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Performance Enhancement for Multihop Cognitive DF and AF Relaying Protocols under Joint Impact of Interference and Hardware Noises: NOMA for Primary Network and Best-Path Selection for Secondary Network

In this paper, we propose and evaluate performance of multihop multipath underlay cognitive radio networks. In a primary network, an uplink nonorthogonal multiple access method is employed to allow primary transmitters to simultaneously transmit their data to a primary receiver. Using an underlay spectrum-sharing method, secondary source and secondary relays must adjust their transmit power to guarantee quality of service of the primary network. Under the limited transmit power, cochannel interference from the primary transmitters, and hardware noises caused by impairments, we propose best-path selection methods to improve the end-to-end performance for the secondary network. Moreover, both multihop decode-andforward and amplify-and-forward relaying protocols are considered in this paper. We derive expressions of outage probability for the primary and secondary networks and propose an efficient method to calculate the transmit power of the secondary transmitters. Then, computer simulations employing the Monte-Carlo approach are realized to validate the derivations.


Introduction
Multihop relaying protocols [1][2][3][4][5] are commonly used to enhance performance for wireless networks, e.g., wireless sensor networks (WSNs) and wireless ad hoc networks (WANs). In WSNs and WANs, the transmit power of wireless devices is often limited, and hence, the source cannot directly communicate with the destination at a far distance. In multihop decode-and-forward protocol (M-DF-P) [6][7][8], intermediate nodes (or relays) decode the received signals, reencode, and send the encoded ones to next hop. In multihop amplify-and-forward protocol (M-AF-P) [9][10][11], relays amplify the received signals and forward the amplified ones to a next node. The implementation of M-DF-P is more complex than that of M-AF-P, but M-DF-P obtains better end-to-end (e2e) performance due to the decoding process at each hop. As shown in [12,13], M-DF-P outperforms M-AF-P over fading and limited cochannel interference (CCI) environments, in terms of outage probability (OP) and bit error rate (BER). References [14,15] evaluated the e2e OP of M-AF-P in a random presence of interference sources. In [16], the authors studied the impact of I/Q imbalance on the performance of M-AF-P. In [17], joint impact of hardware impairments (HI) and CCI on the e2e OP and average channel capacity of M-DF-P was studied. As shown in [17], in fading and CCI environments, the performance of multihop relaying protocols is significantly degraded.
To enhance the performance for multihop relaying protocols over fading environments, various diversity methods have been proposed in [18][19][20][21][22][23][24][25][26]. The authors of [18,19] introduced cooperative multihop protocols, where the relays can exploit the broadcast nature of wireless medium to obtain spatial diversity. Particularly, the relays attempt to receive the signals from all the previous nodes and process them appropriately to enhance reliability of the data decoding. Different with [18,19], diversity-based multihop transmission protocols in [20,21] perform cooperative communication at each hop by using external nodes that are not on the source-destination route. In [22,23], relay selection approaches in cluster-based multihop transmission networks were studied. In the system models proposed in [22,23], the source data is transmitted to the destination via intermediate clusters. Moreover, opportunistic relay selection methods are performed at each cluster to attain the diversity gain. In [24][25][26], the authors introduced ultradense wireless networks in which there exist multiple available paths between a source and a destination. In addition, path selection strategies were proposed to obtain better performance.
Cognitive radio (CR) [27] was first proposed by J. Mitola in 1999 to solve the spectrum scarcity issue and inefficient spectrum usage. In CR, primary users who are used licensed bands at any time can share spectrum to secondary users. Underlay cognitive radio (UCR) [28][29][30] is one of efficient spectrum-sharing methods that can guarantee the continuous data transmission for the secondary network. However, the secondary transmitters in the UCR networks must adjust their transmit power so that quality of service (QoS) of the primary network is not harmful. In [31], the authors considered the performance of multihop UCR AF relaying protocols over the Nakagami-m fading channels. In [32,33], the authors proposed M-DF-P in the UCR networks, where the secondary transmitters can harvest wireless energy from a power beacon deployed in the secondary network. References [34,35] proposed cluster-based multihop UCR protocols using the DF technique and relay selection methods at each cluster. A best-path selection method was studied in [36] to improve the e2e performance of the UCR networks under the interference constraint and presence of an active eavesdropper.
Recently, nonorthogonal multiple access (NOMA) [37][38][39] has gained much attention as a promising method to improve data rate for wireless systems. In downlink NOMA scenarios [40,41], a base station linearly combines signals with different transmit power levels before transmitting them to intended users at the same time, frequency, and code. At these users, the successive interference cancellation (SIC) technique can be performed to obtain their desired signals. For uplink NOMA methods [42,43], the users can simultaneously send their signals to the base station. Similarly, different transmit power levels have to be used at the users so that the base station can realize SIC for decoding the received signals. References [44][45][46] studied the outage performance of dual-hop relaying protocols in the UCR networks. The authors in [47] proposed multihop energy-harvesting WSNs employing NOMA and cooperative jamming methods to enhance the e2e secrecy performance.
In this paper, we propose best-path selection methods to enhance the OP performance for both M-DF-P and M-AF-P in the UCR networks under the joint impact of CCI from primary transmitters and HI. In particular, a secondary source selects one of available paths providing maximum instantaneous e2e channel capacity to send its data to a secondary destination. We also propose an uplink NOMA method to increase the data rate for the primary network, i.e., multiple primary transmitters send their data to a primary receiver at the same time. Different with [45,46], we consider multihop DF and AF UCR networks where the primary network uses NOMA to enhance the system throughput. Unlike [34,35], this paper concerns with path selection strategies in the UCR networks (see Table 1). Although various path selection methods were proposed in [24][25][26], these published works did not consider the UCR networks. Moreover, unlike [24-26, 34, 35], the joint impact of CCI and HI is studied in this paper. Although references [17,48] evaluated the performance of M-DF-P in the presence of the limited interference and hardware noises, the authors in [17] only considered a single-path scheme, and reference [48] did not consider the UCR environment. In addition, our proposed models consider both DF and AF protocols, while only M-DF-P is studied in [17,48]. Unlike [36], we propose the uplink NOMA scheme for the primary network. Furthermore, reference [36] assumed that the transceiver hardware of all wireless devices is perfect, and CCI from the primary transmitter to secondary receivers can be relaxed.
The main contribution of this paper can be listed as follows: (i) Firstly, the NOMA approach is proposed to provide a higher data rate for the primary network. Moreover, we also propose an efficient method to allocate transmit power for the secondary transmitters to guarantee QoS for the primary network, i.e., the secondary transmitters have to adjust their transmit power so that the outage performance of the primary network is lower than a predetermined threshold (ii) Secondly, the best-path selection methods are proposed to enhance the e2e OP for the secondary networks under the joint impact of the interference constraint, CCI from the primary transmitters, and Rayleigh fading channel. In addition, we consider a practical scenario, i.e., the secondary nodes are low-cost devices (e.g., sensor nodes), and hence, their transceiver hardware may be not perfect (iii) Next, we derive an exact closed-form expression of OP for the primary network. The e2e OP of M-DF-P and M-AF-P of the secondary networks is also analyzed. Particularly, we derive the exact and asymptotic closed-form formulas of the e2e OP for M-DF-P, and lower-bound ones of the e2e OP for M-AF-P (iv) Finally, the Monte-Carlo based computer simulations are realized to validate the derived formulas as well as to compare the e2e OP performance of M-DF-P and M-AF-P.

Wireless Communications and Mobile Computing
The rest of this paper is organized as follows. The system model of multihop multipath UCR networks is described in "System Model." Performance evaluation is given in "Performance Analysis." "Numerical Results" presents both simulation and analytical results to compare performance of M-DF-P and M-AF-P. Finally, conclusions are given in "Conclusion." Figure 1 illustrates the system model of the proposed M-DF-P and M-AF-P in the UCR networks. In the primary network, N primary transmitters ðPT n , n = 1, 2, ⋯, NÞ attempt to send their data to a primary receiver (PR). To enhance the throughput for the primary network, the primary transmitters at the same time send their data to the PR node by using NOMA. In addition, the PT nodes use different transmit power levels, and the PR node uses SIC to decode the received data. In the secondary network, a source (S) wants to transmit its data to a destination (D). Because the direct link between S and D does not exist, the multihop relaying technique is employed to relay the source data to the destination. Assume that there are M available paths between S and D ðM ≥ 1Þ, and one of them is selected for the data transmission. Let L m denote the number of relay nodes on the mth path, where m = 1, 2, ⋯, M, L m ≥ 1. As depicted in Figure 1, the relays on the mth path are named by R m,1 , R m,2 ,…, R m,Lm . For ease of presentation and analysis, we also denote S and D by R m,0 and R m,Lm+1 (∀m), respectively. Assume that all the nodes X have only a single antenna and operate on a half-duplex mode, where X ∈ fR m,v , PT n , PRg, v = 0, 1, ⋯, L m + 1.

System Model
Let us denote h X,Y as the Rayleigh fading channel coefficient between X and Y, where ðX, YÞ ∈ fR m,v , PTn, PRg. Hence, channel gain, i.e., φ X,Y = jh X,Y j 2 , is an exponential random variable (RV) whose distributions are given as where F U ð:Þ and f U ð:Þ are cumulative distribution function (CDF) and probability density function (PDF) of a RV U, respectively, λ X,Y = 1/εfφ X,Y g = d PL X,Y [34][35][36], εf•g is an expected operator, d X,Y is distance between X and Y, and PLð2 ≤ PL ≤ 6Þ is a path-loss exponent.
2.1. Hardware Impairments. Considering the data transmission between a transmitter X and a receiver Y, under the impact of HI (assume that there is no CCI), the signal received by Y is written as where P X is transmit power of X, s X is signal of X, η X,Y is the hardware noise caused by the hardware imperfection at X and Y, and n Y is the additive white Gaussian noise (AWGN) at Y with zero-mean. Similar to [49,50], η X,Y can be modeled as a zero-mean Gaussian RV whose variance is κ 2 X,Y ,(κ 2 X,Y is called to be total hardware impairment level of the X ⟶ Y link).
2.1.1. Assumption and Notation. Assume that variance of AWGN at all of the receivers is same, i.e., σ 2 0 . It is assumed that the transceiver hardware of the primary terminals is perfect, i.e., κ 2 PT n ,PR = 0ð∀nÞ. Next, we denote κ 2 ss , κ 2 sp , and κ 2 ps as the total hardware impairment levels of the R m,v ⟶ R m,v+1 , R m,v ⟶ PR, and PT n ⟶ R m,v links, respectively, for all m, n, v.

Operation
Principle. In this subsection, assume that the mth path is selected, and hence, the data transmission is split into ðL m + 1Þ orthogonal time slots. At the first time slot, the  [17] No No No Yes Only DF Reference [24,25] No No Yes No Only DF Reference [26] No No Yes Only HI Only DF Reference [34,35] Yes Only DF Reference [36] Yes No Yes No Only DF Reference [48] No No Yes Yes Only DF 3 Wireless Communications and Mobile Computing source (R m,0 ) sends its data to R m,1 , while the primary transmitters simultaneously transmit their data to PR.
(1) Primary network: under the joint impact of CCI and HI, the received signal at PR in this time slot can be given as In (3), s S is the signal of the source, s ð1Þ PT n is the signal of PT n in the first time slot, P R m,0 is the transmit power of the source, and P PT is the maximum transmit power of all the primary transmitters. Following the NOMA principle, the primary transmitters must use different transmit powers so that PR can decode the received signals. As presented in (3), β n P PT is the transmit power of PT n , where β 1 = 1, β 1 > β 2 > : ⋯ > β N > 0. With this power allocation method, PR will decode the signal s ð1Þ PT n , next uses SIC to remove the compo- PR , and then decodes s ð1Þ PT n+1 . Indeed, if the SIC operation is perfect, the signal used for decoding s ð1Þ PT n+1 can be rewritten as where we note that z Combining (3), (4), and (5), the signal-to-interferenceplus-noise (SINR) obtained by PR for decoding s ð1Þ PT n is obtained as (2) Secondary network: considering the secondary network at the first time slot, the received signal at R m,1 can be expressed as From (7), the SINR received by R m,1 is calculated as Next, we describe the operation of the proposed M-DF-P and M-AF-P models at next time slots.
(3) M-DF-P: in this protocol, R m,1 decodes s S , encodes again, and then forwards it to R m,2 in the second time slot. Generally, at the tth time slot, R m,t−1 sends s S to R m,t , and the received signal at PR and R m,t can be formulated, respectively, as where t = 1, 2, ⋯, L m + 1, s ðtÞ PT n is the signal of PT n in this time slot and P R m,t−1 is the transmit power of R m,t−1 .
Similar to the first time slot, PR uses SIC to decode all the data. Then, the SINR obtained at PR, with respect to s ðtÞ PT n , is shown as Using (10), the SINR at R m,t can be given as follows: Due to usage of the DF relaying method, the e2e channel capacity of the mth path can be expressed as 4 Wireless Communications and Mobile Computing where the factor 1/ðLm + 1Þ implies that the data transmission of the secondary network is split into ðL m + 1Þ time slots.
(4) M-AF-P: in M-AF-P, R m,1 amplifies the received signal y R m,1 , i.e., z R m,1 = G R m,1 y R m,1 , and sends z R m,1 to R m,2 in the second time slot. Because P R m,1 = G 2 R m,1 ε fjy R m,1 j 2 g, using (7), the amplify factor G Rm,1 can be determined as Next, the received signal at PR and R m,2 can be expressed, respectively, as Similar to [49], variances of η Rm,1,PR and η Rm,1,Rm,2 can be calculated, respectively, as Var Combining (15) and (17), the SINR obtained at PR, with respect to s ð2Þ PT n , is given as Next, combining (7), (16), and (18), we can formulate the SINR received at R m,2 by where it is worth noting that γ Rm,0,Rm,1 and γ Rm,1,Rm,2 are given as in (12). Similarly, R m,2 amplifies y Rm,2,AF and then forwards the amplified signal to R m,3 in the third time slot. Using an inductive method, we can obtain an exact formula of the SINR obtained at R m,t at the tth time slot as follows: where 1 < t ≤ L m+1 . Then, the e2e channel capacity of the mth path can be computed as Remark 1. In both M-DF-P and M-AF-P, the SINRs obtained at PR at the tth time slot is the same (see (11)). Therefore, the channel capacity of the PTn ⟶ PR link at the tth hop of the mth path is shown as 2.3. Best-Path Selection Methods. To maximize the e2e channel capacity, we propose the best-path selection methods for M-DF-P and M-AF-P as follows: where bðb ∈ f1, 2, ⋯, MgÞ denotes the best path, and Z ∈ f DF, AFg.

Remark 2.
To select the best path, the secondary nodes on the paths have to estimate the channel state information (CSI) of the data links and the interference links. In addition, the secondary transmitters including source and relays also have to know their transmit power (see "Transmit Power of Secondary Transmitters"). Via operation of the routing set-up and maintenance, the secondary nodes can exchange the estimated CSIs and their transmit power together. Therefore, the source can obtain information of the instantaneous SINR (and instantaneous channel capacity as well) of all the hops so that it can select the best path by using the algorithm given in (24). It is also assumed that the operation of the CSI estimation is perfect, and the channels between the nodes are slow Rayleigh fading, which remain unchanged in each the source-destination data transmission.

Performance Analysis
This section evaluates OP of the primary and secondary networks. We also derive expressions of the transmit power for the secondary transmitters to guarantee QoS for the primary network.
3.1. OP of Primary Network. When the mth path is selected, let us consider the minimum channel capacity between PT n and PR at the tth time slot: We now define OP of the primary network as the probability that C ðtÞ PR,min is lower than a positive threshold, i.e., C P,th , and is expressed as where Equation (26) also implies that the primary network is in outage if one of the signals s ðtÞ PT n cannot be correctly decoded. In particular, employing the independence of RVs γ ðtÞ s PT n , Equation (26) can be rewritten as where Pr ðγ ðtÞ s PT n < θ P,m Þ is the probability that the decoding of s ðtÞ PT n fails.
Combining (11) and (28) yields (29) as where a u,n = β u θ P,m /β n , n < N Substituting CDF and PDF given by (1) into (29), after some manipulation, we obtain where Substituting (31) into (28), OP of the primary network at the tth time slot can be expressed by an exact closed-form expression as follows: 3.2. Transmit Power of Secondary Transmitters. At first, QoS of the primary network in the tth time slot is given as where ε OP is an allowable maximum value of OP at any time slot, and it is an important design parameter of the primary network. Hence, the transmit power of R m,t−1 can be obtained by solving equationOP ðtÞ PR = ε OP . More particularly, we have to find a positive real solution (denoted by τ * m,t−1 ) of the following equation: Then, the transmit power of R m,t−1 is obtained as P R m,t−1 = P PT τ * m,t−1 . In case that Equation (35) has no positive real solution, we have P Rm,t−1 = 0, which means R m,t−1 is not allowed to access the licensed band.

Corollary 4. Assume that Equation
Proof. Let us denote τ 1 as the positive real solution of (35), we have Q N n=1 ðτ 1 + l n Þ = 1/1 − ε OP Q N n=1 g u,n . Therefore, it is straightforward that for the fixed values of l n and g u,n , if ε OP decreases, then τ 1 (or P Rm,t−1 ) also decreases.

Proposition 5. Equation (35) has one positive real solution when
Proof. As proved in Proposition 3, f ðτ m,t−1 Þ is a continuous and increasing function in ½0, +∞Þ. Therefore, it is obvious that if f ð0Þ < 0, Equation (35) always has one real positive solution. Using (38), we obtain (41) and finish the proof.
For example, with N = 2, the condition in (41) can be written as Proposition 6. If Equation (35) has one positive real solution, at high P PT values (i.e., P PT ⟶ +∞), P Rm,t−1 is a linear function of P PT .
Remark 7. Before the secondary source starts the data transmission, the best path is chosen by the source, and all the secondary transmitters on the selected path have to adjust their transmit power. In practice, the value of P Rm,t−1 can be calculated by the primary network and is then sent to the secondary network. In the conventional UCR schemes (see [45,46]), the secondary transmitters have to adjust their transmit power, following the instantaneous CSIs between themselves and the primary receivers. However, these methods are very complex due to requirement of perfect CSI estimation and frequent change of the link channels.

Outage Probability of Secondary
Network. This subsection evaluates the e2e OP of M-DF-P and M-AF-P, which can be defined as where Z ∈ fAF, DFg, C S,th is a predetermined target rate of the secondary network, and OP m,Z = Pr ðC e2e m < C S,th Þ is the e2e OP of the mth path. In the following, we attempt to calculate OP m,Z .
(1) M-DF-P: in this protocol, the e2e OP of the mth path is calculated as follows: (1) M-AF-P: because it is too difficult to find an exact expression of OP m,AF , we attempt to obtain a lowerbound one.

Proposition 8. OP m,DF can be expressed by an exact closedform expression as
Proof. See Appendix A.
Similar to [31], we consider two RVs: X 1,m = min bðL m + 1Þ/2c and bxc is the highest integer that is smaller or equal to x. Then, the e2e SINR Ψ Rm,Lm+1 (see (21) or (22)) can be bounded as in [31]: Our next objective is to find the distributions of X 1,m and X 2,m which are presented in Proposition 10.
Proposition 10. CDF of X 1,m and X 2,m can be given, respectively, as follows: where Proof. See Appendix B.
From (52), if 1 − κ 2 ss x ≤ 0, f X 2,m ðxÞ = 0, and if 1 − κ 2 ss x > 0, PDF of X 2,m can be obtained as Next, we derive the lower-bound expression of OP m,AF as given in Proposition 11. where Proof. See Appendix C.
Remark 12. Firstly, OP LB m,AF can be numerically calculated by using MATLAB or Mathematica. Secondly, as P PT → +∞, the e2e OP of M-DF-P and M-AF-P does not depend on P PT . Indeed, γ R m,t-1 , R m,t in (12) can be approximated as 8 Wireless Communications and Mobile Computing We can observe from (58) that γ R m,t-1 , R m,t at high P PT values do not depend on P PT , and it is the reason why OP m,Z ðOP e2e Z Þ converges to a constant as P PT → +∞S. This also means that the diversity order of both M-DF-P and M-AF-P is zero.
where ðκ DF ss Þ 2 and ðκ AF ss Þ 2 are called as limited hardware impairment levels of M-DF-P and M-AF-P, respectively. Because ðκ DF ss Þ 2 ≥ ðκ AF ss Þ 2 , M-AF-P is more sensitive to HI than M-DF-P.

Proposition 15.
When the transceiver hardware is perfect, i.e., κ 2 ss = κ 2 sp = κ 2 ps = 0, we obtain a lower-bound closed-form expression of OP m,AF by where Proof. See Appendix D.

Numerical Results
This section provides the Monte-Carlo-based computer simulations to verify the analysis as well as to compare the e2e OP of M-DF-P and M-AF-P. Simulation environment is a two-dimensional plane Oxy, in which all the primary and secondary nodes are placed. We assume that there are two primary transmitters, and their coordinates are PT 1 ð0:35, 0:5Þ and PT 2 ð0:6, 0:5Þ, while PR is located at (0.45, 0.5). As mentioned in "System Model," the transmit power of PT 1 is higher than that of PT 2 ; hence, we can set β 1 = 1 and β 2 = 0:2. In the secondary network, assume that there are 03 available paths (M = 3), and the number of relays on the paths is L 1 = 1, L 2 = 2, and L 3 = 3. In addition, the source and destination nodes are located at (0,0) and (0,1), respectively, and the relay node R m,t is placed at ððt/L m + 1Þ, 0Þ , where m = 1, 2, 3 and t = 1, ⋯, L m . To focus on performance trend and performance comparison between M-DF-P and M-AF-P, some system parameters are fixed in all the simulations, as given in Table 2.
Remark 16. Although we do not investigate the impact of the number of paths ðMÞ and the number of the primary transmitters ðNÞ on the e2e OP of the secondary networks, we can guess that the performance of M-DF-P and M-AF-P is better with a higher value of M and a lower value of N. Also, we can expect that the performance of M-DF-P and M-AF-P is better with higher value of ε OP , and vice verse.
In Figure 2, we assume that the third path is selected and presents the transmit power of R 3,0 (Source), R 3,1 , R 3,2 , and R 3,3 as a function of P PT in dB. As seen from Figure 2, the transmit power of all the secondary transmitters is higher than zero when P PT changes from 12 dB to 32 dB. This means that the secondary transmitters are allowed to access the licensed band to transmit the data. As proved in Proposition 6, we also observe that the transmit power linearly increases with the increase of P PT . Also in this figure, the transmit power of R 3,0 is highest, and that of R 3,2 is lowest. It is due to the fact that the distance between R 3,0 and PR is furthest, and R 3,2 is nearest to PR. Figure 3 compares the performance of M-DF-P and M-AF-P when κ 2 ss = 0:05 and κ 2 sp = κ 2 ps = 0:01. We can see that the performance of M-DF-P is better than that of M-AF-P for all values of P PT . As mentioned in Remark 12, the e2e OP of M-DF-P and M-AF-P in Figure 3 converge to a constant when P PT is high enough. In addition, the performance of the proposed protocols is better as C S,th decreases. It is worth noting that for M-DF-P, the simulation results (denoted by Sim) verify the theoretical ones, where Exact denotes the theoretical results obtained by using (46) and Asymp denotes the approximate expression of the e2e OP at high P PT regimes (see (48)). For M-AF-P, there exists gap between the simulation results (Sim) and the theoretical results (LowB) because the usage of (56) for evaluating the outage performance.
In Figure 4, the e2e OP of M-DF-P and M-AF-P presents as a function of P PT in dB when C S,th = 0:03. In this figure, we consider two cases: the transceiver hardware is perfect ðκ 2 ss = κ 2 sp = κ 2 ps = 0Þ and imperfect ðκ 2 ss = κ 2 sp = κ 2 ps = 0:04Þ. As expected, the outage performance of the proposed protocols with perfect transceiver hardware is better. Similar to Figure 3, M-DF-P outperforms M-AF-P, and the OP curves rapidly converge to constants. It is worth noting that  when the hardware impairments are relaxed, the closed-form expression in (63) is used to calculate the lower-bound e2e OP for M-AF-P, instead of (56). Figure 5 compares the performance of M-DF-P and M-AF-P with different values of C S,th . In this figure, the perfect transceiver hardware is assumed, and the transmit power P PT is fixed by 15 dB. We can see from Figure 5 that the e2e OP values of the proposed protocols increase as C S,th increases. Moreover, the performance of M-DF-P is better than that of M-AF-P, and the performance gap is higher as C S,th increases.
In Figure 6, we present the outage performance of M-DF-P as a function of κ 2 ss when P PT = 30 dB, and κ 2 sp = κ 2 ps = κ 2 ss /5. Figure 6 illustrates that the e2e OP rapidly increases as κ 2 ss increases. We also see that when the HI level is high enough, M-DF-P is always in outage. As given in Remark From Figures 3-7, we can see that M-DF-P not only outperforms M-AF-P but also obtains much higher value of the limited HI level.

Conclusion
This paper proposed, evaluated, and compared the e2e OP of M-DF-P and M-AF-P in the UCR networks. The interesting results obtained in this paper can be summarized as follows: (i) the transmit power of the secondary transmitters can

Wireless Communications and Mobile Computing
linearly increase with the transmit power of the primary transmitters, (ii) the OP performance of M-DF-P is significantly better than that of M-AF-P, (iii) there exist error floors that means the diversity gain of M-DF-P and M-AF-P equals to zero, and (iv) M-AF-P is more sensitive to the HI than M-DF-P, i.e., the limited HI level of M-AF-P is much lower than that of M-DF-P. Finally, the OP performance for M-DF-P and M-AF-P can be enhanced by increasing the number of paths between the secondary source and destination and by equipping better transceiver hardware for the secondary nodes (i.e., lower hardware impairment level).

Appendix A. Proof of Proposition 8
Using (13), we have where θ S,m is given by (47).

Data Availability
The simulation data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.