Max-Min Relay Selection for Legacy Amplify-and-Forward Systems with Interference

Transcription

1 1 Max-Min Relay Selection for Legacy Amplify-and-Forward Systems with Interference Ioannis Kriidis, Member, IEEE, John S. Thompson, Member, IEEE, Steve McLaughlin, Senior Member, IEEE, and Norbert Goertz, Senior Member, IEEE Abstract In this paper, an Amplify-and-Forward (AF) cooperative strategy for interference limited networs is considered. In contrast to previously reported wor, where the effect of interference is ignored, the effect of multi-user interference in AF schemes is analyzed. It is shown that the interference changes the statistical description of the conventional AF protocol and a statistical expression is subsequently derived. Asymptotic analysis of the expression shows that interference limits the diversity gain of the system and the related channel capacity is bounded by a stationary point. In addition, it is proven that previously proposed relay selection criteria for multi-relay scenarios become inefficient in the presence of interference. Based on consideration of the interference term, two extensions to the conventional max min selection scheme suitable for different system setups are proposed. The extensions investigated are appropriate for legacy architectures with limitations on their flexibility where the max min operation is pre-designed. A theoretical framewor for selecting when to apply the proposed selection criteria is also presented. The algorithm investigated is based on some welldefined capacity approximations and incorporates the outage probabilities averaged over the fading statistics. Analytical results and simulation studies reveal enhancements of the proposed algorithm. Index Terms Cooperative diversity, relay channel, interference, outage probability, wireless networs. I. INTRODUCTION COOPERATIVE diversity is of considerable interest due to its ability to provide the benefits of spatial-diversity to single antenna systems. It is based on the broadcast nature of the wireless medium and exploits spatial diversity through distributed relays which overhear the transmission and forward a noisy version of the received signal. Since the wor of Sendonaris et al. [1], [2] which introduced the cooperative concept, a number of relaying protocols have been proposed in the literature. Amplify-and-Forward (AF) protocols are of interest since the relay nodes simply retransmit an amplified version of the received signal in the analogue domain and thus have a low complexity. In addition to complexity benefits, it has been shown in [3], [4] that AF asymptotically approaches the Decode-and-Forward (DF) scheme with respect to diversity. Furthermore, as pointed out in [5], in some cases avoiding Manuscript received March 20, 2008; revised August 12, 2008 and December 29, 2008; accepted February 27, The editor coordinating the review of this paper and approving it for publication was Dr. R. Nabar. I. Kriidis, J. S. Thompson and S. McLaughlin are with the Institute for Digital Communications, University of Edinburgh, EH9 3JL, UK ( i.riidis@ed.ac.u; john.thompson@ed.ac.u; steve.mclaughlin@ed.ac.u). N. Goertz is with the Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology, Austria ( norbert.goertz@ieee.org). decoding the signal at the relay nodes can prevent excessive error propagation. Although AF is a well-studied protocol in the literature, existing wor has focused on ideal configurations with no interference present during the cooperation process. Simple three-node topologies where a dedicated relay forwards the signal of one source [6]-[8] are generally assumed. Even though this assumption simplifies theoretical studies and provides useful capacity-bounds, it cannot represent practical scenarios with simultaneous multi-user transmissions. More realistic scenarios concern multi-hop topologies where the source signal arrives at the destination via a sequence of many hops and many relays [9]-[12]. However, this wor also focused on a single source-destination transmission and thus multi-user interference is negligible. Alternatively, multiaccess relay techniques deal with many sources which share the same relay in order to deliver their data to a single destination [13], [14]. However, these approaches analyze the system from an information theory point of view and assume interference cancelation techniques at the destination. The problem of simultaneous independent transmissions for adhoc networs is studied in [15] for two sources and two destinations, but the case considered concerns the DF strategy and uses a full-duplex constraint. There are related wors which concern code-division multiple-access (CDMA) cellular systems where the code-orthogonality simplifies the analysis [16]. Currently there is interest in relay selection schemes for multi-relay environments. In these approaches, selecting one relay based on instantaneous channel conditions offers the same diversity benefits with a lower complexity than distributed space-time codes (DSTC) [17] which require simultaneous relay transmissions [18]-[22]. However, the proposed relay selection criteria lac the flexibility to deal with the presence or absence of interference effects and remain optimal for ideal scenarios without interference. To the best of our nowledge, the statistical behavior of the AF scheme for ad-hoc scenarios with multi-user interference has not yet been reported in the literature. In this paper, the basic AF protocol [3] is considered when external structured interference affects the cooperation process. A similar problem is discussed in [23] but for a different context. The channel model considered corresponds to scenarios where inter-cell power control cannot be performed [24], [25]. This represents practical ad-hoc systems where neighboring clusters would not be perfectly isolated and inter-cluster interference would result in performance degradation. The scope of this paper is to analyze the impact of the out-of-cell structural/unmanaged

2 2 interference in AF ad-hoc networs. More specifically, it is shown that interference modifies the conventional AF statistic expression for interference-free environments [7] and the related system capacity converges to a well-defined static point. Based on this modified asymptotic expression, two novel max min-based relay selection criteria which are optimal for interference-limited systems with many relays are investigated. These criteria tae into account the interference component and are suitable for legacy architectures which have been predesigned based on a conventional max min hardware core [26]-[29]. It is proven that the suitability/optimality of each selection criterion depends on the average system parameters and therefore selection between them is necessary. In order to support this, a theoretical framewor which provides the operation regions for each selection criterion is developed. The related algorithm uses the outage probability criterion and incorporates some well-investigated capacity approximations of the proposed selection policies. It must be emphasized that the wor described so far focuses on the consideration of the interference only during the AF relaying process (at the relays) and does not deal with the interference at the destination. However, it can straightforwardly be extended to the asymmetric problem which assumes severe interference at the destination and negligible interference at the relays. The main contributions of this paper are threefold. 1) The behavior of an Amplify-and-Forward scheme for ad-hoc systems with inter-cluster interference is studied. A modified expression which taes into account the interference component is derived and it is proven that it is asymptotically bounded below a well-defined static point. 2) Two novel max min-based relay selection criteria are investigated which are appropriate for interferencelimited environments with many relays. These criteria are suitable for legacy architectures which have been designed for non-interference environments and thus do not require complicated modifications. Capacity approximations are proposed which simplify significantly the related analysis. 3) As the proposed relay selection criteria are suitable for different system configurations and constraints, a theoretical framewor which decides the best selection scheme is described. The proposed algorithm appropriately incorporates the investigated capacity approximations and uses only an average nowledge of the channel statistics. The paper is organized as follows. Section II introduces the system model and presents a statistical expression for interference-based AF. Section III deals with the relay selection criteria for interference-limited systems and presents their asymptotic analysis. A theoretical framewor with the decision metric for the appropriate relay selection scheme is proposed in Section IV. Numerical results are shown and discussed in Section V, followed by concluding remars in Section VI. II. SYSTEM MODEL As the target of this wor is the analysis of the inter-cluster interference, a simple ad-hoc configuration which consists of ( S relay ) (Cluster C) (S) (D) I I F (S ) Source-Destination lin (Cluster C' ) (D ) Source-Relay lin Fig. 1. The system model; C: cluster of interest (communication via cooperative lin), S: source, D: destination, S relay : relay cluster; C : neighboring cluster (interference via direct lin), S : source, D : destination (final destination or intermediate relay), I INF : interference signal for the -th relay (S ). two neighboring clusters (C, C ) is assumed, the cluster of interest being C without loss of generality. Fig. 1 schematically presents the considered channel model. The cluster of interest (based on the cooperative concept) is composed of one source S, one destination D and a set S relay = {1, 2,..., K of potential K relays. Similar single cluster relay settings have been studied in [11]-[12] for different contexts. For clarity of exposition it is assumed that the neighboring cluster uses direct transmission from one source S to one destination D (could be a relay or the final destination). More complicated interference structures (i.e. relaying at both clusters, asynchronous sources) are beyond the scope of this paper and will be dealt with in future wor. All nodes are half-duplex and thus cannot transmit and receive simultaneously. In order to focus our wor on the relaying process, it is assumed that a direct lin S D is not available in the cluster of interest (i.e. S D channel is deeply shadowed) and communication can be established only via relays [30], [31]. More specifically, the communication in the cluster C is divided into two orthogonal channels (i.e. time slots) in order to support Orthogonal Amplify-and-Forward (OAF) cooperation [3]. In this protocol, the source terminal communicates with the relay terminal in the first orthogonal channel (i.e. time slot). In the second time slot, only the selected relay communicates with the destination terminal. Furthermore, the interference (inter-cluster interference) is assumed to degrade the S lins but is negligible at the destination. This assumption isolates the impact of the structural interference on the AF process and simplifies the analysis. The interference considered can be regarded as a direct lin between the neighboring source S and the relays S relay (S ). The system model can then be described by the following set of equations y = f S, x {{ desired signal + f S, x {{ interference signal +n, y D =f,d(gy ) + n D, (1)

3 3 where y and y D denote the received signal at the -th relay and the destination, respectively; x is the transmitted signal in the cluster of interest with an average transmitted power normalized to unity (x for the neighboring cluster); f A,B denotes the channel coefficient term for the lin A B with the channel reciprocity property f A,B = f B,A ; n A is the additive noise term for at the reception node A; S relay denotes the selected relay for the AF relaying process; G is the AF amplification factor (its definition is presented in the following subsection). Statistically, f A,B is modeled as a zero-mean, independent, circularly-symmetric, complex Gaussian random variable with variance σa,b 2. Furthermore, the additive noise samples n and n D are modeled as zeromean mutually independent, circularly-symmetric, complex Gaussian random variables with variance N 0. Note that, although the transmitted power and noise levels have been parameterized to be symmetric throughout the networ for purposes of exposition, asymmetries in the average signal-tonoise ratio (SNR) and the path-loss can be lumped into the fading variances. Furthermore, for the clarity of exposition, it is supposed that σs, 2 = σ2,d (symmetric hops for the cluster of interest) and the average signal-to-interference power ratio L E(γ S, )/E(γ INF ) = σs, 2 /σ2 S,, where E( ) denotes the expectation of a random variable, γ S, is the SNR of the lin S and γ INF γ S, denotes the interference-tonoise ratio (INR) for the -th relay. However, our analysis can be extended straightforwardly to completely asymmetric configurations (σs, 2 σ2,d ). Note that the parameter L controls the level of interference in the cluster of interest and differentiates the problem under consideration from previously reported studies. A. Amplification factor In the AF case, the relay nodes can not differentiate between source and interference signals. The amplification process is performed in the analogue domain and thus consists of a simple normalization of the total received power without further processing. Therefore, dirty paper coding theory [32] and interference mitigation techniques cannot easily be performed at the relays. The amplification factor can be expressed as P S G = f S, 2, (2) P S + f S, 2 P S + N 0 where G is the AF amplification factor and P S (P S ) denotes the source s transmitted power for the cluster of interest (neighboring cluster). For our system model, it is assumed that the source powers (P S = P S = 1) are normalized to unity and therefore power control issues are not discussed. B. Statistical description Assuming coherent detection with perfect channel estimation, the resulting signal-to-interference and noise ratio (SINR) of the decision variable can be obtained from Eq. (1) as G 2 f S, 2 f,d 2 P S γ D = G 2 f S, 2 f,d 2 P S + f,d 2. (3) G 2 N 0 + N 0 Substituting (2) into (3), we obtain γ D = γ S, γ,d. (4) γ INF (γ,d + 1) + γ S, + γ {{,D + 1 {{ Θ (Interference-based term) Φ (Conventional-based term) Therefore, the impact of the interference during the AF process modifies the statistical description of conventional AF scheme [9] by adding the parameter Θ. This additional term in the denominator of the SINR significantly complicates the process of finding a closed form expression for the probability density function of γ D. However, in the following Sections, some useful approximations and asymptotic bounds which simplify the related analysis are proposed. C. Asymptotic description For high SNRs (SNR ), the denominator terms in Eq. (4) reduce to one term. More specifically, the second order term (included in Θ) becomes significantly higher than the linear terms (i.e. E[γ INF ]E[γ,D] E[γ INF ]+E[γ S, ]+ E[γ,D]) and thus dominates the denominator. Therefore, the SINR expression results in a simplified form which can be written as γ D γ S, γ,d γ INF γ,d = γ S,, (5) γ INF which is the ratio between the SNR of the first hop and the INR of the interference. Therefore, at the high SNRs, the statistical description of the system is independent of the second hop ( D). In this case, the probability density function (PDF) is given in closed form and corresponds to an F distribution (ratio between two exponential random variables). More specifically, the asymptotic description of the system is given as [33] p Γ (γ) = L (L + γ) 2, P Γ(γ) = γ L + γ, (6) where p Γ ( ) and P Γ ( ) denote the PDF and the cumulative distribution function (CDF) of the SINR, respectively. Although this wor focuses on the impact of interference at the relays (I INF ) and therefore assumes negligible interference at the destination (I INF ), it can be extended straightforwardly to the latter problem. More specifically, if it is assumed that the interference at the relays is negligible in comparison with the interference at the destination (I INF I INF ), in a similar manner to the previous analysis, it can be proved that the statistic of this system is described as γ D = γ S, γ,d γ INF (γ S, + 1) + γ S, + γ,d + 1 {{{{ Θ (Interference-based term) Φ (Conventional-based term) γ,d. (For high SNRs) (7) γ INF In this case, the interference parameter affects the second cooperative slot and thus this slot becomes a bottlenec for the performance of the system. This symmetric behavior of the

4 4 system allows one to generalize the following relay selection metrics for AF scenarios with severe interference degradation at the destination. III. RELAY SELECTION FOR AF WITH INTERFERENCE A. Conventional relay selection First consider the conventional relay selection policy which is used in no-interference configurations. The selection policy considered was proposed in [18] and is suitable for distributed implementation. It requires the instantaneous signal strength (SNR) between the lins S and D ( S relay ) and the selected relay is chosen to maximize the minimum between them. The conventional relay selection policy can be expressed as Conv = arg max min, γ,d. (8) S relay The conventional relay selection criterion ensures that the relay with the best end-to-end path between source and destination is used and provides diversity gain on the order of the number of relays [18]. However, this selection criterion has been designed for environments without interference and thus does not tae into account the effects of interference. In the wor presented here, the relay selection criteria investigated can be regarded as an extension of this basic selection scheme. 1) Max-min selection for AF systems: In this wor, we focus on the conventional max min selection criterion for AF systems. This selection policy approximates the optimal AF selection for non-interference environments and optimizes the required computational overhead. More specifically, this fundamental choice is justified based on the following arguments: The max min criterion is an efficient selection metric for both AF and DF techniques. It is the optimal solution for DF protocols and efficiently approximates the performance of the optimal AF selection scheme, which is based on instantaneous AF statistics (maximum harmonic mean). In Appendix A we prove that a max min selection approximates the optimal AF selection. The physical intuition of this basic result has to be expected as the worst path of a relaying lin dominates the end-to-end capacity for both AF and DF strategies. The max min criterion can be adopted as a general selection technique for both AF and DF techniques. The max min criterion has some interesting computational properties, which can simplify the implementation complexity. 1) The selection metric does not involve computational operations (multiplications, additions, and divisions) and thus corresponds to a low complexity hardware core. If the complexity overhead for the computation of the instantaneous AF statistic is equal to 2 multiplications (1 multiplication and 1 division) the total complexity is equal to 2K, where K is the number of relays. 2) It can easily be extended for interference environments without complicated modifications and is amenable to a distributed implementation. In this paper we deal with AF systems, which have been designed based on a conventional max min selection criterion. We assume that the basic min structure follows a legacy architecture and thus cannot deal with potential interference. The optimization of the legacy max min metric for interference environments is the main objective of this wor. B. Asymptotic relay selection The first proposed relay selection criterion is motivated by the simplified expression of the system at high SNRs. As has been seen in Eq. (5) the asymptotic behavior of the system converges to the ratio between S and interference lins. Therefore, from an asymptotic point of view, an appropriate relay selection is to choose the relay which gives the maximum value of this ratio. The asymptotic relay selection criterion can be expressed as Asym = arg max S relay { γs, γ INF. (9) 1) Outage performance: Outage probability is an important tool in the analysis of ergodic (quasi-static) systems. It is defined as the probability that the instantaneous capacity is lower than the required spectral efficiency. Moreover, it has been shown in [34] that the error probability is dominated by the outage probability in the high SNR regime for long codewords. Based on Appendix B, the outage probability for the asymptotic relay selection can be expressed as [ ] K 2 P {C 2R0 1 Asym < R 0 =, (10) L + 2 2R0 1 where C Asym denotes the instantaneous capacity for the asymptotic relay selection and R 0 denotes the spectral efficiency. 2) Distributed implementation: The proposed asymptotic selection criterion is suitable for distributed implementation. It requires only the SNRs of the S lins and the INRs of the interference lins which can be estimated by the relay nodes during the first-slot transmission [18]. This estimation can be performed locally (each relay estimates its corresponding ratio from the received signal) without the need for dedicated channel feedbac. Note that the absence of the second-hop lin ( D) from the selection function gives the proposed asymptotic selection a lower complexity than this one in [18], which requires channel nowledge of the D lins during a feedbac-based set-up period. C. Conventional-based semi-asymptotic relay selection The semi-asymptotic selection criterion combines the previous two criteria in Eq. s (13) and (9). It taes into account the conventional and the asymptotic system behavior and it is suitable for intermediate SNR regions which experience a mixture of conventional and asymptotic SNR characteristics. The conventional semi-asymptotic selection policy is a natural extension of the conventional selection scheme and is motivated by the expression of the general statistics (Eq. (4)), where interference is presented as an additional term (Θ) in

5 5 the denominator of the conventional statistics. It is expressed as a simple ratio between the conventional min operation and the interference term and can be written as { { min γs,, γ,d Semi = arg max S relay γ INF. (11) The semi-asymptotic policy based on the conventional approach does not require complex structural modifications to the conventional mechanism. As the interference term is taen into account independently of the min operation, the mechanism which deals with the instantaneous feedbac in the cluster C remains fixed and equivalent to the noninterference case [18]. Therefore, it is appropriate for legacy relay architectures with adaptivity/reconfigurability limitations where the min operation between the two partial SNRs is predesigned [26]-[29]. Systems that already use the conventional relay selection can easily be updated to the proposed semiasymptotic selection without modifying the min operation. Furthermore, it is suitable for systems with mobility that dynamically and continuously change between interference and non-interference environments and where reconfiguration time becomes critical. The proposed policy efficiently fills the gap between conventional and asymptotic relay selection and is introduced as a general extension of existing selection schemes for interference environments. More specifically, all of the conventional relay selection strategies can be extended to interference environments by a simple division with the instantaneous interference term. However, although this criterion has a very good performance for strong interference environments and scenarios where E[γ,D ] > E[γ S,D ], in general is a suboptimal approach as the interference term is considered outside of the min process. More advanced selection techniques elaborate the interference term into the min mechanism and can give reliable selection for all the interference scenarios. However, in this wor, we focus on this simple extension of the conventional relay selection schemes as it does not change the basic (legacy) structural core of the system and provides some interesting analytical results. 1) Outage performance: From Appendix C the outage probability of semi-asymptotic relay selection is bounded by P {C Semi < R 0 = 1 [ 2 2R 0 ] K [ 2(2 2R 0 ] K 1), 2 L + 2 2R0 1 2 L + 2(2 2R0 1) (12) where C Semi denotes the instantaneous capacity for the semiasymptotic relay selection. 2) Distributed implementation: In contrast to the asymptotic criterion which is independent of the D lins, the semi-asymptotic selection requires their instantaneous estimation. In this case the corresponding distributed implementation becomes equivalent to the solution proposed in [18] which uses the Request-to-Send/Clear-to-Send (RTS/CTS) signaling from the based protocols in order to estimate the source-relay as well as the relay-destination lins at the relays. Furthermore, according to the asymptotic selection, the interference term can be estimated by processing the received signal. However, as mentioned before, the consideration of the interference component outside of the min operation in the relay selection criterion does not change the conventional signaling or selection mechanism. The interference is considered only in the local relay timers in order to access the channel in a distributed fashion. D. An optimal max-min selection for AF with interference In contrast to the previous conventional-based semiasymptotic relay selection, this scheme taes the interference into account in the min operation. This selection criterion is similar to the conventional max min criterion but the instantaneous SNR of the source-relay lin is replaced by the corresponding SINR. Its implementation requires a redesign of the min operation (and the related mechanism) and therefore is suitable for systems with a degree of flexibility. The consideration of the interference only in the source-relay term, overcomes performance limitations of the conventionalbased semi asymptotic selection and avoids misleading relay selections (i.e. scenarios where γ S, > γ,d ). This optimal max min selection criterion can be expressed as Opt = arg max S relay min { γs, γ INF, γ,d. (13) The above elaboration of the interference term can also be applied to other conventional relay selection schemes by replacing the SNR of the source-relay lin with the corresponding SINR. This optimal max min behavior as well as its extension to other relay selection strategies are discussed in Section V based on numerical simulations. IV. SWITCHING ALGORITHM FOR RELAY SELECTION CRITERIA Motivated by the general statistic of Eq. (4), we proposed a logical extension of the max min metric by dividing the conventional min operation with the instantaneous interference term. This metric is an extension of the conventional max min criterion and corresponds to legacy architectures with a pre-designed hardware core. However, the continuous use of this metric is not always the appropriate solution for the system under consideration. Although the semi-asymptotic selection uses the instantaneous quality of all the lins involved, is not an optimal selection metric. It is a suboptimal strategy which allows legacy max min systems to deal with interference. Therefore, although semi-asymptotic selection is an efficient solution for the intermediate SNR, the conventional max min and the asymptotic selection are the suitable choices for the low and the high SNR, respectively. In this Section, the SNR regions (based on average values) are investigated and the switching criteria are defined in relation to them. It is worth noting that the direct use of Eq. (4) (instantaneous statistic) corresponds to the optimal selection for AF with interference but is not applicable for the max min legacy system considered. Therefore it consists of a useful capacity bound and is used for comparison reasons in Figures 8 and 9. Furthermore, except that the proposed max min expressions improve performance according to the average

6 6 TABLE I THE max min EXPRESSIONS AND THE RELATED FEEDBACK FOR EACH RELAY NODE. Feedbac (for each relay) Conventional selection Semi-asymptotic selection Asymptotic selection source-relay interference relay-destination channel, they optimize the complexity overhead of the system. More specifically, at the low SNRs the application of the conventional selection does not need the interference term and thus the related interference estimation is avoided. Accordingly, at high SNRs the asymptotic selection does not need the estimation of the relay-destination lins and thus the complexity is also reduced (also we can avoid the feedbac for the relay-destination lins). Table 1 summarizes the proposed max min techniques and the related complexity (feedbac). A. Low SNRs- Conventional relay selection The conventional criterion is suitable for low SNR regimes where additive white Gaussian noise (AWGN) is the dominant system degradation. At low SNRs, the considered structural interference is negligible in comparison to Gaussian noise and thus can be neglected from the relay selection process. Based on the dominator of Eq. (4), the conventional relay selection criterion (Eq. (13)) can be applied when Φ Θ on average. In this case, the statistics of the system are simplified to the conventional non-interference form and therefore the conventional relay selection is the appropriate criterion. An upper-limit on SNR (of each hop in C) for which the conventional relay selection should be used, is obtained from the point which satisfies the condition Φ = Θ on average. At this point, the structural interference term is equivalent to the conventional term and thus has to be considered in the selection mechanism. For the symmetric configuration under consideration, the proposed upper-limit is equal to E(Φ) = E(Θ) γ INF γ + γ INF = 2 γ + 1 γ INF γ 2 γ γ INF 2 γ Low = 2 L, (14) where E(γ S,) γ INF denotes the average INR of the interference lin, E(γ S, ) γ is the average SNR for each hop in the cluster of interest, L = γ/ γ INF according to the system model under consideration and γ Low is the SNR upper-limit for the conventional relay selection. The above approximation assumes that the term Φ is dominated by the component of the second order. Therefore, the SNR region which corresponds to the conventional relay selection criterion is expressed as ( γ Low ]. Note that the upper-limit of this region is independent of the number of relays and is only a function of the interference parameter (L). B. Intermediate SNRs- Conventional-based semi-asymptotic relay selection In this SNR regime, there is not a dominant degradation term (Φ or Θ) and both components have to be taen into account by the relay selection process. In this case, the proposed semi-asymptotic selection criterion is the appropriate one as it represents the combination of conventional and asymptotic system behavior. Although the lower-limit of this SNR regime is given by the previously defined limit ( γ Low ), the definition of its upper-limit is not straightforward. In order to simplify its definition, an approximation to the system statistic is adopted when the asymptotic selection criterion is always used. 1) Approximation for the asymptotic relay selection: This approximation is based on the observation that the general system statistic in Eq. (4) is a mixture of the conventional and the asymptotic statistic. The proposed approximation decomposes the general statistic into two basic components and assumes each time that one component dominates the general statistic. As a dominant component, it is clearly that with the minimum value. Therefore, the asymptotic relay selection criterion can be approximated as γ Asym = min { γs, γ 0 0,D γ S,, Asym, (15) γ S, 0 + γ 0,D γ INF {{ Asym {{ conventional component asymptotic component where γ Asym denotes the approximation of the asymptotic relay selection statistic and 0 = arg. The proof of the proposed approximation is given in Appendix D. The proposed expression simplifies the computation of the related outage probability, and as will be illustrated in Section V, is an efficient approximation of the statistic at medium SNRs. It should be emphasized that this expression, in addition to the efficient approximation of the outage bound at the high SNRs, also provides a good approximation to the diversity slope at intermediate SNRs. As will be seen in the following discussion, this property is used in order to define the crossover point between the asymptotic selection and the semiasymptotic bound. The outage probability of the asymptotic relay selection based on the approximate expression in Eq. (15) is given as [ { ( 2 2R 0 ) K ] 1 P CAsym R 0 = 1 1 L + 2 2R0 1 K ( ) [ ] K 1 ( + 1)(2 2R0 1)α ( 1) +1, =1 (16) where C Asym is the instantaneous capacity for the approximated asymptotic selection and α = 1/ γ. The derivation of the outage probability can also be found in Appendix D.

7 7 2) Definition of the upper-limit: The upper-limit under question can be calculated by using the outage probabilities. As the SNR upper-limit for the semi-asymptotic selection simultaneously is a SNR lower-limit for the asymptotic selection, it is obvious that their outage probabilities are equivalent at this average SNR. Therefore, the SNR upper-limit is defined as the crossover point between the outage probabilities of the approximated asymptotic selection and the bound of the semi-asymptotic selection. Equating Eq. s (12) and (16) and then solving the system for the average SNR parameter (1/α), results in { P {C Semi R 0 = P CAsym R 0 1)(U γ High = 2(22R0 1 1), (17) U 1 U 2 ( 2 where U 1 = 2R 0 K, ( 1 L+2 2R 0 1) U2 = 2(2 2R 0 K 1) L+2(2 2R 0 1)) and γhigh ( 1/ᾱ High ) denotes the upper-limit of the region where the semi-asymptotic relay selection criterion is applied. Therefore the intermediate SNR region is defined as ( γ Low γ High ]. C. High SNRs- Asymptotic relay selection The third SNR region is related to the high SNRs and corresponds to the asymptotic relay selection criterion. According to the above discussion, this region is defined as ( γ High + ). V. NUMERICAL RESULTS Computer simulations have been carried out in order to validate the performance of the proposed relay selection schemes. The simulation environment follows the system model of Section II. The first simulation results deal with the low SNR regions and focus on the conventional relay selection scheme. Figures 2, 3 compare the Bit Error Rate (BER) performance versus the SNR ( γ) of different selection schemes for the case of K = 2 relays, signal-to-interference power ratio L = 11 and K = 4 relays, L = 4, respectively. An uncoded system with perfect channel estimation and binary phase shift eying (BPSK) modulation is assumed without loss of generality. The conventional selection scheme is compared with the semi-asymptotic and the asymptotic selection policies. The random relay selection (non-selection), the selection of the relay with the minimum interference lin as well as the relay selection with the best S lin are used as reference BER curves. Firstly, it can be seen that the structural interference under consideration bounds the diversity gain of the AF cooperation and yields a convergence region for all of the curves. However, the selection process significantly improves the system performance. The comparison of the selection schemes with the non-selection policy shows that selection is a useful tool in decreasing interference effects and thus increasing the value of the capacity bound. The second important observation is that the three selection criteria considered are the most efficient. Their comparison with the reference selection criteria (minimum interference and best S ) show that they are the most appropriate ones as they have been designed according to the statistical behavior of Bit Error Rate Crossover point correspoinding to γ Low Non selection ] Conventional selection min {γ INF Asymptotic selection ]γ INF Semi asymptotic selection Average E b /N 0 Fig. 2. BER performance for different selection schemes; uncoded system, BPSK modulation, K = 2 relays, L = 11 and γ Low = 13.4 db (Eq. (14)). Bit Error Rate 10 3 Crossover point correspoinding to γ Low Non selection ] Conventional selection min {γ INF Asymptotic selection ] Semi asymptotic selection Average E /N b 0 Fig. 3. BER performance for different selection schemes; uncoded system, BPSK modulation, K = 4 relays, L = 4 and γ Low = 9 db (Eq. (14)). the considered interference-limited AF system. Furthermore, it can be seen that the conventional relay selection criterion outperforms the semi-asymptotic and asymptotic schemes at low SNRs. In this region, AWGN dominates the system degradation and therefore the conventional criterion is the most efficient selection scheme. As far as the upper-limit ( γ Low ) of the corresponding SNR region is concerned, the proposed limit is an efficient estimate. In both cases, the estimated limit which is given by Eq. (14) approaches the real crossover point between the conventional and semi-asymptotic curves. The following simulation results focus on the semiasymptotic and asymptotic selection criteria and the efficiency of their switching algorithm. Figures 4, 5 and 6 present the outage probability performance versus the SNR ( γ) of the semiasymptotic and asymptotic selection policies as well as the proposed theoretical approximations for different interference levels (L = 10, 50, 100). The considered spectral efficiency

8 L=10 Crossover point corresponding to γ High L=10 Outage Probability Crossover point corresponding to γ High L=50 Outage Probability 10 3 L= Real performance (simulation) Approximation (simulation) Approximation (theory) L= Real performance (simulation) Approximation (simulation) max {γ /γ Approximation (theory) S, INF L=100 ] Real performance (simulation) ] Real performance (simulation) ] Asymptotic bound (theory) ] Asymptotic bound (theory) Average E b /N Average E b N 0 Fig. 4. Outage probabilities for asymptotic-based relay selection schemes; K = 2 relays, L = 10, 50, 100, R 0 = 2 BPCU and γ High = [ ] db (Eq. (17)). Fig. 6. Outage probabilities for asymptotic-based relay selection schemes; K = 6 relays, L = 10, 50, 100, R 0 = 2 BPCU and γ High = [ ] db (Eq. (17)) Crossover point corresponding to γ High L= R 0 =5 BPCU Outage Probability Outage Probability Real performance (simulation) L= Approximation (simulation) Approximation (theory) ] Real performance (simulation) ] Asymptotic bound (theory) Average E b /N 0 Real performance (simulation) Approximation (simulation) Approximation (theory) ] Real performance (simulation) ] Asymptotic bound (theory) Crossover point corresponding to γ High R 0 =3 BPCU Aerage E b /N 0 Fig. 5. Outage probabilities for asymptotic-based relay selection schemes; K = 4 relays, L = 10, 100, R 0 = 2 BPCU and γ High = [ ] db (Eq. (17)). Fig. 7. Outage probabilities for asymptotic-based relay selection schemes; K = 5 relays, L = 100, R 0 = 3, 5 BPCU and γ High = [ ] db (Eq. (17)). is R 0 = 2 bit per channel use (BPCU). The plotted curves validate that the semi-asymptotic criterion outperforms the asymptotic selection at the intermediate SNR regions but it is outperformed by the asymptotic selection at the high SNRs. The corresponding gain is increased as the the interference factor is decreased (L is increased). For example, the gain of the semi-asymptotic selection against the asymptotic one is 5 db for a outage probability with K = 4, L = 100 (in Fig. 5). On the other hand, the asymptotic selection is used when the semi-asymptotic selection has already converged and therefore the related gain is not meaningful. Furthermore, it can be seen that the proposed analytical expressions estimate the true performance efficiently. The proposed asymptotic bound for the semi-asymptotic selection policy approaches the real capacity bound for almost all cases. As for the accuracy of the proposed approximation (asymptotic relay selection) is concerned (Eq. (15)), it can be seen that it is a reliable upper-bound which efficiently approaches the true diversity slope and converges to the real capacity bound. The quality of the approximation improves as the interference factor (L) and the number of users (K) are increased (L = 10 vs L = 100 in Fig. 5, K = 2 vs K = 6 in Fig. s 4, 6). These observations are in line with the analysis presented in Appendix D. More specifically, the proposed approximation is a well-defined upper-bound and therefore yields a better outage probability than the true performance. Furthermore, it is based on two simplified expressions which depend on the parameters L, K. More specifically, the approximation given in Eq. (27) becomes more reliable for high L, and this one given in Eq. (28) holds for high values of K.

9 9 Outage Probability 10 0 maxmin max{(γ S, γ,d )/(γ S, +γ,d ) max ] max{ [(γ S, γ,d )/(γ S, +γ,d )] max maxmin max{([γ S, )γ,d ]/[(γ S, )+γ,d ] max{(γ S, γ,d )/(γ S, +γ,d +γ INF (γ,d +1)) Average E b /N 0 Fig. 8. Outage probabilities for conventional max min, conventional max harmonic mean (AF), asymptotic selection, conventional-based semiasymptotic selection, optimal max min selection ( Sec. III.D) and optimal selection (maximization of Eq. (4)); K = 4 relays, L = 10, 50, R 0 = 2 BPCU and E[γ S, ] = E[γ,D ]. Outage Probability 10 0 maxmin max{ (γ S, γ,d )/(γ S, +γ,d ) max ] max{ [(γ S, γ,d )/(γ S, +γ,d )] max maxmin max{([γ S, )γ,d ]/[(γ S, )+γ,d ] max{(γ S, γ,d )/(γ S, +γ,d +γ INF (γ,d +1)) Average E b /N 0 Fig. 9. Outage probabilities for conventional max min, conventional max harmonic mean (AF), asymptotic selection, conventional-based semiasymptotic selection, optimal max min selection (Sec. III.D) and optimal selection (maximization of Eq. (4)); K = 4 relays, L = 10, 50 ( E[γ S, ]/E[γ INF ]), R 0 = 2 BPCU and E[γ S, ] = E[γ,D ] + 10 db. As for the switching upper-limit between the two proposed asymptotic criteria, it can be seen that the proposed method approaches in terms of the corresponding average SNR the true crossover point of the semi-asymptotic and asymptotic curves. As the parameters L, K are increased the theoretical approximations fit perfectly with the true performance which results in an accurate definition of the crossover point under question. In addition to these cases, the proposed algorithm also gives an accurate estimation of this upper-limit for the cases where the theoretical approximations do not match exactly with the true performance (for low K, L-for example see Fig. 4). Also for these cases the crossover point of the proposed theoretical L=10 L=50 L=10 L=50 approximation corresponds to an average SNR which is close to the true crossover points of the curves. Fig. 7 presents the outage behavior of the asymptotic-based relay selection criteria for two indicative spectral efficiencies. More specifically, the considered spectral efficiencies are R 0 = 3, 5 BPCU and the interference factor is equal to L = 100. As can be seen the change in spectral efficiency does not change our previous remars and therefore the curves validate the need for the switching between semi-asymptotic and asymptotic selection as well as the accuracy of the proposed method. Finally, Figures 8 and 9 deal with the extension of the proposed semi-asymptotic methods to other conventional selection schemes. More specifically, we plot the outage probabilities for conventional max min and conventional max harmonic mean [18] ( γ,d /(γ S,, γ,d )) strategies as well as their interference extension. The simulation parameters are K = 4 users, R 0 = 2 BPCU, L = 10, 50 ( E[γ S, ]/E[γ INF ]) and E[γ S, ] = E[γ,D ] + where = 0, 10 db, respectively. The first important observation is that the legacy extension of the max harmonic mean criterion efficiently fills the gap between conventional and asymptotic selection and has a similar behavior with the semi-asymptotic max min selection. This result generalizes our proposed conventional-based semi-asymptotic technique and shows that all the selection strategies can be efficiently extended for interference environments by a simple division with the instantaneous interference term and without further structural modifications. Furthermore, achieving the performance (equivalent to max min case) shows that the proposed switching heuristic can be applied for both cases. Therefore, the switching algorithm is introduced as a general low-complexity heuristic for practical applications which are characterized by legacy implementation constraints. As far as the optimal max min relay selection is concerned (III-D), it can be seen that it approaches the optimal relay selection (maximization of Eq. (4)) for all the cases. The integration of the interference term in the basic structure of the conventional relay selection overcomes misleading selection decisions of the conventional-based semi-asymptotic approach and it is an appropriate solution for systems which can update their min computational core with the SINR term. Moreover, its application to both conventional selection criteria shows that this technique is also introduced as a general extension of selection methods by a simple replacement of the SNR source-relay lin with the corresponding SINR. In addition to these observations, the comparison between Figures 8 and 9 shows that the difference between conventional-based and optimal max min techniques is decreased as L increases (strong interference) or the lin R D becomes stronger than S R (relay cluster is close to the destination). For these two possible operational scenarios, the conventionalbased semi-asymptotic scheme mimics a similar behavior with the optimal max min schemes. VI. CONCLUSION In this paper, the focus has been on the use of Amplifyand-Forward (AF) cooperation policy for interference-limited

10 10 environments. It has been proven that the existence of structural interference during the relaying process changes the conventional statistical description of the system and results in bounded behavior. More specifically, the paper has demonstrated that the diversity slope is bounded by a static point at high SNRs. Furthermore, three extensions of the max min criterion suitable for legacy AF architectures have been studied. These criteria adapt the conventional max min selection to interference environments without complicated computational modifications and are suitable for different SNR regions. In addition to this investigation, a theoretical algorithm which provides an appropriate metric for switching between them has been proposed. This switching scheme is based on some well-defined approximations of the system statistic and uses the outage probabilities. Computer simulations and analytical results have validated the accuracy of the proposed schemes. Techniques for interference mitigation which overcome the presented performance limitation seem to be a promising area for future wor. ACKNOWLEDGEMENTS The wor reported in this paper has formed part of the Wireless Enabling Techniques wor area of the Core 4 Research Programme of the Virtual Centre of Excellence in Mobile and Personal Communications, Mobile VCE, whose funding support, including that of EPSRC, is gratefully acnowledged. Furthermore, the authors would lie to than the anonymous reviewers for their insightful and constructive comments. Finally, the authors wish to than Dr. R. Nabar and Dr. Z. Ding for many helpful discussions. APPENDIX A MAX-MIN SELECTION FOR AF COOPERATION We assume that a and b denote the instantaneous SNR of the lins S and D, respectively. The optimal selection can be approximated as C Asym = 1 ( 2 log γ ) S,Asym 1 ) γ INF 2 log 2 (1 + γ Asym, Asym (19) where the factor of a 1/2 accounts for the fact that information is conveyed to the destination terminal over two time slots. Using the theory of order statistics [35], the CDF of γ Asym corresponds to the selection of the largest random variable among K independent and identically distributed (i.i.d) random variables γ Asym ( = 1...K) with a statistic which is given by Eq. (6). Therefore, the CDF of γ Asym is given by the following expression P γasym (γ) = P γasym1 P γasym2 (γ)... P γasymk (γ) [ K [ ] K γ = P Γ (γ)] =, (20) L + γ where P γasym ( ) = P Γ ( ) denotes the CDF of each random variable. Therefore, the corresponding outage probability is given as P {C Asym < R 0 { 1 = P { (1 + γ Asym ) < R 0 2 log 2 = P γ Asym < 2 2R0 1 =P γasym (2 2R0 1) [ ] K 2 2R0 1 =. (21) L + 2 2R0 1 APPENDIX C SEMI ASYMPTOTIC SELECTION-OUTAGE BOUND The problem here is to analyze the behavior of the ratio γ S, γ INF when is selected according to the semi-asymptotic metric. In order to simplify the approximation of the corresponding outage bound, two cases will be considered. { a b = argmax a + b { 1 = argmax 1 a + 1 b { 1 = argmin + 1 a b { [ 1 argmin max, 1 ] a b { = argmax min [a, b ] where x + y max[x, y] (18) A. The value min Asym, γ Asym,D = γ S, Asym The first case assumes that for the selected node (relay) the minimum between the two hops is the lin S. In this case, the numerator of the ratio of interest becomes the minimum of two i.i.d exponential random variables which can be proved to be exponentially distributed but with a mean divided by two. More specifically, p min (γ) = p γ1 (γ)[1 P γ2 (γ)] + p γ2 (γ)[1 P γ1 (γ)] [ ] = 2p ǫ (γ) 1 P ǫ (γ) [ ] = 2λe λγ e λγ = 2λe 2λγ, (22) APPENDIX B ASYMPTOTIC SELECTION-OUTAGE BOUND The instantaneous capacity of the system is given by where p γ1 ( ) = p γ2 ( ) = p ǫ ( ), P γ1 ( ) = P γ2 ( ) = P ǫ ( ) denote the PDF and the CDF, respectively, of the i.i.d exponential random variables with parameter λ. Based on Eq. s (6) and (22), the CDF of the corresponding ratio is given as

11 11 P Γ(γ) = 2γ L + 2γ. (23) Therefore, the CDF of the semi-asymptotic selection for the case of K relays can be written as P Γ (γ) = [P Γ (γ)]k = B. The value min Asym, γ Asym,D = γ Asym,D [ ] K 2γ. (24) L + 2γ In this case, the minimum between the two hops of the selected relay is the lin D which is not considered in the ratio of interest. However, we can suppose that as the number of nodes increases the selected node also has the maximum asymptotic ratio 1. Therefore, in this case the outage bound can be approximated by the conventional asymptotic relay selection. Based on the above two equiprobable cases, the semiasymptotic outage bound, is given as P γsemi (γ) = 1 2 P Γ(γ) P γ Asym (γ) = 1 [ ] K 2γ + 1 [ ] K γ. (25) 2 L + 2γ 2 L + γ In a similar way with the asymptotic selection, the outage probability for the semi-asymptotic case can be written as P {C Semi < R 0 = P γsemi (2 2R0 1) = 1 2 [ 2 2R0 1 L + 2 2R0 1 ] K + 1 [ 2 2(2 2R0 1) L + 2(2 2R0 1) ] K. (26) APPENDIX D APPROXIMATION FOR THE ASYMPTOTIC RELAY SELECTION X Assume the random variables Z = Y V (Y +1)+X +Y +1 and W = X /V (for = 1,...,K) where X, Y, V are exponentially distributed. The statistic under question corresponds to the random variable Z where = arg max{w. This random variable can be approximated as 1 max{ a b max{a a for all min{b b O, if O n o Pr max{a min{b O 1 and therefore max{ a b max{a min{b, where O is the cardinality of the set O. 1 Z = X + Y + 1 X Y + V X Y + V X + Y X Y (V max max { X + Y X Y + V (Y + 1) X Y V Y, V X + Y 0 0, V X 0 Y 0 { X X for high V, Y ) (27) (28) (arg max{w arg max{x for high K) { Z X 0 min Y { 0, X X 0 + Y 0 V min U, W, (29) where 0 = arg max{x. The CDF in question corresponds to the random variable Z = min{u, W and can be calculated by using the theory of order statistics for the variables U, W. More specifically, the CDF of U has been calculated in [25, Eq. (13)] and is given as P U (u) = 1 K =1 ( K ) e (+1)αu ( 1) +1, (30) where α = 1 γ and γ is the average SNR of each hop in the cluster of interest. Furthermore the CDF of W is given in Eq. (6). Therefore by using basic theory of order statistics [35], the CDF of Z can be written as P Z (z) = P U (z) + P W (z) P U (z) P W (z) K ( ) [ K = 1 e (+1)αz ( 1) +1 1 =1 ( ) ] K z. L + z (31) Therefore, the corresponding outage probability can be written as { P C Asym < R 0 = P γasym (2 2R0 1) K ( K =1 =1 [ 1 ) e (+1)α(22R 0 1) ( 1) +1 ( 2 2R 0 ) K ] 1. L + 2 2R0 1 (32) In order to simplify the above expression and therefore the related definition of the parameter γ High, we approximate the included exponential term with the first two terms of the wellnown Taylor series. More specifically, for high SNRs (α 0), approximate the exponential term as

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Pillai, Probability, random variables and stochastic processes, Mc Grow Hill Editions, 4th Edition, Ioannis Kriidis (S 03-M 07) was born in Athens, Greece, in He received the diploma in Computer Engineering from the Computer Engineering and Informatics Department (CEID) of the University of Patras, Greece, in 2000, and the M.Sc and Ph.D degrees from Ecole Nationale Supérieure des Télécommunications (ENST), Paris, France, in 2001 and 2005, respectively, all in electrical engineering. From 2001 to 2002 he served as a Research Associate at the National Capodistrean University of Athens, Athens, Greece and from 2006 to 2007 he wored, as a Post-Doctoral researcher, with ENST, Paris, France. He is currently a Research Fellow in the School of Engineering and Electronics at the University of Edinburgh, Edinburgh, UK. During summer of 2008, he was visiting researcher at the University of Notre Dame, IN, USA. His current research interests include information theory, wireless communications, cognitive radio and secrecy communications. Dr. I. Kriidis is a member of the Technical Chamber of Greece.

13 13 John S. Thompson received the B.Eng. and Ph.D. degrees from the University of Edinburgh, Edinburgh, U.K., in 1992 and 1996, respectively. From July 1995 to August 1999, he was a Postdoctoral Researcher with the University of Edinburgh, which was funded by the U.K. Engineering and Physical Sciences Research Council and Nortel Networs. In September 1999, he was a Lecturer with the School of Engineering and Electronics, University of Edinburgh, where, since October 2005, he has been a Reader. He has authored approximately 150 papers to date, including a number of invited papers, boo chapters, and tutorial tals; he is currently coauthoring an undergraduate textboo on digital signal processing. He is currently the Editor-in-Chief of the IET Signal Processing Journal. His research interests include signal-processing algorithms for wireless systems, antenna array techniques, and multihop wireless communications. Dr. Thompson was a Technical Program Cochair for the IEEE International Conference on Communications, which was held in Glasgow, U.K., in June Steve McLaughlin (SM 04) was born in Clydeban, Scotland in He received the B.Sc. degree in Electronics and Electrical Engineering from the University of Glasgow in 1981 and the Ph.D. degree from the University of Edinburgh in From 1981 to 1984 he was a Development Engineer with Barr & Stroud Ltd. (Glasgow) involved in the design and simulation of integrated thermal imaging and fire control systems. From 1984 to 1986 he wored on the design and development of high frequency data communication systems with MEL Ltd. In 1986 he joined the Dept. of Electronics and Electrical Engineering at the University of Edinburgh as a research fellow where he studied the performance of linear adaptive algorithms in high noise and nonstationary environments. In 1988 he joined the academic staff at Edinburgh, and from 1991 until 2001 he held a Royal Society University Research Fellowship to study nonlinear signal processing techniques. In 2002 he was awarded a personal Chair in Electronic Communication Systems at the University of Edinburgh. His research interests lie in the fields of adaptive signal processing and nonlinear dynamical systems theory and their applications to biomedical and communication systems. Prof McLaughlin is a Fellow of the Institute of Engineering and Technology, a Senior Member of the IEEE and a Fellow of the Royal Society of Edinburgh. Norbert Goertz (S 97-M 99-SM 03) was born in Mönchengladbach-Rheydt, Germany. He received the Dipl.-Ing. degree from Ruhr-University of Bochum in 1993 and the Dr.-Ing. degree from Christian-Albrechts University, Kiel, Germany, in 1999 for his research wor in coded speech transmission. In 2004 he received the Habilitation degree from Munich University of Technology, Germany, with a thesis in the field of Joint Source-Channel Coding. After a 3-months research visit of the IT department at Lund University, Sweden, he went to Scotland in October 2004 where he was a Senior Lecturer at the Institute for Digital Communications in the School of Engineering and Electronics of The University of Edinburgh. Since September 2008 he has been a full professor for Multimedia Signal Processing at the Institute of Communications and Radio-Frequency Engineering at the Vienna University of Technology. Norbert Görtz has authored a text boo entitled Joint Source-Channel Coding of Discrete-Time Sources with Continuous Amplitudes, World Scientific / Imperial College Press. He is a Senior Member of the IEEE and also a member of VDE and its Information Technology Society (ITG) and a member of EURASIP. Moreover, he is a personal member of the Telecommunications Research Center Vienna (ftw.). His research interests include information theory, source and channel coding, and cross-layer design for wireless networs.