Typical behavior of relays in communication channels

The typical behavior of the relay-without-delay channel under low-density parity-check coding and its multiple-unit generalization, termed the relay array, is studied using methods of statistical mechanics. A demodulate-and-forward strategy is analytically solved using the replica symmetric ansatz which is exact in the system studied at Nishimori’s temperature. In particular, the typical level of improvement in communication performance by relaying messages is shown in the case of a small and a large number of relay units.

Figure 1

The $L$-component relay array. The transmitter sends a codeword $\mathbf{t}$ to the final receiver and to each of the $L$ relays. Each relay receives a message ${\mathbf{r}}_i$ which is a corrupted version of the original codeword subject to the AWGNs ${\mathit{\nu }}_0$ and ${\mathit{\nu }}_i$. It then sends to the final receiver the encoded vector ${\mathbf{t}}_i$. The final receiver receives the original transmitted codeword summed with all the relayed messages ${\mathbf{t}}_i$ and corrupted by the AWGN vectors ${\mathit{\nu }}_0$ and $\mathit{\nu }$.

Figure 2

Schematic drawing of the relay-without-delay setup to be analyzed.

Figure 3

The overlap of retrieved and original codewords for the RWD channel with ${\sigma }_1^2=0.1{\sigma }^2$ is given by the continuous curve. The dashed curve shows the same for the simple Gaussian channel. The vertical line indicates Shannon’s limit for the RWD as calculated by El-Gamal and Hassanpour [16].

Figure 4

Entropy and energies for the RWD channel at Nishimori’s temperature. The upper graph shows the entropy which is given by the difference between the internal energy and the free energy depicted in the bottom graph.

Figure 5

The continuous-dashed line shows the dynamical-thermodynamical transition noise levels of the RWD against $\eta$, the ratio between relay and final receiver noise levels. The upper horizontal line corresponds to Shannon’s limit ${\sigma }^2\approx 8.79$.

Figure 6

Schematic drawing of the classical relay channel setup. The relative gains are all equal to 1 and not shown in the figure.

Figure 7

The continuous-dashed line shows the dynamical-thermodynamical transition noise levels of the RWD against $\lambda$ in the setup of Fig. 6. The continuous line without marked symbols is Shannon’s limit for a CRC with the same noise levels and transmission powers.

Figure 8

Dynamical and thermodynamical transition points for many relays. The exact formula is used to calculate the points $L=1,2,3,4,5$. The horizontal lines represent the large $L$ limit.