We consider a dual-hop cooperative wireless system consisting of wireless nodes: one source node, a set of relay nodes, , and a destination node. We assume that each node has a single antenna and all operate in half-duplex mode. We consider a time-division channel allocation scheme with time slots. In the first time slot, the source broadcasts its message to all the nodes. During the following time slots, the source stops transmitting and the relays of the decoding set will forward the source information to the destination in a scheduled order. denotes the set of indexes of relays which succeed in decoding correctly the source message for a given required SNR threshold, i.e., each relay decides whether to forward the source information or not by comparing the received signal power with a decision SNR threshold. The cardinality of is denoted. In the considered scenario, we assume that the receivers at the relays and destination have exact channel state information but no transmitter channel state information is available at the source or relays [14].

Let us denote, and as the channel gains corresponding to the direct link, links between the source to the relay and the relay to the destination, respectively. Moreover, and, are supposed independent but not necessarily identical Generalized-K distributions with parameters, for and, for, respectively. Let and denote the instantaneous SNR received at the the relay and the destination from the relay, respectively. Thus,

and

where is the transmit energy signal and represents the single sided power spectral density of the AWGN. The corresponding average SNR are denoted

and. Finally, we denote as the contribution of all the relays belonging to the set Q and the direct link using the MRC (Maximum Ratio Combiner) technique.

The small scale multipath fading is usually modeled using Nakagami-m distribution. The envelope of the transmitted signal over the Nakagami fading, , has a probability density function given as follows [21]

where, , is the fading severity parameter called also the Nakagami parameter, , , is the average received power and is the Gamma function.

However, for example in terrestrial and satellite landmobile systems, the link quality is also affected by slow variation of the mean signal level due to the shadowing from terrain, buildings, and trees. Consequently, multipath fading are superimposed on shadowing. Hence, the pdf of the envelope of the received signal will be conditioned on the average power leading to:

Consequently, in combined multipath and shadowing fading, (1) can be rewritten as follows

where is the pdf of the average power which is usually modeled by the Lognormal distribution [21]. To mitigate difficulties to derive closed form of some metric performance measure, using the moment matching method, it has been proposed to use the Gamma distribution to model the average power random variations given as follows

where reflects the shadowing severity and is the local mean power.

Consequently, the pdf of the received signal envelope due to the composite fading (shadowing and multipath) leads to the Generalized-K distribution which is given by

where is the modified Bessel function of the second kind and order and.

In the cooperative system, as mentioned in [2], to capture the path-loss effects, we use

, , where is the distance between the terminals and, and is the pathloss exponent.

Over a Generalized-K fading channel the pdf of the output instantaneous SNR follows a Generalized-K distribution given by [11]

where and is the average SNR over the considered channel.

In the next section, we deal with the outage probability and the ASEP closed form derivations. To this end, in the cooperative case, we deal with the cdf of the sum of Generalized-K random variables. As pointed in [22,23], the computation of the cdf expression is not straightforward and requires the use of approximations and asymptotic expansions or the use of the numerical inversion of the characteristic function. An alternative approach, to avoid the analytical difficulties, is to consider an approximation of the Generalized-K pdf, given by (6), using the moment matching method by a more tractable pdf [22], as shown in the next subsection.

The main goal of this subsection is to give the analytical and main results of the approximation of the Generalized-K pdf by a suitable Gamma pdf. This approximation has been developed in two steps [22]:

• Approximation of the Generalized-K pdf by a Gamma pdf by matching the two first positive moments.

• Introduction of an adjustment factor for the shape and the scale parameters of the approximating Gamma pdf.

If we consider the distribution given by (5), the moment of the Generalized-K distribution can be derived as [22]

The moment of the approximating Gamma distribution can be expressed as

where and denote the scale and shape parameters of the approximating Gamma distribution, respectively.

Matching the two first positive moments of the Generalized-K distribution to those of the approximating Gamma distribution results in

where is the amount of fading in the composite fading channel and, is determined by the smallest physical values of and. However, this approximation results in a poor fitting in the lower and upper tail regions as compared to the original pdf. In order to bypass this inaccuracy, an adjustable form has been introduced for the scale and shape parameters of the region-wise approximating Gamma pdf for the two first moments. These parameters are given as follows

where the adjustment parameter is chosen so that the pdf (cdf) of the approximating Gamma distribution becomes as close as desired to the pdf (cdf) of the Generalized-K distribution in a certain region of interest. We note that. Since, the relevant practical range of AF is from zero (for non-fading channels) to 8 (for severe multipath fading and shadowing conditions corresponding to and), the relevant range of the adjustment factor is.

As aforementioned in Subsection 2.3, the pdf of the sum of N Generalized-K random variables can be approximated by a Gamma pdf using the adjustable form.

where. and are the shape parameters of the Generalized-K random variable.

We note that for the i.i.d case, the expressions in (11) and (12) simplify to

An illustration of the choice of the optimal value of the adjustment factor is given in Section 5.

The outage probability represents the probability of unsatisfactory signal reception. It can be defined as the probability that the received SNR is less than a specified SNR threshold value at the destination, denoted [10]. We consider the direct link composed by the source and the destination node communicating over the composite fading channel. The outage probability denoted is given as [10]

We note that is given by (6). According to [10], we can derive as follows

where, is the cosecant function, and

is the generalized hyper-geometric function

[24].

The ASEP, , for modulated signals over the fading channel can be obtained by averaging its conditional symbol error probability, , under a given SNR in AWGN over the pdf of the received SNR

In this paper, we consider M-QAM modulated signals over Generalized-K fading channel. Thus, can be expressed as [6], where

, and

. By considering the same notations above, the closed form of the ASEP for the direct link is given by [10].

where, is the Meijer’s G-function [24].

In the cooperative case, the outage probability is obtained by the contribution of all the relay nodes belonging to the set Q and the direct link using the MRC (Maximum Ratio Combining) technique. Thus, by using the law of total probability we obtain

where is the probability that nodes cooperate in relaying information to the destination.

is the conditional outage probability in the case of. In the following, we shall derive the expression of and

, respectively.

a) Calculation of

The probability that the relay-node is in the decoding set, is defined as the probability that its SNR, , is larger than the threshold SNR at the relay,. It is given as follows

For simplicity, we choose the same threshold for all the relays, i.e,.

Consequently, is given as follows

where is the probability density of the SNR at the relay. It is given by (6) and its cdf function is given by (16) by setting. Thus, is deduced easily.

Let denotes a combination that of nodes cooperate in relaying information to the destination and as the corresponding probability. Assuming that the relay channels are independent, can be expressed as

Consequently,

b) Calculation of

can be seen as the cdf of

, evaluated at, i.e. the sum of the.

Generalized-K random variables correspond to the active links. As mentioned above in Section 2.4, to avoid analytical complexity, the computation of the cdf expression of can be approximated by the Gamma pdf with the scale and shape parameters given by (11) and (12), respectively. Consequently, and it is expressed as

where.

Consequently is given as follows

where is the incomplete Gamma function. Finally, from (21)-(23) and (25) we derive the outage probability at the destination for the cooperative system as

In the cooperative case, the ASEP is obtained by the contribution of all the relay nodes and the direct link. At the MRC output, by considering the same notation as above, can be expressed as follows

has been calculated in the last subsection given by (23). is the conditional symbol error probability corresponding to the cooperation of the branches and the direct link (source-destination). In the following, we shall derive its expression.

a) Calculation of

can be seen as the probability of occurring an error when relays are active. As mentioned in the last section, it can be expressed as

where is given by (24) and is expressed as

To evaluate this integral, we use the relationship

which leads to a new integral form as follows

where. Using ([6], (5.1), (5.18)), this integral is deduced as follows.

For integer values of, can be written as

For non-integer values of, can be written as

where.

Thus, using (23), (27), (31) or (32), the ASEP can be deduced easily.

To be more accurate in our illustrations, we have simulated the shadowed Nakagami channel by taking into account the Nakagami-m distribution and the Log-normal distribution. For the non cooperative system, Figure 1 depicts the outage probability versus the threshold SNR, , for a transmit SNR of. The Threshold SNR at the relays is chosen. We have considered three different environments with increasing propagation hardness:

• Generalized-K channel with and: for high values of, greater than, we consider that there is no shadowing. It is equivalent to Rayleigh channel. The corresponding plots in Figure 1, show less outage probability compared to the others configurations. Moreover, they are close each other. We note the robustness of the channel simulation compared to the theoretical rayleigh performance. Besides, we deduce the accuracy of the approximated model compared to the exact formula;

• Generalized-K channel with and;

• Generalized-K channel with and.

For the two last configurations, we note similar performances between the Generalized-K cdf and the approximated Gamma cdf with an optimal value of for, and for. More precisely, the optimal value of is obtained for lower tail approximation, since we are interested in small threshold values. We note just a small difference between theoretical results and simulation which is due to the approximation of the Log-normal pdf distribution by a Gamma distribution to model the shadowing fading.

In some cases, it is important to illustrate the outage

probability measure as function of the average SNR for a given threshold SNR. Figure 2 shows this illustration for for the same parameters of the three configurations mentioned above.

In the cooperative communication scenario, we have omitted the propagation environment relative to Rayleigh case; we have considered the two others environments cases corresponding to and, For both cases, we have considered identical Nakagami-m shadowed fading channels for the links of the first hop and those of the second dual hop.

We have considered a number of relay nodes, J = 1, 2, and 3. Similarly to the non cooperative system, we have illustrated:

• The outage probability versus the threshold SNR for a transmit SNR value of for both the first and the second hops. Figures 3 and 4 show

the obtained results.

• The outage probability versus for a given threshold SNR. Figures 5 and 6 show the obtained results.

We note that, since we consider i.i.d channels, the optimal value of is the same for all links, as mentioned in the theoretical analysis. Thus, for m_m = 1, and for. We observe that theoretical results match well simulation results. The little difference is again due to the approximation of the Log-normal distribution by a Gamma distribution. As expected, the use of more relays provides better outage performance.

In this subsection, we have illustrated the results relative

to the ASEP for 16-QAM modulated signals. We plot the performance curves in terms of the analytical and simulation ASEP versus in dB. Similarly to the outage analysis, we consider two scenarii: the non cooperative and the cooperative systems. Figure 7 shows the obtained results for the first scenario.

We note a similar performance between the theoretical Rayleigh curve, the approximated results and the simulation results.

For the second scenario, we have considered identical and independent channels with the following parameters and. We have considered a threshold SNR. To capture the path-loss effects, we have considered the following distances, and. We have chosen a number of relay nodes, J = 1, 2, and 3. Obtained results are depicted in Figure 8. Optimal values of are calculated for each value. They are given by Table 1.

Table 1. versus.

We note a good match between the simulation and theoretical results. The small difference between theoretical results and simulation includes the combination of the two approximations: approximation of the Lognormal pdf distribution by a Gamma distribution and the approximation of the Generalized-K distribution by a Gamma distribution. As expected, the use of more relays provides better ASEP performance.