Impact of Secondary User Interference on Primary Network in Cognitive Radio Systems
IImpact of Secondary User Interferenceon Primary Network in Cognitive Radio Systems
Amit Kachroo and Sabit Ekin
School of Electrical and Computer EngineeringOklahoma State University, Stillwater, OK, USAEmail: amit.kachroo, sabit.ekin { @okstate.edu } Abstract —Most of the research in cognitive radio field isprimarily focused on finding and improving secondary user(SU) performance parameters such as bit error rate, outageprobability and capacity etc. Less attention is being paid towardsthe other side of the network that is the primary network whichis under interference from SU. Also, it is the primary user(PU) that decides upon the interference temperature constraintfor power adaptation to maintain a certain level of quality ofservice while providing access to SUs. However, given the randomnature of wireless communication, interference temperature canbe regulated dynamically to overcome the bottlenecks in entirenetwork performance. In order to do so, we need to analyzethe primary network carefully. This study tries to fill this gapby analytically finding the closed form theoretical expressionsfor signal to interference and noise ratio (SINR), mean SINR,instantaneous capacity, mean capacity and outage probability ofPU, while taking peak transmit power adaptation at SU intopicture. Furthermore, the expressions generated are validatedwith the simulation results and it is found that our theoreticalderivations are in perfect accord with the simulation outcomes.
Index Terms —Cognitive Radio Network, Interference Temper-ature, Mean Capacity, SINR, Outage Probability
I. I
NTRODUCTION
In cognitive radio network, a secondary user (SU) is allowedto access the primary user (PU) spectrum completely if theavailable spectrum is not used by PU (interweave) or con-currently (underlay) with PU. The concurrent transmission isallowed if and only if the SU maintains a certain power thresh-old constraint known as interference temperature [1], [2]. Mostof the studies [1]–[6] that involve this interference temperaturemodel utilize either peak or average transmit power adaptationfor the purpose of analyzing or improving the performance ofsecondary network. However, the impact of SU interference onprimary network is let off completely. Therefore, to asses theperformance and other quality of service (QoS) parameters,closed form expression need to be derived and validated. Thisstudy focuses on the mathematical foundation to derive thesenecessary performance expressions. This theoretical analysisis done by first considering interference from a single SU andthen extended to the case of interference from multiple SUs onprimary network. As per our knowledge, this is the first paperto analytically analyze the effect of SU interference on primarynetwork. The contribution of this paper can be summarized asfollows: • Probability distribution function (PDF) and cumulativedistribution function (CDF) expressions for noise plusinterference, signal to noise and interference ratio (SINR)are derived for both cases of interference from a singleSU and multiple SUs on PU network. • Closed form mean SINR expression, mean capacity andoutage probability expressions are derived. • The expression generated above are validated with sim-ulation results to show the accuracy of the theoreticalexpressions.The rest of the paper is organized as follows. In Section II,system model with underlying assumptions is described in de-tail. Section III presents the extensive theoretical analysis andcomparisons with simulation results and finally, conclusionsare given in Section IV.II. S
YSTEM M ODEL
The cognitive radio network that is under consideration inthis work is shown in Fig. 1. It consists of n SUs (SU network)and k PUs with corresponding SBS and PBS. Since we areanalyzing the effect of SU on PU network, we don’t need toconsider the channel gain between PU and SBS, and also wedon’t need to consider the interference among PUs because oforthogonal resource allocation between them.
Fig. 1. Underlay cognitive network with n -SUs sharing the spectrum withPU network of k -PUs. The channel power gain between any SU-i ( i th user)and PBS is denoted by α i , between any SU-i ( i th user) and SBS by β i andbetween i th -PU and PBS by γ i . In addition, since the channel fading is assumed to beRayleigh distributed, the channel power gains follow an ex-ponential distribution. Considering the peak power adaptation a r X i v : . [ ee ss . SP ] S e p P tx = min (cid:110) p, qα (cid:111) (1)In the following sections, for theoretical analysis pur-pose, we assume that there are n SUs that form the un-derlay cognitive network with primary user, where n = { , , . . . , n } . Furthermore, the thermal additive white Gaus-sian noise (AWGN) in the network is assumed to have circu-larly symmetric complex Gaussian distribution with zero meanand variance as σ , i.e., CN (0 , σ ) .III. T HEORETICAL A NALYSIS
In this section, theoretical expressions for PU performanceparameters with interference from a single SU will be derivedfirst and then extended to the case of interference frommultiple SUs. The interference observed at primary receiverbecause of a single SU and multiple SUs with peak poweradaptation will be given as: I single = αP sec = min { αp, q } ,I multi = min (cid:40) n (cid:88) i =1 α i p, q (cid:41) . (2)Eq. (2) represents a minimum of a random variable and aconstant . From the theory of mixed random variables [7]–[9],a constant c can be modelled as a random variable with PDFequal to δ ( x − c ) and CDF equal to H ( x − c ) , where H ( x ) is a Heaviside function and δ ( x ) is a Dirac Delta function.So with interference from a single SU, the CDF and PDF ofminimum of two independent random variables is then givenby: F I ( x ) = F αp ( x ) + F q ( x ) − F αp ( x ) F q ( x ) ,F I ( x ) = 1 − e − λxp (1 − H ( x − q )) . (3)On including noise CN (0 , σ ) , the CDF of noise plusinterference will be then, F IN ( x ) = 1 − e − λ ( x − σ p (1 − H ( x − σ − q )) . (4)Correspondingly, the PDF of interference and noise isgiven by differentiating the CDF with respect to noise andinterference variable x , i.e. f IN ( x ) = λp e − λ ( x − σ p (cid:16) − H ( x − σ − q )+ pλ δ ( x − σ − q ) (cid:17) , ∀ σ ≤ x ≤ ∞ . (5)For the case of interference from multiple SUs, the distri-bution of of interference given in Eq. (2) will follow Gammadistribution , f ¯ γ ( x, κ, θ ) , where κ and θ represent the shape For illustration purposes, the value’s of peak power p and interferencetemperature q in this paper are chosen to be in linear scale. However, theexpressions derived in this paper hold for any value of p and q for any scale. The distribution of sum of independent exponential random variables withthe same rate parameters follows Gamma distribution. Also, to distinguishbetween the channel γ between PU and PBS, the Gamma distribution isdenoted as ¯ γ in this study. and rate parameter. The PDF and CDF of Gamma distributionis given as f ¯ γ ( x ) = n (cid:88) i =1 α i p = ¯ γ (cid:18) x, n, λ p (cid:19) = ¯ γ (cid:0) x, n, ¯ λ (cid:1) , = ¯ λ n x n − Γ( n, e − ¯ λx , ∀ (cid:8) x ≥ , n > , ¯ λ > (cid:9) ,F ¯ γ ( x ) = 1 − Γ (cid:0) n, ¯ λx (cid:1) Γ( n, , ∀ (cid:8) x ≥ , n > , ¯ λ > (cid:9) , (6)where n is the total number of SUs in the underlay network, ¯ λ = λ /p is the scaled rate parameter between SU and PBSand Γ( a, x ) is an incomplete gamma function defined as: Γ( a, x ) = (cid:90) ∞ a t a − e − t dt, ∀ a > , x ≥ . Following the same mathematical approach that was usedin single SU case, the distribution of noise plus interferencein multiple SUs case is then derived as: F mNI ( x ) = 1 − Γ (cid:0) n, ¯ λ ( x − σ ) (cid:1) Γ( n,
0) + H ( x − σ − q ) × Γ (cid:0) n, ¯ λ ( x − σ ) (cid:1) Γ( n, ,f mNI ( x ) = Γ (cid:0) n, ¯ λ ( x − σ ) (cid:1) Γ( n, δ ( x − σ − q ) + ¯ λ n × ( x − σ ) n − Γ( n, e − ¯ λ ( x − σ ) [1 − H ( x − σ − q )] , (7)where σ is the CN (0 , σ ) . Fig. 2 and Fig. 3 plots the CDFand PDF for the theoretical expression (Eq. (7) ) with thesimulation result of p > q and q < p for different SU densitiesof n = 1 , , . Fig. 2. PDF and CDF of noise and interference for different number of SUs( n = 1 , , ), when p > q , where p = 4 , q = 2 and σ = 1 with supportregion from σ ≤ x ≤ ∞ . A. Instantaneous SINR
The instantaneous SINR at PBS considering the systemmodel (Fig. 1) is given by:SINR = γpσ + I , (8) ig. 3. PDF and CDF of noise and interference for different number of SUs( n = 1 , , ) when p < q , where p = 2 q = 4 and σ = 1 with supportregion from σ ≤ x ≤ ∞ . where I is the interference from SUs given by Eq. (2). Thedistribution of numerator is a scaled exponential distributionand the distribution of denominator is already derived in theprevious section (Eq. (5) and Eq. (7)). Therefore, the PDF ofratio of two independent random variables [10] i.e. z = x/y ,where x = γp and y = σ + I will be given as f z ( z ) = (cid:90) ∞ σ y · f x,y ( yz, y ) dy = (cid:90) ∞ σ y · f x ( yz ) f y ( y ) dy ∀ y ≥ . (9)For the interference from a single SU user, the SINRdistribution will be as follows: f z ( z ) = (cid:90) ∞ σ y · λ e − λ yzp p λ p e − λ y − σ p , × (cid:110) − H ( y − σ − q ) + pλ δ ( y − σ − q ) (cid:111) dy = λ λ p e λ σ p (cid:26)(cid:90) ∞ σ y · e − y ( λ z + λ p dy − (cid:90) ∞ σ + q y × e − y ( λ z + λ p dy + pλ ( σ + q ) e − ( σ q )( λ z + λ p (cid:27) . By using integration by parts and on further simplification,the PDF is reduced to: f z ( z ) = λ λ Λ p (cid:26) e − σ λ zp (cid:16) σ + p Λ (cid:17) + e − σ λ z + q Λ p (cid:18) ( σ + q ) λ zλ − p Λ (cid:19)(cid:27) , where Λ is the scaled and shifted random variable version of z given by Λ = λ + λ z . Under the scenario of λ = λ = λ is the channel rate parameter between PU and PBS, whereas λ is thechannel rate parameter between SU and PBS. , with AWGN as CN (0 , σ = 1) , the PDF can be furthersimplified to f z ( z ) = 1 p ( z + 1) (cid:26) e − zp (cid:18) pz + 1 (cid:19) + e − z + q ( z +1) p × (cid:18) (1 + q ) z − pz + 1 (cid:19)(cid:27) . (10)Following the same analytical framework used for the singleSU case, the distribution with interference from multiple SUswill be: f mz ( z ) = (cid:90) ∞ σ y ¯ λ e − ¯ λ yz (cid:110) f ¯ γ ( y − σ , n, ¯ λ ) + δ ( y − σ − q ) × (1 − F ¯ γ ( y − σ , n, ¯ λ )) − f ¯ γ ( y − σ , n, ¯ λ ) × H ( y − σ − q ) (cid:111) dy. which on further evaluation and simplification reduces to f mz ( z ) = ¯ λ ¯ λ n e − σ ¯ λ z Θ − − n (cid:34) n + (cid:18) σ Θ × (cid:20) − Γ( n, q Θ)Γ( n, (cid:21)(cid:19) − Γ( n + 1 , q Θ)Γ( n, (cid:35) + ¯ λ ( σ + q ) Γ( n, q ¯ λ )Γ( n, e − ¯ λ ( σ + q ) z , (11)where Θ is the scaled and shifted random variable version of z given by Θ = ¯ λ + ¯ λ z . Fig. 4 shows the plot of the derivedtheoretical expression with simulation data for the two casesof p < q and p > q with different SU densities ( n = 1 , , ). z f z ( z ) SimulationTheoreticalSimulationTheoretical Simulation Theoretical z f z ( z ) SimulationTheoretical Simulation Theoretical SimulationTheoretical n=3n=1n=3n=2n=1 n=2
Fig. 4. PDF of SINR for two cases of p < q and p > q for different numberof SUs ( n = 1 , , ). In the following sections, we will look into the crucial per-formance metrics (outage probability and capacity) of underlaycognitive network. The mentioned approach can be extendedto the case of interference from multiple SUs given thatthe important SINR expression Eq. (11) for multiple SUs isalready been derived. However, given the space limitations, thederivations considering multiple SUs are not detailed hereinin coming sections. Nonetheless, the fundamental case ofinterference from single SU case has been presented in detail. Here ¯ λ = λ /p is the scaled rate parameter of SUs and ¯ λ = λ /p isthe scaled rate parameter for PU. . Mean SINR The mean SINR is given as µ = (cid:82) ∞ zf z ( z ) dz where thePDF of SINR f z ( z ) was derived in Eq. (10). Thus, µ = (cid:90) ∞ z (cid:110) p ( z + 1) (cid:110) e − zp (cid:18) pz + 1 (cid:19) + e − ( z + q ( z +1) p ) (cid:18) (1 + q ) z − pz + 1 (cid:19)(cid:111) dz, = (cid:90) ∞ (cid:32) ze − zp p ( z + 1) (cid:33) dz + (cid:90) ∞ (cid:32) ze − zp ( z + 1) (cid:33) dz + (cid:90) ∞ (cid:32) z (1 + q ) e − ( z + q ( z +1) p ) p ( z + 1) (cid:33) dz − (cid:90) ∞ (cid:32) ze − ( z + q ( z +1) p )( z + 1) (cid:33) dz. which on further simplification reduces to µ = e p (cid:26) Γ (cid:18) , p (cid:19) − Γ (cid:18) , qp (cid:19)(cid:27) + pe − qp q . (12)Fig. 5 shows that the change in mean SINR while varying the q Fig. 5. Mean SINR vs Interference Temperature, q , for p = 2 and p = 4 . interference temperature q for a constant peak transmit power p . The higher transmit power (for both PU and SU) with lowerinterference temperature gives better mean SINR than lowertransmit power (for both PU and SU) with high IT constraint. C. Outage Probability of Primary Network
The outage probability is defined as the probability whenthe instantaneous SINR drops below a given threshold. Math-ematically, this is given as:
P r ( γ ≤ ψ ) = F z ( ψ ) , which is nothing but the CDF of SINR. Therefore, F z ( ψ ) = (cid:90) ψ f z ( z ) dz, = (cid:90) ψ (cid:32) e − zp p ( z + 1) (cid:33) dz + (cid:90) ψ (cid:32) e − zp ( z + 1) (cid:33) dz + (cid:90) ψ (cid:32) z (1 + q ) e − ( z + q ( z +1) p ) p ( z + 1) (cid:33) dz − (cid:90) ψ (cid:32) e − ( z + q ( z +1) p )( z + 1) (cid:33) dz. which on further integration and simplification reduces to F z ( ψ ) = 1 − e − ψp ψ + 1 (cid:16) ψe − q ( ψ +1) p (cid:17) F z () SimulationTheoretical pq
Fig. 6. Outage probability of PU for p < q , where p = 2 and q = 4 and for p > q , where p = 4 and q = 2 . It can be directly inferred from Fig. 6 that if q > p , theoutage probability is higher than in the case of p > q . Inaddition to this inference, it can be also observed that thetheoretical expressions derived are in sync with the simulationresults, i.e., increase the spectral efficiency of the network.
D. Instantaneous Capacity of Primary Network
The PDF of instantaneous capacity can be readily foundfrom the PDF of instantaneous SINR by using transformationof random variables method [7], [8]. This can be obtained byusing: f x ( x ) = f z ( z ) (cid:12)(cid:12)(cid:12) dzdx (cid:12)(cid:12)(cid:12) z = e x − , where f z ( z ) is derived in Eq. (10) for the case of interferencefrom single SU on primary network. It can be seen fromFig. 7 that there is a point where the instantaneous capacityfor p < q goes below p > q . It proves the point that theinterference temperature should not be kept constant rathershould be dynamic in nature to exploit full potential of thenetwork. x f x ( x ) SimulationTheoretical p>q p Fig. 7. Instantaneous PDF of Capacity for p < q , where p = 2 and q = 4 and for p > q , where p = 4 and q = 2 . E. Mean Capacity The average capacity from the PDF of instantaneous SINR( f z ( z ) ) is given as: ¯ C = (cid:90) ∞ log (1 + z ) f z ( z ) dz. Substituting Eq. (10) in the above expression and on furtherevaluation. ¯ C = e p p (cid:20) Γ (cid:18) , z + 1 p (cid:19) − Γ (cid:18) , ( q + 1)( z + 1) p (cid:19) × ( p + q + 1) (cid:21) − e − ( q +1)( z +1) p z + 1 (cid:110) e q ( z +1) p × (cid:16) log ( z + 1) (cid:17) + z log ( z + 1) − (cid:111)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ . At z = ∞ , Γ(0 , z ) → and also, e − ( q +1)( z +1) p z +1 → .Therefore, the final mean capacity expression will be evaluatedat z = 0 : ¯ C = 1 − e − qp + e p p (cid:34) ( p + q + 1)Γ (cid:18) , q + 1 p (cid:19) − Γ (cid:18) , p (cid:19) (cid:35) Fig. 8 shows the plot of this theoretical expression withsimulation results for two cases of transmit power p = 2 and p = 4 . Intuitively, high transmit power p = 4 will result inhigh capacity for the network than the low transmit power of p = 2 but when the interference temperature is relaxed, theinterference caused due to secondary user on primary will alsoincrease that in turn will reduce the overall capacity.IV. C ONCLUSION In this paper, the performance of primary network is studiedconsidering interference from the SU network. The analysisis done under peak power adaptation method at secondarytransmitter. Given the importance of dynamic interferencetemperature for network performance, closed form expressions Interference Temperature (q) M ean C apa c i t y SimulationTheoretical p=2p=4 Fig. 8. Theoretical and simulation result plots for capacity at p = 2 and p = 4 with varying interference temperature: q . for the PDF and CDF of interference and noise, SINR forinterference from single and multiple SUs are derived. Further-more, instantaneous capacity with theoretical expressions formean SINR, mean capacity and outage probability are deducedfor simplistic network consisting of interference from singleSU. Finally, the theoretical expressions are validated with thesimulation results. A CKNOWLEDGMENT This work was supported by NASA Oklahoma Space GrantConsortium (EPSCoR) Research Initiation Grant. Also, theauthors would like to thank the TPC and reviewers for theirvaluable feedback and suggestions.R EFERENCES[1] S. Ekin, M. M. Abdallah, K. A. Qaraqe, and E. Serpedin, “Randomsubcarrier allocation in ofdm-based cognitive radio networks,” IEEETransactions on Signal Processing , vol. 60, no. 9, pp. 4758–4774, 2012.[2] L. Musavian and S. A¨ıssa, “Capacity and power allocation for spectrum-sharing communications in fading channels,” IEEE Transactions onWireless Communications , vol. 8, no. 1, pp. 148–156, 2009.[3] H. Tran, M. A. Hagos, M. Mohamed, and H.-J. Zepernick, “Impactof primary networks on the performance of secondary networks,” in Computing, Management and Telecommunications (ComManTel), 2013International Conference on . IEEE, 2013, pp. 43–48.[4] T. W. Ban, W. Choi, B. C. Jung, and D. K. Sung, “Multi-user diversityin a spectrum sharing system,” IEEE Transactions on Wireless Commu-nications , vol. 8, no. 1, pp. 102–106, 2009.[5] S. Ekin, F. Yilmaz, H. Celebi, K. A. Qaraqe, M.-S. Alouini, andE. Serpedin, “Capacity limits of spectrum-sharing systems over hyper-fading channels,” Wiley Wireless Communications and Mobile Comput-ing , vol. 12, no. 16, pp. 1471–1480, 2012.[6] R. Zhang, “On peak versus average interference power constraints forprotecting primary users in cognitive radio networks,” IEEE Transac-tions on Wireless Communications , vol. 8, no. 4, 2009.[7] A. Papoulis and S. U. Pillai, Probability, random variables, and stochas-tic processes . Tata McGraw-Hill Education, 2002.[8] S. Miller and D. Childers, Probability and random processes: Withapplications to signal processing and communications . Academic Press,2012.[9] H. Pishro-Nik, Introduction to probability, statistics, and random pro-cesses . Kappa Research LLC, 2014.[10] J. H. Curtiss, “On the distribution of the quotient of two chancevariables,”