Performance Analysis of Optimum Combiner in Power Limited Cognitive Radios with Multiple Primary Interferers

1Abstract—This paper investigates the performance of power limited Cognitive Radio system with optimum combining at the cognitive user receiver under the influence of interference from multiple primary users’ transmitters ( ) t L in flat Rayleigh fading channels. An approximate analytical result of the probability density function of maximum signal-tointerference ratio at the output of CR-OC receiver is derived. Using this derived PDF, the closed form expressions for the performance metrics viz. Average post processed SIR, Ergodic capacity, Average bit error rate and outage probability of CROC system are derived by taking into account peak interference power constraint denoted by ‘Q’ at PU-Rx. Based on the achieved result, it is concluded that the performance of the proposed system degrades when number of primary interferers exceeds from 3 t L  . Analytical results for CROC system are validated through Monte Carlo simulations also.


I. INTRODUCTION
Current spectrum allocation policies have made a larger part of the licensed spectrum to remain idle at a given time and location, leading to inefficient use of overall radio spectrum [1]. Cognitive Radio (CR) is one of the promising technique for future communication to utilize the scarce spectrum resources in a more efficient and flexible way. In a CR network, there are different spectrum access techniques such as underlay, overlay and interweave. Underlay spectrum sharing is the access policy for secondary users' whereas in overlay strategy, the CR-Tx is allowed to share the spectrum of primary user (PU) [2] simultaneously with PU. Nevertheless, such simultaneous transmissions may result in degradation of the performance of PU. Therefore, the interference power received at the PU-Rx due to CR base station transmissions must be managed to be within some predefined margins [2], [3]. In order to satisfy the constraint, the transmitted power of CR-Tx has to be regulated using different transmit power policies either by considering peak interference received power (PIP) or Manuscript received 14 June, 2017;accepted 29 November, 2017. outage constraint at PU-Rx [4], [5].
Diversity reception with array processing is a powerful technique which suppresses the detrimental effects of interference and fading [6]. Antenna arrays can provide diversity paths to combat multipath fading of the desired signal as well as the interference at the intended receiver. Spatial diversity is an efficient solution, using multiple antennas at one or both sides of the transmission link, to alleviate the effects of multipath fading and enhance system throughput. Hence, to improve system capacity of CR networks different combining techniques such as selection combining (SC), maximal ratio combining (MRC), optimum combining (OC) etc. are studied in the literature. MRC, a combining technique applied in presence of noise and independent fading, is thoroughly studied in MIMO as well in cognitive radios [7]- [15]. The weight vector at each antenna element compensates the effect of phase shift which is proportional to the received signal strength and maximizes the SNR. In [9], the authors analysed MRC in the presence of co-channel interference. The result shows that it maximizes the signal-to-noise-ratio (SNR) at the output and is the most effective choice in noise limited scenario. However, it becomes sub-optimal option in the presence of interference. Whereas in OC [10]- [15], the received signal at different antenna elements are properly weighted and combined to maximize the SINR at the receiver output. In [15], the authors studied and analysed the performance comparison of MRC, EGC and OC in the presence of interference. The study considered MRC with arbitrary number of interferers, whereas in OC the number of interferer sources was larger than the number of antenna elements, such that the array degrees of freedom are not sufficient enough to completely null the interference. However, even a moderate increase in SNR at the Rx output may result in significant improvement in system capacity. MRC is also analysed in the presence of multiple equal power interferers in a Nakagami fading scenario [10]. The authors showed that the MRC is beneficial even in interference limited environment and increasing the order of diversity further improves the system performance. In [16], the authors have examined the analytical performance evaluation of generalized selection combining (GSC) in interference environment in terms of SINR and SNR. This paper considers the two extreme cases i.e. when number of best branches to be combined is c L 1  (SC) and c r L N  (MRC). They also provide the new outage analysis, which gives insight to the GSC reception in the interference limited environment.
To enhance the performance of Cognitive radio network, various diversity combining techniques have been employed [17]- [23]. In [17], the authors have analysed the ergodic capacity of spectrum sharing system employing MRC at the secondary user receiver (SU-Rx). In [18], the authors have studied the GSC in terms of ergodic capacity in a cognitive radio environment under the imperfect CSI. In [19], the author has analysed the spectrum sharing system with MRC diversity in terms of ergodic capacity and symbol error rate (SER), when the proposed system is constrained of transmit power constraint. The impact of multiple PU trans-receivers on the single relay spectrum sharing system has been analysed in [20]. Also, the outage performance of spectrum sharing CR system by employing MRC at SU under the influence of interference from multiple PU's has been examined in [21]. The OC is also studied with transmit antenna selection (TAS) for aggregate interference from multiple secondary users in underlay CR [24]. All this prior work on diversity combining improved our insight into the usefulness of diversity combining schemes in cognitive radios. Motivated by these observations, we observed that MRC mitigates the effects of fading, however it fails to combat interference. OC addresses both the problems of multipath fading and the effect of interference. Thus by considering the advantages of OC over MRC, we have studied and analysed the underlay spectrum sharing CR system by employing Optimum Combining at CR Rx under the impact of multiple PU interferers. Our main contributions in this paper are summarized as following: (i) An approximate analytical expression for the probability density function (PDF) of signal to interference ratio (SIR) at the CR-Rx output is derived, considering the effect from where  is the standard Gamma function and is given as

III. PERFORMANCE ANALYSIS
The OC weight vector that maximizes the SIR at the output of CR-Tx is written as where R denotes the interference covariance matrix [15] conditioned on channel vector of t L interferers and given by where H (.) represent the complex conjugate transpose. Next, we derive PDF for the SIR of CR-OC system in the presence of t L equal power interferers.

A. PDF of Maximum SIR at the CR-Rx
From (2) The PDF of random variable z is given by [15] r t N 1 The PDF in (10) is a modified form of central F -Distribution [15]. The density of the 'z' does not depend upon the form of the covariance matrix R. Thus the performance of the OC is the same regardless whether the fading at each receive antenna is independent or not. However, this is true only for the case t r L N  . Since F Distribution can be converted into Chi-Square distribution [22], therefore (8) can be rewritten as From (6), the marginal PDF for the ratio of two random variables z and CR PU  h is obtained by substituting: The complete solution of above equation is solved in Appendix-A. Now we will derive the PDF of maximum SIR at the output of CR-Rx By using the transformation as in (13), the density of the maximum SIR The PDF in (14) represents the final density function of SIR at the output of CR-Rx of CR-OC system.

B. Average Post Processed SIR
The average post processed SIR or the first moment of CR OC γ  at the output of CR-Rx under the influence of multiple PU interferers is given as pr CR OC N r pr Solving (15) the average post processed SIR of the CR-OC system is expressed as

C. Ergodic Capacity
The Ergodic Capacity   CR OC C  of CR-OC network is defined as the maximum long term achievable rate and determined by averaging over all the channel states of a fading channel. It is approximated using Taylor's series expansion of logarithm function [25] and is given by By further solving (17) where 1 2 F is the hypergeometric function and it is defined as in [25]. Equation (19) represents the final expression for the ergodic capacity of the CR-OC system. The complete solution of (19) is given in Appendix B.

D. Outage Probability
The outage probability is an important statistical measure in the design of spectrum sharing system in fading environment in presence of interference. It is the probability of unsatisfactory reception over the intended coverage area. The outage probability is the probability that the received SIR is below a given threshold required to achieve radio reception in fading environment [14]. It is expressed as where t  is the SIR threshold. Its value depends on the modulation technique used and also on the desired performance criterion [12]. It is also known as cumulative distribution function. Solving (20) the outage probability of CR-OC system is found as

E. Average Bit Error Rate
An average Bit Error Rate is an important parameter for the analysis of performance of CR-OC system. In this section, the ABER of CR-OC system is derived under peak interference power constraint Q at PU-Rx. In case of BPSK

IV. NUMERICAL RESULTS
In this section, we present numerical results to verify simulation counterpart in terms of Average post processed SIR, Ergodic capacity, ABER and Probability of Outage for CR-OC system in flat Rayleigh faded environment. We assume that the number of PU interferers affecting CR network are i.e. t    Furthermore, the average post processed SIR and Ergodic capacity of the CR-OC system improve when 'Q' is increased i.e. the received interference power constraint at PU-Rx is increased which further allows CR-Tx to transmit with increased power. The Fig. 4 and Fig. 5 show the ABER and outage probability of CR-OC system.   We can observe from Fig. 4 and Fig. 5 that the average bit error rate degrades from 0.270 to 0.374 and outage probability increases from 0.75 to 0.97 when the number of PU interferers are increased from t 3 L  to t 6 L  at Q = 5 dB. Thus, the average bit error rate and outage probability of the CR-OC system becomes better when 'Q' is increased and the performance of the proposed system degrades as number of interfering sources are increased.

B. Performance Analysis in terms of Diversity Gain with
Varying Number of Receiver Antennas   r N In this section, we demonstrate the diversity gain in terms of average post processed SIR at CR-OC output and probability of outage of the proposed system. In Fig. 6, the performance analysis of Average post processed SIR at the Optimum combiner output is examined with varying number of r N receive antennas for the proposed system. It can be seen from the figure that the diversity gain is substantially increased, when number of CR-Rx receiver antennas i.e. r N increases from 3 to 6 as effect of channel fading weakens when number of receive antennas increases. The average post processed SIR of CR-OC system rise from 0.237 dB to 1.897 dB. In Fig. 7, the diversity gain of the proposed system is shown in terms of probability of outage for the proposed system with varying number of r N receiver antennas. It can be seen that from the achieved result that outage probability of the CR-OC system drops from 0.970 to 0.469 when the number of r N antennas increase from 3 to 6, respectively.   Table I shows the performance of CR-OC system in terms of Average post processed SIR, Ergodic capacity, ABER and probability of outage. It is shown in the Table I that Average post processed SIR falls when number of PU interferer increases from t 3 L  to 6. Thus it results in the SNR loss of 0.474 dB when t L goes from 3 to 4, 0.158 dB when t L goes from 4 to 5 and 0.079 dB when t L goes from 5 to 6. It is also shown that ergodic capacity gains of the proposed system is achieved as 60 % when t L goes from 3 to 4, 35 % when t L increases from 4 to 5 and 25 % as Lt increases from 5 to 6, respectively.
The ABER of OC-CR system increases from 0.270 to 0.374 as number of PU interferers increases from 3 to 6. As seen from the Fig. 4 We can conclude that there is a power loss of 1 dB, 1.5 dB and 2 dB when number of PU interferer increases from 3 to 6. Hence, the PDF of maximum SIR for the CR-OC system is given in (12 By putting (A2.3) in (A2.1), we obtain the final expression for the ergodic capacity of CR-OC system, which is given in (18).