A General Power Allocation Scheme to Guarantee Quality of Service in Downlink and Uplink NOMA Systems

A General Power Allocaton Scheme to Guarantee Qualty of Servce n Downlnk and Uplnk N Systems Zheng Yang, Student Member, IEEE, Zhguo Dng, Senor Member, IEEE, Pngzh Fan, Fellow, IEEE, and Naofal Al-Dhahr, Fellow, IEEE Abstract In ths paper, a novel dynamc power allocaton scheme s proposed to downlnk and uplnk non-orthogonal multple access N scenaros wth two users for more flexbly meetng varous qualty of servce requrements. The exact expressons for the outage probablty and the average rate acheved by the proposed scheme, as well as ther hgh sgnal-tonose rato approxmatons, are establshed. Compared to exstng works, such as N wth fxed power allocaton and cogntve rado nspred N, the proposed scheme can:. strctly guarantee a performance gan over conventonal orthogonal multple access;. offer more flexblty to realze dfferent tradeoffs between the user farness and system throughput. Monte Carlo smulaton results are provded to demonstrate the accuracy of the developed analytcal results and the performance gan of the proposed power allocaton scheme. Index Terms Non-orthogonal multple access, outage probablty, average rate, dversty order, qualty of servce. I. INTRODUCTION Non-orthogonal multple access N has recently attracted a lot of attenton as a promsng technology to be used n the ffth generaton 5G moble communcaton network [ and [. In addton, a smplfed verson of N, named as mult-user superposton transmsson MUST, has also been appled to Thrd Generaton Partnershp Proect 3GPP Long Term Evoluton LTE downlnk transmsson [3. The key dea of N s to explot the power doman for multple access,.e., multple users can be multplexed at dfferent power levels but at the same tme/frequency/code. A successve nterference canceller SIC s utlzed to separate supermposed messages at the recever sde [ and [4. Therefore, compared to conventonal orthogonal multple access, N can offer a sgnfcant mprovement n both spectrum effcency and user farness. In [5, the authors have nvestgated the mpact of random user deployment on the outage probablty and the average sum rate n downlnk N systems. The performance of uplnk N transmsson was studed n [6. The use of cooperatve N can effectvely mprove the outage performance for the users wth poor channel condtons, snce Z. Yang and P. Fan are wth the Insttute of Moble Communcatons, Southwest Jaotong Unversty, Chengdu 63, P. R. Chna. emal: zyfnu@63.com, p.fan@eee.org. Z. Dng s wth the School of Computng and Communcatons, Lancaster Unversty, LA 4YW, UK. e-mal: z.dng@lancaster.ac.uk. N. Al-Dhahr s wth the Department of Electrcal Engneerng, Unversty of Texas at Dallas, Rchardson, TX 758 USA. e-mal: aldhahr@utdallas.edu. Ths work of Z. Yang and P. Fan was supported by Natonal Natural Scence Foundaton of Chna No. 6473, Natonal Sc & Tech Maor Proect No. 6ZX38-, Natonal Hgh-tech R&D Program of Chna No. 4AAA77, and proect No. --4. the users wth strong channel gans can be used as relays to help the others [7. The work n [8 nvestgated the sum rate for downlnk N wth the coordnated superposton codng scheme. The mpact of power allocaton on the outage performance under the assumptons of avalable nstantaneous and average channel state nformaton CSI has been studed n [9, respectvely. Note that multple-nput multple-output MIMO transmsson can also be ncorporated nto N systems, whch further mproves the performance of N transmsson [. To suppress system overhead, a MIMO- N transmsson scheme wthout requrng CSI at the transmtter was proposed n [ by decomposng a MIMO- N system nto multple separated sngle-nput sngleoutput SISO channels. Power allocaton s the key to the superor spectral effcency of N, snce N uses the power doman for multple access. Most exstng work about N focuses only on the fxed power allocaton strategy, where the power allocaton coeffcents are predefned and are not a functon of channel gans [5- [8, [- [. The drawback of ths type of N, termed N wth fxed power allocaton F-N, s that the users predefned qualty of servce QoS cannot strctly be met. For example, F-N cannot be used f the target data rate for a user wth poor channel condton s large. Another example s that, n F-N, the average rate for the user wth poor channel condtons s much less than that of, partcularly at hgh sgnal-to-nose rato SNR. Recently, n [, the authors have proposed a new power allocaton scheme n downlnk N systems, termed cogntve rado nspred N CR-N, where N s nterpreted as a specal case of CR networks [3. Partcularly, for CR-N, the user wth poor channel condtons can be vewed as a prmary user, whch means that the QoS for ths user can be strctly guaranteed n CR-N. However, CR-N wll sacrfce the performance for the user wth better channel condtons snce ths user s served only after the prmary user s QoS s met. For example, f the target data rate at the worse user s very large, the target sgnal-to-nterferenceplus-nose rato SINR at ths user also tends to large, and hence the transmtter has to allocate all the power to the worse user n order to meet ts demandng rate constrant, whch means that the better user cannot obtan any power and hence cannot be served n ths CR-N scenaro. Therefore, CR- N stll cannot provde suffcent flexblty to guarantee all users QoS requrements. In ths paper, we consder a general N communcatons scenaro wth one base staton and M users, where all nodes are equpped wth a sngle antenna. The man contrbutons of

ths paper are lsted as follows: We propose a general power allocaton scheme whch can be appled to both downlnk and uplnk N scenaros wth two users. Ths power allocaton polcy s obtaned by frst constructng two constrants whch ensure that the ndvdual user rates acheved by N are larger than those of conventonal. By usng these constrants, an explct expresson for the power allocaton factors can be constructed, where these power allocaton factors are dynamcally changng wth the nstantaneous channel gans, unlke F-N. Based on the proposed power allocaton coeffcents, the expressons for the ndvdual user rates are obtaned, and then used for the performance analyss. In partcular, the outage probablty and the average rate are used as the crtera to analyze the performance of the proposed N scheme wth dynamc power allocaton, termed D-N. The exact expressons for the outage probablty and the average rate are developed, and the hgh SNR approxmatons for the outage performance are also carred out n order to fnd the achevable dversty gans. Both analytcal and numercal results are provded to compare D-N wth the exstng N schemes. In partcular, t s shown that D-N can acheve the same dversty gan as F-N, but D-N can ensure that users are served n a balanced way. In other words, D-N can avod the stuaton that the user wth poor channel condtons s served wth a small data rate, a drawback of F-N [5. On the other hand, D-N can acheve a hgher dversty gan at the user wth better channel condtons compared to CR-N. Recall n [ that the dversty gan of CR-N for the user wth better channel condtons s constraned by ts partner s channel qualty. Therefore, we conclude that D-N can realze a more balanced user experence, compared to F-N and CR-N. II. SYSTEM MODEL We consder a downlnk uplnk N system wth M sngle-antenna users and one sngle-antenna base staton. Wthout loss of generalty, assume that the users are ordered as h h h M, where h s the Raylegh fadng channel gan. In ths paper, we assume that the v- th user and the w-th user are selected to perform N, v < w M. The proposed power allocaton scheme can be generalzed to group more than two users for performng N, and a smple example for ths extenson wll be llustrated n Secton V. C. It s worth pontng out that ths s how N s to be mplemented n 3GPP LTE Advanced, where two users are selected for performng N [3. Note that the focus of ths paper s to study the mpact of dfferent choces of v and w on the performance of N, whch wll provde mportant nsghts for the desgn of effectve user groupng clusterng. In practce, users n one cell can be dvded nto small groups, where N s used wthn each group and orthogonal bandwdth resources, such as subcarrers or tme slots, can be used to dstngush N groups [. Ths type of mplementaton can be vewed as a specal case of hybrd N, and the dscusson for ths hybrd N s out of the scope of ths paper. The D-N transmsson mechansm for downlnk and uplnk N s descrbed n the followng two subsectons, respectvely. A. Downlnk N Systems Followng the prncple of N, the rates of user v and user w n downlnk N are gven by [4 Rv,D N α v h v = log α w h v, / and R N w,d = log αw h w, respectvely, where s the transmt SNR, α v and α w are the power allocaton factor for user v and w, respectvely, and α v > α w, α v α w =. Let Rw v,d N denote the rate for user w to decode user v s message,.e., Rw v,d N = log α v h w α w h w. It s easy to verfy that R N w v,d > Rv,D N s always satsfed, snce h v < h w, whch means that user w can always successfully detect user v s message before detectng ts own message,.e., the SIC can be carred out successvely at user w. Therefore, the rate Rw,D N s always achevable at user w. On the other hand, the rate of user n, such as tme dvson multple access TDMA, s gven by R T = log h, v, w. 3 As ponted out n [, N can be vewed as a specal case of CR systems. Followng ths nterpretaton, n ths paper, we frst assume that user w who has better channel condtons s consdered as a prmary user, and assume that the target rate at user w s Rw, T.e., Rw,D N RT w, whch results n the followng constrant: log αw h w log hw α w hw. 4 In addton, we assume that user v wth poor channel condton can also be regarded as a prmary user, and assume that the target rate at user v s Rv T, whch means Rv,D N RT v, then α v h v log α w h v / log hv α w hv. 5 It s mportant to pont out that the upper bound of α w satsfes, whch s mportant to ensure that user v hv < gets more transmsson power [4 and [5. Combnng 4 wth 5 and by ntroducng two constant coeffcents β and β, the proposed power allocaton factor α w can be expressed as follows: α w = β W β V, 6

3 where W = h w, V = h v, β, =,, and β β =. Note that the power allocaton factor α w for D-N becomes a functon of nstantaneous channel gans. The power allocaton factors for F-N are constants, and not changng accordng to the users nstantaneous channel gans, as long as the order of the users channel gans s the same. In [9, the authors have studed the max-mn farness n N, and showed that all users can acheve the same data rate. However, n many practcal communcaton scenaros, users have dfferent QoS requrements. For example, n 5G powered Internet of Thngs IoT, there mght be two types of users. One type can be IoT sensors, whch need to be served tmely but wth low data rates, and the second type can be users whch download moves or multmeda fles. The proposed scheme can ensure that all the users are connected tmely, but wth dfferent rates, n order to meet ther dversfed QoS requrements, as explaned n the followng. The key feature of the proposed power allocaton polcy s that the two users s QoS requrements,.e., performng N yelds a larger data rate than that of, can be met smultaneously, for any choce of β. β s the parameter whch a system desgner can fne tune n order to acheve dfferent tradeoffs between the two users data rates. In partcular, as can be observed from and, Rv,D N s a mono-decreasng functon of α w, and Rw,D N s a mono-ncreasng functon of α w. Furthermore, 6 ndcates that α w s a mono-ncreasng functon of β. Therefore, when decreasng β, Rv,D N wll be ncreased, but Rw,D N wll be decreased. Take an extreme case wth β = as an example. Ths choce treats user v s rate experence wth a hgher prorty, and, among all choces of β, β = yelds the maxmal rate for user v whch s gven by R N,max v,d = log V log V. 7 W Snce the rate for user w, Rw,D N, s an ncreasng functon of β, β = yelds the mnmal rate for user w whch s gven by R N,mn w,d = log W, 8 whch s equal to user w s rate n systems. From 7 and 8, we can obtan the maxmum farness between user v and user w as follows: R gap = R N,mn w,d RN,max v,d = log h w h v. 9 Note that when h w h v s equal to, the pared users can acheve absolute farness n the D-N scheme. On the other hand, when ncreasng β, user w s rate experence s treated wth a hgher prorty, and, among all possble choces of β, β = yelds the largest data rate for user w whch s gven by R N,max w,d = log W V. Compared to the proposed power allocaton polcy, F- N suffers the drawback that user v s rate s very small, partcularly at hgh SNR, and CR-N has the dsadvantage that the QoS requrment at user w s gnored [. Based on and, the sum rate of the N systems s gven by R N sum,d = log h v log αw h w α w h v, whch s an ncreasng functon of α w. It can be observed from 6 that the α w s an ncreasng functon of β. Therefore, β = yelds the maxmal throughput whch s gven by R N,max sum,d = log V V W V. It s worth pontng out that ths work has consdered a dfferent type of user farness compared to that n [9, where ours ensures that users are connected tmely, but wth dfferent rates, n order to meet ther dversfed QoS requrements. B. Uplnk N Systems Recall that the total transmsson power constrant α v α w = s assumed for uplnk N. For many practcal scenaros, the total transmsson power constrant s a crucal crteron. For example, n a cell wth multple users sharng the same bandwdth, the constrant of the total transmsson power wthn ths cell s mportant to manage nter-cell nterference. Another example s hybrd N, where users are pared to perform N and nter-par nterference s cancelled by relyng on conventonal nterference management technques. The use of the total power constrant s therefore useful to measure nter-group nterference [4. For uplnk N, we assume that the decodng order s always from the better user to the worse user, otherwse a sgnfcant amount of transmsson power s to be consumed at user v n order to compensate the severe channel attenuaton. The rates of user w and user v are gven by and Rw,U N = log α w h w α v h v, 3 / R N v,u = log αv h v, 4 respectvely. Smlarly to downlnk N, we frstly consder the constrant R N w,u RT w, whch yelds the followng: log α w h w α v h v / α w log hw h v h v h w. 5 Secondly, we consder that R N v,u RT v, whch leads to the followng: log α v h v log hv hv α w hv. 6

4 Agan combnng 5 wth 6 and ntroducng the tunng parameters β, the proposed power allocaton coeffcent, α w, can be bult as follows: α w = β V V W β V V. 7 Smlar to downlnk N, the values of the two ndvdual user rates can be tuned by adustng β. In addton, when β =, the maxmal rate of user v n uplnk N s gven by R N,max v,u = log V log V W. 8 When β =, the maxmal rate of user w n uplnk N s gven by R N,max w,d = log W. 9 V Comparng 7 to 8 or to 9, one can observe that the users n downlnk and uplnk D-N acheve the same rates, for the two specal cases wth β = and β =. However, t s mportant to pont out that the formulas of the rates are not the same for a general case wth < β <. III. PERFORMANCE ANALYSIS FOR DOWNLINK N In ths secton, we focus on the outage probablty and the average rate for downlnk N wth the dynamc power allocaton factor α w, whch s defned n 6. A. Outage Probablty The overall outage performance and users ndvdual outage probabltes are derved n ths subsecton. Due to the use of SIC, user w needs to decode the sgnal to user v before detectng ts own sgnal. Therefore, the overall outage performance of downlnk D-N can be evaluated as follows: P D total = Pr R N v,d > R v, R N w v,d > R v, R N w,d > R w, where R v and R w are the target rates of user v and user w, respectvely. Recall that the users channels have been ordered as h v < h w, whch means that the condton R N w v,d > RN v,d s always satsfed. Therefore, the outage probablty P D total n can be smplfed as follows: P D total = Pr R N v,d > R v, R N w,d > R w. An exact expresson for the above outage probablty and the dversty order are provded n the followng theorem. Theorem : The overall outage probablty for downlnk D-N s gven by at the top of the next page, where M! ϖ =!w!m w!, ε v = Rv, ε w = Rw, ξ = ε v β ε vβ 4ε vβ /, G x = V ε v β V ε v β, G x = ε v β V ε v β V, G 3 x = β V β /β, = β V 4β β V ε w V, G max x = max G 3 x, G x/g x. The dversty gan acheved by D-N s gven by P D total d = lm = v. 3 log Proof: See Appendx A. Theorem llustrates that the dversty gan based on the overall outage probablty s v, not w. The reason for ths s that the worse user v plays a domnant role n, snce the overall outage event s defned as the event that outage occurs at ether of the two users. In addton to the overall outage probablty, we are also nterested n the ndvdual outage performance for user v and w. Note that when R w =, the probablty n can be degraded as follows: P D v = Pr R N v,d > R v. 4 Followng smlar steps as n the proof of Theorem, we can obtan an exact expresson for the above probablty as follows. Corollary : The ndvdual outage probablty of user v n downlnk D-N systems s gven by P N v,d = ϖ = w = ε M v [e M v e x dx v w M v ε v ξ w e M v G x G x G x. 5 The dversty order of user v n downlnk D-N systems s v. On the other hand, user w frst decodes the message from user v, and then removes ths message before detectng ts own message va SIC. Therefore, the outage probablty for user w can be expressed as follows: P D w = Pr R N w v,d > R v, R N w,d > R w. 6 Note that the above probablty s qute smlar to. Therefore, followng smlar steps as n the proof of Theorem, we can obtan the followng Corollary. Corollary : The ndvdual outage probablty of user w n downlnk D-N systems s gven by P D w = ϖ = w = ε M v [e M v w ε v e x dx ξmax w M v ε w ε v w e M v G 3 x e M v H max x e x dx, 7

5 P D total = ϖ = w = e x dx w M v ε v ξ w e M v G max x [ e M v ε w M v x dx ε w ε v e M v G 3 x. where H max x = maxg 3 x, G 4 x, G 4 x = ε v β V V ε v β, = ε v β V 4εv β V V ε v β, ξ max = maxε v β,. The dversty order of user w s w under the condton ε v β < ; otherwse, the dversty gan s v. Corollary demonstrates that user v acheves a dversty gan of v, and Corollary shows that user w acheves a dversty gan of w f ε v β <, n downlnk D-N systems. Therefore, D-N can acheve the same dversty gan as F-N [5. However, as ponted out n [5, for F- N, the targeted data rates need to be carefully set and satsfy α v > ε v α w ; otherwse, the ndvdual outage probablty wll be always, regardless of how hgh the SNR s. In contrast, the ndvdual outage probablty of D-N wll never be one at hgh SNR, where volatng the condton ε v β < only causes the loss of the dversty gan. On the other hand, CR-N can only realze a dversty gan of v at user w [. The reason for ths s that CR- N treats user v more mportantly, e.g., the transmsson power wll be allocated to user v frst and user w s served only f there s any power left afterwards. Therefore user w s experence n CR-N s not as good as user v s, and partcularly, the rate realzed by CR-N for user w mght be smaller than that n the case of. In contrast, D-N can offer a balanced treatment to both users,.e., both users n D-N can have larger data rates than those n. B. Average Data Rates In ths subsecton, we develop the average ndvdual rates for the users n downlnk D-N systems. Theorem : The average data rate of user v n downlnk D-N s gven by C D v = ϖ ln r r= r em vr/ ϖ y = M v r E w = M v r w e x log β β x w β x y dxe M v y dy, 8 M! where ϖ = exponental ntegral.!m v! and E x = e xt t dt s the Proof: Based on 6, the average rate of user v can be wrtten as follows: [ C D α v h v v = E log α w h v 9 / [ = E log hv [ E log αw h v. Q 4 Q 5 Recall that the probablty densty functon PDF of h v s gven by [5 f hv x = ϖ e x e xm v e x = ϖ r r e M vrx. 3 r= Based on 3, Q 4 n 9 can be calculated as follows: Q 4 = = ϖ ln log x f hv xdx 3 r= r r e M vrx ln xdx. Let t = x and the ntegral n 3 can be evaluated as follows: e M vrx ln xdx = em vr/ M v r = em vr/ M v r E e M vrt/ t M v r dt. 3 Based on 6 and 64, Q 5 n 9 can be calculated as follows: [ Q 5 = E log β β hv β h v hw = ϖ w = = y w w e x log β β x β x y dxe M v y dy. 33 Substtutng 3, 3 and 33 nto 9, the proof s complete. Recall that when β =, user v can acheve the maxmal rate. For ths specal case, the double ntegral Q 5 n 33 can

6 be further evaluated as follows: y log g yx e x dxe M v y dy = ln e M vy log ydy y e x /g y x dxe M v y dy, 34 where g y = y y. Applyng [6, Eq. 3.35. to 34 and substtutng 3 and 34 nto 9, the maxmal average rate of user v n downlnk D-N systems s gven by 35 at the top of the next page. Followng smlar steps as n the proof of Theorem, we can obtan the average rate of user w n downlnk D-N systems as follows: [ C D w = E log αw h w [ β y = E log β β y 36 x = y e x log β β y β y x dxe M v y dy. Furthermore, for the specal case of β =, the maxmal average rate for user w can be obtaned n 37 at the top of the next page. IV. PERFORMANCE ANALYSIS FOR UPLINK N Snce the uplnk and downlnk system models are symmetrcal, we focus only on the dfference between the two scenaros, as shown n the followng two subsectons. A. Outage Probablty The outage probablty based on the sum rate and users ndvdual outage probabltes are derved n ths subsecton. Unlke the downlnk scenaro, the sum rate for the N uplnk transmsson s always the same, no matter whch user s message s decoded frst. Therefore, n ths secton, we frst focus on the outage probablty based on the sum rate. Based on 3 and 4, the sum rate of uplnk D-N systems s gven by Rsum U = log α w h w α v h v log / αv h v = log αv h v α w h w. 38 Then the coverage probablty for the sum rate n uplnk D-N systems s defned as P U sum = Pr log αv h v α w h w > R v R w, 39 and an exact expresson for the outage probablty by usng the above coverage probablty s gven n the followng Theorem. Theorem 3: The outage probablty of the sum rate n uplnk D-N systems s gven by P U sum = ϖ = w = ε M v [e M v w M v ε w exp x M v W / dx, 4 where ε = RvRw, a = β V, b = β V 3 β V V, c = β V β V εv β V, d = β V β V εv 3 V β V 3 β V 4 V, W = b 3a q q p 3 3 /3 q q p 3 3 /3, p = 3a c b, q 3a = 7a d 9a b c b 3. The dversty gan for the 7a 3 sum rate n D-N uplnk s w. Proof: Please see Appendx B. Theorem 3 llustrates that the dversty order for the sum rate n uplnk D-N depends on user w s channel qualty, whch means that user w s channel gan plays an mportant role n the outage probablty at 4, partcularly at hgh SNR. Based on 3 and 7, the coverage probablty of user w n uplnk D-N systems can be expressed as P U w = Pr log α w h w α v h v > R w 4 β V W ε w V = Pr V W β W > ε w β V ε w β V = Pr β W 3 b W c W d >, where b = V β V, c = β ε w ε w V ε w β V, d = ε w β V 3 ε w V ε w β V. Snce the users channels are ordered as h w > h v,.e., W > V. Therefore, P U w n 4 should be rewrtten as follows: P U w = Pr β W 3 b W c W d >, W > V. 4 Based on the relatonshp between β V 3 b V c V d > and W V >, the above probablty can be further expressed as follows: P U w = Pr W > V, V > ε w Pr β W 3 b W c W d >, V < ε w = Pr Pr h w > h v, h v > ε w h w > W, h v < ε w, 43 where W = b 3β q q p 3 3 /3 q q p 3 3 /3, p = 3β c b, and q 3β = 7β d 9β b c b 3. 7β 3

7 C D v,max = ϖ ln E r= r r em vr/ M v r E w [ e M v dy g y E M v M v. M v r ϖ = w = e M v y[ g y E w ln y g y 35 C D w,max = a = exp w = w M v ln w M v x x E x [ e M v E M v M v M v x M v x dx. 37 x Followng smlar steps as n the proof of Theorem, we can obtan an exact expresson for the outage probablty of user w as follows: P U w = ϖ = w = w M v w ε M v w [e ε w exp x M v M v W / dx. 44 Snce SIC s appled at the base staton n uplnk N systems, the base staton has to frst detect the sgnal from user w and then remove t before detectng user v s sgnal. Therefore, based on 4 and 7, the coverage probablty of user v n uplnk D-N systems can be expressed as follows: P U v = Pr log α w h w α v h v > R w, log αv h v > R v = Pr β W 3 b W c W d >, β V β V ε v V W > β V V = Pr β W 3 b W c W d >, W > G 5 V, V > ξ, 45 where ξ = β β 4ε vβ /β and G 5 V = β V V β V β V ε v V. Snce the users channels are sorted as h w > h v,.e., V < W, P U v P U v n 45 can be further rewrtten as follows: = Pr W > V, V > ε w Pr W > W, ε v < V < ε w Pr W > maxw, G 5 V, ξ < V < ε v. 46 Then followng smlar steps to those used n the proof of Theorem, we can obtan an exact expresson for the outage probablty of user v n uplnk D-N as follows: P U v = ϖ = w = ε M v [e M v w ε v e x dx ξ e x dx w M v ε w ε v w e M v W e M v K max x, 47 where K max x = maxw, G 5 x. By usng 44 and 47, t s straghtforward to show that the w-th user and the v-th user also acheve a dversty gan of w and v n uplnk D-N systems, respectvely. On the other hand, the use of F-N n uplnk systems can result n an error floor for the ndvdual outage probabltes at hgh SNR [7. B. Average Data Rate In ths subsecton, we focus on the average data rates for user w and v n uplnk D-N systems. Based on 7, the average data rate for user w n uplnk

8 D-N systems can be expressed as follows: [ Cw U = E log α w h w α v h v 48 [ β W = E log V β β V E [ log β V W V V W β β V β V V V W. Followng smlar steps to those used to get 34, we can obtan the average rate of user w n uplnk D-N systems as follows: C N w,u = ϖ where w = = y Gx, y = β β x w w 49 e x log Gx, ydxe M v y dy, β y x β xy x x y β β x β xx x y Smlarly, based on 7, the average rate for user v n uplnk D-N systems can be expressed as follows: [ Cv U = E log αv h v 5 = E [ log V [ E log β β V β V V V. W By followng smlar steps to the aforementoned ones, we can obtan the average rate of user v n uplnk D-N systems as follows: C U v = ϖ ln r= r ϖ y r e M vr/ M v r = w = M v r E w e x log β β x β xx x y w dxe M v y dy. 5 V. NUMERICAL RESULTS In order to demonstrate the performance of the proposed D-N scheme and verfy the accuracy of the developed analytcal results, Monte Carlo smulaton results are provded n ths Secton. A. Downlnk D-N Fg. shows the overall outage probablty for downlnk D- N systems,.e., P D total = Pr R N v,d > R v, R N w,d > R w, as a functon of SNR. It can be observed from Fg. that, when mprovng user v s channel condtons, the outage performance wll mprove accordngly. Ths observaton s consstent wth our analytcal results n Theorem whch shows that the outage probablty decays as v. Furthermore, one can see from Fg. that the outage performance for. Outage probablty 3 4 v=4, w=5 F N wth α w =.5 [5 F N wth α w =. [5 D N Sm. D N Ana. v=, w=5 5 5 5 3 SNR db Fg.. The overall outage probablty for downlnk D-N systems wth M = 5, β =.5, R v = bts/s/hz, and R w = 3 bts/s/hz. The analytcal results are based on Theorem. D-N systems s always superor to that of for dfferent system parameters, whle F-N fals. Ths s because F-N s senstve to the choce of the parameters, α w, R v and R w. In partcular, for F-N, the condton α v > ε v α w needs to be met; otherwse, the outage probablty s always one as shown n the fgure. In addton, t s worth pontng out that Fg. demonstrates that the analytcal results based on Theorem match the Monte Carlo smulaton results. Outage probablty 3 v=, w=4 F N wth α w =.5 [5 D N Ana. D N Sm. F N wth α w =.5 [5 CR N [ v=, w=3 5 5 5 3 SNR db Fg.. The outage probablty for user v n downlnk D-N systems wth M = 5, β =.4, and R v = bts/s/hz. The analytcal results are based on Corollary. In Fg. and Fg. 3, the outage performance for user v and user w s shown as a functon of SNR, respectvely. As can be seen from the fgures, the analytcal results based on 5 and 7 match the Monte Carlo smulaton results n Fg. and Fg. 3, respectvely. Furthermore, one can observe that the outage probabltes for user v and user w n D-N have the same dversty gan as, but the former offers a constant performance gan over the latter. In addton, outage

9 4 Outage probablty 3 4 v=, w=4 F N wth α w =.5 [5 CR N [ D N Ana. D N Sm. F N wth α w =.5 [5 v=, w=3 5 5 5 3 35 4 45 SNR db a Smulaton vs analytcal results wth β =.4, R v = bts/s/hz and R w = 3 bts/s/hz. Average rate of user v 3.5 3.5.5 D N Sm. D N Ana. SNR= db SNR=5 db SNR= db.5.5.75 β a The average rate of user v. β =.9 4.5 SNR= db Outage probablty 3 β =. R v =R w =4 bts/s/hz R v =R w =3 bts/s/hz R v =R w =4 bts/s/hz R v =R w =3 bts/s/hz 4 5 5 5 3 35 4 SNR db b Impact of β and R v on the dversty gan wth v =, w = 5. Fg. 3. The outage probablty for user w n downlnk D-N systems wth M = 5. The analytcal results are based on Corollary. Average rate of user w 4 3.5 3.5 D N Sm. D N Ana. SNR= db SNR=5 db..4.6.8 β b The average rate of user w Fg. 4. The average rate of user v and user w n dolnk network, where M = 5, w = 5, v = 4. The analytcal results are based on 35 and 36 always occurs for F-N n Fg. and Fg. 3a under the condton α v < ε v α w, and Fg. 3a also demonstrates that user w n CR-N systems only acheves a dversty gan of v, whch s worse than D-N and even. Ths s because n CR-N systems, only after user v s QoS requrement s strctly satsfed, user w s allowed to be admtted nto the system. In contrast, D-N can ensure that both users rates are larger than that n. In addton, t can be seen from Fg. 3b that when ε v β >, the dversty gan of user w s reduced. The man reason s that when the target rate R v s hgh, user w may fal to detect user v s message, whch leads to the mplementaton falure of SIC. Fortunately, when ε v β <, user w can acheve full dversty, and ths condton s easy to be satsfed by tunng the parameter β, as shown n the fgure. In Fg. 4, the average rates of user v and user w n downlnk D-N are shown as a functon of β. As expected, the rate of user w s an ncreasng functon of β, whle the rate of user v s a decreasng functon of β. The use of D-N can always acheve better performance than. The Jan s farness ndex acheve by the proposed D-N scheme and Farness Index.9.8.7.6.5.4.3.. D N CR N.5.5.75 β Fg. 5. Farness comparson between the proposed downlnk D- N and other schemes, where M = 5, w = 5, v =, R v = 3 bts/s/hz, SNR = db.

the comparable schemes s shown n Fg. 5. The Jan s farness ndex s defned as [8 R N v,d Rw,D N J = Rv,D N Rw,D N 5. Note that and CR-N have been regarded as effcent approaches to realze user farness. The fact that the performance of D-N shown n Fg. 5 s smlar to that of CR-N and mples that D-N acheves reasonable user farness. The use of dfferent β enables us to realze dfferent user tradeoffs. It can be seen from Fg. 5 that the proposed D-N can acheve the best farness among the dfferent schemes, especally when β =. Furthermore, the Jan s farness ndex of D-N s a decreasng functon of β, whch s consstent wth our analytcal results, e.g., Rv,D N s a decreasng functon of β, and Rw,D N s an ncreasng functon of β. Outage probablty 3 v=3, w=5 F N wth α w =.4 [6 D N Sm. D N Ana. F N wth α w =.9 [6 v=, w=3 4 5 5 5 3 SNR db Fg. 7. The outage probablty for user w n uplnk N systems wth M = 5, β =.5, and R w = 3 bts/s/hz. The analytcal results are based on 44. v=, w=3 Outage probablty 3 v=4, w=5 F N wth α w =. [6 D N Sm. D N Ana. F N wth α w =.8 [6 4 5 5 5 3 SNR db Fg. 6. The outage probablty based on the sum rate n uplnk D- N systems wth M = 5, β =.5, R v = bts/s/hz, and R w = 3 bts/s/hz. The analytcal results are based on Theorem VI. Outage probablty 3 4 v=3, w=5 F N wth α w =.4 [6 F N wth α w =.9 [6 D N Ana. D N Sm. v=, w=3 5 5 5 3 SNR db Fg. 8. The outage probablty for user v n uplnk N systems wth M = 5, β =.5, R v = bts/s/hz, and R w = 3 bts/s/hz. The analytcal results are based on 47. B. Uplnk D-N In Fg. 6, the analytcal and numercal results for the outage performance based on the sum rate n uplnk D-N systems are shown as a functon of SNR, whch demonstrates that the exact expresson n 4 matches well the Monte Carlo smulaton results. Furthermore, we can see that the outage probablty based on the sum rate for uplnk D-N scheme has a dversty gan of w. In addton, D-N can always acheve better outage performance than, whereas the use of F-N can cause some performance loss compared to. In Fg. 7 and Fg. 8, the analytcal results n 44 and 47 whch are the outage probabltes for user w and user v, respectvely, are compared to smulaton results, wth dfferent system parameters. The two fgures show a perfect match between the analytcal dervatons and the computer smulaton results, for the whole range of the SNR, whch confrms the accuracy of our analytcal expressons. Furthermore, one can observe from Fg. 7 and Fg. 8 that F-N can result n an outage floor for both ndvdual outage probabltes. On the other hand, there s no error floor for the curves of D-N. In addton, D-N acheves the same dversty gan as, and offers a constant outage performance gan over. Smlarly to Fg. 4, Fg. 9 demonstrates that the uplnk D-N can also acheve better performance compared to. C. Extended Downlnk D-N wth An Arbtrary Number of Users Recall that the users channel gans are ordered as h h h M. Followng the proposed D-N approach, we assume user N s consdered as a secondary user, and the others are regarded as the prmary users. Furthermore, assume the QoS requrement at user, N s to

Average rate of user v Average rate of user w 5 4.5 4 3.5 3.5 SNR=5 db D N Sm. D N Ana. SNR= db SNR=5 db.5.5.75 β 5.5 5 4.5 4 3.5 3.5 SNR=5 db a The average rate of user v. SNR= db D N Sm. D N Ana. SNR=5 db.5.5.75 β b The average rate of user w Fg. 9. The average rate of user v and user w n uplnk network, where M = 5, w = 5, v = 3. The analytcal results are based on 49 and 5 acheve a rate no less than that of,.e., α h log N = α h N log h, 53 whch means the power allocaton factor α can be expressed as follows: N α X N X X N α, 54 = where X = h N. The sum rate of downlnk D-N wth N users can be expressed as follows: R sum,d N α h = log N = = α h N N = log h = log α h N = α. h = 55 The dervatve of the R sum,d wth respect to α N s gven as follows: R sum,d α N = N = h h N = α h N = α >, h 56 whch means R sum,d s an ncreasng functon of α N. Based on 54, the maxmal value of the power allocaton factor α N s gven by α max N N = α, 57 = where α = X X N k= α k X N X. Therefore, based on the value of α N, and n 57, the maxmal sum rate of downlnk D-N s gven by α max N Rsum,D max = N log N h log α max N h N. = 58 Average sum rate 3 9 8 7 6 5 4 3 D N N=4 N=4 D N N=3 N=3 D N N= N= 5 5 3 35 SNR db Fg.. The average sum rate for downlnk D-N systems compare to wth dfferent users wth M = 4. Fg. shows the average sum rate of downlnk D-N wth dfferent numbers of users by usng as a benchmark scheme. It can be seen from Fg. that the proposed D- N scheme can always outperform, whch s due to the fact that D-N can acheve a better spectral effcency. It s also mportant to pont out that the performance gan of D-N over can be enlarged by ncludng more users for the mplementaton of N. VI. CONCLUSIONS In ths paper, a dynamc N power allocaton scheme has been proposed. Ths novel D-N power allocaton scheme can be appled to both downlnk and uplnk N scenaros. The analytcal expressons have been derved for

the outage probablty and the average rate n downlnk and uplnk D-N systems. The derved analytcal expressons show that D-N can acheve the same dversty gan as F-N, and avod the stuaton that the rate for the user wth poor channel condtons s smaller than that n. Furthermore, D-N can acheve a larger dversty order compared to CR-N. One promsng future drecton s to combne ths novel D-N power allocaton scheme wth sophstcated user parng/clusterng algorthms n a large scale network. Furthermore, another mportant future drecton s to extend the D-N power allocaton scheme to MIMO scenaros, where t s preferable to consder the use of mperfect CSI at the transmtter. APPENDIX A PROOF OF THEOREM Recall that from, the overall coverage probablty for downlnk N systems,.e., P D total P D total, s gven by P D total = Pr log α v h v α w h v > R v, log αw h w > R w V = Pr ε v β V ε v β W > εv β V ε v β V, can be evaluated as follows: w β W β V W β V ε w V >. Q = ϖ 59 From 59, defne G V = V ε v β V ε v β and ξ = ε v β ε vβ 4ε vβ /. It s easy to verfy that that G V > for V > ξ, and G V < for V < ξ. Therefore, dependng the value of G V, the probablty P D total can be classfed as follows: If G V >,.e., V > ξ, we have P D total = Pr V > ξ, W > G V G V, W > G 3V, 6 where G V = ε v β V ε v β V, G 3 V = β V β /β, and = βv 4β β V ε w V. Recall that the users channels are ordered as h w > h v,.e., W > V. Then the constrant condton of W n P D total can be further rewrtten as follows: G V W > max G V, V, G 3V. 6 Frstly, we focus on the relatonshp between G V G V and V G n 6. It s easy to fnd that when V > ε v, V G V < V, and when ξ < V < ε v, GV G V > V. Therefore, the coverage probablty P N total n 6 can be expressed as follows: P D total = Pr V > ε v, W > max G 3 V, V 6 Pr ξ < V < ε v, W > max G 3 V, G V. G V Secondly, we study the relatonshp between G 3 V and V n 6. It s easy to verfy that when V > ε w, G 3 V < V ; when V < ε w, G 3 V > V. Therefore, P N total n 6 can be further rewrtten as follows: P D total 63 = Pr h v > ε w, h w > h v Q ε Pr v < h v < ε w, h w > G 3 h v Q ξ Pr < h v < ε v, h w > G max h v, where G max h v = max Q 3 G 3 V, G V G V Note that the ont PDF of h v and h w s gven by [5 f hv, h w x, y = ϖ F x F y F x w. F y M w fxfy, < x < y, 64 where fx = e x and F x = e x. Applyng bnomal expanson to 64, the frst factor n 63 = = = ϖ ε w x w = = w w e M v y dye x dx w w e M v ε w M v M v. 65 The second factor n 63 can be evaluated as follows: Q = ϖ w = = ε w ε v = ϖ w = = ε w ε v w G 3 x w e M v y dye x dx w M v w e M v G 3 x x dx. 66 Smlarly, the last term n 63 can be evaluated as follows: Q 3 = ϖ w = = ε v ξ w M v w e M v G max x x dx. 67

3 If G h v <,.e., h v < ξ, there are two followng cases: a If ε v β, t s easy to verfy G x > as well as P D total =. b If ε v β <, only when h v > εβ G x <. However, note that ε v β ε v β εβ, > ξ, whch means G x > wth the constrant condton h v < ξ. Therefore the probablty P D total s zero. Substtutng 65, 66, and 67 nto 63, and usng the fact that the outage probablty P D total = PD total, the frst part of the Theorem s completed. In order to fnd the dversty gan, we focus on the case wth hgh SNR,.e.,. Applyng Taylor expanson of the exponental functons n 65, we have Q = ϖ = l= w = w M v w l ε w l M v l l! l. 68 Based on the PDF of h v and h w n 64, we can obtan the cumulatve dstrbuton functon CDF of h v and h w as follows: F hv, h w x, y 69 = ϖ = ϖ w = = x y m w = = e M vx M v w w e M v n dne m dm w w e M v y e x [ M v. By usng the property of CDF, we obtan the followng: w w w ϖ =. 7 M v M v = = Based on 7 and the bnomal theorem, Q n 68 can be further expressed as follows: w w w l ε w l Q = ϖ = = l k= M v l= l! l l k M v l k k. 7 Recall the followng two sums of the bnomal coeffcents [6, Eq..53 K = and [6, Eq..54 K = K k =, k K, 7 K K = K K!. 73 By usng the above sums, we can observe that the sum of the terms k, k < v, s equal to zero n 7 because of 7. Furthermore, all the factor wth k, k > v, can also be removed, snce the domnant factor s = v. In addton, utlzng 73, we have the followng: w Q ϖ = w w M v ε w v v v. 74 The Gauss-Chebyshev ntegraton [9 can be used to approxmate the ntegral n 66 as follows: ε w ε v e M v G 3 x x dx ε w ε v yne M v G 3 x n xn e M vx n, 75 where x n = ε w ε v y n ε w ε v = τn, y n = cos n N π and N s the number of terms ncluded n the summaton. G Note that when, 3 xn xn. Followng steps smlar to the ones for obtanng 74, we can obtan the followng: Q ϖ ε w v v v π N N sn n N π τn. v 76 n= Smlarly, we have the followng: Q 3 ϖ ε w v v v N sn n N π τn. v 77 n= Combnng 74, 76 wth 77, the result for the dversty order n the theorem can be proved. APPENDIX B PROOF OF THEOREM Based on 7 and 39, the coverage probablty for the sum rate n uplnk D-N systems can be further expressed as follows: P U sum = Pr R sum U > R v R w = Pr β V β V β W V V β V W V V > ε W = Pr a W 3 b W c W d >. 78 Recall that the users channels have been sorted as h w > h v, whch means W > V. Therefore, P U sum n 78 can be rewrtten as follows: P U sum = Pr a W 3 b W c W d >, W > V. 79

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Cambrdge unversty press, 5. [8 R. K. Jan, D.-M. W. Chu, and W. R. Hawe, A quanttatve measure of farness and dscrmnaton for resource allocaton n shared computer systems, DEC Techncal Report 3, Sept. 984. [9 E. Hldebrand, Introducton to Numercal Analyss. Dover, New York, USA, 987. [ R. Wtuła and D. Słota, Cardano s formula, square roots, chebyshev polynomals and radcals, Journal of Mathematcal Analyss and Applcatons, vol. 363, no., pp. 639 647,. Zheng Yang S receved the B.S. and M.S. degrees n mathematcs from Mnnan Normal Unversty, Zhangzhou and Fuan Normal Unversty, Fuzhou, Chna, n 8 and, respectvely. He s currently pursung the Ph.D. degree n the Insttute of Moble Communcatons, Southwest Jaotong Unversty, Chengdu, Chna. He was a Vstng Ph.D. Student at the School of Electrcal and Electronc Engneerng, Newcastle Unversty, Newcastle upon Tyne, U.K., from January 4 to July 4. Hs research nterests nclude 5G networks, cooperatve and energy harvestng networks, sgnal desgn and codng. Zhguo Dng S 3-M 5-SM 5 receved hs B.Eng n Electrcal Engneerng from the Beng Unversty of Posts and Telecommuncatons n, and the Ph.D degree n Electrcal Engneerng from Imperal College London n 5. From Jul. 5 to Aug. 4, he was workng n Queen s Unversty Belfast, Imperal College and Newcastle Unversty. Snce Sept. 4, he has been wth Lancaster Unversty as a Char Professor. From Oct. to Sept. 6, he has also been an academc vstor n Prnceton Unversty. Dr Dng research nterests are 5G networks, game theory, cooperatve and energy harvestng networks and statstcal sgnal processng. He s servng as an Edtor for IEEE Transactons on Communcatons, IEEE Transactons on Vehcular Technology, IEEE Wreless Communcatons Letters, IEEE Communcatons Letters, and Journal of Wreless Communcatons and Moble Computng. He receved the best paper award n IET Comm. Conf. on Wreless, Moble and Computng, 9, IEEE Communcatons Letters Exemplary Revewer, and the EU Mare Cure Fellowshp -4. Pngzh Fan M 93-SM 99-F 5 receved hs PhD degree n Electronc Engneerng from the Hull Unversty, UK. He s currently a professor and drector of the nsttute of moble communcatons, Southwest Jaotong Unversty, Chna. He s a recpent of the UK ORS Award, the Outstandng Young Scentst Award by NSFC, and the chef scentst of a natonal 973 research proect. He served as general char or TPC char of a number of nternatonal conferences, and s the guest edtor-n-chef, guest edtor or edtoral member of several nternatonal ournals. He s the foundng char of IEEE VTS BJ Chapter and IEEE ComSoc CD Chapter, the foundng char of IEEE Chengdu Secton. He also served as a board member of IEEE Regon, IETIEE Councl and IET Asa-Pacfc Regon. He has over research papers publshed n varous academc Englsh ournals IEEE/IEE/IEICE, etc, and 8 books ncl. edted, and s the nventor of granted patents. Hs research nterests nclude hgh moblty wreless communcatons, 5G technologes, wreless networks for bg data, sgnal desgn and codng, etc. He s an IEEE VTS Dstngushed Lecturer 5-7, and a fellow of IEEE, IET, CIE and CIC.

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