Power Allocation and Cooperative Diversity in TwoWay NonRegenerative Cognitive Radio Networks
Abstract
In this paper, we investigate the performance of a dualhop block fading cognitive radio network with underlay spectrum sharing over independent but not necessarily identically distributed (i.n.i.d.) Nakagami fading channels. The primary network consists of a source and a destination. Depending on whether the secondary network which consists of two source nodes have a single relay for cooperation or multiple relays thereby employs opportunistic relay selection for cooperation and whether the two source nodes suffer from the primary users’ (PU) interference, two cases are considered in this paper, which are referred to as Scenario (a) and Scenario (b), respectively. For the considered underlay spectrum sharing, the transmit power constraint of the proposed system is adjusted by interference limit on the primary network and the interference imposed by primary user (PU). The developed new analysis obtains new analytical results for the outage capacity (OC) and average symbol error probability (ASEP). In particular, for Scenario (a), tight lower bounds on the OC and ASEP of the secondary network are derived in closedform. In addition, a closed from expression for the endtoend OC of Scenario (a) is achieved. With regards to Scenario (b), a tight lower bound on the OC of the secondary network is derived in closedform. All analytical results are corroborated using Monte Carlo simulation method.
1 Introduction
In the last decade, a significant growth in wireless cellular network has been observed [[1], [2]]. However, the available technologies and the limited radio spectrum are unable to address the stringent demand for high data rate wireless communication [[3], [4], [5]]. It is desired to have a solution that can mitigate the problem of inadequate cellular spectrum [[6]]. To cope with this, cognitive radio, has been proposed to alleviate the problem of resource congestion via utilization of frequency bands through dynamic spectrum management [[7], [8]]. There are three main cognitive radio protocols: underlay, overlay and interweave. The focus of this paper is on the underlay systems. The underlay paradigm allows secondary user (SU) utilize the spectrum as long as its interference to the primary network remains below a tolerable level [[9]].
On the other hand, cooperative communication has appeared as a powerful spatial diversity technology for wireless communications to overcome the damaging effects of multipath fading [[10]]. Relay technology is one of these methods that are used to enhance the performance of wireless networks. Among various relaying schemes, bestrelay selection scheme has drawn increasing research interests while being spectrally more efficient than repetitionbased schemes [[11]]. Best relay selection [[12]] is an ideal protocol to reach the best performance. There are two sorts of classical relay communication protocols, i.e., nonregenerative and regenerative protocols [[13]].
Twoway relay networks have aroused great interest due to their potential of significantly enhancing the network throughput as well as the spectrally efficient benefits. Thereby two end users exchange information simultaneously with each other in just two time phases via a halfduplex or multiple relays [[14]].
1.1 Literature on Cognitive Relaying Network
Opportunistic relaying has been extensively studied for oneway relaying systems. For instance, in [[15], [16], [17]], the main purpose was to explore the effect of interference in unidirectional cognitive relay networks in terms of outage performance. In [[15]], the outage capacity (OC) of a dualhop cognitive relay network has been explored where the power constraint of the secondary network has been derived with two different strategies. However, in this work the power constraint of the SUs depends on the knowledge of the instantaneous channel state information (CSI) of the link between the SUs and the primary users (PUs). The OC for incremental and DF relaying in underlay spectrum sharing systems over Nakagami fading channels has been discussed in [[16]].
Some studies have extended the opportunistic relaying to twoway relaying systems [[18], [19], [20], [21], [22], [23], [24]]. In [[18]] and [[19]], analog network coding (ANC) and physicallayer network coding (PNC) were adopted in a twoway spectrum sharing relaying network, respectively. In [[20]], a twoway cognitive radio system with a regenerative scheme has been studied in terms of average error rate, average sumrate. The authors in [[22]] investigated an underlay cognitive twoway relaying network in multiple access broadcasting (MABC) and time division broadcasting (TDBC)
Considerable efforts have been spent on addressing the performance of the cognitive radio networks. However, most of the prior works studied the performance of the cognitive radio networks for oneway relaying systems in Nakagami environment or assumed Rayleigh fading channels. This may not be useful in a wide range of fading scenarios that are typical in realistic wireless relay applications. The performance of twoway relaying in spectrum sharing systems over Nakagami fading channels is still much less understood. Moreover, in most of the previous works the power limitation of the secondary network depends on the knowledge of instantaneous CSI of the links between nodes [[26], [27]] while in some practical communication setups with high terminal speed the channel varies rapidly and experiences fast fading. Owing to these facts, the aim of this paper is to investigate the performance of dualhop cognitive radio networks with nonregenerative relay/relays and subject to i.n.i.d. Nakagami fading channels for two Scenarios. To our best knowledge, no similar studies have been reported despite the importance of such an issue. A detailed description of the considered Scenarios follows.
1.2 Considered Scenarios and Technical Contributions
Scenario (a): Scenario (a) is composed of a pair of cognitive sources communicating with each other with the aid of a relay, while sharing the spectrum with a oneway primary network. In cognitive radio network, the source nodes and the relay are affected by all existed interferers and additive white Gaussian noise (AWGN). Scenario (b): Scenario (b) is the extension of Scenario (a) to a cognitive twoway multinonregenerativerelaying dualhop configuration with best relay selection strategy, where the source nodes only suffer from AWGN, while the relays are affected by all existed interferers as well as AWGN. In this Scenario the interference from the PU to the secondary sources is assumed to be neglected as in [[28]]. This can be justified if the primary transmitter is located far away from the secondary sources, or the interference is modeled as the noise term [[28]].

In an attempt to assess the performance, the cumulative distribution function (cdf) of the SINR at primary destination is obtained from which a closedform expressions for the OC is derived over Nakagami fading channels, which build the relationship between the outage performance and the related system parameters. From a realistic point of view, the choice of Nakagami fading includes some other fading channels that are more or less severe than Rayleigh fading via the fading parameter. For example, it includes the Rayleigh fading as a special case. The Nakagami fading also approximates the Hoyt fading, for , and the Rician fading, for . On the basis of the outage performance of the primary network the power allocation of the SUs are presented.

For both Scenarios, the exact equivalent SINR at the secondary sources are provided and upper bounded. Then, the cdfs of the upper bounded SINRs are explored. In addition, the selection of the best relay, , is discussed. According to these results, tight lower bounds on the OC of Scenarios (a) and (b) are achieved. Furthermore, a closedform expression for the endtoend OC of the network in Scenario (a) is obtained.

The tight lower bound closedform expression for the ASEP of Scenario (a) is derived over Nakagami fading channels.
The rest of this paper is organized as: âSection II, introduces channel statistics and system models. The secondary network and its performance is examined in Section IIIâ. Numerical results are provided in Section IVâ and conclusions are given in Section V.
2 Channel Statistics and System Model
Let us consider a spectrumsharing cooperative nonregenerative relaying model. In this model, , , and / represent a primary transmitter, a primary receiver, secondary sources and secondary relay/relays, respectively. Let us consider that there is no direct link between two sources due to poor channel condition [[29]]. Further, we assume that the channels are reciprocal. A timedivision multipleaccess scheme is employed for the secondary nodes which is performed into two consecutive time intervals, namely the multipleaccess (MA) phase and the broadcast (BC) phase. In the first phase, sends its signal , , drawn with equal probability from MPSK constellation of size to with power . Meanwhile sends its information meant for the other with transmit power of under a transmit power constraint and the relay/relays receive(s) the signals. As was previously mentioned, we assume that the instantaneous channel gain of each link () follows the Nakagami distribution, where integer stands for the fading severity parameter. As such, the distribution of the is Gamma random variable (RV), where is scale parameter, while is the mean value of Gamma RVs. Therefore, we have , , , , and , where the symbol denotes ”distributed as”. Then, the signal received by / in Scenarios (a) and (b) and also by in the MA phase can be written as
(1) 
â
(2) 
(3) 
âââwhere â, ââ, ââ, ââ, â and are the channel coefficients of , â and â links, respectively, while and are AWGN at the primary receiver terminal and the relay/relays, respectivelyâ. Noise signals at all nodes are zero mean with power spectral density of . In the second phase, PT, sends another information message to . Meanwhile in the second phase, /, simply scales the received signal before broadcasting to the destination with transmit power . Accordingly, the received signal at and in Scenario (a) and (b) are given respectively by
(4) 


(5) 


(6) 
âwhere â is the channel coefficient of link, while â/, is AWGN at /â and and , are the relay amplification gains defined as
(7) 
(8) 
In the following sections, we will evaluate the network performance in key operation metrics.
â
3 Performance Analysis
â In this section the statistics of the equivalent SINR will be explored, tight lower bounds on the most important performance metrics of the secondary network will be quantified and some useful observations in this realm will be provided.
â
3.1 Primary OC and Power Constraint of the Secondary Nodes
â In this subsection, we present some of statistics for the OC of the primary network. The OC is defined as the probability that the instantaneous SINRs falls below a predetermined rate bits/sec/Hz. Hence the primary OC can be explored by the following theorem:
Theorem 1
The OC of the primary network is obtained in closedform expression as shown in (9),
â


(9) 
where and is the primary transmission rate.
Proof: See Appendix 6
In both Scenarios, by assuming and without loss of generality, the power limitation of are derived under the constraint which guarantees that the interference on does not exceed a threshold. Therefore, the power constraint of is obtained âby solving w.r.t. .
Similar to Theorem 1 and according to (4) the transmit power of th secondary relay regulated alone by the primary outage constraint can be readily achieved by solving (10) w.r.t. . â


(10) 
â
3.2 cdf of the Equivalent SINRs
By substituting (7) into (5) and after some algebraic manipulations of the SINR at in Scenario (a) can be obtained as in Eq. (11) of [[14]]. To simplify the ensuing derivations, and should be addressed in more mathematically tractable forms. As such, we use the tight upper bounds for and as presented in Eqs. (9) and (24) of [[14]]. In the next step we proceed to explore the OC of both Scenarios.
cdf of the SINR of Scenario (a)
The closedform of the cdf of for Scenario (a) can be presented in the following key result.
Theorem 2
The cdf of the SINR of Scenario (a) is obtained in closedform expression as shown in (11)
â






(11) 
ââ
where is the outage threshold.
Proof: See Appendix 7.
EndtoEnd cdf of the SINR of Scenario (a)
The endtoend cdf of the SINR of Scenario (a) is derived as
â
(12) 
ââ where and are given respectively in (3.2.2) and (3.2.2). The terms and can be obtained respectively by replacing appropriately the parameters in and , i.e., , , and .
â








(13) 
ââ
â




(14) 
â
cdf of the SINR of Scenario (b)
In general, by applying the best relay selection strategy, the relay which has the highest value of endtoend SINR is viewed as the ”best” one. Hence, the endtoend received SINR at terminals is given by â
(15) 
â Since the exact cdf of the endtoend SINR is intractable, in the following, we obtain the cdf of the endtoend upper bounded SINR.
Proof: See Appendix 8.
By using (11) and (16), we can now obtain the following lower bounds on the exact OC of the Scenarios (a) and (b) respectively as â
(17)  
(18) 
â
ââ
3.3 Error Probabilityâ
âIn this subsection we concentrate on the ASEP of the secondary network for Scenario (a) as another important performance evaluation metric. âFor several modulation schemes, in practical systems ASEP can be lower bounded as [[30], Eq. (9)]. The following theorem summarizes the ASEP of Scenario (a).
Theorem 4
We derive a closedform expression for the ASEP of Scenario (a) as in (19) shown on the top of the next page,
â






