# Outage Probability of Cognitive DF Relay Systems over Nakagami-m Fading

Authors: Dong Qin
Journal: Journal of Electrical and Computer Engineering
 (2019)

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## Abstract

This paper aims to derive accurate outage probability expressions of a cognitive decode-and-forward (DF) relay system over the Nakagami-m fading channel. A secondary system and a primary system coexist in a spectrum sharing the environment. In order to protect the data transmission of the primary system, the transmission power of the secondary system cannot exceed the tolerable interference threshold of the primary system and its available peak power. In particular, we also consider the impact on the secondary system when the primary system turns on its transmitter. Simulation results align with our theoretical formulas very well.


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## Body

## 1. Introduction

The concept of relaying has been widely applied in cognitive radio networks. For example, a cognitive radio testbed was implemented in [1] for feasibility verification of cooperative relay. Power and channel allocation was investigated in [2] for cognitive radio networks with cooperative relay. Multicarrier modulation techniques were studied in a multihop cognitive radio system [3]. A greedy algorithm for joint bandwidth and power allocation was developed in a cognitive radio hybrid relaying network [4].On the contrary, the outage probability of cognitive relay networks has been studied extensively. The exact outage probability of the secondary user was derived in opportunistic relaying with partial channel state information of the primary user [5]. The outage probability of cognitive relay networks was investigated in [6, 7] when relay selection was applied. A lower bound on outage probability of cognitive amplify-and-forward relay networks was examined in [8] subject to Nakagami-m fading.However, common to the works in [5–8] is that, in their assumption, the primary system is idle and turns off the transmitter. Obviously, the secondary system will not receive additional signals from the primary system, which is an interference from the point of view of the secondary system. But in a spectrum-sharing environment, the primary and secondary systems may use the same spectral band together simultaneously. Therefore, the influence on the secondary system cannot be overlooked when the primary system is busy. Although the busy state of the primary system is studied in [9], only the Rayleigh fading environment was involved. This may not be realistic in practical applications owing to complex fading scenarios.Inspired by the above description, the purpose of this paper is to study the accurate outage probability expression of a cognitive decode-and-forward (DF) relay system subject to Nakagami-m fading channels with the Rayleigh model as its special case. In order to guarantee the quality of service of the primary system, the transmission power of the secondary system is limited by the interference level that the primary user can tolerate and its own peak power constraint. Importantly, the primary system is busy and turns on the transmitter, while the idleness of the primary system can be regarded as a special case where the transmission power is zero. Therefore, the primary system also has a negative impact on the secondary system’s communication, which is often overlooked. When the closed-form outage probability expression is derived, we can immediately evaluate the outage performance of the following three special cases: single interference level constraint, single peak power constraint, and fault tolerant case.


### 2. System Model

Consider a spectrum-sharing environment where an authorized primary system and an unauthorized secondary system share the same spectrum band. The secondary system is composed of one secondary source node SS, one secondary DF relay node SR, and one secondary destination node SD. The primary system consists of one primary source node PS and one primary destination node PD. Denoteh1, h2, h3, g1, g2, and g3 as channel coefficients of links SS ⟶ SR, SS ⟶ PD, PS ⟶ SR, SR ⟶ SD, SR ⟶ PD, and PS ⟶ SD, respectively, as shown in Figure 1. Because the interference on PD cannot exceed the maximum signal-to-noise ratio (SNR) level Qd, the transmission powers at SS and SR are less than or equal to minPs,QdN0/h22 and minPr,QdN0/g22, where Ps and Pr are peak power constraints at SS and SR, respectively. N0 represents noise variance. The instantaneous signal-to-interference-plus-noise ratio (SINR) at SD is given by(1)γsd=minminPs,QdN0/h22h12Pph32+N0,minPr,QdN0/g22g12Ppg32+N0=minγ1,γ2,where(2)γ1=minPs,QdN0/h22h12Pph32+N0,γ2=minPr,QdN0/g22g12Ppg32+N0,and Pp is the transmission power at PS. If PS is idle and closes its transmitter, Pp=0. This case has been widely studied [5–8]. But PS will definitely work, and its transmission power will definitely be greater than zero. Therefore, we consider a more general situation Pp≥0, while the case with Pp=0 can be regarded as a special case.Figure 1
System model.For convenience, letai=hi2/N0andbi=gi2/N0fori=1,2,and3. Because all channels experience Nakagami-m fading, the probability density function (PDF) and cumulative distribution function (CDF) of X, X∈ai,bi, are given by(3)fXx=βmxm−1e−βxΓm,(4)FXx=1−Γm,βxΓm,where Γ⋅ and Γ⋅,⋅ denote the gamma function ([10], equation (8.310.1)) and the upper incomplete gamma function ([10], equation (8.350.2)), respectively. m and β=mEh2/N0 or β=mEg2/N0 are fading and shape parameters, respectively, in which E⋅ represents statistical expectation.


### 3. Outage Probability

The outage probability at SD is given byPout=Prγsd≤γth=1−1−Fγ1γth1−Fγ2γth, where γth is a certain threshold. The CDF of γ1 is given by(5)Fγ1u=PrminPs,Qda2a1Ppa3+1≤u=1−PrminPs,Qda2a1Ppa3+1>u=1−∫0∞∫0Qd/Ps∫uPpz+1/Ps∞fa1xfa2yfa3zdxdydz−∫0∞∫Qd/Ps∞∫uyPpz+1/Qd∞fa1xfa2yfa3zdxdydz.The second term ofFγ1u is denoted as I1, which can be computed as(6)I1=∫0Qd/Ps∫0∞Γma1,βa1uPpz+1/yΓma1fa2y×fa3zdzdy.Applying power series expansion in (6) yields(7)I1=βa3ma3γma2,βa2Qd/PsΓma2Γma3e−βa1u/Ps∑m=0ma1−1∑k=0mCmkPpkPsmm!βa1umk+ma3−1!×Psβa1Ppu+βa3Psk+ma3,where C⋅⋅ and γ⋅,⋅ denote binomial coefficient and the lower incomplete gamma function ([10], equation (8.350.1)).The third term ofFγ1u is denoted as I2. Calculating I2, we encounter one kind of integral form:(8)J1=∫βa1Ppu/Ps+βa3∞e−βa1u+βa2Qdy−βa3/βa1Ppγu×y−βa3βa1Ppum+ma2−1y−k−ma31βa1Ppudy.Using the binomial expansion and equation (3.381.3) in reference [10] in (8) can solve J1. Plugging the value of J1 and finishing some arrangement in I2, we get(9)I2=βa3ma3βa2ma2Qdma2Γma3Γma2∑m=0ma1−1∑k=0m∑n=0m+ma2−1CmkCm+ma2−1nm!×Ppn+1−m−ma2−ma3−βa3m+ma2−n−1×k+ma3−1!eβa3βa1u+βa2Qd/βa1Ppu×βa1u+βa2Qdma3+k−n−1×βa1un+1−k−ma2−ma3Γn−k−ma3+1,βa1Ppu+βa3Psβa1u+βa2QdPsPpβa1u.Substituting (7) and (9) into (5), we can obtain Fγ1u. Then the CDF of γ2, Fγ2u, can easily be derived from Fγ1u by substituting corresponding parameters. Finally, the outage probability can be written as(10)Pout=1−βa3ma3γma2,βa2Qd/PsΓma2Γma3e−βa1γth/Ps∑m=0ma1−1∑k=0mCmkPpkPsmm!βa1γthmk+ma3−1!×Psβa1Ppγth+βa3Psk+ma3+βa3ma3βa2ma2Qdma2Γma3Γma2∑m=0ma1−1∑k=0m∑n=0m+ma2−1CmkCm+ma2−1nm!×Ppn+1−m−ma2−ma3−βa3m+ma2−n−1×k+ma3−1!eβa3βa1γth+βa2Qd/βa1Ppγth×βa1γth+βa2Qdma3+k−n−1×βa1γthn+1−k−ma2−ma3Γn−k−ma3+1,βa1Ppγth+βa3Psβa1γth+βa2QdPsPpβa1γth×βb3mb3γmb2,βb2Qd/PrΓmb2Γmb3e−βb1γth/Pr∑m=0mb1−1∑k=0mCmkPpkPrmm!βb1γthmk+mb3−1!×Prβb1Ppγth+βb3Psk+mb3+βb3mb3βb2mb2Qdmb2Γmb3Γmb2∑m=0mb1−1∑k=0m∑n=0m+mb2−1CmkCm+mb2−1nm!×Ppn+1−m−mb2−mb3−βb3m+mb2−n−1×k+mb3−1!eβb3βb1γth+βb2Qd/βb1Ppγth×βb1γth+βb2Qdmb3+k−n−1×βb1γthn+1−k−mb2−mb3Γn−k−mb3+1,βb1Ppγth+βb3Prβb1γth+βb2QdPrPpβb1γth.Furthermore, we take into account three special cases to detect the influence of different power constraints on the outage performance.Case 1.
Single interference level constraint.
If the powers at SS and SR are just governed by the interference level on PD, this situation is equivalent to infinite peak power at SS and SR. Different from calculating the outage probability in accordance with its definition once again in [8], Pout can simply be written by limitation manipulation as(11)limPs⟶∞,Pr⟶∞Pout=1−βa3ma3βa2ma2Qdma2Γma3Γma2∑m=0ma1−1∑k=0m∑n=0m+ma2−1CmkCm+ma2−1nm!×Ppn+1−m−ma2−ma3×−βa3m+ma2−n−1×k+ma3−1!eβa3βa1γth+βa2Qd/βa1Ppγth×βa1γth+βa2Qdma3+k−n−1×βa1γthn+1−k−ma2−ma3×Γn−k−ma3+1,βa3βa1γth+βa2QdPpβa1γth×βb3mb3βb2mb2Qdmb2Γmb3Γmb2∑m=0mb1−1∑k=0m∑n=0m+mb2−1CmkCm+mb2−1nm!×Ppn+1−m−mb2−mb3×−βb3m+mb2−n−1×k+mb3−1!eβb3βb1γth+βb2Qd/βb1Ppγth×βb1γth+βb2Qdmb3+k−n−1×βb1γthn+1−k−mb2−mb3×Γn−k−mb3+1,βb3βb1γth+βb2QdPpβb1γth.Case 2.
Single peak power constraint.
If we only consider the peak power constraint at SS and SR, this situation corresponds to the case that PD happens to turn off its receiver and be idle. Then, the outage probability is given by(12)limQd⟶∞Pout=1−βa3ma3Γma3e−βa1γth/Ps∑m=0ma1−1∑k=0mCmkPpkPsmm!×βa1γthmk+ma3−1!×Psβa1Ppγth+βa3Psk+ma3×βb3mb3Γmb3e−βb1γth/Pr∑m=0mb1−1∑k=0mCmkPpkPrmm!×βb1γthmk+mb3−1!×Prβb1Ppγth+βb3Psk+mb3.Case 3.
Fault tolerant scheme.
Sometimes, it is difficult to capture accurate instantaneous channel state information, especially in rapid varying channels. In this case, a fault tolerant scheme was proposed in [11], which only knew the statistical knowledge of the varying channel instead of the instantaneous value. The basic idea is the probability that the interference on the primary system from the secondary system beyond the predetermined threshold value does not exceed an acceptable error level. So, the power constraint is readjusted to meet the following conditions:(13)PrPsa2>Qd≤ε,(14)PrPrb2>Qd≤ε,where ε is a tolerated error. In this scenario, outage probability can be expressed by replacing Ps and Pr with minQd/Fa2−11−ε,Ps and minQd/Fb2−11−ε,Pr in (12), respectively, where F−1⋅ is the inverse function of F⋅.


### 4. Simulation Results

In this section, we provide some typical simulation experiments to confirm our theoretical analysis. Without loss of generality, channel gain is normalized to unity, i.e.,Eh2=Eg2=1. Assume ma1=mb1=3,ma2=mb2=2,ma3=mb3=1, and Ps=Pr.Figure2 shows the outage probability versus average SNR Ps/N0 under different tolerance interference level constraints of the primary system Qd=0,2,4,and6 dB, where the SNR is defined as Ps/N0. The PS’s transmission power is Pp = 0 dBW. A perfect match is observed between theoretical analysis and simulation results, which indeed corroborates the correctness of our formula, given in (10). Interestingly, different from the traditional wireless network where the outage probability decreases with the increase of SNR, when the SNR exceeds a certain value, for example, 10 dB in our experiment, the outage probability remains unchanged and constant in the cognitive system. The reason is that SS’s power is governed by the interference level Qd in high SNR regime. The larger the interference that the primary system can tolerate, the lower the outage probability of the secondary system.Figure 2
Outage probability for different interference levels.Figure3 evaluates the impact of PS’s transmission power on the outage probability, where Qd=0 dB. Figure 3 is somewhat similar to Figure 2 due to the interference level Qd. It is observed that the outage probability is deteriorated when PS increases its power. This is just the price that the secondary system pays for using spectrum holes because the secondary system is often unlicensed.Figure 3
Outage probability for different PS’s transmission powers.Finally, Figure4 illustrates the outage probability when the primary system happens to be idle. Although this model may be ideal, it can still provide an insight. In contrast to Figure 3, the curves in Figure 4 have been declining due to Qd⟶∞. Furthermore, all curves are parallel to each other, indicating the diversity order is minma1,mb1.Figure 4
Outage probability for single power constraint.


### 5. Conclusion

The accurate closed form of outage probability is evaluated in a cognitive DF relay system over i.n.i.d. Nakagami-m fading. In particular, our model also incorporates the busy situation of the primary system, which is usually ignored in the existing works. The outage probability expression is very useful and can directly export to three special cases, i.e., single interference level constraint, single power constraint, and fault tolerant scheme.


## 2. System Model

Consider a spectrum-sharing environment where an authorized primary system and an unauthorized secondary system share the same spectrum band. The secondary system is composed of one secondary source node SS, one secondary DF relay node SR, and one secondary destination node SD. The primary system consists of one primary source node PS and one primary destination node PD. Denoteh1, h2, h3, g1, g2, and g3 as channel coefficients of links SS ⟶ SR, SS ⟶ PD, PS ⟶ SR, SR ⟶ SD, SR ⟶ PD, and PS ⟶ SD, respectively, as shown in Figure 1. Because the interference on PD cannot exceed the maximum signal-to-noise ratio (SNR) level Qd, the transmission powers at SS and SR are less than or equal to minPs,QdN0/h22 and minPr,QdN0/g22, where Ps and Pr are peak power constraints at SS and SR, respectively. N0 represents noise variance. The instantaneous signal-to-interference-plus-noise ratio (SINR) at SD is given by(1)γsd=minminPs,QdN0/h22h12Pph32+N0,minPr,QdN0/g22g12Ppg32+N0=minγ1,γ2,where(2)γ1=minPs,QdN0/h22h12Pph32+N0,γ2=minPr,QdN0/g22g12Ppg32+N0,and Pp is the transmission power at PS. If PS is idle and closes its transmitter, Pp=0. This case has been widely studied [5–8]. But PS will definitely work, and its transmission power will definitely be greater than zero. Therefore, we consider a more general situation Pp≥0, while the case with Pp=0 can be regarded as a special case.Figure 1
System model.For convenience, letai=hi2/N0andbi=gi2/N0fori=1,2,and3. Because all channels experience Nakagami-m fading, the probability density function (PDF) and cumulative distribution function (CDF) of X, X∈ai,bi, are given by(3)fXx=βmxm−1e−βxΓm,(4)FXx=1−Γm,βxΓm,where Γ⋅ and Γ⋅,⋅ denote the gamma function ([10], equation (8.310.1)) and the upper incomplete gamma function ([10], equation (8.350.2)), respectively. m and β=mEh2/N0 or β=mEg2/N0 are fading and shape parameters, respectively, in which E⋅ represents statistical expectation.


## 3. Outage Probability

The outage probability at SD is given byPout=Prγsd≤γth=1−1−Fγ1γth1−Fγ2γth, where γth is a certain threshold. The CDF of γ1 is given by(5)Fγ1u=PrminPs,Qda2a1Ppa3+1≤u=1−PrminPs,Qda2a1Ppa3+1>u=1−∫0∞∫0Qd/Ps∫uPpz+1/Ps∞fa1xfa2yfa3zdxdydz−∫0∞∫Qd/Ps∞∫uyPpz+1/Qd∞fa1xfa2yfa3zdxdydz.The second term ofFγ1u is denoted as I1, which can be computed as(6)I1=∫0Qd/Ps∫0∞Γma1,βa1uPpz+1/yΓma1fa2y×fa3zdzdy.Applying power series expansion in (6) yields(7)I1=βa3ma3γma2,βa2Qd/PsΓma2Γma3e−βa1u/Ps∑m=0ma1−1∑k=0mCmkPpkPsmm!βa1umk+ma3−1!×Psβa1Ppu+βa3Psk+ma3,where C⋅⋅ and γ⋅,⋅ denote binomial coefficient and the lower incomplete gamma function ([10], equation (8.350.1)).The third term ofFγ1u is denoted as I2. Calculating I2, we encounter one kind of integral form:(8)J1=∫βa1Ppu/Ps+βa3∞e−βa1u+βa2Qdy−βa3/βa1Ppγu×y−βa3βa1Ppum+ma2−1y−k−ma31βa1Ppudy.Using the binomial expansion and equation (3.381.3) in reference [10] in (8) can solve J1. Plugging the value of J1 and finishing some arrangement in I2, we get(9)I2=βa3ma3βa2ma2Qdma2Γma3Γma2∑m=0ma1−1∑k=0m∑n=0m+ma2−1CmkCm+ma2−1nm!×Ppn+1−m−ma2−ma3−βa3m+ma2−n−1×k+ma3−1!eβa3βa1u+βa2Qd/βa1Ppu×βa1u+βa2Qdma3+k−n−1×βa1un+1−k−ma2−ma3Γn−k−ma3+1,βa1Ppu+βa3Psβa1u+βa2QdPsPpβa1u.Substituting (7) and (9) into (5), we can obtain Fγ1u. Then the CDF of γ2, Fγ2u, can easily be derived from Fγ1u by substituting corresponding parameters. Finally, the outage probability can be written as(10)Pout=1−βa3ma3γma2,βa2Qd/PsΓma2Γma3e−βa1γth/Ps∑m=0ma1−1∑k=0mCmkPpkPsmm!βa1γthmk+ma3−1!×Psβa1Ppγth+βa3Psk+ma3+βa3ma3βa2ma2Qdma2Γma3Γma2∑m=0ma1−1∑k=0m∑n=0m+ma2−1CmkCm+ma2−1nm!×Ppn+1−m−ma2−ma3−βa3m+ma2−n−1×k+ma3−1!eβa3βa1γth+βa2Qd/βa1Ppγth×βa1γth+βa2Qdma3+k−n−1×βa1γthn+1−k−ma2−ma3Γn−k−ma3+1,βa1Ppγth+βa3Psβa1γth+βa2QdPsPpβa1γth×βb3mb3γmb2,βb2Qd/PrΓmb2Γmb3e−βb1γth/Pr∑m=0mb1−1∑k=0mCmkPpkPrmm!βb1γthmk+mb3−1!×Prβb1Ppγth+βb3Psk+mb3+βb3mb3βb2mb2Qdmb2Γmb3Γmb2∑m=0mb1−1∑k=0m∑n=0m+mb2−1CmkCm+mb2−1nm!×Ppn+1−m−mb2−mb3−βb3m+mb2−n−1×k+mb3−1!eβb3βb1γth+βb2Qd/βb1Ppγth×βb1γth+βb2Qdmb3+k−n−1×βb1γthn+1−k−mb2−mb3Γn−k−mb3+1,βb1Ppγth+βb3Prβb1γth+βb2QdPrPpβb1γth.Furthermore, we take into account three special cases to detect the influence of different power constraints on the outage performance.Case 1.
Single interference level constraint.
If the powers at SS and SR are just governed by the interference level on PD, this situation is equivalent to infinite peak power at SS and SR. Different from calculating the outage probability in accordance with its definition once again in [8], Pout can simply be written by limitation manipulation as(11)limPs⟶∞,Pr⟶∞Pout=1−βa3ma3βa2ma2Qdma2Γma3Γma2∑m=0ma1−1∑k=0m∑n=0m+ma2−1CmkCm+ma2−1nm!×Ppn+1−m−ma2−ma3×−βa3m+ma2−n−1×k+ma3−1!eβa3βa1γth+βa2Qd/βa1Ppγth×βa1γth+βa2Qdma3+k−n−1×βa1γthn+1−k−ma2−ma3×Γn−k−ma3+1,βa3βa1γth+βa2QdPpβa1γth×βb3mb3βb2mb2Qdmb2Γmb3Γmb2∑m=0mb1−1∑k=0m∑n=0m+mb2−1CmkCm+mb2−1nm!×Ppn+1−m−mb2−mb3×−βb3m+mb2−n−1×k+mb3−1!eβb3βb1γth+βb2Qd/βb1Ppγth×βb1γth+βb2Qdmb3+k−n−1×βb1γthn+1−k−mb2−mb3×Γn−k−mb3+1,βb3βb1γth+βb2QdPpβb1γth.Case 2.
Single peak power constraint.
If we only consider the peak power constraint at SS and SR, this situation corresponds to the case that PD happens to turn off its receiver and be idle. Then, the outage probability is given by(12)limQd⟶∞Pout=1−βa3ma3Γma3e−βa1γth/Ps∑m=0ma1−1∑k=0mCmkPpkPsmm!×βa1γthmk+ma3−1!×Psβa1Ppγth+βa3Psk+ma3×βb3mb3Γmb3e−βb1γth/Pr∑m=0mb1−1∑k=0mCmkPpkPrmm!×βb1γthmk+mb3−1!×Prβb1Ppγth+βb3Psk+mb3.Case 3.
Fault tolerant scheme.
Sometimes, it is difficult to capture accurate instantaneous channel state information, especially in rapid varying channels. In this case, a fault tolerant scheme was proposed in [11], which only knew the statistical knowledge of the varying channel instead of the instantaneous value. The basic idea is the probability that the interference on the primary system from the secondary system beyond the predetermined threshold value does not exceed an acceptable error level. So, the power constraint is readjusted to meet the following conditions:(13)PrPsa2>Qd≤ε,(14)PrPrb2>Qd≤ε,where ε is a tolerated error. In this scenario, outage probability can be expressed by replacing Ps and Pr with minQd/Fa2−11−ε,Ps and minQd/Fb2−11−ε,Pr in (12), respectively, where F−1⋅ is the inverse function of F⋅.


## 4. Simulation Results

In this section, we provide some typical simulation experiments to confirm our theoretical analysis. Without loss of generality, channel gain is normalized to unity, i.e.,Eh2=Eg2=1. Assume ma1=mb1=3,ma2=mb2=2,ma3=mb3=1, and Ps=Pr.Figure2 shows the outage probability versus average SNR Ps/N0 under different tolerance interference level constraints of the primary system Qd=0,2,4,and6 dB, where the SNR is defined as Ps/N0. The PS’s transmission power is Pp = 0 dBW. A perfect match is observed between theoretical analysis and simulation results, which indeed corroborates the correctness of our formula, given in (10). Interestingly, different from the traditional wireless network where the outage probability decreases with the increase of SNR, when the SNR exceeds a certain value, for example, 10 dB in our experiment, the outage probability remains unchanged and constant in the cognitive system. The reason is that SS’s power is governed by the interference level Qd in high SNR regime. The larger the interference that the primary system can tolerate, the lower the outage probability of the secondary system.Figure 2
Outage probability for different interference levels.Figure3 evaluates the impact of PS’s transmission power on the outage probability, where Qd=0 dB. Figure 3 is somewhat similar to Figure 2 due to the interference level Qd. It is observed that the outage probability is deteriorated when PS increases its power. This is just the price that the secondary system pays for using spectrum holes because the secondary system is often unlicensed.Figure 3
Outage probability for different PS’s transmission powers.Finally, Figure4 illustrates the outage probability when the primary system happens to be idle. Although this model may be ideal, it can still provide an insight. In contrast to Figure 3, the curves in Figure 4 have been declining due to Qd⟶∞. Furthermore, all curves are parallel to each other, indicating the diversity order is minma1,mb1.Figure 4
Outage probability for single power constraint.


## 5. Conclusion

The accurate closed form of outage probability is evaluated in a cognitive DF relay system over i.n.i.d. Nakagami-m fading. In particular, our model also incorporates the busy situation of the primary system, which is usually ignored in the existing works. The outage probability expression is very useful and can directly export to three special cases, i.e., single interference level constraint, single power constraint, and fault tolerant scheme.



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