Performance of a cooperative multiplexing scheme with opportunistic user and relay selection over Rayleigh fading channels
- Dileep Kumar Verma^{1} and
- Shankar Prakriya^{1}Email author
https://doi.org/10.1186/1687-1499-2012-345
© Verma and Prakriya; licensee Springer. 2012
Received: 16 February 2012
Accepted: 27 August 2012
Published: 21 November 2012
Abstract
In this article, we analyze the performance of a cooperative spatial multiplexing scheme using amplify and forward relays over Rayleigh fading channels. We propose a simple user and relay selection scheme, and demonstrate that gains in spectral efficiencies are possible by such opportunistic selection. Using a bound on the outage performance, we develop a power control mechanism. We develop a bound on the ergodic rate of the proposed scheme. Computer simulation results are presented to demonstrate the performance of the scheme, and to validate the analytical expressions.
Introduction
Several new standards for cellular communication are now incorporating amplify and forward (AF) or decode and forward (DF) relays [1]. Cooperative communication protocols have the ability to exploit these relays to harness the spatial features of the network for reliable data transmissions [1–3]. A variety of protocols has been proposed to obtain higher spectral efficiencies, better outage performance, or to save transmitter power. Typically, the relays are half-duplex due to practical constraints, and this leads to loss of spectral efficiency. Various protocols have been proposed to overcome this limitation, and their performance has been analyzed [4].
It is well known that enormous gains in spectral efficiencies are possible through frequency reuse by exploiting the spatial features of the network [5–8]. Yomo and Carvalho [5] discuss the spectral efficiency improvement of the cellular system where two mobile subscribers uplink their respective data to base station (BS) while an AF relay facilitates communication to a shadowed subscriber. For downlink transmission, a similar type of spectrally efficient scheme to obtain gain in spectral efficiency for two users is proposed in [6]. Thai and Popovski [7] proposed an alternative to two-way relaying, and use a half-duplex relay to obtain significant improvement in spectral efficiency. Cooperative multiplexing protocols [9–11] proposed in recent years help realize these promised gains through frequency reuse. In addition, they also help to overcome the loss in spectral efficiency due to half-duplex nature of practical relays. In [10], a two-user cooperative multiplexing and scheduling (CMS) protocol using AF relays has been proposed, where a BS multiplexes the data of two selected users. In the first phase, the BS transmits the signal intended for the second user to the relay. In the second phase, the BS transmits the signal intended for the first user, while the relay retransmits the signal intended for the second user. In [11], interference cancellation and precoding techniques are proposed (assuming DF relays) to deal with the interference due to simultaneous transmissions in the second phase. This interference (in the second phase) makes the study of power distribution between the BS and the RS important. While Shi et al. [10] do not address this issue, an optimum power distribution is derived in [11] for the case when DF relays are utilized. In both [10, 11], the focus is on the achievable sum data rates, though some simplified outage analysis is also presented assuming Rayleigh fading channels. Performance with these protocols is limited by the link between the BS and the relay. Although CMS protocols employing relay selection are consequently of interest, they have not been considered so far. To the best of the authors’ knowledge, outage and ergodic rate performance of CMS protocols with AF relays have not been analyzed for Rayleigh fading channels.
In this article, we consider an opportunistic user and relay selection-based CMS system that enables the BS to multiplex data to a (selected) user, and another (selected) user in the shadowed region. AF relays are utilized for transmission to the user in the shadowed region. A sub-optimal distributed users’ selection scheme is proposed. For relay selection the sub-optimal scheme [12] is employed to facilitate distributed implementation. However, we propose a sub-optimal selection strategy based on [12] for the CMS network, in which the node is selected that maximizes the gain of the corresponding channel for desired signal, unlike the optimal node selection scheme in which interference channel should also be considered. Assuming Rayleigh fading channels, an analysis of the outage probability and ergodic performance of the user(s) and relay selection scheme-based CMS system is presented. Since the exact analysis is mathematically intractable, we present bounds of outage and ergodic performances. Moreover, we develop a power control strategy for the second phase of the multiplexing scheme. In summary, the contributions of the article are as follows:
- 1.
We demonstrate that a simple user and relay selection scheme can be implemented in a distributed manner and yield good performance.
- 2.
Unlike other work on cooperative multiplexing [9–11], we consider both user and relay selection. Also, we consider AF relays, analysis for which is complicated by the noise amplification in cooperative multiplexing scenarios.
- 3.
We evolve a simple power control scheme for the second phase of the multiplexing scheme.
The rest of the article is organized as follows. The system model is described in “System model description” section. In “Outage performance & power control” section, a power control scheme is discussed based on a bound on the outage probability. A bound for the ergodic sum-rate is presented in “Ergodic sum-rate” section. Performance of the proposed protocol is analyzed through computer simulations in “Numerical and simulation results” section, where the derived analytical expressions are validated. Conclusions are drawn in “Conclusion” section.
Notations: The probability density function of gain (|h|^{2}) of Rayleigh fading channel coefficient (h) is given by $\left(\right)close="">{f}_{|h{|}^{2}}\left(x\right)=exp(-x/\Omega )/\Omega $ and cumulative distribution function Pr(|h|^{2}≤x)=1−exp(−x/Ω) where $\left(\right)close="">\Omega =\mathbb{E}\left[\right|h{|}^{2}]$, and $\mathbb{E}[\xb7]$ denotes expectation of random variables. $\left(\right)close="">\Gamma \left(m\right)=\underset{0}{\overset{\infty}{\int}}{t}^{m-1}{e}^{-t}\phantom{\rule{2.77695pt}{0ex}}\mathrm{dt}$ is the Gamma function, and ${\mathrm{E}}_{n}\left(x\right)(=\underset{1}{\overset{\infty}{\int}}\frac{exp(-\mathrm{xt})}{{t}^{n}}\phantom{\rule{2.77695pt}{0ex}}\mathrm{dt})$ represents the exponential integral of order n.
System model description
We assume that the fading channels (from any node) to nodes in each cluster are statistically independent and identically distributed (i.i.d.). The Rayleigh fading channel coefficients are denoted by h_{j,k} where source j∈{BS,l}, destination k∈{l,m,n}, l∈{1,2,…,L}, m∈{1,2,…,M}, and n∈{1,2,…N}. The inter node distance (nodes in separate cluster) is approximated to inter cluster distance, as nodes in cluster are close to each other as compared to distance between clusters. Therefore, the average channel gains of various links are denoted by $\left(\right)close="">{\Omega}_{j,k}=\mathbb{E}\left[\right|{h}_{j,k}{|}^{2}]=\frac{1}{{d}_{j,k}^{\alpha}}$, where d_{j,k} is the distance between nodes/clusters j and k, and α is the pathloss exponent. Note, we use subscripts BS,A,R, and B for alternatively representing the BS, selected user from $\mathcal{A}$, selected relay from RS, and selected user from $\mathcal{B}$, respectively.
In the first phase (time slot), the BS broadcasts unit energy symbols (s_{B}) with power P meant for user $n\in \mathcal{B}$. The l th relay and m th user receive symbols ${y}_{l}=\sqrt{P}\phantom{\rule{0.3em}{0ex}}{h}_{\text{BS},l}\phantom{\rule{0.3em}{0ex}}{s}_{\text{B}}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{w}_{l}$ and $\left(\right)close="">{y}_{m}^{\text{I}}=\sqrt{P}\phantom{\rule{0.3em}{0ex}}{h}_{\text{BS},m}\phantom{\rule{0.3em}{0ex}}{s}_{\text{B}}+{w}_{m}^{\text{I}}$, respectively, where superscript “I” represents the phase number (first, in this case), and w_{ l } and $\left(\right)close="">{w}_{m}^{\text{I}}$ are zero mean additive white Gaussian noises with variance $\left(\right)close="">{\sigma}_{w}^{2}$ at the l th relay and m th user, respectively. User $n\in \mathcal{B}$ in the shadowed region ignores the weak signal from the BS in this phase. The m th user of cluster $\mathcal{A}$ detects the symbols s_{B}, and utilizes it in the subsequent phase to mitigate interference caused by simultaneous transmission from BS and RS.
It is clear from (1) and (2) that relay selection (choice l) plays an important role in determining performance. It is noted that the influence of interference due to the term $\left(\right)close="">{\mathcal{I}}_{n}$ can readily be minimized independently by minimizing $\left(\right)close="">{\mathcal{I}}_{n}$. For this reason, it is proposed to choose n to minimize I_{ n }. This is discussed in detail in “Outage performance & power control” Section. Estimation of the channel gains, and selection of users m^{⋆}, n^{⋆}, and relay l^{⋆} is facilitated by a training phase that precedes the actual data transmission. Ideally, the best users and best relay should be selected jointly so as to minimize the outage probability. However, such joint selection, though ideal, is difficult to implement. Note that this users and relay selection scheme are important in a distributed manner. We therefore propose to use the sub-optimal scheme such that a distributed timer-based scheme for cooperative communication network is used to opportunistically select the best relay node as given in [12, 14]. Making use of the fact that the term $\left(\right)close="">{\mathcal{I}}_{n}$ is common to all γ_{n,l}(for different relays to n th node), we directly use [12] for the selection of the best relay (${l}^{\star}=arg\underset{l\in \mathcal{L}}{max}\phantom{\rule{2.77695pt}{0ex}}{\gamma}_{n,l}$) as explicit mechanism of selection is already discussed in [12]. For ease of presentation, we first consider the single user set $\mathcal{B}$, i.e., N=1, and later in “Outage performance & power control” and “Ergodic sum-rate” sections show how the case of general N can be accommodated.
With the chosen relay, the m th user receives $\left(\right)close="">{y}_{m}^{\mathrm{II}}=\sqrt{{P}_{\text{BS}}}\phantom{\rule{0.3em}{0ex}}{h}_{\text{BS},m}x{s}_{\text{A}}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\sqrt{{P}_{\text{RS}}}\phantom{\rule{0.3em}{0ex}}{h}_{{l}^{\star},m}\phantom{\rule{0.3em}{0ex}}\beta \phantom{\rule{0.3em}{0ex}}{y}_{{l}^{\star}}+\phantom{\rule{0.3em}{0ex}}{w}_{m}^{\mathrm{II}}$ in the second phase, where $\left(\right)close="">{w}_{m}^{\mathrm{II}}$ is additive noise of variance $\left(\right)close="">{\sigma}_{w}^{2}$. The interfering symbols s_{B}at users in cluster $\left(\right)close="">\mathcal{A}$ can be cancelled using the symbols decoded in the first phase [11].
It is noted that the selection of user $\left(\right)close="">m\in \mathcal{A}$ has to take into consideration the degradation in SNR due to noise amplification at the relay. This degradation performance is a consequence of the use of AF relays. However, AF relays possess several advantages in implementations, and introduce less delay as compared to DF relays.
Relay selection in (4) ensures that $\left(\right)close="">{h}_{\text{BS},{l}^{\star}}$ is large. Further, user selection using (7) ensures that Γ in (5) is close to unity. For this reason, when L>1 and M>1, $\left(\right)close="">{\gamma}_{{m}^{\star}}^{\text{UB}}$ is almost always close to $\left(\right)close="">{\gamma}_{{m}^{\star}}$.
Using the relay selection as per (4), and user selection as per (7), we analyze the performance of the cooperative multiplexing scheme in the following sections.
Outage performance & power control
In this section, we investigate the outage performance of the considered system, and demonstrate how powers P_{BS} and P_{RS}can be selected to minimize the outage performance (which is a relevant QoS parameter in the cellular scenario).
Since, the signaling mechanism signals to two users together, such an overall outage probability is more meaningful than a single-user outage, and has been used by other authors in the context of cooperative multiplexing [10, 11]. From (1) and (5), it can be seen that the SNRs at the two terminals $\left(\right)close="">{\gamma}_{{m}^{\star}}$ and $\left(\right)close="">{\gamma}_{n,{l}^{\star}}$ are statistically dependent, making analysis of the exact outage performance intractable. In what follows, we derive a lower bound $\left(\right)close="">{p}_{o}^{\text{LB}}(=Pr\{min(\underset{{m}^{\star}}{\overset{\text{UB}}{\gamma}},\underset{n,{l}^{\star}}{\overset{\text{UB}}{\gamma}})\le {\gamma}_{\mathrm{th}}\left\}\right)$ on the outage probability.
where Ω_{BS,A} is an average channel gain between BS and cluster $\mathcal{A}$.
through numerical techniques. Note that this requires knowledge of the channel variances only.
Special case
It can be seen that as N becomes large, the outage performance improves and outage probability does not saturate as N→∞.
Ergodic sum-rate
where $\left(\right)close="">C=\frac{{\gamma}_{\text{RS}}{\Omega}_{\text{R,A}}}{{\gamma}_{\text{BS}}{\Omega}_{\text{BS,A}}}$ and $\left(\right)close="">D=\frac{1}{\gamma {\Omega}_{\text{BS,R}}}+\frac{1}{{\gamma}_{\text{RS}}{\Omega}_{\text{R,B}}}$. Proof is presented in Appendix.
Special case
As we demonstrate in the next section, the ergodic rate (due to interference cancellation) continuously improves with SNR. Also, the ergodic rate can be improved by increasing L, M, and N. This makes the use of cooperative multiplexing with user/relay selection advantageous.
Numerical and simulation results
Outage sub-optimal power allocation factor ( ζ _{ BS } ) for various combinations of L and M ({ L , M })
γ | {1,3} | {2,3} | {3,3} | {3,2} | {3,1} |
---|---|---|---|---|---|
in dB | ζ _{BS} | ζ _{BS} | ζ _{BS} | ζ _{BS} | ζ _{BS} |
5 | 0.26 | 0.32 | 0.35 | 0.42 | 0.50 |
10 | 0.18 | 0.22 | 0.29 | 0.33 | 0.48 |
15 | 0.11 | 0.17 | 0.22 | 0.26 | 0.40 |
20 | 0.06 | 0.12 | 0.15 | 0.21 | 0.38 |
25 | 0.04 | 0.07 | 0.10 | 0.16 | 0.29 |
Conclusion
We analyzed the performance of a cooperative spatial multiplexing scheme with opportunistic user and relay selection over Rayleigh fading channels. Bounds were derived for the outage probability and ergodic sum-rate with the proposed scheme. A simple power control scheme was suggested to improve outage performance. Performance of the protocol was validated by computer simulations and numerical analysis, and the suggested protocol was shown to be advantageous.
Appendix
This completes the proof.
Declarations
Acknowledgment
This study was funded by the Department of Science and Technology, Govt. of India (Project no. SR/S3/EECE/031/2008).
Authors’ Affiliations
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