On the Design of MIMO-NOMA Downlink and Uplink Transmission

Transcription

1 On the Design of MIMO-NOMA Downlink and Uplink Transission Zhiguo Ding, Meber, IEEE, Robert Schober, Fellow, IEEE, and H. Vincent Poor, Fellow, IEEE Abstract In this paper, a novel MIMO-NOMA fraework for downlink and uplink transission is proposed by applying the concept of signal alignent. By using stochastic geoetry, closedfor analytical results are developed to facilitate the perforance evaluation of the proposed fraework for randoly deployed users and interferers. The ipact of different power allocation strategies, such as fixed power allocation and cognitive radio inspired power allocation, on the perforance of MIMO-NOMA is also investigated. Coputer siulation results are provided to deonstrate the perforance of the proposed fraework and the accuracy of the developed analytical results. I. INTRODUCTION Non-orthogonal ultiple access NOMA has been recognized as a spectrally efficient ultiple access MA technique for the next generation of obile networks [] and []. For exaple, the use of NOMA has been recently proposed for downlink scenarios in long-ter evolution LTE systes as well as in fifth generation 5G obile networks [3] [5]. The key idea of NOMA is to exploit the power doain for ultiple access, which eans ultiple users can be served concurrently at the sae tie, frequency, and spreading code. The perforance of NOMA in a network with randoly deployed single-antenna nodes was investigated in []. User fairness in the context of NOMA was addressed in [6], where power allocation was optiized under different assuptions regarding the channel state inforation CSI. The application of ultiple-input ultiple-output MIMO techniques to NOMA is iportant since the use of MIMO provides additional degrees of freedo for further perforance iproveent. In [7], the ultiple-input single-output MISO scenario, where the base station had ultiple antennas and users were equipped with a single antenna, was considered. Ref. [8] focused on a special case of MIMO-NOMA with two users, where a throughput axiization proble was forulated and two algoriths were proposed to solve the optiization proble. In any practical scenarios, it is preferable to serve as any users as possible in order to reduce user latency and iprove user fairness. Following this rationale, in [9], users were first grouped into sall-size clusters, where NOMA was ipleented for the users within each cluster and MIMO detection was used to cancel inter-cluster interference. Siilar to [0], this ethod does not need CSI at the base station; however, unlike [0], it avoids the use of rando beaforing which can cause uncertainties for the quality of service QoS experienced by the users. This paper considers a general MIMO-NOMA counication network where a base station is counicating with ultiple users using the sae tie, frequency, and spreading Z. Ding and H. V. Poor are with the Departent of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA. Z. Ding is also with the School of Coputing and Counications, Lancaster University, LA 4WA, UK. R. Schober is with the Institute for Digital Counications, University of Erlangen-Nurnberg, Gerany. code resources, in the presence of randoly deployed interferers. The contributions of this paper are listed as follows. Firstly, a general MIMO-NOMA fraework which is applicable to both downlink and uplink transission is proposed, by applying the concept of signal alignent, originally developed for ulti-way relaying channels in [] and []. By exploiting this fraework, the considered ulti-user MIMO-NOMA scenario can be decoposed into ultiple separate single-antenna NOMA channels, to which conventional NOMA protocols can be applied straightforwardly. Secondly, since the choice of the power allocation coefficients is key to achieve a favorable throughput-fairness tradeoff in NOMA systes, two types of power allocation strategies are studied in this paper. The fixed power allocation strategy can realize different QoS requireents in the long ter, whereas the cognitive radio inspired power allocation strategy can ensure that users QoS requireents are et instantaneously. Finally, exact expressions and asyptotic perforance results are developed in order to obtain an insightful understanding of the proposed MIMO-NOMA fraework. In particular, the outage probability is used as the perforance criterion since it not only bounds the error probability of detection tightly, but can be also used to calculate the outage capacity/rate. The ipact of the rando locations of the users and the interferers is captured by applying stochastic geoetry, and the diversity order is coputed to illustrate how efficiently the degrees of freedo of the channels are used by the proposed fraework. II. SYSTEM MODEL Consider a MIMO-NOMA downlink uplink counication scenario in which a base station is counicating with ultiple users. The base station is equipped with M antennas and each user is equipped with N antennas. In this paper, we consider the scenario N > M in order to ipleent the concept of signal alignent, an assuption ore general than the one used in [9]. This assuption is applicable to various counication scenarios, such as sall cells in heterogenous networks and 5G cloud radio access networks. The users are assued to be uniforly deployed in a disc, denoted by D, i.e., the cell controlled by the base station. The radius of the disc is r, and the base station is located at the center of D. In order to reduce the syste load, several existing studies on NOMA have proposed to pair two users for the ipleentation of NOMA, and have deonstrated that it is ideal to pair two users whose channel conditions are very different [3], [4]. Based on this insight, we assue that the disc is divided into two regions. The first region is a saller disc, denoted by D, with radius r r < r and the base station located at its origin. The second region is a ring, denoted by D, constructed fro D by reoving D. Assue that M pairs of users are selected, where user, randoly located in D, is paired with user, randoly located in D.

2 Hence, the users are randoly scheduled and paired together. The use of ore sophisticated schedulers can further iprove the perforance of the proposed MIMO-NOMA fraework of course, but this is beyond the scope of this paper. In addition to the essages sent by the base station, the downlink NOMA users also observe signals sent by interference sources which are distributed in R according to a hoogeneous Poisson point process PPP Ψ I of density λ I [5]. The sae assuption is ade for the uplink case. In practice, these interferers can be cognitive radio transitters, WiFi access points in LTE in the unlicensed spectru LTE- U, or transitters fro different tiers in heterogenous networks. In order to obtain tractable analytical results, it is assued that the interference sources are equipped with a single antenna and use identical transission powers, denoted by ρ I. Consider the use of a coposite channel odel with both quasi-static Rayleigh fading and large scale path loss. In particular, the channel atrix fro the base station to user is H = G, where G denotes an N M atrix Ld whose eleents represent Rayleigh fading channel gains, d denotes the distance fro the base station to the user, and the path loss is odelled as follows: { d α Ld =, if d > r 0 r0 α,, otherwise where α denotes the path loss exponent and paraeter r 0 avoids a singularity when the distance is sall. It is assued that r r 0 in order to siplify the analytical results. For notational siplicity, in case of uplink transission, the channel atrix fro user to the base station is denoted by H H, where H denotes the Heritian operator. We denote the transpose operator by T. Global CSI is assued to be available at the users and the base station. The proposed MIMO-NOMA fraework for downlink and uplink transission is described in the following two subsections, respectively. A. Downlink MIMO-NOMA Transission The base station sends the following [ M inforation-bearing vector ] s = T α s + α s α M s M + α M s M, where s is the signal intended for the -th user, α is the power allocation coefficient, and α + α =. The choice of the power allocation coefficients will be discussed later. Without loss of generality, we focus on user, whose observation is given by y = G Ld Ps + w I + n, where P is the M M precoding atrix to be defined at the end of this subsection, w I denotes the overall co-channel interference received by user, and n denotes the noise vector. Following the classical shot noise odel in [6], the co-channel interference, w I, can be expressed as follows: ρi LdIj, N, w I j Ψ I where denotes an all-one vector, and d Ij, denotes the distance fro user to the j-th interference source. Note that sall scale fading has been oitted in the interference odel, since the effect of path loss is ore doinant for interferers located far away. In addition, this siplification will facilitate the developent of tractable analytical results. User applies a detection vector v to its observation, and therefore the user s observation can be re-written as follows: G vy H = v H pαs + α s 3 Ld G + v H p iα i s i + α i s i + vw H I + n, i Ld } {{ } interference including inter pair interference + noise where p denotes the -th colun of P. In order to reove inter-pair interference, the following constraint has to be et: [ ] v H G v H G p i = 0, i, 4 where 0 n denotes the n all zero atrix. Without loss of generality, we focus on p which needs to satisfy the following constraint: [ G H v G H v GH M v M G H M v M ] H p = 0. 5 It is easy to verify that for general v a non-zero vector p i satisfying 5 does not exist. To overcoe this proble, in this paper, the concept of interference alignent is applied, which eans the detection vectors are designed to satisfy the following constraint [7], [8] vg H = v H G, 6 or equivalently [ ] [ ] G H G H v = 0 M. Define U v as the N N M atrix containing the N M right singular vectors of [ ] G H G H corresponding to its zero singular values. Therefore, the detection vectors at the users are designed as follows: [ ] v = U x, 7 v where x is a N M vector to be defined later. We noralize x to, i.e., x =, due to the following two reasons. First, the uplink transission power has to be constrained as shown in the following subsection. Second, this facilitates the perforance analysis carried out in the next section. It is straightforward to show that the choice of the detection vectors in 7 satisfies [ ] G H G H U x = 0 M. The effect of the signal alignent based design in 6 is the projection of the channels of the two users in the sae pair into the sae subspace. Define g G H v as the effective channel vector shared by the two users. As a result, the nuber of rows in the atrix in 5 can be reduced significantly. In particular, the constraint for p i in 5 can be rewritten as follows: [ ] H g g i g i+ g M pi = 0 M. 8

3 3 Note that [ g g i g i+ g M ] H is an M M atrix, which eans that a p i satisfying 8 exists. Define G [ g g M ] H. A zero forcing based precoding atrix at the base station can be designed as follows: P = G H D, 9 where D is a diagonal atrix to ensure power noralization at the base station, i.e., D = diag{ G G H,,, G G H M,M }, where A, denotes the -th eleent on the ain diagonal of A. As a result, the transission power at the base station can be constrained, i.e., tr { PP H} ρ = Mρ, where ρ denotes the transit signal-to-noise ratio SNR. With the design in 6 and 9, the signal odel for user can now be written as follows: v H y = α s + α s LdG G H, + v H w I + n. 0 For notational siplicity, we define h = and h Ld G G H =., Ld G G H, The use of the signal alignent based precoding and detection atrices decoposes the ulti-user MIMO-NOMA channels into M pairs of single-antenna NOMA channels. Therefore, it is iportant to point out that h and h share the sae sall scale fading gain with different distances. Recall that two users belonging to the sae pair are selected fro D and D, respectively, which eans that d < d. Therefore, the two users fro the sae pair are ordered without any abiguity, which siplifies the design of the power allocation coefficients, i.e., α α, following the NOMA principle. User decodes its essage with the following signal-to-interference-plus-noise ratio SINR ρ h α SINR = ρ h α + v + v H, N I where the interference ter is given by I = ρ I L. d Ij, j Ψ I User carries out successive interference cancellation SIC by first reoving the essage to user with SINR, ρ h SINR, = α ρ h α + v + v H, and then decoding its own essage with N I SINR SINR = which becoes the SNR if ρ I = 0. B. Uplink MIMO-NOMA Transission ρ h α v + v H N I, 3 For the NOMA uplink case, user will send out an inforation bearing essage s, and the signal transitted by this user is denoted by α v s. Because of the reciprocity between uplink and downlink channels, v, which was used as a downlink detection vector, can be used as a precoding vector for the uplink scenario. Siilarly, P will be used as the detection atrix for the uplink case. In this paper, we assue that the total transission power fro one user pair is noralized as follows: α v + α v ρ. 4 The base station observes the following signal: M G H y BS = α v s + GH α v s Ld Ld = + w I + n BS, where w I is the interference ter defined as follows w I j Ψ I ρi 5 L M. 6 d Ij,BS Here, d Ij,BS denotes the distance between the base station and the j-th interferer, and the noise ter is defined siilarly as in the previous sub-section. The base station applies a detection atrix P to its observations and the sybols fro the -th user pair can be detected based on p H y BS =p H + p H G H α v s + GH α v s Ld Ld i + p H w I + n BS. G H i α iv i s i + GH i α i v i s i Ldi Ldi 7 In order to avoid inter-pair interference, the following constraint needs to be et p H G H i α iv i s i + GH i α i v i s i = 0, i. 8 Ldi Ldi i Again applying the concept of signal alignent, the constraint that G H v = G H v is iposed on the precoding vectors v. Therefore, the sae design of v as shown in 7 can be used. The total transission power within one pair is constrained, i.e., ρα v +ρα v ρ, which eans that the use of the precoding vector in 7 ensures that the total transission power of one user pair is constrained. Following steps siilar to those used in the previous subsection, one can observe that the use of the proposed precoding/detection atrices can decopose the original MIMO- NOMA channels into separated SISO-NOMA channels. Due to the space liitation, we will only focus on the downlink case in the rest of this paper. More details about the uplink case can be found in the journal version of this paper [9]. III. PERFORMANCE ANALYSIS FOR DOWNLINK MIMO-NOMA TRANSMISSION Two types of power allocation policies are considered in this section. One is fixed power allocation and the other is inspired by the cognitive radio concept, as illustrated in the following two subsections, respectively. Recall that the precoding vectors v and v are deterined by x as shown in 7. In this paper, a rando choice of x is considered.

4 4 A. Fixed Power Allocation In this case, the power allocation coefficients α and α are constant and not related to the instantaneous realizations of the fading channels. We will first focus on the outage perforance of user. The outage probability of user to decode its inforation is given by P o = P log + SINR < R, where Px < a denotes the probability of the event x < a. The correlation between v and h akes the evaluation of the above outage probability very challenging. Hence, we focus on the following odified expression for the outage probability ρ h P α = P log + ρ h α < R, + + δi where δ 0. Since v + v =, we have v and v. In addition, because N N n= x n N N n= x n, v H N N v. Therefore, we have P o P, 9 for δ N, which eans that P provides an upper bound on P if δ N. Note that a choice of δ = is sufficient to ensure that P provides a very tight approxiation to P as shown in [9]. In addition, the use of P will be sufficient to identify the achievable diversity order of the proposed MIMO- NOMA schee. Given a rando choice of x, the following lea provides an exact expression for P as well as its high SNR approxiation. Lea. If α α ϵ, the probability P =, where ϵ = R. Otherwise, the probability P can be expressed as follows: P = r r where ϕ = e πλ I β x α γ α, β x r 0 α r e ϕ α x r φ I xxdx, 0 ϵ ρα ρα ϵ, φ Ix =, β x = ϕ δρ I L x α, and γ denotes the incoplete Gaa function. If ρ I is fixed and transit SNR ρ approaches infinity, the outage probability can be approxiated as follows: P ϕ + θ r r r α+ r α+ α +, where θ α = πλ I δρ I r 0. When ρ I = 0, P siplifies to P = e ϕ rα r e ϕ rα r ϕ α r r r r γ α +, ϕ rα γ α +, ϕ rα. Proof: Please refer to Appendix A in [9], which is the journal version of this paper By using the high SNR approxiation in Lea and also the fact that both ϕ and θ are at the order of ρ, the following corollary can be obtained straightforwardly. Corollary. If α > α ϵ, the diversity order achieved by the proposed MIMO-NOMA fraework for user is one. On the other hand, user first decodes the essage for user before decoding its own essage via SIC. Again, we focus on a odified expression for the outage probability as follows: ρ h P α = P log + ρ h α < R δi + P log + ρ h α < R, log + + δi ρ h α ρ h α > R + + δi, which is an upper bound for δ N as explained in the proof for Lea. The following lea provides an exact expression for this probability as well as its high SNR approxiation. Lea. If α α ϵ, the probability P =. Otherwise, the probability P can be expressed as follows: P = r r r r 0 r0 0 e ϕ r α 0 φi r 0 xdx 4 e α ϕ x φ I xxdx, where ϕ = ax{ϕ, ϕ } and ϕ = ϵ ρα. If ρ I is fixed and the transit SNR ρ approaches infinity, the outage probability can be approxiated as follows: P ϕ + θ αr α+ r α r α+, 5 where θ was defined in Lea. Proof: Please refer to Appendix B in [9]. B. Cognitive Radio Power Allocation In this section, a cognitive radio inspired power allocation strategy is studied. In particular, assue that user is viewed as a priary user in a cognitive ratio network. With orthogonal ultiple access, the bandwidth resource occupied by user cannot be reused by other users, despite its poor channel conditions. In contrast, with NOMA, one additional user, i.e., user, can be served siultaneously, under the condition that the QoS requireents of user can still be et. In particular, assue that user needs to achieve a target data rate of R, which eans that the power allocation coefficients of NOMA need to satisfy the following constraint ρ h α ρ h α + v + v H N I > ϵ, 6 which leads to the following choice for α α = ax 0, ρ h ϵ v + v H N I + ϵ ρ h. 7 It is straightforward to show that ρ h ϵ v + v H N I +ϵ ρ h is always less than one. An outage at user eans here that all power is allocated to user, but outage still occurs. As a result, the outage probability of user is exactly the sae as that in conventional orthogonal ultiple access systes. Therefore, in this

5 5 section, we only focus on the outage probability of user which can be expressed as follows: P o =P h < ax { ϕ v + v H N I, 8 ϕ v + v H N I }, if α > α ϵ, otherwise outage always occurs. It can be verified that α α ϵ is equivalent to α = 0, in the context of cognitive radio power allocation. Analyzing this outage probability is very difficult due to the following two reasons. First, h and v are correlated, and second, the users experience different but correlated co-channel interference, i.e., I I. Therefore, in this subsection, we only focus on the case without co-channel interference, i.e., ρ I = 0. In particular, we focus on the following outage probability P =P h < ax { ϕ, ϕ }, 9 where ϕ = ϵ ϵ ρᾱ, ϕ = ρᾱ ρᾱ ϵ, and ᾱ = ρ h ax 0, ϵ +ϵ ρ h. Siilarly to the case with fixed power allocation, the outage probability P tightly bounds P o. The following lea provides the expression for the outage probability P. Lea 3. When ρ I expressed as follows: P = Υ ϵ ρ = 0, the outage probability can be ϵ + ϵ Υ ρ, 30 where Υ y = r r e yr α r e yrα r + γ α +, yrα γ α +, yrα, and r Υ z = 0 e zrα 0 + r r e zr α r e zrα 0 r 0 + z α r γ α +, zrα γ α +, zrα 0. At high SNR, the outage probability can be approxiated as follows: y α r r P 4ϵ r α+ ρ + αr r r α+ + r+α 0 ϵ + ϵ ρr + 4ϵ + ϵ ρ + αr 3 r α+ r0 α+. Proof: Please refer to Appendix C in [9]. By using the above lea, it is straightforward to show that a diversity gain of one is still achievable at user i.e., there is no error floor, and it is iportant to point out that this is achieved when user experiences the sae outage perforance as if it solely uses the channel. Therefore, by using the proposed cognitive radio NOMA, one additional user, user, is introduced into the syste to share the spectru with the priary user, user, without causing any perforance degradation at user. IV. NUMERICAL STUDIES In this section, the perforance of the proposed NOMA fraework is investigated by using coputer siulations. The perforance of three benchark schees, tered MIMO- OMA without precoding, MIMO-OMA with precoding, and MIMO-NOMA without precoding, is shown in Fig.. The Outage Su rate: P R + PR MIMO OMA without precoding MIMO OMA with precoding 0.5 MIMO NOMA without precoding SA MIMO NOMA Transission Power in db Outage Probabilities: P and P P, R = 0.5 BPCU a Outage Su Rate MIMO OMA without precoding MIMO OMA with precoding MIMO NOMA without precoding SA MIMO NOMA P, R = 5 BPCU Transission Power in db b Outage Probabilities Fig.. Perforance coparison for downlink transission. R = 5 bits per channel use BPCU and R = 0.5 BPCU. r = 0 and r = 0. M = N = 3. r 0 =. a = 3. The path loss exponent is α = 3. The 4 noise power is 30dB and the interference power is ρ I = 0. design for the two schees without precoding can be found in [9]. The MIMO-OMA schee with precoding serves M users during each orthogonal channel use, e.g. one tie slot, whereas M users are served siultaneously by the proposed schee. The fraework proposed in this paper is tered SA-MIMO- NOMA. The size of D and D is deterined by r = 0, and r = 0. The paraeter for the bounded path loss odel is set as r 0 =. Since the benchark schees were proposed for the interference-free scenario, Fig. shows the perforance coparison of the four schees for ρ I = 0. In Fig. a, the downlink outage su rate, defined as R P +R P, is shown as a function of transission power, and the corresponding outage probabilities are studied in Fig. b. As can be seen fro the figures, the two NOMA schees can achieve larger outage su rates copared to the two OMA schees, which deonstrates the superior spectral efficiency of NOMA. In Fig. b, the two schees with precoding can achieve better outage perforance than the two schees without precoding, due to the efficient use of the degrees of freedo at the base station. Coparing the proposed SA-MIMO-NOMA schee with the one proposed in [9], one can observe that their outage su rate perforances are siilar, but SA-MIMO-NOMA can offer uch better reception reliability, particularly for high transission power. In ters of individual outage probability,

6 Outage Probabilities 0 0 Solid lines: siulations Dash-dotted lines: analytical results P, ρ I = ρ 0 P, ρ I =0dB P, ρ I = ρ 0 P, ρ 0 3 I = 0dB Transission Power in db Fig.. Outage probabilities P and P for downlink transission. λ I = 0 4, δ =, r = 0, r = 0, M = N =, r 0 =, and a = 3 4. R = R = BPCU. The path loss exponent is α = 3 and the noise power is 30dB. SA-MIMO-NOMA can ensure a lower outage probability at user, i.e., a saller P, copared to the MIMO-OMA schee with precoding, but results in perforance degradation for the outage probability at user, i.e., an increase of P. This is consistent with the finding in [4] which shows that the NOMA user with poorer channel conditions will suffer soe perforance loss due to the co-channel interference fro its partner. In Fig., the accuracy of the analytical results developed in Leas and for downlink transission is verified. As can be seen fro Fig., the exact expression developed in Lea perfectly atches the coputer siulations, and the asyptotic results developed in Lea are also accurate at high SNR, as shown in Fig.. The accuracy of Lea can be confired siilarly. Note that error floors appear when increasing ρ I in Fig., which is expected due to the strong co-channel interference caused by the randoly deployed interferers. In Fig. 3, the perforance of the cognitive radio power allocation schee proposed in Section III-B is studied. In particular, given the target data rate at user, the power allocation coefficients can be calculated opportunistically according to 7. As can be seen fro the figure, the probability for this NOMA syste to support the secondary user, i.e., user, but with a target data rate of R approaches one at high SNR. Note that with OMA, user cannot be aditted into the channel occupied by user, and with cognitive radio NOMA, one additional user, user, can be served without degrading the outage perforance of the priary user. V. CONCLUSIONS In this paper, we have proposed a signal alignent based fraework which is applicable to both MIMO-NOMA downlink and uplink transission. By applying tools fro stochastic geoetry, the ipact of the rando locations of the users and interferers has been captured, and closed-for expressions for the outage probability achieved by the proposed fraework have been developed to facilitate perforance evaluation. REFERENCES [] Y. Saito, A. Benjebbour, Y. Kishiyaa, and T. Nakaura, Syste level perforance evaluation of downlink non-orthogonal ultiple access NOMA, in Proc. IEEE Annual Syposiu on Personal, Indoor and Mobile Radio Counications PIMRC, London, UK, Sept. 03. 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