II. SYSTEM MODEL. A. Link and path model

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1 HARQ I. INTRODUCTION ARQ (automatic repeat-requet i a link layer protocol ued for packet error detection and retranmiion. Errordetection bit (uch a CRC bit are attached and tranmitted along with the meage packet. When the received packet i detected to be erroneou, a negative acknowledgment (NACK i generated and ent back. The meage packet will then be retranmitted. HARQ (hybrid automatic repeat-requet i a variation of ARQ and ha two type. In Type- HARQ, FEC (forward-error correction bit are additionally added with error-detection bit. Retranmiion i requeted only when the received packet i corrupted and cannot be corrected, detected to be erroneou. Type- HARQ reduce the number of retranmiion when the channel condition i poor. A fixed through-put offet alway exit due to the fixed FEC redundancy. Type-2 HARQ i propoed a a cheme to relax the fixed redundancy by introducing redundancy incrementally. In Type-2 HARQ, only data and error-detection bit are ent in the firt tranmiion attempt. When the received packet i detected to be erroneou, additional FEC parity bit are ent. They will be combined with the corrupted received packet and to be corrected. When it i till in error, additional FEC parity bit are ent again. The proce will continue until the received packet i corrected or all the FEC parity bit are tranmitted. Type-2 HARQ take advantage of good channel condition through generating little redundancy and work a well a Type- HARQ when the channel condition i poor. Note that Type-2 HARQ will have longer decoding delay compared with Tyep- HARQ due to ucceively FEC parity bit tranmiion. Alo, all the above ARQ cheme have bi-directional link aumption. In typical wirele ad hoc network, due to aymmetric tranmiion power and different interference level at receiver, unidirectional link may exit. In cognitive radio network (CRN, unidirectional link become more common a avoiding interference with primary uer. A tranmiion opportunity may exit in one direction but not the other. The link level ARQ cheme baed on feed-back channel may not be uitable in CRN. Note that communication between ource and detination i an end-to-end activity. We want to provide a HARQ coding cheme at eion level (end-to-end rather than at link level. A forward (backward path from ource (detination to detination (ource conit of unidirectional and/or bi-directional link. Within a path, link-by-link fully decodeand-forward i not poible due to the unidirectional characteritic of a link and amplify-and-forward i ued intead. Alo, there may exit many forward and backward path. We take the advantage of patial diverity by uing different path between ource and detination. So, the idea of our propoed cheme i a follow. We generate a coded packet conited of data bit, errordetection bit and FEC bit from a meage packet at the ource and divide the coded packet into many coded ub-packet. Then, they are ent over different forward path exploiting patial diverity. Each intermediate node

2 2 within a path amplifie and forward the coded ub-packet to next hop. Decoding i only performed at detination by combining coded ub-packet gathering from different path. A eion level ACK (NACK i only generated by the detination paing through backward path to the ource provided that the origin meage can (cannot be uccefully recovered. In ection II-A, link and path model are preented, epecially, the relationhip between the ignal to noie ratio (SNR of a path and the SNR of it compoed link i dicued. It will be ued for performance analyi of the coding cheme in ection II-B. In ection III, we will dicu how to decide the ratio of the coded packet allocated to each path. It i formulated a an optimization problem. In ection IV, the impact of network delay i addreed. In ection V, a rate adaption cheme i introduced. Note that traditional Type-2 HARQ a a rate adaption cheme generalize Type- HARQ. Incremental redundancy i ent ucceively a needed. Decoding delay will be everal link delay. It will be tranformed into everal end-to-end path delay in our cae if we imply imitate the concept of Type-2 HARQ. Such long decoding delay may not be uitable for ome application. Our propoed rate adaption cheme work in a backward ene. All redundancy i tranmitted at firt attempt, next time the rate will change according to the information piggybacked along the acknowledge packet of the firt attempt. Section VI i the imulation reult. A. Link and path model II. SYSTEM MODEL We aume that there exit L link-dijoint path between the ource and the detination a hown in figure. Each path i, i L, ha M i intermediate node. Each link between a pair of node i modeled a an independent low flat Rayleigh fading channel. The received ignal at node R i,j i y i,j = h i,j x i,j + z i,j. Channel gain h i,j i Rayleigh ditributed. x i,j i the ignal generated from the previou hop R i,j. z i,j i independent zero-mean additive white Gauian noie with variance N 0 /2. Each intermittent node amplifie and forward the received ignal ubject to the ame ignal energy contraint E b. The ignal generated by node R i,j will be x i,j = α i,j y i,j, α 2 i,j = E b h 2 i,j E b + N 0 ( α i,j i the amplifying coefficient. The per hop received SNR i defined a γ i,j = h 2 i,j E b/n 0. It i exponential ditributed with mean γ i,j. The equivalent end-to-end path SNR γ eqi of path i i [2], M i ( γ eqi = +, i L (2 γ i,j At high SNR, (2 reduce to j= M i ( γ eqi =, i L (3 j= γ i,j It i more tractable and correpond to the amplifying coefficient α i,j = /h i,j. The above equation ha related the path SNR with the SNR of it compoed link.

3 3 Fig.. Netwoek topology B. Coding cheme We aume the rate of the FEC code i R. Meage packet ize i K bit. Coded packet ize i N = K/R bit. We divide the coded packet into L coded ub-packet. They are tranmitted over L different path. The L coded ub-packet are received, combined and decoded at the detination. The pairwie error probability (PEP between two codeword can be evaluated a ( 2d P (d γ eq = Q γ eq + 2d 2 γ eq d L γ eql γ eq = (γ eq, γ eq2,..., γ eql. d i the Hamming ditance between the tranmitted codeword and the codeword obtained after decoding. d i i the portion of Hamming ditance contributed from the i th coded ub-packet tranmitted through path i, i L, and d = d + d d L. The unconditional pairwie error probability can be evaluated a P (d = P (d γ eq f(γ eq dγ eq (5 γ eq f(γ eq = f(γ eq...f(γ eql i the probability denity function of γ eq. In high SNR, uing the parameterization method introduced in [3], (5 can be evaluated a P (d L (2i L2 L+ L M i d i γ j= i,j (4 (6 We can ee that diverity gain of order L i achieved. Alo, note that the denominator of (6 ha a term L d i. The coding gain i maximized when d i d/l, i.e., the total Hamming ditance i ditributed evenly among each coded ub-packet. The reult can be hown in an alternative way by noticing that in high SNR, the performance of the weaket link dominate the path, γ eqi γ bi min (γ i,, γ i,2,..., γ i,mi, i L (7

4 4 γ bi i exponential ditributed with mean γ bi = γ i, + γ i, , i L (8 γ i,mi Uing the moment generating function method, (5 can be implified and bounded a P (d = π ( 2 + d γ b...( π 0 in 2 + d Lγ bl θ in 2 dθ θ ( ( d γ b + d L γ bl < L M i (9 2 d i It i the ame a (6 with a caling factor. γ j= i,j For convolutional code, the bit error rate and packet error rate can be bounded a BER min γ eq 2, N K wa w,d P (d γ eq f(γ eq dγ eq (0 k P ER min γ eq d=d f w=, N d=d f w= K a w,d P (d γ eq B f(γ eq dγ eq ( where k i the number of information bit per trelli tage. d f i the free ditance of the code. a w,d i the number of codeword correponding to input weight w and output weight d. B i the number of trelli tage in the codeword. III. CODED PACKET DIVISION PROBLEM In thi ection, we want to dicu how to decide the ratio of the coded packet allocated to each path, in other word, the ize ditribution of coded ub-packet. Our objective i to chooe a diviion trategy o a to minimize the outage probability given a fixed ource information rate. The trategy we will chooe alo depend on how much ide information (path tatitic available at the ource. Firt, we conider the cae that the ource doe not know any information about the path tatitic. A reaonable trategy i that the coded packet evenly divided into the L path. In other word, the ource ue different path in an equal time-haring fahion. log 2 ( + γ eqi, i L i the end-to-end capacity of path i. The ource i aumed to have information rate R. In the high SNR regime, the outage probability P out i ( L P out = P L log 2( + γ eqi < R = P log 2 ( + γ eqi < LR L P {log 2 ( + γ eqi < R} = P (log 2 ( + γ eqi < R = In the low SNR regime, it become ( L = P P out L P ( γ eqi < 2 R L ( 2 R γ bi ( L log L 2( + γ eqi < R P γ eqi < LR log 2 e (2 (3

5 5 Next, we conider the cae that each ource ha ide information of mean tatitic of path between itelf and it detination. w i i the ratio of a coded packet of admitted to it ith path (the relative ize of it coded ub-packet, i w i =. Here, we preent thi problem in a network with multiple ource-detination pair uing the network utility maximization framework [4]. Each ource ha fixed information rate R and utility U (ρ which i a concave function in ρ. ρ i the reliability of and it equal P out. w i we want to find uch that the um utility of ource i maximized. are the deciion variable The network tructure i decribed a follow. It ha E link and N ource-detination pair. Each ource ha L path between itelf and it detination. (We do not aume link-dijoint path. A matrix Hei i defined for each ource. H ei = when path i of ource ue link e, otherwie it equal 0, N, i L, e E. Before cating the optimization problem, we addre that even in the high/low SNR regime, the outage probability P out i not jointly convex in w i, i L. In the high SNR regime, P out In the low SNR regime, P out = P = P The optimization problem i, wi log 2 ( + γeq i < R P wi log 2 ( + γeq i < R P maximize ubject to It can be olved only numerically. P out U ( P out = P (γeq i w i < 2 R wi γeq i < R log 2 e wi log 2 ( + γeq i < R, Heiw i, e i wi =, i 0 w i,, i (4 (5 N, i L, e E (6 IV. NETWORK DELAY In thi ection, we want to analyi the impact of network delay. We model each multi-hop path a an M/M/ tandem network. Suppoe there exit an information flow with Poion arrival rate λ between the ource and the detination. The flow i divided into L ub-flow with Poion rate λ i = w i λ, i L. (w i can be choen a decribed in ection III. Each node R i,j ha own background traffic with Poion arrival rate b i,j and the proceing

6 6 time i exponential ditributed with mean /µ i,j. The probability denity function of packet delay (queueing time plu proceing time for R i,j i f wi,j (t = µ i,j ( ρ i,j e µi,j( ρi,jt (7 where ρ i,j = (λ i + j k= b i,k/µ i,j. The probability denity function of end-to-end packet delay for path i i the M i -fold convolution of f wi,j (t, j =,..., M i f wi (t = f wi,+...+w i,mi (t = [ Mi µ i,l ( ρ i,l l= ] Mi j= e µi,j( ρi,jt Mi k=,k j (µ i,k( ρ i,k µ i,j ( ρ i,j The probability that the end-to-end packet delay of path i i maller than t can be computed a P i (t P (w i t = t 0 f w i (xdx. If there ha an end-to-end packet delay threhold T th, i.e., the packet are conidered to be lo when the end-to-end delay i greater than the threhold. For each path i, it happen with probability P i (T th. The performance degradation due to network delay can be characterized by the expected number of received coded ub-packet. It change form L to L P i(t th. (8 V. CODING SCHEME WITH RATE ADAPTION A decribed in introduction, if we follow the deign of conventional Type-2 HARQ for rate adaptation, we may face a long decoding delay. It i becaue that we will forward one out of L coded ub-packet at a time and wait an end-to-end round trip time to ee if decoding ucceed. Then we decide whether we will forward the next coded ub-packet. The decoding delay will be erval end-to-end round trip time until decoding ucceed. There exit a trade-off between decoding delay and ability of fine adaption of code rate. Here we propoe a two phae rate adaption cheme. In the following, the coded packet i aumed to be evenly divided into coded ub-packet. In probe phae, the ource ha no idea of the condition of path between itelf and the detination. Thi may be due to the firt attempt of tranmiion or after abrupt change of condition of path indicated by previou decoding failure. The ource will tranmit all L coded ub-packet in thi phae. (I.e., it ue the lowet code rate to probe the path. Thi coare rate etting will lead to high probability of ucceful decoding and thu low decoding delay. At the detination, we firt conider the cae that the receiver receive and combine all the L coded ub-packet and decoding ucceed. We want to inference how many coded ub-packet actually needed facilitating the ucceful decoding.we firt try to combine and decode the L coded ub-packet coming from path having highet path SNR. If decoding till ucceed, we throw away the leat favor one and combine the L 2 coded ub-packet having highet equivalent path SNR. Thi proce continue until we fail to decode at combining M bet coded ubpacket. At thi time, we know that combining the M bet coded ub-packet i jut enough (It correpond to code rate R L M. and the remaining L M coded ub-packet are not neceary. Note that each relay along a path need to include the channel ide information γ i,j in it header to facilitate calculation of equivalent path SNR at the detination a in (2. The header of a coded ub-packet i aumed protected by a eparated powerful low rate code and can be recovered individually. The detination will generate an acknowledgement of ucce

7 7 decoding to the ource attaching the information of the combining ratio L M and the path travering by the M bet coded ub-packet. The ource can ue thi information to adapt the code rate and chooe the appropriate path for ucceive packet tranmiion. I.e., the ource will end M out of L coded ub-packet through the reporting M path. Thi come the beginning of ettling phae. In ettling phae, we continue trying to reduce the number of coded ub-packet tranmitted at each ucceive tranmiion. The ettling phae end when decoding failure occur at the detination (due to abrupt change of condition of path. A NACK i generated at the detination requeting tranmiion of all remaining coded ub-packet. After that, all (L received coded ub-packet are combined to decode. We are now in probe phae again. In thi rate adaptive cheme, in a looe ene, there are regeneration point correponding to the detination having all L coded ub-packet, i.e., beginning of probe phae. The regenerative cycle conit of the interlacing probe phae and ettling phae. We aume that the regenerative cycle ha average period W in unit ene-to-end round trip time (RTT. The two phae interlace with each other of length and W in unit RTT. Note that, if condition of path change abruptly within one round trip time, we are alway in probing phae (W =, i.e., we alway ue the lowet code rate correponding to total L coded ub-packet tranmiion. A. Performance analyi We ue the approximation (7 in high SNR regime in the following performance analyi. In the probing phae, the PEP can be derived a (9. In the ettling phae, the PEP change due to now chooing the M bet path from the L path rather than M independent path from the L path. The joint probability denity function of path tatitic will become f ( γ b(,..., γ b(m, the upper M order tatitic of γb,..., γ bl. Here γ b(... γ b(l. In the following derivation, we aume γ b,.., γ bl have equal mean γ b for implicity, the PEP i P M (d γ b(,..., γ b(m = Q ( 2d γ b( d M γ b(m (9 with change of variable, x i = γ b(i γ b(i+, i =,..., L x L = γ b(l or, γ b(i = x i x L, i =,..., L (20 x i, i =,..., L, are independent and ditributed a exponential with mean γ b i. Subtitute (20 in (9, it become P M (d γ b(,..., γ b(m ( = Q 2d x + 2(d + d 2 x (d + d d M x M (d + d d M x L (2

8 8 The unconditional PEP can be computed a P M (d =... P M (d γ b(,..., γ b(m f(γ b(,..., γ b(m dγ b(...dγ b(m =... P M (d γ b(,..., γ b(m f(x...f(x L dx...dx L = π ( 2 + d γ b π 0 in 2 θ ( ( 2 + d γ b ( + (d + d 2 γ b 2 in 2 θ + (d + d 2 γ b 2 We can oberve that diverity gain of order L i till achieved.... ( (... + (d + d d M γ ( b M in (d + d d M γ b L θ in 2 dθ θ ( + (d + d d M γ... b M + (d + d d M γ (22 b L For convolutional code, the packet error probability after combining M coded ub-packet can be bounded a B N K P PER (M... min, a w,d P M (d γ b(,..., γ b(m f(γ b(,..., γ b(m dγ b(,..., dγ b(m d=d f w= (23 B. Throughput analyi We define the normalized throughput a T = K l avg (24 where l avg i the average number of total tranmitted bit per K information bit. In the probe phae of the rate adaption cheme, we have l p avg = N. In the ettling phae, the throughput i the average number of code ubpacket ued to facilitate ucceful decoding. P PER (i i the packet error probability combining i coded ub-packet defined in (23 So, we have l avg L j= j jn L ( P PER(j P PER (i + N L P PER (i (25 T W lp avg + W W l avg (26 VI. SIMULATION We aume that each packet ha 20 information bit. It i coded with convolutional code with rate /2 in figure 2, otherwie with rate /3, and the contraint length i 7. Modulation cheme i BPSK. Link are low Rayleigh fading channel. In figure 2, a packet i encoded, divided into L coded ub-packet and tranmitted through L different path, L = 2, 4, 6, 8, 0. We can ee that a L increae, packet error rate decreae due to higher path diverity. In the imulation, the number of relay of a path i uniformly ditributed between and 3. γ i the average SNR of a link and i ame for all link. In figure 3 to 5, we can ee that when condition of path become better, we only need to combine ubet of L = 0 received coded ub-packet o a to maintain a fixed packet error rate (0 3. In figure

9 9 Fig. 2. Packet error rate with L = 2, 4, 6, 8, 0 correpond to different degree of path diverity. Fig. 3. Here we have L = 0, we only need to combine ubet of received coded ub-packet out of L to maintain fixed packet error rate 0 3 when path equivalent SNR Increae. 4, the exact number of coded ub-packet needed out of L = 0 i hown. In figure 5, throughput increae due to reult of rate adaption.

10 0 Fig. 4. The number of coded ub-packet out of L = 0 needed with regard to figure 3. Fig. 5. Throughput increae due to rate adaption with regard to figure 4. REFERENCES [] Hunter, T.E.; Noratinia, A., Diverity through coded cooperation, Wirele Communication, IEEE Tranaction on, vol.5, no.2, pp , Feb. 2006

11 [2] Hana, M.O.; Alouini, M.-S., End-to-end performance of tranmiion ytem with relay over Rayleigh-fading channel, Wirele Communication, IEEE Tranaction on, vol.2, no.6, pp. 26-3, Nov [3] Ribeiro, A.; Xiaodong Cai; Giannaki, G.B., Symbol error probabilitie for general Cooperative link, Wirele Communication, IEEE Tranaction on, vol.4, no.3, pp , May 2005 [4] Mung Chiang; Low, S.H.; Calderbank, A.R.; Doyle, J.C., Layering a Optimization Decompoition: A Mathematical Theory of Network Architecture, Proceeding of the IEEE, vol.95, no., pp , Jan. 2007

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