On the Sum Rate of MISO Rate Splitting with Spatial Randomness

—Efﬁcient and intelligently designed multiple access schemes constitute a promising technology for achieving massive connectivity, required by the forthcoming ﬁfth generation and beyond networks. In this work, we investigate the performance of the sophisticated rate splitting (RS) scheme for a multiple-input single-output broadcast channel with two receivers. We consider a distance disparity among the receivers and incorporate a simple precoding which reveals the gains of RS. By taking into account spatial randomness, we provide closed form expressions for the coverage probabilities of each receiver and derive the average sum rate achieved with the employment of RS. We present numerical results that validate our analysis and show that RS brings signiﬁcant performance gains compared to the well-known non-orthogonal multiple access scheme, while it consists of a general and more ﬂexible scheme.


I. INTRODUCTION
With the development of the Internet of Things (IoT), massive machine type networks coexisting with the cellular network consist of the new paradigm of the fifth generation (5G) and beyond networks [1]. Efficient resource management is therefore required for the forthcoming systems, in order to provide coverage to a tremendous amount of devices. Towards this direction, the employment of intelligently designed multiple access schemes is a promising technology to achieve massive connectivity [2]. Previous generations have successfully adopted orthogonal multiple access (OMA) schemes to allocate the available resources to multiple users. Such schemes include, among others, the orthogonal frequency division multiple access (OFDMA), adopted by the 4G systems, and the space division multiple access (SDMA), which is realized with the utilization of multiple antennas and appropriate precoding at the transmitter.
Over the recent years, non-orthogonally designed schemes have attracted significant attention by both industry and academia, since they exploit the available resources more efficiently, increasing the network's throughput. A very well known scheme that falls under this category constitutes the power-domain non-orthogonal multiple access (NOMA).
This work was co-funded by the European Regional Development NOMA is employed with superposition coding at the transmitter and successive interference cancellation at the receivers' side, taking place in an ascending order of their channel gain [3]. In the presence of channel gain disparity and with proper parameters design, it is shown that the single-antenna NOMA outperforms OMA, in terms of both the achievable sum rate [4] and network fairness [5]. Motivated by the gains associated with the single-antenna setups, the research community has shown interest in investigating NOMA with multiple antennas. Recently though, NOMA in multiple antennas setups was shown to be inefficient while experiencing performance loss in comparison with the rate splitting (RS) scheme [6].
RS was first introduced in [7] and more recently, the research community shed light on the gains of RS as a general scheme. More precisely, it is shown in [8] that, RS in a multiple-input single-output (MISO) broadcast channel is a generalization of the well known schemes NOMA and SDMA, while outperforming both in terms of spectral efficiency. The investigation of RS as a general scheme is also presented in [9] for a two-user multi-antenna broadcast channel, where the authors indicate the optimal scheme that RS can specialize to, in regards to the channel disparities and directions. Moreover, a system level analysis on RS is derived in [10], for a single antenna setup which highlights the flexibility of RS as well as the gains brought in terms of network fairness compared to NOMA. Similar gains are also presented in [11], where the authors take into account channel state information errors and investigate a RS-based design for a downlink multiuser MISO system. Furthermore, a degrees of freedom analysis is applied in [12], where the authors show the fairness gains of RS in multiple multicast groups. RS assisted multicast and unicast transmission systems are also studied in [13], where the superiority of RS is demonstrated, in terms of both spectral and energy efficiency, against NOMA and multi-user linear precoding.
Motivated by the above, in this paper we investigate the performance of RS in a MISO setup by taking into account spatial randomness. Similar to [9], we focus on a two-receiver network, while considering a distance disparity among the receivers. We provide a system level analysis for a simple distance-based precoding, which compensates the receivers' disparity. By using tools from stochastic geometry, we derive in closed-form-expressions the coverage probabilities of the two receivers and provide the average performance of RS in terms of sum rate. Our results show that RS can provide significant gains in terms of the average sum rate against NOMA, which is usually employed in the presence of distance disparity [14]. Notation: P [X] represents the probability of the event X with expected value E [X]; bold lower and upper case letters represent vectors and matrices, respectively; superscripts T and H correspond to the transpose and Hermitian operations, respectively and γ(·, ·) and Γ(·, ·) denote the lower and upper incomplete Gamma functions, respectively.

A. System Model
Consider a single cell downlink network consisting of a transmitter equipped with two antennas 1 scheduling two single-antenna receivers. The scheduled receivers are selected such that there is distance disparity among them [14]. Let r 0 , r 1 , and r 2 denote the radii of three disks centered at the transmitter where r 0 < r 1 < r 2 . The first receiver is uniformly distributed within the disk of radius r 0 , while the second receiver is distributed within the annulus shaped by the difference of disks with radii r 1 and r 2 . We consider that all the radio links are affected by small-scale block Rayleigh fading. We denote the channel between the transmitter and the k-th receiver, k ∈ {1, 2}, by h k ∈ C 1×2 with complex Gaussian entries i.e., CN (0, 1), and we assume that the transmitter has knowledge of the channel. Furthermore, we take into account the large-scale path-loss attenuation and consider a powerlaw distance attenuation model expressed by ( where α > 2. Finally, we take into account the additive white Gaussian noise with variance σ 2 .

B. Rate Splitting
The transmitter employs the RS scheme in order to simultaneously deliver the messages m 1 and m 2 to the first and second receiver, respectively. The two messages are split into the so-called common and private parts i.e., The parts m c 1 and m c 2 are encoded and transmitted in a common stream s 0 and the parts m p 1 and m p 2 are transmitted over two private streams s 1 and s 2 , respectively [8]. The streams denoted with s = [s 0 , s 1 , s 2 ] T are then linearly precoded with W = [w 0 , w 1 , w 2 ], where w n ∈ C 2×1 corresponds to the precoding vector for the stream s n , n ∈ {0, 1, 2}. Let P denote the total transmit power. Then, the power allocated to each stream is given by p 0 = βP, p 1 = (1 − β)ρP and p 2 = (1 − β)(1 − ρ)P , respectively, where β, ρ ∈ [0, 1]. As such, the precoding vector can be expressed by w n = √ p nŵn , whereŵ n represents the direction (unitnorm) vector. Therefore, the signal at the output of the transmitter is The superimposed signal is decoded by the receivers in two steps. In particular, each receiver decodes firstly the common 1 We consider two antennas for the sake of simplicity while the analytical framework can serve as a guideline for more complex configurations. stream s 0 , by treating the private streams as noise [9]. As such, the received signal-to-interference-plus-noise ratio (SINR) at the k-th receiver for decoding the common stream is given by where k = i. This step is also known as partial interference cancellation since once the common stream is decoded, it is extracted from the received signal and then the receivers decode their corresponding private stream without interference from the common stream. Furthermore, since s 0 should be decoded by both receivers, the rate of the common stream is given by R 0 = min{log 2 1 + η 0 1 , log 2 1 + η 0 2 }, which is then split and allocated to each of the two receivers [9].
Once the common stream is extracted, the private stream s k is decoded by the k-th receiver with SINR given by and the rate of each receiver for decoding s k is given R k = log 2 (1 + η k ).

C. Precoding 1) Common Stream:
Recall that the common stream is dedicated to both receivers with rate R 0 = min{log 2 1 + η 0 1 , log 2 1 + η 0 2 }. In order to compensate for the distance disparity among the receivers, we consider maximal ratio transmission (MRT) to the direction of the receiver with the larger distance from the transmitter 2 . As such, the precoding direction vector is given byŵ 0 = h H 2 ||h2|| . Note that, when h 1 ⊥ h 2 , then η 0 1 = 0. However, if indeed the two channels are orthogonal, then the optimal scheme is SDMA [9], which can be simply achieved by setting β = 0 as it constitutes a special case of the RS scheme. Besides, in this work, we consider the above precoding and focus on the average performance of the network, averaging over all network realizations.
2) Private streams: For the private streams we consider zero-forcing beamforming (ZFB) similar to [9]. That is, the precoding vector is given by the projection of the desired channel onto the null space of the interference term to be eliminated.
where I is the 2 × 2 identity matrix and i ∈ {1, 2}. Then the direction vector for the stream Note that, since we consider ZFB, then h k ⊥ŵ i and ||h kŵi || 2 = 0 eliminating in both (1) and (2), the interference which occurs from the private stream s i . The special case where β = 0 with ρ = 1 or ρ = 0 corresponds to scheduling a single receiver (OMA). Since there is no interference, then MRT can be used instead. Therefore, in this case the direction vector for the stream k is given byŵ k = h H k ||h k || . 2 MRT provides to η 0 2 a balance for the loss due to the distance disparity. Besides, MRT is also the basis of more intelligently designed precoding such as linear combination of MRTs to the direction of each receiver [9].
Based on the proposed precoding and by taking into account distance disparity among the receivers, we evaluate in the following the average sum rate for the MISO RS.

III. AVERAGE SUM RATE
We now derive the average sum rate which is our key performance metric. For this purpose, we first obtain the average rate of each stream by utilizing the coverage probability of the received SINRs. That is, the probability that the received SINR η exceeds a predefined threshold t, denoted by π(t).
Note that η 0 1 is upper-bounded by lim P →∞ η 0 1 = p0(1−δ) p1δ and is not bounded when p 1 δ = 0. Then, the coverage probability is evaluated by the following proposition. Proposition 1. The coverage probability of the first receiver, decoding the common stream, is given by for t < x y , and is zero for t > x y , where x = p 0 (1 − δ) and y = p 1 δ.

Proof. See Appendix A.
We can now derive the average rate R 0,1 which is given as follows where (8) occurs from the fact that π 0 1 (t) is the complementary cumulative distribution function (ccdf) of η 0 1 , and by making use of the upper bound of η 0 1 ; then the expectation is taken over δ ∈ [0, 1] in order to evaluate the average over the channel directions. Note that when p 1 δ = 0, the upper limit in the inner integral becomes infinity.
We now focus on deriving R 0,2 . By following a similar procedure as above, we can express η 0 2 as follows which is upper-bounded by lim P →∞ η 0 1 = p0 p2δ , while it becomes not bounded when p 2 δ = 0. Then, the coverage probability π 0 2 (t) is given in the following proposition. Proposition 2. The coverage probability of the second receiver decoding s 0 is evaluated by for t < x y , and is zero for t > x y , where x = p 0 and y = p 2 δ. Proof. The proof is similar to the one provided for Proposition 1, with the substitution of f d1 with f d2 (u) = 2 By utilizing Proposition 2, we evaluate the rate R 0,2 in a similar way to R 0,1 and is given by Provided with R 0,1 and R 0,2 we evaluate the average rate of the common stream by substituting in (3).

B. Private stream rate
We now focus on deriving the average rate of each private stream. For that purpose, we need to derive the coverage probabilities for the two cases where ZFB or OMA are employed. We first consider ZFB, and provide in the following proposition the coverage probability of the two receivers for decoding their corresponding private stream. Proposition 3. When ZFB is employed, the coverage probability of the first receiver decoding s 1 is given by and the coverage probability of the second receiver decoding s 2 is given by Proof. See Appendix B.
In the case where a single receiver is scheduled (OMA), then the coverage probability of the scheduled receiver is evaluated by utilizing the following lemma. Lemma 1. When β = 0 and ρ = 1, then only the first receiver is scheduled. In this case, the coverage probability for decoding s 1 is evaluated by (11) with the substitution of x = P and y = 0.
When β = 0 and ρ = 0, then only the second receiver is scheduled and the coverage probability for decoding s 2 is evaluated by (7) with the substitution of x = P and y = 0.
Proof. See Appendix C.
Since the common stream is extracted and there is no interference term in η k , k ∈ {1, 2}, then η k is not bounded and the average rate is independent from the angle between the two channels. Therefore, by making use of the coverage probability π k (t), the average rate of the private stream s k is evaluated by Then, the average sum rate is given by R s = R 0 +R 1 +R 2 .

IV. NUMERICAL RESULTS
We now provide numerical results in order to evaluate the performance of the MISO RS scheme and the proposed precoding for receivers with distance disparity. We focus on the high signal-to-noise-ratio (SNR) regime and unless otherwise stated, we consider the following P = 1 W, σ 2 = 10 −6 , α = 3, r 0 = 10 m, r 1 = r 0 + g m and r 2 = r 1 + g m, where g is the minimum distance disparity among the receivers.
In Fig. 1 we plot the average sum rate with respect to ρ which corresponds to the portion of power allocated to each private stream. We plot the sum rate for β = {0, 0.2, 0.6} in order to compare with SDMA (β = 0) and for g = {5, 10} in order to compare between smaller and larger distance disparity among the receivers, respectively. As can be seen our analytical results indicate that (3) consists of a tight approximation. The OMA scheme is depicted at β = 0, ρ = 0 and β = 0, ρ = 1 corresponding to serving only the second and the first receiver, respectively. In all plots, we can see that the lower rate is achieved when solely the second receiver receives a private stream i.e., ρ = 0. This is due to the larger path-loss attenuation that occurs from the larger euclidean distance from the transmitter. As ρ increases, both receivers' private streams rates contribute to the sum rate which is subsequently increasing as well. On the other hand, when more power is allocated to s 1 , then R 2 decreases hence the sum rate has mainly contribution by the first receiver's rate. As such, the sum rate decreases again reaching the rate achieved when solely the first receiver receives a private stream. This point corresponds to the NOMA scheme which is clearly outperformed by the RS scheme. Furthermore, it is clear that with g = 10, the second receiver has larger distance from the transmitter, which results in a lower sum rate. Finally, we can see that with 20% of the available power dedicated to the common stream, RS can also provide gains in comparison to SDMA. In Fig. 2, we plot the average sum rate with respect to β which corresponds to the power allocated to the common stream. We plot the sum rate for ρ = {0.5, 1} in order to further compare with the NOMA scheme. As can be seen, RS achieves significant gains compared to NOMA (ρ = 1), while the two schemes converge to multicasting when β = 1 which results in the lowest sum rate. Furthermore, the average sum rate rate achieved by NOMA, is dominated by the private stream s 1 . As a result, the distance disparity brings negligible impact on the sum rate and the highest sum rate is achieved when only the first receiver is scheduled i.e., the one closer to the transmitter. On the other hand, RS outperforms both OMA and NOMA by transmitting three non-orthogonal streams. With equal power allocation among the two private streams i.e., ρ = 0.5 RS outperforms NOMA for both g = 5 and g = 10. However, we can see that, as β increases, after a certain point, a lower sum rate occurs, since less power is allocated to the private streams. This is expected since the rate of the common stream is limited by min{R 0,1 , R 0,2 }, while on the other hand, the private streams can be decoded without interference. Therefore, a careful power allocation for the three streams can bring significant gains to the sum rate in comparison to NOMA.

V. CONCLUSIONS
In this paper we studied the performance of a MISO RS by taking into account spatial randomness. We investigated a simple precoding scheme, appropriate for scenarios with distance disparity among the receivers. By using tools from stochastic geometry, we derived in closed-form expressions the coverage probability of decoding each stream at the receivers and provided the average sum rate. We presented numerical results to validate our analysis and we showed that the proposed distance-based precoding reveals the gains of RS providing significant sum rate enhancements in comparison to the NOMA scheme.

B. Proof of Proposition 3
When β ∈ (0, 1) i.e., ZFB is employed, then the coverage probability of the first receiver decoding s 1 is expressed by π 1 (t) = P ||h 1ŵ1 || 2 > tσ 2 (1+d α 1 ) p1 . It follows from the ZFB precoding that ||h 1ŵ1 || 2 ∼ exp(1). As such, π 1 (t) is evaluated as follows where in (17), we make use of f d1 (u) in a similar way to Proposition 1. Then the integral is solved with the help of [15, 3.326.4]. By following a similar procedure we derive the coverage probability of the second receiver decoding s 2 by substituting f d1 with f d2 (u) = 2 r 2 2 −r 2 1 .
Similarly, we can evaluate the coverage probability of the second receiver when β = 0 and ρ = 0 by utilizing (11) with x = P and y = 0.