# On the Performance of Multiple Antenna Cooperative Spectrum Sharing
Protocol under Nakagami- Fading^{†}^{†}thanks: Part of this work was supported by Deity (Department of Electronics
and Information Technology), Government of India, under SRP-43 project
grant.

###### Abstract

In a cooperative spectrum sharing (CSS) protocol, two wireless systems operate over the same frequency band albeit with different priorities. The secondary (or cognitive) system which has a lower priority, helps the higher priority primary system to achieve its target rate by acting as a relay and allocating a fraction of its power to forward the primary signal. The secondary system in return is benefited by transmitting its own data on primary system’s spectrum. In this paper, we have analyzed the performance of multiple antenna cooperative spectrum sharing protocol under Nakagami-m Fading. Closed form expressions for outage probability have been obtained by varying the parameters and of the Nakagami- fading channels. Apart from above, we have shown the impact of power allocation factor () and parameter on the region of secondary spectrum access, conventionally defined as critical radius for the secondary system. A comparison between theoretical and simulated results is also presented to corroborate the theoretical results obtained in this paper.

## I Introduction

Cooperative spectrum sharing (CSS) have attracted a great deal of attention among researchers in the past few years due to its dual utilization of cooperative diversity for reliable communication and cognitive abilities to utilize the spectrum more efficiently [1, 2]. The concept of CSS, to employ the secondary transmitter (ST) as a relay to forward the information of the primary system and get spectrum access in exchange, can be utilized in cellular and ad-hoc networks [1]-[3].

Considerable work has been done to validate the performance of CSS protocols in Rayleigh faded channels, however to the best of our knowledge very few literature is publicly available to demonstrate the performance of these protocols on Nakagami faded channels. Many experimental works show that Nakagami distribution, as compared to Rayleigh distribution, is often more accurate for modeling the urban multipath channels [4]. Although Rayleigh fading models are frequently utilized in modeling the non light-of-sight channels however it is better fit for the signals propagating within small areas, as it does not gauge for large-scale propagation effects like shadowing by buildings, bridges and other obstructions which are typically encountered in mobile communication channel. Hence, Nakagami fading models are usually preferred in modeling long distance fading effects, specifically with respect to mobile communications [4].

Moreover, by tuning the fading severity parameter, m , Nakagami-m fading can be used to represent a wider class of fading channel conditions. For instance, m=1 represents Rayleigh fading whereas m = 0.5 represents one-sided Gaussian fading [5].

The authors in [6] have analyzed the performance of classical decode and forward (DF) cooperative communications over Nakagami- fading channels. They have measured the performance in terms of symbol error rate (SER) for different modulation schemes. By varying the parameters of the fading channel the authors are able to enhance the cooperation performance between primary and secondary system. In [7, 8] the outage performance of an underlay system with cognitive decode and forward (DF) and amplify and forward (AF) relaying schemes has been investigated. The authors in their system model have used relays for transmitting secondary system’s data. Secondary transmitter and its relay limits their transmit power so that the interference on the primary system do not exceed a certain threshold.

Compared to the previous work proposed in the literature on Nakagami fading channels, in this paper we have considered an overlay model in which there is no limitation on the secondary transmit power. On a contrary, depending on the power allocation factor, , the performance of the primary and secondary system may increase with an increase in the secondary transmit power. This paper can also be seen as an extension of the work done for Rayleigh fading channels in [1, 2], however, in the proposed work the results have been obtained for independent Nakagami- fading channels. Furthermore, unlike [1], we believe that cognitive system is going to be an advance system that utilizes the multiple antenna functionality [3] such as IEEE 802.11n, IEEE 802.16m or 3GPP LTE - Advanced [9]. Hence, in the proposed work, it is assumed that ST is equipped with multiple antennas.

In our system, when the target rate of the primary system drops below
a particular threshold (), it seeks cooperation
from the neighboring terminals. The secondary system which disguises
itself as relay, cooperates with the primary system, promising better
performance to primary system in exchange for the spectrum access
in the operating frequency band of primary system. Once ST is confirmed
as a relay, spectrum access for secondary system is obtained by adopting
the following two-phase transmission protocol. In phase 1, the data
broadcasted from primary transmitter (PT) is received by primary receiver
(PR), secondary transmitter (ST) and secondary receiver (SR). The
data received at PR in phases 1 and 2 is decoded using maximum ratio
combining (MRC) to get the desired data, considering secondary data
as noise. At SR, after successful decoding of primary signal in phase
1, the interference component can be canceled out in phase 2 to obtain
the desired secondary data [1, 2]^{1}^{1}1Interested readers may refer to [1] for further details
on the control protocol. . Our proposed model has been quantified by obtaining closed form
expressions for the outage probability. Apart from above, we have
calculated the critical region of ST which helps in determining the
maximum distance within which ST can achieve spectrum access.

## Ii System Model

The system model for the and phase is shown in fig 1.

The system consists of PT, PR, ST and SR. The channel between all the links, i.e. PT-PR, PT-, PT-…PT-, -PR, -SR, where n is antenna selected randomly, and PT-SR are described by Nakagami- distribution with channel coefficients and respectively. The probability density function (PDF) of a Nakagami random variable is given by

where is the variance of , is the Nakagami fading figure and is the Gamma function. Generally, when the above PDF reduces to the PDF of well-known Rayleigh fading model. For , the fading is Nakagami which is more severe than that of Rayleigh fading.

The parameter where is the path loss component and is the normalized distance between the respective transmitters and receivers. This normalization is done with respect to distance between PT and PR, i.e. . Primary and secondary signals are denoted by and respectively, with zero mean and . and are the target rates for and respectively. We denote the transmit power at PT and ST as and , respectively. The additive white Gaussian noise (AWGN) at each receiver is denoted by where represents the transmission phase and represents the respective channel link, assumed to have identical variance . In the following sections we will analyze the performance of cooperative spectrum sharing based on DF protocol under Nakagami- fading channels.

## Iii Outage Performance of Primary System

In phase 1, PT will broadcast the signal . This signal is overheard by PR, , , …, and SR. The received signal at PR is denoted by , which is given by

where, . The received signal at ST is denoted by

where, . denote the signals coming from channel , respectively. Also,

and The signals thus received at ST is decoded for . The rate at ST is given by,

(1) |

where the factor in the above equation accounts for the fact that the transmission is being divided into two phases. In phase 2, if is decoded successfully, ST will transmit along with its own data . The signals received at PR is given by

where

and The signals in phase 1 and 2, and , are then combined at PR using MRC. The achieved rate is then derived as in [1] is given by

(2) |

On the other hand if ST is not able to decode in phase 1 then it will not transmit in phase 2. In such a case PR can still receive through a direct link from PT to PR with achievable rate of

Thus, the outage probability of the primary signal transmission with target rate is given as

(3) |

Assuming as in [1], [2] we obtain

(4) |

where , , is the Gamma function and indicate the incomplete Gamma function.

(5) | ||||

(6) | ||||

(7) |

Substituting (4), (5), (6) and (7) in (3). We get,

(8) |

(9) |

(10) |

## Iv Region for secondary spectrum access

In this section, we are going to define the region, within which the secondary system can access primary’s spectrum without compromising the performance of primary system. This region has been conventionally defined as critical radius in [1]. To calculate a critical region for such a system, the outage probability of primary system with cooperation i.e. , must be less than the outage probability without cooperation, i.e. . The outage probability of direct transmission (without cooperation) is given as

(11) |

where and indicate the incomplete Gamma function. From (9), (10), we can observe that not only depends on the but it also varies with change in the value of . Therefore, there are two cases which describes the successful spectrum access of the secondary system. i.e for and . The theoretical values of after solving is given as below

(12) |

where indicate the inverse incomplete Gamma function. The for is given as

(13) |

where

and

We can note that for (12), (13) reduces to the results given in [1, 2] for Rayleigh flat fading.

## V Outage Performance of Secondary System

In phase 1, received signals at secondary receiver is given by

where . The rate at SR for the direct transmission from PT is given by

(14) |

At SR, an estimate of is obtained as

The achievable rate at ST is given in (1). In phase 2, signal received at SR is given by

where

and is the AWGN. The estimate is used to cancel the interference component, to obtain

The achieved rate between ST and SR, conditioned on successful decoding of at both ST and SR in the first phase, is given by

(15) |

Outage is declared if ST and SR are not able to decode , and therefore the outage probability of the secondary signal transmission with target rate is given as

(16) | |||||

(17) |

(18) |

where . Substituting (6), (17) and (18) in (16) we get the outage probability as

(19) |

## Vi Simulation Results and Discussions

In this section, we discuss the performance of a cooperative spectrum sharing protocol for Nakagami-m fading. Target rates of primary as well as secondary systems are chosen to be . The value of is taken as , which measures the depth of fading envelope. PT, ST, SR, PR nodes are assumed to be collinear as in [1], [10]. The node ST is equipped with N antennas. For simulation, we have taken the value of N=2 and N=4. The distance between PT and PR is normalized and taken as . The distance between PT and ST is denoted by and the respective distances between different nodes is calculated in terms of . The distance between ST and PR is , ST to SR and PT to SR is . We have taken and . The , is the path loss component.

Fig. 2 shows the outage probability performance of primary system w.r.t. the power allocating factor for different values of for N=4 and for N=2. These set values of are calculated from , where the last values in both the sets are the critical values calculated from (12). It can be seen from the figure that as we increase the value of the outage probability tends to decrease.

For a particular value of between PT and ST, when
the outage probability drops below the outage probability of direct
link and spectrum access can be achieved by secondary system. As we
increase then for a particular
between PT and ST much lower outage probability can be achieved by
the primary system. The outage probability performance of the system
under Nakagami fading reduces to Rayleigh fading for .^{2}^{2}2In fig. 2, theoretical results are plotted by assuming ,
however, for small values of the approximation
might not hold and there would be a slight gap between the simulation
and theoretical results.

Fig. 3 shows reasonably good outage probability of secondary system w.r.t. . The theoretical results are exactly matching with the simulation results, authenticating the analytical results obtained for the outage probability of secondary system. We can observe from figure that the outage probability has a constant value for almost all values of and tends to 1 as .

Fig. 4 shows the outage probability of the primary as well as the secondary system w.r.t the fading coefficient . It can be observed from the figure that as the value of increases i.e. the fading effect of the channel decreases, the outage probability of the overall system decreases, which is quite obvious from the fact that as there is no fading in the channel the data can be transmitted smoothly and efficiently to the destination. From fig. 3 and fig. 4 it can be inferred that there is good agreement between theoretical and simulating results thus validating the analysis done in this paper.

## Vii Conclusions

In this paper, we analyzed the performance of cooperative spectrum sharing scheme over Nakagami- fading. A cognitive relay, equipped with multiple antennas decodes the message from primary transmitter and forwards, by means of DF relaying, it to the destination by randomly selecting one antenna in order to achieve the target rate of primary system, getting the spectrum access for secondary system in exchange. It was shown that, even in presence of Nakagami- fading CSS protocol with multiple antennas at ST can help in considerable improvement in the performance of primary system. From above observations, we can conclude that as the value of increases the severity of fading decreases and performance of outage probability improves. The excellent agreement between the simulated results and the analytically obtained closed form expressions authenticates the theoretical analysis presented in this paper.

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