A class of Bi-directional multi-relay protocols

Transcription

1 1 A class of Bi-directional multi-relay protocols ang Joon Kim, Natasha Devroye, and Vahid Tarokh Astract In a i-directional relay channel, two nodes wish to exchange independent messages over a shared wireless halfduplex channel with the help of relays. ecent work has considered information theoretic limits of the i-directional relay channel with a single relay. In this work we consider idirectional relaying with multiple relays. We derive achievale rate regions and outer ounds for half-duplex protocols with multiple decode and forward relays and compare these to the same protocols with amplify and forward relays in an additive white Gaussian noise channel. We consider three novel classes of half-duplex protocols: the (m, 2) 2 phase protocol with m relays, the (m, 3) 3 phase protocol with m relays, and general (m, t) Multiple Hops and Multiple elays (MHM) protocols, where m is the total numer of relays and 3 < t m +2 is the numer of temporal phases in the protocol. Finally, we provide a comprehensive treatment of the MHM protocols with decode and forward relaying and amplify and forward relaying in Gaussian noise, otaining their respective achievale rate regions, outer ounds and relative performance at different Ns. The (m, m + 2) DF MHM protocol achieves the largest rate region under simulated channel conditions. Index Terms i-directional communication, achievale rate regions, decode and forward, amplify and forward, multiple relays I. INTODUCTION In i-directional channels, two terminal nodes (a and ) wish to exchange independent messages. In wireless channels or mesh networks, this communication may take place with the help of m other nodes r i, i 1, 2, m termed relays. This two-way channel [2] was first considered in [9], in which full-duplex operation where nodes could transmit and receive simultaneously, was assumed. ince full-duplex operation is, with current technology, of limited practical significance, in this work we assume that the nodes are half-duplex, i.e. at each point in time, a node can either transmit or receive symols, ut not oth. Our main goal is to determine the limits of i-directional communication with multiple relays. To do so, we propose and determine the achievale rate regions, as well as outer ounds otained using several protocols. The protocols we propose for the multiple-relay i-directional channel may e descried in terms of two parameters: the numer of relays, m, and the numer of temporal phases t, called hops. Throughout this work, phases and hops are used interchangealy. We also define an intermediate hop as a hop in which only relays transmit (and not the terminal nodes). Note that our protocols are all composed of a numer of temporal phases/hops due to the half-duplex nature of the channel. We denote our proposed ang Joon Kim and Vahid Tarokh are with the chool of Engineering and Applied ciences, Harvard University, Camridge, MA s: sangkim@fas.harvard.edu, vahid@deas.harvard.edu. Natasha Devroye is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicage, IL devroye@ece.uic.edu. protocols as (m, t) MHM (Multiple Hops and Multiple elays) protocols, for general positive integers m 2 and t 2. For the special case of two hops (t =2), the terminal nodes may simultaneously transmit in phase 1 as in the MABC (Multiple Access Broadcast Channel) protocol of [5], while the relays transmit the decoded messages to the terminal nodes in phase 2. For the special case of three hops (t =3) the terminal nodes may sequentially transmit in the first two phases as in the TDBC (Time Division Broadcast Channel) protocol of [5], after which the relays transmit in phase 3. In the TDBC protocol, side information may e exploited to improve the rate region. By side information we mean information otained from the wireless channel in a particular phase which may e comined with information otained in different stages to potentially improve decoding or increase transmission rates. For each of the MHM protocols, the relays may process and forward the received signals differently. tandard forwarding techniques include decode-and-forward, amplify-andforward, compress-and-forward, and de-noise and forward. We consider only the first two relaying schemes. ome similar protocols and relaying schemes have een previously considered. In [6], the DF TDBC protocol with a single relay is considered. There, network coding in Z k 2 is used to encode the message of relay r from the estimated messages w a and w. The works [7] and [8] consider the MABC protocol with multiple hops, where an amplification and denoising relaying scheme is introduced. In [5], achievale rate regions and outer ounds of the MABC protocol and the TDBC protocol for a single DF relay are derived. In [1], a comprehensive analysis of the AF scheme in large networks is provided. This paper is structured as follows: in ection II, we introduce our notation. In ection III, we introduce novel (m, t) MHM protocols. In ection IV we derive achievale rate regions for the (m, t) MHM protocols with DF relaying. In ection V we derive outer ounds for the MHM protocols. In ection VI, we numerically compute these ounds in the Gaussian noise channel and compare the results for different powers and channel conditions. II. PELIMINAIE Nodes a and are the two terminal nodes and := r 1, r 2,, r m is the set of relays which aid the communication etween nodes a and. For convenience of analysis we define r 0 := a, r m+1 := and use these notations interchangealy. Also define := a, = r 0, r 1,, r m+1. We use i,j for the transmitted data rate from node i to node j. In our case, two terminal nodes denoted a and exchange their messages W a and W at the rates a := a, and :=,a, that is, the two messages W a and W are

2 2 taken to e independent and uniformly distriuted in the set of 0,, 2 n a 1 := a and 0,, 2 n 1 :=. Each node i has channel input alphaet Xi = X i and channel output alphaet Yi = Y i. Because of the half-duplex constraint, not all nodes transmit/receive during all phases and we use the dummy symol to denote that there is no input or no output at a particular node during a particular phase. The half-duplex constraint forces either X (l) i = or Y (l) i = for all l phases. The channel is assumed to e discrete memoryless. For convenience, we drop the notation from entropy and mutual information terms when a node is not transmitting or receiving. During phase l we use X (l) i to denote the input distriution and Y (l) i to denote the distriution of the received signal of node i and similarly X (l) := X (l) i i, a set of input distriutions during phase l. i is the phase duration of phase i when n. Lower case letters x i denote instances of the upper case X i which lie in the calligraphic alphaets Xi. N X i denotes the Cartesian product of the alphaets X i, i =1, 2,,N for non-negative integer N. Boldface x i represents a vector indexed y time at node i. For the (m, 2), (m, 3) DF MHM protocols we define A (resp. B) as the set of relays which are ale to decode w a (resp. w ). We define I min(x(l) i ; Y s (l) ) := min s I(X (l) i ; Y s (l) ). For example, IA min(x(1) a ; Y r (1) ) = min r A I(X a (1) ; Y r (1) ), i.e. the minimum mutual information etween node a and a relay in the set of relays which can decode w a. We also define Iø min (X; Y ) = 0. denotes the cartesian product, i.e., 3 X i = X 1 X 2 X 3. III. POTOCOL We next descrie a class of i-directional multiple-relay protocols which we term (m, t) DF MHM (Decode and Forward, Multiple Hop Multiple elay) protocols, where m is the numer of relays and t is the numer of hops. A protocol is a series of temporal phases through which i-directional communication etween nodes a and is enaled. In ection VI we consider Amplify and Forward relaying in the Gaussian channel and use the term (m, t) AF MHM protocol. We rename MHM protocols for some special cases as follow: MABC MHM protocol : (m, 2) MHM protocol TDBC MHM protocol : (m, 3) MHM protocol 1 regular MHM protocol : each hop has the same numer of relays, only valid when m mod (t 2) = 0. 2 We first descrie the (m, m + 2) MHM protocol. From the (m, m + 2) MHM protocol the (m, t) for 3 < t < m + 2 protocol and corresponding achievale rate regions readily follow. r 0 (= a) r 1 r 2 r m r m+1 (= ) is one possile graphical representation of our multi-hop network with m relays. A simple naïve protocol for the aove 1 The relaying protocols for the MABC and TDBC protocols are descried in [4] and we do not state them here due to space constraints. The protocols are a simple extension of the single relay protocols in [5]. 2 For example, the (8, 6) regular MHM protocol consists of two relays in each of the four hops. TABLE I [ALGOITHM] - (m, m + 2) DF MHM POTOCOL Preparation a and divide their respective messages into B su-messages, one for each lock. Thus, a has the message set,,w a,(b 1). Likewise has w,(0),w,(1),,w,(b 1). Initialization 01: For i =0to m 1 02: For j =0to i 03: r i j transmits x ri j (w a,(j) ) 04: r i j+1 decodes w a,(j) 05: end 06: end Main routine 01: For i =0to B m 1 02: r m+1 transmits x rm+1 (w,(i) ) 03: r m decodes w,(i) and generates x rm (w a,(i) w,(i) ) 04: For j =0to m 1 05: r m j transmits x rm j (w a,(i+j) w,(i) ) 06: r m j 1 decodes w,(i) and generates x rm j 1 (w a,(i+j+1) w,(i) ) 07: r m j+1 decodes w a,(i+j) 08: end 09: r 0 transmits x r0 (w a,(m+i) ) 10: r 1 decodes w a,(m+i) 11: end Termination 01: For i = B m to B 1 02: r m+1 transmits x rm+1 (w,(i) ) 03: r m decodes w,(i) and generates x rm (w a,(i) w,(i) ) 04: For j =0to m 1 05: r m j transmits x rm j 06: r m j 1 decodes w,(i) and generates xrm j 1 (w a,(i+j+1) w,(i) ), if i + j B 2 x rm j 1 (w,(i) ), otherwise 07: r m j+1 decodes w a,(i+j) if i + j B 1 08: end 09: end example network is : r 0 r 1 r m r m+1 and then r m+1 r m r 1 r 0. This is one possile (m, 2m+2) MHM protocol which may e spectrally inefficient as the numer of phases is large. Intuitively, spectral efficiency may e improved y comining phases through the use of network coding. We reduce the numer of phases needed from 2m +2 to m +2 y the algorithm in Tale I. After initialization, relay r i has the following messages from node a:,,w a,(m i) (1 i m). In other words message w a,(i) has reached r m i at the end of the initialization. In the main routine, which, when the numer of locks B makes up the majority of this protocol, w,(i) travels along the path r m+1 r m r 1 r 0 in the i th loop. During the same loop, as the single su-message from node travels to node a, the stream of messages from node a sitting in the each of the relays are all shifted to the right y one through the use of network coding. Overall then, we require 2 transmissions from the terminal nodes, and m relay transmissions to transfer two individual su-messages. When node a finishes sending its all su-messages to r 1, the termination step starts. The remaining w a,(i) s in the relays and w,(i) s in node are processed in this step. The numer of transmissions in the main routine depends only on the numer of locks B while the numer of transmissions in

3 3 $%&'!%()'**+,-)./-', 0&/%!(-,-/-.1-2./-', 3.-,(!'+/-,%( 0&/%!(/%!*-,./-',! ". 4! # w,(0) w,(0) wa,(0) x r1 (w a,(1) w,(0) ) w,(0),w a,(2) w,(0) w,(0) x r2 ( w,(0) ) w,(0) w,(0) w,(0),w,(1),w,(2) w,(0),w,(1),w,(2)! ". 4! # Fig. 1. Illustration of the (m, m + 2) DF MHM protocol with B=3, m=2. Grey denotes the su-messages of w a at the nodes, lue denotes the su-messages of w at the nodes, and green denotes the current transmission. Dotted lines denote the path taken during initialization and termination phases. the initialization and termination steps are a function of the hop size m. We can easily show that y increasing the lock size B, our algorithm asymptotically results in m +2 phases. A graphical illustration for the case when B =3and m =2 is shown in Fig. 1. IV. ACHIEVABLE ATE EGION emark : Due to space constraints, the proofs for the Theorems 1, 2, 3, and Corollary 4 are provided in [4]. A. (m, 2) DF MABC protocol Theorem 1: An achievale region of the half-duplex idirectional channel under the (m, 2) DF MABC protocol is the closure of the set of all points ( a, ) a < min 1 IA B min (X(1) a ; Y r (1) X (1),Q), < min 1 IA\B min (X(1) a ; Y r (1), 2 I(X (2) 1 IA B min (X(1) ; Y r (1) X a (1),Q), 1 IB\A min (X(1) ; Y r (1), 2 I(X (2) A ; Y (2) B ; Y a (2) a + < 1 IA B min (X(1) a,x (1) ; Y r (1) (3) over all joint distriutions p(q)p (1) (x a q)p (1) (x q) p (2) (x A B q)p (2) (x A\B q)p (2) (x B\A q) with Q 3m +2 m+1 over the restricted alphaet for all possile A, B. B. (m, 3) DF TDBC protocol Theorem 2: An achievale region of the half-duplex idirectional channel under the (m, 3) DF TDBC protocol is the closure of the set of all points ( a, ) a < min < min 1 IA min (X(1) a ; Y r (1), 1 I(X (1) a ; Y (1) + 3 I(X (3) 2 IB min (X(2) ; Y r (2), 2 I(X (2) ; Y (2) a + 3 I(X (3) A ; Y (3) B ; Y a (3) over all joint distriutions p(q)p (1) (x a q)p (2) (x q) p (3) (x A B q)p (3) (x A\B q)p (3) (x B\A q) with Q 2m +2 m+1 over the restricted alphaet for all possile A, B. C. (m, t) DF MHM protocol First, we recall that in the (m, m + 2) MHM protocol a single relay transmits in each hop. We then extend the ideas of the (m, m + 2) MHM protocol to derive achievale rate regions for general (m, t) protocols with 3 < t < m +2. Theorem 3: An achievale rate region of the half-duplex i-directional multi-hop relay channel under the (m, m + 2) DF MHM protocol (m >1) is the closure of the set of all points ( a, ) k a < min 1 k m+1 k < min 1 k m+1 m+3 i I(X (m+3 i) r i 1 i I(X (i) r m+2 i ; Y (i) r m+1 k ; Y r (m+3 i) k over all joint distriutions p(q) m+2 p(i) (x rm+2 i q) with Q 2m +2 over the restricted. Corollary 4: An achievale rate region of the half-duplex i-directional channel in the (m, t) DF MHM protocol for 3 < t < m +2 is the closure of the set of all points ( a, ) k a < min min t+1 i I(X (t+1 i) ; Y (t+1 i) i 1 r k (8) 1 k t 1 r k k < min min 1 k t 1 r t 1 k t 1 k k i I(X (i) t i ; Y (i) r t 1 k over all joint distriutions p(q) t p(i) (x t i q) with Q 2m+2 over the restricted, for all possile i such that i j = for all i, j [0,t 1], where 0 = a and t 1 =. (1) (2) (4) (5) (6) (7) (9)

4 4 V. OUTE BOUND emark : We derive outer ounds using the cut-set ound lemma in [5]. Due to space constraints, the proofs for the Theorems 5, 6, 7, and Corollary 8 are provided in [4]. A. (m, 2) MABC protocol Theorem 5: (Outer ound) The capacity region of the halfduplex i-directional relay channel with the (m, 2) MABC protocol is outer ounded y the set of rate pairs ( a, ) a min 1 I(X a (1) ; Y (1) X (1),Q)+ 2 I(X (2) ; Y (2) X (2),Q) min 1 I(X (1) ; Y (1) X a (1),Q)+ 2 I(X (2) ; Y (2) a X (2),Q) (10) for all choices of the joint distriution p(q)p (1) (x a q) p (1) (x q)p (2) (x q) with Q 2 m+1 over the restricted for all possile. B. (m, 3) TDBC protocol Theorem 6: (Outer ound) The capacity region of the halfduplex i-directional relay channel with the (m, 3) TDBC protocol is outer ounded y the set of rate pairs ( a, ) a min 1 I(X a (1) ; Y (1),Y (1) + 3 I(X (3) ; Y (3) X (3),Q) min 2 I(X (2) ; Y (2),Y a (2) + 3 I(X (3) ; Y (3) a X (3),Q) for all choices of the joint distriution p(q)p (1) (x a q) p (2) (x q)p (3) (x q) with Q 2 m+1 over the restricted for all possile. C. (m, t) MHM protocol Theorem 7: (Outer ound) The capacity region of the half-duplex i-directional multi-hop relay channel under the (m, m+2) MHM protocol (m >1) is outer ounded y the set of rate pairs ( a, ) a min min r i a r i m+2 i I(X (m+2 i) r i m+2 i I(X (m+2 i) r i for all choices of the joint distriution p(q) m+2 p(i) (x rm+2 i q) with Q 2 m+1 over the restricted for all possile. Corollary 8: (Outer ound) The capacity region of the halfduplex i-directional channel in the (m, t) MHM protocol for 3 < t < m +2 is outer ounded y the set of rate pairs ( a, ) a min min t 1 i=0 t 1 i=0 t i I(X (t i) i ( a) ; Y (t i) t i I(X (t i) i ( ) ; Y (t i),y (t i) \ i,y \ a (t i) i X (t i) i,q) (16) X (t i) i,q) (17) for all choices of the joint distriution p(q) t p(i) (x t i q) with Q 2 m+1 over the restricted, for all possile i such that i j = for all i, j [0,t 1], where 0 = a and t 1 = for all possile. VI. NUMEICAL ANALYI A. The Gaussian relay network In this section, we apply the ounds otained in the previous section to a Gaussian relay network. The corresponding mathematical channel model is, for each channel use k : Y[k] =HX[k]+Z[k] (18) (11) where Y[k], X[k] and Z[k] are in (C ) (m+2) 1, and H C (m+2) (m+2). In phase l, if node r i is in transmission mode X ri [k] follows the input distriution X r (l) i N (0,P ri ). Otherwise, X ri [k] =, which means that the input symol does not exist in the aove mathematical channel model. In each phase, the total transmit power is ounded y P. While ideally the per-phase power of P could e distriuted amongst the nodes in l aritrarily, as a first step, we allocate equal power P/ l for each relay in l. Equal power allocation etween participating nodes may also e simpler to implement. Then, in each phase, we allow for cooperation etween relays which have the same messages. h (12) i,j is the effective channel gain etween transmitter i and receiver j. We assume the channel is reciprocal (h i,j = h j,i ) and that each (13) node is fully aware of the channel gains, i.e., full CI. The noise at all receivers is independent, of unit power, additive, white Gaussian, complex and circularly symmetric. As a comparison point for the DF MHM protocols, we derive an achievale region of the same temporal protocols in which the relays use a simple amplify and forward relaying scheme rather than a decode and forward scheme. imple means that there is no power optimization in each phase, i.e. each node during phase l has equal transmit power P/ l. Also, in the amplify and forward scheme, all phase durations are equal since relaying is performed on a symol y symol ; Y (m+2 i),y (m+2 i) asis. Thus, l = 1 t, where t is the numer of phases and l (14) [1,t]. Furthermore, relay r scales the received symol y Pr P yr to meet the transmit power constraint. 3 ; Y (m+2 i),y (m+2 i) a (15) B. ate region comparisons with one to two relays In this section we numerically evaluate the rate regions in the Gaussian relay network with two relays. We use the following channel gain matrix 4 : H = (19) In the proposed protocols, the (2,4) DF MHM protocol achieves the largest rate region in most scenarios. In the high 3 Due to space constraints, achievale rate regions for AF MHM protocols are provided in [4]. 4 If other channel gains are chosen, the numerical results may change.

5 5 N regime, the (2,2) AF MABC protocol may achieve rates slightly etter than the (2,4) DF MHM protocol, as noise amplification is less of an issue. However, in most cases multiple hops with DF relaying dominates in this i-directional half-duplex channel. In the DF relaying protocols, the (2,4) MHM protocol outperforms the other protocols at oth low and high N. This improved performance may e attriuted to this protocol s effective use of side information. During each phase, every node which is not transmitting can receive the current transmission which it may employ as side information to aid decoding during later stages. There is naturally a tradeoff etween the numer of phases and the amount of information roadcasted in each phase. However, as seen y our simulations in this particular channel, the effect of reducing the numer of phases to 2 or 3 does not outweigh the effect of roadcasting information. The inner and outer ounds differ for a numer of reasons, with the prevailing one eing that our inner ounds use a DF scheme. For the MABC scheme using DF relaying, every relay contriutes to enlarging the outer ound regions, while only the suset of relays A B are used in determining the achievale regions. At low N, when A B is relatively small, the gaps, shown in Fig. 2 (top) are larger than the gaps at high Ns shown in Fig. 2 (ottom), where the numer of relays in A B are relatively larger. In addition to simply having more relays contriute to the outer ound regions, their effect is summed up outside of the logarithm for the outer ound, and inside of it for the inner ounds. Lastly, the achievale rate regions for DF relaying are significantly reduced y the necessity of having all relays decode the message(s) w a or w individually, resulting in the min function which significantly diminishes the region. This requirement to decode all messages is not present in the outer ounds. The inner ounds for the AF relaying schemes are relatively small as (a) noise is carried forward, () no power optimization is performed and (c) no phase-length optimization is performed. The inner ounds may e improved through the use of compress and forward relaying [3] or de-noising, which may e ale to capture the optimal tradeoff etween eliminating the noise and requiring the messages to e decoded. The exploration of different relaying schemes as well as the analytical impact of different channel gain matrices is left for future work. Due to space constraints we defer the interested reader to [4] for a rate region comparison in which 8 relays are placed uniformly on a line. There, the (8, 10) DH MHM protocol dominates the other protocols oth in the low N and high N regime. This may e attriuted to the roadcasted side information: while increasing the numer of phases means that less information may e transmitted during each time phase, the accumulated side information and improved channel gains (shorter distances) for each hop outweigh these detrimental effects, yielding higher overall rates. VII. CONCLUION In this paper, we proposed protocols for the half-duplex idirectional channel with multiple relays. We derived achievale rate regions as well as outer ounds for 3 half-duplex idirectional multiple relay protocols with decode and forward (2,2) MABC (2,3) TDBC (2,4) MHM DF AF Outer Bound a (2,2) MABC (2,3) TDBC (2,4) MHM DF AF Outer Bound a Fig. 2. Comparison of achievale regions of AF and DF and outer ounds with P =0dB (top) and P = 20 db (ottom). relays. We compared these regions to those achieved y the same protocols with amplify and forward relays in the Gaussian noise channel. Numerical evaluations suggest that the (m, m + 2) DF MHM protocol achieves the largest rate region under simulated channel conditions. EFEENCE [1]. Borade, L. Zheng, and. Gallager, Amplify-and-forward in wireless relay networks: ate, diversity, and network size, IEEE Trans. Inform. Theory, vol. 53, no. 10, pp , Nov [2] T. Cover and J. Thomas, Elements of Information Theory, 2nd ed. New York:Wiley, [3]. J. Kim, N. Devroye, P. Mitran, and V. Tarokh, Comparison of idirectional relaying protocols, in Proc. IEEE arnoff ymposium, [4]. J. Kim, N. Devroye, and V. Tarokh, Bi-directional half-duplex protocols with multiple relays, umitted to IEEE Trans. Inform. Theory, [5]. J. Kim, P. Mitran, and V. Tarokh, Performance ounds for idirectional coded cooperation protocols, IEEE Trans. Inform. Theory, vol. 54, no. 11, pp , Nov [6] P. Larsson, N. Johansson, and K.-E. unell, Coded i-directional relaying, in Proc. IEEE Veh. Technol. Conf. - pring, 2006, pp [7] P. Popovski and H. Yomo, The anti-packets can increase the achievale throughput of a wireless multi-hop network, in Proc. IEEE Int. Conf. Commun., 2006, pp [8], Bi-directional amplification of throughput in a wireless multi-hop network, in Proc. IEEE Veh. Technol. Conf. - pring, 2006, pp [9] C. E. hannon, Two-way communications channels, in 4th Berkeley ymp. Math. tat. Pro., Chicago, IL, June 1961, pp