Channel Statistics-Based Rate Splitting with Spatial Randomness

Multiple access schemes based on non-orthogonality are potential solutions to the requirements of the future wireless communications for enhancing the spectral efficiency. This paper investigates the performance of a generalization of the well-known power domain non-orthogonal multiple access (NOMA) scheme; the so-called rate splitting (RS). We consider fixed power allocation and by following a probabilistic approach, we provide an analytical framework on the RS scheme. Specifically, we capture spatial randomness and based on the channel statistics, we provide closed-form expressions for the distributions of the signal-to-interference-plus-noise ratio at the receivers. Furthermore, we derive the average rate achieved at each receiver and we show how the different design parameters impact the individual rates. Our results highlight the flexibility of RS against NOMA in terms of fairness performance.


I. INTRODUCTION
With the ever increasing population of cellular subscribers and the rapid development of the Internet of Things (IoT), the 5G wireless networks are expected to cover a tremendous amount of data demands while serving several types of devices [1]. The development of massive machine type networks overlaid with the cellular network impose the need for multiple access schemes that manage to simultaneously offer service to several subscribers. With the evolution of communication systems various orthogonal multiple access (OMA) were applied consisting orthogonality in several dimensions e.g., space division multiple access (SDMA) while the current 4G systems have introduced orthogonal frequency division multiple access. However, the spectral efficiency requirements of the foreseen 5G and beyond systems, necessitate the employment of multiple access schemes which are non-orthogonal designed [2], in order to enable simultaneous access of the channel resources.
Recently, both industry and academia have presented fundamental efforts on the power domain non-orthogonal multiple access (NOMA). This scheme employs superposition coding in order to serve two receivers simultaneously and successive interference cancellation (SIC) is applied at the receiver with the best channel statistics. NOMA with appropriate power allocation has been proved to improve the performance of OMA in terms of achievable sum rate [3]. Furthermore, it has This work was co-funded by the European Regional Development been shown that the NOMA scheme significantly improves the performance of the user with the worst channel statistics, hence further outperforming OMA in terms of fairness [4]. Due to its superiority against OMA, NOMA has been already included in the 3GPP-LTE (as multi user superposition transmission) [5] and is a promising scheme for future wireless communications.
The NOMA scheme though, is a special case of the rate splitting (RS); a multiple access scheme which consists of an extra degree of freedom in the power domain and allows partial interference cancellation at both receivers. The concept of RS was first introduced in [6] and even though the scheme is known for a while, recent studies emerge the benefits and its applications in different setups by optimizing the power allocation coefficients. Specifically, the authors in [7] consider a multiple-input single-output (MISO) broadcast channel and show that RS provides a smooth transition between SDMA and NOMA and outperforms them in terms of spectral efficiency with lower computational complexity. As an extension to this work, in [8] the authors investigate the energy efficiency maximization problem and show that the energy efficiency achieved by RS is equal or higher than the one achieved by SDMA or NOMA.
Furthermore, the MISO system with bounded channel state information errors at the transmitter is considered in [9], where the authors present the gains of RS in terms of max-min fairness. A similar objective is also investigated in [10], where the authors consider transmit beamforming in multiple multicast groups and present the benefits of RS through degrees of freedom analysis. Additionally, the enhancements of RS in terms of spectral and energy efficient are also shown in [11], where the authors study a RS-assisted non-orthogonal unicast and multicast transmission system. Although the current literature shed lights on the efficiency of the RS, its performance from a fixed power allocation perspective, similar to the approach in [3] for NOMA, is missing from the literature.
Motivated from the above, this paper investigates the performance of the RS scheme with fixed power allocation by following a probabilistic approach. We focus on single antenna configuration and by considering spatial randomness, we draw an analytical framework on RS based on the channel statistics. Specifically, we use tools from stochastic geometry and derive closed form expressions for the distributions of the signal-tointerference-plus-noise (SINR) at each receiver. By taking into account a minimum rate threshold, we provide the average rates achieved at each receiver and we show how the different resource coefficients impact the performance of each rate. Finally, we present a comparison with the NOMA scheme in order to demonstrate the gains brought by the flexibility of RS against NOMA in terms of fairness performance.

A. System Model
Consider a single cell downlink network consisting of a transmitter and two receivers i.e., U 1 and U 2 . The transmitter is located at the centre of the cell, denoted by O and allocates its transmit power P according to the RS scheme in order to serve both receivers over a single channel use. The scheduled receivers are chosen in a way that enables a distance based ordering 1 i.e., d 1 < d 2 , where d n denotes the distance of U n from the transmitter. Let D(c, r) denote a disk with centre c and radius r. Then, the first receiver i.e., U 1 is randomly located within D(O, R 0 ) while the second receiver i.e., U 2 is located within the annulus shaped by the difference of disks D(O, R 1 ) and D(O, R 2 ), where R 0 < R 1 < R 2 and thereafter establishing d 1 < d 2 . The annulus shaped by the difference of disks D(O, R 0 ) and D(O, R 1 ) i.e., the guard zone, serves as a receive power separator [12]. The topology of the network is depicted in Fig. 1. Furthermore, we consider that all the wireless signals suffer from both largescale path-loss and small-scale block Rayleigh fading. As such, the channel between the transmitter and the n-th receiver, n ∈ {1, 2}, is given by h n (1 + d α n ) −1 , where h n ∼ exp(1) and α is the path loss propagation exponent. Finally, we take into account additive white Gaussian noise with variance σ 2 .

B. Rate splitting & power allocation
The transmitter employs the RS scheme to broadcast the messages m 1 and m 2 to the receivers U 1 and U 2 , respectively. Each of the two messages is separated in two parts i.e., m 1 = {m 1 1 , m 12 1 }, m 2 = {m 2 2 , m 12 2 } in order for m 12 1 and m 12 2 to be encoded together and transmitted in a common stream s 12 [6]. On the other hand, the parts m 1 1 and m 2 2 are transmitted in two private streams s 1 and s 2 , respectively. Therefore, the transmitter transmits the superposition of the three streams by allocating the available power to s 12 , s 1 and s 2 with βP , and ρ ∈ [0, 1]. For decoding each stream, the received signal to interference plus noise ratio (SINR) at U n , n ∈ {1, 2}, should exceed a predefined threshold, denoted with ζ for s 12 and ξ for s n . Based on the power allocation described above, the SINR at the receiver U n for decoding s 12 is given by If the common stream is successfully decoded, then U n attempts to decode the private stream s n with SINR η p n , where 1 The ordering in RS is not necessary; however in this paper is considered for purposes of comparison with NOMA [12].
On the other hand, if s 12 is not successfully decoded, then U n attempts to decode the private stream with SINR η pI n , where

C. Decoding rates
Following from above, the average rates achieved for decoding each stream are defined as follows • If both receivers successfully decode the common stream s 12 i.e., η c 1 > ζ and η c 2 > ζ, then s 12 has rate R c where the min operation establishes that the common stream's rate is achievable by both receivers. Hence, the rate R c 12 is allocated to the receivers U 1 and U 2 with uR c 12 and (1 − u)R c 12 , respectively; where u ∈ [0, 1]. • If solely the receiver U n , successfully decodes s 12 i.e., η c n > ζ and η c k < ζ, k = n, then R c

III. ACHIEVABLE RATES BASED ON CHANNEL STATISTICS
In this section we evaluate the total average rate achieved at each receiver i.e, R 1 at U 1 and R 2 at U 2 . Note that the SINR expressions provided in Section II are upper bounded when P → ∞, hence we will make use of the following set , where the i-th element of the set is denoted with ϑ i ; and the equations (1) to (5) are upper bounded by ϑ 1 , ϑ 2 , ϑ −1 2 , ϑ 3 and ϑ 4 , respectively. Note that, when the denominator of ϑ i becomes zero, then In what follows, we provide the coverage probability π η and the probability density function (PDF) g η of the SINR η. Proposition 1. The coverage probability for the receiver U 1 with SINR η is given by and the PDF of η is given by where for η = η c 1 , we set θ = ϑ 1 and s η c 1 = Proposition 2. The coverage probability for the receiver U 2 with SINR η is given by and the PDF of η is given by where for η = η c 2 , we set θ = ϑ 1 and s η c 2 = s η c 1 ; for η = η p 1 , we set θ = ϑ −1 2 and s η p 2 = σ 2 t (β−1)P (ρt+ρ−1) ; for η = η pI 2 , we set θ = ϑ 4 and s η pI .

A. Average Stream Rates
Provided with the above we now derive the analytical expressions for the average rates defined in Section II-C. We first derive the common rate R c 12 for the case where both received SINRs at U 1 and U 2 achieve the minimum threshold ζ. As such, we first express the instantaneous common rate as Since η c 1 and η c 2 are independent then their joint PDF is given by g η c 1 (t)g η c 2 (t) and the average rate is given by which is evaluated as where g η c 1 (t) and g η c 2 (t) are provided in Propositions 1 and 2, respectively. Note that, R c 12 accounts for the case where both U 1 and U 2 successfully decode s 12 . When U n decodes s 12 while U k not, n = k, then due to independence between η c 1 and η c 2 the rate for decoding s 12 at U n is given by As such, the average rate for decoding s 12 at U 1 is and respectively the average rate for decoding s 12 at U 2 is We now focus on the average rate for decoding the private stream s n which is given by R sn = R p n + R pI n . The rate R p n is the achieved rate when s 12 is successfully decoded and is provided in the following Proposition.
Proposition 3. The average rate for decoding the stream s n , n ∈ {1, 2}, when s 12 is successfully decoded is given by On the other hand, when s 12 is not successfully decoded, the stream s n has average rate R pI n which is provided below. Proposition 4. The average rate for decoding the stream s n , n ∈ {1, 2}, when s 12 is not successfully decoded is given by if ζ > ϑ 1 , ϕ 1 > 0, ξ < θ and by and set ϕ 1 = 1 + ρ(β − 1), Provided with the individual rates for each stream we can now express the average rate achieved at each receiver by R n = R s12 n + R sn , and the sum rate is expressed as R s = R 1 + R 2 . For the case where P → ∞, β ∈ (0, 1), ρ ∈ (0, 1) the average rates become deterministic and are given as and where 1{x} = 1 if x is true, otherwise gives 0.

B. Comparison with NOMA
For the purpose of comparison, we also provide the average rates when the NOMA scheme is employed. In this case, all the information of m 1 is encoded in s 1 and all the information of m 2 is encoded in s 12 i.e., ρ = 1, while the receiver U 1 employs SIC solely to remove the interference from s 12 and hence R s12 1 = 0. Since we consider minimum required rates for the common and private streams in RS, in order to compare with NOMA, we consider that the minimum required rates at U 1 , U 2 are u log 2 (1 + ζ) + log 2 (1 + ξ) and (1 − u) log 2 (1 + ζ) + log 2 (1 + ξ), respectively. Therefore, the average rate at the receivers U 1 and U 2 are given by R 1 = R p 1 (w 1 , w 2 ) + R pI 1 (w 1 , w 2 ) and R 2 = R c 2 (w 2 ), respectively; where w 1 = (1 + ζ) u (1 + ξ) − 1 and w 2 = (1 + ζ) (1−u) (1 + ξ) − 1.
In Fig. 2, we present the average rates R 1 , R 2 and the common rate R c 12 with respect to β, for u = {0.3, 0.5}. As can be seen, the common rate R c 12 increases with β since more power is allocated to the stream s 12 . In addition, while β < ζ 1+ζ then R c 12 = 0 due to the upper limit ϑ 1 , as explained in Propositions 1 and 2. Similarly, we can see that R 1 and R 2 decrease for values 0 < β < ζ 1+ζ , while after β = ζ 1+ζ , both rates are increased. This is due to the fact that before that critical value of β, the total rate at U n is obtained solely from the contribution of R sn , which decreases as β increases since less power is allocated to the private streams. On the other hand, when R c 12 > 0 we can see that R 1 and R 2 perform better, as expected. An interesting observation is that for β = 1, R 1 becomes lower while R 2 keeps performing better. The reason for that is on one hand the zero rate from the private stream and on the other hand the rate R s12 1 is limited by η 2 i.e., d 1 < d 2 as shown in (11). Furthermore, with higher β, higher π η c n (ζ) is achieved and for U 1 means lower R c 1 (see (14)). However, for U 2 , due to d 1 < d 2 the higher β the higher rate is achieved since the partial interference cancellation from decoding s 12 is more critical. Finally, we can see that with lower u, we can boost the low performance of U 2 and as in the case of u = 0.3, with proper design we can have R 1 = R 2 .
In Fig. 3, we plot R 1 , R 2 and the common rate R c 12 with respect to ρ for β = {0.6, 0.9}, while indicating the minimum rate achieved for each value of β . It can be observed that R 1 increases with ρ while R 2 decreases. This is expected since (1−β)ρ is the power allocated to s 1 , while the power allocated to s 2 is (1−β)(1−ρ). From the upper bound ϑ 2 we can see that R s1 is zero for ρ < ξ 1+ξ and R 2 is zero for ρ > 1 1+ξ . As such, for ρ < ξ 1+ξ , R 1 is obtained solely from the common rate's contribution where for higher ρ, R 1 increases proportionally with ρ due to the contribution of both; common and private rates. On the other hand, we can see that R 2 decreases with ρ until ρ = 1 1+ξ . After that point R s1 = 0, hence R 1 becomes independent of ρ and remains constant. Furthermore we can see that a higher β is beneficiary for U 1 at low values of ρ and more beneficiary for U 2 at high values of ρ. This is expected since in those cases β offers a balance for the rate that was lost due to the decrease in R sn . Finally, we can see that the maximum value obtained from the min{R 1 , R 2 } is higher with higher β.
In order to further investigate the behavior of the minimum achieved rate, we plot in Fig. 4, the maximum value of min{R 1 , R 2 } which is obtained for each β, and we compare with the NOMA scheme. Note that, in the case of NOMA, β accounts for the power allocated to U 2 while the remaining power (1 − β) is allocated to U 1 . Due to that, we can see that the max-min achieved by NOMA is zero for β < w2 1+w2 which follows from the upper limit ϑ 1 and the substitution of ζ with w 2 . On the other hand, when β > w2 1+w2 , U 2 successfully decodes its message and with higher β more power is assigned to U 2 implying that the max-min is obtained from the performance of U 1 . Different from NOMA, the flexibility of the RS scheme, enables non-zero rates of the max-min performance for more values of β. This is expected since with RS each receiver can decode either the common or the private stream, or both as explained above. Furthermore, along with ρ and β, in RS we can also adjust u based on the system's requirements in order to boost the performance of either receiver, while achieving non-zero rate at the one with the worst performance.
Finally, in Fig. 5 we plot the sum rate R s with respect to P for different power allocations and thresholds. As can be seen while the power is increasing, the sum rate also increases until reaching the asymptotic bound. Furthermore, we can see that with higher β a higher rate is achieved since the partial interference cancellation is employed with higher probability. Furthermore, we can see that by increasing ρ we achieve higher R s , since we further improve the performance of U 1 which due to d 1 < d 2 performs better than U 2 .

V. CONCLUSIONS
In this paper we studied the performance of a fixed power allocation RS following a probabilistic approach. By considering spatial randomness and based on the channel statistics, we derived the average rates at each receiver obtained from decoding the private and the common stream. Within our expressions we captured the impact of the different design parameters on the individual rates. Our analysis was validated with our numerical results and we discussed the effect of the resource coefficients. We showed that the RS allows a more flexible design against NOMA for serving both receivers with non-zero rate.

A. Proof of Proposition 1
Consider the SINR given by (1) for n = 1 i.e., for receiver U 1 . The coverage probability is evaluated as follows Based on the upper bound ϑ 1 given in (6), we can see that for t > β 1−β , the coverage probability is zero. Hence, for t < ϑ 1 , where (24) follows from h 1 ∼ exp(1) and f d1 (x) = 1/πR 2 0 is the PDF of d 1 . We follow the same procedure for the SINRs η p 1 and η pI 1 and the final general expression for the coverage probability is obtained from [13, 3.326.4].
Since the coverage probability π η (t) is the complementary cumulative distribution function (CCDF) of SINR, we can use it to derive the PDF of η i.e., g η (t) = d (1 − π η (t)) /dt. Hence, by using (26), the PDF of η c 1 is evaluated as follows where the integral is solved with the help of [13, 3.326.4].

B. Proof of Proposition 3
The average rate for s 1 , when s 12 is successfully decoded, is given by We can deduce from η c 1 > ζ and η p 1 > ξ that respectively. When , the average rate for s 1 reaches its upper limit which is ϑ 2 , and the conditional private rate is evaluated as On the other hand, when , and so by solving the inequality with respect to ξ, the upper limit of the conditional rate E [log 2 (1 + η p 1 ) | η c 1 < ζ, η p 1 > ξ], is given by ξ < (1−β)ρζ βρζ+β−ρζ . Hence, in this case, R p 1 can be evaluated as For the derivation of R p 2 , we substitute η c 1 and η p 1 with η c 2 and η p 2 , respectively, and follow similar procedure as above.

C. Proof of Proposition 4
The average rate for s 1 , when s 12 adds interference to the received SINR, is given by If ζ > ϑ 1 then the rate becomes independent of the condition η c 1 < ζ and thus R pI 1 = E log 2 (1 + η pI 1 ) | η pI 1 > ξ = ϑ3 ξ log 2 (1 + t)g η pI On the other hand, if ζ < ϑ 1 , we follow similar procedure as in Appendix B by solving the inequalities with respect to h 1 , which gives For the case i.e., (35) we evaluate the upper limit of R pI 1 , by solving with respect to ξ, and is given by ξ < (1−β)ρζ β(ρζ+ζ+1)−ρζ . Finally, when , we have η c 1 > ζ. In this case, s 12 is successfully decoded and removed, which implies s 1 is decoded with rate R p 1 . As a result, R pI 1 = 0.