ot stsf ddtvt oet d does ot tt vle fo. To ovecoe these dffcltes Bh d ood 993 oosed d chctezed the followg two eses of sefl foto: d log log t be oted tht these eses hve vle whe fo ech d stsf ddtvt oet. d Coesodg to the codtol sefl foto ese c be defed s log / whee d e ot tlt d obblt dstbtos of d esectvel d s the codtol obblt dstbto. Let the we hve log O sl les we c defe etc. 22
11 23 Net we cosde log o log t les log log Sll we c ove tht Fo d togethe we hve o Let... 2 be set of t lhbet wth lettes d be set of ott lhbet wth lettes.let d d be the obblt dstbto fcto defed o d esectvel. Let be tlt coesodg wth fo ech. The vege sefl tl foto c be defed s
12 So the chel cct wth tltes o weghted chel cct of eoless chel s gve b C M A Dscete Meoless Chel d ts Chel Cct wth tlt A geel Method fo deteg the cct of dscete chel ws sggested b Mog 953. B lg ths ethod Te d Sh 995 coted the weghted cct of DMC de weghted efoce costts. ths secto we eset ew detestc odel fo weghted chel cct of dscete eoless chel tes of tl foto ese gve b A dscete eo less chel wth ts d otts sbols s descbed b stochstc t... : A :... A [ ] 2... d 2... whee d. Now we ove theoe fo vege tl foto of dscete eoless chels wth tlt. Theoe 6.5. The vege sefl tl foto ocessed b chel s cove fcto of the t obbltes. Poof: Let... 2 e o-egtve bes sch tht... defe t dstbto. Let s P P
13 25 the we shll ove tht the vege sefl tl foto coesodg to P stsfes whee s the vege sefl tl foto coesodg to t dstbto P. Let K the K ] [ we ow log Sce s st s of the chel obbltes thee foe we hve o K whee K log Eqto togethe wth edces to K. t les tht K log
14 26 log log o log log log log B the geelzed Sho s eqlt we hve h log log wth eqlt ol f fo ech t les log log Ths togethe wth gves. Sce s lws ostve.ece Theoe 6.5. s oved. Net we detee the chel cct wth tltes the followg theoe: Theoe Let stochstc t A whch descbes eo less chel be sqe d o sgl d q be the eleets th ow d th col of A - =2 the the chel cct wth tltes s gve b C q 6.5.7
15 27 ovde = costt = s. Poof: Mese c lso be wtte s We ze sbect to. Fo tht we sse tht the solto does ot volve d l Lgge s ethod of ltle. Let P N L Dffeettg L w.. t. d eqtg to zeo we hve. N L Sce theefoe 6.5. we ow log log K whee d K. Net dffeettg w..t. we hve ] log [ K 6.5.
16 Ths the eqto togethe wth 6.5. d 6.5. edces to [ log ] K KN. Sce theefoe c be wtte s { [ log ] KN} K we c wte t fo s follows: A t [ log ] NK 2 [ log : : [ log 2 ] NK ] NK K K : : K 2 Sce A s o- sgl t theefoe ts vese ests. Mltlg both sdes b t t A A we hve [ log ] NK K q Ag ltlg both sdes of b d sg ove we get N q Sce L s the s of cove d le fctos theefoe t s cove o the set of o-egtve bes. t les tht fo the gve N we c fd bsolte of the fcto L ove the do d efe to Ash 996.Ths the solto elds bsolte fo the foto ocessed. 28
17 29 f we ltl both sdes of b d sg ove we hve K KN log o N o N t les N M C fo d togethe we get q C whch s the eqed eslt. ece the oof of the Theoe s coleted. 6.6 Coclso ths chte we hve dscssed the chel cct d dffeet tes of chel cct. We hve lso dscssed the Dscete Meoless chel d ts clssfcto. We hve obted odel fo the chel cct of dscete eoless chel wth tlt d oved two theoes. theoe f we t oe o oe of eql to zeo we e essetll edcg the be of ossble otts. The edced chel t s o loge sqe so tht the eses of theoe do ot l. The geel oble of cotg chel cct s oble ecl lss best teted b cove ogg ethod.
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