Multi-layer coding strategy for multi-hop block fading channels with outage probability

Abstract

This paper considers communication over a multi-hop Rayleigh block-fading channel, where there is no direct link between the transmission ends, and communication is carried out through the use of multiple cascaded relays, employing the Decode and Forward (DF) strategy. It is assumed that the relays cannot do buffering or apply coding over consecutive transmission blocks, and they can merely re-encode the retried information and forward an encoded version of information to the next hop. In this case, assuming the channel information of each hop is not available at the corresponding transmitter and considering the destination either receives a minimum rate R ₀ or declare an outage event, a multi-layer coding strategy in addition to a single-layer code of the minimum rate R ₀ are employed, where the optimal power allocation policy across code layers is derived, leading to the maximum average achievable rate for a given outage probability.

Appendices

Appendix A

This section aims at addressing the numerical solution of Eq. 47 in more details. To this end, it should be noted that $R_{r_{j}}(x_{j})= {\int }_{\gamma _{th_{j}}}^{x_{j}} \!\! \frac {-aI^{\prime }_{r_{j-1}}(a\mid R_{r_{j-1}}(x_{j-1}))da}{1+aI_{r_{j-1}}(a\mid R_{r_{j-1}}(x_{j-1}))}$ can be discretized as:

$$\begin{array}{@{}rcl@{}} R_{r_{j}}(\gamma(m))=\!\!\left\{ \!\begin{array}{ll} 0 & 0 \leq m \leq1\\ {\sum}_{i=1}^{m} \frac{ -\gamma(i) {\Delta} I_{r_{j-1}}(\gamma(i)) } {1+\gamma(i)I_{r_{j-1}}(\gamma(i))} & 1\leq m\end{array} \right. \end{array} $$

(57)

As a result, the partial differential Eq. 47 can be approximated as:

$$\begin{array}{@{}rcl@{}} &&R_{avg_{j}}^{\prime}\!(R_{r_{j}}(\gamma(m)))\left[\sum\limits_{i=1}^{m} \frac{1-\gamma(i)-\gamma(i)^{2}I_{r_{j-1}}(\gamma(i))}{(1+\gamma(i)I_{r_{j-1}} (\gamma(i))^{2}}\right] \\ &&-R_{avg_{j}}^{\prime\prime}\!(R_{r_{j}}(\gamma(m))) \left[\left(\frac{-\gamma(m) {\Delta} I_{r_{j-1}}(\gamma(m))}{1+\gamma(m) I_{r_{j-1}}(\gamma(m)) }\right) \right. \\ &&\qquad\qquad\qquad\quad\qquad\left.\times{\sum}_{i=1}^{m} \frac {-\gamma(i)}{1+\gamma(i)I_{r_{j-1}}(\gamma(i)) } \right] \\&&+\frac{ \lambda e^{\gamma(m)-\gamma_{th_{j}}} } {(1+\gamma(m)I_{r_{j-1}} (\gamma(m)))^{2}} =0 \end{array} $$

(58)

where $R_{r_{j}}(\gamma (m))$ can be calculated from Eq. 57 and $R_{avg_{j}}^{\prime }\!(R_{r_{j}}(\gamma (m)))$ and $R_{avg_{j}}^{\prime \prime }\;\!(R_{r_{j}}(\gamma (m)))$ denote, respectively, the first and the second derivatives of the conditional average rate associated with the (j+1)^th hop. It is worth mentioning that the objective is to find the discretized function $I_{r_{j-1}}(\gamma (m))$ associated with the j ^th hop and noting the fact that multistage optimization in Eq. 14 is tackled from the M ^th hop to the first hop, thus for the j ^th hop the conditional power allocation policy for the (j+1)^th hop is known, hence, $R_{avg_{j}}\!(R_{r_{j}}(\gamma (m)))$ and its derivatives is known when attempting to solve Eq. 58.

However, finding the optimal function $I_{r_{j-1}}(\gamma (m))$ is still a difficult task to deal with as the values of γ(m) are not known. Thus, instead of finding $I_{r_{j-1}}(\gamma (m))$, $I_{r_{j-1}}(.)$ is discretized into N equal-distance points $[I_{r_{j-1}}(\gamma (1)),..., I_{r_{j-1}}(\gamma (N))]$ with the boundary conditions $I_{r_{j-1}}(\gamma (1))=P_{r_{j-1}}-P_{0_{j-1}}$, $I_{r_{j-1}}(\gamma (N))=0$ and the best values γ(m), m = 1,…, N, which satisfy the differential equation in Eq. 58, are derived. To this end, for m = 1, noting $R_{avg_{j}}^{\prime }\!(R_{r_{j}}(\gamma (m)))$ is a linear function of the unit slope for small values of $R_{r_{j}}(\gamma (m))$, thus, $R_{avg_{j}}^{\prime }\!(R_{r_{j}}(\gamma (1)))$ and $R_{avg_{j}}^{\prime \prime }\!(R_{r_{j}}(\gamma (m)))$ in Eq. 58 can be well approximated by one and zero, respectively. Consequently, for the starting point of γ(1), it follows:

$$\begin{array}{@{}rcl@{}} &&\frac{1-\gamma(1)-\gamma(1)^{2}(P_{r_{j-1}}-P_{0_{j-1}})}{(1+\gamma(1)(P_{r_{j-1}}-P_{0_{j-1}}) )^{2}} \gamma(1)\ {} \\ &&\qquad\qquad+\frac{ \lambda e^{\gamma(1)-\gamma_{th_{j}}} } {(1+\gamma(1)(P_{r_{j-1}}-P_{0_{j-1}}) )^{2}} =0 \end{array} $$

(59)

Accordingly, N−1 values of γ(i) for i = 2,…, N are recursively calculated such that Eq. 58 holds for each value. Finally, the power allocation density function can be numerically computed from $\rho _{r_{j-1}}(x_{j} \mid R_{r_{j-1}}(x_{j-1}))=\frac {-dI_{r_{j-1}}(x_{j} \mid R_{r_{j-1}}(x_{j-1}))}{dx_{j}}$. It is worth mentioning that the early values of γ(m) may fall below $\gamma _{{th}_{j}}$. In this case, the power assigned to these layers is transferred to a single-layer at the threshold level.

Appendix B

This section aims at exploring the numerical solution of Eq. 50. To do this, I _s (.) is discretized into N discrete points [I _s (γ(1)), I _s (γ(2)),..., I _s (γ(N))] with the boundary conditions I _s (γ(1)) = P _s −P ₀, I _s (γ(N))=0, and $I_{s}(\gamma (i))=(P_{s}-P_{0})\frac {N-i}{N-1}$, where $P_{s_{0}}$ is the power assigned to a single-layer code to carry the minimum rate R ₀ over the first hop. Accordingly, N−1 values of γ(i) for i = 2,…, N are recursively derived through discretizing (50) as follows:

$$\begin{array}{@{}rcl@{}} &\;&R_{avg_{1}}^{\prime}(R_{r_{1}}(\gamma(m)))\left[\sum\limits_{i=1}^{m} \frac{1-\gamma(i)-\gamma(i)^{2}I_{s}(\gamma(i))}{(1+\gamma(i)I_{s}(\gamma(i)))^{2}}\right]\\ &\;& -R_{avg_{1}}^{\prime\prime}(R_{r_{1}}(\gamma(m)))\left[\left(\frac{-\gamma(m) {\Delta} I_{s}(l(m))}{1+\gamma(m)I_{s}(\gamma(m))}\right) \right. \\ &\;&\left. \quad\qquad\qquad\qquad\qquad\times\! \sum\limits_{i=1}^{m}\! \frac {-\gamma(i)}{1+\gamma(i)I_{s}(\gamma(i))}\right]\,=\,0 \end{array} $$

(60)

where $R_{r_{1}}(\gamma (m))$ can be computed as:

$$\begin{array}{@{}rcl@{}} R_{r_{1}}(\gamma(m))=\!\!\left\{ \!\begin{array}{ll} 0 & 0 \leq m \leq1\\ {\sum}_{i=1}^{m} \frac{ -\gamma(i) {\Delta} I_{s}(\gamma(i)) } {1+\gamma(i)I_{s}(\gamma(i))} & 1\leq m \end{array} \right. \end{array} $$

(61)

where again, using the approximations $R_{avg_{1}}^{\prime }(R_{r_{1}}(\gamma (m)))\approx 1$ and $R_{avg_{1}}^{\prime }(R_{r_{1}}(\gamma (m)))\approx 0$, it turns out that the starting point of γ(1) is the solution of the following equation:

$$ \frac{1-\gamma(1)-\gamma(1)^{2}(P_{s}-P_{0_{1}})}{(1+\gamma(1)(P_{s}-P_{0_{1}}) )^{2}} =0~. $$

(62)

It should be noted that if the early points of γ(m) for m=1,2,… become lower than $\gamma _{{th}_{1}}$, they are replaced with a single-layer code with power equal to the commutative power assigned to these layers.