Location-Aware Transmission for Two-Cell Wireless Networks With Caching

A novel downlink transmission selection scheme based on the locations of users for network coded cache-enabled, Multiple-Input Multiple-Output (MIMO) systems is proposed in this paper. Caches are available at both base stations (BSs) and mobile terminals (MTs) and network coding is employed in order to reduce power consumption. We compare the analysis of the proposed scheme with a simple direct transmission scheme in terms of achievable sum rate and outage probability. It is shown that the proposed scheme is preferred for high transmit signal-to-noise ratio (SNR) and/or high cross-link channel gain ratio scenarios, while the simple direct transmission scheme is better for low SNR and/or low cross-link channel gain ratio. Motivated by this analytical results, we present a selection algorithm for location-aware transmission scheme for two-cell networks. Numerical results confirm our theoretical analysis.


I. INTRODUCTION
Caching at the edge of mobile networks (e.g., at base stations (BSs), access points (APs), or mobile terminals (MTs)) is a promising key technology for 5G and beyond wireless networks [1]. The benefits of caching popular content in the BSs or APs, such as backhaul load relief, low access latency and lower energy consumption, as well as creating more cooperative MIMO opportunities, have been studied in, e.g., [2]- [10]. The authors in [11] have proposed a novel coded caching scheme in which both local and global caching gains can be achieved by designing a particular pattern in cache placement at the receivers and exploiting coding in delivery. A cache-aided interference management (CAIM) scheme for general networks with caches at both transmitters and receivers was discussed in [12]. The authors have designed a scheme which loads the caches judiciously in order to maximize the opportunity not only for zero-forcing the outgoing interference from the transmitters but also for interference cancellation at the receivers due to the cached The associate editor coordinating the review of this manuscript and approving it for publication was Zheng Chang . content at the receivers. In [13], the comparison between different caching strategies has been addressed.
Inspired by [12], we in this paper first design a network coded caching scheme (''Scheme I''). The popular contents are divided into different files and cached at BSs and MTs separately. When two MTs require the same or different contents, network coding is implemented between files at the same BS before transmission. The MTs will recover the desired content by using the original cached files and the network coded files received from different BSs. Our scheme differs from the work in [12] in that network coded files are transmitted in different time slots in order to reduce the energy consumption.
We then compare the proposed scheme with an alternative approach (''Scheme II'') which to cache the popular contents at the BSs only (i.e., no need for MT caching). In this paper, we analyze and compare these two different schemes in terms of sum rate and outage probability and then propose a caching scheme selection algorithm based on the MT location information (or channel conditions).
The rest of the paper is organized as follows. In Section 2, an introduction to the system definition and the novel VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ caching scheme will be presented. Section 3 will include the sum rate and outage probability analysis. In Section 4, an optimal caching methods selection algorithm will be proposed. Numerical results and discussion will be presented in Section 5, followed by the conclusions in Section 6.

II. SYSTEM MODEL
In this section, we introduce the downlink system model which involves 2 MTs and 2 BSs. Please note that the proposed cashing scheme can be extended to a larger network with multiple MTs and BSs [11], [12]. The focus of this paper is to evaluate the schemes in the simplest scenario applicable. Different scenarios are considered according to the locations of two MTs. A detailed channel model is derived followed by the introduction of two cache and delivery schemes.

A. THE CHANNEL MODEL
We consider the downlink of a cluster of two cooperating BSs and two MTs as shown in Fig. 1. The channel from the j th BS to the i th MT is modeled in the following form where i, j ∈ {0, 1}, g ij ∼ CN (0, 1) is a complex normally distributed random variable, α ≥ 2 denotes the path-loss exponent, and r ij is the distance between the j th BS and the i th MT. In Fig. 1, four cases of the downlink scenario based on the locations of the two MTs are illustrated. In Case (a), two MTs are located in the overlapped edge area of two cells, while one MT located in the overlapped edge area of two cells while another is near a cell center in Case (d). In Cases (b) and (c), none of the MTs are located in the overlapped cell edge area. The MTs are located near the centers of different cells in Case (b), and in Case (c), both MTs are located near the center of the same cell. In order to obtain explicit expressions and clear boundaries between the different cases, we define three parameters ω, κ 0 and κ 1 , given by ω = (r 00 /r 11 ) α , κ 0 = (r 00 /r 01 ) α , κ 1 = (r 11 /r 10 ) α . (2) The above three parameters are the ratios of the mean cross-link channel gains, which characterize the balances of the cross-links. Without loss of generality, we set cell 0 in Fig. 1 as the reference, and thus the values of ω and κ 0 are between 0 and 1. The expressions for different cases are given in Table 1. It is worth noting that Case (a) is a special case of Cases (b), (c) and (d) in which κ = 1, ω = 1 and ω = κ 0 = 1, respectively. Under such a setup and assuming no channel state information (CSI) at the transmitter (i.e., BSs), two transmission schemes can be employed as described below.

B. SCHEME I: CACHING AT BOTH BSs AND MTs
In Scheme I, as in [12], both the transmitters and receivers are equipped with caches. We define file A and B as two files stored in the library. Each BS and MT can cache at most a fraction µ T and µ R of the library content respectively, where µ T , µ R ∈ [0, 1] is defined as the fractional cache size. For Scheme I, the BS and MT cache sizes are µ T = 0.5 and µ R = 0.5, respectively. In the prefetching phase, each file in the library is divided into 4 disjoint subfiles with the same size and then stored at the caches of the BSs and MTs. For example, file A is divided into A 0,0 , A 0,1 , A 1,0 , A 1,1 , where A j,i is stored at both the j th BS and the i th MT [12], i, j ∈ {0, 1} (see Fig. 2(a)). File B is partitioned and stored in the same way.
Without loss of generality, we assume that MT 0 and MT 1 requests file A and B, respectively. Then the delivery phase is split into two time slots and in each delivery slot, one BS broadcasts a network coded signal while the other BS keeps silent. One example is shown in Fig. 2(b). In the first time slot, BS 0 transmits the content A 0,1 ⊕ B 0,0 and BS 1 keeps silent, where ⊕ stands the bit-wise XOR operation. In the second slot, BS 0 delivers the content A 1,1 ⊕B 1,0 and BS 0 keeps silent. Other cases work in a similar fashion. The delivery scheme in [12] is illustrated in Fig. 2(c) where both BSs transmit contents at both time slots.
Let x (k) j denote the transmitted signal from the j th BS at the k th time slot, then the received signal at the i th MT in the k th time slot can be expressed as where z (k) i is the additive white Gaussian noise (AWGN) with independent real and imaginary parts of equal variance N 0 /2.
In (3), different BSs transmit a network coded signal of different subfiles in different time slots, which indicates that x = 0. This is very different from the scheme proposed in [12], where x j in the proposed scheme is obtained by simple XOR of the subfiles, and the desired subfile for different MTs can be recovered by using the cached subfiles. Another significant difference compared with the algorithm in [12] (see Fig. 2(c) for example) is that in the proposed scheme, one BS keeps silent while the other BS is transmitting, which means that the energy consumption of the proposed delivery scheme is half of that of the scheme in [12].
The instantaneous received signal-to-noise ratio (SNR) of the i th BS at the k th time slot can be expressed as where E s denotes the average energy per symbol of the non-zero transmitted signal. The average value of γ ij is given byγ By substituting (2) into (5), we havē where γ 0 r −α 00 E s /N 0 . Following [14], the normalized SNR is defined as ρ ij = γ ij /γ ij = |g ij | 2 . Since g ij ∼ CN (0, 1), the probability density function (PDF) of ρ ij is given by C. SCHEME II: CACHING AT BSs In Scheme II, only BSs are equipped with caches. An MT is only served by its nearest BS, and each BS has cached all the files, (i.e., the BS and MT cache sizes are µ T = 1 and µ R = 0, respectively), which means that there is no cooperation between the BSs. With this assumption, the signal from the non-nearest BS will be treated as unknown interference [15]. We assume that the i th MT is served only by the i th BS. Then, the expression of the i th MT's received signal is the same as in (3) except that x j and x j are both non-zero. The instantaneous received signal-to-interference-plus-noise ratio (SINR) for the i th MT is given by where where and E 1 (x) is the exponential integral function [16]. Proof: Please refer to Appendix A. It is worth noting that the cache and transmission strategies of Schemes I and II do not need channel state information at the transmitters (CSIT). The system performance can be improved by utilizing the instantaneous CSIT at very short time scales. However, in this work, the transceiver strategy is designed by using the location information of users at longer time scales [17], [18].

III. PERFORMANCE ANALYSIS
In this section, the achievable average sum rate and outage probability of Scheme I and II are given. The four cases illustrated in Fig. 1 have been considered in both schemes in order to address the effect of different locations of the MTs to the schemes.

A. ACHIEVABLE SUM RATE ANALYSIS
For Scheme I, the average achievable sum rate is defined as where f ij (x) is the PDF of ρ ij andγ ij is the mean value of γ ij . Substituting (5) and (7) into (11) and using the equality For Scheme II, the average achievable sum rate is defined as where q i (x) is the PDF of β i andγ i is the mean value of γ i . R 2 can be expressed as Proof: Please refer to Appendix B. The function ψ(x) is important to the average achievable sum rates of Scheme I and Scheme II derived in (12) and (14): it can be defined as the ergodic capacity of Rayleigh fading channels [19], [20]. ψ(x) includes the non-elementary function E 1 (x), which makes it difficult to obtain much insight about the sum-rate. Estimations of ψ(x) based on series expansions can be found in [19], [20], while polynomial and rational approximations of ψ(x) can be found in [21]. For further discussions, we need to investigate the monotonicity of ψ(x) and its related functions, and the main results are summarized in the following lemma.
Lemma 1: The function ψ(x) is monotonically increasing in x. For the following ψ(x) related functions θ 1 (x) and θ 4 (x) are monotonically decreasing in x, θ 2 (x) is monotonically increasing in x, and θ 3 (x) is monotonically increasing and decreasing in x for a > b and a < b, respectively. Proof: Please refer to Appendix C. With the general results of average achievable sum-rates for Scheme I and Scheme II as shown in (12) and (14), we give the sum-rate expressions for Cases (a) to (d) as follows.
Case (a): In this case, both ω and κ i are assumed to be constants. By substituting ω = 1 and κ 0 = κ 1 = 1 into (12) and (14), we have According to the monotonicity of ψ(x) and θ 1 (x) established in Lemma 1, both R In this case, we assume that the MTs' locations are symmetrical with respect to the cell edge. We also assume that 0 < κ < 1. If κ = 1, Case (b) reduces to Case (a). By substituting ω = 1 and κ 0 = κ 1 = κ into (12) and (14), According to the monotonicity of ψ(x), θ 3 (x) and θ 4 (x) derived in Lemma 1, we conclude that R 1 is monotonically increasing in both γ 0 and κ, while R (b) 2 is monotonically increasing in γ 0 and decreasing in κ.
Case (c): In this case, we assume 0 < ω < 1. If ω = 1, it transforms to Case (a). Substituting κ 0 = ω and κ 1 = ω −1 into (12) and (14) yields Interestingly, the only difference between the sum-rate expression in Case (c) and that in Case (b) is the use of the parameters ω or κ.

B. OUTAGE PROBABILITY ANALYSIS
The outage probability is another important performance indicator in order to evaluate communication over fading channels; it indicates the probability that the instantaneous SNR or SINR falls below a certain threshold γ th [14], [22]. The average outage probabilities of Schemes I and II are given by In general, P 1 is monotonically decreasing in γ 0 , κ i and ω, while P 2 is monotonically decreasing in γ 0 and ω and increasing in κ i . With Table 1, we can obtain the expressions of the outage probabilities for Cases (a)-(d). We omit here for brevity.

IV. LOCATION-AWARE DELIVERY SCHEME SELECTION ALGORITHM
Based on the analysis in the previous section, clearly, a key question is: which caching/transmission scheme should be used under which conditions? As such, this section will discuss the conditions under which Scheme I or II should be selected. The caching/transmission scheme selection criteria are designed based on the comparison of sum rates and outage probabilities of Schemes I and II. It is worth pointing out that the conditions are determined by the user locations (r ij or equivalently ω and κ i , i, j ∈ {0, 1}) and the transmit SNR.
Proposition 1: According to the locations of MTs described in Cases (a) to (d), the conditions that Scheme I achieves a higher sum rate than Scheme II are given by 2 if and only if the condition (29) (κ replaced by ω) is satisfied.
Special cases for Case (b): (i) If the value of γ 0 is large, then the condition (29) will always hold for all κ > 0, which means that Scheme I achieves higher sum rate than Scheme II when the SNR is high. (ii) If κ approaches 1, then the condition (29) will always hold, which means that Scheme I achieves higher sum rate than Scheme II when the cross-link channel gain is strong enough. (iii) If κ approaches 0, then the condition (29) will not hold, which means that Scheme II is preferred when the cross-link channel gain is weak.
The scheme selection criteria according to outage probability for Cases (a) to (d) are contained in the following proposition.
Proposition 2: Considering the locations of MTs in Section II-A, the conditions that Scheme I achieves lower outage probabilities than Scheme II in Cases (a) to (d) are given by 2 if and only if the condition that 2 for all γ 0 > 0 and 0 < ω < 1. 2 if and only if the following condition is satisfied. Proof: For Cases (a) and (c), by respectively substituting ω = κ 0 = κ 1 = 1 and ω = κ 0 = κ −1 1 into (27) and (28), we have Thus, P Proposition 2 indicates that the outage probability of Scheme I is always lower than that of Scheme II for grouped users (Cases (a) and (c)), no matter where the group of users are located in a cell.
We now discuss some extreme cases for the condition as shown in (31). (i) The condition (31) will always hold if γ 0 approaches infinity, while it will not hold if γ 0 approaches zero. This property means that Scheme I and Scheme II should be selected to achieve the lower outage probability in high and low SNR regions, respectively. (ii) The condition (31) will always hold if κ approaches 1, while it will not hold if κ approaches 0. Hence, Scheme I should be selected for strong cross-links, while Scheme II should be selected for weak cross-links. (iii) For Case (d), if ω approaches 0, then the condition in (32) reduces to the condition in (31). In addition, (32) will always hold if ω approaches 1 and γ th ≥ 1. Fig. 3 shows the achievable sum rates for Case (b). 1 It is worth pointing out that Case (a) is a special case of Case (b) with κ = 1. Fig. 3(a) indicates the average sum rate as a function of γ 0 with different κ, while Fig. 3(b) shows the average sum rate as a function of κ with different γ 0 . The curves with and without markers depict the Monte-Carlo and analytical results, respectively. Clearly, our theoretical analysis matches the numerical results very well. According to Fig. 3(a), there is a switching point of γ 0 for a given value of κ. For example, if κ = −12 dB, then the average sum rate of Scheme I will 1 If we replace the parameter κ in Fig. 3 with ω, then this figure becomes the achievable sum rates for Case (c). outperform that of Scheme II when γ 0 ≥ 18 dB. Similarly, Fig. 3(b) illustrates a switching point of κ for a given value of γ 0 . For example, if γ 0 = 15 dB, then the average sum rate of Scheme I will outperform that of Scheme II when κ ≥ −10 dB. Fig. 4 displays the achievable sum rates for Case (d) in terms of γ 0 , κ 0 and ω. For each of these three figures, one of these variables changes over the whole range, while the remaining two variables take two typical and fixed values. From Fig. 4(a), we have the following observations: (i) Scheme I is always better than Scheme II when κ 0 = 0 dB. (ii) If κ 0 = −10 dB, then Scheme I will be better than Scheme II when γ 0 > 1 dB and γ 0 > 11dB for ω = 0 dB and ω = −10 dB, respectively. Fig. 4(b) shows that: (i) Scheme I is always better than Scheme II when γ 0 = 30 dB. (ii) If γ 0 = 15 dB, then Scheme I will be always better than Scheme II for ω = 0 dB, and is conditionally better than Scheme II for ω = −10 dB when κ 0 > −14 dB.    Fig. 5 shows the outage probabilities of Case (b). 2 The SNR threshold γ th is set to 0 dB. We observe that: (i) Schemes I and II will be always better for all γ 0 > 0 when κ is big enough (e.g., 0 dB) and small enough (e.g., −24 dB), respectively. However, for moderate values of κ (e.g., −12 dB), there is a switching point (γ 0 = 21 dB) between Schemes I and II. (ii) If γ 0 = 15 dB, then Scheme I is better than Scheme II only when κ > −9 dB. (iii) The switching point of κ will decrease when the SNR γ 0 increases. Fig. 6 shows the outage probabilities of Case (d) with respect to γ 0 , κ 0 and ω. We observe from Fig. 6(a) that: (i) Scheme I is always better than Scheme II when the SNR γ 0 is high enough, or κ 0 = 0 dB or ω = 0 dB. (ii) In case of κ 0 = ω = −10 dB, the switching point of Scheme I and Scheme II is γ 0 = 10 dB. Fig. 6(b): (i) Scheme I is than Scheme II when the ω is big enough, or γ 0 = 30 dB or κ 0 = 0 dB. (ii) In case of γ 0 = 15 dB and ω = −10 dB, the switching point between Schemes I and II is ω = −18 dB.

C. SWITCHING REGION BETWEEN SCHEMES I AND II
The switching regions between Schemes I and II based on achievable sum rate and outage probability are plotted in Fig. 7(a) and Fig. 7(b), respectively. The figures clearly mark out the regions where inequality in Proposition 1 or Proposition 2 is satisfied, meaning that Scheme I is the better strategy; while the region where the inequality is reversed implies that Scheme II is better. In both figures, the Y-axis variable x stands for the ratio of cross-link channel gain. By replacing x with κ, ω and κ 0 , we obtain the results for Cases (b), (c) and (d), respectively. Note that the optimal strategy depends on both the average transmitted SNR γ 0 and the cross-link channel gain ratio x. In general, for high SNR and/or high cross channel gain ratio, the better strategy is Scheme I, while for low SNR and/or low cross channel gain ratio, Scheme II is the better strategy.

VI. CONCLUSION
We have proposed a novel network coded caching scheme for two-cell downlink networks. In order to investigate its performance, the achievable sum-rate and outage probability have been derived in closed-form. A comparison between the proposed scheme and a simple direct transmission scheme has also been conducted and a location-based cache/transmit scheme selection algorithm has been explored. The proposed selection scheme allows BSs to select cache and transmit scheme for different scenarios based on different locations of MTs in order to achieve higher sum rate or lower outage probability.

APPENDIX A PDF OF THE NORMALIZED SINR β i
We first derive the PDF of the instantaneous SINR for the i th MT. From (8), we see that γ i is a ratio of two independent random variables X i and Y i , both of which are the linear transform of chi-squared variables with 2 degrees of freedom, and their distributions are respectively given by Then, the ratio distribution can be obtained from integration of the following form [23] By substituting (35) and (36) into (37), we have The mean of the γ i is Thus, the PDF of normalized SINR β i can be expressed as Substituting (38) and (39) into (40) yields (9). This completes the proof.

APPENDIX B DERIVATION OF (14)
From (9), the integration of q i (x) can be written in the following closed-form µ i x, then the i th summation term of (13) can be calculated as The integration of (43) can be calculated by using the identities [16, 3.252.4] and [16, 3.353.3] for the conditions that κ i = 1 and κ i = 1, respectively, which yields (14). This completes the proof.

APPENDIX C PROOF OF LEMMA 1
According to the definition of ψ(x) as in (10), we have The right-hand side (RHS) of (44) is a well studied function in the literature [21], [24], [25]. By simple transformation of the inequalities of [21, 5.1.19, 5.1.20], we have the following upper and lower bounds of ψ(x): With (46), we can easily obtain the limits of ψ(x) and its related functions, as x approaches 0 or ∞. However, we omit the derived results here for brevity. Now we begin to prove the monotonicity and convexity of ψ(x), which is useful for proving the monotonicity of functions related to ψ(x). The first and second order derivatives of ψ(x) are respectively given bẏ Using the upper bound of ψ(x) in (45), we see thatψ(x) > 0 for x > 0. Thus, ψ(x) is monotonically increasing in x.
In order to determine the sign ofψ(x), we first define an auxiliary function The first order derivative of φ(x) can be expressed aṡ This shows that φ(x) is monotonically decreased with respect to x, so φ(x) < φ(0 + ) = 0, or equivalently Substituting (51) into (48) yieldsψ(x) < 0, which means that ψ(x) is a concave function with respect to x. It is worth pointing out that (51) is a new upper bound of ψ(x), which is tighter than that of (45) for all x > 0 and is tighter than that of (46) only for small positive x. We can then prove the monotonicities of ψ(x) related functions. The first order derivatives of θ 1 (x), θ 2 (x) and θ 3 (x) are respectively given bẏ where the signs of (52) and (53) are determined by using (45) and (51), respectively. Thus, θ 1 (x) and θ 2 (x) are monotonically decreasing and increasing in x, respectively. Since θ 1 (x) is monotonically decreasing in x, the sign of (54) is the same as that of a − b. Hence, θ 3 (x) is monotonically increasing and decreasing in x for a > b and a < b, respectively. The function θ 4 (x) defined in (18) is proportional to the slope of the line segment between fixed point (b, ψ(b)) and changeable point (bx, ψ(bx)). From the property of concave function, θ 4 (x) is monotonically decreasing in x if and only if ψ(x) is concave. This completes the proof of Lemma 1.