# 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

Authors: Tran Trung Duy; Pham Thi Dan Ngoc; Phuong T. Tran
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

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-and-forward 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.


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## Body

## 1. Introduction

Multihop relaying protocols [1–5] 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) [6–8], 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) [9–11], 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 [18–26]. 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 [24–26], 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) [28–30] 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) [37–39] 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 [44–46] 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 [24–26], 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.Table 1
Main differences with the published works related to multihop relaying networks.
ReferencesUnderlay CRNOMA-based primary transmissionPath selectionJoint impact of CCI and HIDF/AF protocolProposed schemeYesYesYesYesBothReference [17]NoNoNoYesOnly DFReference [24, 25]NoNoYesNoOnly DFReference [26]NoNoYesOnly HIOnly DFReference [34, 35]YesNoNoNoOnly DFReference [36]YesNoYesNoOnly DFReference [48]NoNoYesYesOnly DFThe 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.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.”


## 2. System Model

Figure1 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 PTn,n=1,2,⋯,N 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 M≥1, and one of them is selected for the data transmission. Let Lm denote the number of relay nodes on the mth path, where m=1,2,⋯,M, Lm≥1. As depicted in Figure 1, the relays on the mth path are named by Rm,1, Rm,2,…, Rm,Lm. For ease of presentation and analysis, we also denote S and D by Rm,0 and Rm,Lm+1 (∀m), respectively. Assume that all the nodes X have only a single antenna and operate on a half-duplex mode, where X∈Rm,v,PTn,PR, v=0,1,⋯,Lm+1.Figure 1
System model of multihop DF/AF relaying protocols in the UCR networks.Let us denotehX,Y as the Rayleigh fading channel coefficient between X and Y, where X,Y∈Rm,v,PTn,PR. Hence, channel gain, i.e., φX,Y=hX,Y2, is an exponential random variable (RV) whose distributions are given as
(1)FφX,Yx=1−exp−λX,Yx,fφX,Yx=λX,Yexp−λX,Yx,where FU. and fU. are cumulative distribution function (CDF) and probability density function (PDF) of a RV U, respectively, λX,Y=1/εφX,Y=dX,YPL [34–36], ε∙ is an expected operator, dX,Y is distance between X and Y, and PL2≤PL≤6 is a path-loss exponent.


### 2.1. Hardware Impairments

Considering the data transmission between a transmitterX and a receiver Y, under the impact of HI (assume that there is no CCI), the signal received by Y is written as
(2)yY=PXhX,YsX+ηX,Y+nY,where PX is transmit power of X, sX is signal of X, ηX,Y is the hardware noise caused by the hardware imperfection at X and Y, and nY is the additive white Gaussian noise (AWGN) at Y with zero-mean. Similar to [49, 50], ηX,Y can be modeled as a zero-mean Gaussian RV whose variance is κX,Y2,(κX,Y2 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.,σ02. It is assumed that the transceiver hardware of the primary terminals is perfect, i.e., κPTn,PR2=0∀n. Next, we denote κss2, κsp2, and κps2 as the total hardware impairment levels of the Rm,v⟶Rm,v+1, Rm,v⟶PR, and PTn⟶Rm,v links, respectively, for all m,n,v.


### 2.2. Operation Principle

In this subsection, assume that themth path is selected, and hence, the data transmission is split into Lm+1 orthogonal time slots. At the first time slot, the source (Rm,0) sends its data to Rm,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(3)yPR1∑n=1NβnPPThPTn,PRsPTn1+PRm,0hRm,0,PRsS+ηRm,0,PR+nPR1.In (3), sS is the signal of the source, sPTn1 is the signal of PTn in the first time slot, PRm,0 is the transmit power of the source, and PPT 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), βnPPT is the transmit power of PTn, where β1=1,β1>β2>.⋯>βN>0. With this power allocation method, PR will decode the signal sPTn1, next uses SIC to remove the component βnPPThPTn,PRsPTn1 from yPR1, and then decodes sPTn+11. Indeed, if the SIC operation is perfect, the signal used for decoding sPTn+11 can be rewritten as
(4)zPR,n+11=∑k=n+1NβkPPThPTk,PRsPTk1+PRm,0hRm,0,PRsS+ηRm,0,PR+nPR1,where we note that zPR,11≡yPR1.For the last signalsPTN1, Equation (4) reduces to
(5)zPR,N1=βNPPThPTN,PRsPTN1+PRm,0hRm,0,PRsS+ηRm,0,PR+nPR1.Combining (3), (4), and (5), the signal-to-interference-plus-noise (SINR) obtained by PR for decoding sPTn1 is obtained as
(6)γPR,sPTn1=βnPPTφPTn,PR∑u=n+1NβuPPTφPTu,PR+1+κsp2PRm,0φRm,0,PR+σ02,n<NβNPPTφPTN,PR1+κsp2PRm,0φRm,0,PR+σ02,n=N.(2)
Secondary network: considering the secondary network at the first time slot, the received signal atRm,1 can be expressed as(7)yRm,1=PRm,0hRm,0,Rm,1sS+ηRm,0,Rm,1+∑n=1NβnPPThPTn,Rm,1sPTn1+ηPTn,Rm,1+nRm,1.From (7), the SINR received by Rm,1 is calculated as
(8)γRm,0,Rm,1=PRm,0φRm,0,Rm,1κss2PRm,0φRm,0,Rm,1+1+κps2∑n=1NβnPPTφPTn,Rm,1+σ02.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,Rm,1 decodes sS, encodes again, and then forwards it to Rm,2 in the second time slot. Generally, at the tth time slot, Rm,t−1 sends sS to Rm,t, and the received signal at PR and Rm,t can be formulated, respectively, as(9)yPR,DFt=∑n=1NβnPPThPTn,PRsPTnt+PRm,t−1hRm,t−1,PRsS+ηRm,t−1,PR+nPRt,(10)yRm,tDF=PRm,t−1hRm,t−1,Rm,tsS+ηRm,t−1,Rm,t+∑n=1NβnPPThPTn,Rm,tsPTnt+ηPTn,Rm,t+nRm,t,where t=1,2,⋯,Lm+1,sPTnt is the signal of PTn in this time slot and PRm,t−1 is the transmit power of Rm,t−1.Similar to the first time slot, PR uses SIC to decode all the data. Then, the SINR obtained at PR, with respect tosPTnt, is shown as
(11)γsPTnt=βnPPTφPTn,PR∑u=n+1NβuPPTφPTu,PR+1+κsp2PRm,t−1φRm,t−1,PR+σ02,n<NβNPPTφPTN,PR1+κsp2PRt−1φRm,t−1,PR+σ02,n=N.Using (10), the SINR at Rm,t can be given as follows:
(12)γRm,t−1,Rm,t=PRm,t−1φRm,t−1,Rm,tκss2PRm,t−1φRm,t−1,Rm,t+1+κps2∑n=1NβnPPTφPTn,Rm,t+σ02.Due to usage of the DF relaying method, the e2e channel capacity of themth path can be expressed as
(13)Cme2e,DF=1Lm+1log21+mint=1,2,⋯,Lm+1γRm,t−1,Rm,t,where the factor 1/Lm+1 implies that the data transmission of the secondary network is split into Lm+1 time slots.
(4)
M-AF-P: in M-AF-P,Rm,1 amplifies the received signal yRm,1, i.e., zRm,1=GRm,1yRm,1, and sends zRm,1 to Rm,2 in the second time slot. Because PRm,1=GRm,12εyRm,12, using (7), the amplify factor GRm,1 can be determined as(14)GRm,1=PRm,11+κss2PRm,0φRm,0,Rm,1+1+κps2∑n=1NβnPPTφPTn,Rm,1+σ02.Next, the received signal at PR andRm,2 can be expressed, respectively, as
(15)yPR,AF2=∑n=1NβnPPThPTn,PRsPTn2+hRm,1,PRzRm,1+ηRm,1,PR+nPR2,(16)yRm,2,AF=hRm,1,Rm,2zRm,1+ηRm,1,Rm,2+∑n=1NβnPPThPTn,Rm,2sPTn2+ηPTn,Rm,2+nRm,2.Similar to [49], variances of ηRm,1,PR and ηRm,1,Rm,2 can be calculated, respectively, as
(17)VarηRm,1,PR=κsp2GRm,12εyRm,12=κsp2PRm,1,(18)VarηRm,1,Rm,2=κss2GRm,12εyRm,12=κss2PRm,1.Combining (15) and (17), the SINR obtained at PR, with respect to sPTn2, is given as
(19)γsPTn2=βnPPTφPTn,PR∑u=n+1NβuPPTφPTu,PR+1+κsp2PRm,1φRm,1,PR+σ02,n<NβNPPTφPTN,PR1+κsp2PRm,1φRm,1,PR+σ02,n=N.Next, combining (7), (16), and (18), we can formulate the SINR received at Rm,2 by
(20)ΨRm,2=γRm,0,Rm,1γRm,1,Rm,2γRm,0,Rm,1+γRm,1,Rm,2+1,,where it is worth noting that γRm,0,Rm,1 and γRm,1,Rm,2 are given as in (12).Similarly,Rm,2 amplifies yRm,2,AF and then forwards the amplified signal to Rm,3 in the third time slot. Using an inductive method, we can obtain an exact formula of the SINR obtained at Rm,t at the tth time slot as follows:
(21)ΨRm,t=∏k=1tγRm,k−1,Rm,k∏k=1t1+γRm,k−1,Rm,k−∏k=1tγRm,k−1,Rm,k,where 1<t≤Lm+1. Then, the e2e channel capacity of the mth path can be computed as
(22)Cme2e,AF=1Lm+1log21+ΨRm,Lm+1=1Lm+1log21+∏k=1Lm+1γRm,k−1,Rm,k∏k=1Lm+11+γRm,k−1,Rm,k−∏k=1Lm+1γRm,k−1,Rm,k.Remark 1.
In both M-DF-P and M-AF-P, the SINRs obtained at PR at thetth 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
(23)CsPTnt=1Lm+1log21+γsPTnt.


### 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:(24)Cbe2e,Z=maxm=1,2,⋯,MCme2e,Z,where bb∈1,2,⋯,M denotes the best path, and Z∈DF,AF.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.


## 2.1. Hardware Impairments

Considering the data transmission between a transmitterX and a receiver Y, under the impact of HI (assume that there is no CCI), the signal received by Y is written as
(2)yY=PXhX,YsX+ηX,Y+nY,where PX is transmit power of X, sX is signal of X, ηX,Y is the hardware noise caused by the hardware imperfection at X and Y, and nY is the additive white Gaussian noise (AWGN) at Y with zero-mean. Similar to [49, 50], ηX,Y can be modeled as a zero-mean Gaussian RV whose variance is κX,Y2,(κX,Y2 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.,σ02. It is assumed that the transceiver hardware of the primary terminals is perfect, i.e., κPTn,PR2=0∀n. Next, we denote κss2, κsp2, and κps2 as the total hardware impairment levels of the Rm,v⟶Rm,v+1, Rm,v⟶PR, and PTn⟶Rm,v links, respectively, for all m,n,v.


## 2.1.1. Assumption and Notation

Assume that variance of AWGN at all of the receivers is same, i.e.,σ02. It is assumed that the transceiver hardware of the primary terminals is perfect, i.e., κPTn,PR2=0∀n. Next, we denote κss2, κsp2, and κps2 as the total hardware impairment levels of the Rm,v⟶Rm,v+1, Rm,v⟶PR, and PTn⟶Rm,v links, respectively, for all m,n,v.


## 2.2. Operation Principle

In this subsection, assume that themth path is selected, and hence, the data transmission is split into Lm+1 orthogonal time slots. At the first time slot, the source (Rm,0) sends its data to Rm,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(3)yPR1∑n=1NβnPPThPTn,PRsPTn1+PRm,0hRm,0,PRsS+ηRm,0,PR+nPR1.In (3), sS is the signal of the source, sPTn1 is the signal of PTn in the first time slot, PRm,0 is the transmit power of the source, and PPT 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), βnPPT is the transmit power of PTn, where β1=1,β1>β2>.⋯>βN>0. With this power allocation method, PR will decode the signal sPTn1, next uses SIC to remove the component βnPPThPTn,PRsPTn1 from yPR1, and then decodes sPTn+11. Indeed, if the SIC operation is perfect, the signal used for decoding sPTn+11 can be rewritten as
(4)zPR,n+11=∑k=n+1NβkPPThPTk,PRsPTk1+PRm,0hRm,0,PRsS+ηRm,0,PR+nPR1,where we note that zPR,11≡yPR1.For the last signalsPTN1, Equation (4) reduces to
(5)zPR,N1=βNPPThPTN,PRsPTN1+PRm,0hRm,0,PRsS+ηRm,0,PR+nPR1.Combining (3), (4), and (5), the signal-to-interference-plus-noise (SINR) obtained by PR for decoding sPTn1 is obtained as
(6)γPR,sPTn1=βnPPTφPTn,PR∑u=n+1NβuPPTφPTu,PR+1+κsp2PRm,0φRm,0,PR+σ02,n<NβNPPTφPTN,PR1+κsp2PRm,0φRm,0,PR+σ02,n=N.(2)
Secondary network: considering the secondary network at the first time slot, the received signal atRm,1 can be expressed as(7)yRm,1=PRm,0hRm,0,Rm,1sS+ηRm,0,Rm,1+∑n=1NβnPPThPTn,Rm,1sPTn1+ηPTn,Rm,1+nRm,1.From (7), the SINR received by Rm,1 is calculated as
(8)γRm,0,Rm,1=PRm,0φRm,0,Rm,1κss2PRm,0φRm,0,Rm,1+1+κps2∑n=1NβnPPTφPTn,Rm,1+σ02.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,Rm,1 decodes sS, encodes again, and then forwards it to Rm,2 in the second time slot. Generally, at the tth time slot, Rm,t−1 sends sS to Rm,t, and the received signal at PR and Rm,t can be formulated, respectively, as(9)yPR,DFt=∑n=1NβnPPThPTn,PRsPTnt+PRm,t−1hRm,t−1,PRsS+ηRm,t−1,PR+nPRt,(10)yRm,tDF=PRm,t−1hRm,t−1,Rm,tsS+ηRm,t−1,Rm,t+∑n=1NβnPPThPTn,Rm,tsPTnt+ηPTn,Rm,t+nRm,t,where t=1,2,⋯,Lm+1,sPTnt is the signal of PTn in this time slot and PRm,t−1 is the transmit power of Rm,t−1.Similar to the first time slot, PR uses SIC to decode all the data. Then, the SINR obtained at PR, with respect tosPTnt, is shown as
(11)γsPTnt=βnPPTφPTn,PR∑u=n+1NβuPPTφPTu,PR+1+κsp2PRm,t−1φRm,t−1,PR+σ02,n<NβNPPTφPTN,PR1+κsp2PRt−1φRm,t−1,PR+σ02,n=N.Using (10), the SINR at Rm,t can be given as follows:
(12)γRm,t−1,Rm,t=PRm,t−1φRm,t−1,Rm,tκss2PRm,t−1φRm,t−1,Rm,t+1+κps2∑n=1NβnPPTφPTn,Rm,t+σ02.Due to usage of the DF relaying method, the e2e channel capacity of themth path can be expressed as
(13)Cme2e,DF=1Lm+1log21+mint=1,2,⋯,Lm+1γRm,t−1,Rm,t,where the factor 1/Lm+1 implies that the data transmission of the secondary network is split into Lm+1 time slots.
(4)
M-AF-P: in M-AF-P,Rm,1 amplifies the received signal yRm,1, i.e., zRm,1=GRm,1yRm,1, and sends zRm,1 to Rm,2 in the second time slot. Because PRm,1=GRm,12εyRm,12, using (7), the amplify factor GRm,1 can be determined as(14)GRm,1=PRm,11+κss2PRm,0φRm,0,Rm,1+1+κps2∑n=1NβnPPTφPTn,Rm,1+σ02.Next, the received signal at PR andRm,2 can be expressed, respectively, as
(15)yPR,AF2=∑n=1NβnPPThPTn,PRsPTn2+hRm,1,PRzRm,1+ηRm,1,PR+nPR2,(16)yRm,2,AF=hRm,1,Rm,2zRm,1+ηRm,1,Rm,2+∑n=1NβnPPThPTn,Rm,2sPTn2+ηPTn,Rm,2+nRm,2.Similar to [49], variances of ηRm,1,PR and ηRm,1,Rm,2 can be calculated, respectively, as
(17)VarηRm,1,PR=κsp2GRm,12εyRm,12=κsp2PRm,1,(18)VarηRm,1,Rm,2=κss2GRm,12εyRm,12=κss2PRm,1.Combining (15) and (17), the SINR obtained at PR, with respect to sPTn2, is given as
(19)γsPTn2=βnPPTφPTn,PR∑u=n+1NβuPPTφPTu,PR+1+κsp2PRm,1φRm,1,PR+σ02,n<NβNPPTφPTN,PR1+κsp2PRm,1φRm,1,PR+σ02,n=N.Next, combining (7), (16), and (18), we can formulate the SINR received at Rm,2 by
(20)ΨRm,2=γRm,0,Rm,1γRm,1,Rm,2γRm,0,Rm,1+γRm,1,Rm,2+1,,where it is worth noting that γRm,0,Rm,1 and γRm,1,Rm,2 are given as in (12).Similarly,Rm,2 amplifies yRm,2,AF and then forwards the amplified signal to Rm,3 in the third time slot. Using an inductive method, we can obtain an exact formula of the SINR obtained at Rm,t at the tth time slot as follows:
(21)ΨRm,t=∏k=1tγRm,k−1,Rm,k∏k=1t1+γRm,k−1,Rm,k−∏k=1tγRm,k−1,Rm,k,where 1<t≤Lm+1. Then, the e2e channel capacity of the mth path can be computed as
(22)Cme2e,AF=1Lm+1log21+ΨRm,Lm+1=1Lm+1log21+∏k=1Lm+1γRm,k−1,Rm,k∏k=1Lm+11+γRm,k−1,Rm,k−∏k=1Lm+1γRm,k−1,Rm,k.Remark 1.
In both M-DF-P and M-AF-P, the SINRs obtained at PR at thetth 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
(23)CsPTnt=1Lm+1log21+γsPTnt.


## 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:(24)Cbe2e,Z=maxm=1,2,⋯,MCme2e,Z,where bb∈1,2,⋯,M denotes the best path, and Z∈DF,AF.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.


## 3. 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 themth path is selected, let us consider the minimum channel capacity between PTn and PR at the tth time slot:
(25)CPR,mint=minn=1,2,⋯,NCsPTnt=1Lm+1log21+minn=1,2,⋯,NγsPTnt.We now define OP of the primary network as the probability thatCPR,mint is lower than a positive threshold, i.e., CP,th, and is expressed as
(26)OPPRt=PrCPR,mint<CP,th=Prminn=1,2,⋯,NγsPTnt<θP,m,where
(27)θP,m=2Lm+1CP,th−1.Equation (26) also implies that the primary network is in outage if one of the signals sPTnt cannot be correctly decoded. In particular, employing the independence of RVs γsPTnt, Equation (26) can be rewritten as
(28)OPPRt=1−∏n=1N1−PrγsPTnt<θP,m,where PrγsPTnt<θP,m is the probability that the decoding of sPTnt fails.Combining (11) and (28) yields (29) as
(29)PrγsPTnt<θP,m=∫0+∞⋯∫0+∞FφPTn,PR∑u=n+1Nau,nxu+bnτm,t−1y+cn×fφRm,t−1,PRydy∏u=n+1NfφPTu,PRxudxu,where
(30)au,n=βuθP,m/βn,n<N0,n=N,bn=1+κsp2θP,mβn,cn=σ02θP,mβnPPT,τm,t−1=PRm,t−1PPT.Substituting CDF and PDF given by (1) into (29), after some manipulation, we obtain
(31)PrγsPTnt<θP,m=1−gu,nln+τm,t−1.where
(32)gu,n=∏u=n+1NλPTu,PRλPTu,PR+au,nλPTn,PRλRm,t−1,PRbnλPTn,PRexp−λPTn,PRcn,ln=λRm,t−1,PRbnλPTn,PR.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:
(33)OPPRt=1−∏n=1Ngu,nln+τm,t−1.


### 3.2. Transmit Power of Secondary Transmitters

At first, QoS of the primary network in thetth time slot is given as
(34)OPPRt≤εOP,where εOP 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 ofRm,t−1 can be obtained by solving equationOPPRt=εOP. More particularly, we have to find a positive real solution (denoted by τm,t−1∗) of the following equation:
(35)∏n=1Nτm,t−1+ln=11−εOP∏n=1Ngu,n.Then, the transmit power ofRm,t−1 is obtained as PRm,t−1=PPTτm,t−1∗. In case that Equation (35) has no positive real solution, we have PRm,t−1=0, which means Rm,t−1 is not allowed to access the licensed band.For example, whenN=2, τm,t−1∗ is a solution of (36):
(36)τm,t−12+l1+l2τm,t−1+pu=0,where pu=l1l2−gu,1gu,2/1−εOP. Moreover, if Equation (36) has a positive real solution, we have
(37)τm,t−1∗=−1l1+l2+l1+l22−4pu2..Proposition 3.
Equation (35) cannot have two distinct positive real solutions.Proof.
Firstly, we consider the following function:(38)fτm,t−1=∏n=1Nτm,t−1_ln−11−εOP∏n=1Ngu,n.
Assume that there exist at least two distinct positive real solutions, e.g.,τ1 and τ2, 0<τ1<τ2; we then have
(39)fτ1=fτ2=0.
Differentiatingfτm,t−1 with respect to τm,t−1 yields
(40)∂fτm,t−1∂τm,t−1=∑n=1N∏v=1,v≠nNτm,t−1_lv.
Since∂fτm,t−1/∂τm,t−1>0 when τm,t−1>0, fτm,t−1 increases in interval 0,+∞. This means that fτ1<fτ2, and hence, we cannot obtain (39).Corollary 4.
Assume that Equation (35) has one positive real solution; the transmit power PRm,t−1 is lower with higher QoS of the primary network (or lower value of εOP).Proof.
Let us denoteτ1 as the positive real solution of (35), we have ∏n=1Nτ1+ln=1/1−εOP∏n=1Ngu,n. Therefore, it is straightforward that for the fixed values of ln and gu,n, if εOP decreases, then τ1 (or PRm,t−1) also decreases.Proposition 5.
Equation (35) has one positive real solution when
(41)∏n=1Nln−11−εOP∏n=1Ngu,n<0.Proof.
As proved in Proposition3, fτm,t−1 is a continuous and increasing function in 0,+∞. Therefore, it is obvious that if f0<0, Equation (35) always has one real positive solution. Using (38), we obtain (41) and finish the proof.For example, withN=2, the condition in (41) can be written as
(42)l1l2<gu,1gu,21−εOP⇔pu<0.Proposition 6.
If Equation (35) has one positive real solution, at high PPT values (i.e., PPT⟶+∞), PRm,t−1 is a linear function of PPT.Proof.
AsPPT⟶+∞, we have
(43)gu,n≈PPT→+∞∏u=n+1NλPTu,PRλPTu,PR+au,nλPTn,PRλRm,t−1,PRbnλPTn,PR.⏟gu,n∗
Therefore, at highPPT values, Equation (35) becomes
(44)∏n=1Nτm,t−1+ln=11−εOP∏n=1Ngu,n∗.
Ifτm,t−1∗∗ is a positive real solution of (44), τm,t−1∗∗ does not depend on PPT. Hence, PRm,t−1 becomes a linear function of PPT, i.e., PRm,t−1≈PPT→+∞τm,t−1∗∗PPT.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 ofPRm,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.


### 3.3. Outage Probability of Secondary Network

This subsection evaluates the e2e OP of M-DF-P and M-AF-P, which can be defined as(45)OPZe2e=PrCbe2e<CS,th=∏m−1MOPm,Z,where Z∈AF,DF, CS,th is a predetermined target rate of the secondary network, and OPm,Z=PrCme2e<CS,th is the e2e OP of the mth path. In the following, we attempt to calculate OPm,Z.
(1)
M-DF-P: in this protocol, the e2e OP of themth path is calculated as follows:(1)
M-AF-P: because it is too difficult to find an exact expression ofOPm,AF, we attempt to obtain a lower-bound one.Proposition 8.
OPm,DF can be expressed by an exact closed-form expression as
(46)OPm,DF=1,if1−κss2θS,m≤01−∏t=1Lm+1∏n=1NλPTn,Rm,tλPTn,Rm,t+λRm,t−1,Rm,tρ1,nexp−λRm,t−1,Rm,tρ2,1−κss2θS,m,where
(47)θS,m=2Lm+1CS,th−1,ρ1,n=1+κps2θS,mβnPPT1−θS,mκss2PRm,t−1,ρ2=σ02θS,m1−θS,mκss2PRm,t−1.Proof.
See AppendixA.Corollary 9.
If1−θS,mκss2>0, OPm,DF at high PPT values can be approximated as
(48)OPm,DF≈PPT→+∞∏t=1Lm+1∏n=1NλPTn,PRλPTn,PR+λRm,t−1,Rm,tρ1,n∗,where
(49)ρ1,n∗=1+κps2θS,mβn1−θS,mκss2τm,t−1∗∗.Proof.
SubstitutingPRm,t−1≈PPT→+∞τm,t−1∗∗PPT and exp−λRm,t−1,Rm,tρ2≈PPT→+∞1 into (46), we obtain (48).Similar to [31], we consider two RVs: X1,m=minq=1,⋯,T1λRm,q−1,Rm,q and X2,m=minr=T1+1,⋯,Lm+1λRm,r−1,Rm,r, where T1,m=Lm+1/2 and x is the highest integer that is smaller or equal to x. Then, the e2e SINR ΨRm,Lm+1 (see (21) or (22)) can be bounded as in [31]:
(50)ΨRm,Lm+1≤X1,mX2,mX1,m+X2,m+1=Φm.Our next objective is to find the distributions ofX1,m and X2,m which are presented in Proposition 10.Proposition 10.
CDF ofX1,m and X2,m can be given, respectively, as follows:
(51)FX1,mx=1,1−κss2x≤01−exp−ρ4ϑx∑q=1T1∑n=1Nεq,nϑx+χq,n,1−κss2x>0,(52)FX2,mx=1,1−κss2x≤01−exp−ρ6ϑx∑r=T1+1Lm+1∑l=1Nεr,lϑx+ωr,l,1−κss2x>0,where
(53)ρ3,n=βn1+κps2PPTPRm,q−1,ρ4=∑q=1T1λRm,q−1,Rm,qσ02PRm,q−1,ϑx=x1−κss2x,χq,n=λPTn,Rm,qλRm,q−1,Rm,qρ3,n,εq,n=χq,n∏a=1T1∏b=1a,b≠q,nNχa,bχa,b−χq,n,(54)ρ5,l=βl1+κps2PPTPRm,r−1,ρ6=∑r=T1+1Lm+1λRm,r−1,Rm,rσ02PRm,r−1,ωr,l=λPTl,Rm,rλRm,r−1,Rm,rρ5,l,ξr,l=ωr,l∏a=T1+1Lm+1∏b=1a,b≠r,lNωa,bωa,b−ωr,l.Proof.
See AppendixB.From (52), if 1−κss2x≤0, fX2,mx=0, and if 1−κss2x>0, PDF of X2,m can be obtained as
(55)fX2,mx=∑r=T1+1Lm+1∑l=1Nξr,lϑx+ωr,l21−κss2x2exp−ρ6ϑx+∑r=T1+1Lm+1∑l=1Nξr,lρ6ϑx+ωr,l21−κss2x2exp−ρ6ϑx.Next, we derive the lower-bound expression ofOPm,AF as given in Proposition 11.Proposition 11.
If1−kSS4+2kSS2θS,m≤0, then OPm,AF=1, and if 1−kSS4+2kSS2θS,m>0,OPm,AF is bounded by
(56)OPm,AF≥1‐∫v1v21−FX1,mθS,mx+θS,mx−θS,mfX2,mxdx,where
(57)v1=1+kSS2θS,m1+kSS2θS,m,v2=1kSS2.Proof.
See AppendixC.Remark 12.
Firstly,OPm,AFLB can be numerically calculated by using MATLAB or Mathematica. Secondly, as PPT→+∞, the e2e OP of M-DF-P and M-AF-P does not depend on PPT. Indeed, γRm,t‐1,Rm,t in (12) can be approximated as
(58)ΨRm,Lm+1=1∏t=1Lm+1Wt+1+kSS2−1≤11+kSS2Lm+1−1.
We can observe from (58) that γRm,t‐1,Rm,t at high PPT values do not depend on PPT, and it is the reason why OPm,ZOPZe2e converges to a constant as PPT→+∞S. This also means that the diversity order of both M-DF-P and M-AF-P is zero.Corollary 13.
OPm,AF always equals to 1 if 1−1+kSS2Lm+1−1θS,m≤0.Proof.
At first, we can rewrite the e2e SINRΨRm,Lm+1 under the following form:
(59)ΨRm,Lm+1=1∏t=1Lm+11+1/γRm,t−1,Rm,t−1.
Combining (12) and (59), it is straightforward that
(60)ΨRm,Lm+1=1∏t=1Lm+1Wt+1+κss2−1≤11+κss2Lm+1−1,where
(61)Wt=1+κps2∑n=1NβnPPTφPTnRm,t+σ02PRm,t−1φRm,t−1,Rm,t.
Therefore, if1−1+κss2Lm+1−1θS,m≤0, then OPm,AF=1.Remark 14.
From Proposition8, we can see that if κss2≥κssDF2=maxm=1,2,⋯,M1/θS,m, all of the paths in M-DF-P are in outage, and hence, M-DF-P is always in outage, regardless of the values of the other system parameters. For the AF method, from Corollary 13, M-AF-P is always in outage if
(62)κss2≥κssAF2=maxm=1,2,⋯,M1+θS,mθS,m1/Lm+1−1,where κssDF2 and κssAF2 are called as limited hardware impairment levels of M-DF-P and M-AF-P, respectively. Because κssDF2≥κssAF2, M-AF-P is more sensitive to HI than M-DF-P.Proposition 15.
When the transceiver hardware is perfect, i.e.,κss2=κsp2=κps2=0, we obtain a lower-bound closed-form expression of OPm,AF by
(63)OPm,AF≥1−∑q=1T1∑n=1N∑r=T1+1Lm+1∑l=1Nμ1μ3−μ2+μ1μ2μ2−μ3logeμ2μ3,where
(64)χq,n∗=λPTn,Rm,q−1λRm,q−1,Rm,qβnPPT,εq,n∗=χq,n∗∏a=1T1∏b=1a,b≠q,nNχa,b∗χa,b∗−χq,n∗,ωr,l∗=λPTl,Rm,rPRm,r−1λRm,r−1,Rm,rβlPPT,εr,l∗=ωr,l∗∏a=T1+1Lm+11∏b=1a,b≠r,lNωa,b∗ωa,b∗−ωr,l∗,μ1=εq,n∗ξr,l∗θS,m+χq,n∗,μ2=θS,m2+θS,mθS,m+χq,n∗,μ3=θS,m+ωr,l∗.Proof.
See AppendixD.


## 3.1. OP of Primary Network

When themth path is selected, let us consider the minimum channel capacity between PTn and PR at the tth time slot:
(25)CPR,mint=minn=1,2,⋯,NCsPTnt=1Lm+1log21+minn=1,2,⋯,NγsPTnt.We now define OP of the primary network as the probability thatCPR,mint is lower than a positive threshold, i.e., CP,th, and is expressed as
(26)OPPRt=PrCPR,mint<CP,th=Prminn=1,2,⋯,NγsPTnt<θP,m,where
(27)θP,m=2Lm+1CP,th−1.Equation (26) also implies that the primary network is in outage if one of the signals sPTnt cannot be correctly decoded. In particular, employing the independence of RVs γsPTnt, Equation (26) can be rewritten as
(28)OPPRt=1−∏n=1N1−PrγsPTnt<θP,m,where PrγsPTnt<θP,m is the probability that the decoding of sPTnt fails.Combining (11) and (28) yields (29) as
(29)PrγsPTnt<θP,m=∫0+∞⋯∫0+∞FφPTn,PR∑u=n+1Nau,nxu+bnτm,t−1y+cn×fφRm,t−1,PRydy∏u=n+1NfφPTu,PRxudxu,where
(30)au,n=βuθP,m/βn,n<N0,n=N,bn=1+κsp2θP,mβn,cn=σ02θP,mβnPPT,τm,t−1=PRm,t−1PPT.Substituting CDF and PDF given by (1) into (29), after some manipulation, we obtain
(31)PrγsPTnt<θP,m=1−gu,nln+τm,t−1.where
(32)gu,n=∏u=n+1NλPTu,PRλPTu,PR+au,nλPTn,PRλRm,t−1,PRbnλPTn,PRexp−λPTn,PRcn,ln=λRm,t−1,PRbnλPTn,PR.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:
(33)OPPRt=1−∏n=1Ngu,nln+τm,t−1.


## 3.2. Transmit Power of Secondary Transmitters

At first, QoS of the primary network in thetth time slot is given as
(34)OPPRt≤εOP,where εOP 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 ofRm,t−1 can be obtained by solving equationOPPRt=εOP. More particularly, we have to find a positive real solution (denoted by τm,t−1∗) of the following equation:
(35)∏n=1Nτm,t−1+ln=11−εOP∏n=1Ngu,n.Then, the transmit power ofRm,t−1 is obtained as PRm,t−1=PPTτm,t−1∗. In case that Equation (35) has no positive real solution, we have PRm,t−1=0, which means Rm,t−1 is not allowed to access the licensed band.For example, whenN=2, τm,t−1∗ is a solution of (36):
(36)τm,t−12+l1+l2τm,t−1+pu=0,where pu=l1l2−gu,1gu,2/1−εOP. Moreover, if Equation (36) has a positive real solution, we have
(37)τm,t−1∗=−1l1+l2+l1+l22−4pu2..Proposition 3.
Equation (35) cannot have two distinct positive real solutions.Proof.
Firstly, we consider the following function:(38)fτm,t−1=∏n=1Nτm,t−1_ln−11−εOP∏n=1Ngu,n.
Assume that there exist at least two distinct positive real solutions, e.g.,τ1 and τ2, 0<τ1<τ2; we then have
(39)fτ1=fτ2=0.
Differentiatingfτm,t−1 with respect to τm,t−1 yields
(40)∂fτm,t−1∂τm,t−1=∑n=1N∏v=1,v≠nNτm,t−1_lv.
Since∂fτm,t−1/∂τm,t−1>0 when τm,t−1>0, fτm,t−1 increases in interval 0,+∞. This means that fτ1<fτ2, and hence, we cannot obtain (39).Corollary 4.
Assume that Equation (35) has one positive real solution; the transmit power PRm,t−1 is lower with higher QoS of the primary network (or lower value of εOP).Proof.
Let us denoteτ1 as the positive real solution of (35), we have ∏n=1Nτ1+ln=1/1−εOP∏n=1Ngu,n. Therefore, it is straightforward that for the fixed values of ln and gu,n, if εOP decreases, then τ1 (or PRm,t−1) also decreases.Proposition 5.
Equation (35) has one positive real solution when
(41)∏n=1Nln−11−εOP∏n=1Ngu,n<0.Proof.
As proved in Proposition3, fτm,t−1 is a continuous and increasing function in 0,+∞. Therefore, it is obvious that if f0<0, Equation (35) always has one real positive solution. Using (38), we obtain (41) and finish the proof.For example, withN=2, the condition in (41) can be written as
(42)l1l2<gu,1gu,21−εOP⇔pu<0.Proposition 6.
If Equation (35) has one positive real solution, at high PPT values (i.e., PPT⟶+∞), PRm,t−1 is a linear function of PPT.Proof.
AsPPT⟶+∞, we have
(43)gu,n≈PPT→+∞∏u=n+1NλPTu,PRλPTu,PR+au,nλPTn,PRλRm,t−1,PRbnλPTn,PR.⏟gu,n∗
Therefore, at highPPT values, Equation (35) becomes
(44)∏n=1Nτm,t−1+ln=11−εOP∏n=1Ngu,n∗.
Ifτm,t−1∗∗ is a positive real solution of (44), τm,t−1∗∗ does not depend on PPT. Hence, PRm,t−1 becomes a linear function of PPT, i.e., PRm,t−1≈PPT→+∞τm,t−1∗∗PPT.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 ofPRm,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.


## 3.3. Outage Probability of Secondary Network

This subsection evaluates the e2e OP of M-DF-P and M-AF-P, which can be defined as(45)OPZe2e=PrCbe2e<CS,th=∏m−1MOPm,Z,where Z∈AF,DF, CS,th is a predetermined target rate of the secondary network, and OPm,Z=PrCme2e<CS,th is the e2e OP of the mth path. In the following, we attempt to calculate OPm,Z.
(1)
M-DF-P: in this protocol, the e2e OP of themth path is calculated as follows:(1)
M-AF-P: because it is too difficult to find an exact expression ofOPm,AF, we attempt to obtain a lower-bound one.Proposition 8.
OPm,DF can be expressed by an exact closed-form expression as
(46)OPm,DF=1,if1−κss2θS,m≤01−∏t=1Lm+1∏n=1NλPTn,Rm,tλPTn,Rm,t+λRm,t−1,Rm,tρ1,nexp−λRm,t−1,Rm,tρ2,1−κss2θS,m,where
(47)θS,m=2Lm+1CS,th−1,ρ1,n=1+κps2θS,mβnPPT1−θS,mκss2PRm,t−1,ρ2=σ02θS,m1−θS,mκss2PRm,t−1.Proof.
See AppendixA.Corollary 9.
If1−θS,mκss2>0, OPm,DF at high PPT values can be approximated as
(48)OPm,DF≈PPT→+∞∏t=1Lm+1∏n=1NλPTn,PRλPTn,PR+λRm,t−1,Rm,tρ1,n∗,where
(49)ρ1,n∗=1+κps2θS,mβn1−θS,mκss2τm,t−1∗∗.Proof.
SubstitutingPRm,t−1≈PPT→+∞τm,t−1∗∗PPT and exp−λRm,t−1,Rm,tρ2≈PPT→+∞1 into (46), we obtain (48).Similar to [31], we consider two RVs: X1,m=minq=1,⋯,T1λRm,q−1,Rm,q and X2,m=minr=T1+1,⋯,Lm+1λRm,r−1,Rm,r, where T1,m=Lm+1/2 and x is the highest integer that is smaller or equal to x. Then, the e2e SINR ΨRm,Lm+1 (see (21) or (22)) can be bounded as in [31]:
(50)ΨRm,Lm+1≤X1,mX2,mX1,m+X2,m+1=Φm.Our next objective is to find the distributions ofX1,m and X2,m which are presented in Proposition 10.Proposition 10.
CDF ofX1,m and X2,m can be given, respectively, as follows:
(51)FX1,mx=1,1−κss2x≤01−exp−ρ4ϑx∑q=1T1∑n=1Nεq,nϑx+χq,n,1−κss2x>0,(52)FX2,mx=1,1−κss2x≤01−exp−ρ6ϑx∑r=T1+1Lm+1∑l=1Nεr,lϑx+ωr,l,1−κss2x>0,where
(53)ρ3,n=βn1+κps2PPTPRm,q−1,ρ4=∑q=1T1λRm,q−1,Rm,qσ02PRm,q−1,ϑx=x1−κss2x,χq,n=λPTn,Rm,qλRm,q−1,Rm,qρ3,n,εq,n=χq,n∏a=1T1∏b=1a,b≠q,nNχa,bχa,b−χq,n,(54)ρ5,l=βl1+κps2PPTPRm,r−1,ρ6=∑r=T1+1Lm+1λRm,r−1,Rm,rσ02PRm,r−1,ωr,l=λPTl,Rm,rλRm,r−1,Rm,rρ5,l,ξr,l=ωr,l∏a=T1+1Lm+1∏b=1a,b≠r,lNωa,bωa,b−ωr,l.Proof.
See AppendixB.From (52), if 1−κss2x≤0, fX2,mx=0, and if 1−κss2x>0, PDF of X2,m can be obtained as
(55)fX2,mx=∑r=T1+1Lm+1∑l=1Nξr,lϑx+ωr,l21−κss2x2exp−ρ6ϑx+∑r=T1+1Lm+1∑l=1Nξr,lρ6ϑx+ωr,l21−κss2x2exp−ρ6ϑx.Next, we derive the lower-bound expression ofOPm,AF as given in Proposition 11.Proposition 11.
If1−kSS4+2kSS2θS,m≤0, then OPm,AF=1, and if 1−kSS4+2kSS2θS,m>0,OPm,AF is bounded by
(56)OPm,AF≥1‐∫v1v21−FX1,mθS,mx+θS,mx−θS,mfX2,mxdx,where
(57)v1=1+kSS2θS,m1+kSS2θS,m,v2=1kSS2.Proof.
See AppendixC.Remark 12.
Firstly,OPm,AFLB can be numerically calculated by using MATLAB or Mathematica. Secondly, as PPT→+∞, the e2e OP of M-DF-P and M-AF-P does not depend on PPT. Indeed, γRm,t‐1,Rm,t in (12) can be approximated as
(58)ΨRm,Lm+1=1∏t=1Lm+1Wt+1+kSS2−1≤11+kSS2Lm+1−1.
We can observe from (58) that γRm,t‐1,Rm,t at high PPT values do not depend on PPT, and it is the reason why OPm,ZOPZe2e converges to a constant as PPT→+∞S. This also means that the diversity order of both M-DF-P and M-AF-P is zero.Corollary 13.
OPm,AF always equals to 1 if 1−1+kSS2Lm+1−1θS,m≤0.Proof.
At first, we can rewrite the e2e SINRΨRm,Lm+1 under the following form:
(59)ΨRm,Lm+1=1∏t=1Lm+11+1/γRm,t−1,Rm,t−1.
Combining (12) and (59), it is straightforward that
(60)ΨRm,Lm+1=1∏t=1Lm+1Wt+1+κss2−1≤11+κss2Lm+1−1,where
(61)Wt=1+κps2∑n=1NβnPPTφPTnRm,t+σ02PRm,t−1φRm,t−1,Rm,t.
Therefore, if1−1+κss2Lm+1−1θS,m≤0, then OPm,AF=1.Remark 14.
From Proposition8, we can see that if κss2≥κssDF2=maxm=1,2,⋯,M1/θS,m, all of the paths in M-DF-P are in outage, and hence, M-DF-P is always in outage, regardless of the values of the other system parameters. For the AF method, from Corollary 13, M-AF-P is always in outage if
(62)κss2≥κssAF2=maxm=1,2,⋯,M1+θS,mθS,m1/Lm+1−1,where κssDF2 and κssAF2 are called as limited hardware impairment levels of M-DF-P and M-AF-P, respectively. Because κssDF2≥κssAF2, M-AF-P is more sensitive to HI than M-DF-P.Proposition 15.
When the transceiver hardware is perfect, i.e.,κss2=κsp2=κps2=0, we obtain a lower-bound closed-form expression of OPm,AF by
(63)OPm,AF≥1−∑q=1T1∑n=1N∑r=T1+1Lm+1∑l=1Nμ1μ3−μ2+μ1μ2μ2−μ3logeμ2μ3,where
(64)χq,n∗=λPTn,Rm,q−1λRm,q−1,Rm,qβnPPT,εq,n∗=χq,n∗∏a=1T1∏b=1a,b≠q,nNχa,b∗χa,b∗−χq,n∗,ωr,l∗=λPTl,Rm,rPRm,r−1λRm,r−1,Rm,rβlPPT,εr,l∗=ωr,l∗∏a=T1+1Lm+11∏b=1a,b≠r,lNωa,b∗ωa,b∗−ωr,l∗,μ1=εq,n∗ξr,l∗θS,m+χq,n∗,μ2=θS,m2+θS,mθS,m+χq,n∗,μ3=θS,m+ωr,l∗.Proof.
See AppendixD.


## 4. 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 arePT10.35,0.5 and PT20.6,0.5, while PR is located at (0.45, 0.5). As mentioned in “System Model,” the transmit power of PT1 is higher than that of PT2; hence, we can set β1=1 and β2=0.2. In the secondary network, assume that there are 03 available paths (M=3), and the number of relays on the paths is L1=1, L2=2, and L3=3. In addition, the source and destination nodes are located at (0,0) and (0,1), respectively, and the relay node Rm,t is placed at t/Lm+1,0, where m=1,2,3 and t=1,⋯,Lm. 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 pathsM and the number of the primary transmitters N 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 εOP, and vice verse.Table 2
Values of the system parameters.
System parametersValuesσ021PL3CP,th0.2εOP0.05In Figure2, we assume that the third path is selected and presents the transmit power of R3,0 (Source), R3,1, R3,2, and R3,3 as a function of PPT in dB. As seen from Figure 2, the transmit power of all the secondary transmitters is higher than zero when PPT 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 PPT. Also in this figure, the transmit power of R3,0 is highest, and that of R3,2 is lowest. It is due to the fact that the distance between R3,0 and PR is furthest, and R3,2 is nearest to PR.Figure 2
Transmit power of the secondary transmitters on the third pathL3=3 as a function of PPT in dB when κsp2=0.01.Figure3 compares the performance of M-DF-P and M-AF-P when κ2ss=0.05 and κ2sp=κ2ps=0.01. We can see that the performance of M-DF-P is better than that of M-AF-P for all values of PPT. As mentioned in Remark 12, the e2e OP of M-DF-P and M-AF-P in Figure 3 converge to a constant when PPT is high enough. In addition, the performance of the proposed protocols is better as CS,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 PPT 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.Figure 3
End-to-end outage probability of M-DF-P and M-AF-P as a function ofPPT in dB when κss2=0.05, κsp2=κps2=0.01.In Figure4, the e2e OP of M-DF-P and M-AF-P presents as a function of PPT in dB when CS,th=0.03. In this figure, we consider two cases: the transceiver hardware is perfect κss2=κsp2=κps2=0 and imperfect κss2=κsp2=κps2=0.04.Figure 4
End-to-end outage probability of M-DF-P and M-AF-P as a function ofPPT in dB when CS,th=0.03.As expected, the outage performance of the proposed protocols with perfect transceiver hardware is better. Similar to Figure3, 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).Figure5 compares the performance of M-DF-P and M-AF-P with different values of CS,th. In this figure, the perfect transceiver hardware is assumed, and the transmit power PPT is fixed by 15 dB. We can see from Figure 5 that the e2e OP values of the proposed protocols increase as CS,th increases. Moreover, the performance of M-DF-P is better than that of M-AF-P, and the performance gap is higher as CS,th increases.Figure 5
End-to-end outage probability of M-DF-P and M-AF-P as a function ofCS,th when PPT=15dB, κss2=κsp2=κps2=0.In Figure6, we present the outage performance of M-DF-P as a function of κ2ss when PPT=30dB, and κsp2=κps2=κss2/5.Figure 6
End-to-end outage probability of M-DF-P as a function ofκss2 when PPT=30dB, κsp2=κps2=κss2/5.Figure6 illustrates that the e2e OP rapidly increases as κss2 increases. We also see that when the HI level is high enough, M-DF-P is always in outage. As given in Remark 14, the limited hardware impairment levels of M-DF-P in this figure can be obtained as
(65)κssDF2=13.93,CS,th=0.056.72,CS,th=0.1.Figure7 depicts the performance of M-AF-P as a function of κ2ss when PPT=30dB, and κsp2=κps2=κss2/5. Similar to M-DF-P in Figure 6, the performance of M-AF-P is worse when κ2ss increases. Also, M-AF-P is always in outage if κss2≥κssAF, where κssAF2 is calculated by using (62):
(66)κssAF2=2.86,CS,th=0.051.77,CS,th=0.1.Figure 7
End-to-end outage probability of M-AF-P as a function ofκss2 when PPT=30dB,κsp2=κps2=κss2/5.From Figures3–7, we can see that M-DF-P not only outperforms M-AF-P but also obtains much higher value of the limited HI level.


## 5. 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 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).



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