Throughput performance of parallel and repetition coding in incremental decode-and-forward relaying

Abstract

The throughput performance of cooperative repetition and parallel coding in incremental decode-and-forward is investigated. Four transmission methods are considered: parallel coding with and without distributed space-time coding (PC-ST and PC, respectively); and repetition coding with and without Chase combining at the destination (RC and SC, respectively). The analysis is based on the mutual information seen at the receiver for each scheme. Exact expressions for the outage probability and throughput for all methods are derived. Both ad-hoc and infra-structured relaying scenarios are investigated. Results show that SC can perform very close to RC, PC and PC-ST in terms of throughput, specially in the case of infra-structured relaying or adequate power and rate allocation. The conclusion is that SC would be a better option in practice, since it requires a simpler receiver than PC-ST, PC, and RC.

Appendix: The case of N retransmissions

1.1 Outage formulation

Allowing for additional retransmissions from either the sourcr the relay in the case of SC is not useful since we consider a long-term quasi-static channel and there is no symbol combining at the receiver. Therefore, next we consider only the cases of RC, PC, and PC-ST.

The RC outage probability in (7) can be easily generalized to the case of N relay retransmissions as:

$$ {\mathcal{P}}_{RC_{N}}=\frac{N \hbox{SNR}_{RD}\left( e^{\frac{1-2^{R}}{N {\rm SNR}_{RD}}}-1\right) -\hbox{SNR}_{SD}\left( e^{\frac{1-2^{R}}{{\rm SNR}_{SD}}}-1\right) }{\hbox{SNR}_{SD}-N \hbox{SNR}_{RD}}, $$

(25)

while for the cases of PC and PC-ST, (12) and (15) can be generalized as, respectively:

$$ \begin{aligned} {\mathcal{P}}_{PC_{N}} &= {\mathcal{P}}\left\{ I_{PC_{N}}<R\right\} =\frac{1}{N \hbox{SNR}_{SD}\hbox{SNR}_{RD}} \\ &\times \int\limits_{1}^{2^{R}}\int\limits_{0}^{z-1}\frac{e^{\frac{1}{{\rm SNR}_{RD}}-\frac{w}{{\rm SNR}_{SD}}-\frac{1}{{\rm SNR}_{RD}} \left(\frac{z}{1+w}\right) ^{\frac{1}{N}}}}{\left(1+w\right)^{\frac{1}{N}} z^{\frac{1}{N}-1} }dwdz, \end{aligned} $$

(26)

and

$$ \begin{aligned} {\mathcal{P}}_{PC-ST_{N}} &= \frac{1}{\hbox{N} \hbox{SNR}_{RD}\hbox{SNR}_{SD}} \\ & \times \int\limits_{1}^{2^{R}}\int\limits_{0}^{-1+z^{\frac{1}{N+1}}}\frac{e^{-\left( \frac{1}{{\rm SNR}_{RD}}\left( \frac{z}{w+1}\right) ^{\frac{1}{N}}+\frac{\left( {\rm SNR}_{RD}-{\rm SNR}_{SD}\right) w}{{\rm SNR}_{RD}{\rm SNR} _{SD}}-\frac{\hbox{1}}{{\rm SNR}_{RD}}\right) }}{(w+1)^{1/\hbox{N}}z^{1-\frac{1}{\hbox{N}}}}dw dz, \\ \end{aligned} $$

(27)

when κ = 1. Note that $\mathcal{P}_{RC_{N}}$ in (25) for N = 1 is the same as $\mathcal{P}_{RC}$ defined in (7). Moreover, we use the notation $\mathcal{P}_{RC_{1}}$ and $\mathcal{P}_{RC}$ interchangeably. The same holds for the cases of PC and PC-ST.

The generalization of the overall outage probabilities after N retransmissions (therefore including the effect of the SR channel) is quite straightforward and can be easily obtained following the principles used in (9). We skip that here for the sake of brevity.

1.2 Throughput formulation

The throughput for RC when N retransmissions are allowed can be written as:

$$ \begin{aligned} {\mathcal{T}}_{RC_N} &= R \left(1-{\mathcal{P}}_{SD}\right) + \frac{R}{2} \cdot {\mathcal{P}}_{SD} \cdot {\mathcal{P}}_{SR} \cdot \left( 1 - \frac{{\mathcal{P}}_{SD_2}}{{\mathcal{P}}_{SD}} \right) \\ &\quad+ \frac{R}{2} \cdot {\mathcal{P}}_{SD} \cdot \left(1-{\mathcal{P}}_{SR}\right) \cdot \left(1 - \frac{{\mathcal{P}}_{RC}}{{\mathcal{P}}_{SD}}\right) \\ &\quad+ \cdots \frac{R}{N+1} \cdot {\mathcal{P}}_{SD} \cdot {\mathcal{P}}_{SR} \cdot \left(\frac{{\mathcal{P}}_{SD_2}}{{\mathcal{P}}_{SD}} \right) \cdots \left(\frac{{\mathcal{P}}_{SD_{N}}}{{\mathcal{P}}_{SD_{N-1}}} \right) \left( 1 - \frac{{\mathcal{P}}_{SD_{N+1}}}{{\mathcal{P}}_{SD_{N}}} \right) \\ &\quad+ \frac{R}{N+1} \cdot {\mathcal{P}}_{SD} \cdot \left(1-{\mathcal{P}}_{SR}\right) \cdot \left(\frac{{\mathcal{P}}_{RC}}{{\mathcal{P}}_{SD}}\right) \cdot \left(\frac{{\mathcal{P}}_{RC_2}}{{\mathcal{P}}_{RC}}\right) \cdots \left(\frac{{\mathcal{P}}_{RC_{N-1}}}{{\mathcal{P}}_{RC_{N-2}}}\right) \cdot \left(1-\frac{{\mathcal{P}}_{RC_N}}{{\mathcal{P}}_{RC_{N-1}}}\right), \end{aligned} $$

(28)

where $\mathcal{P}_{SD_{N}}$ is the outage probability in the SD link after N consecutive transmissions from the source when Chase combining is applied at the destination. Moreover, the above throughput formulation can be rewritten as:

$$ \begin{aligned} {\mathcal{T}}_{RC_N} &=R \left(1-{\mathcal{P}}_{SD}\right) + \frac{R}{2} \cdot {\mathcal{P}}_{SD} \cdot \left(1-{\mathcal{P}}_{SR}\right) \cdot \left(1 - \frac{{\mathcal{P}}_{RC}}{{\mathcal{P}}_{SD}}\right)\\ &\quad+ {\mathcal{P}}_{SR} \sum_{n=1}^{N} \left\{\frac{R}{n+1} \cdot \left[ {\mathcal{P}}_{SD_{n}} - {\mathcal{P}}_{SD_{n+1}} \right] \right\} \\ &\quad+ \left( 1- {\mathcal{P}}_{SR} \right) \sum_{n=2}^{N} \left\{\frac{R}{n+1} \cdot \left[ {\mathcal{P}}_{RC_{n-1}} - {\mathcal{P}}_{RC_n} \right] \right\}. \end{aligned} $$

(29)

The cases of PC and PC-ST can be easily obtained from above.