Estimate and forward

The estimate-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.

This protocol is also known as Compress-and-Forward (as first introduced by Cover and Gamal ) or Quantize-and-Forward. At the relay, a transformation is applied to the received signal, which provides an estimate of the source signal. This estimate is also known as soft information, and it is forwarded to the destination.

EF Relay Function
For the general case the relay function is given by:
 * $$\ f_{DF}(y_{SR}) = \widehat{x}_{ef} $$

where, $\widehat{x}_{ef}$ denotes the estimate the relay forms from the received signal $y_{SR}$. Also, the EF relay function can be denoted by any type of side information or extra information the relay can forward to the destination to help the communication.

From the broad range of possible relay functions, only few improve the overall performance of EF. In their paper, Abou-Faycal and Medard showed that the Lambert $W$ function minimizes the probability of error in a system without the direct link. Others, such as Gomadam and Jafar, proved that the MMSE maximizes the receiver SNR and numerically showed that it is capacity optimal, again for a system without the direct link. These two functions are very similar in form. Log-likelihood ratio (LLR) functions have also been used as soft information, as an approximation of the MMSE estimate, but proved to encounter difficulties at the detector located at destination. Hu and Lilleberg used a weighted type of soft information which mimics the behavior of the MMSE estimate.

A special interest was attracted by the MMSE estimate as the relay function because of the properties this transformation has, and because it efficiently incorporates information about the quality of the S to R channel.

MMSE estimate
One type of relay function used in the EF protocol is the minimum mean squared error (MMSE) estimate, which is the conditional expectation of the source sent symbol $ x $ given the received signal at the relay $ y_{SR} $
 * $$\ 	f_{EF}(y_{SR}) = k \mathrm{E}[x|y_{SR}] = k \sum_{x} x \mathrm{p}(x|y_{SR}) . $$

After applying Bayes' rule, we get
 * $$\ f_{EF}(y_{SR}) = k \sum_{x} x \frac{\mathrm{p}(y_{SR}|x)\mathrm{p}(x)}{\mathrm{p}(y_{SR})}  = k  \frac{  \underset{x}{\sum} x \mathrm{p}(y_{SR}|x)p(x)}{ \underset{x}{\sum} p(y_{SR}|x)p(x)}, $$

where $k$ is the average transmit power constraint coefficient for the relay. Therefore, for an average transmit power at the relay $ P_R $ we can obtain the value of $ k $ similar to the AF case:
 * $$\ E[|k*E[x|y_{SR}]|^2] \leq P_R $$
 * $$\ k \leq \sqrt{\frac{P_R}{E[|E[x|y_{SR}]|^2]}} $$

where $ | \cdot | $ represents the absolute value. Next, we investigate its characteristics by applying it to different modulation schemes and by comparing it to the common relay functions used for amplify-and-forward (AF) and decode-and-forward (DF). The conditional density function of $ y_{SR} $ given the source symbol $ x $ had a similar form as for the DF and AF relay protocols, that is $\mathrm{p}(y_{SR}|x) = \mathcal{N}(h_{SR}x,1) $. Introducing the conditional probability density function of $ y_{SR} $ in the given form of the MMSE estimate, the EF relay function is further expanded as
 * $$\ f_{EF}(y_{SR}) = k ~ \frac{\underset{x}{\sum}   x  \mathcal{N}(h_{SR} x, 1) \mathrm{p}(x)  }{   \underset{x} {\sum} \mathcal{N}(h_{SR} x , 1) \mathrm{p}(x)  } \label{eq:efCondExp1}  $$


 * $$\ \quad \quad k ~\frac{\underset{x}{\sum}   x  \frac{1}{\sqrt{2 \pi }} \exp \left\lbrace - \frac{(y-h_{SR}x)^2}{2} \right\rbrace \mathrm{p}(x)  }{   \underset{x} {\sum} \frac{1}{\sqrt{2 \pi }} \exp \left\lbrace - \frac{(y-h_{SR}x)^2}{2} \right\rbrace \mathrm{p}(x)  } $$

BSPK modulation


Following we show the MMSE estimate for BPSK modulation for a specific system network, in which each channel is degraded by additive white Gaussian noise (AWGN).

When the source uses antipodal signals, $ x \in \left\lbrace -1,1 \right\rbrace $, also known as BPSK modulation, the soft information of the EF protocol becomes \begin{eqnarray} f_{EF}(y_{SR}) = k~ \frac{\mathrm{e}^{h_{SR} y_{SR}} -\mathrm{e}^{- h_{SR} y_{SR}}} { \mathrm{e}^{h_{SR} y_{SR}} + \mathrm{e}^{- h_{SR} y_{SR}} } \\ f_{EF}(y_{SR})= k~ \tanh \left(y_{SR} h_{SR}\right) \end{eqnarray}

The EF relay function specific for BPSK modulation is shown in Figure Relay functions for BPSK modulation, where both the AF and DF relay functions are also plotted. The MMSE as the EF relay function is denoted as EF - MMSE. One important observation made on this graph is that the EF relay function is in between the AF and DF ones.

We noticed that the quality of the S to R link has great influence on the performance of the EF protocol, as seen in Figure MMSE tanh function. We define $ SNR $ to represent signal to noise ratio at the receiver. For high values of the $ SNR_{SR} $, when the relay is closer to the source, the MMSE estimate becomes very steep and similar to the DF hard decision relay function.

While, for low values of the $ SNR_{SR} $ (the relay is closer to the destination), the EF relay function is similar to the AF one.

Thus, we expect that for high SNR on the S to R link, EF will perform as good as DF and for a degraded S to R channel, the performance of the system will be similar to AF.

With this type of soft information, we expect the EF relay protocol to perform as the best of AF and DF. Furthermore, the MMSE estimate inherits one important property from the DF protocol: the instantaneous transmit power at the relay is slightly higher than for DF, but it is much lower than the one the AF protocol requires. This can be observed in the Figure Relay functions for BPSK modulation; for $ y_{SR} \geq  1.3$ the AF relay protocol requires more instantaneous transmit power than all other schemes.