In what follows, we will proceed to calculate the distributions of relaying gains, assuming different levels of available CSI. The principle of selecting the transmission path is to minimize the required transmission power for the delivery of a data packet with unit received signal power, or alternatively, to maximize the received signal power under a unit transmission power. The two normalizations give the same result and can be transformed easily to each other. Depending on different CSI assumptions, it may be easier to solve the problem using one than the other. We will use the normalization which makes it easier to find a solution. For unified presentation of the results, we always show them with a normalization to unit transmission power.
The channel models with and without fast fading have the same form except that the mean and the standard deviation differ. For ease of discussion, we consider the channel model without fast fading from now on. Whenever needed, it is straightforward to add fast fading by changing the mean value and the standard deviation.
5.1. Full CSI Relay Selection
In this subsection, we consider a full CSI scenario where the CSI of all involved links is used for selecting the best relaying path.
We follow the same notation as in Section 3 for the received power and the distance corresponding to direct link, relaying links, and access links. We assume that all transmitters use unit transmission power. The received signal power is therefore , where is a realization of a zero-mean Gaussian random variable, , modeling log-normal shadow fading. Since there are candidate RSs for the MS, an additional index is used to denote different candidate RSs. Therefore, for the relaying path, we denote the received power of the inferior of the access link and the relaying link by .
Out of candidate relaying paths, we select the one with maximum as the final candidate relaying path and denote it as . Due to the independence assumption between MSs, the statistics of 's conditioned on the position of the MS, , is i.i.d. We express this quantity by .
The transmission strategy is to use the direct path when the number of available candidate RSs is or when the received power of the direct path is no worse than half of that of the relaying path . On the other hand, we use the relaying path when the received power of the relaying path is at least twice that of the direct path . The factor of accounts for the relay multiplexing loss due to the half-duplex and two-hop assumptions of relaying. This means that whenever a decision for direct transmission is made, two sequential transmissions of the same message are conducted by the source. Thus the received power during a time period of completing a relaying path is
It should be noted that when comparing the distribution of with the distribution of a pure direct transmission case (i.e., PDF of ), there is a 3dB difference in the distribution of , because of the definition of . The derivation of the conditional CDF of can be found in Appendix . The result is
where denotes the unit step function, and is the shadow fading sample experienced in the direct link. The received signal power of the direct link is
The received signal power distribution with full CSI, , is obtained by integrating over and weighted by their corresponding PDFs. Note that these integrations give the cumulative distribution of the received power with the considered relaying protocol, taking the stochastic geometry and the complete channel model into account.
5.2. Partial CSI Relay Selection
In this subsection, we assume that only the CSI of the links ending at the BS is known to the BS, that is, the CSI of the direct link and the access links. In a modern cellular network, this CSI information is typically available at the BS for all active MSs, so that no extra effort is required for collecting it.
Now we require unit received signal power at the BS and derive the distribution of the required transmission power. The transmission power with unit-received power is expressed as , where is the required transmission power, and is as defined below (16). Similar to the notation in Subsection 5.1, we denote the transmission power with respect to the direct link by , the transmission power with respect to the relaying link by and the transmission power with respect to the access link by . Since we do not have information of , the RS with the best channel condition in its access link is chosen. We denote the chosen relay by so that . The joint distribution of the transmission power set is needed before we can investigate the transmission power distribution.
The joint distribution of differs from that of because obeys an order statistic in 's. It is also different from the order statistics in Subsection 5.1 because involves order statistics in 's and 's. The distribution of conditioned on is the same for all in our system model. Therefore, the joint distribution of is
where and are defined in (B.5) and (B.6). In the last step of (18), the fact that the distributions of and do not depend on is applied. Using (18), we may derive the transmission power distribution under the following strategy:
The derivation of the distribution of , , can be found in Appendix . The result is
where and is the required transmission power of the direct link:
The transmission power distribution with partial CSI is obtained by integrating over and weighted by their corresponding PDFs.
5.3. Partial CSI with AoA Assistance Relay Selection
The relaying gain provided by totally ignoring the channel condition on the relaying link is degraded compared to the full CSI scenario. To compensate for this deterioration without requiring significant amounts of feedback information as in the full CSI scenario, we incorporate Angle of Arrival (AoA) information into the path selection. Thus we assume that the BS knows the angles shown in Figure 3. In a macrocellular environment, this is possible to estimate if there is an antenna array at the BS. Requiring unit-received signal power, the path selection is made based on , , and .
A distribution of the relaying link power can be derived based on AoA information and an assumed channel statistic. This distribution provides information about the channel condition of the relaying link and can be used for the decision making. For any given realization , , we may calculate the conditional distributions and . Since is determined by the realization of , , and through (13), the conditional distribution and thus, can be obtained.
As shown in Appendix , the conditional distribution of is
where and .
The conditional distribution of is used to determine the relaying strategy. In the following, we simply select the estimate of to be the value which achieves an assumed CDF goal denoted by . The estimated value is used to determine the relaying strategy. Specifically, we now have the transmission power set for the relaying path. The rest of the work is similar to that in Subsection 5.2. The required transmission power is therefore expressed as
where the best relaying path is , the estimated transmission power for the best relaying path is , and the needed transmission power on the best relaying path is .
The estimation of above involves inverting the conditional CDF , which is nonalgebraic. To proceed with formulating the statistics of would require dealing with complicated nonlinear functions. Therefore, a semi-numerical approach, where we calculate according to (22) for each channel realization and collect the numerical samples of according to the path selection strategy (23) for the CDF distribution, is more efficient.