# Cache Aided Decode-and-Forward Relaying Networks: From the Spatial View

Authors: Junjuan Xia; Fasheng Zhou; Xiazhi Lai; Haochuan Zhang; Hongbin Chen; Qinghai Yang; Xin Liu; Junhui Zhao
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

We investigate cache technique from the spatial view and study its impact on the relaying networks. In particular, we consider a dual-hop relaying network, where decode-and-forward (DF) relays can assist the data transmission from the source to the destination. In addition to the traditional dual-hop relaying, we also consider the cache from the spatial view, where the source can prestore the data among the memories of the nodes around the destination. For the DF relaying networks without and with cache, we study the system performance by deriving the analytical expressions of outage probability and symbol error rate (SER). We also derive the asymptotic outage probability and SER in the high regime of transmit power, from which we find the system diversity order can be rapidly increased by using cache and the system performance can be significantly improved. Simulation and numerical results are demonstrated to verify the proposed studies and find that the system power resources can be efficiently saved by using cache technique.


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## Body

## 1. Introduction

With the rapid progress in wireless big data, a lot of wireless techniques have been proposed to enhance the transmission quality and reliability [1–10]. Cooperative relaying, such as amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols, is an effective means to enhance network reliability as well as security [11, 12], extend coverage region [13, 14], and reduce the impact of interference [15–17]. Therefore, cooperative relaying is a significant technique in the current and future generations wireless networks, and the research on relaying networks has been active. The outage performance of multirelay networks was studied in [18, 19], and the exact outage probabilities were derived. In [20], the authors turned to investigate the performance of relaying networks in terms of symbol error rate (SER). In [21, 22], the performance of relaying networks in the high signal-to-noise ratio (SNR) regime was analyzed and the diversity order was revealed.To improve the transmission performance relaying networks, cache technique has been proposed to enhance the user experience quality and reduce the latency. In particular, the authors in [23] studied the fundamental limits of caching and found an information-theoretic formulation of the caching problem by focusing on its basic structure. Then the authors in [24] applied the coding technique into cache and studied the fundamental limits by proposing an improved delivery rate-cache capacity tradeoff. Further, the authors in [25] investigated content-centric sparse multicast beamforming for cache-enabled cloud RAN and revealed that caching strategy had a significant impact on the system performance. The application of cache into relaying networks has been studied in [26], where optimization of hybrid cache placement for collaborative relaying was investigated. In this paper, the signal transmission cooperation gain and content delivery diversity were achieved to enhance the system transmission performance. There are some other works on the application of cache in network security to ensure the secure information transmission [27–30]. However, to the best of the authors’ knowledge, there has been little work on the spatial view of cache on performance improvement of the relaying networks.In this work, we study the spatial view of cache on the DF relaying networks, where the source communicates with the destination with several DF relays. If the cache is used, the source can prestore the data among the nodes around the destination. In this way, the destination can directly access the data from the nearby nodes, instead of having to communicate with the source. For the traditional dual-hop relaying and the DF relaying assisted by the cache, we study the system transmission performance by deriving the analytical expression of outage probability and SER. We also provide the asymptotic expressions of outage probability and SER in the high region of transmit power, from which we find that the cache can efficiently improve the system performance by increasing the transmission diversity order. Numerical and simulation results are demonstrated to verify the proposed studies in this paper.Notations 1.
We useCN(0,σ2) to represent a circularly symmetric complex Gaussian random variable (RV) with zero mean and variance σ2. We use fX(·) and FX(·) to represent the probability density function (PDF) and cumulative density function (CDF) of the random variable (RV) X, respectively. In addition, the notation X~Naka(m,α) represents that the random variable X is subject to Nakagami-m distribution, where the PDF of variable of Y=|X|2 is fY(y)=mmym-1/αmΓ(m)e-my/α. We use Pr[·] to denote probability.


## 2. System Model

Figure1(a) shows the traditional dual-hop relaying networks without cache, where a single decode-and-forward relay node R assists the two-phase data transmission from the source S to the destination D. In contrast, Figure 1(b) demonstrates the cache aided dual-hop relaying networks, where the data can be prestored at N nodes {Rn∣1≤n≤N} (the relays can obtain the data from the source during nonpeak time, and these relays can work in either the cooperative or noncooperative ways [31]) around the destination D. Then the data can be directly transmitted from the relays to the destination, instead of through the traditional dual-hop relaying links. Due to the size limitation, each node has a single antenna only [32–35], and we assume that each node operates in a time-division half-duplex mode [36–39]. In the following, we describe the data transmission for the relaying networks without and with cache, respectively.Figure 1
Decode-and-forward relaying networks.
(a)
Without cache (b)
With cacheFor the traditional data transmission without cache as shown in Figure1(a), let hSR~Naka(m1,α) and hRD~Naka(m2,β) denote the channel parameter of the S-R and R-D links, respectively. In the first phase, the source S sends the normalized signal s to the relay, and then the relay receives(1)yR=PhSRs+nR, where P is the transmit power of the source and nR~CN(0,σ2) is the additive white noise at the relay. If the relay R can correctly decode the signal from the source, it will forward the signal to the destination in the second phase, and accordingly the destination D receives(2)yD=PhRDs+nD, where nD~CN(0,σ2) is the additive white noise at the destination. From (1)-(2), we can obtain the received end-to-end SNR at the destination D as(3)SNRD=Pσ2min⁡u,v, where u=hSR2 and v=hRD2 denote the channel gains of the first and second hops, respectively.For the cache aided DF relaying networks demonstrated in Figure1(b), the source data can be firstly prestored at the N relay nodes around the destination, and then the data transmission can be directly performed from the relays to the destination, instead of through the traditional two-hop relaying links. Specifically, when the nth relay Rn is used for data transmission, the received signal at the destination D is given by(4)yD=PhRn,Ds+nD, where hRn,D~Naka(m2,β) is the channel parameter of the second-hop relaying links associated with the nth relay. Accordingly, the received end-to-end SNR at the destination D is given by(5)SNRD=Pσ2vn, where vn=hRn,D2 is the instantaneous channel gain of the Rn-D link. Among these N relays, one best relay Rn∗ can be selected to assist the data transmission, and the selection criterion is given by(6)n∗=argmax1≤n≤Nvn.


## 3. Performance Evaluation

### 3.1. Outage Probability

In this part, we investigate the data transmission performance by deriving the analytical expression of outage probability as well as the asymptotic expression with high transmit power. LetPoutwo and Poutw denote the outage probability of the DF relaying networks without and with cache assistant, respectively.


#### 3.1.1. Without Cache

For the traditional dual-hop relaying networks without cache, the outage event occurs when the transmission data rate is below a target data rateRt [40, 41],(7)12log21+SNRD<Rt, where the term 1/2 comes from the two-phase data transmission. This equation is equivalent to(8)SNRD<γ1t, where γ1t=22Rt-1 is the SNR threshold without cache. Accordingly, we can write the system outage probability Poutwo as(9)Poutwo=PrSNRD<γ1t=Prminu,v<γ1tσ2P=1-Prminu,v≥γ1tσ2P. By considering that the variables of u and v are independent of each other, we can rewrite Poutwo as(10)Poutwo=1-Pru≥γ1tσ2PPrv≥γ1tσ2P. By applying the PDFs of fu(u)=m1m1um1-1/αm1Γ(m1)e-m1u/α and fv(v)=m2m2vm2-1/βm2Γ(m2)e-m2v/β, we can compute the analytical outage probability of the DF relaying networks without cache as(11)Poutwo=1-∫γ1tσ2/P∞m1m1um1-1αm1Γm1e-m1u/αdu×∫γ1tσ2/P∞m2m2vm2-1βm2Γm2e-m2v/βdv(12)=1-e-γ1tσ2/Pm1/α+m2/β∑k1=0m1-1∑k2=0m2-11k1!k2!×m1γ1tσ2Pαk1m2γ1tσ2Pβk2, where [42, eq. (3.351.2)] is applied in the last equality. The analytical outage probability in the above equation contains elementary functions only, and hence it is easy to be evaluated.To obtain more insights on the system, we propose to derive the asymptotic expression of outage probability for the DF relaying networks without cache, in the high transmit power regime. Since the approximation ofex≃∑n=0Nxn/n! holds for small value of |x| [42], we apply this approximation in (12) and can obtain the asymptotic outage probability of the DF relaying networks without cache as(13)Poutwo≃m1m1-1Γm1γ1tσ2Pαm1+m2m2-1Γm2γ1tσ2Pβm2.By considering the relationship between m1 and m2, we can further specify the asymptotic outage probability as(14)
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#### 3.1.2. With Cache

For the cache aided DF relaying networks, the destination can directly access the data from the relays, and the outage event occurs when then∗th relay is selected to assist the data transmission,(15)log21+SNRD<Rt, which is equivalent to(16)SNRD<γ2t, where γ2t=2Rt-1 is the SNR threshold with cache. From this expression, we can write the system outage probability as(17)Poutw=PrSNRD<γ2t=Prv1<γ2tσ2P,…,vN<γ2tσ2P. By considering that vn values are independent of each other and are identically distributed, we can further write Poutw as(18)Poutw=Pr⁡v1<γ2tσ2P︸IN. To compute the probability I, we apply the PDF of fv1(v)=m2m2vm2-1/βm2Γ(m2)e-m2v/β as(19)I=∫0γ2tσ2/Pm2m2vm2-1βm2Γm2e-m2v/βdv=1-e-m2γ2tσ2/Pβ∑k=0m2-11k!m2γ2tσ2Pβk. Accordingly, we obtain the analytical expression of outage probability for the cache aided DF relaying networks as(20)Poutw=1-e-m2γ2tσ2/Pβ∑k=0m2-11k!m2γ2tσ2PβkN.Note that the above equation includes some elementary functions only and is also easy to be evaluated.We now derive the asymptotic outage probability for the considered system, to obtain more insights on the system design. By applying the approximation ofex≃∑n=0Nxn/n! into I, we can simplify the expression of I as(21)I≃m2m2-1Γm2γ2tσ2Pβm2. From the asymptotic I, we can obtain the asymptotic expression of the outage probability for the cache aided DF relaying networks as(22)Poutw≃m2m2-1Γm2Nγ2tσ2Pβm2N.From this equation, we can find that the system diversity order is equal to m2N, indicating that the system transmission performance can be rapidly improved by increasing the number of cache relays. Moreover, the system performance is not dependent on the first relaying links, as the destination can directly access the data from the relays.


### 3.2. SER

In this part, we derive the analytical expression of symbol error rate for the DF relaying networks, as well as the asymptotic expression in the high regime of transmit power. We usePewo and Pew to denote the SER of the system without or with cache, respectively. For the linear modulation scheme, the SER can be computed from the expression of outage probability [43](23)Pe=12π∫0∞FSNRDx2λe-x2/2dx,where FSNRD(x) is the CDF of the received SNR at the destination D, which can be obtained from the analytical expression of outage probability in (12) and (20). And λ is a constant depending on the specific modulation scheme. For example, λ is equal to 2 when the BPSK modulation is used.


#### 3.2.1. Without Cache

For the traditional dual-hop relaying networks without cache, we can write the system SRR as(24)Pewo=12π∫0∞1-e-x2σ2/Pλm1/α+m2/β∑k1=0m1-1∑k2=0m2-11k1!k2!m1x2σ2Pαλk1m2x2σ2Pβλk2e-x2/2dx(25)=12-∑k1=0m1-1∑k2=0m2-11k1!k2!m1σ2Pαλk1m2σ2Pβλk2×12π∫0∞x2k1+k2e-1/2+σ2/Pλm1/α+m2/βx2dx=12-∑k1=0m1-1∑k2=0m2-11k1!k2!m1σ2Pαλk1m2σ2Pβλk2×bk1+k21+2σ2/Pλm1/α+m2/βk1+k2+1/2, where [42, eq. (3.321.3)] and [42, eq. (3.461.2)] are applied and bn is(26)bn=1,Ifn=02n-1!!,Ifn≥1. Note that the analytical expression of SER in (25) contains some elementary functions only, and hence we can easily evaluate the system SER for the traditional DF relaying networks without cache aid. The details about the numerical computation can be found in [44–46].We now propose to derive the asymptotic SER for the traditional DF relaying networks without cache, in the high regime of transmit powerP. By combining (13) and (23), we can write the asymptotic SER with high transmit power P as(27)Pewo≃12π∫0∞m1m1-1Γm1x2σ2Pαλm1+m2m2-1Γm2x2σ2Pβλm2e-x2/2dx=m1m1-12m1-1!!2Γm11Pαλm1+m2m2-12m2-1!!2Γm21Pβλm2, where [42, eq. (3.461.2)] is used in the last equality. From this asymptotic SER, we can find that the system diversity order is equal to min(m1,m2), indicating that the weaker hop between dual hops becomes the bottleneck of the whole data transmission. In particular, the bottleneck is the first hop link if m1<m2 and the second hop otherwise.


#### 3.2.2. With Cache

To compute the analytical expression of SER for the cache aided DF relaying networks, we first re-express the analytical expression in (12) by using the binomial expansion [42],(28)Poutw=∑~c1nc2ne-jm2γ2tσ2/Pβm2γ2tσ2Pβc3n, where(29)∑~=∑j=0N∑n1=0j∑n2=0n1⋯∑nm2-1=0m2-2,c1n=jn1n1n2⋯nm2-2nm2-1,c2n=-1j2!n2-n33!n3-n4⋯×m2-2!nm2-2-nm2-1m2-1!nm2-1c3n=n1+n2+⋯+nm2-1. From this expression, we write the system SER for the cache aided relaying networks as(30)Pew=∑~c1n2πc2nm2σ2Pβλc3n∫0∞e-1/2+jm2σ2/Pβλx2x2c3ndx=∑~c1n2c2nm2σ2Pβλc3nbc3n1+2jm2σ2/Pβλc3n+1/2, where [42, eq. (3.321.3)] and [42, eq. (3.461.2)] are used. From this analytical expression, we can easily evaluate the system SER for the DF relaying networks without cache.We now propose to derive the asymptotic SER for the cache aided DF relaying networks, where we assume that the transmit power is large. By applying the asymptotic outage probability of (22) in (23), we can obtain the asymptotic SER as(31)Pew≃12π∫0∞m2m2-1Γm2Nx2σ2Pβλm2Ne-x2/2dx=2m2N-1!!2m2m2-1Γm2N1Pβλm2N. From this asymptotic SER, we can find that the system diversity order is equal to m2N, indicating that the system transmission performance can be rapidly improved by increasing the fading parameter m2 or the number of relays.


## 3.1. Outage Probability

In this part, we investigate the data transmission performance by deriving the analytical expression of outage probability as well as the asymptotic expression with high transmit power. LetPoutwo and Poutw denote the outage probability of the DF relaying networks without and with cache assistant, respectively.


### 3.1.1. Without Cache

For the traditional dual-hop relaying networks without cache, the outage event occurs when the transmission data rate is below a target data rateRt [40, 41],(7)12log21+SNRD<Rt, where the term 1/2 comes from the two-phase data transmission. This equation is equivalent to(8)SNRD<γ1t, where γ1t=22Rt-1 is the SNR threshold without cache. Accordingly, we can write the system outage probability Poutwo as(9)Poutwo=PrSNRD<γ1t=Prminu,v<γ1tσ2P=1-Prminu,v≥γ1tσ2P. By considering that the variables of u and v are independent of each other, we can rewrite Poutwo as(10)Poutwo=1-Pru≥γ1tσ2PPrv≥γ1tσ2P. By applying the PDFs of fu(u)=m1m1um1-1/αm1Γ(m1)e-m1u/α and fv(v)=m2m2vm2-1/βm2Γ(m2)e-m2v/β, we can compute the analytical outage probability of the DF relaying networks without cache as(11)Poutwo=1-∫γ1tσ2/P∞m1m1um1-1αm1Γm1e-m1u/αdu×∫γ1tσ2/P∞m2m2vm2-1βm2Γm2e-m2v/βdv(12)=1-e-γ1tσ2/Pm1/α+m2/β∑k1=0m1-1∑k2=0m2-11k1!k2!×m1γ1tσ2Pαk1m2γ1tσ2Pβk2, where [42, eq. (3.351.2)] is applied in the last equality. The analytical outage probability in the above equation contains elementary functions only, and hence it is easy to be evaluated.To obtain more insights on the system, we propose to derive the asymptotic expression of outage probability for the DF relaying networks without cache, in the high transmit power regime. Since the approximation ofex≃∑n=0Nxn/n! holds for small value of |x| [42], we apply this approximation in (12) and can obtain the asymptotic outage probability of the DF relaying networks without cache as(13)Poutwo≃m1m1-1Γm1γ1tσ2Pαm1+m2m2-1Γm2γ1tσ2Pβm2.By considering the relationship between m1 and m2, we can further specify the asymptotic outage probability as(14)
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### 3.1.2. With Cache

For the cache aided DF relaying networks, the destination can directly access the data from the relays, and the outage event occurs when then∗th relay is selected to assist the data transmission,(15)log21+SNRD<Rt, which is equivalent to(16)SNRD<γ2t, where γ2t=2Rt-1 is the SNR threshold with cache. From this expression, we can write the system outage probability as(17)Poutw=PrSNRD<γ2t=Prv1<γ2tσ2P,…,vN<γ2tσ2P. By considering that vn values are independent of each other and are identically distributed, we can further write Poutw as(18)Poutw=Pr⁡v1<γ2tσ2P︸IN. To compute the probability I, we apply the PDF of fv1(v)=m2m2vm2-1/βm2Γ(m2)e-m2v/β as(19)I=∫0γ2tσ2/Pm2m2vm2-1βm2Γm2e-m2v/βdv=1-e-m2γ2tσ2/Pβ∑k=0m2-11k!m2γ2tσ2Pβk. Accordingly, we obtain the analytical expression of outage probability for the cache aided DF relaying networks as(20)Poutw=1-e-m2γ2tσ2/Pβ∑k=0m2-11k!m2γ2tσ2PβkN.Note that the above equation includes some elementary functions only and is also easy to be evaluated.We now derive the asymptotic outage probability for the considered system, to obtain more insights on the system design. By applying the approximation ofex≃∑n=0Nxn/n! into I, we can simplify the expression of I as(21)I≃m2m2-1Γm2γ2tσ2Pβm2. From the asymptotic I, we can obtain the asymptotic expression of the outage probability for the cache aided DF relaying networks as(22)Poutw≃m2m2-1Γm2Nγ2tσ2Pβm2N.From this equation, we can find that the system diversity order is equal to m2N, indicating that the system transmission performance can be rapidly improved by increasing the number of cache relays. Moreover, the system performance is not dependent on the first relaying links, as the destination can directly access the data from the relays.


## 3.1.1. Without Cache

For the traditional dual-hop relaying networks without cache, the outage event occurs when the transmission data rate is below a target data rateRt [40, 41],(7)12log21+SNRD<Rt, where the term 1/2 comes from the two-phase data transmission. This equation is equivalent to(8)SNRD<γ1t, where γ1t=22Rt-1 is the SNR threshold without cache. Accordingly, we can write the system outage probability Poutwo as(9)Poutwo=PrSNRD<γ1t=Prminu,v<γ1tσ2P=1-Prminu,v≥γ1tσ2P. By considering that the variables of u and v are independent of each other, we can rewrite Poutwo as(10)Poutwo=1-Pru≥γ1tσ2PPrv≥γ1tσ2P. By applying the PDFs of fu(u)=m1m1um1-1/αm1Γ(m1)e-m1u/α and fv(v)=m2m2vm2-1/βm2Γ(m2)e-m2v/β, we can compute the analytical outage probability of the DF relaying networks without cache as(11)Poutwo=1-∫γ1tσ2/P∞m1m1um1-1αm1Γm1e-m1u/αdu×∫γ1tσ2/P∞m2m2vm2-1βm2Γm2e-m2v/βdv(12)=1-e-γ1tσ2/Pm1/α+m2/β∑k1=0m1-1∑k2=0m2-11k1!k2!×m1γ1tσ2Pαk1m2γ1tσ2Pβk2, where [42, eq. (3.351.2)] is applied in the last equality. The analytical outage probability in the above equation contains elementary functions only, and hence it is easy to be evaluated.To obtain more insights on the system, we propose to derive the asymptotic expression of outage probability for the DF relaying networks without cache, in the high transmit power regime. Since the approximation ofex≃∑n=0Nxn/n! holds for small value of |x| [42], we apply this approximation in (12) and can obtain the asymptotic outage probability of the DF relaying networks without cache as(13)Poutwo≃m1m1-1Γm1γ1tσ2Pαm1+m2m2-1Γm2γ1tσ2Pβm2.By considering the relationship between m1 and m2, we can further specify the asymptotic outage probability as(14)
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. From this asymptotic expression, we can find that the weaker hop between the dual hops becomes the bottleneck of the system whole transmission, and the system diversity order is equal to min(m1,m2). In particular, if the first hop is weaker with m1<m2, the first hop regulates the system transmission performance and the diversity order is m1; if the second hop is weaker with m2<m1, the second hop becomes the bottleneck of the system whole transmission and the diversity order is m2.


## 3.1.2. With Cache

For the cache aided DF relaying networks, the destination can directly access the data from the relays, and the outage event occurs when then∗th relay is selected to assist the data transmission,(15)log21+SNRD<Rt, which is equivalent to(16)SNRD<γ2t, where γ2t=2Rt-1 is the SNR threshold with cache. From this expression, we can write the system outage probability as(17)Poutw=PrSNRD<γ2t=Prv1<γ2tσ2P,…,vN<γ2tσ2P. By considering that vn values are independent of each other and are identically distributed, we can further write Poutw as(18)Poutw=Pr⁡v1<γ2tσ2P︸IN. To compute the probability I, we apply the PDF of fv1(v)=m2m2vm2-1/βm2Γ(m2)e-m2v/β as(19)I=∫0γ2tσ2/Pm2m2vm2-1βm2Γm2e-m2v/βdv=1-e-m2γ2tσ2/Pβ∑k=0m2-11k!m2γ2tσ2Pβk. Accordingly, we obtain the analytical expression of outage probability for the cache aided DF relaying networks as(20)Poutw=1-e-m2γ2tσ2/Pβ∑k=0m2-11k!m2γ2tσ2PβkN.Note that the above equation includes some elementary functions only and is also easy to be evaluated.We now derive the asymptotic outage probability for the considered system, to obtain more insights on the system design. By applying the approximation ofex≃∑n=0Nxn/n! into I, we can simplify the expression of I as(21)I≃m2m2-1Γm2γ2tσ2Pβm2. From the asymptotic I, we can obtain the asymptotic expression of the outage probability for the cache aided DF relaying networks as(22)Poutw≃m2m2-1Γm2Nγ2tσ2Pβm2N.From this equation, we can find that the system diversity order is equal to m2N, indicating that the system transmission performance can be rapidly improved by increasing the number of cache relays. Moreover, the system performance is not dependent on the first relaying links, as the destination can directly access the data from the relays.


## 3.2. SER

In this part, we derive the analytical expression of symbol error rate for the DF relaying networks, as well as the asymptotic expression in the high regime of transmit power. We usePewo and Pew to denote the SER of the system without or with cache, respectively. For the linear modulation scheme, the SER can be computed from the expression of outage probability [43](23)Pe=12π∫0∞FSNRDx2λe-x2/2dx,where FSNRD(x) is the CDF of the received SNR at the destination D, which can be obtained from the analytical expression of outage probability in (12) and (20). And λ is a constant depending on the specific modulation scheme. For example, λ is equal to 2 when the BPSK modulation is used.


### 3.2.1. Without Cache

For the traditional dual-hop relaying networks without cache, we can write the system SRR as(24)Pewo=12π∫0∞1-e-x2σ2/Pλm1/α+m2/β∑k1=0m1-1∑k2=0m2-11k1!k2!m1x2σ2Pαλk1m2x2σ2Pβλk2e-x2/2dx(25)=12-∑k1=0m1-1∑k2=0m2-11k1!k2!m1σ2Pαλk1m2σ2Pβλk2×12π∫0∞x2k1+k2e-1/2+σ2/Pλm1/α+m2/βx2dx=12-∑k1=0m1-1∑k2=0m2-11k1!k2!m1σ2Pαλk1m2σ2Pβλk2×bk1+k21+2σ2/Pλm1/α+m2/βk1+k2+1/2, where [42, eq. (3.321.3)] and [42, eq. (3.461.2)] are applied and bn is(26)bn=1,Ifn=02n-1!!,Ifn≥1. Note that the analytical expression of SER in (25) contains some elementary functions only, and hence we can easily evaluate the system SER for the traditional DF relaying networks without cache aid. The details about the numerical computation can be found in [44–46].We now propose to derive the asymptotic SER for the traditional DF relaying networks without cache, in the high regime of transmit powerP. By combining (13) and (23), we can write the asymptotic SER with high transmit power P as(27)Pewo≃12π∫0∞m1m1-1Γm1x2σ2Pαλm1+m2m2-1Γm2x2σ2Pβλm2e-x2/2dx=m1m1-12m1-1!!2Γm11Pαλm1+m2m2-12m2-1!!2Γm21Pβλm2, where [42, eq. (3.461.2)] is used in the last equality. From this asymptotic SER, we can find that the system diversity order is equal to min(m1,m2), indicating that the weaker hop between dual hops becomes the bottleneck of the whole data transmission. In particular, the bottleneck is the first hop link if m1<m2 and the second hop otherwise.


### 3.2.2. With Cache

To compute the analytical expression of SER for the cache aided DF relaying networks, we first re-express the analytical expression in (12) by using the binomial expansion [42],(28)Poutw=∑~c1nc2ne-jm2γ2tσ2/Pβm2γ2tσ2Pβc3n, where(29)∑~=∑j=0N∑n1=0j∑n2=0n1⋯∑nm2-1=0m2-2,c1n=jn1n1n2⋯nm2-2nm2-1,c2n=-1j2!n2-n33!n3-n4⋯×m2-2!nm2-2-nm2-1m2-1!nm2-1c3n=n1+n2+⋯+nm2-1. From this expression, we write the system SER for the cache aided relaying networks as(30)Pew=∑~c1n2πc2nm2σ2Pβλc3n∫0∞e-1/2+jm2σ2/Pβλx2x2c3ndx=∑~c1n2c2nm2σ2Pβλc3nbc3n1+2jm2σ2/Pβλc3n+1/2, where [42, eq. (3.321.3)] and [42, eq. (3.461.2)] are used. From this analytical expression, we can easily evaluate the system SER for the DF relaying networks without cache.We now propose to derive the asymptotic SER for the cache aided DF relaying networks, where we assume that the transmit power is large. By applying the asymptotic outage probability of (22) in (23), we can obtain the asymptotic SER as(31)Pew≃12π∫0∞m2m2-1Γm2Nx2σ2Pβλm2Ne-x2/2dx=2m2N-1!!2m2m2-1Γm2N1Pβλm2N. From this asymptotic SER, we can find that the system diversity order is equal to m2N, indicating that the system transmission performance can be rapidly improved by increasing the fading parameter m2 or the number of relays.


## 3.2.1. Without Cache

For the traditional dual-hop relaying networks without cache, we can write the system SRR as(24)Pewo=12π∫0∞1-e-x2σ2/Pλm1/α+m2/β∑k1=0m1-1∑k2=0m2-11k1!k2!m1x2σ2Pαλk1m2x2σ2Pβλk2e-x2/2dx(25)=12-∑k1=0m1-1∑k2=0m2-11k1!k2!m1σ2Pαλk1m2σ2Pβλk2×12π∫0∞x2k1+k2e-1/2+σ2/Pλm1/α+m2/βx2dx=12-∑k1=0m1-1∑k2=0m2-11k1!k2!m1σ2Pαλk1m2σ2Pβλk2×bk1+k21+2σ2/Pλm1/α+m2/βk1+k2+1/2, where [42, eq. (3.321.3)] and [42, eq. (3.461.2)] are applied and bn is(26)bn=1,Ifn=02n-1!!,Ifn≥1. Note that the analytical expression of SER in (25) contains some elementary functions only, and hence we can easily evaluate the system SER for the traditional DF relaying networks without cache aid. The details about the numerical computation can be found in [44–46].We now propose to derive the asymptotic SER for the traditional DF relaying networks without cache, in the high regime of transmit powerP. By combining (13) and (23), we can write the asymptotic SER with high transmit power P as(27)Pewo≃12π∫0∞m1m1-1Γm1x2σ2Pαλm1+m2m2-1Γm2x2σ2Pβλm2e-x2/2dx=m1m1-12m1-1!!2Γm11Pαλm1+m2m2-12m2-1!!2Γm21Pβλm2, where [42, eq. (3.461.2)] is used in the last equality. From this asymptotic SER, we can find that the system diversity order is equal to min(m1,m2), indicating that the weaker hop between dual hops becomes the bottleneck of the whole data transmission. In particular, the bottleneck is the first hop link if m1<m2 and the second hop otherwise.


## 3.2.2. With Cache

To compute the analytical expression of SER for the cache aided DF relaying networks, we first re-express the analytical expression in (12) by using the binomial expansion [42],(28)Poutw=∑~c1nc2ne-jm2γ2tσ2/Pβm2γ2tσ2Pβc3n, where(29)∑~=∑j=0N∑n1=0j∑n2=0n1⋯∑nm2-1=0m2-2,c1n=jn1n1n2⋯nm2-2nm2-1,c2n=-1j2!n2-n33!n3-n4⋯×m2-2!nm2-2-nm2-1m2-1!nm2-1c3n=n1+n2+⋯+nm2-1. From this expression, we write the system SER for the cache aided relaying networks as(30)Pew=∑~c1n2πc2nm2σ2Pβλc3n∫0∞e-1/2+jm2σ2/Pβλx2x2c3ndx=∑~c1n2c2nm2σ2Pβλc3nbc3n1+2jm2σ2/Pβλc3n+1/2, where [42, eq. (3.321.3)] and [42, eq. (3.461.2)] are used. From this analytical expression, we can easily evaluate the system SER for the DF relaying networks without cache.We now propose to derive the asymptotic SER for the cache aided DF relaying networks, where we assume that the transmit power is large. By applying the asymptotic outage probability of (22) in (23), we can obtain the asymptotic SER as(31)Pew≃12π∫0∞m2m2-1Γm2Nx2σ2Pβλm2Ne-x2/2dx=2m2N-1!!2m2m2-1Γm2N1Pβλm2N. From this asymptotic SER, we can find that the system diversity order is equal to m2N, indicating that the system transmission performance can be rapidly improved by increasing the fading parameter m2 or the number of relays.


## 4. Simulation and Numerical Results

In this section, we provide some numerical and simulation results to validate the proposed studies. All links in the network experience Nakagami-m fading, and we setα=β=1 without loss of generality. The power of the additive white noise at the receiver is set to unity with σ=1. The target data rate Rt is set to 1 bps/Hz, so that the two SNR thresholds γ1t and γ2t are associated with 3 and 1, respectively. For the SER computation, the BPSK modulation scheme is used, so that the modulation constant λ is set to 2.Figure2 depicts the outage probability of the traditional dual-hop relaying networks versus the transmit power P, where m1=2 and m2 varies from 1 to 3. As observed from this figure, we can find that, for various values of P and m2, the analytical result matches well with the simulation value and the asymptotic result converges to the exact one in the high regime of transmit power P. This validates the effectiveness of the derived analytical and asymptotic expressions of outage probability for the traditional dual-hop relaying networks without cache. Moreover, the curve slope of the outage probability linearly increases with m2, when m2≤m1, since the second relaying link becomes the bottleneck of the system data transmission. In further, the curves with m2=2 and m2=3 have the same slope, indicating that the system diversity order remains unchanged.Figure 2
Outage probability versus the transmit powerP without cache.Figure3 demonstrates the symbol error rate of the traditional dual-hop relaying networks without cache, where λ=2, the transmit power P varies from 0 dB to 30 dB, and m2 varies from 1 to 3. We can see from this figure that the analytical SER fits well with the simulated SER and the asymptotic SER converges to the exact SER in the high region of transmit power P. This verifies the derived analytical and asymptotic expressions of SER for the traditional dual-hop relaying networks without cache. Moreover, the curve slope increases with m2 when m2≤m1, since the second relaying link becomes the bottleneck of the system whole data transmission. Furthermore, the curve slope with m2=2 is the same as that of m2=3, indicating that the system diversity order remains unchanged.Figure 3
Symbol error rate versus the transmit powerP without cache.Figure4 illustrates the outage probability of the cache aided DF relaying networks, where N=2, P varies from 0 dB to 30 dB, and m2 varies from 1 to 3. As seen in Figure 4, we can find that the analytical outage probability tallies with the simulation probability and the asymptotic outage probability converges to the exact Pout with large value of P. This validates the correctness of the derived analytical and asymptotic expressions of outage probability. Moreover, the curve slopes are in parallel with the value of m2, indicating that the system diversity order increases with the value of m2. By comparing Figures 2 and 4, we can obtain that, for achieving a target level of outage probability 10-4, the transmit power can be saved by about 17.5dB when m2=3, indicating that the cache technique can save the power resource efficiently.Figure 4
Outage probability versus the transmit powerP with cache.Figure5 provides the symbol error rate of the cache aided DF relaying networks versus the transmit power P, where λ=2 and m1=2 and m2 varies from 1 to 3. We can see from Figure 5 that the analytical SER is identical to the simulation one, and the asymptotic SER converges to the exact value in the high regime of transmit power P. This verifies the derived analytical and asymptotic expressions of SER for the cache aided DF relaying networks. Moreover, the curve slopes are in parallel with the value of m2, indicating that the system diversity order increases linearly with m2. By comparing Figures 3 and 5, we can find that, to achieve a target SER level of 10-4, using cache can save about 10dB, indicating that cache will be very useful in the scarce power resource scenarios.Figure 5
Symbol error rate versus the transmit powerP with cache.Figures6 and 7 show the impact of number of relays N on the performance of cache aided relaying networks versus the transmit power P, where m1=m2=2 and N varies from 1 to 3. Specifically, Figures 6 and 7 are associated with outage probability and SER, respectively. We can find that, for different values of N, the analytical performance is in good agreement with the simulation value and the asymptotic value converges to the exact one when the transmit power is large, which validates the effectiveness of the derived analytical and asymptotic expressions of outage probability and SER. Moreover, the curve slopes are in parallel with N, indicating that the system diversity order increases linearly with the number of relays. For the target outage probability and SER around 10-4, we can find that the cache technique with N=3 can save about 18.5dB and 11dB, respectively, indicating that cache can be used to save power resource.Figure 6
Impact of relay numberN on the cache aided relaying networks: outage probability.Figure 7
Impact of relay numberN on the cache aided relaying networks: symbol error rate.


## 5. Conclusions

This paper studied a two-hop DF relaying network, where cache could be used to prefetch the data from the source to the nodes around the destination. In this way, the destination could directly access the data from the relay nodes, instead of communicating with the source. For the DF relaying networks without and with cache, the system performances were studied by deriving the analytical expressions of outage probability and symbol error rate (SER). The asymptotic outage probability and SER were also provided in the high regime of transmit power, from which we could find the system diversity order is rapidly increased by using cache and the system performance is significantly improved. Simulation and numerical results were demonstrated to verify the proposed studies, and the transmit power could be saved by about18.5dB and 11dB for achieving outage probability and SER level of 10-4.



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*Source: 5963584-2018-04-23.xml*