 Research
 Open Access
Relay selection in cooperative networks with frequency selective fading
 Qingxiong Deng^{1}Email author and
 Andrew G Klein^{1}
https://doi.org/10.1186/168714992011171
© Deng and Klein; licensee Springer. 2011
 Received: 6 April 2011
 Accepted: 16 November 2011
 Published: 16 November 2011
Abstract
In this article, we consider the diversitymultiplexing tradeoff (DMT) of relayassisted communication through correlated frequency selective fading channels. Recent results for relays in flat fading channels demonstrate a performance and implementation advantage in using relay selection as opposed to more complicated distributed spacetime coding schemes. Motivated by these results, we explore the use of relay selection for the case when all channels have intersymbol interference. In particular, we focus on the performance of relaying strategies when multiple decodeandforward relays share a single channel orthogonal to the source. We derive the DMT for several relaying strategies: best relay selection, random relay selection, and the case when all decoding relays participate. The best relay selection method selects the relay in the decoding set with the largest sumsquared relaytodestination channel coefficients. This scheme can achieve the optimal DMT of the system under consideration and generally dominates the other two relaying strategies which do not always exploit the spatial diversity offered by the relays. Different from flat fading, we found special cases when the three relaying strategies have the same DMT. We further present a transceiver design which is proven to asymptotically achieve the optimal DMT. Monte Carlo simulations are presented to corroborate the theoretical analysis and to provide a detailed performance comparison of the three relaying strategies in channels encountered in practice.
Keywords
 cooperative communication
 relay selection
 opportunistic relaying
 diversitymultiplexing tradeoff
 outage probability
 frequency selective fading
 intersymbol interference
1 Introduction
Cooperative relay networks have emerged as a powerful technique to combat multipath fading and increase energy efficiency [1, 2]. To exploit spatial diversity in the absence of multiple antennas, several spatially separated singleantenna nodes can cooperate to form a virtual antenna array. Such systems usually employ halfduplex relays and come in two flavors [3–6]: those where the relays transmit on orthogonal channels so that transmission from the source and each relay is received separately at the destination, or those where a single nonorthogonal channel is shared between the source and relays so that all nodes may transmit on the same common channel at the same time. Here, we focus on the former class of systems which employ orthogonal relay channels, where the orthogonality is often accomplished through time division.
Cooperative relay systems with orthogonal channels typically either employ multiple orthogonal relay subchannels in conjunction with repetition coding, or all relays use a single orthogonal relay channel along with distributed spacetime coding (DSTC) [7]. While the use of repetition codes is attractive for its simplicity, this approach requires relay scheduling and dedicated orthogonal channels for each relay which uses up precious system resources. On the other hand, when using a single orthogonal relay channel with DSTC, the scheduling of relays is of no concern, but DSTC requires synchronization between relays which is very difficult in distributed networks. Asynchronous forms of spacetime coding have been proposed (e.g. [8]), but the decoding complexity may still be prohibitively complex to permit their use in lowcost wireless ad hoc networks. Furthermore, the nonlinearity of most existing RF frontends poses additional implementation challenges for DSTCbased approaches [9].
More recently, relay selection schemes have been proposed [10, 11] which use simple repetition coding, very simple scheduling, and a single relay channel. Remarkably, these schemes can achieve the same diversitymultiplexing tradeoff (DMT) [12] as DSTC relaying, and can even outperform DSTC systems in terms of outage probability [11, 13]. Using relay selection is an attractive alternative to avoid the spectral inefficiency of repetition coding and the increased decoding complexity required for DSTC.
Most existing cooperative diversity research assumes that the fading channels have flat frequency responses. In high datarate wireless applications, however, the coherence bandwidth of the channels tends to be smaller than the bandwidth of the signal, resulting in frequency selective fading [14]. For such high rate communication in cooperative relay networks, existing techniques for flat fading channels need to be adapted, or new techniques need to be designed for frequency selective fading channels. In [15], the authors considered a system with a single amplifyandforward (AF) relay over frequency selective channels, and proposed three DSTCs. In [16], the authors consider a multipleAFrelay OFDM system and proposed a distributed spacefrequency code. The three DSTCs in [15] and the distributed spacefrequency code in [16] can achieve both cooperative diversity and frequency diversity where the frequency diversity through a relay is up to the minimum of the sourcerelay channel length and the relaydestination channel length. Simpler, nonDSTC approaches that employ relay selection have been proposed for communication through frequencyselective fading channels. For example, in [17, 18], uncoded OFDM is studied, and it was shown that if relay selection is done on a persubcarrier basis, full spatial diversity can be achieved. However, neither of these OFDMbased relay selection methods were able to exploit the frequency diversity of the ISI channel [19]. A linearly precoded OFDM system was proposed in [20] which uses multiple amplifyforward relays with linear transmit precoding; a simulationbased study showed that two relay selection schemes exhibited a coding gain improvement compared to an orthogonal roundrobbin relaying scheme.
This article investigates the performance limits of relay selection with frequency selective fading, and focuses on the DMT for singlecarrier systems without transmit channel state information (CSI) and transmit precoding. We analyze three different relay selection methods, including best relay selection, random relay selection, and alldecodingrelay participation. The relays in these three methods use a single orthogonal subchannel with repetition coding. We derive the DMT for the relay selection methods and then propose a practical lowcomplexity system which asymptotically attains the DMT by using uncoded QAM with guard intervals between blocks along with linear zeroforcing (ZF) equalizers.
2 System model
2.1 Channel model
While the assumption of equal node powers and equal noise variances may seem impractical, the case of unequal powers and variances does not change the asymptotic highSNR analysis which follows since these constants disappear in the derivation; consequently, we make this simplifying assumption to aid the clarity of the exposition.
2.2 Diversitymultiplexing tradeoff
The DMT has proven to be a useful theoretical tool that has considerably advanced the design of codes in the MIMO context. By restricting attention to system behavior in the highSNR regime, DMT analysis permits a mathematically tractable comparison of various transmission and relaying schemes.
2.3 Upper bound on the DMT
where (3) follows from log A + log B = log(AB), (4) follows from the fact that Pr(a + b < c) ≤ Pr(a < c) for any a, b, c ≥ 0, (5) follows from the fact that ${\xi}_{jk,min}^{2}{\u2225{\mathit{\delta}}_{jk}\u2225}^{2}\le {\u2225{\mathit{h}}_{jk}\u2225}^{2}\le {\xi}_{jk,max}^{2}{\u2225{\mathit{\delta}}_{jk}\u2225}^{2}$, and (6) holds as ${\u2225{\mathit{\delta}}_{sd}\u2225}^{2}+{\sum}_{{\mathbf{R}}_{i}\in \left(\mathcal{R}\backslash \mathcal{V}\right)}{\u2225{\mathit{\delta}}_{s{r}_{i}}\u2225}^{2}+{\sum}_{{\mathbf{R}}_{i}\in \mathcal{V}}{\u2225{\mathit{\delta}}_{{r}_{i}d}\u2225}^{2}$ is chisquare distributed with ${L}_{sd}+{\sum}_{{\mathbf{R}}_{i}\in \left(\mathcal{R}\backslash \mathcal{V}\right)}{L}_{s{r}_{i}}+{\sum}_{{\mathbf{R}}_{i}\in \mathcal{V}}{L}_{{r}_{i}d}$ degrees of freedom.
as the minimum is attained when relay R_{ i }is in $\mathcal{V}$ if ${L}_{{r}_{i}d}<{L}_{s{r}_{i}}$ and is not in $\mathcal{V}$ otherwise.
3 Outage probability analysis of decodeandforward relay system
 1.
In phase one, the source broadcasts the message to the destination and the relays, and each relay attempts to decode the message.
 2.
In phase two, the source is silent. Depending on the relay selection strategy, some or all of the relays that successfully decoded the message (if any) forward the message to the destination.
We continue to use the MFB to derive the upper bound on outage probability for the three relaying strategies and assume that a single symbol x[0] is sent by the source.
Next, in phase two, each relay attempts to decode the message. Those relays which are able to successfully decode the message comprise the decoding set $\mathcal{D}$ where $\mathcal{D}\subseteq \left\{{\mathbf{R}}_{\mathsf{\text{1}}},\dots ,{\mathbf{R}}_{K}\right\}$. Depending on the relay selection strategy that is employed, some nodes in the decoding set will participate in the relaying.
Referring back to (12), we now need to calculate $Pr\left[I<R\mathcal{D}\right]$, which depends on the particular choice of relay selection strategy. Next, we complete the outage probability and DMT derivation for each of the three selection strategies.
3.1 Best relay selection DMT

Centralized selection: In turn, each decoding relay transmits some known information to the destination, and the destination estimates each relaytodestination channel. The destination chooses the relay with the largest sumsquared relaytodestination channel coefficients, and feeds back this decision to the relays. The feedback requires $\left\mathcal{D}\right$ bits and is assumed to be fed back reliably.

Distributed selection: The relaytodestination channel and the destinationtorelay channel are assumed to be the same due to reciprocity. The destination broadcasts some known information to all the relays, each of which individually estimate its relaytodestination channel. Each relay waits for a time duration which is inversely proportional to its sumsquared relaydestination channel coefficients before sending its signal to the destination, so the relay with the largest sumsquared relaytodestination channel will be the first to send its signal to the destination. Other relays do not start transmission if they overhear any signal from the best relay. The detailed process for this distributed relay selection is discussed in [10].
The system designer may choose which of these two approaches to adopt depending on the application. The centralized selection might consume more time since the channels between relays and destination would need to be estimated sequentially. Centralized selection also puts more estimation load on the destination. Distributed selection, on the other hand, is more spectrally efficiently since relays concurrently estimate the channels; however, the relays need to resolve collisions which may complicate the implementation. The practical details of the selection process itselfsuch as the overhead in performing the selection, as well as the possibility of poor channel estimates that result in a suboptimal relay selectionare beyond the scope of the present study. Throughout our analysis, we assume that the best relay is always selected with negligible overhead.
where (18) follows by applying (16), (19) follows from [26, equation 2.321], (20) follows from the change of variable y = αb with 0 ≤ α ≤ 1, (21) comes by dropping terms in the polynomial of b with order higher than ${L}_{{r}_{i}d}$, and (22) follows from the fact that the integration in (21) is not a function of b.
where (24) follows as the minimum in (23) is attained when relay R_{ i }is in decoding set if ${L}_{{r}_{i}d}<{L}_{s{r}_{i}}$ and is not in decoding set otherwise. We see that full spatial diversity is achieved by this relay selection method since there are K + 1 terms in (24), but the achieved frequency diversity through each relay is the minimum of the length of the sourcetorelay and relaytodestination channels.
3.2 Random relay selection DMT
Comparing (29) with (23), we find that the DMT offered by the random selection method is dominated by the best relay selection method. The random selection method cannot always fully exploit the spatial diversity due to the presence of the min in (29) which results in a diversity bottleneck, though we will consider some cases in Section 3.4 where random selection can exploit full spatial diversity.
3.3 Alldecodingrelay DMT
Next, we analyze the outage of a scheme where all relays in the decoding set participate. Since all decoding relays participate in the forwarding, no overhead, no feedback, and no CSI is needed to perform selection. We assume perfect symbol synchronization now and will comment on this later.
where (34) follows by applying (32), (35) follows by applying (33), and (36) follows as ${\u2225{\stackrel{\u0304}{\mathit{\delta}}}_{rd}\u2225}^{2}+Y$ is chisquare distributed with L_{ sd }+ L_{ rd }degrees of freedom. From (36), we see that within the decoding set, dividing power among transmit antennas without phase alignment does not offer spatial diversity and only offers frequency diversity where the diversity order equals the longest delay length.
While we assume perfect symbol synchronization, we note that imperfect symbol synchronization has the effect of artificially increasing the channel lengths by adding zeros (or delays) to the front of the impulse responses. The use of intentional asynchronization to induce delay diversity was studied in [27] for the case of flat fading channels. A similar approach could be used in ISI channels; by artificially adding zeros to the front of each component relaytodestination channel, the effective sum channel from all relays to the destination can be made to have ${L}_{rd}={\sum}_{{\mathbf{R}}_{i}\in \mathcal{D}}{L}_{{r}_{i}d}$ independent paths so that the alldecodingrelay scheme can attain performance equal to the best relay selection if the symbollevel asynchronization is chosen appropriately.
3.4 Summary
DMT of each selection scheme for r ∈ [0,1/2]
Selection  d(r)  d(r) when $\forall i,\phantom{\rule{2.77695pt}{0ex}}{L}_{{r}_{i}d}={L}_{rd},\phantom{\rule{2.77695pt}{0ex}}{L}_{s{r}_{i}}={L}_{sr}$ 

Best  $\left(12r\right)\left({L}_{sd}+{\sum}_{i=1}^{K}min\left\{{L}_{{r}_{i}d},{L}_{s{r}_{i}}\right\}\right)$  (1  2r)(L_{ sd }+ min{KL_{ rd }, KL_{ sr }}) 
Random  $\left(12r\right)\left({L}_{sd}+min\mathcal{D}\left\{\left({min}_{{\mathbf{R}}_{i}\in \mathcal{D}}\phantom{\rule{2.77695pt}{0ex}}{L}_{{r}_{i}d}\right)+\left({\sum}_{{\mathbf{R}}_{i}\notin \mathcal{D}}{L}_{s{r}_{i}}\right)\right\}\right)$  (1  2r)(L_{ sd }+ min{L_{ rd }, KL_{ sr }}) 
All  $\left(12r\right)\left({L}_{sd}+min\mathcal{D}\left\{\left({min}_{{\mathbf{R}}_{i}\in \mathcal{D}}\phantom{\rule{2.77695pt}{0ex}}{L}_{{r}_{i}d}\right)+\left({\sum}_{{\mathbf{R}}_{i}\notin \mathcal{D}}{L}_{s{r}_{i}}\right)\right\}\right)$  (1  2r)(L_{ sd }+ min{L_{ rd }, KL_{ sr }}) 
Comparing each of these expressions with the DMT upper bound in (9), we see that the best relay selection method is the only one which can always achieve the DMT bound. Table 1 also includes the special case when all sourcetorelay channels have identical length L_{ sr }, and all relaytodestination channels have identical length L_{ rd }. We note that our theoretical diversity expressions agree with results reported in elsewhere in the literature. For example, in the special case of flatfading, our results coincide with those of [10, 11] which showed that the best relay selection protocol can achieve diversity equal to K + 1. Another example is that in [15], with a single relay K = 1, a system employing STBC can achieve diversity equal to the expression we found for all the three relaying schemes. Additionally, the diversity achieved when using multiple orthogonal relay subchannels in an OFDM system with precoding [20] is identical to the one achieved here by the best relay selection scheme.
It is interesting to note that even random relay selection can achieve the same diversity as best relay selection in some cases. For example, looking at the last column of Table 1, we see that all schemes have an equivalent DMT when L_{ rd }> KL_{ sr }. This situation could arise when there is significant scattering and dispersion in the relaytodestination channel (due to a high density of large buildings, for example) when compared with the sourcetorelay channel (which may have a lower density of reflecting structures and terrain). Thus, when the relaytodestination channel is sufficiently rich, the lower overhead of random relay selection is attractive. This is different from the situation in flat fading channels, since with L_{ sr }= L_{ rd }= 1, best relay selection is the only scheme which can exploit spatial diversity.
The outage probability and DMT bounds derived here are based on the MFB. As the MFB effectively ignores the intersymbol interference, these results provide an optimistic bound on the attainable outage probability and DMT. We now consider a transceiver design for attaining the bound for best relay selection.
4 OptimalDMTachieving transceiver
In the previous section, we proved that best relay selection can achieve the optimal DMT, the DMT upper bound derived in Section 2. We now propose a specific transmission and reception scheme for best relay selection and we will prove that it can asymptotically achieve the optimal DMT.
4.1 Transceiver description
After transmission of N symbols, a guard interval of length M_{max} 1 zeros follows. The choice of Q or r' here is to make sure the total transmission rate is still R = r log ρ with the guard interval. M_{max} is essentially an upper bound on the length of all channels in the system. In practice, it is unrealistic for the source node to have knowledge of the lengths of all channels in the system. The system designer needs only choose the parameter M_{max} to be greater than or equal to the largest channel length expected in the transmission environment. The insertion of guard time eliminates the possibility of interblock interference, but intersymbol interference is still present. Due to the insertion of guard time between alternating phases of source/relay transmission, we see from (38) that the system incurs a rate penalty that can be made arbitrarily small by increasing the block length N.
We assume channel state information at the receiver (CSIR) is perfect, but that no channel state information at the transmitter (CSIT) is needed. We also assume perfect frame synchronization though in practice the system can accommodate modest symbollevel synchronization errors since they can be lumped into the FIR channel model. Each relay and the destination uses a ZF equalizer prior to detection to compensate for the intersymbol interference.
4.2 Outage analysis
Using a proof identical to [28, Lemma IV.1], it can be shown that ${\stackrel{\u0304}{\lambda}}_{sd}>0$ and ${\stackrel{\u0304}{\lambda}}_{{r}_{m}d}>0$, therefore $\stackrel{\u0304}{\lambda}>0$.
which shows that the proposed scheme can asymptotically achieve the DMT for best relay selection.
We also point out that since minimum mean squarederror (MMSE) and decision feedback equalizer (DFE) performance dominates ZF equalizers [30], MMSE equalizers and DFEs should attain the same DMT curve. In practice, a system designer may prefer a MMSE or DFE equalizer for their improved BER performance.
5 Numerical results
This section presents numerical examples of the performance of the proposed relay selection methods developed in Sections 3 and 4. In evaluating performance over finite SNRs, the diversity measured as the negative slope of each outage curve often does not coincide exactly with the predicted maximal diversity [31, 32]; the predicted diversity assumes that the SNR grows arbitrarily large to permit the analysis to be mathematically tractable. We now compare the performance of the three selection schemes in a variety of scenarios at finite SNR, and show that the schemes follow the general trends predicted by the DMT results.
Simulation scenarios
Scenario  K  L_{sd}  L_{sr}  L_{rd}  d_{max,best}  d_{max,random}  d_{max,all} 

1  1  2  2  2  4  4  4 
2  1  2  2  3  4  4  4 
3  1  2  3  2  4  4  4 
4  1  2  4  4  6  6  6 
5  1  4  2  2  6  6  6 
6  2  2  2,2  2,2  6  4  4 
7  2  2  2,2  3,3  6  5  5 
8  2  2  3,3  2,2  6  4  4 
9  2  2  4,4  4,4  10  6  6 
10  2  4  2,2  2,2  8  6  6 
When as many as K = 10 relays are available, as shown in Figure 7, the diversity order of the best relay selection may be significantly larger than the other two methods. In examining the power gain of best relay selection over the other two relaying strategies, we note an interesting trend. When the fading channels contain L = 2 taps, the power gain of the best relay selection is about 6 dB at a bit error rate of 10^{6}. When L increases to 4, however, the power gain of best relay selection is only about 2dB. Thus, when there is already sufficient frequency diversity in the system, the improvement in using sophisticated selection schemes which better exploit the spatial diversity is not as pronounced. Again, a system designer may favor one of the simpler selection schemes if it is known that the transmission environment has sufficient frequency diversity.
where the underlying independent fading coefficients α∈ ℂ^{4} are complex Gaussian with variance given by the GSM typical rural power delay profile. For fair comparison, we normalize the total average power in the underlying independent fading coefficients to 1. The resulting sampled channel with four independent path arrivals gives rise to a correlated discrete channel with 19 taps. As shown in Figure 8, the simulated performance demonstrates that at finite SNR the receiver is not able to exploit the diversity offered by all four independent paths since the last path which has a power of 20 dB contributes very little to the received signal, an effect masked by the highSNR analysis of the DMT. Nevertheless, the choice of relay selection method still has significant impact on system performance when the number of relays K is relatively large.
6 Conclusion
In this article, we have considered the relay selection problem for the orthogonal decodeandforward system where correlated frequency selective fading is present. We analyzed the outage performance and derived the DMT for three relay methods: best relay selection, random relay selection, and the alldecodingrelay method. Our analysis shows that best relay selection performance dominates the other two schemes with respect to outage. We further proposed a transceiver to realize the DMT offered by best relay selection with minimal complexity; the proposed scheme uses uncoded QAM transmission with guard times and uses ZF equalization at each node. The analysis and simulation results show that the proposed scheme asymptotically achieves the DMT.
While the diversity offered by relay systems in flat fading channels is fairly well understood, the deployment of relay systems in ISI channels requires consideration of a variety of new issues in order to best exploit the available diversity. For example, we presented cases where random relay selection and the alldecodingrelay method can achieve the same diversity as best relay selection, which runs counter to the situation in flat fading channels where best relay selection is always superior. We also found that only when the number of relays in the system is relatively large, the best relay selection offers a significant performance advantage over the other relaying strategies, though this tends to diminish with increased frequency diversity in the system. As the overhead of random relay selection is lower than that of the best relay selection, system designers may favor random relaying depending on the application and transmission environment.
The analytical study presented here focuses on the highSNR regime and is an important step toward understanding the diversity offered by relay systems in frequency selective fading channels. The relaying strategies presented in this article do not require sophisticated spacetime coding, they have relaxed synchronization requirements, and are spectrally efficient; these advantages make the relay selection methods ready for implementation in today's distributed networks. Future study may consider the use of alternate forwarding protocols (such as amplifyandforward or equalizeandforward) as well as the overhead tradeoff of the various relay selection methods.
Declarations
Authors’ Affiliations
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