Low-Density Graph Codes for slow fading Relay Channels

1 Lo w-Density Graph Codes for Coded Cooperation on Slo w Fading Relay Channels Dieter Duyck, Joseph J. Boutros, and Marc Moeneclaey Abstract W e study Low-Density Parity-Check (LDPC) codes with iter ativ e deco ding on block- fading (BF) Relay Channels. W e co nsider two u sers that employ coded c ooperatio n, a variant of d ecode-an d- forward with a smaller outage pro bability than th e latter . A n outage prob ability analy sis fo r discrete constellations shows th at full diversity can be achieved only wh en th e coding rate do es not exceed a maximum value that depends on the le vel of cooperation. W e derive a new cod e structure by extending the previously pu blished f ull-diversity root-LDPC code, designed for the BF point-to- point chan nel, to exhibit a rate-com patibility prop erty which is necessary for coded coo peration. W e estimate the asymptotic performanc e th rough a new d ensity evolution analysis and the word error rate perform ance is deter mined for finite length codes. W e show tha t o ur code co nstruction exhibits near-outage limit perfor mance for all b lock len gths and f or a rang e of coding rates up to 0.5, which is the highest possible coding r ate for tw o co operatin g users. Index T erms Block fading channels, density evolution, low-density parity-check code, mu tual information , relay channels. Dieter Duyck and Marc Moeneclae y wish to ackno wl edge the acti vity of the Network of Excellence in W ireless COMmuni- cations NEWCOM++ o f the Euro pean Commission (con tract no. 2167 15) that moti vated th is wo rk. T he wo rk of Josep h Boutros and part of the work of Dieter Duyck were supported by the B roadband Communications Systems project funded by Qatar T elecom (Qtel) Dieter Duyck and Marc Moeneclaey are wi th the Department of T elecommunications and Information processing, Ghent Univ ersity , St-Pietersnieuwstraat 41, B-9000 Gent, Belgium, { dduyck,mm } @telin.ugent.be. Joseph J. Boutros is with T exas A&M Univ ersity at Qatar, PO Box 23874 Doha, Qatar , boutros@tamu.edu c  2009 IEE E. Personal use of this material is permitted. Permission from IEEE must be ob tained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collectiv e works, for resale or redistribution to servers or lists, or reuse of any copy righted componen t of this work in other works. July 30, 2021 DRAFT 2 I . I N T RO D U C T I O N When communicating ov er fading channels, W ord Error Rate (WER) performances as well as power savings are dramatically i mproved through transmi t div ersity , i.e., transmitti ng sig nals carrying the same information over different path s in time, frequency or sp ace. Recently , a new network protocol called Cooperative Communicatio n [11], [26 ], [32 ], [40], [41] yields transmit div ersi ty using single-antenna d e vices in a multi-user en vironment by taki ng advantage of the broadcast nature of wireless transmission. The most elementary example of a cooperativ e network is the relay channel, introduced by van d er Meul en [31]. In a relay channel, a relay helps the s ource in transmitti ng its d ata to a dest ination by relaying t he messages sent by the source s o that the received ener gy at the destinati on i s increased. This relay channel can be generalized to a cooperative Multi ple Access Channel (MAC ) [26], where two users transm itting data to a s ingle receive r cooperate by alternately bei ng the relay for the other user , as indi cated in Fig. 1. Further generalization to more users is possible, but t his will not be discussed here for sim plicity . 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 0000 0000 0000 0000 1111 1111 1111 1111 Destination User 2 User 1 Fig. 1. A Cooperati ve Mu ltiple Access Chann el (MA C). Arrows between two nodes illustrate that both node s communicate between each other . A challengin g channel model is the BF [3] frequency non-s electiv e Singl e-Input Single-Output (SISO) channel. When the fading g ain is constant over a codew ord and n o cooperation is u sed, the resulting word error rate curve (displaying the logarithm of the error rate versus the av erage signal-to-noise rati o (SNR) i n dB) has the same high -SNR slop e as for uncoded transmissi on: the July 30, 2021 DRAFT 3 corresponding di versity order 1 equals one. The potential di versity increa se brought by cooperati ve techniques allows to save much transmit ener gy at a giv en error rate. BF channels are a realistic model for a number of channels affected by slowly v aryin g fading and flat fading is assum ed in order to isolate the ef fect of cooperati ve dive rsity . The specific task of the relay i s determined by the strategy or protocol. In the case Decode and Forwar d (DF), the relay first decodes and then re-encodes the message before sending it to the destination. A var iant of DF is coded cooperation, where the relay decodes the message recei ved from the source, and then t ransmits additional p arity bits of the message, resulting in a more spectral effi cient strategy [22], compared to a traditional D F p rotocol. Instead of SNR accumulation (logarithm ic ris e of mutual information with receive d power from the relay) at the destination, we get information accumulat ion (linear rise of m utual information with received power from the relay) [46]. It has been sho wn in [23] that the o utage probabi lity [3], [33] of coded cooperation for half-duplex BF channels is smaller than for repeti tion-based proto cols. Moreover , the concept of cod ed coop eration can be used in mo re complex strategies, s uch as Amplify-Decode-Forward [2], where the relay can choose between DF and AF . So finally , replacing the decode-and-repeat part in any protocol by this more intelligent “information adding” strategy improves the outage probabil ity performance. As a consequence, constructi ng a near - outage channel code for a coded cooperation s cenario results in a competitive error-correc t ing code in terms of error-ra te performance vs. SNR for a gi ven rate R . Up till no w , coded cooperation has mainl y been imp lemented using rate-compati ble conv o- lutional codes [22]. The main d rawback of these codes is that the WER increases wi th the logarithm of block length t o the power d where d is the diversity order [6], [7]. The WER of practical near-outage codes should be i ndependent of the block length in order t o app roach the outage probability limi t [16], [17]. The sol ution is t o use capacity-achie ving cod es, for example LDPC codes [36] . LDPC codes desi gned for the special case of a cooperative channel hav e been reported for the Gauss ian channel by Razaghi and Y u [34], [35] and by Chakrabarti et al. [9]. 1 Here, di versity order is defined as the ratio of the high-SNR slopes of the error rate curv es of the con si dered system and of the uncoded system, respectiv ely . Alternative ly , diversity order can be defined as the slope of the error-rate curve of the considered system. The div ersit y depends on the fading gain distribution i n t he latter definition, but not in the f ormer definition. Both definitions are equal in the case of Rayleigh fading. July 30, 2021 DRAFT 4 For the block-fading channel howe ver , t here is st ill a lack of a near -outage LDPC code. Hu et al. [20] also designed LDPC codes for the Gaussi an relay channel, whereafter they applied this random LDPC code to a BF relay channel. Unfortun ately , a random code does not perform very well on a BF relay channel, because it has n ot the structure to achiev e full div ersity , as shown by Boutros et al. [5] and as wi ll be explained in the rest o f the paper . In Section III, this paper analyzes the outage probabili ty for binary phase s hift ke ying (BPS K) modulation s and deri ves a coding ra t e limitation that is necessary for th e protocol to ha ve diversity two, valid for all di screte alphabets. Deriving a code structure for coded cooperation will b e treated in the s econd part of the paper . The ai m of coded cooperation is to send a codew ord over two independent fading paths and the relay mus t be able to decode after recei ving the first part of the codeword. An error-correcting code must th erefore exhibit two prop erties: full-diversity and rate-compatibility . This paper deriv es a new code structure satisfyi ng both properties. Often [13]– [15], perfect source-relay channels are assumed when designing error -correcting codes. These codes can be extended im mediately to codes for cooperative syst ems with non-perfect so urce- relay channels us ing th e proposed rate-compatibl e structure from this paper . W e also determine density e volution equations t o obtain a lower bou nd on t he WER o f the LDPC ensemble. The density e volution analysis can also be used to optimize the degree distributions, whi ch will be discussed briefly , but th is is not the topic of the p aper . Channel-State Informati on (CSI) is ass umed at the decoder . W e consider half-duplex d e vices, assuming that simu ltaneously receiving and t ransmittin g data in the same frequency-band is too complicated due t o the lim ited isolatio n of directional couplers. In additi on, we also restrict the protocol to be orthogonal since we transmi t at low rates (we use Binary Phase-Shift Ke yi ng (BPSK)). The proposed code constructio n can ne vertheless be used in more complex no n- orthogonal protocols , where one can achieve more coding gai n i n high-rate scenarios [1]. I I . S Y S T E M M O D E L A N D N OT A T I O N As mentioned in the i ntroduction, the devices are half-dupl ex and users transm it in non- overlapping tim e slo ts. The transmis sion of a codeword is or ganized in t wo frames which constitute one block. W e denote the transmis sion of user u , u = 1 , 2 , in frame m , m = 1 , 2 , by X u,m . The pair ( C u, 1 , C u, 2 ) denotes the codew ord of user u . In the first frame of a block, July 30, 2021 DRAFT 5 each user broadcasts the first part of it s encoded data to th e other user and t o the desti nation. In the second frame, users either cooperate or send add itional parity b its related to th eir o wn information message, dependin g o n wh ether th ey are able to decode th e transm issions in the first frame. The decoding failure is detected b y the relaying user vi a a Cyclic Redundancy Check (CRC) code o r any other i ntelligent detection scheme. There are 4 cases to be disting uished, as summarized in Fig. 2: in case 1, both users hav e successfull y decoded the information from the other user; in case 2, no ne o f th e us ers has been able to decode the informatio n from the other user; in case 3 (case 4), only us er 2 (user 1) has s uccessfully decoded the i nformation from the other user . M ethods are known allowing the destination to detect which of these 4 cases has occurred [21]. Frame 1 Frame 2 User 2 User 1 X 1 , 1 = C 1 , 1 X 2 , 1 = C 2 , 1 X 1 , 2 = C 2 , 2 X 2 , 2 = C 1 , 2 (a) Case 1. Both interuser transmissions are successfully decoded. Each user cooperates in the second frame. Frame 1 User 1 User 2 Frame 2 X 1 , 1 = C 1 , 1 X 1 , 2 = C 1 , 2 X 2 , 1 = C 2 , 1 X 2 , 2 = C 2 , 2 (b) Case 2. Both interuser communications failed. Each user sends its o wn parity bits in t he second frame. Frame 1 Frame 2 User 2 User 1 X 1 , 1 = C 1 , 1 X 2 , 1 = C 2 , 1 X 1 , 2 = C 1 , 2 X 2 , 2 = C 1 , 2 (c) Case 3. User2-to-User1 commu nication failed. In the second frame, user 1 sends its o wn parity bits and user 2 cooperates with user 1. Frame 1 Frame 2 User 1 User 2 X 1 , 1 = C 1 , 1 X 1 , 2 = C 2 , 2 X 2 , 1 = C 2 , 1 X 2 , 2 = C 2 , 2 (d) Case 4. User1-to-User2 communicatino failed. In the second frame, user 2 sends its o wn parity bits and user 1 cooperates with user 2. Fig. 2. The 4 cases encountered in coded cooperation are l isted abo ve. A code word will consequently be s plit over 2 frames. W e con sider codewords to have a t otal length equal to N binary digit s, where N = N 1 + N 2 , and N 1 and N 2 denote the length of the first and second part of the codew ord. W e d efine the level of cooperation, β , as t he ratio N 2 / N . W e denote the transmitt er of a frame, which can be user 1 or u ser 2, by s and the recei ver of a frame, which can be user 1, user 2 or the destination, by r . T ransm itted sym bols of user 1 will be denoted x 1 [ i ] where i is the symb ol time index, i ∈ { 1 , . . . , N } . Similarly , t ransmitted symbols of user 2 are denoted x 2 [ i ] . The transmitted symbols are chosen from a BPSK alphabet, x s [ i ] ∈ { 1 , − 1 } . Receive d s ymbols will be denoted y sr [ i ] for receiv ed sy mbols from transm itter s to receiv er r . Th e receiv ed sym bol is given by July 30, 2021 DRAFT 6 y sr [ i ] = α sr x s [ i ] + z r [ i ] , (1) where z r [ i ] ∼ N (0 , σ 2 ) are ind ependent noi se sampl es and α sr ∈ R + is th e Rayleigh dist ributed fading gain betw een sender s and receiv er r , with normalized second order m oment, E [ α 2 sr ] = 1 . The fading coefficient α sr is assumed to be con stant during 2 frames. Note th at th is channel model is memoryless [10] and satisfies the channel symmetry condition, p ( y sr [ i ] | α sr , x s [ i ] = 1) = p ( − y sr [ i ] | α sr , x s [ i ] = − 1) . Each termi nal is t ransmittin g at a constant enery per sym bol E s , which is related to the ener gy p er i nformation bit E b by E s = R c E b (BPSK). The total ener g y per inform ation bit-to-noi se ratio is specified b y E b / N 0 . W e focus on binary LDPC codes C [ N , K ] 2 with block length N , dimension K , and cod ing rate R c = K / N . Regular LDPC ensembl es are characterized by the p air ( d b , d c ) , where d b is the maximum bitnode degree and d c is the maximu m checknode degree. Irregularity is introduced through the standard p olynomials λ ( x ) and ρ ( x ) [38]: λ ( x ) = d b X i =2 λ i x i − 1 , ρ ( x ) = d c X i =2 ρ i x i − 1 . where λ ( x ) and ρ ( x ) are the left and ri ght degree distributions from an edg e perspecti ve. In Section V the pol ynomials ˚ λ ( x ) and ˚ ρ ( x ) , which are the left and right dist ributions from a nod e perspectiv e, will also be ado pted: ˚ λ ( x ) = d b X i =2 ˚ λ i x i − 1 , ˚ ρ ( x ) = d c X i =2 ˚ ρ i x i − 1 . In th is paper , not all bit nodes and check nodes in the T anner graph will be treated equally . T o elucidate the diffe rent classes of b it nodes and check nodes, a compact representation of the T anner graph, adopted from [8] and also known as protograph representation [42], will be us ed. In this com pact T anner graph, bit nodes and check nodes of t he same class are merged into one node. July 30, 2021 DRAFT 7 Definition 1 The diversity or der attained by a code C is defined as d = − lim γ →∞ log P e log γ , wher e P e is the wor d err or rate aft er d ecoding. Definition 2 An err or-corr ecting code is sai d to have full diversity if d = N u , wher e N u is the number of cooperating u sers. Notice that t he above definition assumes Rayleigh distributed single antenna channels. Ac- cording to t he blockwise Singleton bound [25], [30], the coding rate for an n -order full-diversity code is upper b ounded by R cmax = 1 /n . Hence, in a 2-user scenario we g et R c ≤ 0 . 5 . I I I . O U TAG E P RO B A B I L I T Y A NA L Y S I S The word error rate of practical systems is , in the lim it of large blo ck length, lower bounded by the info rmation outage pr ob ability P out = P  I ( α , γ ) < R  , where I ( α , γ ) i s t he i nstantaneous mutual informatio n as a function of a certain fading gain α and average SNR γ , γ = E s N 0 = 1 2 σ 2 , where E s is the symbol energy . This definition remains valid for a channel m odel as described i n (1), but t hen α is the set of fading gains over a codew ord and γ is the set of average received SNRs. The rate R is the spectral ef ficiency of a us er , only taking into account its tim eslots, hence not t he average spectral effic iency 2 . T he diversity order of the outage probabil ity l imit is the s ame as the order attai ned by a full-diversity channel code [16]. It is our aim in this paper to approach th e outage probabil ity limit for a range of values of the spectral ef ficiency R . Since we use BPSK signal ing, the spectral efficienc y R i s identical to R c . The outage probabili ty analysis of coded cooperation with a Gauss ian alphabet has been made in [23]. Here, the analysis considers BPS K si gnaling, leading to an important conclu sion in Corollary 1 at the end of thi s s ection. The stated corollary is also valid for l ar ger discrete 2 This is, i n our opinion, necessary for a fair comparison between multiple user networks with a dif ferent number of users. July 30, 2021 DRAFT 8 alphabets. The a verage mutual information of a SISO channel with received signal y , condi tioned o n the channel realization α , is determin ed by t he following well-known formula [44]: I ( X ; Y | α ) = 1 − E Y | α  log 2  1 + exp  − 2 y α σ 2  , (2) where E Y | α is the mathematical expectation over Y given α . The o utage eve nt of a poi nt-to-point link is defined by the mutual information of that link being less than i ts transmissi on rate. T he outage ev ent E o of the relay channel is determined by a specific region in the multidi mensional space of instantaneous signal-to -noise ratio s. Next, we give the exact definition of E o for coded cooperation with BPSK m odulation. W e shorten the no tation I ( X i ; Y j | α ij ) to I ij . Pr oposition 1 In coded coop eration for a two-user MAC wit h BPSK sign aling, the outage event E o r elated t o user 1 is e xpr ess ed as follo ws: E o a ) =  I 12 > R 1 − β  ∩  I 21 > R 1 − β  ∩ ( I 1 d (1) < R )  ∪  I 12 < R 1 − β  ∩  I 21 < R 1 − β  ∩ ( I 1 d (2) < R )  ∪  I 12 > R 1 − β  ∩  I 21 < R 1 − β  ∩ ( I 1 d (3) < R )  ∪  I 12 < R 1 − β  ∩  I 21 > R 1 − β  ∩  I 1 d (4) < R 1 − β  , wher e I 12 b ) = 1 − E Y | α 12  log 2  1 + e xp  − 2 y 12 α 12 σ 2 12  , (3) I 21 b ) = 1 − E Y | α 21  log 2  1 + e xp  − 2 y 21 α 21 σ 2 21  , (4) and wher e I 1 d (1) is I 1 d in case i . F or each of the cases consider ed in Fig . 2, the mutual informati on I 1 d can be calcul ated as follows: Case 1: I 1 d (1) c ) = 1 − ( 1 − β ) E⋫ Y | α 1 d  log 2  1 + exp  − 2 y 1 d α 1 d σ 2 1 d  − β E Y | α 2 d  log 2  1 + exp  − 2 y 2 d α 2 d σ 2 2 d  . (5) July 30, 2021 DRAFT 9 Case 2: I 1 d (2) c ) = 1 − E Y | α 1 d  log 2  1 + exp  − 2 y 1 d α 1 d σ 2 1 d  . (6) Case 3: I 1 d (3) c ) = 1 − ( 1 − β ) E Y | α 1 d  log 2  1 + exp  − 2 y 1 d α 1 d σ 2 1 d  − β E Y ′ | α 1 d α 2 d  log 2  1 + exp  − 2( y ′ )( α 2 1 d + α 2 2 d ) 3 / 2 σ 2 1 d α 2 1 d + σ 2 2 d α 2 2 d  , (7) y ′ = ( α 1 d y 1 d + α 2 d y 2 d ) p α 2 1 d + α 2 2 d . Case 4: I 1 d (4) c ) = 1 − E Y | α 1 d  log 2  1 + exp  − 2 y 1 d α 1 d σ 2 1 d  . (8) Pr oof: a) is the union of four events associated to the four cases con sidered i n Fig . 2. Each case in E o in volves t he intersection wi th an outage e vent wh ere the mutual information between a user and t he destinati on is below the rate R , except for case 4, where on ly the first frame is dedicated to user 1. b) follows directly from (2). c) uses the f act that t he two frames in a block behave as p arallel Gaussian channels whose capacities add together . Of course, both frames tim eshare a time-interval, which gives a weight to each capacity term [10, Section 9.4], [43, Section 5.4.4]. (7) follows from m aximum ratio combin ing [43] at the destination during the second frame. The outage probability is obtained by integrating the joint probabilit y distribution p ( α 12 , α 21 , α 1 d , α 2 d ) over the volume defined by E o : P out = Z Z Z E o p ( α 12 , α 21 , α 1 d , α 2 d ) d α 12 d α 21 d α 1 d d α 2 d . Just as for the Gauss ian mo dulation, there is only one free parameter β because R and γ are fixed by the protocol and t he physical en vironment. Hence, giv en R and γ , one can optimi ze the value of β . For example, notice that for a low-SNR interuser channel, the outage probabili ty improves July 30, 2021 DRAFT 10 while taking β s maller than 0 . 5 due to the enhanced prot ection of t he source-relay chann el. On the other hand, a β smaller than 0 . 5 results in lo w er achie vable coding rates, as proved in Corollary 1. Th e op timization of β , as already undertaken in [23] for Gaussi an m odulations , is not within the sub ject of this paper . There is an im portant conclusion to d raw from th e analys is of Prop. 1: Corollary 1 In coded cooperation over a block-fading channel for th e 2-user MAC with a cooperation level β , transmitting at a codin g rate gr eater th an min ( β , (1 − β )) r enders a si ngle or der di versity . Pr oof: A necessary condit ion for coded coo peration to achiev e full diversity over a block- fading channel, is t hat it achiev es ful l dive rsity over a Block Erasure Channel (BEC) [27], because a BEC is an extremal case of a blo ck-fading channel. W e will show t hat this conditi on is not satisfied for coding rates greater than min ( β , (1 − β )) . In a BEC, the fading gain α takes two possi ble values { 0 , + ∞} . An outage event on a point-to -point channel is defined by th e fading gain α being zero. As a consequence, t he pos sible values of th e BPSK capacity on a BEC are confined to zero or one. Hence, for the two-user MA C, the mutual in formation I 1 d related to case 1 belongs to { 1 , β , (1 − β ) , 0 } . A doubl e diver sity ord er is equiv alent to stating that two ou tage eve n ts are necessary to lo se the t ransmitted codeword. T ake the s cenario wh ere the user1-to-destination channel has fading gain zero and the user2-to-destinati on channel has fading gain ∞ . In this scenario, the mutual i nformation I 1 d is equal to β . All codi ng rates higher than β wil l limit th e diversity order of the outage probabi lity t o one, since only one channel in outage is enough to lose the code word. From a sim ilar reasoning, it is sho wn that R c must be smaller than (1 − β ) . This corollary is also valid for si gnaling strategies with M constellat ion points. In th e sequel, if not otherwise stated, we assum e a rate equal t o R c = 1 3 . From Corollary 1, we know that the level o f coop eration must at l east belong to β ∈ [ 1 3 , 2 3 ] . W e st ress on the fact that the propos ed code const ruction is very flexible in parameters such as the b lock length and the coding rate. W e will use β = 0 . 5 throug hout this paper , which allows t he b roadest range of coding rates according to Corollary 1. W e il lustrate this in the numerical resul ts b y showing the WER performance of an LDPC code whose coding rate R c approaches 1 / 2 . July 30, 2021 DRAFT 11 I V . F U L L - D I V E R S I T Y L O W - D E N S I T Y C O D I N G F O R C O D E D C O O P E R A T I O N Code words in coded cooperatio n are split over 2 frames. The first part of a codeword, transmitted durin g the first frame sho uld protect information on the nois y source-relay channel. Consequently , a channel cod e, com patible with two d istinct rates is to be devised. In non - cooperativ e communication s, this property is known as rate-compatibil ity where parity bits o f higher rate codes are emb edded in those of lo w er rate codes [19]. The advantage is that all codes can be encoded/decoded using a si ngle encoder/decoder . Rate-compatibility in the context of LDPC codes was first introduced by Li et al. [29] and Ha et al. [18] and further elaborated for example in [45]. T wo techniques have been used: puncturing and extending. A fraction of parity bits of a m other code could be punctured to o btain higher rate codes. Howe ver , the resulting rate range is limited because the deletion o f too many bits has a negati ve ef fect on decoding via belief propagati on. T o obt ain a more dynamic range in rates, the technique of extending has been us ed. The extension is made by adding e x tra parity bits as illustrated in Fig. 3, wh ere the overall code is the intersection o f two constituent codes defined by H 2 and H 1 padded with zeros on the right. 0 i 1p 2p 0 Frame 2 Frame 1 H 2 H 1 Fig. 3. Parity-check matrix of a rate-compatible LDP C code obtained by the exten si on of higher rate codes. Symbols are split into three classes: i for the information bits, 1 p and 2 p for two cl asses of parity bits. The classes i and 1 p are transmitted by the source in frame 1. Parity bits 2 p are transmitted in the second frame, for example by the relay after successful decod ing of the first frame. For simplicity , we on ly us ed the technique of extending t o acquire rate-comp atibility , b ut th is may be further optim ized by combining pun cturing and extending via known techni ques [18], [29], [45]. July 30, 2021 DRAFT 12 A. Full-di versity LDPC codes In coded cooperation, 4 cases occur depending on the success of the transmi ssion in the first frame. In each of the cases, the d estination has other log-likelihood ratios at the input of the decoder . In the following propositi on, we will s how that i t i s s uffi ci ent to g uarantee t hat t he decoder at the destination achieves full diversity in case 1. Pr oposition 2 In coded cooperation on a cooperative MAC, a code C atta ins full diversity , if and only if f ull diversity is attain ed i n cas e 1. Pr oof: The WER after decoding P e can be spli t as fo llows P e = 4 X i =1 P ( case i ) P ( e | case i ) . (9) The prob ability that a certain case occurs, depends on the success o f decodi ng two poin t-to-point channels, so that i t is easy to derive that: P ( case 1 ) = (1 − c γ )(1 − c γ ) (10) P ( case 2 ) = ( c γ )( c γ ) (11) P ( case 3 ) = (1 − c γ )( c γ ) (12) P ( case 4 ) = ( c γ )(1 − c γ ) , (13) where c is a positive constant. T o hav e P e ∝ 1 γ 2 , the fol lowing condi tions app ly: P ( e | case 1 ) ∝ 1 γ 2 , (14) P ( e | case 2 ) ∝ 1 , (15) P ( e | case 3 ) ∝ 1 γ , (16) P ( e | case 4 ) ∝ 1 γ . (17) Eqs. (15), (16) and (17) are automatically satisfied, so that the only nessecary and sufficient condition is (14). July 30, 2021 DRAFT 13 Due to Proposition 2 , we will assum e in the fol lowing analysis the o ccurrence of case 1 where the transmission on the interuser channel in the first frame has been successful and both us ers are coop erating i n th e second frame. Full-div ersity codin g on a relay channel must cope wi th block erasures. Consider the coding structure plotted in Fig. 3. If all parit y bits 2 p are erased due to deep fading in frame 2, then the decoder should be capable t o retri e ve information bits i thanks to H 1 and possibl y recompu te 2 p thanks to H 2 . Unfortunately , under deep fading in frame 1, a structure wi th a randomly generated H 2 , as in Fig. 3, cannot guarantee the retriev al of the i nformation bits t hrough H 2 . The aim of this section is to explain how H 2 can be tuned in order to have full div ersity for any left and right degree distri bution and for any block length . T o the des tination, it appears as if one source has s ent its codew ord over a point-to -point BF channel i n case 1. Therefore, we take the constit uent code defined by H 2 to be a full-di versity LDPC code (referred to as root-LDPC code) as constructed by Bout ros et al. i n [8], [5] for non- cooperativ e single-antenna channels with two or more fading states per code word. The T anner graph no tation for the root-LDPC code is given in Fig. 4. This notation is essential for the analysis because we seek full di versity under iterati ve decoding. Full diversity of a root-LDPC 00000 00000 00000 00000 11111 11111 11111 11111 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 Parity node transmitted in frame 2. Check node. connected to a rootcheck. connected to a rootcheck. Information node transmitted in frame 2, Parity node transmitted in frame 1. Information node transmitted in frame 1, Fig. 4. Notation for the T anner graph of a full-div ersity LDPC code. structure is created by r ootchec ks , a special type of checknodes i n the T anner g raph. As s hown in Fig. 5, the root and the leaves o f this special checknode d o not belong to t he same frame. When the rootbit is i n frame 1 , the leav ebit s are in frame 2, and vice versa. Usi ng th e l imiting case of a Block-Erasure Channel, it i s easy to verify that a rootbit is determ ined via it s root check when its o wn frame is erased. The complete root -LDPC st ructure is built after s plitting information July 30, 2021 DRAFT 14 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111 0000 0000 0000 0000 1111 1111 1111 1111 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 Fig. 5. T wo types of rootcheck s. On the left-hand side, the rootbit belong s to frame 1 and the leav ebits belong to frame 2. The symmetric case where chann el states are switched is shown at the right-hand side. bits into two classes, denoted 1 i and 2 i , and p arity bits into two classes, denoted 1 p and 2 p . The checknodes are cut into two classes denoted 3 c and 4 c 3 . Th e classes 3 c and 4 c consist of rootchecks for information bits 1 i and 2 i respectively . The comp lete root-LDPC structure including all t ypes of nodes is illustrated in Figs. 6 and 7. Rootchecks are translated into two identity matrices (or permutation matrices in general) i nside the parity -check matrix i n Fig. 7. 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 4c 3c 2p 2i 1 2 3 3 2 1 1i 1p Frame 1 Frame 2 N 4 N 4 N 4 N 4 N 4 N 4 Fig. 6. T anner graph of a full-diversity LDPC code of length N and rate 1 2 . This compact graph representation has been adopted from [8], [5], i t is also kno wn as protog r aph representation [42]. The integ ers labeling the edges of the T anner graph indicate the degree of a node along those edges for a regular (3,6) root-LDPC code. The binary elements are split into four classes of each N 4 bits. The checkno des are cut into two classes of N 4 checks. The proof of full-div ersity for block-Rayleigh fading can be found in [ 8]. Note th at the diver sity order of the root-LDPC code does not depend on the righ t or left degree dis tributions. For 3 The checkno de notation 1 c and 2 c is reserve d for H 1 in the cooperati ve code as described in the next subsection . July 30, 2021 DRAFT 15 simplicit y , we only showed a regular (3,6) structure in Fig. 6. 1 1 1 1 1 1 1 1 2i 1p 2p 0 0 4c 3c 1i Frame 1 Frame 2 H 1 p H 1 i H 2 i H 2 p Fig. 7. Parity-check matr ix of a rate 1 2 root-LDPC code. Note that although t his code is natural for the p oint-to-poin t BF channel, it is n’t for the cooperativ e M A C. The source is s ending only half of its informati on bits to the relay , who is supposed to d ecode all the i nformation bit s. This sounds cou nter-intuiti ve and we are the first to apply this concept in coo perativ e comm unications. Although it is counter-intuitive, it is necessary to achiev e full diversity wi th iterative d ecoding, as explained above. For asympt otic code lengths, multi-edge type messages propagate in the root-LDPC graph [39]. One has t o choose between t wo differe nt root-LDPC ensembles. If we refer to t he T anner graph in Fig. 6, the two ensembl es are di stinguish ed as follows: (i) The first ensemble is built by t wo random edge permutations (edge interleave rs) connecting 3 c to ( 2 i , 2 p ) and 4 c to ( 1 i , 1 p ) respectively . This is equiv alent to t he random generation of t wo low-density matrices ( H 2 i , H 2 p ) and ( H 1 i , H 1 p ) in the parity-check matri x shown in Fig. 7. (ii) The second ensemble is built b y four random edge permutati ons 3 c − 2 i , 3 c − 2 p , 4 c − 1 i , and 4 c − 1 p . In the root-LDPC parity-check matrix , this is equiv alent to building seperately the four sub matrices H 2 i , H 2 p , H 1 i , and H 1 p . For simplicity reasons, mainl y i n the densi ty e volution (DE) analysis, we adopt the first root-LDPC ens emble as p art of the full-diversity cooperativ e code proposed in the ne x t subsection. B. Rate-compati ble full -diversity LDPC codes The dif ference with [8] is that our code cons truction must take into account the p rotocol of coded coo peration, i.e., the 4 different cases, t o perform well on this channel. Furthermore, the July 30, 2021 DRAFT 16 optimized degree distributions of our code construction will be dif ferent from [8], because of the multi-edge type structure [39] of this code construction. The structure of an L DPC ensemble for cod ed cooperation is deriv ed by joini ng the rate-compatibility property and the ful l-div ersity property . The global parity-check matrix i s obtained by em bedding t he root-LDPC matrix (Fig. 7) into the rate-compatible matrix (Fig. 3). This l eads to an asymmetric code where class 1 i may hav e a higher cod ing gain t han class 2 i . Therefore, we p ropose an extension to the “extending” technique, due to the fac t that we split the information bits ov er two frames, which is a ne w phenomenon. T o get a b alanced st ructure, we replace the zero-padded H 1 by the direct s um of two rate R 1 codes defined by H 1 s and H 1 r as il lustrated in Fig. 8. Thus, t he const ituent code H 1 s protects bit s 1 i and 1 p via e x tra parity bits p ′ 1 . Sim ilarly , in the second frame, extra p arity bit s p ′ 2 are generated from 2 i and 2 p . The bot tom o f the global parity-check matrix simply in cludes the root-LDPC structure, connecting ( 1 i , 1 p ) to ( 2 i , 2 p ). For sim plicity we can assume th at H 1 s and H 1 r belong to the same rate R 1 random LDPC ensemble, defined by the degree di stributions ( λ 1 ( x ) , ρ 1 ( x )) . Hence, if the degree distribution of the root-LDPC is ( λ 2 ( x ) , ρ 2 ( x )) , we refer to the rate-compatib le root-LDPC (RCR-LDPC) as a ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) code. The T anner graphs of a regular (3 , 9 , 3 , 6) LDPC code and an irregular ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) code are shown in Figs. 9 and 10. Since we g uarantee full di versity via a root-LDPC with a fixed rate 1 2 , the gl obal codin g rate of the RCR-LDPC code observed at the dest ination is R c = R 1 2 . As a consequence, the global coding rate R c can be easily var ied throug h R 1 and is upp er l imited by 0 . 5 . 1 1 1 1 1 1 1 1 1i 1p 0 0 1c 2c 3c 4c 0 0 0 Frame 2 0 2i 2p 0 0 0 0 0 0 Frame 1 H 1 p H 1 i p ′ 1 H 1 s H 2 i H 2 p p ′ 2 H 1 r Fig. 8. Parity-check matrix of a RC R-LDPC code for coded cooperation. The upp er coding rate associated to H 1 s and H 1 r is R 1 = 2 3 , the bottom root-LDPC coding rate is 1 2 , and the ov erall coding rate i s R c = R 1 2 = 1 3 . July 30, 2021 DRAFT 17 2i 0000 0000 0000 0000 1111 1111 1111 1111 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1c 2i 3 3 3 1 2 3 3 2 9 6 6 4c 2c 3c 3 3 9 3 1 1i 1p 2p Frame 1 Frame 2 N 6 N 6 N 6 N 6 N 6 N 6 p ′ 2 p ′ 1 N 6 N 6 N 6 N 6 Fig. 9. T anner graph of a regular (3 , 9 , 3 , 6) RCR-LDPC code for coded coope ration. W e see that the av erage bit degree is ¯ d b = 5 and the av erage check degree is ¯ d c = 15 2 which results in R c = 1 − ¯ d b ¯ d c = 1 3 . 2i 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 1c 2i 4c 2c 3c 1 1 1i 1p 2p Frame 1 Frame 2 R 1 N 4 R 1 N 4 R 1 N 4 R 1 N 4 R 1 N 4 R 1 N 4 p ′ 2 p ′ 1 λ 1 ( x ) λ 1 ( x ) ρ 1 ( x ) ρ 2 ( x ) λ 1 ( x ) λ 1 ( x ) ˜ λ 2 ( x ) λ 2 ( x ) λ 2 ( x ) λ 1 ( x ) ˜ λ 2 ( x ) ρ 2 ( x ) ρ 1 ( x ) λ 1 ( x ) (1 − R 1 ) N 2 (1 − R 1 ) N 2 R 1 N 4 R 1 N 4 Fig. 10. T anner graph of an irregular RCR-LDPC code for coded cooperation. The binary elements are split i nto six classes, p ′ 1 and p ′ 2 of each (1 − R 1 ) N 2 bits and 1 i , 1 p , 2 i , and 2 p of each R 1 N 4 bits. The checknodes are cut i nto four classes of R 1 N 4 checks. Due to the ident ity m atrices inside the parity-check matrix, new p olynomials ˜ λ 2 ( x ) app ear in Fig. 10 in the connections 1 i − 4 c and 2 i − 3 c , as illustrated in Fig. 11. July 30, 2021 DRAFT 18 1 bits bits λ ( x ) ˜ λ ( x ) Fig. 11. T ransition from a traditional representation, characterized by an edge distribution polynomial λ ( x ) , towards a representation where one edge per bitnode i s isolated resulting in a ne w de gree distribution ˜ λ ( x ) . Pr oposition 3 In a T anner graph with a left de gr ee dis tribution λ ( x ) , isolati ng one edge per bitnode yields a ne w left de gr ee distribution d escribed by the p olynomial ˜ λ ( x ) : ˜ λ ( x ) = X i ˜ λ i x i − 1 , ˜ λ i − 1 = λ i ( i − 1) /i P j λ j ( j − 1) /j . (18) Pr oof: Let us define T bit, i as the numb er of edges connected to a bi tnode of degree i . Similarly , the number of all edges is denoted T bit . From Section II, we know that λ ( x ) = P d bmax i =2 λ i x i − 1 expresses the left degree distribution, where λ i is the fraction of all edges in the T anner graph, connected to a bi tnode of degre e i . So finally λ i = T bit, i T bit . A similar reasoning can be followed to determi ne ˜ λ i : ˜ λ i − 1 a ) = T bit, i − λ i i T bit T bit − P j λ j j T bit b ) = λ i T bit − λ i i T bit T bit − P j λ j j T bit = λ i − λ i i P j λ j j j − P j λ j j = λ i i ( i − 1) P j λ j j ( j − 1) . a) P j λ j j T bit is equal to t he number of edges that are removed which is equal to the number of bits . b) λ i T bit is equal to the num ber of edges connected to a bi t of degree i . In Section V, we will also use ˜ ρ ( x ) , which is defined sim ilarly as ˜ λ ( x ) . Pr oposition 4 Consider a ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) RCR-LDPC code for coded cooperation July 30, 2021 DRAFT 19 transmitted on a 2-user block-fading cooperative MA C. Then, und er iterative belief pr opagati on decoding, th e RCR-LDPC code has full diversity . Pr oof: Let Λ a i , i = 1 . . . d c − 1 deno te the inpu t log -ratio probabi listic mess ages to a checknode Φ of degree d c . The out put m essage Λ e for belief propagati on is [37] Λ e = 2 th − 1 d c − 1 Y i =1 th  Λ a i 2  ! , where th ( x ) denotes th e hyperbolic-tangent function. Superscripts a and e stand for a pri ori and ex trinsic , respective l y . T o sim plify the proof, we show that the sub optimal min-sum decoder yields a d iv ersi ty order 2 . For a m in-sum decoder , t he outp ut message produced by a checknod e Φ is now Λ e = min ( | Λ a i | ) d c − 1 Y i =1 sign (Λ a i ) . An information bit ϑ of class 1 i of degree d b has Λ 0 = 2 α sr y sr σ 2 where Λ 0 is the log-li kelihood ratio coming from t he likelihood p ( y sd | ϑ ) . It also recei ves d b messages: Λ e 1 ,i , i = 1 . . . d b 1 and Λ e 2 ,i , i = 1 . . . d b 2 , d b = d b 1 + d b 2 , from its neighbourin g checknodes in th e constitu ent codes H 1 s and H 2 respectiv ely . The to tal a po steriori message corresponding to ϑ is Λ = Λ 0 + P d b 1 i =1 Λ e 1 ,i + P d b 2 i =1 Λ e 2 ,i . In [8] it is proven t hat full-diversity is achiev ed if and onl y if Λ behav es as aα 2 1 d + bα 2 2 d , where a, b > 0 . The addition of P d b 1 i =1 Λ e 1 ,i cannot degrade the error probability P e (1 i ) because t he con v olution with the dens ity of messages from H 1 s can only phys ically up grade the result ing density . Thu s, it is sufficient to prove that message Λ 0 + P d b 2 i =1 Λ e 2 ,i exhibits full div ersi ty , i.e., behaves as aα 2 1 d + bα 2 2 d , which is proven in [8]. V . D E N S I T Y E VO L U T I O N O N T H E B L O C K - F A D I N G R E L A Y C H A N N E L Richardson and Urbanke [36], [37] establ ished that, if t he block l ength is large enough, (almost) all codes in an ensembl e of codes 4 beha ve alike, so the d etermination of the average 4 The ensemble of all LDP C-codes that sati sfy the left degree distribution λ ( x ) and right degree distribution ρ ( x ) is considered. The ensemble is equipped with a uniform probab ility distribution. July 30, 2021 DRAFT 20 beha vior is suf ficient to characterize a particular code beha vio r . This ave rage beha vior con ver ges to the cycle-free case if the block length augments and it can be found in a determinis tic way through density ev ol ution (DE). The ev oluti on trees represent the local neighb orhood of a bitnod e in an in finite l ength code whose graph has no cycles, hence incoming messages to ev ery node are independent . A. Interuser channel T o determi ne th e density of m essages propagating in the graph of the constituent code H 1 s , the following notation is u sed: d m sr ( x ) = densit y o f message from a bitnode to a checknode in the m th iteration . µ sr ( x ) = densit y o f the likelihood of the source-relay channel. Let X 1 ∼ p 1 ( x ) and X 2 ∼ p 2 ( x ) be two independent real rando m v ariables. The dens ity functi on of X 1 + X 2 is obtained by con volving t he t wo origi nal densit ies, writt en as p 1 ( x ) ⊗ p 2 ( x ) . The notation p ( x ) ⊗ n denotes the con volution of p ( x ) w ith itself n ti mes. Let X 1 ∼ p 1 ( x ) and X 2 ∼ p 2 ( x ) be two independent real rando m v ariables. The dens ity functi on p ( y ) of the variable Y = 2 th − 1  th  X 1 2  th  X 2 2  , obtain ed through a checknode wit h X 1 and X 2 at the input, i s obtained through the R-con vol ution [37], written as p 1 ( x ) ⊙ p 2 ( x ) . The notation p ( x ) ⊙ n denotes the R-con volution o f p ( x ) wit h i tself n t imes. T o simplify the notations, we use the fol lowing definiti ons: λ ( p ( x )) = X i λ i p ( x ) ⊗ i − 1 , ρ ( p ( x )) = X i ρ i p ( x ) ⊙ i − 1 . In the next subsection we wi ll also use th e foll owing definitions: July 30, 2021 DRAFT 21 ρ ( p ( x ) , t ( x )) = X i  ρ i p ( x ) ⊙ i − 1 ⊙ t ( x )  , λ ∗ ( p ( x )) = λ ( p ( x )) ⊗ ( p ( x )) , ρ ∗ ( p ( x )) = ρ ( p ( x )) ⊙ ( p ( x )) . The first definition is necessary because of the non-linearity of the R-con volution. Therefore, t he first equation is n ot equal to t ( x ) ⊙ ρ ( p ( x )) . The next sub section will also u se the polyn omials ˚ ρ ∗ ( x ) and ˚ λ ∗ ( x ) which are d efined by comb ining the two transformati ons, deno ted by ˚ ( . ) (see introduction) and ( . ) ∗ . Fig. 1 2 illu strates the local neighb orhood of a bi tnode in the cons tituent code H 1 s . 1c 1c bit− node 1c 1p 1i 1p 1i p ′ 1 ρ 1 ( x ) λ 1 ( x ) p ′ 1 ρ 1 ( x ) Fig. 12. Local neighb orhood of a bitnode in the constituent code H 1 s . T his tr ee is used to determine the e volution of density d sr ( x ) of messages from a bitnode to a checknode. The DE equation in the neighbo rhood of the bitnode for a ( λ 1 ( x ) , ρ 1 ( x )) LDPC code [36] is, for all m , d m +1 sr ( x ) = µ sr ( x ) ⊗ λ 1  ρ 1  d m sr ( x )   . (19) The threshold of a code is the minim um SNR at which a codew ord can be decoded perfectly [36]. Com paring the recei ved si gnal-to-noise ratio with thi s threshol d, the relay and the source can determi ne wheth er the interuser transmis sions can be d ecoded su ccessfully and consequently decide what to transmit in the second frame. July 30, 2021 DRAFT 22 B. Overall cooperative MAC The proposed ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) root-LDPC code h as 6 v ariable node types and 4 checknode types. Consequentl y , the ev olution of mess age d ensities under i terativ e decoding has to be described through mul tiple e volution trees. Figs. 13, 15 and 16 show the local neighborhood of a bit node of t he class 1 i . The local neig hborhoods of bit nodes of th e class es 1 p , and p ′ 1 can be deri ved similarly . The lo cal n eighborhood of classes 2 i , 2 p , and p ′ 2 are equiv alent because of code symm etry . T o determ ine the density o f messages, the following not ation is used: a m 1 ( x ) , a m 2 ( x ) = density o f message from 1 i to 1 c and 2 i t o 2 c respectively , at t he m th iteration , f m 1 ( x ) , f m 2 ( x ) = density o f message from 1 i to 3 c and 2 i t o 4 c respectively , at t he m th iteration , g m 1 ( x ) , g m 2 ( x ) = density o f message from 1 i to 4 c and 2 i t o 3 c respectively , at t he m th iteration , k m 1 ( x ) , k m 2 ( x ) = density o f message from 1 p to 1 c and 2 p to 2 c respectively , at the m th iteration , l m 1 ( x ) , l m 2 ( x ) = density o f message from 1 p to 4 c and 2 p to 3 c respectively , at the m th iteration , q m 1 ( x ) , q m 2 ( x ) = density o f message from p ′ 1 to 1 c and p ′ 2 to 2 c respectively in the m th iteration , µ i ( x ) = density of the likelihood of the channel in the i ’th frame . Note that µ 2 ( x ) depends on the s uccess or the failure of t he transmis sions in the first frame. Pr oposition 5 The DE equations in th e neighborhoo d of 1 i for a ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) July 30, 2021 DRAFT 23 00000 00000 00000 00000 11111 11111 11111 11111 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 1 1 1c ˜ ρ 2 ( x ) f 1 p 1 c 1 p p ′ 1 1 i ρ 1 ( x ) f 1 i 1 c f 1 i 4 c f 1 p 4 c 2 i 1 p 1 i 2 i 2 p f 2 i 3 c ˚ ρ 2 ( x ) f 2 p 3 c 1 c 4 c 3 c ˚ λ 2 ( x ) λ 1 ( x ) 1 i f p ′ 1 1 c Fig. 13. Local neighborho od of bitnode 1 i . This tree is used to determine the ev olution of the density of messages 1 i → 1 c . a m +1 1 ( x ) = µ 1 ( x ) ⊗ ˚ λ 2  ˜ ρ 2  f 1 i 4 c g m 1 ( x ) + f 1 p 4 c l m 1 ( x ) , f m 2 ( x )   ⊗ λ 1  ρ 1  f 1 i 1 c a m 1 ( x ) + f 1 p 1 c k m 1 ( x ) + f p ′ 1 1 c q m 1 ( x )   ⊗ ˚ ρ 2  f 2 i 3 c g m 2 ( x ) + f 2 p 3 c l m 2 ( x )  , (20) f m +1 1 ( x ) = µ 1 ( x ) ⊗ ˚ λ ∗ 1  ρ 1  f 1 i 1 c a m 1 ( x ) + f 1 p 1 c k m 1 ( x ) + f p ′ 1 1 c q m 1 ( x )   ⊗ ˚ λ 2  ˜ ρ 2  f 1 i 4 c g m 1 ( x ) + f 1 p 4 c l m 1 ( x ) , f m 2 ( x )   , (21) g m +1 1 ( x ) = µ 1 ( x ) ⊗ ˚ λ ∗ 1  ρ 1  f 1 i 1 c a m 1 ( x ) + f 1 p 1 c k m 1 ( x ) + f p ′ 1 1 c q m 1 ( x )   ⊗ ˜ λ 2  ˜ ρ 2  f 1 i 4 c g m 1 ( x ) + f 1 p 4 c l m 1 ( x ) , f m 2 ( x )   ⊗ ˚ ρ 2  f 2 i 3 c g m 2 ( x ) + f 2 p 3 c l m 2 ( x )  , (22) RCR-LDPC ensemble for coded cooperation, for all m , a r e given in Eqs. (20), (21) a nd (22) July 30, 2021 DRAFT 24 2i 00000 00000 00000 00000 11111 11111 11111 11111 4c 2i 1 1p 1i R 1 N 4 ˜ ρ 2 ( x ) T 1 p T 1 i T R 1 N 4 R 1 N 4 R 1 N 4 Fig. 14. Part of the compact graph representation of t he T anner graph of a root-LDPC for coded cooperation. The number of edges connecting ( 1 i , 1 p ) to 4 c is T . the number of edg es connecting 1 p to 4 c is T 1 p . The number of edge s connecting 1 i to 4 c is T 1 i . wher e f 1 p 4 c = P i ˜ ρ 2 i /i P i λ 2 i /i , (23) f 1 i 4 c = P i ˜ ρ 2 i /i P i ˜ λ 2 i /i , (24) f 1 p 1 c = P i ρ 1 i /i P i λ 1 i /i , (25) f 1 i 1 c = f 1 p 1 c , (26) f p ′ 1 1 c = 1 − f 1 i 1 c − f 1 p 1 c , (27) f 2 i 3 c = f 1 i 4 c , (28) f 2 p 3 c = f 1 p 4 c . (29) Pr oof: E quations (20)-(29) are directly deri ved from the local neighborhood trees. T o obtain the proportionality factors (23)-(29), it i s important to remark that we use the first ensemble of root-LDPC codes, as explained at t he end of Section IV -A. Let T denote the total num ber of edges between the var iable nodes (1 i − 1 p ) and the checknodes 4 c . Fig. 14 il lustrates how f 1 p 4 c and f 1 i 4 c are obtained: July 30, 2021 DRAFT 25 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 1i 3c 1 ˚ λ 2 ( x ) ˚ λ 1 ( x ) ∗ x 1 c 1 p p ′ 1 1 i f 1 p 1 c f p ′ 1 1 c ρ 1 ( x ) f 1 i 1 c 4 c f 1 i 4 c ˜ ρ 2 ( x ) f 1 p 4 c 1 i 1 p 2 i Fig. 15. Local neighborho od of bitnode 1 i . This tree is used to determine the ev olution of the density of messages 1 i → 3 c . T a ) = R 1 N/ 4 P i ˜ ρ 2 i /i (30) T 1 p a ) = R 1 N/ 4 P i λ i /i (31) T 1 i a ) = R 1 N/ 4 P i ˜ λ i /i (32) f 1 p 4 c b ) = T 1 p T (33) f 1 i 4 c b ) = T 1 i T . (34) a) The num ber of checknod es connected to i edges of T is ˜ ρ 2 i i T . A Similar reasoning proves equatio ns (31) and (32). b) The fraction of edges T connecting 1 p to 4 c is f 1 p 4 c . The fraction of edges T connecting 1 i to 4 c is f 1 i 4 c . The DE equations in the neighb orhood of 1 p and p ′ 1 for a ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) RCR- LDPC ensemble for coded cooperation can be deri ved sim ilarly . Proposition 5 can b e used for multipl e pu rposes. First of all, it is used to estimate the asymptotic performance. For a fixed fading set ( α 12 , α 21 , α 1 d , α 2 d ) , it is possib le to determi ne whether the bit error probability con ver ges to 0 or not. W e refer t o the event where the bit error July 30, 2021 DRAFT 26 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111 1i 4c 1 1 ˚ λ 1 ( x ) ∗ x ˜ λ 2 ( x ) 1 c 1 p p ′ 1 1 i f 1 p 1 c f p ′ 1 1 c ρ 1 ( x ) f 1 i 1 c 4 c f 1 i 4 c ˜ ρ 2 ( x ) f 1 p 4 c 1 i 1 p 2 i 2 i 2 p f 2 i 3 c 3 c ˚ ρ 2 ( x ) f 2 p 3 c Fig. 16. Local neighborho od of bitnode 1 i . This tree is used to determine the ev olution of the density of messages 1 i → 4 c . probability does not conv erge to 0 by Densi ty Evolutio n Outage ( D E O ). Thus, at a fixed SNR, it is possible to determine the probabi lity of a Density Evolution Out age P D E O by av eraging over a suffic ient number of fading instances. Now , it is po ssible to write the word error probabilit y P ew of the ensem ble as P ew = P ew | D E O × P D E O + P ew | C ON V × (1 − P D E O ); (35) where P ew | D E O is the word error probabi lity g iv en a DEO ev ent, P ew | D E O = 1 , and P ew | C ON V is the word error p robability when DE con verges. The probabili ty P ew | C ON V depends on the speed of conv ergence of density ev ol ution and the popul ation expansion of the ensemble with the number of decoding it erations [24], so that P D E O ≤ P ew . (36) Thus, the performance estim ated via densi ty e volution is a lower bound for the word error probability . Secondly , Proposition 5 can be used to determi ne t he threshold of C on an ergodic channel. This does n ot directly s erve th e performance analysis for th e BF channel. Howe ver , an analysis in the real space of the fading coef ficients has sho wn that this can be used to increase the coding July 30, 2021 DRAFT 27 gain on a BF relay channel [12]. But the optimization of the coding gain is outs ide the scope of this paper and here w e will only u se Propos ition 5 in the applicatio n o f Eq. (36). V I . N U M E R I C A L R E S U L T S In this section we estimate the asym ptotic performance of RCR-LDPC codes through DE and verify Eq. (36) th rough finite length s imulation s. W e studied d iffe rent scenarios: 1) S cenario 1: • Th e av erage SNR of the independent interuser channels is 5dB higher t han the average SNR on the source-destinati on link. • Th e a verage SNR of the relay-destination lin k is equal t o that o n the source-destination link. • Th e codi ng rate is R c = 1 3 and the coop eration lev el is β = 0 . 5 . For th is scenario, we have tested two cod e ensembles: a regular (3,9,3,6) RCR-LDPC code and an irregular ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) RCR -LDPC code with left and right degree distributions giv en by t he p olynomials λ 1 ( x ) = 0 . 1 989 x + 0 . 23 05 x 2 + 0 . 00 68 x 5 + 0 . 27 74 x 6 +0 . 142 67 x 19 + 0 . 13 35 x 20 + 0 . 01 02 x 21 , ρ 1 ( x ) = x 12 , λ 2 ( x ) = 0 . 2 2767 x + 0 . 2033 3 x 2 + 0 . 21 45 x 5 +0 . 011 048 x 6 + 0 . 34 346 x 19 , ρ 2 ( x ) = 0 . 5 x 7 + 0 . 5 x 8 . 2) S cenario 2: • Th e a verage SNR of the independent interuser channels is 12d B higher than the a verage SNR on the source-destinati on link. • Th e average SNR of th e relay-desti nation link is 4dB hig her th an t he av erage SNR o n the source-destination link. • Th e codi ng rate is R c = 0 . 45 and the cooperation level is β = 0 . 5 . July 30, 2021 DRAFT 28 Here, we imit ated the chann el con ditions used in [20] 5 . T he av erage SNR o f the interuser channels is high wi th respect to the uplink channels , allowing a hi gh coding-rate for t he source- relay channel. W e used an irregular ( λ 1 ( x ) , ρ 1 ( x ) , λ 2 ( x ) , ρ 2 ( x )) RCR-LDPC ensemble wi th left and right degree distributions given by the polyno mials λ 1 ( x ) = 0 . 1 581 x + 0 . 26 48 x 2 + 0 . 11 16 x 5 + 0 . 13 54 x 6 +0 . 330 1 x 14 , ρ 1 ( x ) = x 43 , λ 2 ( x ) = 0 . 2 3441 3 x + 0 . 2 1392 x 2 + 0 . 1237 11 x 5 + 0 . 12 5548 x 6 +0 . 302 41 x 19 , ρ 2 ( x ) = 0 . 7 1875 x 7 + 0 . 28 125 x 8 . The coding rate for the interuser channel subcode H 1 is equal to 0 . 9 . A. Density Evolut ion Out age W e ev alu ated t he asymptotic performance of RCR -LDPC codes by applying DE on the proposed code construction. The probability of Density Evolution Ou tage P D E O , which is a lower bound of the WER, for both scenarios is illus trated in Fig. 17. Note that the outage probability for both rates i s, by coi ncidence, too clos e to dis tinguish. The sim ulated RCR-LDPC code ensemb les all perform with in 1 . 5 dB from the outage probabil ity limit, whereas th e irregular RCR-LDPC code ensembles are within 1 dB from the o utage probability li mit. This distance is respected for many variations of the chann el cond itions, such as other interuser channel conditi ons or uplink channel condi tions. Note that o ur code construction can be appl ied on a full-duplex channel, doubling the overall s pectral ef ficiency . As mention ed before, the coding rate is adjustable by var ying the number o f parit y bits p ′ 1 and p ′ 2 , which is ill ustrated in scenario 2. In thi s work, we mainly focussed on the diversity order achieve d by the code construction. In more recent work [12] we opt imized the degree distribution using the analysis of Section V. Another metho d is based on density ev olu tion wi th a modified Gaussi an approximati on that 5 W e use t he same distribution of the fading and t he same averag e SNR . Howe ver , in [20], the source keeps transmitting in the second frame, so that a direct comparison between our code and the performance of the code proposed in [20] is not possible. July 30, 2021 DRAFT 29 10 -5 10 -4 10 -3 10 -2 10 -1 6 9 12 15 18 21 Probability of Density Evolution Outage E s /N 0 (dB) BPSK Outage Probability Irregular LDPC code R c =0.45 Irregular LDPC Code R c =0.33 Regular LDPC Code R c =0.33 Fig. 17. Density Evolution Outage proba bility of RC R-LDPC codes with coding rates R c = 1 3 (scenario 1) and R c = 0 . 45 (scenario 2) with iterative decoding on a cooperati ve MA C with two users. E s / N 0 is the av erage symbol energ y-to-noise ratio on the source-destina tion link. takes int o account the SNR variation in one recei ved code word as well as th e rate-compat ibility constraint [28]. B. F ini te Length LDPC Codes It is interesting to ev aluate the finite length performance of the propo sed RCR-LDPC codes. Not on ly to app rove the asym ptotic performance, but also to see how to generate an inst ance of the parity-check matrix , given by Fig. 8. Before showing the result s, w e will first discuss the practical generation of th is parity-check matrix . Consider case 1 from Fig. 2. For the decoding process, the d estination will apply the sum - product algorithm on the overall graph including H 1 s , H 1 r , and H 2 . For the encoding process, it is easier to determine the parity bits p ′ 1 , p ′ 2 , and (1 p, 2 p ) with the parity-check matrices H 1 s , H 1 r , and H 2 respectiv ely . As with standard LDPC encoding, these matrices will t hen b e systemi zed to determine the parity bits. An important con straint for the decoding process is the ali gnment in the overall parity-check matrix o f common bit nodes in both constituent codes. This can July 30, 2021 DRAFT 30 be achiev ed by prohibit ing colum n permutat ions du ring the systemization o f H 1 s , H 1 r and H 2 . Except for case 4 , w hich only decodes on H 1 s , the other cases need t he same constraints . 1) Generation of H 1 s and H 1 r : H 1 s and H 1 r are randomly generated satisfying the degree distribution ρ 1 ( x ) for its ro ws and the degree dist ribution λ 1 ( x ) for its columns. A sufficient condition to prohibi t column permutation s during the systemi zation of H 1 s and H 1 r is im posing on H p ′ 1 and H p ′ 2 to be full-rank. H p ′ 1 ( H p ′ 2 respectiv ely) is the most right squ are matrix of H 1 s ( H 1 r respectiv ely). 2) Generation of H 2 : T he generation of H 2 can be split in the generation of H 4 c and H 3 c , where H 3 c ( H 4 c resp.) is the upper part (resp. lower part) of the parit y-check matrix H 2 . H 3 c is th e con catenation of an identity m atrix (permut ation matrix), zeros and a randomly generated matrix ( H 2 i , H 2 p ) . The rows o f ( H 2 i , H 2 p ) satisfy the degree distribution ˜ ρ 2 ( x ) , th e colu mns of the m ost l eft square matrix H 2 i satisfy the degree dist ribution ˜ λ 2 ( x ) and the columns of the m ost right square matrix H 2 p satisfy the degree d istribution λ 2 ( x ) . This is equivalent t o g enerating a random graph with two classes of b itnodes at the left side and one class of checknodes at th e right side of the graph. If n 3 c is the n umber of checknodes at the right side, then a random graph with n 3 c P i ˜ ρ 2 i edges is generated. A fraction P i ˜ ρ 2 i /i P i ˜ λ 2 i /i of th e edges is connected to bit nodes of the class 2 i , whereas a fraction P i ˜ ρ 2 i /i P i λ 2 i /i of the edges i s connected to bit nodes of t he class 2 p . In the end, the i dentity matrix is sim ply added. H 4 c is generated simi larly . For the encoding process, we hav e to sy stemize th is matri x. One s olution is t o switch t he columns associated wi th the 1 i bit node class and the 2 p bit node class. T he most left square matrix of H 2 will then be block-diagonal with H 2 p and H 1 p on its diagonal. Having H 2 p and H 1 p full-rank is consequently a suf ficient condition to exclude col umn permu tations during t he systemization of this matrix. After the generation of (2 p, 1 p ) , all the bits are put i n the required order 1 i − 1 p − 2 i − 2 p by swit ching back the bits o f the classes 1 i and 1 p . 3) WER performance of finit e length LDPC codes: The probability of Density Evolution Outage P D E O is a lo wer bound of the WER of LDPC ensembles without cycles in its T anner graph, wh ich is illustrated in Fig. 18 for irregular codes and in Fig. 19 for the regular code of scenario 1. In the latter , we augment t he bl ocklength to sho w th at the WER of LDPC codes is July 30, 2021 DRAFT 31 independent of the block length. The results shows that inequality (36) is very tig ht in this case. 10 -5 10 -4 10 -3 10 -2 10 -1 10 13 16 19 22 25 Word Error Rate E b /N 0 (dB) DEO Irregular LDPC code R c =0.45 Irregular LDPC code R c =0.45 N=2000 DEO Irregular LDPC Code R c =0.33 Irregular LDPC Code R c =0.33 N=2000 Fig. 18. Comparison of Density E volution Outage (DEO) probability of irregular RC R-LDPC codes wi th coding rates R c = 1 3 (scenario 1) and R c = 0 . 45 (scenario 2) wi th iterative decoding on a cooperati ve MAC with two users. E b / N 0 is the average information bit energ y-to-noise ratio on the source-destination link. 10 -5 10 -4 10 -3 10 -2 10 -1 10 13 16 19 22 25 Word Error Rate E b /N 0 [dB] BPSK Outage Probability Regular Finite Length LDPC Code, N=500 Regular Finite Length LDPC Code, N=5000 DEO Regular LDPC Code Fig. 19. Comparison of RCR-LDPC codes for different block lengths wi th iterative decoding on a cooperativ e MA C for two users, coding rate R c = 1 / 3 . The ratio E b / N 0 is the av erage information bit energy-to-no ise ratio on the source-destination link. C. Comparison with Pre vious W ork As mentioned in the introduction , especially rate-compati ble punctured con volutional codes (RCPC) hav e been used in coded cooperatio n. The main drawback of these codes i s that the WER increases with the logarithm of the block length to the power d where d is the di versity order [6], [7], whereas the WER of near -outage codes s hould be independent of the block length. This can be seen clearly on Fig. 20, where we show the WER o f two rate-compatible non -recursi ve July 30, 2021 DRAFT 32 non-systematic (75 ,53,47) con volutional codes with block l ength 500 and 5000 respectively . W e used the same channel conditi ons and coding rate as in scenario 1. W e also compared with another protocol, Decode and F orward (DF), us ing near -outage LDPC codes for this p rotocol. Despi te t he fact that thi s impl ementation has near -outage performance, the WER performance i s worse than that of our code construction. The reason is that the ou tage probability limit of DF is higher than that o f coded cooperation. 10 -5 10 -4 10 -3 10 -2 10 -1 12 15 18 21 24 27 Word Error Rate Eb/N0 [dB] Regular RCR-LDPC Code, N=5000 Regular LDPC Code for DF, N=5000 Convolutional Code, N=500 Convolutional Code, N=5000 Fig. 20. Comparison of RCR-LDP C codes for cod ed coo peration with other work on a coop erativ e MA C for two users. W e simulated LD PC codes for Decode and Forward under iterative decoding and an implementation of rate-compatible con volutional codes [21]. The ratio E b / N 0 is the av erage information bit energ y-to-noise rati o on the source-destination link. D. Comparison with fully random LDPC codes Finally , a comparison with random LDPC codes is made. In Sec. IV -B, the global parity -check matrix is obtained by embeddin g the root-LDPC matrix (Fig. 7) int o the rate-compati ble matrix (Fig. 3). When using codes t hat are fully random generated, i.e., no special rootchecks are used, then the global parity-check matrix is ob tained by embeddin g a random LDPC matrix into the rate-compatible matrix (Fig. 3 ), see Fig. 21, where H 1 and H 2 are randomly generated. W e sim ulated the same scenarios from the previous subsections, u sing t he s ame code for H 1 and us ing th e degree di stribution of pre vio usly publis hed excellent LDPC codes for the Gaussi an channel for the random generation o f H 2 . July 30, 2021 DRAFT 33 0 i 1p 2p Frame 2 Frame 1 0 H 2 H 1 Fig. 21. Parity-check matri x of a rate-compatible L DPC code obtained by the extension of higher rate codes. S ymbols are split into three classes: i for the information bits, 1 p and 2 p for two cl asses of parity bits. The classes i and 1 p are transmitted by the source in frame 1. Parity bits 2 p are transmitted in the second frame, for example by the relay after successful decod ing of the first frame. Matrix H 1 is used to protect the information bits on the source channel. The parity bits generated by the relay provid e an extra protection through the code H 2 . 1) S cenario 1: λ 2 ( x ) = 0 . 18 9 x + 0 . 177 x 2 + 0 . 13 6 x 4 + 0 . 12 6 x 5 + 0 . 02 7 x 6 +0 . 037 x 11 + 0 . 00 6 x 13 + 0 . 07 6 x 21 + 0 . 22 5 x 28 , ρ 2 ( x ) = 0 . 15 3 x 4 + 0 . 12 5 x 5 + 0 . 04 0 x 6 + 0 . 26 1 x 7 +0 . 149 x 8 + 0 . 17 8 x 9 + 0 . 04 1 x 10 + 0 . 05 5 x 11 , where the coding rate of ( λ 2 ( x ) , ρ 2 ( x )) is R c 2 = 0 . 4 , so that the ov erall coding rate i s R c = 1 / 3 . The comparison wi th a regular (3 , 9 , 3 , 6) RCR-LDPC code is s hown in Fig. 22. 2) S cenario 2: λ 2 ( x ) = 0 . 23 0 x + 0 . 164 x 2 + 0 . 14 9 x 5 + 0 . 12 6 x 6 + 0 . 02 7 x 7 +0 . 037 x 15 + 0 . 00 6 x 16 + 0 . 24 3 x 17 + 0 . 01 8 x 23 , ρ 2 ( x ) = 0 . 15 3 x 5 + 0 . 42 5 x 7 + 0 . 14 9 x 8 + 0 . 27 3 x 9 , where the coding rate of ( λ 2 ( x ) , ρ 2 ( x )) is R c 2 = 9 / 19 , so th at the overa ll coding rate is R c = 0 . 45 . The comparison with an irregular RCR-LDPC code is shown in Fig. 23. In scenario 1, the threshold of ( λ 2 ( x ) , ρ 2 ( x )) is E b / N 0 = 0 . 1 dB which is 0 . 338 d B from t he Shannon limit; and in scenario 2, the threshold of ( λ 2 ( x ) , ρ 2 ( x )) is E b / N 0 = 0 . 4 dB which is July 30, 2021 DRAFT 34 Fig. 22. Comparison of RCR-LDPC codes with rate-compatible random LDPC codes for coded cooperation on a cooperati ve MA C for two users, coding rate R c = 1 / 3 . The ratio E b / N 0 is t he average information bit energy-to-noise ratio on the source-destation link. Fig. 23. Comparison of RCR-LDPC codes with rate-compatible random LDPC codes for coded cooperation on a cooperati ve MA C for two users, coding rate R c = 0 . 45 . The ratio E b / N 0 is the av erage information bit energy-to-noise ratio on the source-destination link. 0 . 33 dB from th e Shannon limit. Despite the excellent th resholds of t he codes in both scenarios, full-dive rsity is no t achieved. From these two examples, it is clear that rootchecks are necessary to hav e full-div ersity . V I I . C O N C L U S I O N W e have s tudied LDPC codes for relay channels in a slowly varying f adi ng en vironment under iterativ e decoding. W e ha ve introduced the ne w family of rate-compatible root-LDPC codes, which com bines the rate-compatibil ity property with the full-div ersit y property for any coding rate R c ≤ R cmax = min ( β , 1 − β ) , where β is the cooperation level. 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Ungerboeck, “Channel coding with multilevel/pha se signals, ” IE EE T rans. Inf. Theory , vol. IT -28, no. 1, pp. 55-67, 1982. [45] M. Y azdani and A.H. Banih ashemi, “Irregu lar rate-compatible L DPC cod es for cap acity-approach ing hybrid-ARQ schemes, ” Cana dian Conf. on Electrical and Computer Engineering , 2004. [46] B. Zhao and M.C. V alenti, “Some new adaptiv e protocols for the wir eless relay channel, ” A llerton confer ence on communication contr ol and computing , vol. 41, no. 3, pp. 1588-1589, 2003. Dieter Duyc k (S’09) r eceived the M.S. degre e in electrical en gineerin g in 2007 fr om the Kath olieke Univ er siteit Leuven (KUL), L euven, Belgium. In 20 06, he spent one year with the Com municatio ns and Electro nics Departme nt, at the E cole Natio nale Sup ´ erieure d es T ´ el ´ ec ommun ications (E NST , T elecom ParisT ec h), Paris, France. In 2007, h e started h is Ph.D. research at the Departm ent of T elecommu nications and Inf ormation Processing ( TELIN), Ghen t University , Gen t, Belgium. From Oct. 20 07 until p resent, he c onduc ted his Ph.D. research. He has held visiting appo intments with Ecole Nationale Sup ´ er ieure des T ´ el ´ ecommu nications (ENST), Paris, France; and T exas A& M Un iv ersity at Qatar, Do ha, Qatar . His research inter ests are in comm unication theory , inform ation theory , channel cod ing, joint ne twork-chann el coding, digital m odulatio n and space-time coding. M. Sc. Dieter Duy ck r eceived the first Y oung Researcher A ward for [13] awarded by the A ward Committee, formed by all Advisory Board Me mbers of the European Newcom++ ( Network o f Excellence in W ireless COMmu - nications). He also received t he b est studen t paper award at the IEEE Symp osium on Com municatio ns and V eh icular T ech nolog y in the Benelux (SCVT) in 2010. Joseph Jean Boutros (M’94 , SM’09 ) received the M.S. de g ree in electrical engineer ing in 1992 and the Ph.D. degree in 19 96, both f rom Eco le Nationale Sup ´ er ieure des T ´ el ´ ecomm unications ( ENST , T eleco m ParisT ech) , Paris, France. July 30, 2021 DRAFT 38 From 19 96 to 200 6, he was with the Communicatio ns and Electronics Departm ent, ENST , as an Associate Professor . He was also a member of th e research unit UMR-514 1 of the French National Scientific Research Cente r (CNRS). In 2007, he join ed T exas A& M Uni versity at Qatar (T AMUQ) as a full Pro fessor in the electr ical e ngineerin g progr am. He h as been a scientific con sultant for Alcatel Espace, Philips Research , and Motorola Semicondu ctors, and was a member of the Dig ital Signal Proce ssing team of Ju niper Networks Cable. His fields of interest are codes on graphs, iter ativ e decodin g, joint source- channel codin g, space -time coding, and lattice sphere packings. Marc Moeneclaey (M’93 , SM’9 9, F’02) received the diplom a of electr ical eng ineering and th e Ph.D. degree in electrical engineering fr om Ghent Uni versity , Gent, Belgium, in 1978 and 198 3, resp ectiv e ly . He is Professor at the Departmen t of T elecommun ications and Inf ormation Pro cessing ( TELIN), Gent University . His main research interests ar e in statistical commun ication theory , (iterative) estimation an detection , carrier and symbo l synchroniz ation, band width-efficient mod ulation an d cod ing, spread-spectr um, satellite a nd mobile commun ication. He is the author of mor e than 400 scientific pap ers in internation al jour nals and conferen ce proceed ings. T o gether with Prof. H. Meyr (R WTH Aachen) an d D r . S. Fech tel (Siemens AG), he co- authors the book D igital co mmunicatio n receivers Sy nchron ization, channel e stimation, and signal pr ocessing. (J. W iley , 1 998) . He is co -recipien t of the Mannesmann I nnovations Prize 20 00. During th e period 1992-1 994, he was E ditor fo r Sy nchron ization, for the IEEE T ran sactions o n Co mmunicatio ns. He served as co-guest ed itor f or special issues of the W ireless Personal Commun ications Jou rnal ( on E qualization and Synch ronization in Wi reless Com munication s) and the IEEE Jou rnal on Selected Areas in Communication s (on Signal Synchronizatio n in Digital Transmission Systems) in 199 8 and 2001 , respectively . July 30, 2021 DRAFT