NETWORK CODING GAIN OF COOPERATIVE DIVERSITY

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1 NETWORK COING GAIN OF COOPERATIVE IVERITY J Nicholas Laneman epartment of Electrical Engineering University of Notre ame Notre ame, Indiana jlaneman@ndedu ABTRACT Cooperative diversity allows a collection of radio terminals that relay signals for each other to emulate an antenna array and exploit spatial diversity in wireless fading channels Analysis of performance in terms of information-theoretic outage probability leads to two key parameters for these systems, namely, network diversity order and network coding gain, corresponding to the slope and intercept, respectively, in a plot of log-outage versus signal-to-noise ratio (NR) in decibels (db) This paper examines the network coding gains of various cooperative diversity protocols, studying the impact of transmission rate and network geometry on this important parameter In particular, we observe that amplify-and-forward generally outperforms repetition decode-and-forward except when all the relays are much closer to the source than to the destination Coding gains for amplify-and-forward relative to repetition decode-and-forward are determined and shown to match simulation results I INTROUCTION Cooperative diversity [] [3] allows wireless terminals to obtain improved reliability and substantial energy savings by relaying messages for each other in order to propagate redundant signals over multiple paths in the network This redundancy allows the ultimate receivers to essentially average spatial channel variations resulting from physical channel effects such as fading and shadowing or from interference caused by intentional jamming Previous work develops various algorithms for cooperative diversity that have the relays () simply amplify what each receives, or (2) fully decode, re-encode, and retransmit each other s messages In addition, algorithms based upon repetition codes [], [2], channel coding [4], [5], or more general space-time codes [3] have been evaluated, as well as algorithms with and without limited feedback from the ultimate receivers It is demonstrated that cooperative diversity can provide full spatial diversity, as if each terminal had as many transmit antennas as the entire set of cooperating terminals uch diversity gains translate into greatly improved robustness to fading for the same transmit power, or substantially reduced transmit power for the same level of performance Much of the previous work on cooperative diversity has focused on coherent scenarios and Rayleigh fading More recently, the results for coherent scenarios have been extended to more general fading distributions [6] These results develop simple and accurate approximations to the outage probability [7], as a function of the signal-to-noise ratio NR and spectral efficiency R, of the form P out (NR,R) (c(r) NR) d () for large NR, where d > 0 corresponds to the diversity order of the scheme, and c(r) corresponds to the coding gain of the scheme imilar results have been developed for bit-error rates in [8], [9] The results developed in [2], [3], [6] emphasize the fact that cooperative diversity achieves full diversity order Coding gains have been only partially addressed For the case of two cooperating terminals, [2] examines the coding gains in some detail For the case of more than two cooperating terminals, [3] derives the coding gains, but does not fully interpret them or compare protocols; this is the aim of the present paper ifferent coding gains arise depending upon the spacetime coding format as well as the location and/or selection of relay terminals ince this coding gain arises from coding spatially across a network of relays, we call it network coding gain Using high NR approximations to outage performance from [3], [6], this paper studies network coding gains for small networks, eg, cooperative networks with one or two relays, and develops a recursion for computing the coding gains for larger networks Although both amplify-and-forward and repetition decodeand-forward offer the same diversity benefits, amplifyand-forward provides better network coding gains except when the relays are much closer to the source than to the destination Coding gains are derived for amplifyand-forward relative to repetition decode-and-forward and The approximation () is in the sense of (c(r) NR) d P out(nr,r) as NR of 7

2 Transmits 2 Transmits Frequency 3 Transmits Phase I Phase II m Transmits Time Illustration of the two-phases of cooperative diversity algo- Fig rithms Fig 2 Non-cooperative medium-access control Example source allocations among m transmitting terminals across orthogonal frequency channels compared to simulations that illustrate the accuracy of the analysis even for moderate NR Just as classical channel codes can be designed to maximize the coding gain, cooperative diversity scenarios can be designed to maximize the network coding gain; toward this end, the results developed in this paper can be employed as part of higher-level relay location or selection/routing algorithms Frequency PHAE I PHAE II Transmits 2 Repeats 3 Repeats m Repeats 2 Transmits Repeats 2 3 Repeats 2 m Repeats 2 3 Transmits Repeats 3 2 Repeats 3 m Repeats 3 m Transmits Repeats m 2 Repeats m m Repeats m II YTEM MOEL This section highlights the system model that we employ The model is identical to the one developed in [3], allowing generally for repetition-based cooperative diversity as well as space-time coded cooperative diversity In this paper, we focus on the case of repetition-based cooperative diversity for simplicity of exposition The cooperative protocols we study consist of two transmission phases, as in [2], [3] Fig illustrates these two phases In the first phase, the source broadcasts to its destination and all potential relays uring the second phase of the algorithms, the other terminals relay to the destination, either by amplifying what they receive, or by decoding and suitably re-encoding and retransmitting the source message Narrowband transmissions suffer the effects of frequency nonselective Rayleigh fading and additive white Gaussian noise We consider the scenario in which the receivers can accurately measure the realized fading coefficients in their received signals, but the transmitters either do not possess or do not exploit knowledge of the realized fading coefficients As in [3], we focus on the case of slow fading and measure performance by outage probability [7] to isolate the benefits of space diversity We utilize a baseband-equivalent, discrete-time channel model for the continuous-time channel A Medium-Access Control We consider a network consisting of m cooperating terminals, denoted by the set M = {,2,,m} For medium-access control, terminals transmit on essentially orthogonal channels as in many current wireless networks Time Fig 3 Repetition-based medium-access control Example source channel allocations across frequency and relay subchannel allocations across time for repetition-based cooperative diversity among m terminals As a baseline for comparison, Fig 2 illustrates example channel allocations for non-cooperative transmission, in which each transmitting terminal utilizes a fraction /m of the total degrees of freedom in the channel For cooperative diversity transmission, the mediumaccess control protocol also manages orthogonal relaying to ensure that terminals operate in half-duplex mode, ie, they do not transmit and receive simultaneously on the same subchannel Note that these are the same basic restrictions on medium-access control protocols described in [2], [3] Fig 3 illustrates example channel and subchannel allocations for repetition-based cooperative diversity, in which relays either amplify what they receive or fully decode and repeat the source signal, as in [2], [3] In order for the destination to combine these signals and achieve diversity gains, the repetitions must occur on essentially orthogonal subchannels For simplicity, Fig 3 shows channel allocations for different source terminals across frequency, and subchannel allocations for different relays across time More generally, for a given source s and destination d(s), the relays M {s} can repeat in any pre-determined order Arbitrary permutations of these allocations in time and frequency do not alter the conclusions to follow, as long as causality is preserved and each of the subchannels contains a fraction /m 2 of the total degrees of freedom in the 2 of 7

3 channel As in non-cooperative transmission, transmission between source s and destination d(s) utilizes a fraction /m of the total degrees of freedom in the channel imilarly, each cooperating terminal transmits in a fraction /m of the total degrees of freedom B Equivalent Channel Models Under the above orthogonality constraints, we can now conveniently, and without loss of generality, characterize our channel models ue to symmetry of the channel allocations, we focus on transmission of a message from source s to its destination d(s) using terminals M {s} as relays uring the first phase, each potential relay r M {s} receives y r [n] = a s,r x s [n] + z r [n], (2) in the appropriate subchannel, where x s [n] is the source transmitted signal and y r [n] is the received signal at r We use the notation (s) to denote the set of decoding relays for source s For decode-and-forward transmission, if the NR is sufficiently large for relay r to decode the source transmission, then relay r serves as a decoding relay for the source s, so that r (s) For amplify-and-forward transmission, we view (s) as being the entire set of relays for source s, ie, (s) = M {s} The destination receives signals during both phases uring the first phase, we model the received signal at d(s) as y d(s) [n] = a s,d(s) x s [n] + z d(s) [n], (3) in the appropriate subchannel uring the second phase, the destination receives separate retransmissions from each of the relays, ie, for r M {s}, we model the received signal at d(s) as y d(s) [n] = a r,d(s) x r [n] + z d(s) [n], (4) in the appropriate subchannel, where x r [n] is the transmitted signal of relay r In (2) (4), a i,j captures the effects of path-loss, shadowing, and frequency nonselective fading, and z j [n] captures the effects of receiver noise and other forms of interference in the system Note that all the fading coefficients are constant over the example time and frequency axes shown in Figures 2 3 tatistically, we model a i,j as zero-mean, independent, circularly-symmetric complex Gaussian random variables with variances /λ i,j, so that the magnitudes a i,j are Rayleigh distributed Furthermore, we model z j [n] as zero-mean mutually independent, circularly-symmetric, complex Gaussian random sequences with variance N 0 C Parameterizations Two important parameters of the system are the transmit signal-to-noise ratio NR and the spectral efficiency R It is natural to define these parameters in terms of standard quantities in the continuous-time channel with noncooperative transmission (cf Fig 2) as a baseline For a continuous-time channel with total bandwidth W Hz available for transmission, the discrete-time model contains W two-dimensional symbols per second (2/s) If the transmitting terminals have an average power constraint in the continuous-time channel model of P c Joules/s, we see that this translates into a discrete-time power constraint of P := mp c /W Joules/2, since each terminal transmits in a fraction /m of the available degrees of freedom for non-cooperative transmission (cf Fig 2) and repetition-based cooperative diversity (cf Fig 3) Thus, the channel model is parameterized by the random variables NR a i,j 2, where NR := mp c N 0 W = P N 0 is the NR without fading, or transmit NR In addition to NR, transmission schemes are further parameterized by the spectral efficiency R b/s/hz attempted by the transmitting terminals Note that throughout this paper, R is the transmission rate normalized by the number of degrees of freedom utilized by each terminal under non-cooperative transmission, not by the total number of degrees of freedom in the channel III OUTAGE PROBABILITY This section summarizes the outage results of [2], [3], and sets the stage for studying the network coding gains in depth in the sequel As we will see, the coding gains depend upon two factors: the fading statistics, parameterized by λ i,j in our model; and the rate R A Amplify-and-Forward Relaying For amplify-and-forward transmission, the source broadcasts its signal to the destination and all potential relays The relays amplify what they receive, subject to a power constraint, and retransmit in the appropriate subchannel The destination employs maximum ratio combining before decoding the source message This scheme can be viewed as repetition coding from separate transmitters, except that the relay transmitters amplify their own receiver noise As shown in [2], the mutual information obtained by amplify-and-forward is given by I af (NR) = m log( + NR a s,d(s) 2 + f(nr a s,r 2,NR a r,d(s) 2 )), r (s) 3 of 7

4 where (s) = M {s} and f(x,y) = (xy)/(x + y + ) An approximation to the outage probability, suitable for high NR, is [2], [6] P af out (NR,R) := Pr[I af(nr) < R] (c af (R) NR) m, (5) where the network coding gain is given by [ λs,d(s) r c af (R) := (λ ] s,r + λ r,d(s) ) /m (2 mr ) m! (6) B Repetition ecode-and-forward Relaying For repetition decode-and-forward transmission, the source broadcasts its signal to the destination and all potential relays The relays that can decode reliably, ie, those that do not experience outage, are denoted by the decoding set (s) The relays in (s) re-encode and retransmit the information in the appropriate subchannels For simplicity of exposition in this paper, we focus on the case of repetition coding Again, the destination employs maximum ratio combining before decoding the source message As developed in [3], since (s) is a random set, we utilize the total probability law to express the outage probability as P rdf out(nr,r) := Pr[I rdf (NR) < R] = (s)pr[(s)] Pr[I rdf (NR) < R (s)], where, conditioned on a particular (s), the channel mutual information is given by I rdf (NR) = m log( + NR a s,d(s) 2 + NR r (s) a r,d(s) 2 ) An approximation to the outage probability, suitable for high NR, is [3] P rdf out (NR,R) (c rdf(r) NR) m, (7) where the network coding gain is given by c rdf (R) := λ s,d(s) r (s) λ s,r r (s) λ r,d(s) ( (s) + )! (s) /m (2 mr ) (8) Comparing (8) with (6), we observe that the impact of the rate R on the coding gain is identical under the two protocols As a result, the only differences between the performance of amplify-and-forward and repetition decode-and-forward lie in the dependence of the coding gain on the fading average powers λ i,j We will explore these differences in more depth in the sequel We stress that codes more powerful than repetition coding can be utilized within the decode-and-forward relaying framework in order to maximize the coding gain For example, [3] develops outage results for decode-andforward relaying based upon distributed space-time codes In that setting, the outage approximation at high NR has a form very similar to (8): the dependence upon the fading statistics is identical, but the dependence upon the rate is much more involved IV NETWORK COING GAIN As (5) and (7) clearly indicate, cooperative protocols based upon either amplify-and-forward or repetition decode-and-forward relaying achieve full network diversity order m, the number of cooperating terminals As a result, the only differences in outage performance between the two protocols lies in their network coding gains, (6) for amplify-and-forward and (8) for repetition decode-andforward To compare the network coding gains, we specialize to networks of size m = 2,3 We then develop a recurrence relation that allows for computation of the network coding gain for repetition decode-and-forward for general m Finally, we derive closed-form expressions for the symmetric case, ie, λ s,r = λ r,d(s) for each r M {s} For a given source s, we at times order the relays and index them by n =,2,,m, for notational convenience only A One-Relay Case If m = 2, there is one relay, denoted by r, and two possible decoding sets, namely, (s) = or (s) = {r} The coding gain (8) for this case is simply c rdf (R) = [λ s,d(s) (2λ s,r + λ r,d(s) )/2)] /2 (2 2R ) (9) Comparing to amplify-and-forward (6) in this case, c af (R) = [λ s,d(s) (λ s,r +λ r,d(s) )/2] /2 (2 2R ), (0) we see that decode-and-forward has an asymmetry in the emphasis of the source-relay and the relay-destination channels In particular, the statistic λ s,r is weighted twice as much as λ r,d(s) in (9), whereas in (0) both terms are weighted equally Taking the ratio of (0) and (9), we can readily determine that c af(r) c rdf (R) 2, () with the lower bound being met if λ s,r /λ r,d(s) 0, and the upper bound being met if λ s,r /λ r,d(s) This 4 of 7

5 essentially says that, when the source-relay average power is much larger than the relay-destination average power, the two protocols have essentially the same coding gain; on the other hand, when the relay-destination average power is much larger than the source-relay average power, amplifyand-forward has a gain of as much as roughly 5 decibels (db) gain relative to repetition decode-and-forward We stress that this gap can be closed if the source, upon learning that the relay cannot decode, continues its own transmission, as in [2] B Two-Relay Case If m = 3, there are two relays denoted r and r 2 The four possible decoding sets are (s) = (s) = {r } (s) = {r 2 } (s) = {r,r 2 } The coding gain (8) in this case becomes c rdf (R) =[λ s,d(s) (λ s,r {λ s,r2 + λ r2,d(s)/2}+ λ r,d(s){λ s,r2 /2 + λ r2,d(s)/6)}] /3 (2 3R ) (2) Comparing to amplify-and-forward (6) in this case, c af (R) =[λ s,d(s) (λ s,r + λ r,d(s)) (λ s,r2 + λ r2,d(s))/6] /3 (2 3R ), (3) we again see certain asymmetries in the fading statistics Moreover, taking the ratio of (3) to (2), we can easily check that c af(r) c rdf (R) 6/3, (4) with the lower bound being met if λ s,rn /λ rn,d(s) 0, n =,2, and the upper bound being met if λ s,rn /λ rn,d(s), n =,2 In this case, the widest gap of roughly 26 db occurs when both relay-destination average powers are much larger than their corresponding source-relay average powers On the other hand, if λ s,r /λ r,d(s) 0, we have c af(r) c rdf (R) 3/3, (5) with the lower bound being met if λ s,r2 /λ r2,d(s) 0, and the upper bound being met with equality if λ s,r2 /λ r2,d(s) Identical bounds arise if, instead λ s,r2 /λ r2,d(s) 0 and λ s,r /λ r,d(s) varies Here the widest gap of roughly 6 db between the protocols occurs if one source-relay average power is much larger than its corresponding relaydestination average power, and vice versa for the other relay λ s,r λ s,r2 λ s,r3 C 3 () λ r3,d(s) C 2 () C 2 (2) λ r2,d(s) λ s,r2 λ r2,d(s) C () C (2) C (2) C (3) λ r,d(s) 2 λ s,r 2 λ r,d(s) 6 λ s,r 2 λ r,d(s) λ s,r Fig 4 Tree diagram for computation of C 3() in (8) C Recursion for the General Case 6 6 λ r,d(s) For general m, further examination of the decode-andforward coding gain (8) suggests a recursion for its computation pecifically, let C (k) :=λ s,r /(k)! + λ r,d(s)/(k + )! (6) C n (k) :=λ s,rn C (n ) (k) + λ rn,d(s)c (n ) (k + ) (7) for k > 0 and n = 2,3,,m both integers Then (8) becomes c rdf (R) = [ λ s,d(s) C (m ) () ] /m (2 mr ) (8) The intuition behind (6)-(8) is as follows The parameter k counts the number of terminals involved in diversity combining at the destination; the recursion starts with k = in (8) to account for the source-destination transmission The recursion step (7) handles relay r n, incrementing k by one for the case in which the relay can decode The base case (6) treats the first relay r and normalizes the statistics by the appropriate factorial: k! if the relay cannot decode, and (k +)! if the relay can decode We note that, although the notation of the recursion (6)-(7) suggests an ordering of the relays, the coding gain is unchanged under permutations of the indices n =,2,,m For illustration purposes, Fig 4 shows a tree diagram for computation of C 3 () in (8) for the case of m = 4 tarting at leaf nodes, nodes are multiplied by the coefficients along the edges, and the results added, to form the ancestor nodes From this we observe that generally, the tree corresponding to C (n ) () is a subtree of the tree corresponding to C n () for n > Note also that the corresponding computation for the network coding gain of amplify-and-forward (6) can also be viewed as a tree, except that the coefficient at each of the leaf nodes is simply /m! 24 5 of 7

6 m = l = 0 5 db 26 db 34 db 42 db 48 db 0 db 6 db 27 db 36 db 43 db 2-0 db 5 db 26 db 35 db db 4 db 25 db db 3 db db TABLE I MAXIMUM NETWORK COING GAIN FOR AMPLIFY-AN-FORWAR RELATIVE TO REPETITION ECOE-AN-FORWAR GAIN ARE IN APPROXIMATE ECIBEL (B) m = db 7 db 24 db 30 db 35 db TABLE II NETWORK COING GAIN FOR AMPLIFY-AN-FORWAR RELATIVE TO REPETITION ECOE-AN-FORWAR WITH YMMETRIC RELAY, ie, λ s,r = λ r,d(s) FOR EACH r M {s} GAIN ARE IN APPROXIMATE ECIBEL (B) 9 8 Using the tree interpretation, we can easily see that, if there are only l relays, 0 l < m, satisfying λ s,r /λ r,d(s) 0, ie, the source-relay average power is much larger than its corresponding relay-destination average power, then c [ ] af(r) c rdf (R) m! /m, (9) (l + )! with the lower bound being met if when all the remaining relays have λ s,r /λ r,d(s) 0, and with the upper bound being met if all the remaining relays have λ s,r /λ r,d(s) Table I contains maximum network coding gains for amplify-and-forward relative to repetition decode-andforward for up to six cooperating terminals Exact Expression for the ymmetric Case For general m, we now consider the symmetric case in which λ s,r = λ r,d(s), ie, the source-relay and relaydestination average powers are the same, for each potential relay r M {s} Using the tree interpretation of (8), we see that the calculation can be performed by first factoring the term m n= λ rn,d(s) to the root of the tree and multiplying by the sum of the coefficients at the leaf nodes of the tree Thus, the network coding gain for repetition decode-and-forward (8) has C (m ) () = m m λ rn,d(s) n= k=0 ( ) m k (k + )! (20) Recognizing the summation as a hypergeometric series, we have m ( ) m k (k + )! = F ( m;2; ), (2) k=0 cenario # 9 cenario #3 5 cenario #2 cenario #4 Fig 5 Network scenarios with two relays (m = 3) employed for simulations in Fig 6 where F (a;b;z) is the confluent hypergeometric function, or Kummer function, of the first kind [0, Chapter 3] Combining (20) and (2) into (8), we obtain [ ] m /m c rdf (R) = λ s,d(s) λ rn,d(s) F ( m;2; ) n= (2 mr ) (22) in the symmetric case Comparing (22) to the network coding gain for amplify-and-forward (6) in the symmetric case, we obtain c af (R) c rdf (R) = [ ] m! F ( m;2; ) /m (23) 2 (m ) Table II contains network coding gains for amplify-andforward relative to repetition decode-and-forward for up to six cooperating terminals in the symmetric case V NUMERICAL REULT In this section, we incorporate standard path-loss models by setting λ i,j d α i,j, where d i,j is the distance between terminals i and j, and α is the path-loss exponent Then larger average powers correspond to smaller distances between terminals We consider a cooperating group of m = 3 terminals, so that there are two relays Fig 5 depicts the four network geometries we utilize to illustrate the results The simulation results are shown in Fig 6 We observe that (a) the protocols all achieve full network diversity order three and (b) the relative network coding 5 6 of 7

7 irect RF, 3 RF, 2 RF, ; AF, 3 RF, 4 AF, 4 readily developed for other combining algorithms, such as selection combining Furthermore, the protocols do not exploit power control or more general space-time codes among the cooperating terminals; both of these directions represent natural extensions of the results of this paper P out NR (db) Fig 6 imulation results for two relays (m = 3) for the scenarios -4 illustrated in Fig 5 with R = /2 gains predicted by Tables I and II are accurate at moderate to high NR VI CONCLUION This paper determines and compares network coding gains for two classes of low-complexity cooperative diversity protocols, namely, amplify-and-forward and repetition decode-and-forward uch algorithms may be practical in sensor networks, or other networks where terminal complexity must be kept at a minimum Although both classes of algorithms offer full diversity advantage, the results in this paper indicate that amplify-and-forward offers several db of additional coding gain for many network geometries, and performs at least as well for those geometries in which the relays are much closer to the source than to the destination We note that the protocols under consideration employ maximum ratio combining, but similar results could be ACKNOWLEGMENT This work has been supported in part by the tate of Indiana through the Twenty-First Century Research and Technology Fund and by NF through grant EC REFERENCE [] A endonaris, E Erkip, and B Aazhang, User Cooperation iversity, Parts I & II, IEEE Trans Commun, vol 5, no, pp , Nov 2003 [2] J N Laneman, N C Tse, and G W Wornell, Cooperative iversity in Wireless Networks: Efficient Protocols and Outage Behavior, IEEE Trans Inform Theory, 2003, accepted for publication [Online] Available: jnl/pubs/it2002 pdf [3] J N Laneman and G W Wornell, istributed pace-time Coded Protocols for Exploiting Cooperative iversity in Wireless Networks, IEEE Trans Inform Theory, vol 49, no 0, pp , Oct 2003 [4] T E Hunter and A Nosratinia, Cooperation iversity through Coding, in Proc IEEE Int ymp Information Theory (IIT), Lausanne, witzerland, July 2002, p 220 [5] A tefanov and E Erkip, Cooperative Information Transmission in Wireless Networks, in Proc Asian-European Information Theory Workshop, Breisach, Germany, June 2002 [6] J N Laneman, Limiting Analysis of Outage Probabilities for iversity chemes in Fading Channels, in Proc IEEE Global Comm Conf (GLOBECOM), an Francisco, CA, ec 2003 [7] L H Ozarow, hamai (hitz), and A Wyner, Information Theoretic Considerations for Cellular Mobile Radio, IEEE Trans Veh Technol, vol 43, no 5, pp , May 994 [8] A Ribeiro, X Cai, and G B Giannakis, ymbol Error Probabilities for General Cooperative Links, IEEE Trans Wireless Commun, 2005, to appear [9] Z Wang and G B Giannakis, A imple and General Parameterization Quantifying Performance in Fading Channels, IEEE Trans Commun, vol 5, no 8, pp , Aug 2003 [0] M Abramowitz and I A tegun, Eds, Handbook of Mathematical Functions New York: over, June of 7