Scaling Laws and Techniques in Decentralized Processing of Interfered Gaussian Channels

Scaling La ws and T echniques in Decentrali zed Processing of Interfered Gauss ian Channels Amichai Sanderovich, Mi chae l Pele g and Shlomo Shamai (Shitz) T echnion, Haifa, Israe l Email: { amichi@tx,michae l@ee,sshlomo@e e } .technion.ac .il Abstract The scaling laws of the achiev able communication rates and the corresponding upper bounds of distributed reception in the presence of an interfering signal are in vestigated . The scheme includes one transmitter communicating to a remote destination via two relays, which forward messages to the remote destination through reliable links with finite capacities. The relays recei ve the transmission along with some unkno wn interference. W e focus on three common settings for distributed reception, wherein the scaling laws of the capacity (the pre-log as the po wer of the transmitter and the i nterference are taken to infinity) are completely characterized. It i s shown in most cases that in order to overcome the i nterference, a definite amount of information about t he i nterference needs to be forwarded along with the desired message, to the destination. It is ex emplified in one scenario that the cut-set upper bound is strictly loose. T he results are deriv ed using the cut-set along with a new bounding technique, which relies on multi letter expressions. Furthermore, lattices are found to be a useful communication technique in this setting, and are used to characterize the scaling l aws of achie vable rates. I . I N T R O D U C T I O N In th is p aper , we treat the prob lem of decentralize d detection, with an in terfering signal. Decen tralized detectio n [1] is an in teresting and timely setting, w ith many applications such as th e emerging 4G networks [2], [3], smart-du st and r emote inf erence to n ame just a f ew . This setting consists of a tr ansmitter wh ich co mmunicates to a d istant d estination via inter mediate r elay/s which facilitate the commu nication, wh ere no direct link between the transmitter and destination is provided . The model includes reliable links with fixed capa cities be tween the relays and the destina tion. Such mo del is fur ther extend ed in [4] to incorporate a fading chan nel between the transmitter and the relays. The setting also includes an in terference, which is modeled as a Ga ussian white signal (no encoding is assumed). The interfering sign al is un known to either th e tr ansmitter , th e d estination o r th e relays. Such setting suits numerous real-world scenario s such as airpo rt tower commu nication wh ich need to h av e mo re than one reception p oint for increased security against jamm ing, hot-spo ts operating in a dense interferen ce e n viron ment and c ellular network with a strong in terference . This model is somewhat different than that treatin g the jamming problem as a minmax op timization, where the jammer is optim ized to b lock the communicatio n of the transmitter, which in turn is o ptimized to maximize the reliably conv eyed rate [ 5], [6]. Some recent pape rs that de al with similar yet different settings are [7],[8] and [9] September 7, 2021 DRAFT 1 and also [10]. T he con cept of generalized degr ees of freedom , used in [11] is intimately related to the scaling laws defined in th is p aper . Th e on ly r emedy o ffered in this paper is the exploitation of the spatial co rrelation, where different relays are r eceiving the same jammin g signal. In order to efficiently overcome th e jammer in a d istributed manner, we use lattice codes, which enab le ef ficient mo dulo-like o peration, filtering out some of the unde sired interferen ce. Lately , several contributions suggested scen arios in wh ich structure d codes, an d lattices in particular dem onstrated to outperfor m best known rand om coding strategies [12],[13] [14] an d [15]. Th ese works apply the new advances in the un derstanding of lattice co des [16],[17] and [ 18], and nested lattices [19], [20], and specifically th eir ability to perform well as both source an d chan nel codes. Sp ecifically , [21] used lattice codes for interf erence channel. Additional work s relev ant in this respec t are [22] which uses lattices to overcome known in terference (by a remote helper node) , and is a special case of [2 3], w hich discusses the cap acity of th e doubly dirty MAC channel. These two works deal with a helper node which aids the tran smitter to overcome an inter ference. Although there the interferenc e is known to the helper node, this is still similar to our model, where instead o f a channel, the helper node has a reliab le link with finite capacity to the destin ation. Sev eral other works are relev ant here. The capac ity for distributed computatio n of the mod ulo function with binary symmetric variables is given in [ 24], while distributed computatio n with more general assumptions is p rovided in [25]. A deterministic approach to wireless n etwork, which specifically fits the scaling law m easure, is gi ven in [ 26]. W e will focus on thre e main scen arios, each o ne m odels different constraints imposed on the r esources av ailable for communicatio n. Each of these models rev eals another aspect of the general problem, while their joint contribution is dem onstrating the same p rinciples. The scaling laws derived fo r th ese scen arios r ev eal tha t a definite amou nt of inform ation about the interferenc e is req uired at th e destination if reliable co mmunication is established . This con clusion is a n extension of a special case of the results of Kr ¨ oner an d Marton [ 24], for two separated in dependen t binary sour ces X , Y , where a remo te destination wishes to retrieve X ⊕ Y , and needs to get bo th X , Y . The d ifference bein g that in [24], random coding strategies sufficed to demonstrate this principle, while here we resort to lattice codes. Further, for dependent b inary sources X , Y , [24] demonstrated the advantages of lattice codes ov er r andom codes necessitating th e knowledge of both ( X , Y ) . All the ach iev able rates in this pa per are derived, fo cusing on scaling laws. Th ese are by no m eans optimal in any o ther sense. Each rate expression was gi ven f or both the case of P X > P J and P X < P J , by using a simple minimum oper ation, for simplicity an d brevity ( P X being the av erag e power o f the transmitter an d P J being the av erage p ower o f the interference) . This paper is divided into four sections, the first section contains the gen eral setting and the basic definitions, the second p resents all central results a long with some discussion, the third section contain s pro ofs of the necessary and sufficient co nditions. W e then conclud e with some final no tes. Some proofs are relegated to the appendices. September 7, 2021 DRAFT 2 Relay R 2 Relay R 1 Destination D 3 C 1 C 2 Y 1 Y 2 X Transmitter a S 0 b + + N 1 N 2 1 1 Interference J Fig. 1. The system setting , with transmitter S 0 recei ved by R 1 and R 2 with channe l transfe r coe fficie nts a and b , respecti vely . L ossless links with respecti ve capacit ies of C 1 and C 2 betwee n relays R 1 , R 2 and destination D 3 . I I . M O D E L A N D D E FI N I T I O N S W e d enote by X i the random variable at the i-th position an d by X the vector of random variables ( X 1 , . . . , X n ) . W e consider the channel as it appear s in Figu re 1, where a source S 0 wishes to transmit the me ssage W ∈ { 1 , . . . , 2 nR } to the destination D 3 by transmitting X to the channel, w here R de signates the commu nication rate. The transmitter is limited b y an average power co nstrain P n k =1 E[ X 2 k ] ≤ P X . The cha nnel has two outp uts, Y 1 and Y 2 , wh ere Y 1 = aX + J + N 1 (1) Y 2 = bX + J + N 2 . (2) The additi ves J, N 1 , N 2 are Gaussian memoryless independent p rocesses, with zero mean and v ariances of P J , P N 1 , P N 2 , respectively , and a, b ∈ R are two fixed coef ficients to be addressed later . The tw o channe l outputs Y 1 , Y 2 are recei ved by two distinct r elay un its, R 1 and R 2 , respecti vely . Th ese relay s u se sepa rate pr ocessing on the recei ved signals Y 1 and Y 2 , and then forward the resultant signal to the destination D 3 . It is noted that the destination is only interested in the message W . The r elays ca n forward m essages to the d estination over reliable link s with finite cap acities of C 1 and C 2 bits per chan nel use. The de stination then d ecides o n th e tran smitted message ˆ W . A commun ication rate R is said to be achiev able, if the a verage err or pr obability at the destination Pr { W 6 = ˆ W } is arbitrarily clo se to zero for sufficiently large n . The communication system is therefor e completely characterized by four deterministic functions ( I n , [1 , . . . , 2 n ] ) X ( W ) = φ S 0 ( W ) : I nR → R n (3) V 1 ( Y 1 ) = φ R 1 ( Y 1 ) : R n → I nC 1 (4) V 2 ( Y 2 ) = φ R 2 ( Y 2 ) : R n → I nC 2 (5) ˆ W ( V 1 , V 2 ) = φ D 3 ( V 1 , V 2 ) : I nC 1 × I nC 2 → I nR . (6) W e can divide the genera l case of (1)-(2) into three p ossible option s: September 7, 2021 DRAFT 3 1) a = 0 o r b = 0 , a 6 = b . 2) a, b 6 = 0 , a 6 = b . 3) a = b . The last case is of no r eal interest since the scalin g behavior is obviou s and remain s th e same for one o r two r elays, so this paper will fo cus only o n the fir st two. Similarly , for C 1 , C 2 we have three ca ses, where C → ∞ means that C goes to infinity mu ch faster than P X , P J : 1) C 1 → ∞ or C 2 → ∞ . 2) Neither C 1 → ∞ nor C 2 → ∞ . 3) C 1 → ∞ and C 2 → ∞ . The last case resu lts with full cooper ation. I n this case the achievable rate and the upper boun d are iden tical and equal to the Shannon capa city , given by (for example for a = 1 , b = − 1 ): R = 1 2 log 2 (1 + 2 P X ) . (7) This rate is ach iev ed by using maximal ratio comb ining of th e two r eceptions, completely eliminating the in terfer- ence. So the g eneral case in th is paper reduces to th ree main scenarios, in which we inv estigate the scaling laws o f C 1 and C 2 , as a f unction of the achiev able r ate R . T hese thr ee cases consist of the f our po ssible option s describ ed above for a, b, C 1 , C 2 , where th e option of C 1 → ∞ or C 1 → ∞ while ab 6 = 0 was drop ped since it is very similar to the case where either C 1 → ∞ or C 1 → ∞ when a = 0 or b = 0 wh ile a 6 = b . Since the channel between the transmitter and the relays can support a rate with scaling of up to lim P X →∞ R log 2 ( P X ) = 1 2 , we inv estigate the requ ired cap acities of the links to achieve this scaling, an d also the degradatio n of R when they are smaller . W e d enote b y scaling the pre- log coefficient de fined by the limit scal ing = lim P X →∞ R log 2 ( P X ) , and write it fo r the sake of brevity as R ∼ scali ng log 2 ( P X ) . Similarly , the relation lim P X →∞ R log 2 ( P X ) ≥ 1 2 is designated by R & 1 2 log 2 ( P X ) . Case A: Relayin g the In terfer e nce The scen ario her e specializes to a = 1 (8) b = 0 (9) P N 1 = 1 (10) P N 2 = 0 (11) C 1 → ∞ . (12) The last condition simply states that the destination receives the channel o utput, which is composed of the transmission plus some u nknown interfere nce, which m ay degrade or even prevent any reliab le decoding . The September 7, 2021 DRAFT 4 interferen ce is received in tact in R 2 , and then relayed to th e de stination to ena ble r eliable decoding. W e conside r a fixed a = 1 fo r simp licity . Th e results for any a 6 = 0 ar e ob tained f rom the results for the fixed a = 1 alm ost verbatim. This model can describe a situation wher e an additi ve jam mer is known to a relay wh ich can assist th e d estination with resolving the transmission from S 0 . No tice th at for an in finite Jammer power ( P J P X → ∞ ), a link to the r elay R 2 is nec essary to achieve any positive rate. Case B: Relayin g the In terfer e d Sign al and th e I nterfer ence The on ly ch ange comp ared to the pr e viou s Scenar io A is that here C 1 is finite. T he adde d limitatio n extends the d ecentralization inher ent in the scheme, which mo dels practical systems whe re the d estination is not collocated near eith er relays. Case C: Relaying 2 Interfered Signals In this scenario we con sider the case wher e a = 1 (13) b = − 1 (14) P N 1 = 1 (15) P N 2 = 1 . (16) In this case, the signals a re in an ti-phase and hen ce joint processing ( C 1 , C 2 → ∞ ) via sub straction Y 1 − Y 2 would completely r emove the interference , and would allow for reliable rate of R = 1 2 log 2 (1 + 2 P X ) . As in the p revious cases, the solution to a = 1 , b = − 1 , gives the same scaling b ehavior as tak ing any a 6 = b, a 6 = 0 , b 6 = 0 . Notice that for scaling, the destination still wants to cancel the interference, and thus still n eeds to perfo rm Y 1 − Y 2 , r egardless of th e actual a, b , as long as a 6 = b . Also the finite P X , P J results f or th is gen eral ca se are readily deriv ed following the same steps. I I I . M A I N R E S U LT S The main results d escribed in this Section are divided into the th ree scenarios d etailed above, and includ e both the achiev able rates and outer bo unds. The resulting scaling laws of the inner and ou ter bou nds co incide. Case A: Relayin g the In terfer e nce Pr o position 1: 1) The following rate for Case A is a chievable R ≤ max    1 2 log 2  1 + P X 1 + min n 1 + P X , P J P X P X +1 o 2 − 2 C 2  , 0    . (17) September 7, 2021 DRAFT 5 2) An up per bou nd for all achievable rates for Ca se A is (cut-set bo und) R ≤ min { C 2 + I ( X ; Y 1 ) , I ( X ; Y 1 | J ) } (18) which for the Gau ssian channel r eads, R ≤ min  C 2 + 1 2 log 2  1 + P X P J + 1  , 1 2 log 2 (1 + P X )  . (19) The p roof of Proposition 1 ap pears in Section IV, where the ach iev able rate is just a special case o f the achiev able rate of Case B. It is u nderstood that the ach ie vable rate (17) is n ot optim al, but it d oes p rove the scaling laws. From Pro position 1 the scaling laws can be der i ved. Pr o position 2: T o achieve a scaling of R ∼ 1 2 log 2 ( P X ) , when r elaying the interferer (Ca se A), the suf ficien t and necessary lossless link capa city scaling is C 2 & 1 2 log 2  P X P J P X + P J  . (20) Furthermore , the gap b etween the achievable rate an d the upper bo und is no more than 1 bit. This Proposition is proved in Appendix I. The n ext corollary is a special case of Pro position 2: Cor o llary 1: A scaling of C 2 ∼ 1 2 log 2 ( P X ) is sufficient, for a ny interferer, regardless o f its power an d statistics. This Corollary holds, since the proo f for th e ach ie vable r ate in section IV uses random dithering, which achiev es the same perfo rmance for any J which is ind ependen t of the transmitted signal. For example, wh en the interf erer is anoth er transmitter, the robustness is with respect to the the applied co de, modulatio n technique and interf erence power . Howe ver , the exact phase, between th e reception at Y 1 and Y 2 is still required . Case B: Relayin g the In terfer e d Sign al and th e I nterfer ence Now also C 1 is finite. Pr o position 3: 1) The following rate for Case B is a chievable R < max    1 2 log 2   (1 + P X )(2 2 C 1 − 1) P X + 2 2 C 1 + min n 1 + P X , P J P X P X +1 o 2 − 2 C 2 (2 2 C 1 − 1)   , 0    . (21) 2) An up per bou nd for all achievable rates of Case B is (cu t-set boun d) R ≤ min { C 1 , C 2 + I ( X ; Y 1 ) , I ( X ; Y 1 | Y 2 ) } (2 2) which for the un derlying Gaussian chan nel turns out to b e R ≤ min  C 1 , C 2 + 1 2 log 2  1 + P X P J + 1  , 1 2 log 2 (1 + P X )  . (23) The proo f of Propo sition 3 appears in Sectio n IV. It is unde rstood that the achiev able rate ( 21) is not optimal, but it do es prove the scaling laws. Next, we quantify the necessary scaling of the link capacity C 1 . September 7, 2021 DRAFT 6 Pr o position 4: T o achieve a scalin g of R ∼ 1 2 log 2 ( P X ) , when r elaying the interfer e d signal and the interfer ence (Case B ), sufficient and necessary lossless links capacities scale as C 1 & 1 2 log 2 ( P X ) (24) C 2 & 1 2 log 2  P X P J P X + P J  . (25) Furthermore , the gap b etween the achievable rate an d the upper bo und is no more than 1.29 bits. The p roof app ears in Ap pendix I. This Proposition establishes tha t C 1 , that is the cap acity o f the link fr om the relay that receives the signal and the interference to the destination , scales the same as if there was no interf erence. Cor o llary 2: A scalin g of C 1 ∼ 1 2 log 2 ( P X ) suffices to ach ie ve robustness again st any in terference, regardless of its power or statistics, as lo ng as it remains indepen dent of X . In Figure 2, the scaling of the ac hiev able ra te of Case B is drawn as a function of C 1 + C 2 for P J < P X . For the sake of th e achiev able rate C 1 , C 2 were selected such that C 1 + C 2 is fixed a nd the achiev able r ate is m aximized. The cu t-set up per b ound in Case B is m et by an ac hiev able rate along the entire rang e in Figure 2. Specifically , from the point C 1 + C 2 = 0 to C 1 + C 2 = 1 2 log 2 ( P X /P J ) ( P 2 ), an a chiev able scheme wh ich uses simple lo cal decoding at R 1 is optima l. This is since usin g the d ecoded info rmation rate eq uals the cu t-set bound ( C 1 ≥ R ), wh ich is there fore tight even for finite P X , P J . The slope of the cu rve is 1, since on ly in formation b its are fo rwarded. Such local d ecoding is op timal as long as R ≤ 1 2 log 2 (1 + P X P J +1 ) . Higher su m-links-rate b enefit by d ev oting some bandwidth also to th e message from R 2 . The achiev able rate of Prop osition 3, equ ation (21) outp erforms local decodin g. In such scheme both relay s basically f orward the received sign als to the destination, wh ere the signa ls are subtr acted a t the destinatio n, which eliminates th e interfer ence. Th us every additional for warded info rmation bit r equires also on e bit for for warding the inte rference. Th is mean s that the rate increases on ly as 1 2 ( C 1 + C 2 ) . The o uter boun d for the r ange b etween P 2 and C 1 + C 2 = 1 2 log 2 ( P X P J ) ( P 1 ) is due to t he diagona l cu t- set upper bounds ( C 2 & R − 1 2 log 2  1 + P X P J  ). The maximal r ate is 1 2 log 2 ( P X ) , which is re ached on ly wh en C 1 + C 2 & 1 2 log 2 ( P X P J ) at P 1 . Case C: Relaying 2 Interfered Signals In this case th e d esired signal is received b y both relays alo ng with the co mmon interf erence. Pr o position 5: 1) An achievable rate for Case C is R > max    1 2 log 2   P X P X P X +1 + P X 2 2 C 1 − 1 + min { P X , P J P 2 X ( P X +1) 2 } 2 − 2 C 2   , 0    . (26) and this hold s also with the indices 1 a nd 2 interc hanged. 2) An up per bou nd for all achievable rates for Ca se C is ( cut-set bou nd), R ≤ min { C 1 + C 2 , C 1 + I ( X ; Y 2 ) , C 2 + I ( X ; Y 1 ) , I ( X ; Y 1 , Y 2 ) } (27) September 7, 2021 DRAFT 7 which for the un derlying Gaussian chan nel turns out to b e R ≤ min  C 1 + C 2 , C 1 + 1 2 log 2  1 + P X P J + 1  , C 2 + 1 2 log 2  1 + P X P J + 1  , 1 2 log 2 (1 + 2 P X )  . (28) Another upper boun d for a ll achievable rates for Case C is ( Modulo bo und) R ≤ 1 2  C 1 + C 2 + 1 2 log 2  1 + P X P J  + 1 4 log 2 (8 π e ) . (29) The proo f f or Proposition 5 a ppears in Sectio n IV. The ach ie vable r ate (26) is not op timal, b ut it does p rove the scaling laws. Note that (2 9) states an u pper bound for any R , in cluding fin ite rates. Howe ver , the bo und is interesting only in the case of large P X , P J , b ecause of the added 1.55 ( 1 4 log 2 (8 π e ) ) bits per channel use. Pr o position 6: Nece ssary an d sufficient co nditions o n C 1 , C 2 , to achieve the scaling of R when P X , P J ar e taken to infinity ar e      C 1 + C 2 & max n 2 R − 1 2 log 2  1 + P X P J  , R o C 1 , C 2 & R − 1 2 log 2  1 + P X P J  (30) Furthermore , the gap between the achievable rate and the ou ter b ound in the a symptotic re gime when R ∼ 0 . 5 log 2 ( P X ) is bou nded to 2.816 bits. The p roof of Proposition 6 appear s in appen dices II and I. The resulting scaling of the rate r egion is p resented in Figure 3, wher e the required scaling of C 1 , C 2 , so that the achiev able rate has the scaling of R is filled. The bound B 1 in Figu re 3 stands for the bounds on C 1 + C 2 such that C 1 + C 2 & max n 2 R − 1 2 log 2  1 + P X P J  , R o , while B 2 stands f or the diagon al boun ds, which sepa rately limit C 1 , C 2 such that C 1 , C 2 & R − 1 2 log 2  1 + P X P J  . It is e vide nt that any in crease of C 1 or C 2 can indeed only h elp, and th e rate-region is c on vex. Achieving the p oints P 1 an d P 2 , allows to ac hiev e any other po int in th e inter ior rate-region, throu gh time sha ring. In Figure 2, the scaling o f the achiev able rate is drawn as a fun ction o f the scaling of C 1 + C 2 , when C 1 = C 2 , letting P J < P X . From the p oint (0 , 0) to P 2 , an ac hiev able scheme uses simple local decoding at the relay s. Since this scheme uses all the links’ bandw idth to for ward only decode d infor mation, the cut-set boun d is tight, and the slope of the curve is 1. Such local decod ing is p ossible as long as R . 1 2 log 2 (1 + P X P J ) . Achieving h igher rates requ ires more than local decoding , and the achiev able rate of Proposition 5, equation (26) is used. Th e o uter bound for th is range is due to the modu lo o uter boun d ( 29). This scheme basically for wards the received signals to the destinatio n, where the signals are subtr acted to eliminate the in terference. As in Case B, we g et rid of the interferen ce only at the destination , which means that the rate scaling increases only as th e scaling of 1 2 ( C 1 + C 2 ) . The maximal rate scaling is 1 2 log 2 ( P X ) , which is reach ed o nly when C 1 + C 2 & 1 2 log 2 ( P X P J ) at P 1 . The modulo bound (29) determin es th e be havior b etween the points P 2 and P 1 . Cor o llary 3: The cu t-set upp er bound is strictly loose for the in terference chann el of Case C. Pr o of: T ake P J = √ P X and C 1 = C 2 = 1 4 log 2 ( P X ) . Th en th e cut-set bou nd f rom Equ ation ( 27) for the scaling reads R & R cut = 1 2 log 2 ( P X ) , while the m odulo bo und of Equation (29) fo r th e scaling reads R & R mod = 1 4 log 2 ( P X ) . So we showed th at R cut > R mod + ǫ for some ǫ > 0 . September 7, 2021 DRAFT 8 1/2log 2 (P x /P J ) C 1 +C 2 R 1/2log 2 (P x ) P1 1/2log 2 (P x /P J ) 1/2log 2 (P x P J ) P2 Cut-set Diagonal cut-set (case B) Modulo bound (case C) Cut-set Fig. 2. The achie va ble scaling of the rate R as a function of C 1 + C 2 , for cases B and C. Upper bounds are drawn with dotted li nes, while the full li ne is the achie va ble rate. It is evi dent that the cut -set bound in Case C is not ti ght, and the modulo bound wa s nee ded to charact erize the scaling laws. For Case B, the cut-set bound is tight for the whole region. B1 B2 C 1 C 2 B1 B2 P1 P1 Fig. 3. The scalin g of C 1 , C 2 as a functi on of the achie vable rate R , where B 1 is the scaling of max n 2 R − 1 2 log 2 “ 1 + P X P J ” , R o and B 2 is the scalin g of R − 1 2 log 2 “ 1 + P X P J ” . The filled are a denotes C 1 and C 2 which ena ble communica tion at rate R . September 7, 2021 DRAFT 9 Remark 1: For cases A an d B, wh en consid ering the o uter boun d due to the u nderlyin g Gau ssian chann el, R ≤ 1 2 log 2 (1 + P X ) and comb ining propositions 2 and 4 we g et that for large P X , P J , 1 n H ( V 2 ) & 1 2 log 2  P X P J P X + P J  . By ad ding also that H ( V 2 | J ) = 0 since V 2 is a deterministic function of J , it follows that 1 n I ( V 2 ; J ) & 1 2 log 2  P X P J P X + P J  . (31) So that in order to achieve a reliable rate R ∼ 1 2 log 2 ( P X ) , a defined amou nt of info rmation ab out the interf erer is required at the d estination with scaling 1 2 log 2 min { P X , P J } , fo r large P X , P J . I V . P RO O F S F O R B A S I C P R O P O S I T I O N S In th is section we p rove the basic propo sitions, not the p ropositions d ealing with the scaling laws, which appear in appendices I-II. A. Pr oofs for the Outer Bou nds In this section we present the p roofs of the ne cessary condition s o f th e prop ositions in Section III. Pr o of of Outer Bo unds for Cases A ,B a nd C in eq uations ( 18),(22) an d (27): The cut-set outer b ound [2 7] is simply the minimum a mong all th e communication r ates between any two cu ts of th e network [27] as is reflected in the th ree cases under study . Let us show the cut-set f or one su ch cu t, for the sake of concisen ess, where th e rest of the cuts read ily f ollow . T ake the cut such that one set inclu des the transmitter with relay R 1 and therefo re th e other set includ es R 2 and the destination . From [27], Theorem 14.10 .1: The achie vable r ate must be less than or equal to I ( X ( S ) ; Y ( S c ) | X ( S c ) ) for some single letter joint proba bility distribution P ( X ( S ) , X ( S c ) ) . In our setting with the chosen c ut, X ( S ) = ( X , V 1 ) , Y ( S c ) = ( Y 2 , V 1 ) and X ( S c ) = V 2 , sinc e the destination can n ot transmit anything. The under lying ch annel is P ( Y ( S c ) | X ( S c ) , X ( S ) ) = P ( Y 2 , V 1 | X , V 1 , V 2 ) = P ( Y 2 | X ) . T he r ight-most equ ality is since Y 2 is not affected by V 1 or V 2 , a nd the resulting Mar kov chain is V 1 − X − Y 2 . T o ge t the upper bound we n eed to m aximize I ( X, V 1 ; Y 2 , V 1 | V 2 ) over P ( X, V 1 , V 2 ) . The mutual info rma- tion I ( X, V 1 ; Y 2 , V 1 | V 2 ) is determin ed o nly by P ( X , V 1 , V 2 ) an d by the given chan nel P ( Y 1 , Y 2 | X ) such that I ( X , V 1 ; Y 2 , V 1 | V 2 ) = I ( X , V 1 ; Y 2 , V 1 ) . Sinc e V 1 − X − Y 2 is a Markov chain: I ( X , V 1 ; Y 2 , V 1 ) = H ( V 1 , X ) − H ( X | V 1 , Y 2 ) = H ( V 1 | X ) + I ( X ; V 1 , Y 2 ) = = H ( V 1 | X ) + I ( Y 2 ; X ) + H ( V 1 | Y 2 ) − H ( V 1 | Y 2 , X ) (32) Due to the Markov chain above, H ( V 1 | Y 2 , X ) = H ( V 1 | X ) . So that R ≤ I ( Y 2 ; X ) + H ( V 1 | Y 2 ) ≤ I ( Y 2 ; X ) + C 1 . (33) Considering all the cut-sets, equa tion (18) fo llows fro m R ≤ min { C 2 + I ( X ; Y 1 ) , I ( X ; Y 1 , Y 2 ) } = min { C 2 + I ( X ; Y 1 ) , I ( X ; Y 1 | Y 2 ) + I ( X ; Y 2 ) } = min { C 2 + I ( X ; Y 1 ) , I ( X ; Y 1 | J ) } (34) September 7, 2021 DRAFT 10 where the last equality follows since Y 2 = J . The complete proof s for the inequalities in equations (22) and (28) are om itted here since th ey are pr oved exactly the same way . Pr o of of the Modulo Outer Bound in Case C (29): The basis of the p roof is the representatio n of the tr ansmitted sign al ( X ) b y two co mponen ts, one is an integer which is basically known at the re lays (with high pro bability), and the oth er is a heavily interfer ed real signal. Definition: For any X , X − , X mo d √ P J and X + , j X √ P J k 1 . Assuming, without loss of any gen erality , that H ( V 1 | X + , V 2 ) ≤ H ( V 2 | X + , V 1 ) . (35) If (35) is no t satisfied, replace the ind ices of 1 and 2 in th e following. Using Fano’ s inequ ality , wh ere ǫ > 0 is arbitrary sma ll, fo r sufficiently large n , we get that nR ≤ I ( X ; V 1 , V 2 ) + nǫ (36) = I ( X + ; V 1 , V 2 ) + I ( X − ; V 1 , V 2 | X + ) + nǫ (37) = I ( X + ; V 1 , V 2 ) + I ( X − ; V 1 | X + , V 2 ) + I ( X − ; V 2 | X + ) + nǫ (38) ≤ I ( X + ; V 1 , V 2 ) + H ( V 1 | X + , V 2 ) + I ( X − ; Y 2 | X + ) + nǫ (39) ≤ I ( X + ; V 1 , V 2 ) + 1 2 [ H ( V 1 | X + , V 2 ) + H ( V 2 | X + )] + h ( Y 2 | X + ) − h ( Y 2 | X ) + nǫ (40) ≤ I ( X + ; V 1 , V 2 ) + 1 2 H ( V 1 , V 2 | X + ) + h ( − X − + J + N 2 ) − h ( J + N 2 ) + nǫ (41) = I ( X + ; V 1 , V 2 ) + 1 2 [ H ( V 1 , V 2 ) − I ( X + ; V 1 , V 2 )] + I ( X − ; − X − + J + N 2 ) + nǫ (42) ≤ 1 2 I ( X + ; V 1 , V 2 ) + 1 2 H ( V 1 , V 2 ) + n 2 log 2  1 + P J 1 + P J  + nǫ (43) ≤ n 4 log 2  2 π e  P X P J + 1 12  + n 2 ( C 1 + C 2 ) + n 2 log 2  1 + P J 1 + P J  + nǫ. (44) Where ( 39) is since H ( V 1 | X , V 2 ) ≥ 0 , and th e data proce ssing Lemma V 2 = φ R 2 ( Y 2 ) , (40) fo llows from (35) and by wr iting th e mutu al inform ation as the difference b etween two entropies, (41) is since h ( X − + J + N 2 | X + ) ≤ h ( X − + J + N 2 ) and h ( Y 2 | X ) = h ( J + N 2 ) , ( 42) is by noticing that h ( J + N 2 ) = h ( X − + J + N 2 | X − ) an d sim ply writing the difference between the en tropies a s mutu al info rmation, ( 43) is beca use E | X − | 2 ≤ P J and finally (44) is since H ( V 1 , V 2 ) ≤ C 1 + C 2 and I ( X + ; V 1 , V 2 ) ≤ H ( X + ) , wh ere E | X + | 2 ≤ P X P J , an d using Th eorem 9.7. 1 from [2 7]. B. Pr oofs for the Achievable Ra te Pr o of for Cases B and C (Pr opo sitions 3 a nd 5): Here we av oid reconstru cting th e whole J at th e destination by utilizing a lattice code and reducing the signals into 1 ⌊ X ⌋ rep resents the la rgest int eger , which is no greater than X September 7, 2021 DRAFT 11 its V oron oi cell b y a modulo oper ation. Our scheme is an ad aptation of the MLAN channel tech nique from [17]. First define th e lattice cod e C 2 which is a go od sou rce P X -code, which means that it satisfies, fo r a ny ε > 0 and ad equately large lattice dimension n log(2 π eG ( C 2 )) < ε P X = R ν 2 || x || 2 d x V o n . (45) Where G ( C 2 ) , ν 2 and V o are the normalized second mom ent, the V oro noi c ell and the V oronoi cell volume of th e lattice associate d with C 2 , r espectiv ely . Such codes are known to exist [16]. 1) T ransmission Scheme : Transmit the inform ation W as a codeword from a codebo ok, where e very codeword in this co debook is ran domly and ind ependently gener ated b y dividing it into m any (multi-letter) entries, each generated unif ormly i.i.d. over the V or onoi region o f C 2 , ν 2 . Defin e the transmitted codeword as V . Add a pseudo r andom ditherin g − U , which is uniformly gen erated over ν 2 and k nown to all par ties, to ge t: X = V − U mo d C 2 (modu lo V o ronoi region of C 2 ). This d ither is r equired for the analysis, to ensure indepen dence of the modulo no ise with respect to the message index. 2) Relaying Scheme : Both relays R 1 and R 2 multiply the received signals by α > 0 , apply mod C 2 , and quantize the received signal using stand ard inf ormation theo ry techniqu es into W 1 = αY 1 mo d C 2 + D 1 and W 2 = αY 2 mo d C 2 + D 2 . The qu antization is gi ven in Appen dix II I, wh ere U , Y in Ap pendix III are W 1 = αY 1 mo d C 2 + D 1 and αY 1 mo d C 2 , respectively . Th e underlyin g single letter distortions D 1 and D 2 in W 1 , W 2 are Gaussian with zer o mean and ar e ind ependen t with a ny other random variable. A Slepian W olf encod ing is then used on th e two v ector qua ntized signals, befo re tra nsmission to the destination. 3) Decoding at Destination : No w the de stination decodes W 1 and W 2 and calcula tes W 1 − W 2 . From the result, it fu rther subtrac ts the known pseu do rando m dither U , and applies again modulo C 2 (see e quation (4 8)). It then find s the vector ˆ V wh ich is jo intly typical with the resulting outcomes of the modulo operatio n. The decoded message is th e co rrespond ing message index W , if decoding is successful. 4) Analysis of P erformance : T he inde pendent distortion variance P D 1 correspo nds to what is promised b y th e rate distortion functio n for any ran dom v ariable with variance of P X , and in particular, to αY 1 mo d C 2 (See Append ix III for th e co mplete proo f). The r ate for independ ent distor tion for Y 1 is I ( W 1 ; αY 1 mo d C 2 ) = 1 2 log P X + P D 1 P D 1 ≤ C 1 . (46) Notice that taking D 1 such that P D 1 fulfills (4 6) allow us to chose D 1 to b e distributed indep endently o f αY 1 mo d C 2 , r egardless o f α . F or Pr oposition 3: Dependin g on wh ether α 2 P J < P X or α 2 P J > P X , using th e result in Appendix III, we get for P D 2 I ( W 2 ; αY 2 mo d C 2 ) = 1 2 log min { P X , α 2 P J } P D 2 ≤ C 2 . (47) September 7, 2021 DRAFT 12 Since X = V − U , we can write αY 1 − αY 2 + U = αX + αN 1 + U = X + U − (1 − α ) X + αN 1 = V − (1 − α ) X + αN 1 , a nd the following equalities h old Y = W 1 − W 2 + U mod C 2 (48) = αY 1 + D 1 − αY 2 − D 2 + U mo d C 2 (49) = V − (1 − α ) X + αN 1 + D 1 − D 2 mo d C 2 . (50) Define N eq = αN 1 + D 1 − D 2 − (1 − α ) X . (51) with P N eq = α 2 P N 1 + (1 − α ) 2 P X + P D 1 + P D 2 . (52) Encodin g accor ding to V wh ich is un iformly distributed over C 2 giv es an achiev able rate of (See Inflate d Lattice Lemma in [17]) R > 1 n  log 2 V o G ( C 2 ) − H ( N eq )  − δ = 1 2 log 2 (2 π eP X ) − 1 2 log 2  2 π eP N eq  − δ. (53) Setting α to maximize the achiev able rate α = P X P X + P N 1 , (54) along with (46) and (4 7) results in R > 1 2 log 2   P X P X P N 1 P X + P N 1 + P D 1 + P D 2   − δ = 1 2 log 2   P X P X P N 1 P X + P N 1 + P X 2 2 C 1 − 1 + min { P X , P J P 2 X ( P X + P N 1 ) 2 } 2 − 2 C 2   − δ. (55) Considering th at P N 1 = 1 , after som e simple algebra, (5 5) becom es (21). F or Pr opo sition 5: On one hand 1 n H ( W 2 | W 1 ) ≤ 1 n H ( W 2 ) ≤ 1 2 log  1 + P X P D 2  , (56) on th e oth er hand 1 n H ( W 2 | W 1 ) = 1 n H ( W 1 + W 2 | W 1 ) ≤ 1 n H ( W 1 + W 2 ) ≤ 1 2 log  1 + α 2 (4 P J + P N 1 + P N 2 ) + P D 1 P D 2  . (5 7) So for a successful Slep ian-W o lf deco ding we req uire that the minimum between the right hand side s of equations (56) and (57) be smaller than C 2 . T his br ings us to 1 2 log  1 + min { P X , α 2 (4 P J + P N 1 + P N 2 ) + P D 1 } P D 2  ≤ C 2 . (58) As in (50), here we hav e Y = V − (1 − 2 α ) X + α ( N 1 + N 2 ) + D 1 + D 2 mo d C 2 . (59) and P N eq in this case, is P N eq = α 2 ( P N 1 + P N 2 ) + (1 − 2 α ) 2 P X + P D 1 + P D 2 . (60) September 7, 2021 DRAFT 13 T ak ing α = 2 P X 4 P X + P N 1 + P N 2 , (61) we g et (con sidering that P N 1 = P N 2 = 1 ) R > 1 2 log 2   P X P X ( P N 1 + P N 2 ) 4 P X + P N 1 + P N 2 + P D 1 + P D 2   − δ ≥ 1 2 log 2    P X 1 2 + P X 2 2 C 1 − 1 + min { P X , 4 P J +2+ P X 2 2 C 1 − 1 } 2 2 C 2 − 1    − δ. (62) The p roof can be f urther replica ted also wh en inter changing the indices 1 and 2 . V . C O N C L U S I O N S A N D D I S C U S S I O N S In this paper, we derive bo th inner and outer b ounds of the commu nication rate, for three comm on distributed reception scenar ios, with u nknown interference. The th ree scenarios c haracterize the very lo w noise c ase of the more general case of distributed reception o f wanted signal plus unk nown interfere nce. The inner boun ds rely on lattice coding, since stan dard r andom coding tec hniques do not provide satisfactory results, in ge neral. Ou ter bounds b ased on the c ut-set techniq ue ar e d eriv ed, and ad ditional tig hter boun d is derived by using mu lti-letter techniqu es, for a case where the cut-set b ound d oes no t suffice. T his ca se inc ludes two relays, which receive both the de sired signa l and the interf erence. Th e gen erally lo ose inner and outer bou nds wh ich coin cide at asymptotically large powers o f the tr ansmitter and th e inter ferer, are used to der i ve th e scaling laws. These scaling laws r e veal that in ord er to overcome inter ference, a d efined amou nt o f info rmation about the interferen ce m ust be k nown at the d estination. The p roposed schem e for the inner bound , is also ro bust aga inst the interference statistics, code, mod ulation etc. The m odel is intimately related to the ca se of tw o independ ent transmitters. Then the transmission o f on e transmitter can be treated as interferen ce with p ower P J as in this paper . This appro ach is beneficial when the rate in which the interfer ing transmitter R J is h igh, so codebo ok knowledge is useless. If in add ition the power of the in terfering transmitter is h igh P J > P X , th en the achievable r ates in this paper provide a better appr oach than the standard compress-an d-forward. A C K N O W L E D G M E N T This research was sup ported by the NEWCOM++ network of excellence and the ISRC Co nsortium. A P P E N D I X I P RO O F S F O R T H E A S Y M P T O T I C G A P S Define the gap betwe en the achiev able rate and the outer bo und as ∆ . A. Case A For Case A, where C 2 = 1 2 log 2 (1 + P X ) an d P J > P X we g et ∆ = C 2 + 1 2 log 2  1 + P X P J + 1  − 1 2 log 2  1 + P X 1 + min n 1 + P X , P J P X P X +1 o 2 − 2 C 2  ≤ 1 2 log 2  1 + P X P J + 1  + 1 2 log 2 (2) ≤ 1 2 log 2 (1 . 5 × 2) = 0 . 7925 , (63) September 7, 2021 DRAFT 14 whereas f or P J < P X and C 2 = 1 2 log 2 ( P J ) we get ∆ = C 2 + 1 2 log 2  1 + P X P J + 1  − 1 2 log 2  1 + P X 1 + min n 1 + P X , P J P X P X +1 o 2 − 2 C 2  ≤ 1 2 log 2 ( P J + P X ) − 1 2 log 2  1 + P X 1 + P X P X +1  ≤ 1 2 log 2 (2 ∗ 2) = 1 . (64) So overall for Case A, ∆ ≤ 1 . B. Case B For Case B, wher e C 1 = 1 2 log 2 (1 + P X ) , C 2 = 1 2 log 2 ( P J ) an d P X > 1 , we g et th at min  1 + P X , P J P X P X + 1  2 − 2 C 2 (2 2 C 1 − 1) ≤ 1 + P X . (65) Which gives ∆ = C 2 + 1 2 log 2  1 + P X P J + 1  − 1 2 log 2   (1 + P X )(2 2 C 1 − 1) P X + 2 2 C 1 + min n 1 + P X , P J P X P X +1 o 2 − 2 C 2 (2 2 C 1 − 1)   ≤ 1 2 log 2 ( P J + P X ) − 1 2 log 2  (1 + P X ) P X P X + (1 + P X ) + (1 + P X )  ≤ 1 2 log 2 (2) + 1 2 log 2 ( P X ) − 1 2 log 2  P X 3  ≤ 1 . 29 . (66 ) So overall for Case B, ∆ ≤ 1 . 29 . C. Case C For Case C, wher e P J < P X < (1+ P X ) 2 P X , th e m odulo upper b ound is relev ant. So we have ∆ = 1 2  C 1 + C 2 + 1 2 log 2  1 + P X P J  + 1 4 log 2 (8 π e ) − 1 2 log 2 P X P X P X +1 + P X 2 2 C 1 − 1 + P J P X ( P X +1) 2 2 − 2 C 2 ! . ( 67) W e ev aluate the gap for corner case where we take C 1 = 1 2 log 2 (1 + P X ) an d C 2 = 1 2 log 2 ( P J ) . ∆ = 1 2  1 2 log 2 (1 + P X ) + 1 2 log 2 ( P J ) + 1 2 log 2  1 + P X P J  + 1 4 log 2 (8 π e ) − 1 2 log 2 P X P X P X +1 + P X P X + P X ( P X +1) 2 ! . ( 68) Since P X P X +1 + 1 + P X ( P X +1) 2 = P X (1+ P X )+(1+ P X ) 2 + P X ( P X +1) 2 ≤ 3 , (68) usin g P J > 1 giv es ∆ ≤ 1 4 log 2 (4) + 1 2 log 2 ( P X ) + 1 4 log 2 (8 π e ) − 1 2 log 2  P X 3  = 1 4 log 2 (8 × 4 × 9 π e ) = 2 . 81 6 . (69) September 7, 2021 DRAFT 15 For P J > (1+ P X ) 2 P X , the r elev ant upper bo und is the c ut-set boun d. W e fin d the gap fo r C 1 = C 2 = 1 2 log 2 (1 + P X ) , which g i ves the correct scaling . So here we have (also using P X > 1 ): ∆ = C 1 + 1 2 log 2  1 + P X P J + 1  − 1 2 log 2 P X P X P X +1 + P X 2 2 C 1 − 1 + P X 2 − 2 C 2 ! ≤ C 1 + 1 2 log 2 (2) − 1 2 log 2 P X P X P X +1 + 1 + P X 1+ P X ! ≤ 1 2 log 2 (2 × 4) = 1 . 5 . (70) So overall, in the low noise power limit, when R ∼ 0 . 5 , f or cases A,B an d C the gap between the achiev able rate and th e outer bound is bound ed to 1,1. 29 an d 2.8 16 bits, respectiv ely . A P P E N D I X I I P RO O F F O R S C A L I N G L AW S O F C A S E C Pr o of: Necessary conditio ns: The outer boun d in ( 28),(29) is written as a r ate region for C 1 , C 2 in ( 30), su ch th at fo ur constraints are met, where two constraints limit C 1 + C 2 and the other two co nstraints limit C 1 and C 2 separately . Sufficient cond itions: The outer boun d (30) consists o f th ree inequ alities, which leads to two intersections p oints (see Figure 3). Thus the entire r egion is achievable, for example by using time sharing , provid ed the point where the c apacities o f the links are C 1 ∼ max n R, 1 2 log 2  1 + P X P J o , C 2 ∼ R − 1 2 log 2  1 + P X P J  (P1 in Figure 3) correspo nds to a scheme with the same scaling of the reliab le r ate as R . Th e pro of is th en co mpleted by repeating the same arguments for th e seco nd po int (P2 in Figure 3). In case R . 1 2 log 2  1 + P X P J +1  , use P X to tran smit so that the me ssage will be sep arately d ecoded at the agents, where C 1 = R and C 2 = 0 . Since the agen ts recei ve the transmitted signa l with signal to noise plus interferen ce ratio of P X P J +1 , dec oding is reliable. I n case R & 1 2 log 2  1 + P X P J  , use th e scheme fr om Proposition 5, wh ich achieves the rate o f (26), with C 1 ∼ R and C 2 ∼ R − 1 2 log 2  1 + P X P J  . T his rate is 1 2 log 2  P X P X 2 − 2 C 1 + min { P X , P J } 2 − 2 C 2  = 1 2 log 2 2 2 R P X P X + min { P X , P J } (1 + P X P J ) ! ∼ R. (71) The b ounded gap between the achiev able rate and the upper b ound is e valuated in Appendix I. A P P E N D I X I I I P RO O F F O R C O M P R E S S I O N Proof that R > I ( Y ; U ) is sufficient for the relay wh ich received Y to forward U to th e final de stination. For any ǫ > 0 , 1) Pr eliminaries As is comm only done ( see [ 27], section 13. 6), define the ǫ -typical set T ǫ of vecto rs a 1 , 2 , with relation to the probability density function P a 1 , 2 as T ǫ , ( a 1 , 2 : ∀S ⊆ { 1 , 2 } ,     − 1 n log 2  P n a S ( a S )  − h ( P a S )     < ǫ, ) (72) September 7, 2021 DRAFT 16 where P n a S ( a S ) = Q n i =1 P a S (( a S ) i ) , and h ( P a S ) is the differential entropy of the probability density function P a S , wh ere S = [1 , 2 , { 1 , 2 } ] . Lemma 1: (AEP) For any ǫ > 0 , there exist n ∗ such tha t for all n > n ∗ and a 1 , 2 ∼ Q P A 1 , 2 we h av e P ( a 1 , 2 ∈ T ǫ ) ≥ 1 − ǫ. (73) Pr o of: See [ 27] Theo rem 9.2.2 . Lemma 2: Let a 1 , 2 be gen erated accor ding to a 1 , 2 ∼ n Y i =1 P a 1 (( a 1 ) i ) P a 2 (( a 2 ) i ) . (74) Then we have Pr( a 1 , 2 ∈ T ǫ ) = Pr( a 1 , 2 ∈ T ǫ S a 1 ∈ T ǫ S a 2 ∈ T ǫ ) Pr( a 1 ∈ T ǫ ) P r( a 2 ∈ T ǫ ) ≥ 2 − n [ h ( a 1 )+ h ( a 2 ) − h ( a 1 , 2 )+ ǫ 1 ] = 2 − n [ I ( a 1 ; a 2 )+ ǫ 1 ] (75) where ǫ 1 → 0 as ǫ → 0 . 2) Code gene ration Randomly gen erate 2 nI ( Y ; U ) codewords U , acco rding to i.i.d. distribution Π n i =1 P U ( U i ) . Index these co dew ords by z ∈ [1 , 2 nI ( Y ; U ) ] . Th e code book is made available to the relay and the d estination. 3) Compr ession Af ter receiving th e vector Y , the relay search es for z such that { U ( z ) , Y } ∈ T ǫ . I f no such z is fo und, the relay sends z = 1 . 4) Err or Analysis T he probab ility of two ind ependen t random variables U , Y to be jointly typical is lower bound ed b y 2 − n [ I ( Y ; U )+ ǫ ] . (76) Thus th e pr obability that no such z is jointly typical is upper bo unded by  1 − 2 − n [ I ( Y ; U )+ ǫ ]  2 nR , (77) which ten d to zero a s n gets large as long as R > I ( Y ; U ) + ǫ . 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