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Proportional fair scheduling with superposition coding in a cellular cooperative relay system

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Abstract

Many works have tackled on the problem of throughput and fairness optimization in cellular cooperative relaying systems. Considering firstly a two-user relay broadcast channel, we design a scheme based on superposition coding (SC) which maximizes the achievable sum-rate under a proportional fairness constraint. Unlike most relaying schemes where users are allocated orthogonally, our scheme serves the two users simultaneously on the same time-frequency resource unit by superposing their messages into three SC layers. The optimal power allocation parameters of each SC layer are derived by analysis. Next, we consider the general multi-user case in a cellular relay system, for which we design resource allocation algorithms based on proportional fair scheduling exploiting the proposed SC-based scheme. Numerical results show that the proposed algorithms allowing simultaneous user allocation outperform conventional schedulers based on orthogonal user allocation, both in terms of throughput and proportional fairness. These results indicate promising new directions for the design of future radio resource allocation and scheduling algorithms.

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Acknowledgments

This work was supported by the Grants-in-Aid for Scientific Research nos. 10595739 and 24560457 from the Ministry of Education, Science, Sports, and Culture of Japan.

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Correspondence to Megumi Kaneko.

Appendix

Appendix

We derive the achievable rate and optimal power allocation for the 2L-PF-SC scheme.

  1. Step 1

    For the decoding at the RS,

    $$ R_{\mathrm{b1}} \leq C \left(\dfrac{(1-\alpha) \gamma_{\mathrm{R}}}{1+\alpha\gamma_{\mathrm{R}}} \right) = R_{\mathrm{b1}}^{\mathrm{A}}, $$
    (64)
    $$ R_{\mathrm{2}} \leq C (\alpha\gamma_{\mathrm{R}} ), $$
    (65)

    which ensure that the RS can decode \({\bf x}_{\mathrm {b1}}\), \({\bf x}_{\mathrm {2}}\), respectively.

  2. Step 2

    For the decoding at MS\(_1\), we get

    $$ R_{\mathrm{R2}} \leq C (\gamma_{\mathrm{R1}} ), $$
    (66)
    $$ R_{\mathrm{b1}} \leq C ((1-\alpha) \gamma_{\mathrm{D1}} ) = R_{\mathrm{b1}}^{\mathrm{B}} , $$
    (67)

    which ensure that MS\(_1\) can decode \({\bf x}_{\mathrm {R2}}\) (\({\bf x}_{\mathrm {2}}\)), \({\bf x}_{\mathrm {b1}}\). For the decoding at MS\(_2\), we obtain

    $$\begin{array}{@{}rcl@{}} R_{\mathrm{R2}} \leq & C (\gamma_{\mathrm{R2}}) , \end{array} $$
    (68)

    which ensures that MS\(_2\) can decode \({\bf x}_{\mathrm {R2}}\). Equation 66 is satisfied a fortiori given that \(\gamma _{\mathrm {R2}}<\gamma _{\mathrm {R1}}\), so only (68) will be considered for \(R_{\mathrm {R2}}\). \(R_{\mathrm {b1}}^{\mathrm {A}}\), \(R_{\mathrm {b1}}^{\mathrm {B}}\) denote the two constraints on \(R_{\mathrm {b1}}\) to be satisfied.

In Step 1, the BS transmits \(M(\min (R_{\mathrm {b1}}^{\mathrm {A}},R_{\mathrm {b1}}^{\mathrm {B}})+R_{\mathrm {2}})\) bits, where \(M\) is the number of symbol times. In Step 2, the RS forwards \(MR_{\mathrm {2}}\) bits. The sum rate can thus be written

$$\begin{array}{@{}rcl@{}} R_{\text{2L-PF-SC}} = \frac{\min(R_{\mathrm{b1}}^{\mathrm{A}},R_{\mathrm{b1}}^{\mathrm{B}})+R_{\mathrm{2}}}{1 + \frac{R_{\mathrm{2}}}{R_{\mathrm{R2}}}}, \end{array} $$
(69)

and the PF constraint between MS\(_1\) and MS\(_2\) is given by

$$ \frac{\min(R_{\mathrm{b1}}^{\mathrm{A}},R_{\mathrm{b1}}^{\mathrm{B}})}{\bar{R}_1}=\frac{R_2}{\bar{R}_2}. $$
(70)

Next, we consider the different cases arising from the conditions on \(R_{\mathrm {b1}}\).

Case 1

\(R_{\mathrm {b1}}^{\mathrm {A}} \leq R_{\mathrm {b1}}^{\mathrm {B}} \Leftrightarrow \alpha \geq \alpha ^*=\frac {1}{\gamma _{\mathrm {D1}}}-\frac {1}{\gamma _{\mathrm {R}}}\). In this case, \(R_{\mathrm {b1}}=R_{\mathrm {b1}}^{\mathrm {A}}\). Condition (70) becomes

$$\begin{array}{@{}rcl@{}} \bar{R}_2\log_2\left(\frac{1+\gamma_{\mathrm{R}}}{1+\alpha\gamma_{\mathrm{R}}}\right)=\bar{R}_1\log_2(1+\alpha\gamma_{\mathrm{R}}), \end{array} $$
(71)

which gives by straightforward calculations,

$$\begin{array}{@{}rcl@{}} \alpha=\frac{(1+\gamma_{\mathrm{R}})^{\frac{\bar{R}_2}{\bar{R}_1+\bar{R}_2}}-1}{\gamma_{\mathrm{R}}}. \end{array} $$
(72)

From (72) and (69), we obtain the achievable rate in this case,

$$ R_{\text{2L-PF-SC}}^{(1)} = \frac{\log_2(1+\gamma_{\mathrm{R}})}{1+\frac{\bar{R}_2}{\bar{R}_1+\bar{R}_2}\frac{\log_2(1+\gamma_{\mathrm{R}})}{\log_2(1+\min{(\gamma_{\mathrm{R}1},\gamma_{\mathrm{R}2})})}}, $$
(73)

Inserting (72) into condition \(\alpha \geq \alpha ^*\), the rate expression is valid for

$$\begin{array}{@{}rcl@{}} \gamma_{\mathrm{D1}}\geq \frac{\gamma_{\mathrm{R}}}{(1+\gamma_{\mathrm{R}})^{\frac{\bar{R}_2}{\bar{R}_1+\bar{R}_2}}}. \end{array} $$
(74)

Case 2

\(R_{\mathrm {b1}}^{\mathrm {A}} > R_{\mathrm {b1}}^{\mathrm {B}} \Leftrightarrow \alpha < \alpha ^*\). In this case, \(R_{\mathrm {b1}}=R_{\mathrm {b1}}^{\mathrm {B}}\). Condition (70) becomes

$$\begin{array}{@{}rcl@{}} \bar{R}_2\log_2(1+(1-\alpha\gamma_{\mathrm{D1}}))&=&\bar{R}_1\log_2(1+\alpha\gamma_{\mathrm{R}}),\\ \Longleftrightarrow 1+(1-\alpha)\gamma_{\mathrm{D1}}&=&(1+\alpha\gamma_{\mathrm{R}})^{\frac{\bar{R}_1}{\bar{R}_2}}. \end{array} $$
(75)

We denote \(f(\alpha )=1+(1-\alpha )\gamma _{\mathrm {D1}}\) and \(g(\alpha )=(1+\alpha \gamma _{\mathrm {R}})^{\frac {\bar {R}_1}{\bar {R}_2}}\). \(f(\alpha )\) is a monotonically decreasing function of \(\alpha \) over \([0,1]\) with \(f(0)=1+\gamma _{\mathrm {D1}} \geq 1\) and \(f(1)=1\). \(g(\alpha )\) is a monotonically increasing function of \(\alpha \) over \([0,1]\) with \(g(0)=1\) and \(g(1)=(1+\gamma _{\mathrm {R}})^{\frac {\bar {R}_1}{\bar {R}_2}}\geq 1\). Thus, there exists a unique value \(\tilde {\alpha }\) in \([0,1]\) such that \(f(\tilde {\alpha })=g(\tilde {\alpha })\), which can be easily determined numerically. Thus, the achievable rate (69) is expressed as

$$ R_{\text{2L-PF-SC}}^{(2)} = \frac{\log_2(1+(1-\tilde{\alpha})\gamma_{\mathrm{D1}})+\log_2(1+\tilde{\alpha}\gamma_{\mathrm{R}})}{1+\frac{\log_2(1+\tilde{\alpha}\gamma_{\mathrm{R}})}{\log_2(1+\min{(\gamma_{\mathrm{R}1},\gamma_{\mathrm{R}2})})}}, $$
(76)

in the case \(\gamma _{\mathrm {D1}} < \frac {\gamma _{\mathrm {R}}}{(1+\gamma _{\mathrm {R}})^{\frac {\bar {R}_2}{\bar {R}_1+\bar {R}_2}}}\).

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Kaneko, M., Hayashi, K., Popovski, P. et al. Proportional fair scheduling with superposition coding in a cellular cooperative relay system. Ann. Telecommun. 68, 525–537 (2013). https://doi.org/10.1007/s12243-012-0337-4

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