Noisy Channel-Output Feedback Capacity of the Linear Deterministic Interference Channel

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1 Nosy Channel-Output Feedback Capacty of the Lnear Determnstc Interference Channel Vctor Quntero, Samr M. Perlaza, Jean-Mare Gorce arxv:.4649v6 [cs.it] Jan 6 Abstract In ths paper, the capacty regon of the two-user lnear determnstc (LD) nterference channel wth nosy output feedback (IC-NOF) s fully characterzed. Ths result allows the dentfcaton of several asymmetrc scenaros n whch mplementng channel-output feedback n only one of the transmtterrecever pars s as benefcal as mplementng t n both lnks, n terms of achevable ndvdual rate and sum-rate mprovements w.r.t. the case wthout feedback. In other scenaros, the use of channel-output feedback n any of the transmtter-recever pars benefts only one of the two pars n terms of achevable ndvdual rate mprovements or smply, t turns out to be useless,.e., the capacty regons wth and wthout feedback turn out to be dentcal even n the full absence of nose n the feedback lnks. Index Terms Capacty, Lnear Determnstc Interference Channel, Nosy Channel-Output Feedback. I. INTRODUCTION Perfect channel-output feedback (POF) has been shown to dramatcally enlarge the capacty regon of the two-user nterference channel (IC) [], [], [], [4], []. More recently, the same observaton has been made wth a larger number of transmtter-recever pars n the IC [6]. In general, when a transmtter observes the channel-output at ts ntended recever, t obtans a nosy verson of the sum of ts own transmtted sgnal and the nterferng sgnals from other transmtters. Ths mples that, subject to a fnte feedback delay, transmtters know at least partally the nformaton transmtted by other transmtters n the network. Ths nduces an mplct cooperaton between transmtters that allows them to use nterference as sde-nformaton [], [], [], [8], [9]. A more explct cooperaton s also observed n the case n whch one of the transmtter-recever pars acts as a relay for the other transmtter-recever par by provdng an alternatve path: transmtter recever j transmtter j recever []. These types of cooperaton, even when t s not explctly desred by both transmtter-recever pars, play a fundamental role n enlargng the capacty regon. Interestngly, ths holds also n the case of fully decentralzed networks n whch each transmtter-recever par seeks exclusvely to ncrease ts The authors are wth the CITI Laboratory of the Insttut Natonal de Recherche en Informatque et en Automatque (INRIA), Unversté de Lyon and Insttut Natonal de Scences Aplquées (INSA) de Lyon. 6 Av. des Arts 696 Vlleurbanne, France. ({Vctor.Quntero-Florez, Samr. Perlaza, Jean- Mare.Gorce}@nra.fr). Vctor Quntero s also wth Unversdad del Cauca, Popayán, Colomba. Samr M. Perlaza s also wth the Department of Electrcal Engneerng at Prnceton Unversty, Prnceton, NJ. Ths research was supported n part by the European Commsson under Mare Sklodowska-Cure Indvdual Fellowshp No. 696 (CYBERNETS); the Doctoral School n Electroncs, Electrcal Engneerng and Automaton (EEA) of Unversté de Lyon; Unversdad del Cauca, Popayán, Colomba; and the Admnstratve Department of Scence, Technology and Innovaton of Colomba (Colcencas), fellowshp No. 6-. ndvdual rate. That s, channel-output feedback ncreases both the capacty regon and the Nash equlbrum (NE) regon []. Despte the vast exstng lterature, the benefts of feedback are unfortunately less well understood when the channeloutput feedback lnks are mpared by addtve nose. The capacty regon of the LD-IC wth nosy channel-output feedback (NOF) s known only n the two-user symmetrc case, see []. The converse regon n [] nherts exstng outer bounds from the case of POF, the cut-set outer bounds and ncludes two new outer bounds. The outer-bounds nherted from the POF are those of the ndvdual rates and the sumrate n []. The new outer-bounds are of the form R + R and R + R j. The achevablty regon n [] s obtaned usng a partcularzaton of the achevablty scheme presented n [], whch holds for a more general model,.e., nterference channel wth generalzed feedback. In ths paper, the results presented n [] are generalzed for the asymmetrc case and the correspondng capacty regon of the two-user LD-IC-NOF s fully characterzed. Ths generalzaton s acheved by usng the same tools as n [], however, t s far from trval due to the number of parameters that descrbe ths channel model: two forward sgnal to nose ratos (SNRs) n, n, two feedback SNRs n, n and two forward nterference to nose ratos (INRs) n, n. The new converse regon also nherts exstng outer bounds from the case of POF, the cut-set outer bounds and ncludes two new outer bounds on R + R and R + R j. These new bounds generalze those presented n []. The achevablty regon s obtaned by usng a codng scheme that combnes a three-part message splttng, superposton codng and backward decodng. Despte the fact that ths codng scheme s bult usng the exact number of requred message-splttng parts for the IC-NOF, t can stll be consdered as a specal case of the general scheme presented n []. Fnally, ths paper s concluded by a dscusson n whch numercal examples are presented to hghlght the benefts of NOF. At the same tme, examples n whch NOF s absolutely useless n terms of capacty regon mprovement are also presented. Due to space constrants, the achevablty and the converse parts of the proof are presented n []. II. LINEAR DETERMINISTIC INTERFERENCE CHANNEL WITH NOISY-CHANNEL OUTPUT FEEDBACK Consder the two-user LD-IC-NOF, wth parameters n, n, n, n, n and n descrbed n Fg.. n, {, }, s a non-negatve nteger used to represent the sgnal to nose rato (SNR) at recever ; n j, {, } and j {, }\ {}, s a non-negatve nteger used to represent the nterference

2 n n! n! n X, X, X, X,4 X, X, X, X, X,4 TX TX RX RX X, X, X, L X, X,4 L X, X, L X, X, X, L X, X, L X, X, L X,4 X,4 L X, Sgnal Interference Feedback Fg.. Two-user lnear determnstc nterference channel wth nosy channeloutput feedback (LD-IC-NOF). n n calculated as follows p = M M {ˆb,l b,l}. () l= A rate par (R, R ) R + s sad to be achevable f t satsfes the followng defnton. Defnton (Achevable Rate Pars): The rate par (R, R ) R + s achevable f there exsts at least one par of codebooks X and X wth codewords of length N, wth the correspondng encodng functons f (),..., f (N) and f (),..., f (N) such that the average bt error probablty can be made arbtrarly small by lettng the block length N grows to nfnty. The followng secton determnes the set of all the rate pars (R, R ) that are achevable n the LD-IC-NOF wth parameters n, n, n, n, n and n. to nose rato (INR) at recever from transmtter j; and n, {, }, s a non-negatve nteger used to represent the sgnal to nose rato (SNR) at transmttter n the feedback lnk from recever. At transmtter, wth {, }, the channel-nput X (n) at channel use n, wth n {,..., N}, ä T, s a q-dmensonal bnary vector X (n) = Ä X (n),,..., X(n),q wth q = max ( n, n, n, n ) and N the block-length. At recever, the channel-output Y (n) a q-dmensonal bnary vector Y (n) at channel use n s also = Ä (n) Y,,..., ä T. Y (n),q The nput-output relaton durng channel use n s gven as follows (n) Y =S q n X (n) + S q nj X (n) j, () and the feedback sgnal avalable at transmtter at the end of channel use n s: (n) Y =S (q n ) + Y (n d), () where d s a fnte feedback delay, addtons and multplcatons are defned over the bnary feld, and S s a q q lower shft matrx. Ä n and n j correspond to äù SNR and log (INR j ) res- The parameters ö n, log Ä äù ö SNR, log pectvely, where SNR, SNR and INR j are parameters of the Gaussan nterference channel (G-IC). Transmtter sends M nformaton bts b,,..., b,m by sendng the codeword Ä ä X (),..., X (N). The encoder of transmtter can be modeled as a set of determnstc mappngs f (),..., f (N), wth f () : {, } M {, } q and n {,..., N}, f (n) : {, } M {, } q(n ) {, } q, such that ( ) b,,..., b,m and () X () =f () X (n) =f (n) ( b,,..., b,m, Y (),..., Y (n ) ). (4) At the end of the block, recever uses the sequence Y (),..., Y (N) to generate the estmates ˆb,,..., ˆb,M. The average bt error probablty at recever, denoted by p, s III. MAIN RESULTS Denote by C( n, n, n, n, n, n ) the capacty regon of the LD-IC-NOF wth parameters n, n, n, n, n and n. Theorem (on top of next page) fully characterzes the capacty regon C( n, n, n, n, n, n ). A. Proofs Theorem fully characterzes the capacty regon of the LD-IC-NOF. That s, the converse and achevable regons are dentcal. In the converse regon, the nequaltes (6) and (8) are nherted from the converse regon of the LD-IC-POF n []. The nequalty n () s a smple cut-set bound whose proof s presented n []. The nequaltes (9) and () are new and generalze those presented n []. The correspondng proofs are presented n []. The achevablty regon s obtaned usng a codng scheme that combnes a three-part message splttng, superposton codng and backward decodng, as frst suggested n [], [], []. Ths codng scheme s fully descrbed n [] and t s specally desgned for the IC-NOF. However, t can also be obtaned as a specal case of the more general scheme,.e., nterference channel wth generalzed feedback, presented n []. The relevance of ths new achevablty scheme s that t plays a key role n the achevablty of the NE regon, subject to the ncluson of random messages, as suggested n [], []. Nonetheless, the analyss of the achevablty of the NE regon [4] s out of the scope of ths paper. B. Connectons to Exstng Results In the case n whch channel-output feedback s not avalable,.e., n = n =, n = n, for all {, }, Theorem reduces to the capacty regon of the LD-IC wthout feedback,.e., Lemma 4 n []. Conversely, when perfect channel-output feedback lnks are avalable.e., n = max ( n, n ) and n = max ( n, n ), Theorem reduces to the capacty of the LD-IC-POF, that s, Corollary n [] n the correspondng notaton.

3 Theorem : The capacty regon C( n, n, n, n, n, n ) of the two-user LD-IC-NOF s the set of non-negatve rate pars (R, R ) that satsfy {, } and j {, } \ {}: R mn (max ( n, n j ), max ( n, n j )), (6) R mn Ä max ( n, n j ), max Ä n, n jj ( n jj n j ) +ää, () R +R mn Ä max ( n, n ) + ( n n ) +, max ( n, n ) + ( n n ) +ä, (8) R +R max Ä ( ä Ä ä n n ) +, n + max ( n n ) +, n (9) ( Ä + mn ( n, max ( n, n )) ( n n ) +ä + (n n ) + mn ( n, n ) + mn Ä ( ä ) + n n ) +, n Ä +( mn ( n, max ( n, n )) ( n n ) +ä + (n n ) + mn ( n, n ) + mn Ä ( ä ) +, n n ) +, n R +R j max ( n jj, n j ) + max ( n, n j ) + ( n n j ) + mn Ä ( ä n jj n j ) +, n j () Ä +( mn ( n jj, max ( n jj, n j )) ( n jj n j ) +ä +! (nj n jj ) + mn ( n jj, n j ) + mn Ä ( ä ) +, n n jj n j ) + =,! n =,n =,n =, n, = n j, n = 4 () In the symmetrc case n whch channel-output feedback lnks are mpared by nose,.e., n = n = n, n = n = m and n = n = l, Theorem reduces to the capacty of the symmetrc LD-IC-NOF,.e., Theorem n []. Note also that n the case n whch the nterference channel s symmetrc, Theorem partally generalzes Theorem 4. n [9]. For nstance, note that when there exsts only one perfect channel-output feedback,.e., n = n = n, n = n = m, n = max ( n, n ), and n =, Theorem reduces to Theorem 4. (model ) n [9]. In the two-user IC-NOF, a transmtter sees a nosy verson of the sum of ts own transmtted sgnal and the nterferng sgnal from the other transmtter. Hence, subject to a fnte delay, one transmtter knows at least partally the nformaton transmtted by the other transmtter n the network. Ths observaton hghlghts the connectons between the IC wth feedback and the IC wth source cooperaton studed n []. C. Dscusson Ths secton provdes a set of numercal examples n whch partcular scenaros are hghlghted to show that channeloutput feedback can be strongly benefcal for enlargng the capacty regon of the two-user LD-IC even when channeloutput feedback lnks are mpared by nose. At the same tme, t also hghlghts other examples n whch channel-output feedback does not brng any beneft n terms of the capacty regon. These benefts are gven n terms of the followng metrcs: (a) ndvdual rate mprovements and ; and (b) sum-rate mprovement Σ. In order to formally defne, and Σ, consder an LD-IC-NOF wth parameters n, n, n, n, n and n. The maxmum mprovement ( n, n, n, n, n, n ) of the ndvdual rate R due to the effect of channel-output feedback wth respect to R = R +R = R +R = = R + R = 4 R + R = Fg.. Capacty regon C(,,,,, ) wthout feedback (thck red lne) and C(,,,,, 4) wth nosy channel-output feedback (thn blue lne) of the example n Sec. III-C. Note that (,,,,, 4) = bts/ch.use, (,,,,, 4) = bts/ch.use and Σ(,,,,, 4) = bts/ch.use. the case wthout feedback s: = ( n, n, n, n, n, n ) = () max R j> sup (R, R j ) C (R, R j) C R R, and the maxmum sum rate mprovement Σ( n, n, n, n, n, n ) wth respect to the case wthout feedback s: Σ( n, n,n, n, n, n ) = () sup (R, R ) C (R, R ) C R + R (R + R ), where C = C( n, n, n, n, n, n ) and C = C( n, n, n, n,, ) are the capacty regon wth nosy channel-output feedback and wthout feedback, respectvely. The followng descrbes partcular scenaros that hghlght some nterestng observatons. ) Example : Only one channel-output feedback lnk allows smultaneous maxmum mprovement of both ndvdual

4 !! n = ;! n n ; n = ; n =. = ;! n = ; n = ; n =.! n = ;! n = ; n = ; n =.! n = ;! n = ; n = ; n =8.! n = ;! n = ; n = ; n =8.! n = ;! n = ; n = ; n = n n! n n 4 n =,! n =,n =,n =8, n =9, n =4. n 4! n = ;! n = ; n = ; n =8. 8 n n n 8 9 Fg.. Maxmum mprovement of ndvdual rates of the example n Sec. III-C = R +R = R +R = = R + R = X = R + R = 8 9 n! n n =,! n =,n =6,n 4 =, n =, n = Fg.. Maxmum mprovement of one ndvdual rate and the sum rate of the example n Sec. III-C R + R = R + R = R + R = Fg. 4. Capacty regon C(,,, 8,, ) wthout feedback (thck red lne) and C(,,, 8, 9, 4) wth nosy channel-output feedback (thn blue lne) of the example n Sec. III-C. Note that (,,, 8, 9, 4) = bt/ch.use, (,,, 8, 9, 4) = bt/ch.use and Σ(,,, 8, 9, 4) = bt/ch.use. R + R = 4 =. R + R = = rates: Consder the case n whch transmtter-recever pars and are n weak and moderate nterference regmes, wth n =, n =, n =, n =. In Fg. the capacty regon s plotted wthout channel-output feedback and wth nosy channel-output feedback ( n =, n = 4). In Fg., (,,,, n, n ) s plotted for both = and = as a functon of n and n. Theren, t s shown that: (a) Increasng parameter n beyond threshold n = allows smultaneous mprovement of both ndvdual rates ndependently of the value of n. Note that n the case of perfect channel-output feedback,.e., n = max ( n, n ), the maxmum mprovement of both ndvdual rates s smultaneously acheved even when n =. (b) Increasng parameter n beyond threshold n = provdes smultaneous mprovement of both ndvdual rates. However, the mprovement on the ndvdual rate R strongly depends on the value of n. (c) Fnally, the sum rate does not ncrease by usng channel-output feedback n ths case. ) Example : Only one channel-output feedback lnk allows maxmum mprovement of one ndvdual rate and the sum-rate: Consder the case n whch transmtter-recever pars and are n very weak and moderate nterference regmes, wth n =, n =, n =, n = 8. In Fg. 4 the capacty regon s plotted wthout channel-output feedback and wth nosy channel-output feedback( n = 9, n = 4). In Fg., (,,, 8, n, n ) s plotted for both = and = as a functon of n and n. Theren, t s shown that: (a) Increasng n beyond threshold n = 8 or ncreasng n beyond threshold n = allows smultaneous mprovement of both ndvdual rates. Fg. 6. Capacty regon C(,, 6,,, ) wthout feedback (thck red lne) and C(,, 6,,, ) wth nosy channel-output feedback (thn blue lne) of the example n Sec. III-C. Note that (,, 6,,, ) =. bts/ch.use, (,, 6,,, ) = bts/ch.use and Σ(,, 6,,, ) = bts/ch.use. Nonetheless, maxmum mprovement on R s acheved by ncreasng n. (b) Increasng ether n or n beyond thresholds n and n! n, allows maxmum mprovement of = ;! n = ; n = 6; n =.! the sum rate (see Fg. ). n = ;! n = ; n = 6; n =. 4 n 8 n Fg.. Maxmum mprovement of one ndvdual rate of the example n Sec. III-C ) Example : At least one channel-output feedback lnk does not have any effect over the capacty regon: Consder the case n whch transmtter-recever pars and are n the weak nterference regme, wth n =, n =, n = 6, n =. In Fg. 6 the capacty regon s plotted wthout channel-output feedback and wth nosy channeloutput feedback ( n =, n = ). In Fg., (,, 6,, n, n ) s plotted for both = and = as a functon of n and n. Theren, t s shown that: 6 4 n 8 n

5 ! n =,! n =8,n =,n =, n =, n =9 R = = R + R = IC wth and wthout channel-output feedback are dentcal,.e., nether n nor n enlarges the capacty regon. In ths specfc example, t s not possble to take advantage of the feedback lnks for usng nterference as sde nformaton or to generate an alternatve path. Fg. 8. Capacty regon C(, 8,,,, ) wthout feedback (thck red lne) and C(, 8,,,, 9) wth nosy channel-output feedback (thn blue lne) of the example n Sec. III-C4. Note that (, 8,,,, 9) = bts/ch.use, (, 8,,,, 9) = bts/ch.use and Σ(, 8,,,, 9) = bts/ch.use. (a) Increasng parameter n does not enlarge the capacty regon, ndependently of the value of n. (b) Increasng parameter n beyond threshold n = 8 allows smultaneous mprovement! of both ndvdual rates. (c) Fnally, none of the parameters n = ;!! n = ;! n = 8; n = ; n =. n or n = 8; n = ; n =. n ncreases the sum-rate n ths case. 6 n n = n 8 n Fg. 9. Maxmum mprovement of one ndvdual rate of the example n Sec. III-C4 4) Example 4: The channel-output feedback of lnk exclusvely mproves R j : Consder the case n whch transmtter-recever pars and are n the very strong and strong nterference regmes, wth n =, n = 8, n =, n =. In Fg. 8 the capacty regon s plotted wthout channel-output feedback and wth nosy channel-output feedback ( n =, n = 9). In Fg. 9, (, 8,,, n, n ) s plotted for both = and = as a functon of n and n. Theren, t s shown that: (a) Increasng parameter n beyond threshold n = 8 exclusvely mproves R. (b) Increasng parameter n beyond threshold n = exclusvely mproves R. (c) None of the parameters n or n has an mpact over the sum rate n ths case. Note that these observatons are n lne wth the nterpretaton of channel-output feedback as an altrustc technque, as n [], [4]. Ths s bascally because the lnk mplementng channel-output feedback provdes an alternatve path to the nformaton sent by the other lnk, as frst suggested n []. ) Example : None of the channel-output feedback lnks has any effect over the capacty regon: Consder the case n whch transmtter-recever pars and are n the very weak and strong nterference regmes, wth n =, n = 9, n =, n =. Note that the capacty regon of the LD- IV. CONCLUSIONS In ths paper, the nosy channel-output feedback capacty of the lnear determnstc nterference channel has been fully characterzed by generalzng exstng results. Based on specfc asymmetrc examples, t s hghlghted that even n the presence of nose, the benefts of channel-output feedback can be sgnfcantly relevant n terms of achevable ndvdual rate and sum-rate mprovements wth respect to the case wthout feedback. Nonetheless, there also exst scenaros n whch these benefts are totally nexstent. REFERENCES [] D. 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