Mixed Noisy Network Coding

Transcription

1 Mixed Noisy Networ Coding Arash Behboodi Dept of elecommunication Systems, U Berlin Berlin, Germany behboodi@tntu-berlinde Pablo Piantanida Dept of elecommunications, SUPELEC 9119 Gif-sur-Yvette, France pablopiantanida@supelecfr Abstract Noisy Networ Coding NNC was recently introduced, generalizing Compress-and-Forward CF to multiterminal networs In this paper, we present Mixed Noisy Networ Coding scheme as the generalization of NNC where part of the nodes are allowed to select Decode-and-Forward DF as their cooperative strategy while all nodes without exception transmit the compressed version of their observations he compressed version of relays is exploited at each destination to decode the intended message It is shown that Mixed NNC scheme performs potentially better than NNC In particular, for AWGN networs it achieves a tighter constant gap with respect to the cut-set bound, provided that DF relays are chosen properly I INRODUCION he relay channel is conceived as the building bloc of multiterminal cooperative networs he main cooperative strategies for the single relay channel were first developed by Cover and El Gamal in 1979 [1, the attempts were made recently to develop similar cooperative strategies for multiterminal networs Various configurations have been studied during years, including multiple access relay channel and broadcast relay channel [ Kramer et al developed an inner bound for a point-to-point general networ using Decode-and-Forward DF which achieves the capacity of the degraded multicast networ [3 heir scheme assumes certain hierarchy amount the relays that must be available at the source to generate the code Whereas it is neither preferable nor liely that a certain hierarchy be always preserved during the communication given the heterogeneous nature of future wireless networs he authors also present a generalized version of Compress-and- Forward CF for multiple relay networs On the other hand, El Gamal et al developed an alternative version of CF in [4, later baptized as Noisy Networ Coding Although the scheme performs as well as conventional CF in single relay channel, it is strictly better when generalizing to multiple relay channel Lim et al in [5 proposed the Noisy Networ Coding NNC scheme for the general multicast networ which includes bounds in [6, [7 hese bounds are shown to be tight only for some specific cases lie deterministic and erasure networ but rarely for wireless models herefore, the authors in [5 showed that NNC inner bound achieves the cut-set bound within a constant gap that is independent of channel gains NNC scheme utilizes repetitive encoding, meaning that the same long message is transmitted in all communication blocs he idea of sending different short messages in each bloc without performance loss was first used in [8 and formalized in [9 to then referred to Short Message Noisy Networ Coding SNNC SNNC can be developed in different ways while in general the proof requires bacward decoding For instance, the transmission in [9 is done in B + L blocs but to achieve the largest rate both B and L should tend to infinity On the other hand in [10, L can be finite during which the last compression index is decoded NNC and SNNC schemes have since then been exploited in various scenarios as in [9 [11 amount many others SNNC scheme opens up the possibility of combining DF strategy with NNC his path has been taen in previous contributions he authors in [10 proposed Mixed Noisy Networ Coding MNNC where relays are divided into two sets, relays in the first set use NNC while those in the second one use DF Nevertheless, DF relays cannot help each other in decoding source messages and the destination can only benefit from the compression index of part of the relays since DF relays do not employ NNC strategy his issue introduces considerable difficulty when attempting to compare this scheme with the original NNC In this paper, we develop an achievable rate that generalizes NNC to the case of mixed coding strategy, where DF relays exploit the help of both NNC and DF relays Mainly DF relays superimpose the compressed version of their observation on the source message ransmission is done in B + L blocs, where relays retransmit the compression index of bloc B+ in the last L blocs, and bacward decoding is used at the destination We derive a novel gap expression for MNNC for which it is shown that whenever DF is used, the gap respect to the cut-set bound cannot be made independent constant of the channel gains However, provided that DF relays are chosen properly, the constant Gap is improved directly affected by the number of active DF relays his paper is organized as follows Section II presents definitions while Section III introduces MNNC theorem, and the setch of proof is relegated to Section IV Finally, the constant gap for AWGN networs is presented in Section V II MAIN DEFINIIONS he unicast memoryless multiple relay networ consists of a source X, a destination Y 1 and relays N denoted by pairs X i, Z i with i N = 1,, N, and transition probability W = P Y1Z 1Z N XX 1X N : X X 1 X N Y 1 Z 1 Z N We shall denote X S = X i : i S for any S N

2 X DF relays CF relays w b l,b 1 X j, Z j Xi, Z i X, Z l j,b 1 l j,b 1, w b 1 l i,b 1 l,b 1, w b 1 Fig 1 Mixed Noisy Networ Coding MNNC Definition 1 code and achievability: A code-cn, M n, ɛ n for the unicast multiple relay channels consists of: An encoder mapping ϕ : M n X n, A decoder mapping φ : Y1 n M n, n A sequence of relay functions f i : Z i 1 X i=1 for N and the average error probability is defined as ɛ n Pr φy n 1 W A rate R is said to be achievable for the unicast multiple relay networ if there exists a code-cn, M n, ɛ n satisfying lim inf n 1 n log M n R, and lim sup ɛ n = 0 n he supremum of all achievable rates is the capacity of the unicast multiple relay networ III MIXED NOISY NEWORK CODING Consider the unicast multiple relay networ described in Fig 1 Nodes are divided in two groups, V and V c, where: 1 Relay nodes in set V c employ partly DF and partly CF schemes as their cooperative strategy eg nodes j, Relay nodes in set V use only CF eg lie i As it will be clarified later, the destination selects for decoding only the help of a subset of nodes N ransmission is done in B+L blocs where DF relays in each bloc b B+ forward the source message of the bloc b, as it is shown for nodes j, in Fig 1 Notice that this is slightly different from conventional DF scheme where the relay forwards the message of the previous bloc he reason is that each relay in the networ is transmitting the compressed version of its observation which introduces the possibility of full cooperation amount the networ nodes Particularly, DF relays can use the compression index of other relays to decode the source message he compression index of the bloc b is transmitted only in bloc b + 1, so DF relays have to wait until the end of bloc b + 1 to decode the compression index and the message of the bloc b herefore, they can forward the message of bloc b only after bloc b+1 Similarly to the destination, the -th DF relay in V c exploits only the help of a selected subset Y 1 of relays he following theorem provides the achievable rate corresponding to this communication strategy heorem 1 Mixed Noisy Networ Coding: For the multiple relay networ, the following rate is achievable: R max P P max V N min where max Υ N min V c R S IXX S ; ẐS cy 1 X S cq min R S, V S max Υ N min R S S 1 IẐS; Z S XX Ẑ S cy 1 Q R S IX; Ẑ Z V X X Q + IX S ; Z V X X S cq IẐS; Z S V X X Ẑ S cz Q 3 where S c S in and S c S in 3 Moreover, N, and N, and V c = N V Also Υ N and Υ N are defined as follows: Υ N N : S, Q S 0, 4 Υ N N : S, Q S 0 5 with Q S and Q S defined by Q S IX S ; ẐS cy 1 V XX S cq IẐS; Z S V XX Ẑ S cy 1 Q, Q S IX S; Z V X X S cq IẐS; Z S V XX X Ẑ S cz Q he set of all admissible distributions P is given by P = P QV XXN Z N Ẑ N Y 1 = P Q P V Q P X V Q P Y1Z N XX N Q P Xj V QPẐj V X jz jq P Xj QPẐj X jz jq j V c j V he proof of this theorem is provided in next section It can be shown using the same technique in [9 that the optimization of R S in 1 can be performed over N instead of Υ N heorem 1 reduces to SNNC by choosing V = N as in [8, [9 which is also equivalent to NNC [5 On the other hand, heorem 1 differs from the cooperative mixed NNC in [10, where the relays who use DF are not allowed to use CF so cannot help each other when decoding messages herefore, heorem 1 generalizes and includes all previous NNC schemes and it provides a potentially large achievable rate It is worth mentioning that for the single degraded relay channel, it achieves the capacity which is not the case in NNC Note that because DF relays has to use forward decoding, R S is smaller compared to the situation where bacward decoding is employed he reason for this, as stated in [8, is that the gain in NNC is achieved by delaying the decoding until the last bloc But postponing decoding to the last bloc is not possible in DF relays which require decoding to reencode in each bloc, introducing the rate loss we discussed

3 IV OULINE OF HE PROOF OF HEOREM 1 he relays in the networ are divided into two sets V and V c = N V hose relays in V are using CF while the others are using both CF and DF he DF relays transmit also the compressed version of their observation, superimposed over their DF code he DF relay V c decodes the source message of bloc i using the compressed version of observation of other relays Because the relays transmit the compression index of the bloc i in the bloc i + 1, the -th relay has to wait until the end of bloc i + 1 to decode it and therefore DF relays has to wait until the bloc i+ to forward the source message of the i-th bloc Moreover the relay exploits only the compression index of relays in N Similarly the destination decodes the compression index of the relays in N he reason for this choice is that the performance may be degraded if the compression index of all relays is used A set of relays contribute to augmenting the rate only if certain condition is satisfied Each and consist of DF and CF relays For simplicity, we adopt the following notation: DF Code generation: V c, CF V, DF V c, CF V 1 Randomly and independently generate nr sequences v drawn iid from PV nv = n j=1 P V v j Index them as vr with index r [ 1, nr For each V c and each vr, randomly and independently generate n ˆR sequences x drawn iid from n PX n V x vr = P X V x,j v j r j=1 Index them as x r, r, where r [1, n ˆR for ˆR = IZ ; Ẑ X, V + ɛ 3 For each V, randomly and independently generate n ˆR sequences x drawn iid from PX n x = n j=1 P X x,j Index them as x r, where r [1, n ˆR for ˆR = IZ ; Ẑ X + ɛ 4 For each vr, randomly and conditionally independently generate nr sequences x drawn iid from PX V n x vr = n j=1 P X V x j v j Index them as xr, w, where w [ 1, nr 5 For each V c and each x r, r, randomly and conditionally independently generate n ˆR sequences ẑ each with probability P ṋ ẑ Z X x r, r = n j=1 P Ẑ X ẑ,j x,j r, r Index them as ẑ r, r, ŝ, where ŝ [1, n ˆR 6 For each V and each x r, randomly and conditionally independently generate n ˆR sequences ẑ each with probability P ṋ Z X ẑ x r = n j=1 P Ẑ X ẑ,j x,j r Index them as ẑ r, ŝ, where ŝ [1, n ˆR Encoding part: 1 In every bloc i = [1 : B, the source sends w i using x w i, w i w0 = w 1 = 1 Moreover, for blocs i = [B+1 : B+L, the source sends the dummy message w i = 1 nown to all users For every bloc i = [1 : B + L, and each V c, the relay nows w i by assumption and w 0 = w 1 = 1, so it pics up v w i For each i = [1 : B +, the relay after receiving z i, searches for at least one index l,i with l,0 = 1 such that vwi,x w i, l,i 1, z i, ẑ w i, l,i 1, l,i A n ɛ [V X Z Ẑ he probability of finding such l,i goes to one as n goes to infinity due to the choice of ˆR 3 For i = [1 : B + and V c, relay nows from the previous bloc l,i 1 and w i and it sends x w i, l,i 1 Moreover, relay repeats l,b+ for i = [B + 3 : B + L, ie for L blocs 4 For each i = [1 : B +, each V, the relay after receiving z i, searches for at least one index l,i with l,0 = 1 such that x l,i 1, z i, ẑ l,i 1, l,i A n ɛ [X Z Ẑ he probability of finding such l,i goes to one as n goes to infinity due to the choice of ˆR 5 For i = [1 : B + and V, relay nows from the previous bloc l,i 1 and it sends x l,i 1 Moreover, relay repeats l,b+ for i = [B+3 : B+L, ie for L blocs Decoding part: 1 After transmission of bloc i = [1 : B + 1 and for each V c, the relay decodes the message w i and the compression index l,i, the compression index of relays in for the bloc i, with the assumption that all messages and compression indices up to bloc i 1 have been correctly decoded Note that there are two ind of relays inside, those who employ DF and those who are using CF he relay nows the message w i, w i 1 and so v w i and v wi 1 Let s define the sequences: E ŵ b, ˆl,b = xw b, ŵ b, vw b, x w b, l,b 1, z b, x l,b 1, ẑ l,b 1, ˆl,b x w b, l,b 1, ẑ w b, l,b 1, ˆl,b CF DF E ˆl,b = vw b 1, x w b 1, l,b, z b + 1, x w b 1, ˆl,b, x ˆl,b DF CF,

4 he -th relay searches for the unique index ŵ b, ˆl,b by looing at two consecutive blocs b and b + 1 such that E ŵ b, ˆl,b A n ɛ [V XX X Ẑ Z and E ˆl,b A n ɛ [V X X Z he error probability can be made arbitrary small provided that IẐS; Z S V XX X Ẑ S cz < IX S ; Z V X X S c, R < IX; Ẑ Z V X X + IX S ; Z V X X S c IẐS; Z S V X X Ẑ S cz 6 Decoding at the destination is done bacwardly After the last bloc, the decoder jointly searches for the unique indices ˆl,B+ : such that for all b [B + 3 : B + L, x ˆl,B+1 : CF and x 1, ˆl,B+1 : DF are jointly typical with x1, 1, v1 and y 1 b he probability of error goes to zero as n goes to infinity provided that for all S : IẐ; Z X + IẐ; Z X V + S CF S DF L IX S ; V XX S cy 1 3 After finding correctly l,b+ for all and since w B+1 = 1, the destination decodes jointly the message and all the compression indices w b, l,b+1 for each b = [1 : B where l,b = l,b he decoding is done bacwardly, assuming that w b+, l,b+ have been correctly decoded Define the following sequence: Eŵ b,ˆl,b+1 = xŵ b, w b+, vŵ b, y 1 b +, x ˆl,b+1, ẑ ˆl,b+1, l,b+, CF x ŵ b, ˆl,b+1, ẑ ŵ b, ˆl,b+1, l,b+ CF he destination finds the unique pair of indices ŵ b, ˆl,b+1 such that Eŵ b, ˆl,b+1 A n ɛ [V XX Ẑ Y 1 he probability of error goes to zero as n, similar to [10 provided that IX S ; ẐS cy 1 XX S cv IẐS; Z S V XX Ẑ S cy 1 > 0 R < IXX S ; ẐS cy 1 X S c IẐS; Z S XX Ẑ S cy 1 By choosing finite L but large enough, the previous inequalities along with 6 prove heorem 1, where the rate is achieved by letting B, n tend to infinity At the end, a time sharing random variable Q can be added V CONSAN GAP FOR GAUSSIAN NEWORKS In this section, we study constant gaps for AWGN networs Before proceeding to the general case, we first consider the gap of DF rate for the single AWGN relay channel CF constant gap follows the same line as [5 Consider the Gaussian relay channel with average power inputs E[X P, E[X1 P and channel outputs defined as follows: Y1 = h 3 X + h X 1 + N 1 Z 1 = h 1 X + N, Ẑ1 = Z 1 + ˆN 7 It is easy to chec that the second term in the DF rate, corresponding to namely MAC part of the channel, appears untouched in the cut-set bound CB and the gap is zero between the term for MAC part and thus only the difference between first two terms affects the gap his can be bounded as follows β is the correlation coefficient: CB, DF = 1 log 1 + βv 1P + βv 3 P 1 log 1 + βv 1P N N 1 log 1 + v 3 v 1 Observe that, unlie NNC, the gap for DF cannot be independent of the channel gains v 1 and v 3 Now we move to the unicast multiple relay networ o this end, consider an AWGN Gaussian networ with N relays, single source and destination of N + nodes he relays are indexed as usual with index in the set N = 1,, N For simplicity, we associate the source with the index 0, ie X 0 = X and the destination with the index N +1 herefore, there is bijection from the set of nodes to the set 0, 1,, N, N +1 he transmitters set is denoted by M = 0, 1,, N and the receivers set is denoted by D = 1,, N, N + 1 By g ij we denote the channel gains from the i-th node to the node j he noise at the node j is denoted by N j which assumed to zero mean of unit variance complex Gaussian random variable he input and output relation are defined as follows: Y D = GD, XM + ND ẐN = ZN + ˆNN 8 where Y D = [Z 1 Z N Y 1 GD, is the channel gain matrix where evidently g ii = 0 Let GS 1, S denote the set [g ij : i S 1, j S and similarly for any A, AS 1 = [a i : i S 1 All nodes transmit with power less or equal than P By ˆNS we denote the compression noise vector which is assumed to be Gaussian with unit variance he covariance matrix of the channel inputs is KS = [P ρ ij for i, j S o evaluate the constant gap for MNNC from heorem 1, we assume = N with all inputs chosen as Gaussian Given a set of channel gains, assume all relays in V c use DF while those in V use CF he constant gap for the conventional NNC has been evaluated in [5, S CB, NNC max S N + min S, Sc log S N + log4n Proposition 1 constant gap for MNNC: For unicast multiple relay networs, if the source-relay channel is good enough for DF relays, the constant gap can be stated as follows: CB, MNNC max N V S N [ S min S, Sc log 4 V S c When only one relay is using DF, the gap is bounded by N+ 4 log4n 4 which is clearly less than NNC scheme

5 o prove the proposition, we first find an upper bound on the cut-set bound using the following lemma: Lemma 1: Suppose that A = [ A11 A 1 A 1 A and B = [ A A are positive definite matrices then B A o prove the lemma, it is enough to see that B A is positive definite Now we move to bounding cut-set bound Suppose that S c = S c N + 1, V c = V c 0 and S CF = S V denoting the CF relays he cut-set bound reads as IXX S ; Z S cy 1 X S c = 1 log [ ISc + G KV c KV c, S CF KS CF, V c KS CF Now the determinant can be bounded as follows: [ KV ISc + G c KV c, S CF KS CF, V c KS CF [ a KV ISc + S CF G c 0 0 P IS CF 10 where a comes from Lemma 1, and from rs CF IS CF KS CF By rewriting 10 similar too [5, we obtain R 1 log IS c + 1 [ KV G c 0 G 0 P IS CF + min Sc + 1, S + 1 log 4 S CF 11 he term IẐS; Z S XX N Ẑ S cy 1 is lower bounded by IẐS; Z S XX N = S / Notice in the first part of R N S, unlie NNC, the relays from V c are not independent and hence IXX S ; ẐS cy 1 X S c IXX S ; ẐS cŷ 1 X S c = 1 log IS c + 1 G [ KV c 0 0 P IS CF where the matrix KV c, is the one that maximizes CB So the gap between R N S and its corresponding CB becomes: [ S 1 CB,MNNC max N V S N + min S c + 1, S + 1 log 4 S CF 1 In order to bound the other part of the rate related to DF relays, note that Y 1 is absent in the rate expression as for DF in the single AWGN relay channel, while it is always present in the CB hus, the gap between the rate and the CB, no matter how good we manage to bound it, will depend on the channel gains between Y 1 and inputs For sae of clarity, let us assume that each DF relay is decoding the source message alone, ie, without using the compression index of other relays is reduced to R DF = IX; Z V X First, observe that Z = g 0 X + g i X i + g i X i + N, i V i V c yielding = Given this choice, the rate R where similarly V is the set of users using CF From which the mutual information can be stated as follows: IX; Z V X = 1 log [ 1 + g 0 1 ρ 0 P i V g i P + i V g i 1 ρ c i P + 1 he cut-set bound corresponding to this rate is calculated as: IXX N \ ; Z Y 1 X = 1 log I + GKM\G As it is expected, this gap, say CB, MNNC, is not independent of the channel gains he final gap CB, MNNC would be the maximum of 1 CB, MNNC and CB, MNNC and it is dependent on the channel gains However, if DF relays are chosen correctly then they should not contribute to increasing the gap and the dominant term in the gap is 1 CB, MNNC which is independent of the channel gain Now the question is to find the proper condition for choosing the DF relays One solution is to choose the relays with g 0 enough large compared to other gains which indicates that the channel quality of source-relay is very good ACKNOWLEDGMEN he authors are grateful to Prof Abbas El Gamal for his valuable advice at the early stage of this wor hey are also grateful to Prof Gerhard Kramer for valuable comments and suggestions REFERENCES [1 Cover and A El Gamal, Capacity theorems for the relay channel, Information heory, IEEE rans on, vol I-5, pp , 1979 [ Y Liang and G Kramer, Rate regions for relay broadcast channels, Information heory, IEEE ransactions on, vol 53, no 10, pp , Oct 007 [3 G Kramer, M Gastpar, and P Gupta, Cooperative strategies and capacity theorems for relay networs, Information heory, IEEE ransactions on, vol 51, no 9, pp , Sept 005 [4 A El Gamal, M Mohseni, and S Zahedi, Bounds on capacity and minimum energy-per-bit for AWGN relay channels, IEEE rans Information heory, vol 5, no 4, pp , Apr 006 [5 S H Lim, Y-H Kim, A El Gamal, and S-Y Chung, Noisy networ coding, Information heory, IEEE ransactions on, vol 57, no 5, pp , may 011 [6 A Dana, R Gowaiar, R Palani, B Hassibi, and M Effros, Capacity of wireless erasure networs, Information heory, IEEE ransactions on, vol 5, no 3, pp , march 006 [7 A Avestimehr, S Diggavi, and D se, Wireless networ information flow: A deterministic approach, Information heory, IEEE ransactions on, vol 57, no 4, pp , april 011 [8 X Wu and L-L Xie, On the optimal compressions in the compressand-forward relay schemes, Arxiv:, Feb 011 [Online Available: [9 J Hou and G Kramer, Short message noisy networ coding for multiple sources, in Information heory Proceedings ISI, 01 IEEE International Symposium on, july 01, pp [10 A Behboodi and P Piantanida, Selective coding strategy for unicast composite networs, in Information heory Proceedings ISI, 01 IEEE International Symposium on, 01 [11 J Du, M Xiao, M Soglund, and S Shamai, Short-message noisy networ coding with partial source cooperation, in Information heory Worshop IW, 01 IEEE, sept 01, pp