Selective Coding Strategy for Unicast Composite Networks

Transcription

1 Selective Coding Strategy for Unicast Composite Networs Arash Behboodi Dept of elecommunications, SUPELEC Gif-sur-Yvette, France Pablo Piantanida Dept of elecommunications, SUPELEC Gif-sur-Yvette, France arxiv: v1 [csi] 22 May 2012 Abstract Consider a composite unicast relay networ where the channel statistic is randomly drawn from a set of conditional distributions indexed by θ Θ, which is assumed to be unnown at the source, fully nown at the destination and only partly nown at the relays Commonly, the coding strategy at each relay is fixed regardless of its channel measurement A novel coding for unicast composite networs with multiple relays is introduced his enables the relays to select dynamically based on its channel measurement the best coding scheme between compress-and-forward CF and decode-and-forward As a part of the main result, a generalization of Noisy Networ Coding is shown for the case of unicast general networs where the relays are divided between those using and CF coding Furthermore, the relays using scheme can exploit the help of those based on CF scheme via offset coding It is demonstrated via numerical results that this novel coding, referred to as Selective Coding Strategy SCS, outperforms conventional coding schemes I INRODUCION Multiterminal networs are the essential part of modern telecommunication systems Recently, these were studied from various aspects he cutset bound for the general multicast networ was established in [1] Networ coding theorem for the graphical multicast networ was investigated in [2] where the max-flow min-cut theorem for networ information flow was presented for the point-to-point communication networ Whereas Lim et al proposed the Noisy Networ Coding NNC scheme for the general multicast networ, which includes most of the existing bounds on multiterminal networs [3] Kramer et al developed an inner bound for a point-topoint general networ using decode-and-forward which achieves the capacity of the degraded multicast networ [4] For the above mentioned scenarios, the probability distribution PD of the networ is supposed to be fixed during the communication and hence available to all nodes beforehand However, wireless channels are essentially time-varying due to fading and user mobility, and hence the terminals do not have full nowledge of all channel parameters involved in the communication In particular, without feedbac channel state information CSI cannot be available to the encoders end During years, an ensemble of research activities has been dedicated to both theoretical and practical aspects of communication in presence of channel uncertainty From an information-theoretic viewpoint, the compound channel, first he wor of P Piantanida is partially supported by the ANR grant FIREFLIES INB introduced by Wolfowitz [5] is one of the most important models that deals with channel uncertainty, and continues to attract much attention from researchers see [6] and references therein Composite models are more appropriate to deal with wireless scenarios since unlie compound models they deal with channel uncertainty by introducing a PD P θ on the channel selection hese models consist of a set of conditional PDs from which the current channel index θ, which can be a vector of parameters, is drawn according to P θ and remains fixed during the communication An example of this model can be slowly fading channels Capacity for this class of channels has been widely studied beforehand see [7] and references therein, for wireless scenarios via the well-nown notion of outage capacity see [8] and references therein and oblivious cooperation over fading Gaussian channels in [9] [11] In this paper, we study the composite multiple relay networ where the channel index θ Θ is randomly drawn according to P θ he index θ = θ r, θ d remains fixed during the communication but is unnown at the source, fully nown at the destination and partly nown θ r at the relays end Although a compound approach can guarantee asymptotically zero-error probability regardless of θ, it would be not an adequate choice for most of wireless models As a different approach, the coding rate r is selected regardless of the current index Hence the encoder cannot necessarily guarantee arbitrary small error probability In this case the asymptotic error probability becomes the measure of interest, characterizing the reliability function Moreover, it turns out that depending on the channel draw, there may not be a unique set of relay functions that minimizes the error probability In other words, the relay function should be chosen based on the channel parameters However, since full CSI is not available to all nodes, the relay functions are usually chosen regardless of their channel measurements which becomes the bottlenec of the coding We present a novel coding strategy from which the relays can select, based on their measurements, an adequate coding strategy o this purpose an achievable region that generalizes NNC to the case of mixed coding strategy, where relays exploit the help of CF relays, is derived Section II presents definitions and Section III introduces main results, and the setch of proofs is relegated to Section IV Finally, numerical evaluation over the slow-fading Gaussian two relay channel is presented in Section V

2 X X 1 1, X2 1 X 1 2 Fig 1, θ r / D 1 X 2 1, X2 2 X 1 2, θ r D 2 W θ X 1 i, X2 i X 1 j, θ r D i X 1 i X 2 j, X2 j, θ r / D j Composite Multiple Relay Networ II PROBLEM DEFINIION he composite multiple relay channel consists of a set of multiple relay channels, as depicted in Fig 1 and denoted by W n θ = P Y n 1,θ Zn 1θr Zn 2θr Zn Nθr Xn X n 1θr Xn 2θr Xn Nθr n=1 where X denotes the channel input, X θr and Z θr the relay inputs and outputs and Y 1θ the channel output We assume a memoryless multiple relay channel with N relays but single source and destination he channels are indexed by vector of parameters θ = θ d, θ r with θ d Θ d, θ r Θ r, where θ r denotes all parameters affecting the relays output and θ d are the remaining parameters involved in the communication Let P θ be a joint probability measure on Θ he channel parameters affecting relay and destination outputs θ = θ r, θ d are drawn according to the joint PD P θ and remain fixed during the communication However, the specific draw of θ is assumed to be unnown at the source, fully nown at the destination and partly nown θ r at the relays end Assume that N = 1,, N and for any S N, X S = X i : i S Definition 1 code and achievability: A code-cn, M n, r for the composite multiple relay channel consists of: An encoder mapping ϕ : M n X n, A decoder mapping φ θ : Y1 n M n, A set of relay functions for N Only partial CSI at the relay is assumed which is mainly related to the -th source-to-relay channel An error probability 0 ɛ < 1 is said to be r-achievable, if there exists a code-cn, M n, r with rate satisfying lim inf n and average error probability lim sup n f iθ r Y 1θ : Z i 1 X n i=1 1 n log M n r 1 E θ [ Pr φθ Y n 1θ W θ ] ɛ 2 he infimum of all r-achievable EPs ɛr is defined as ɛr = inf 0 ɛ < 1 : ɛ is r-achievable 3 We emphasize that for channels satisfying the strong converse property and with unique best code word, eg, Gaussian slowfading single user channel, 3 coincides with the definition of the outage probability for the unique best codeword In the present setting, we assume that the source is not aware of the specific draw θ P θ and hence, the coding rate r and the coding strategy or CF scheme must be chosen independent of the channel draw Furthermore, both remain fixed during the communication regardless of the channel measurement at the relays end We aim to characterize the smallest possible average error probability as defined by 2, as a function of the coding rate r It can be shown that ɛr can be bounded as follows P θ r S θ ɛr inf C P θr / R θ C, 4 where R θ is any achievable rate for the unicast networ with a given θ, and S θ is the infimum of all rates such that every code with such rate yields error probability tending to one, and C as all codes It can be shown that S θ can be replaced with max-flow min-cut bound A special case of this result has been proved in recent wor [11] for the relay channel III COMPOSIE MULIPLE RELAY NEWORKS Consider the composite unicast networ with multiple relays and parameters θ he rate is fixed to r and so is the source code he goal is to minimize the expected error probability he common option is that each relay fixes its coding strategy, namely or Compress-and-Forward CF, regardless of θ In other words, the relays with index in V N will use CF scheme For instance, to evaluate the expected error probability we first present an achievable rate for the multiple relay networ where NNC is generalized to networs with mixed cooperative strategy Part of relays are using coding while the reminding relays use CF scheme Moreover, relays exploit the help of CF relays to decode the source message Using this theorem, an achievable rate can be obtained for every set V of relay nodes heorem 1 Cooperative Mixed NNC: For the multiple relay channel, the following rate is achievable R max max min max min R S, P P V N Υ V S min max S 5 with min R V c Υ V S R S = IXX V cx S ; ẐS cy 1 X S cq IZ S ; ẐS XX V cẑ S cy 1 Q S c = S, R S = IX; Ẑ Z X V cx Q + IX S ; Z X Vc S cq IẐS; Z S X V c Ẑ S cz Q S c = S, for, V N and V c = N V Moreover Υ V and Υ V are defined by Υ V = V : for all S, Q S 0, 6 Υ V = V : for all S, Q S 0, 7

3 while Q S and Q S are given by Q S = IX S ; ẐS cy 1 XX Sc V cq IZ S ; ẐS XX V cẑ S cy 1 Q, 8 Q S = IX S; Z X Vc S cq IẐS; Z S XX V c ẐS cz Q 9 And P is the set of all admissible distributions: P = P QXXN Z N Ẑ N Y 1 = P Q P XXV c QP Y1Z N XX N j V P Xj QPẐj X jz jq Notice that R S condition of correct decoding at the destination is in general better than R S condition of correct decoding at the relay his is because the destination uses bacward decoding Moreover using the same technique as [12], it can be shown that the optimization in 5 can be done over V instead of Υ V In other words the relays in V c can increase the rate only if they satisfy 7 It can be seen that for each, if there is A such that Q A < 0, then we can remove the relays in A from and the rate is improved by not using their compression, which is easier to manipulate in composite setting By choosing V = N, the region of heorem 1 is reduced to the same region, as in [12], [13], which is equivalent to NNC region [3] So heorem 1 generalizes and includes the previous NNC scheme and it provides a potentially larger region Indeed, for the degraded single relay channel, it is capacity achieving while NNC is strictly suboptimal In fact, relay nodes are divided into two groups he first group is V c which are using coding and the second group V which are using CF scheme Now consider the composite setting again Before starting the communication, the source nows that the relays with index in V use CF while the others use scheme For each θ and V, the rate r is achievable if it belongs to the region in the previous theorem and otherwise an error is declared All that CF relays can do in this case is to choose their distribution based on θ r such that it minimizes the error probability hus the expected error probability for the composite multiple relay channel with partial CSI θ r at the relays is bounded by ɛr inf V N E θr min px,x V c,q min P [ j V pxj qpẑj xjzjq θ θ r r > ICMNNC V ] θr where for all θ = θ r, θ d, I MNNC V = min max min R S, θ, min max V S min R V c Υ V S R S, θ = IXX V cx S ; ẐS cy 1θ X S cq, S, θ, 10 IZ Sθr ; ẐS XX V cẑ S cy 1θ QS c = S, R S, θ = IX; Ẑ Z θr X V cx Q +IX S ;Z θr X V c S cq IẐS; Z Sθr X V c Ẑ S cz θr Q for, V N and V c = N V Similarly Υ V is Υ V = V : for all S, Q S, θ r 0 11 where Q S, θ r is defined as follows: Q S, θ r =IX S ; Z θr X Vc S cq IẐS; Z Sθr XX Vc ẐS cz θ r Q In the preceding scheme all relays fix coding regardless of the available CSI However, it is possible that the relays select and change their coding based on its CSI o this purpose each relay generates many codeboos and sends one of them which fits the best to the channel with the parameter θ r More precisely, each relay has a decision region D such that for all θ r D, the relay uses scheme and otherwise it uses CF scheme For each V N, define D V as follows: D V = V c D V D c If θ r D V, then θ r / D for all V, and θ r D for all / V So the -th relay, for each V uses CF and the relay for V c uses he ensemble of decision regions of relays will thus provide the regions D V which are mutually disjoint and all together form a partitioning over the set Θ r Now if θ r D V, we have a multiple relay networ where the relays in V are using CF he achievable rate corresponding to this case is nown from heorem 1 As shown in Fig 1, each relay has two set of codewords: X 1 and X 2 he first code X1 is transmitted when θ r D his code is based on strategy so the - th relay decodes the source message and transmits it to the destination However the source, not nowing whether the - th relay is sending X 1 or not, uses superposition coding and superimpose its code over X 1 If the -th relay sends X1 then this will become relaying he source, oblivious to the relays decision, generates its own code by using superposition coding and X is superimposed over X 1 N, ie, all possible relay inputs his does not affect R S, θ by applying the proper Marov chain, but changes R S, θ, as we will see On the other hand, if θ r / D then CF scheme is used Note that unlie, the code which is used for CF X 2, is independent of the source code and so its PD can be chosen adaptively based on θ r he optimum choice for D V will potentially give a better outage probability than the case that each relay is using a fixed coding for all θ r his provides a non-formal proof of the next theorem heorem 2 SCS with partial CSI-cooperative relays: he average error probability of the composite multiple relay channel with partial CSI θ r at the relays can be upper bounded by ɛr min 1 px,x N,q inf D V,V N ΠΘ r,n E θr min V N j V px2 j qpẑ j x 2 j z jq P θ θr [ r > ICMNNC V, θ r D V θ r ], 12

4 Π Θ r, N is the set of all partitioning over Θ r into at most 2 N disjoint sets he relay inputs X is chosen from X 1, X2 such that X is equal to X 1 if θ r D and equal to X 2 if θ r / D Indeed, for θ r D V the next Marov chain holds: X 1 V, X2 Vc X, X1 V, c X2 V Y 1θ, Z N θr, where I MNNC V and Υ V are defined by expressions 10 and 11 with the difference that in R S, θ and Q S, θ r, X V c is replaced with X 1 N IV SKECH OF HE PROOF OF HEOREM 1 Consider first the two relay networ Relay 1 uses scheme to help the source so it has to decode the source messages successively and not bacwardly, and Relay 2 uses CF scheme However, relay 1 wants to exploit the help of relay 2 to decode the source message So it does not start decoding until it retrieves the compression index o this end, relay 1 uses offset decoding which means that it waits two blocs instead of one to decode the source message and the compression index In bloc b = 2, the relay 1 decodes the compression index l 1 and the message w 1 Equally, the source code at bloc b + 2 is correlated with relay 1 code from the bloc b and not bloc b + 1 his comes at the expense of one bloc of delay he source has to wait until b = B +L to start bacward-decoding he compression index l B+2 is repeated until the bloc B + L Fix P, V, and s such that they maximize the right hand side of 5 Assume a set M n of size 2 nr of message indices W to be transmitted, again in B + L blocs, each of them of length n At the last L 2 blocs, the last compression index is first decoded and then all compression indices and transmitted messages are jointly decoded Relays in V c start to decode after bloc 2 Code generation: i Randomly and independently generate 2 nr sequences x V c drawn iid from PX n V c x V c = n P XV c x V c j Index them as x V cr with index r [ 1, 2 nr] ii For each x V cr, randomly and conditionally independently generate 2 nr sequences x drawn iid from PX X n x x V c V cr = n P X XV c x j x V c j Index them as xr, w, where w [ 1, 2 nr] iii For each V, randomly and independently generate 2 n ˆR sequences x drawn iid from PX n x = n P X x j Index them as x r, where r [1, 2 n ˆR ] for ˆR = IZ ; Ẑ X + ɛ iv For each V and each x r, randomly and conditionally independently generate 2 n ˆR sequences ẑ each with probability n P ṋ ẑ Z X x r = PẐ X ẑ j x j r Index them as ẑ r, ŝ, where ŝ [1, 2 n ˆR ] Encoding part: i In every bloc i = [1 : B], the source sends w i using x w i 2, 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 ii For every bloc i = [1 : B + L], and each V c, the relay nows w i 2 by assumption and w 0 = w 1 = 1, so it sends x wi 2 iii For each i = [1 : B + 2], 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 iv For i = [1 : B + 2] and V, relay nows from the previous bloc l i 1 and it sends x l i 1 Moreover, relay repeats l B+2 for i = [B+3 : B+L], which means for L 2 blocs Decoding part: i After the transmission of the bloc i = [1 : B + 1] and for each V c, with the assumption that all messages and compression indices up to bloc i 1 have been correctly decoded, the -th relay searches for the unique index ŵ b, ˆl b by looing at two consecutive blocs b and b + 1 such that: xw b 2, ŵ b, x V cw b 2, z b, x l b 1, ẑ l b 1, ˆl b A n ɛ [XX V cẑ Z ] and xv cw b 1, z b + 1, x ˆl b A n ɛ [X V cz ] Probability of error goes to zero as n if R IX; Ẑ Z X V cx + IX S ; Z X V cx S c IẐS; Z S X Vc Ẑ S cz 13 0 IZ ; X S X V c S c IẐS; Z S XX V c Ẑ S cz 14 Given the fact that Υ V, the last inequality holds for each S ii he destination jointly searches for the unique indices ˆl B+2 such that for all b [B + 3 : B + L], x ˆl B+1, x1, 1, x V c1, y 1 b belongs to A n ɛ [XX X V cy 1 ] he probability of error goes to zero as n provided for all subsets S : IẐ; Z X L 2IX S ; XX Sc V cy 1 S iii With the assumption that w b+2, l b+2 have been correctly decoded, the destination finds bacwardly the unique pair of indices ŵ b, ˆl b+1 such that: xŵ b, w b+2, x V cŵ b, y 1 b + 2, x ˆl b+1, ẑ ˆl b+1, l b+2 A n ɛ [XX V cẑ Y 1 ]

5 Z 1 : X 1 h 12, h 21 Z 2 : X 2 h 01 h 02 X h 23 h 13 Y 1 h 03 Fig 2 Fading Gaussian two-relay channel εr average error probability full full CF Mixed Coding SCS CB Cutset Bound he probability of error goes to zero as n if: 0 < IẐS cy 1; X S XX V c S c R IXX V cx S ; Y 1 Ẑ S c X S c IẐS; Z S XX V c ẐS cy 1, IẐS; Z S XX Vc ẐS cy 1 Using the previous inequalities, and by choosing finite L but large enough, by letting B, n and adding the time-sharing random variable Q the proof is finished V GAUSSIAN SLOW-FADING NEWORKS Consider the Gaussian fading two-relay networ, depicted in Fig 2, which is defined by the following relations: Z 1 = h 01 d α X + h 21X 2 + N 1, Z 2 = h 02 X + h 12 X 1 + N 2, Y 1 = h 03 X + h 13 X 1 + h 23 X 2 + N 3 Define N i s to be additive noises, iid circularly symmetric complex Gaussian RVs with zero-mean; let h ij s be independent zero-mean circularly symmetric complex Gaussian RVs d is the random path-loss he average power of the source and relay inputs X, X 1 and X 2 must not exceed powers P, P 1 and P 2, respectively Compression is obtained by adding additive noises Ẑ1 = Z 1 + ˆN 1, Ẑ 2 = Z 2 + ˆN 2 It is assumed that the source is not aware of the fading coefficients, the relays now all fading coefficients except h i3 s and the destination is fully aware of everything he possibilities to choose the proper cooperative strategy are as follows: i both relays use scheme to transmit the information full case, ii both relays use CF scheme to transmit the information full CF case, where the destination can consider one or both relays as noise to prevent the performance degradation, and iii one relay uses scheme and the other uses CF scheme Mixed Coding case Finally, the relays can select their coding strategy based on the channel parameters SCS case Fig 3 presents numerical analysis of these strategies We assume all fading coefficients are of unit variance and so are the noises d is chosen with uniform distribution between 0 and 01, which means the first relay is always around the source Given this assumption, we suppose that the first relay uses in case of mixed coding while the other uses CF scheme he r coding rate [bits/symbol] Fig 3 Asymptotic error probability ɛr vs coding rate r source and relay powers are respectively 1 and 10 It can be seen that none of the non-selective strategies lie full, full CF and Mixed Coding is not in general the best regardless of fading coefficients However, if one lets the relay select their strategy given its channel measurement, this SCS will lead to significant improvement compared to the other strategies and becomes close to the cutset bound REFERENCES [1] P Elias, A Feinstein, and C Shannon, A note on the maximum flow through a networ, Information heory, IRE ransactions on, vol 2, no 4, pp , 1956 [2] R Ahlswede, N Cai, S-Y Li, and R Yeung, Networ information flow, Information heory, IEEE ransactions on, vol 46, no 4, pp , jul 2000 [3] 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 2011 [4] 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 2005 [5] J Wolfowitz, Simultaneous channels, Arch Rat Mech Anal, vol 4, pp , 1960 [6] A Lapidoth and P Narayan, Reliable communication under channel uncertainty, IEEE rans Information heory, vol 44, pp , October 1998 [7] M Effros, A Goldsmith, and Y Liang, Generalizing capacity: New definitions and capacity theorems for composite channels, Information heory, IEEE ransactions on, vol 56, no 7, pp , 2010 [8] E Biglieri, J Proais, and S Shamai, Fading channels: informationtheoretic and communications aspects, Information heory, IEEE ransactions on, vol 44, no 6, pp , oct 1998 [9] M Katz and S Shamai, Cooperative schemes for a source and an occasional nearby relay in wireless networs, Information heory, IEEE ransactions on, vol 55, no 11, pp , nov 2009 [10] A Behboodi and P Piantanida, Broadcasting over the relay channel with oblivious cooperative strategy, in Communication, Control, and Computing Allerton, th Annual Allerton Conference on, oct 1, pp [11], On the asymptotic error probability of composite relay channels, in Information heory Proceedings ISI, 2011 IEEE International Symposium on, 2011 [12] G Kramer and J Hou, On message lengths for noisy networ coding, in IEEE Information heory Worshop, October 2011, pp [13] X Wu and L-L Xie, On the optimal compressions in the compress-and-forward relay schemes, Feb 2011 [Online] Available: