Abstract
We investigate the performance of adaptive decode-forward (ADF), amplify-forward (AF), and hybrid decode-amplify-forward (HDAF) relaying protocol. Specifically, we derive the closed-form expression of symbol error probability (SEP) for these three relaying protocols. For the AF protocol, we derive a simpler and more tractable final result than previously published work. Furthermore, analyzing the high signal-to-noise ratio (SNR) regime, we derive an asymptotically tight approximation for SEP. The approximation is used to assess an important aspect between the three different relaying protocols, i.e., the SEP performance gain of HADF over the two conventional relaying schemes ADF and AF. It is shown that the performance gain of HDAF over ADF and AF significantly depends on the relative value of channel conditions between source-to-relay and relay-to-destination links. Interestingly, it is shown that HDAF has no benefit compare to the two others if the relay is located close to the source. In contrast, the performance gains of HDAF over the ADF and AF scheme are remarkably large as the relay moves nearby the destination. Specifically, the numerical results given as well as our analysis indicate that for QPSK modulation, HDAF outperforms ADF and AF with a performance improvement by 5 dB and 6 dB, respectively.
) fixed decode-forward (FDF): the relay always forwards its received message, where potential propagating errors could lead to wrong decision at the destination, and (
) adaptive decode-forward (ADF): the relay will not be active if the signal from the source is corrupted. In [
], are derived independently they can be expressed in a similar form, only with the difference at the scale value. Interestingly, we demonstrate that our SEP approximation for the AF protocol is tighter than the work given in [
regime. Numerical results are provided in Section 4 to verify our analysis. Some discussion and future work have been addressed in Section 5. Finally, Section 6 concludes the paper.
to satisfy the power constraint at the relay (i.e., equal power allocation between S and R). In this paper, to simplify the analysis we only consider the orthogonal AF (OAF) protocol, that is, the source S remains silent in the relaying phase. In contrast, a nonorthogonal AF protocol (the source S continues to transmit in the relaying phase) which improves the performance compared to the OAF scheme has been investigated in [
is the transmitted signal at the source with the average transmit power per symbol 
. Depending on the relaying operation during the second-hop transmission, the received signals at the destination for ADF and AF mode, respectively, are given as follows:
, 
, and 
, 
, represent the fading channel magnitude with the channel mean power 
, received signals, and additive white Gaussian noise (AWGN) with variance 
, respectively. It is noted that all random variables 
, where 
denotes the complex Gaussian random variable with zero mean and variance 
, and 
are statistically independent. At the destination, the received signals from the source and relay are combined together using the MRC technique.
of the MRC output is given by
and 
are the instantaneous 
of source-to-destination link and relay-to-destination link, respectively. Otherwise, the relay remains silent, hence, in this case we have
is chosen based on the power constraint between the two phases. In this paper, without considering the power optimization problem, we assume the relay consumes the same amount of power as the source does in the first hop. In case of available instantaneous CSI at R (CSI-assisted relay), an amplifying parameter 
can be formulated as [
is the common 
of each link without fading [
after MRC for CSI-assisted AF protocol is expressed as [
-PSK modulation with three relaying protocols, that is, ADF, AF, and HDAF, by applying the well-known MGF approach [
regime. In practice, the decoding decision at the relay is determined by checking the cyclic redundancy check (CRC) of a frame. However, in this paper, it is assumed that this decision is made symbol by symbol for mathematical tractability of SEP derivation. Denoting 
as the probability that the relay correctly decodes the symbol, we have [
and 
is the MGF of 
defined as
is the expectation operator over the random variable 
.
and 
as the average SEP associated with the events that the relay correctly and incorrectly decodes the symbol transmitted from the source, respectively, the average SEP of the ADF protocol can be written as
and 
given in (3) and (4), respectively, along with the fact that 
and 
are statistically independent, we have
, 
, it is obvious that 
obeys an exponential distribution with parameter 
and as a consequence the MGF of 
can be written as
regime, that is, 
, the MGF of 
, by omitting the unit value in (15), can be approximated by
is obtained as
and 
are assumed to be statistically independent, the average SEP of the AF protocol is given by
and 
is derived in the appendix. In [
given in (A.8) and [
], provide the identical numerical result. It is worth mentioning that our final expression, contains only elementary functions and therefore is much simpler than [
]. This finding helps us to derive the asymptotically tight approximation of SEP for both AF and HDAF relay protocols. Moreover, using the asymptotic approximation of 
in (A.10), a tight approximation of 
in the high 
can be obtained as follows:
given in (7). Specifically, for 
-PSK modulation and Rayleigh fading channels for all links, the SEP approximation for the AF protocol derived in [
]:
as the average SEP associated with the event that the relay incorrectly decode the source's symbol, hence, the SEP in this protocol can be expressed as
are statistically independent with 
and 
, from (3) and (7), we obtain
and the approximation of 
given in (A.10) of the appendix, we can tightly asymptotically approximate 
as
regime by analyzing their approximations. As can clearly be seen from (17), (20), and (26), the three protocols result in a diversity order of two since the SEP expressions are inversely proportional to 
. In other words, the related three SEP curves plotted in log-log scales are parallel with the slope of order two in the high 
regime as illustrated in Figure 
. Here, 
with 
is the SEP performance gain of HDAF compared to the 
protocol. As shown in Figure 
is obtained in the high 
regime. In this context, we now calculate 
and 
. Substituting (17) and (26) in (27) and performing some elementary manipulations, 
can be expressed as
to be considered here is given by
, then 
. For example, a path-loss exponent of 
corresponds to a typical nonline-of-sight propagation scenario. This geometrical model has been widely used in the context of relay networks (see, e.g., [
or 
, respectively. As the relay is located half-way between the source and destination, we have 
.
is constant, 
depends only on the ratio between channel mean power of the relay-to-destination link and that of source-to-relay link.
leading to 
. We can intuitively explain this result as follows. As the channel of source-to-relay link is very good, the probability that the relay correctly decodes the source's signal is high. As a result, the HDAF scheme mostly acts in the DF mode. However, there may be some rare situations in which the relay cannot decode the source's message and HDAF will act in the AF mode to assist the direct communication. That is the reason HDAF still achieves some very small gain, which also can be neglected, in this particular case. Furthermore, for the AF scheme, the compound source-relay-destination path can be approximated by the inferior channel between source-to-relay and relay-to-destination links. Hence, the AF mode now provides a similar performance as ADF. In other words, the three protocols almost result in identical performance.
. We can similarly explain this result as in the case the relay is close to the source.
(deduced from (18), with 
(equality occurs for BPSK modulation), we can easily see that the global maximum value of 
is 2/3. Consequently, we have 
. In other words, the HDAF scheme always provides more gain over AF than over ADF. This again confirms a well-known result that ADF always outperforms AF scheme.
, we introduce several specific examples for QPSK modulation (substituting 
in (18) yields 
and 
).
, we have 
dB and 
dB.
, 
goes to zero. In this special case, the HDAF scheme has no benefit compared to ADF and AF.
, 
becomes remarkably large. Specifically, in case of 
, HDAF achieves an increase in SEP performance of 5.03 dB and 6.15 dB compared to ADF and AF scheme, respectively.
for the considered three relay protocols taking into account the relay's location as the relay is placed halfway between the source and destination, the relay is close to the source, and the relay is close to the destination, respectively.
.
.
.
]. As can clearly be seen from these figures, our numerical results exactly match with those calculated from [
]. However, with the complex expression of the MGF of the harmonic mean of two exponential random variables given in [
], one may not be able to perform the tight approximation of SEP leading to relaying gain 
as our result in (A.6) does. The results shown also reveal that the approximation of SEP becomes very tight in the high 
regime.
], in separated examples. We compare the tightness of two SEP approximations, that is, our expression given in (20) and one derived in [
] which is again shown in (21). For this purpose, we compute the relative error of two SEP approximations and exact SEP as follows:
regime of Figure 
Given in (8)
can be given by
is always greater than or equal to 
(since 
, it is easy to see that 
). Taking the differentiation in (A.1) with respect to 
, the PDF of 
is given by
and for the sake of simplicity denote 
with 
. Then, (A.2) can be rewritten as follows:
can be expressed as
and 
, we have
)] and [
. Our final expression in (A.8) is simpler and more tractable than [
], which contains the Gauss hypergeometric function. Both formulas, that is, (A.8) and [
], lead to the same numerical result but (A.8) is more helpful to analyze the SEP performance in the high 
regime by providing the asymptotically tight approximation of 
.
is high, that is, 
goes to infinity, we have
can be tightly approximated as follows: