In this section, we derive closedform expressions for SEP of PSK modulation with three relaying protocols, that is, ADF, AF, and HDAF, by applying the wellknown MGF approach [15, 16]. We further show very tight approximations for SEP, induced from the numerical results given in Section 4, which help to assess asymptotic behavior of SEP in the high 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 [15, 16]
where and is the MGF of defined as
where is the expectation operator over the random variable .
3.1. Adaptive DecodeForward Relay Protocol
By defining 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
From the expressions of and given in (3) and (4), respectively, along with the fact that and are statistically independent, we have
Moreover, by substituting (9) in (12), we obtain the SEP of ADF as follows
By assessing the SEP expression of ADF given in (13) at large SNR, we observe that the negative term is much smaller compared to two positive terms as there are three MGF values included in the last term. Therefore, eliminating the last term (negative term) of (13) leads to the SEP approximation
Since , , it is obvious that obeys an exponential distribution with parameter and as a consequence the MGF of can be written as
Also, at the high regime, that is, , the MGF of , by omitting the unit value in (15), can be approximated by
Eventually, from (14) and (16) the approximation of is obtained as
where
3.2. AmplifyForward Relay Protocol
From (7) and using the fact that and are assumed to be statistically independent, the average SEP of the AF protocol is given by
where and is derived in the appendix. In [10], the closedform expression of SEP for AF relay is given in the form of hypergeometric function. We will show in the next section that both formulas, that is, our final expression of given in (A.8) and [10, ], provide the identical numerical result. It is worth mentioning that our final expression, contains only elementary functions and therefore is much simpler than [10, ]. 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:
Remark 1.
In [11], the SEP of the AF protocol has been approximated in the high SNR regime using the McLaurin series expansion of the probability distribution function (PDF) of given in (7). Specifically, for PSK modulation and Rayleigh fading channels for all links, the SEP approximation for the AF protocol derived in [11] is rewritten in terms of our notations as follows [11, ]:
Examining the two expressions of approximated SEP given in (20) and (21), we can see that although both formulas are derived independently they produce a similar form, only with the difference at the scale value. In the next section, we numerically show that our approximation (20) is tighter than (21) deduced in [11].
3.3. Hybrid DecodeAmplifyForward Relay Protocol
With the HDAF relaying scheme, the relay operates in DF mode if it can correctly decode the message from the source, otherwise the relay acts in AF mode. Let us denote 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
where the first term in (22) corresponds to the ADF mode and the second term indicates the AF mode. Since are statistically independent with and , from (3) and (7), we obtain
We next substitute (9) in (23) and the above formula can be rewritten as
Similarly as in the ADF protocol, by eliminating the last term in (24), the SEP approximation of HDAF can be determined by
Using the fact that and the approximation of given in (A.10) of the appendix, we can tightly asymptotically approximate as
3.4. Performance Gain of HDAF over ADF and AF
We now assess the behavior of SEP performance for the considered three relay protocols in the high 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 loglog scales are parallel with the slope of order two in the high regime as illustrated in Figure 2. Intuitively, the HDAF scheme outperforms both ADF and AF. These observations inspired us to deduce the performance gain that can be achieved with HDAF compared to the two conventional protocols ADF and AF. To answer this question, we adapt the concept of relaying gain . Here, with is the SEP performance gain of HDAF compared to the protocol. As shown in Figure 2, we have
The limit operation in (27) implies that the gain 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
Substituting (20) and (26) in (27), the gain to be considered here is given by
Regarding the channel mean power of each link, we also assume that the relay is placed in between the source and destination. The path loss of each link is assumed to follow an exponentialdecay model. As such, if the distance between the source S and destination D is given as , then . For example, a pathloss exponent of corresponds to a typical nonlineofsight propagation scenario. This geometrical model has been widely used in the context of relay networks (see, e.g., [11, 17]). According to this physical model, when the relay is located close to the source or the destination we have or , respectively. As the relay is located halfway between the source and destination, we have .
In view of (28) and (29), the following general observations can be made for the respective gains.

(i)
With a fixed modulation scheme, that is, is constant, depends only on the ratio between channel mean power of the relaytodestination link and that of sourcetorelay link.

(ii)
As the relay moves closely to the source, we have leading to . We can intuitively explain this result as follows. As the channel of sourcetorelay 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 sourcerelaydestination path can be approximated by the inferior channel between sourcetorelay and relaytodestination links. Hence, the AF mode now provides a similar performance as ADF. In other words, the three protocols almost result in identical performance.

(iii)
In contrast, the gain is significant when the relay is located nearby the destination, that is, . We can similarly explain this result as in the case the relay is close to the source.

(iv)
Examining the function (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 wellknown result that ADF always outperforms AF scheme.
Next, regarding the relaying gain , we introduce several specific examples for QPSK modulation (substituting in (18) yields and ).

(i)
For a symmetric cooperative system, that is, , we have dB and dB.

(ii)
For an asymmetric cooperative system, where the relay is close to the source, that is, , goes to zero. In this special case, the HDAF scheme has no benefit compared to ADF and AF.

(iii)
For an asymmetric cooperative system, where the relay is located nearby the destination, that is, , 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.