Amplify and forward

The amplify-and-forward relay protocol is a protocol defined for wireless cooperative communications. An example of a wireless communication network in which cooperation improves the performance of the system is the relay network.

In this case, the relay just amplies its received signal, maintaining a fixed average transmit power.

AF relay function


The AF relay function is an amplification of the received signal,
 * $$\ f_{AF}(y_{SR})=\beta y_{SR} $$

with $\beta$ the relay transmit average power constraint coefficient. The coefficient $ \beta $ ensures that the average transmit power at the relay is constant and equal to $ P_R $, therefore $ \beta $ is derived in a similar way to the one obtained by Laneman :


 * $$\ E[ |f(y_{SR})|^2 ] \leq P_R $$
 * $$\ E[|\beta y_{SR}|^2] \leq P_R $$
 * $$\ \beta \leq \sqrt{\frac{P_R}{ h_{SR}^2 E[|x|^2] + E[|n_{SR}|^2]}}  $$

and for Gaussian noise (AWGN) with unit variance and $ Es $, the energy of the symbol becomes
 * $$\ \beta \leq \sqrt{\frac{P_R}{ h_{SR}^2 Es + 1}}. $$

The form of the AF relay function is independent of the modulation scheme of the source symbol. Since the transmitted signal at the relay is linear in $ y_{SR} $, as seen in Figure Relay functions for BPSK modulation, for higher values of $ y_{SR} $ the relay requires more instantaneous transmit power than the case when the DF relay function is used. The probability density function of $ y_{SD} $ is the same as in the DF relaying protocol, while the signal $ y_{RD} $ is
 * $$\ y_{RD} = h_{RD} \beta y_{SR} + n_{RD} = h_{RD} \beta (h_{SR}x+n_{SR}) +n_{RD} $$
 * $$\ \quad \quad = \beta h_{SR} h_{RD}  x + h_{RD} \beta n_{SR} + n_{RD} $$

Therefore, given the independence of the noise and using the linear property of Gaussian random variables, the conditional probability density function of $ y_{RD} $ is
 * $$\ p(y_{RD}|x) = \mathcal{N}(\beta h_{SR} h_{RD} x, h_{RD}^2 \beta^2 + 1) $$

This probability density function together with the one for $ y_{SD} $, are replaced in the equation of the ML detector and a simplified formulation is obtained.
 * $$\ \text{ML: } ~ \widehat{x}_D = \underset{x}{\max} ~ \mathrm{p}(y_{SD}|x) \mathrm{p}(y_{RD}|x) $$

Note, that until now, all the results shown for AF are independent on the modulation order.

BPSK relay function
For example, for BPSK modulation, replacing $p(y_{RD}|x)$ in the ML detector, we obtain that the detector at the destination is given by
 * $$\ h_{SD} y_{SD} \underset{H_0}{\overset{H_1}{\gtrless}} \ln \left(    \frac{    \frac{1}{\sqrt{2 \pi (h_{RD}^2 \beta^2  + 1)} } \exp \left\lbrace -\frac{(y_{RD} + \beta h_{SR} h_{RD} )^2}{2 h_{RD}^2 \beta^2  + 1} \right\rbrace}			{\frac{1}{\sqrt{2 \pi (h_{RD}^2 \beta^2  + 1)} } \exp \left\lbrace -\frac{(y_{RD} - \beta h_{SR} h_{RD} )^2}{2 (h_{RD}^2 \beta^2  + 1)} \right\rbrace} 	   \right) $$:$$\ h_{SD} y_{SD} \underset{H_0}{\overset{H_1}{\gtrless}} \ln \left(	 \exp \left\lbrace - \frac{4\beta h_{SR} h_{RD} y_{RD}}{2 (h_{RD}^2 \beta^2  + 1)} \right\rbrace \right) $$:$$\ h_{SD} y_{SD} + \frac{ \beta h_{SR} h_{RD} }{h_{RD}^2 \beta^2  + 1} y_{RD} \underset{H_0}{\overset{H_1}{\gtrless}}  0  $$

In addition to the decode-and-forward (DF) decision rule, for which the destination is required to know only S to D and R to D link characteristics, AF requires extra information about the S to R link. This implies that for the AF relay protocol the destination decision rule takes into consideration the quality of the channel between S and R. Therefore, one might assume that AF should perform better than the DF protocol. However this is not always true; the performance of different relay protocols is directly influenced by the quality of the channels, and more precisely by the position of the relay. One argument for this behavior is that, for the AF protocol, when the relay amplifies the received signal it also amplifies the noise. For very good channel conditions, the DF protocol makes the right decision, eliminating the noise, while the AF relay protocol, amplifies the noise. Thus, for high SNR on the S to R channel, a clean signal as the one provided by the DF protocol is preferred at the destination, rather than the noisy signal resulted from the AF relay function.