SUCCESSIVE INTERFERENCE CANCELLATION DECODING FOR THE K -USER CYCLIC INTERFERENCE CHANNEL

Transcription

1 Joural of Theoretical ad Applied Iformatio Techology 31 st December 212 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwatitorg E-ISSN: SCCESSIVE INTERFERENCE CANCELLATION DECODING FOR THE -SER CYCLIC INTERFERENCE CHANNEL 12 YONG LI 14 BO ZHANG 3 IAOIA SONG BAOMING BAI 1 1 State ey Lab of ISN idia iversity i a 7171 Shaaxi Chia 2 School of Educatio Sciece & Techology Shaxi Datog iversity Datog 379 Shaxi Chia 3 School of Physical & Electroics Shaxi Datog iversity Datog 379 Shaxi Chia 4 No No 9641 troop Baoi 7217 Shaaxi Chia ABSTRACT The -user cyclic iterferece chael ca model oe sceario i vehicular etwors where all the base statios are arraged alog a highway The th user is iterfered oly by the (( 1) mod ) th user i the -user cyclic iterferece chael The Ha-obayashi scheme based rate regio the best ier boud ow to date for the two-user iterferece chael is exteded to the -user cyclic iterferece chael which is obtaied by simultaeously decodig the iteded messages ad a part of the iterfered messages at the receiver I this paper we show that the simple successive iterferece cacellatio decodig ca also achieve this rate regio eywords: Iterferece Chael Cyclic Iterferece Chael Achievable Rate Regio Successive Iterferece Cacellatio Decodig Multi-level Ecodig 1 INTRODCTION The iterferece chael first itroduced by Shao i 1961 [1] models a commuicatio sceario where all user pairs share the same physical medium ad the commuicatio of each user pair will be iterfered with that of aother user pairs A user pair is usually amed a user for short The iterferece chael is useful for determiig the performace limits of may practical systems such as the wireless etwors However the capacity regio of the two-user iterferece chael is still uow except for the followig several cases For the two-user discrete memoryless iterferece chael its capacity regio is ow if the itesity of the iterferece is strog [234] if the chael outputs are some determiistic fuctios of the chael iputs [5] ad if the chael satisfies a class of degraded form [67] For the two-user Gaussia iterferece chael the capacity regios of very strog ad strog iterferece chaels [489] ad the same capacities of oisy iterferece [11112] mixed iterferece [1113] ad oe-sided [14] iterferece chaels are ow So far the best achievable rate regio for the two-user iterferece chael is provided by Ha ad obayashi (H) [8] usig superpositio codig i couctio with simultaeous decodig A simplified descriptio of the H rate regio amed the CMG rate regio is give i [15] via multi-level ecodig ad simultaeous decodig I this paper we refer to these ier bouds as the H scheme based ier bouds Recetly a special iterferece chael amed the cyclic iterferece chael was itroduced to study the codig problem of a special sceario may appear i vehicular etwors where all the base statios are arraged alog a highway or high-speed rail The cyclic iterferece chael is motivated by the modified Wyer model that is used to depict the soft hadoff sceario of a cellular etwor [16] I the -user cyclic iterferece chael depicted i Figure 1 the commuicatio of the th user pair is iterfered oly by that of the (( 1) mod )th user pair I [17] the H scheme based ier bouds for the two-user iterferece chael are exteded to the -user cyclic iterferece chael ad its all operatig poits are poited out to be achieved by successive iterferece cacellatio decodig via taig a geometric viewpoit ad usig rate-splittig argumets Ispired by this we also focus o the -user 961

2 Joural of Theoretical ad Applied Iformatio Techology 31 st December 212 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwatitorg E-ISSN: discrete memoryless cyclic iterferece chael i this paper ad show that the H scheme based ier boud of the -user cyclic iterferece chael ca be achieved by successive iterferece cacellatio decodig via a rigorous ad complete achievability proof usig radom codig argumets The ey idea of the proof is motivated by the equivalece of simultaeous decodig ad successive iterferece cacellatio decodig i achievig the capacity regio of the multiple access chael (MAC) Throughout the paper we use x ad to deote a radom variable its realizatio ad rage respectively Moreover a sequece is described = = as ( 1 2 ) 1 ( ) ad = ( ) CHANNEL MODEL The -user discrete memoryless cyclic iterferece chael is show i Figure 1 which cosists of iput ad output alphabets Y = ad a collectio of coditioal probability mass fuctios p ( y y x x ) o L Y Y Specially the margial distributios of the chael outputs satisfy p ( y x x 1 ) = p ( y x x ( 1) ) mod It is assumed that trasmitter ( = ) wats to sed a idepedet message W uiformly distributed over (( 2 R ) ) 2 R 1 W to its iteded receiver A code for the -user discrete memoryless cyclic iterferece chael cosists of R message sets W = { 122 } ecodig fuctios f : W performed by the correspodig ecoder ad decodig fuctios g : Y W performed by the correspodig decoder The average probability of error over the codeboo is defied P = Pr W ˆ Wˆ W W A rate as e {( ) ( )} R R 1 is said to be achievable for the -user discrete memoryless cyclic iterferece chael if there exists a sequece of tuple (( 2 R ) ) 2 R 1 codes with P e as The capacity regio of the -user discrete memoryless cyclic iterferece chael is the closure of the set of achievable rate tuples ( R R ) W W 1 W 1 Ecoder Ecoder 1 Ecoder -1 1 Y Y1 Y Decoder Decoder 1 W 2 2 Y 2 Ecoder -2 Decoder -2 Decoder -1 Wˆ Wˆ 1 ˆ 2 W ˆ W Figure 1: The -ser Cyclic Iterferece Chael 3 MAIN RESLT I this sectio we will show that the H scheme based ier boud of the -user discrete memoryless cyclic iterferece chael ca be achieved by multi-level ecodig i couctio with successive iterferece cacellatio decodig We rewrite the achievable rate regio of the - user discrete memoryless cyclic iterferece chael i [17] as follows Give a -user discrete memoryless cyclic iterferece chael the radom variables tae values i Q respectively Ad let 1 Y Y 1 tae values i chael iput alphabets ad chael output alphabets Y Y respectively Let P be the set of all probability distributios o ( Q 1 1) that factor as ( q u x ) ( q) ( u q) ( x u q) = p p p p = Give a fixed value p P cosider ( q u 1 1 ) ( 1 1 ) ( 1 1 x y q u x = y x ) p p p ( Q Y ) o ( ) ( ) ( ) rate tuples ( p c (( 1) mod ) c ) R ad let R p be the set of R R R that satisfy R I ( (1) p ( 1) mod R R I ( (2) c p ( 1) mod ( ; ) R I Y Q p (( 1) mod ) c ( 1) mod (3) R R R I ( c p (( 1) mod ) c ( 1) mod (4) 962

3 Joural of Theoretical ad Applied Iformatio Techology 31 st December 212 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwatitorg E-ISSN: Let R ( p) deote the rate tuples R R such that R = R R where ( ) p c R R R(( 1) mod ) R for each = 1 Lemma 1: The set p c c p = R R p P ( p) is a achievable rate regio for the -user discrete memoryless cyclic iterferece chael Proof: We use the coded time sharig techique multi-level ecodig ad successive iterferece cacellatio decodig The formal proof usig radom codig argumets is give below: a) Radom codeboo costructio p q p u x q p u x q Fix ( ) ( ( 1) mod ( 1) mod ) Radomly geerate a sequece accordig to p ( q ) For i= 1 Q i q draw ad = ( 1) mod radomly ad q coditioally idepedetly geerate R 2 R sequeces u ( w ) w 1 : 2 each accordig to i= 1 p ( u q ) For each u ( w ) Q radomly ad coditioally idepedetly geerate R 2 p i R sequeces x ( w w p ) w 1 : 2 p p each i= 1 Q accordig to ( i i ) i p x u w q b) Ecodig To sed w =( w w p ) trasmitter = ( 1) mod seds the correspodig codeword x ( w w ) p c) Decodig At each receiver a virtual three-user MAC is formed because there are three messages icludig its ow commo messages w private messages w ad the rival s commo p messages w (( 1) mod ) c to be decoded It is performed i three steps via successive iterferece cacellatio decodig So we have six decodig orders to decode these three messages Four decodig orders are left because we ca get rid of two meaigless decodig orders where the rival s commo messages are decoded at last at the receiver Also due to the use of the multi-level ecodig techique we have to decode the commo message of each user before decodig its private message So we oly have two decodig c orders ie the rival s commo messages the user s commo messages ad its ow private messages ( w(( 1) mod ) w w ) ad the user s commo c c p messages the rival s commo messages ad the user s private messages ( wc w(( 1) mod ) c w p ) Each decodig order determies a rate regio amed sub-regio The covex hull of the uio of these two sub-regios costitutes the rate regio of the virtual three-user MAC at each receiver Ad the rate regio of the -user discrete memoryless cyclic iterferece chael is the itersectio of the rate regios of these virtual three-user MACs Without loss of geerality we oly give the proof of oe decodig order w(( 1) mod ) w w The c c p proof of the other decodig order w w w is similar c (( 1) mod ) c p ŵ The decoder declares that (( 1) mod ) c is set if it is the uique message such that ( q u ( wˆ ) y ) T ò c otherwise it declares a error If such ŵ (( 1) mod ) c is foud the decoder fids the uique message w ˆ c such that ( u ( wˆ ) u ˆ ( w ) y ) c c q T ò otherwise it declares a error If such ŵ (( 1) mod ) c ad wˆ c are foud the decoder fids the uique message w ˆ p such that ( ˆ u ˆ ( 1) mod ( w(( 1) mod ) ) x ( w w ) y ) c c p q T ò otherwise it declares a error d) Aalysis of the probability of error We boud the probability of error averaged over codeboos ad messages By symmetry of the radom code geeratio the probability of error does ot deped o which codeword was set So without loss of geerality we assume that the message 11 W were set pairs W = ad ( 1) mod = 11 The the coditioal probability of a evet give 11 W were set is that W = ad defied as P ( 1) mod = 11 ( ( 1) mod ) = P = ( 11 ) = ( 11) ε ε W W A decodig error occurs oly if {( Q (11) (1) Y ) } ò T ε 1 963

4 Joural of Theoretical ad Applied Iformatio Techology 31 st December 212 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwatitorg E-ISSN: ( c ) Tò Q ˆ ( w ) Y 2 (( 1) mod ) c ( c ) Tò Q ( wˆ ) (1) Y 3 c Q (1 wˆ ) (1) Y 4 p ( p ) Tò The by the uio boud of evets we P ε P ε P ε P ε P ε By the have ( 1) ( 2 ) ( 3 ) ( 4 ) law of large umbers (LLN) ( ε1) By the pacig lemma [18] P ( ε 2 ) as ad (( 1) mod ) c < ( ( 1) mod ; δ P ( ε3 ) if Rc I( ( 1) mod Q ) δ ( ò ) ad ad ( ε 4 ) ( ; δ P as R I Y ò hold p ( 1) mod P as if R I Y ò Hece if the followig iequalities are satisfied the total average probability of decodig error P ( ε ) as Actually that describes a rate sub-regio whe Q is fixed: R I ( (5) p ( 1) mod R (6) c ( 1) mod R(( 1) mod ) c ( 1) mod (7) R R I ( (8) c p ( 1) mod Rc R(( 1) mod ) c ( 1) mod (9) R R I ( p (( 1) mod ) c ( 1) mod I ( ( 1) mod (1) R R R I ( c p (( 1) mod ) c ( 1) mod c (( 1) mod ) c p (11) Similarly accordig to the other decodig order w w w we have the other rate subregio for a giveq : R I ( (12) p ( 1) mod R (13) c R(( 1) mod ) c ( 1) mod (14) R R c p I ( ( 1) mod (15) Rc R(( 1) mod ) c ( 1) mod (16) R R p (( 1) mod ) c ( 1) mod (17) R R R I ( c p (( 1) mod ) c ( 1) mod (18) e) Rate regio Accordig to the covex hull of the uio of these two sub-regios we obtai a rate regio of a virtual three-user MAC at receiver The rate regio of the -user discrete memoryless cyclic iterferece chael is the itersectio of these rate regios which ca be described as follows R I ( (19) p ( 1) mod R (2) c ( 1) mod R(( 1) mod ) c ( 1) mod (21) R R I ( (22) c p ( 1) mod Rc R(( 1) mod ) c ( 1) mod (23) R ; p R(( 1) mod ) c ( 1) mod Y Q ) (24) R R R I ( c p (( 1) mod ) c ( 1) mod where = (25) Accordig to [15] we get rid of those redudat iequalities ad the have the simplified iequalities as below R R I ( (26) p ( 1) mod R R I ( (27) c p ( 1) mod ( ) R I Q p (( 1) mod ) c ( 1) mod (28) R R R I ( c p (( 1) mod ) c ( 1) mod (29) With this we complete the proof of the achievability of the rate regio for the -user discrete memoryless cyclic iterferece chael via multi-level ecodig i couctio with successive iterferece cacellatio decodig 4 CONCLSION The cyclic iterferece chael models a special sceario that may appear i vehicular etwors where all the base statios are arraged alog a highway or high-speed rail To determie the performace limit of this commuicatio system the -user cyclic iterferece chael is ofte cosidered by the iformatio theory researchers It 964

5 Joural of Theoretical ad Applied Iformatio Techology 31 st December 212 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwatitorg E-ISSN: is o woder that the Ha-obayashi scheme based ier boud is exteded to the -user cyclic iterferece chael ad this Ha-obayashi scheme based ier boud is obtaied by superpositio codig or multi-level ecodig i couctio with simultaeous decodig I this paper we give a rigorous achievability proof usig radom codig argumets to show that successive iterferece cacellatio decodig ca also achieve the Ha-obayashi scheme based ier boud of the -user cyclic iterferece chael which is helpful for us to uderstad ad determie the Ha- obayashi scheme based ier boud ACNOWLEDGEMENTS This wor was supported i part by the NSFC of Chia uder grat the 973 Program of Chia uder grat 212CB31613 ad the PCSIRT of Chia uder grat IRT852 REFERENCES: [1] CE Shao Two-way commuicatio chaels Proceedigs of the 4th Bereley Symp Mathematical ad Statistical Probability Bereley CA 1961 pp [2] H Sato O the capacity regio of a discrete two-user chael for strog iterferece IEEE Tras If Theory Vol 24 No 3 May 1978 pp [3] M H M Costa ad A A E Gamal The capacity regio of the discrete memoryless iterferece chael with strog iterferece IEEE Tras If Theory Vol 33 No 5 Sept 1987 pp [4] A B Carleial A case where iterferece does ot reduce capacity IEEE Tras If Theory Vol 21 No 5 Sept 1975 pp [5] A A E Gamal ad M H M Costa The capacity regio of a class of determiistic iterferece chaels IEEE Tras If Theory Vol 28 No 2 Mar 1982 pp [6] N Liu ad S luus The capacity regio of a class of discrete degraded iterferece chaels IEEE Tras If Theory Vol 54 No 9 Sept 28 pp [7] R Bezel The capacity regio of a class of discrete additive degraded iterferece chaels IEEE Tras If Theory Vol 25 No 2 Mar 1979 pp [8] T Ha ad obayashi A ew achievable rate regio for the iterferece chael IEEE Tras If Theory Vol 27 No 1 Ja 1981 pp 49-6 [9] H Sato The capacity of the gaussia iterferece chael uder strog iterferece IEEE Tras If Theory Vol 27 No 6 Nov 1981 pp [1] Shag G ramer ad B Che A ew outer boud ad the oisy-iterferece sumrate capacity for Gaussia iterferece chaels IEEE Tras If Theory Vol 55 No 2 Feb 29 pp [11] A S Motahari ad A hadai Capacity bouds for the Gaussia iterferece chael IEEE Tras If Theory Vol 55 No 2 Feb 29 pp [12] V S Aapureddy ad V V Veeravalli Gaussia iterferece etwors: sum capacity i the low-iterferece regime ad ew outer bouds o the capacity regio IEEE Tras If Theory Vol 55 No 7 July 29 pp [13] D Tuietti ad Y Weg O gaussia mixed iterferece chaels Proceedigs of the IEEE Iteratioal Symposium Iformatio Theory Toroto Caada July 28 [14] I Saso O achievable rate regios for the gaussia iterferece chael IEEE Tras If Theory Vol 5 No 6 Jue 24 pp [15] HFChog MMotai HGarg ad H El Gamal O the Ha obayashi regio for the iterferece chael IEEE Tras If Theory Vol54 No7 28 pp [16] OSomeh BMZaidel ad SShamai Sum rate characterizatio of oit multiple cell-site processig IEEE Tras If Theory Vol 53 No 12 Dec 27 pp [17] ESasoglu Successive cacellatio for cyclic iterferece chaels Proceedigs of Iformatio Theory Worshop Porto May 28 pp 36-4 [18] A A El Gamal ad YH imlecture Notes o Networ Iformatio Theory