k+1$.

![Single relay placement, total power constraint, exponential path-loss: $\frac{y_{1}^{*}}{L×}$ and optimum $\frac{P_{0,1}}{P_{T}×}$ versus $\lambda$.[]{data-label="fig:single_relay_total_power"}](single-relay-total-power-position-power.pdf){height="3cm" width="8cm"}

Optimal Placement of a Single Relay Node {#subsection:optimal_placement_single_relay_sum_power}
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In the following theorem, we derive the optimal power allocation, node location and data rate when a single relay is placed.

\[theorem:single\_relay\_total\_power\]

For the single relay node placement problem with sum power constraint and exponential path-loss model, the normalized optimum relay location $\frac{y_{1}^{*}}{L×}$, power allocation and optimized achievable rate are given as follows:[^5]

1.  [*For $\lambda \leq \log 3$*]{}, $\frac{y_{1}^{*}}{L×}=0$, $P_{0,1}=\frac{2P_{T}}{e^{\lambda}+1×}$, $P_{0,2}=P_{1,2}=\frac{e^{\lambda}-1}{e^{\lambda}+1×}\frac{P_{T}}{2×}$ and $R^{*}=C \left(\frac{2P_{T}}{(e^{\lambda}+1)\sigma^{2}×}\right)$.

2.  [*For $\lambda \geq \log 3$*]{}, $\frac{y_{1}^{*}}{L×}=\frac{1}{\lambda×} \log \left(\sqrt{e^{\lambda}+1}-1\right)$, $P_{0,1}=\frac{P_{T}}{2×}$, $P_{0,2}=\frac{1}{\sqrt{e^{\lambda}+1}×}\frac{P_{T}}{2×}$, $P_{1,2}=\frac{\sqrt{e^{\lambda}+1}-1}{\sqrt{e^{\lambda}+1}×}\frac{P_{T}}{2×}$ and $R^{*}=C \left( \frac{1}{\sqrt{e^{\lambda}+1}-1×}\frac{P_{T}}{2 \sigma^{2}×} \right)$

See Appendix \[appendix:proof\_of\_single\_relay\_sum\_power\_results\].

[*Discussion:*]{} It is easy to check that $R^{*}$ obtained in Theorem \[theorem:single\_relay\_total\_power\] is strictly greater than the AWGN capacity $C \left(\frac{P_{T}}{\sigma^{2}×}e^{-\lambda}\right)$ for all $\lambda>0$. This happens because the source and relay transmit coherently to the sink. $R^{*}$ becomes equal to the AWGN capacity only at $\lambda=0$. At $\lambda=0$, we do not use the relay since the sink can decode any message that the relay is able to decode.

The variation of $\frac{y_{1}^{*}}{L×}$ and $\frac{P_{0,1}}{P_{T}}$ with $\lambda$ has been shown in Figure \[fig:single\_relay\_total\_power\]. We observe that (from Figure \[fig:single\_relay\_total\_power\] and 