Long-Term CSI Based Multi-User Scheduling for Collaborative. Spatial Multiplexing in Mobile-WiMAX

Transcription

1 Log-Term CSI Based ulti-user Schedulig for Collaborative Spatial ultiplexig i obile-wiax Ji-Woo Lee ad Yog-Hwa Lee School of Electrical Egieerig ad INC Seoul Natioal Uiversity, Seoul, Korea jiu@ttl.su.ac.kr ad ylee@su.ac.kr Abstract I this paper, we propose a multi-user schedulig scheme to maximize the ergodic sum-rate of the uplik of a multi-user multiple-iput multiple-output wireless system. The proposed schedulig scheme oly utilizes the receive correlatio iformatio of users without istataeous chael state iformatio (CSI). It ca be applied to the collaborative spatial multiplexig mode specified i the IEEE 80.6e mobile-wiax system, where the umber of users i trasmissio is limited to the umber of receive ateas of the BS at each time slot. A upper-boud aalysis ad umerical results show that the proposed schedulig scheme outperforms covetioal radom schedulig scheme i terms of the ergodic sum-rate without loss of opportuity fairess. oreover, the proposed schedulig scheme ca provide a ergodic sum-rate comparable to full-csi based schedulig scheme i the presece of user mobility, while sigificatly reducig the feedback sigalig overhead. Idex Terms Collaborative spatial multiplexig (CS), ergodic sum-rate, feedback sigalig overhead, obile-wiax system, receive correlatio, schedulig complexity. I. INTRODUCTION The sum-rate of the uplik of multi-user multiple-iput multiple-output (IO) systems ca be icreased by usig multiple receive ateas at the base statio (BS) [] [4]. For example, the use of two receive ateas at the BS ca almost double the sum-rate compared to the use of a sigle receive atea, eve whe users use oly a sigle trasmit atea [3], [4]. The sum-rate capacity of a multi-user uplik

2 IO system ca be achieved by employig a miimum mea-squared error-successive iterferece cacellatio (SE-SIC) receiver that decodes the receive sigal from all users i a sequetial maer [3], [4]. However, it may ot be applicable to practical systems due to huge pilot sigalig overhead for the chael estimatio of all users [4], [5]. This problem ca be alleviated by limitig the umber of users i trasmissio to the umber of receive ateas of the BS (i.e., the umber of available degrees of freedom) at each time slot [4] [7]. As a cosequece, much effort has bee devoted to multi-user schedulig methods to maximize the sum-rate. The BS ca maximize the sum-rate by schedulig users with the use of istataeous chael state iformatio (CSI) of all users [8], [9]. However, these full-csi based schedulig schemes may require a large amout of feedback or traiig overhead [0] [], ad may also suffer from so-called chael mismatch problem i the presece of user mobility [], [3], makig it impractical to be employed. Therefore, the use of radom schedulig is ofte cosidered at the expese of sum-rate reductio [6], [7]. The use of spatial correlatio iformatio has recetly bee cosidered for multi-user schedulig i the uplik of multi-user IO systems [4], sice the spatial correlatio characteristics ofte remai uchaged for a relatively log duratio. I fact, the spatial correlatio is little chaged i a iterval of about 00 ms eve whe the user is i a mobility of 000 km/h [5]. By usig this property, [4] ca remarkably improve the sum-rate performace with low feedback sigalig overhead. However, this work oly cosiders the use of the trasmit correlatio (i.e., spatial correlatio at the user) for multi-user schedulig assumig o receive correlatio (i.e., spatial correlatio at the BS). Although this assumptio may be appropriate i idoor eviromets, but ot i outdoor eviromets where few scatters exist ear the BS ad may scatters are ear the users [4], [6]. oreover, the receive correlatio is more meaigful tha the trasmit correlatio i the uplik sice the use of multi-atea cofiguratio is more feasible at the BS tha the user due to the termial complexity issues [6], [7]. I this paper, we propose a ew multi-user schedulig scheme that utilizes the receive correlatio to maximize the ergodic sum-rate of the uplik i multi-user IO systems. We assume that the umber of users i trasmissio is limited to the umber of receive ateas of the BS at each time slot ad that each user has a sigle trasmit atea. I fact, this is cosidered as oe of key multi-user IO techiques i the uplik of the IEEE 80.6e obile-wiax system, so-called collaborative spatial multiplexig (CS) [6]. To optimize the multi-user schedulig i the CS mode, we first aalyze the impact of the

3 receive correlatio o the ergodic sum-rate i the CS mode. The, we fid the coditio to maximize the ergodic sum-rate assumig that the BS oly kows the receive correlatio iformatio of all users. Fially, we propose a multi-user schedulig scheme that ca maximize the ergodic sum-rate i the CS mode. It is show that the proposed schedulig scheme outperforms the radom schedulig scheme i terms of the ergodic sum-rate without loss of opportuity fairess. Furthermore, it is show that the proposed multi-user schedulig scheme provides a ergodic sum-rate comparable to full-csi based multiuser schedulig scheme i the presece of user mobility, while oticeably reducig the feedback sigalig overhead. The remaider of this paper is orgaized as follows. Sectio II describes the system ad chael model i cosideratio. The impact of the receive correlatio o the ergodic sum-rate i the CS mode is aalyzed i Sectio III. Sectio IV describes the proposed receive correlatio-based multi-user schedulig scheme that maximizes the ergodic sum-rate i the CS mode. Sectio V verifies the performace of the proposed multi-user schedulig scheme i terms of the ergodic sum-rate, feedback sigalig overhead, ad schedulig complexity via aalysis ad computer simulatio. Fially, coclusios are give i Sectio VI. II. SYSTE AND CHANNEL ODEL Cosider the uplik of a multi-user IO wireless system, where the BS uses ateas for the receptio ad each of N users employs a sigle atea for the trasmissio. I the CS mode [6], the BS receives the sigal from users at each time slot as illustrated i Fig.. Without loss of geerality, it ca be assumed that user, user,, user are scheduled to trasmit the sigal at ay time slot. Let x = [ x x ] T be the trasmitted sigal vector from the scheduled users ad h = [ h h ] T be the chael vector from user to the BS, where =,,, ad the superscript T deotes traspose. The chael matrix from the scheduled users to the BS ca be represeted as H = [ h h ]. The, the received sigal vector at the BS ca be represeted as [3] y = h x + w = = Hx + w () where w = [ w w ] T deotes the additive oise vector. Here, w m are idepedet ad idetically

4 distributed (i.i.d.) zero-mea Gaussia radom variables with the same variace N 0 m =,,., where Whe the chael is spatially correlated, the chael matrix H ca be geerated usig a complex w w w white Gaussia radom matrix H = [ h h ] whose etries are i.i.d. zero-mea Gaussia radom variables with uit-variace, i.e., [7] H = R H () / w vec( ) vec( ) h R h h BS R h R h h User N User User User (a) User User User User User User 7 User 4 User N User 9 User 6 N / Time slot (b) Fig.. odel of the CS mode. (a) Sigal trasmissio at ay time slot. (b) A example of multi-user schedulig. where vec( H ) deotes a operator that stacks matrix H ito a vector columwise (i.e., vec( H) = [ h h ] ) ad T T T / R deotes the Hermitia positive defiite square root of the chael correlatio matrix (i.e., / / R = R R ). Here, the superscript deotes cojugate traspose. Sice users are ot physically co-located i real eviromets, it ca be assumed that there is o trasmit correlatio betwee the users [8] []. I other words, the colums of H are ucorrelated as

5 E { h h } = O, for, (3) where O deotes a zero matrix whose all etries are zero. Thus, the CS mode has a chael correlatio matrix R represeted as [0] R = E{vec( H)vec( H) } E{ hh } E{ hh } = E{ } E{ } h h h h R O = O R (4) where R deotes the receive correlatio matrix of user defied as R = E{ h h } Ehh { } Ehh { } =. Eh { h } Eh { h} (5) III. ERGODIC SU-RATE IN THE CS ODE To aalyze the impact of the receive correlatio o the ergodic sum-rate i the CS mode, we assume that user, user,, user are scheduled at time slot t without loss of geerality ad that the BS perfectly estimates the chael of the scheduled users (i.e., user, user,, user ) [6]. The, the BS ca get a ergodic sum-rate by meas of a SE-SIC process, give by [3], [4] γ Ct () = E log det I + hh = HΓH = E log det I + (6) where I is a idetity matrix, Γ is a diagoal matrix whose -th diagoal elemet γ represets the average sigal-to-oise ratio (SNR) of user, which equals P / N 0, ad P deotes the average trasmit power of user for =,,. From the cocave property of the logarithmic fuctio ad after some mathematical hadlig, it ca be show that the ergodic sum-rate (6) is upper bouded by

6 (refer to Appedix I) ( ) (0!) {(+ γtr( R)/ ) (+ γtr( R)/ )} (!) (0!) γγ tr( )(+ γ3tr( 3) / ) (+ γtr( ) / ) RR R R + + ( ) + γ γtr( R R)(+ γtr( R) / ) (+ γ tr( R ) / ) 3 (!) (0!) γγγ 3tr( RRR 3)(+ γ4tr( R4) / ) (+ γtr( R) / ) + 3 γ γ γtr( )(+ γtr( ) / ) (+ γ 3tr( 3) / ) + R R R R R Ct () log + ( ) 4 (!) (0!) γγγγ 3 4tr( RR )tr( RR 3 4)(+ γ5tr( R5) / ) (+ γ tr( R )/ ) + (7) γ 3γ γ γtr( R 3R)tr( R R )(+ γtr( R) / ) (+ γ 4tr( R 4) / ) + + (( )!) + ( ) γγ γ{tr( RR R)} Ct (). Cosider a simple case where = for ease of mathematical tractability. The, (7) ca be simplified to Ct ( ) log (( + γ tr( R ) / )( + γ tr( R ) / ) γ γ tr( R R ) / 4). (8) where ρ R = ρ. (9) j for =,. Here, ρ e θ = α deotes the receive correlatio coefficiet of user, where α ( 0 α < ) ad θ (0< θ π ) deote the amplitude ad the phase of the receive correlatio coefficiet, respectively. Sice tr( R) = tr( R ) = ad tr( ) ρρ RR = + + ρρ, (8) ca be rewritte as γγ Ct () log + γ + γ + ( ρρ ρρ). 4 (0) Notig that ρ = α for =,, we have e j θ γγ Ct () log + γ + γ + ( αα cos( θ θ)). () Fig. depicts the ergodic sum-rate i the CS mode with the upper-boud () for a average SNR of 0 db (i.e., γ = γ = 0 db ). It ca be see that the ergodic sum-rate i the CS mode highly depeds o the receive correlatio coefficiet of the two scheduled users (i.e., user ad user ). Notice that the ergodic sum-rate icreases as the phase differece approaches to π (i.e., θ θ π ), ad the product

7 of the two amplitudes approaches to oe (i.e., αα ). It ca also be see that although the simulatio results ad the aalytic upper-boud () have some discrepacy due to Jese s iequality loss (i.e., the ergodic sum-rate differece betwee (33) ad (34) i Appedix II), they have a tedecy very similar to each other. Thus, the aalytic upper-boud ca be applied to the multi-user schedulig to maximize the ergodic sum-rate i the CS mode. I other words, the BS ca maximize the ergodic sumrate by fidig a pair of users whose receive correlatio coefficiets have a phase differece close to π ad a amplitude product close to oe Ergodic Sum-Rate (bps/hz) θ -θ =π (simulatio) θ -θ =π/ (simulatio) θ -θ =0 (simulatio) 4.5 θ -θ =π (upper-boud) γ = γ = 0 db θ -θ =π/ (upper-boud) θ -θ =0 (upper-boud) α = α Fig.. Ergodic sum-rate accordig to receive correlatio coefficiet whe =. IV. PROPOSED ULTI-USER SCHEDULING IN THE CS ODE Cosider multi-user schedulig that utilizes oly the receive correlatio iformatio without the use of istataeous CSI of all users i the CS mode. To provide fair schedulig opportuity to all N users, it is assumed that each of N users is scheduled oce every N/ time slots (i.e., times every N time slots) i a average sese. Uder the above assumptio, there ca be N C schedulig cases at time slot, where x C y x! () ( x y)! y! ad x! x ( x ) 3. At time slot, there ca be N C schedulig cases sice the

8 scheduled users at time slot are excluded for the schedulig. I this maer, there ca be N t + C schedulig cases at time slot t. Thus, there ca be a total S umber of schedulig cases i N/ time slots, where N / N t+ C S =. (3) ( N / )! t = Sice N / N / C = C, it ca be show that N t+ t t= t= N! S =. (4) ( N ) N / (!) /! Let C () t s be the aalytic upper-boud (7) correspodig to the s -th schedulig case at time slot t, where s =,, S, ad t =,, N /. The, the optimum schedulig case ca be foud by N / sopt = arg max C s ( t). (5) s {,..., S} N t = As a simple example to clarify the above procedure, cosider the applicatio of the proposed multi-user schedulig to a CS mode with =, where 4 users have the same average SNR (i.e., γ the receive correlatio coefficiets whose amplitudes are the same (i.e., α = γ ), ad = α ) ad whose phases are equally scattered o (0, π ] (e.g., θ = π / N for =,, 4 ). The, there ca be 3 possible cases for the schedulig of 4 users i N / ( = ) time slots as illustrated i Fig. 3 (i.e., {(, ), (3, 4)}, {(, 3), (, 4)}, {(, 4), (, 3)}, where the umbers deote the user idex). The upper-boud of the ergodic sum-rate correspodig to these three schedulig cases ca be represeted as γ C () t = log + γ +, (6) t = γ C () t = log + γ + ( + α ), (7) t = γ C 3() t = log + γ +. (8) t = Thus, it ca easily be foud that the secod schedulig case (i.e., {(, 3), (, 4) } i Fig. 3 (b)) is the optimum case (i.e., s opt = ) that maximizes the ergodic sum-rate i this CS mode.

9 User User User User User 3 User User User User User User 4 User 3 User 4 User 4 Time slot Time slot (a) (b) (c) User 3 Time slot Fig. 3. Three schedulig cases for = ad N = 4. (a) s =. (b) s =. (c) s = 3. The proposed multi-user schedulig scheme ca maximize the ergodic sum-rate by oly utilizig the receive correlatio iformatio without loss of opportuity fairess. However, it may ivolve schedulig complexity. This complexity problem ca be alleviated by sub-optimizig the schedulig by meas of user sheddig-based exhaustive search: At time slot, the BS schedules users that maximize the ergodic sum-rate based o the aalytic upper-boud (7) amog N users. At time slot, the BS selects users that maximize the aalytic upper-boud (7) amog ( N ) users, sheddig the scheduled users at time slot. I this maer, all N users are scheduled i N/ time slots. This suboptimum method ca sigificatly reduce the schedulig complexity without oticeable sum-rate performace degradatio which will be verified i Sectio V. V. PERFORANCE EVALUATION I this sectio, we verify the performace of the proposed multi-user schedulig scheme i terms of the ergodic sum-rate, feedback sigalig overhead, ad schedulig complexity. For referece, the performace of covetioal multi-user schedulig schemes (i.e., full-csi based ad radom multi-user schedulig scheme) is also evaluated. To make the performace compariso tractable, we cosider a simple case where the BS has two receive ateas (i.e., = ). We assume that all users have the same average SNR (i.e., γ amplitudes are the same (i.e., = γ ), the same speed of v, ad the receive correlatio coefficiets whose α = α ) ad whose phases θ are uiformly distributed o (0, π ]. A. Ergodic Sum-Rate Gai We aalyze the ergodic sum-rate gai of the proposed schedulig scheme over the radom schedulig scheme by usig the aalytic upper-boud (). Based o the above assumptio, the aalytic upper-boud

10 () ca be simplified to γ Ct () Ct () = log + γ + ( α cos( θ θ )) (9) where θ ad ad, at time slot t. θ are the phase of the receive correlatio coefficiets of the two scheduled users, user The radom multi-user schedulig scheme radomly picks up a pair of users at each time slot. The correspodig ergodic sum-rate ca be represeted as π π EC { } Ctd ( ) θ dθ 4π r 0 0 γ = + + 4π EC { }. π π 0 log 0 γ ( α cos( θ θ )) d d θ θ r (0) Note that (0) correspods to the esemble average of the ergodic sum-rate of the radom multi-user schedulig scheme. It ca be show from the defiitio of the ergodic sum-rate [] that (0) ca be approximated i terms of the time averaged sum-rate as S N / EC { } C( t). r SN s = t = s () Ulike the radom schedulig scheme, the proposed schedulig scheme assigs users to maximize the ergodic sum-rate amog S umber of schedulig cases. The correspodig ergodic sum-rate of the proposed schedulig scheme ca be represeted as N / EC { p} C ( t) sopt N t = EC { }. p () The ergodic sum-rate gai of the proposed schedulig scheme over the radom schedulig scheme ca be represeted as N / S N / EC { } EC { } = C () t C() t. (3) p r s s opt N t= S N s= t= Sice (3) caot be represeted i a closed form, we cosider a asymptotic ergodic sum-rate gai as the

11 umber of users icreases to ifiity (i.e., N ). I the proposed scheme with this assumptio, it is possible to choose a pair of users whose receive correlatio coefficiet has a phase differece close to π (i.e., θ θ π ) at each time slot. The correspodig asymptotic ergodic sum-rate of the proposed schedulig scheme ca be represeted as γ lim EC { p} log + γ + ( + α ) N. (4) It ca be show that (refer to Appedix II) k ( k ) γ α γ!! EC { r } = log + γ + π l k = + 4γ + γ k k!! (5) where!! deotes the double factorial defied by [3] ( ) ( ) x x 5 3 x > 0 odd x!! x x 6 4 x > 0 eve (6) x =, 0. Thus, the asymptotic gai by the proposed multi-user schedulig scheme ca be represeted as k αγ αγ + 4γ + γ k = + 4γ + γ ( k )!! lim EC { p} EC { r} log. N + + π l k k!! (7) It ca be see that the asymptotic ergodic sum-rate gai by the proposed schedulig scheme icreases as γ ad α icrease. It ca be see that the proposed schedulig scheme has a asymptotic ergodic sumrate gai of up to about.45 bps/hz as N, γ, ad α, i.e., p r α γ N π ( k )!! lim lim lim EC { } EC { } = + l k = k k!!.45 bps/hz (8)!!/!!.5. sice ( k ) ( k k ) k = B. Feedback Sigalig Overhead We aalyze the feedback sigalig overhead of the proposed, full-csi based ad radom multi-user schedulig schemes accordig to N. The full-csi based schedulig scheme requires N feedback sigals at each time slot sice all users should report their istataeous CSI at each time slot. Thus, the

12 amout of feedback sigalig overhead i the full-csi based schedulig scheme icreases i liear proportio to N [0] []. O the other had, the proposed schedulig scheme oly requires N feedback sigals i N / time slots (i.e., feedback sigals at each time slot) as illustrated i Sectio IV. Thus, the amout of feedback sigalig overhead for the proposed schedulig scheme is fixed to regardless of N, remarkably reducig the feedback sigalig overhead over the full-csi based scheme. The radom schedulig scheme ca work without feedback sigal, but it yields the worst sumrate performace as will be show i Sectio V-D. C. Schedulig Complexity We compare the schedulig complexity of the proposed, full-csi based, ad radom multi-user schedulig schemes. I the full-csi based schedulig scheme, the BS eeds to calculate the sum-rate for N C schedulig cases at each time slot to fid out the optimum schedulig case. O the other had, i the proposed schedulig scheme, the BS eeds to calculate the ergodic sum-rate for N / N!/{(!) ( N / )!} schedulig cases i / N time slots (i.e., N / ( N )! /{(!) ( N / )!} at each time slot). This complexity ca be reduced by employig a sheddig-based exhaustive search N / N t+ C t = scheme, requirig the calculatio for /( N / ) schedulig cases at each time slot. For example, whe =, the schedulig complexity of the proposed scheme is much larger tha that of the full-csi based scheme whe N is larger tha 8 (e.g., 9305 vs. 9 at each time slot whe N = 4 ). However, the schedulig complexity of the proposed schedulig with sheddig-based exhaustive search is always lower tha that of the full-csi based multi-user schedulig scheme sice N t + C N C (e.g., 36 vs. 9 at each time slot whe N = 4 ). D. Simulatio Results We verify the aalytic desig ad the sum-rate performace of the proposed schedulig scheme by computer simulatio. The system parameters for the simulatio are take from those of the uplik of IEEE 80.6e obile-wiax system, where the carrier frequecy f =.3 GHz, the frame legth c T = 5ms, the umber of BS receive ateas is two (i.e., = ) ad users employ a sigle trasmit atea [6]. We cosider the sigal trasmissio over a spatially-correlated chael represeted as [4]

13 h () t = εhˆ ( t T) + ε R h (9) / w where h () t deotes the chael vector from user to the BS at time slot t, h ˆ ( t T ) deotes the estimated CSI of user at the previous frame, w h is a complex white Gaussia radom vector whose etries are i.i.d. zero-mea Gaussia radom variables with uit-variace, ad ε is the time correlatio coefficiet associated with the user mobility v. The time correlatio coefficiet ε i spatiallycorrelated chael eviromets ca be represeted as [5] πtfc v πtfc v ε = J0 + J c c (30) where c is the speed of light ( kid defied by [6] 8 = 3 0 m/s) ad J ( x ) is the l -th order Bessel fuctio of the first l k l+ k ( ) ( x /) Jl ( x), (3) k! Γ ( l+ k + ) k = 0 ad Γ ( x) is the gamma fuctio defied by [6] Γ ( ) x z x z e dz. 0 (3) I the full-csi based schedulig scheme, the BS first selects a pair of users maximizig the sum-rate based o the CSI reported from N users at the previous frame time (i.e., at ( t T )) ad the it iforms them the schedulig iformatio. Fially, the two scheduled users trasmit the data sigal to the BS at time slot t.

14 Ergodic Sum-Rate (bps/hz) Full-CSI based schedulig Proposed schedulig (optimum) Proposed schedulig (sub-optimum) Radom schedulig v = 0 km/h v = 30 km/h v = 60 km/h v = 00 km/h = 8 α = γ (db) Fig. 4. Ergodic sum-rate accordig to γ whe N = 8 ad α = 0.9. Fig. 4 depicts the ergodic sum-rate i the CS mode accordig to the average SNR γ with the use of the three multi-user schedulig schemes for N = 8 ad α = 0.9. It ca be see that the proposed suboptimum scheme provides performace comparable to the proposed optimum scheme, while reducig the schedulig complexity by 3.75 at each time slot. It ca also be see that the full-csi based schedulig scheme outperforms the proposed schedulig scheme, especially i low user mobility. However, the proposed scheme provides almost the same performace as the full-csi based scheme whe the user mobility is higher tha 60 km/h. This is maily due to the chael mismatch problem i the full-csi based schedulig scheme [], [3]. The proposed scheme outperforms the radom schedulig scheme without loss of opportuity fairess. oreover, the ergodic sum-rate gap betwee the two schedulig schemes icreases as γ icreases, which meas that the proposed scheme is effective i high SNR regime. Fig. 5 depicts the ergodic sum-rate of the three schemes accordig to the umber of users whe γ = 0 db ad α = 0.9. It ca be see that as N icreases, the full-csi based scheme outperforms the other schemes i low mobility eviromets by exploitig so-called multi-user diversity gai [7], [8], but it suffers from the chael mismatch problem. It ca also be see that the proposed scheme works almost idifferetly from user mobility ad its ergodic sum-rate gai icreases as the umber of users icreases. It ca also be see that both the proposed optimum ad sub-optimum schemes provide similar performace. Thus, it is quite practical to employ the proposed sub-optimum scheme as N icreases.

15 For example, whe N = 4, the schedulig complexity of the two schemes is 9305 ad 36 at each time slot, respectively. Notice that the radom schedulig scheme caot exploit ay multi-user diversity gai. Ergodic Sum-Rate (bps/hz) Full-CSI based schedulig Proposed schedulig (optimum) Proposed schedulig (sub-optimum) Radom schedulig v = 0 km/h v = 30 km/h v = 60 km/h v = 00 km/h γ = 0 db α = N Fig. 5. Ergodic sum-rate accordig to N whe γ = 0 db ad α = Full-CSI based schedulig Proposed schedulig (optimum) Proposed schedulig (sub-optimum) Radom schedulig v = 0 km/h v = 30 km/h v = 60 km/h v = 00 km/h Ergodic Sum-Rate (bps/hz) γ = 0 db N = α Fig. 6. Ergodic sum-rate accordig to α whe γ = 0 db ad N = 8.

16 Fig. 6 depicts the ergodic sum-rate of the three schedulig schemes accordig to the variatio of α whe applied to the CS mode with N = 8 at γ = 0 db. It ca agai be see the full-csi based scheme works well i low user mobility ad its performace somewhat icreases as α icreases as does the proposed scheme. It ca also be see that the performace of the proposed ad radom schedulig schemes is the same whe α = 0, but the performace of the radom scheme decreases as α icreases. This is maily due to the fact that the radom schedulig scheme has high probability of choosig pairs of users whose receive correlatio coefficiets have a phase differece close to 0, ad the correspodig ergodic sum-rate rapidly decreases as α icreases as show i Fig.. As a cosequece, the proposed schedulig scheme is quite effective compared to the radom schedulig scheme i highly correlated chael eviromets. VI. CONCLUSION We have cosidered multi-user schedulig i the CS mode specified i the IEEE 80.6e obile- WiAX system. To miimize the feedback sigalig overhead ad improve the sum-rate performace i mobile eviromets, the proposed multi-user schedulig scheme maximizes the ergodic sum-rate by oly utilizig the receive correlatio iformatio of users. The performace of the proposed schedulig scheme has bee aalyzed ad optimized i terms of the ergodic sum-rate i the CS mode by usig a upper boud. The schedulig complexity of the proposed scheme ca be reduced by employig a user sheddig-based exhaustive search method without oticeable sum-rate performace degradatio. The simulatio results show that the proposed multi-user schedulig scheme is quite effective i the presece of receive chael correlatio ad user mobility. This work ca be exteded to trasmit correlatio-based user schedulig i the dowlik of multi-user IO systems. This may require the use of a multibeamformig techique based o the trasmit correlatio iformatio of users. It ca be show from det ( + ) = det ( + ) APPENDIX I DERIVATION OF UPPER-BOUND (7) I AB I BA [] that (6) ca be rewritte as ΓH H Ct () = E log det I +. (33)

17 Applyig Jese s iequality, it ca be show that (33) is upper-bouded by ΓH H Ct () log E det I +. (34) Or, (34) ca explicitly be represeted as + γhh / γhh / Ct () log E det. γ / + γ / h h h h (35) Sice h = R h from (), (35) ca be rewritte as / w w w w / / w + γh Rh / γh R R h / Ct () log E det. w / / w w w γ / + γ / h R R h h R h (36) w w Sice E{ h Ah } = tr( A ), E{ H A H H A H H H H A H } = tr( A A A ) w w w w w w w w 3 3, we ca have E{ H A H H A H } = tr( A A ),, w w w w ( ) (0!) {(+ γtr( R)/ ) (+ γtr( R)/ )} (!) (0!) γγ tr( )(+ γ3tr( 3) / ) (+ γtr( ) / ) RR R R + + ( ) + γ γtr( R R)(+ γtr( R) / ) (+ γ tr( R ) / ) 3 (!) (0!) γγγ 3tr( RRR 3)(+ γ4tr( R4) / ) (+ γtr( R) / ) + 3 γ γ γtr( )(+ γtr( ) / ) (+ γ 3tr( 3) / ) + R R R R R Ct () log + ( ). (37) 4 (!) (0!) γγγγ 3 4tr( RR )tr( RR 3 4)(+ γ5tr( R5) / ) (+ γ tr( R )/ ) γ 3γ γ γtr( R 3R)tr( R R )(+ γtr( R) / ) (+ γ 4tr( R 4) / ) + + (( )!) + ( ) γγ γ{tr( RR R)}

18 ( ) (0!) {(+ γtr( R)/ ) (+ γtr( R)/ )} (!) (0!) γγ tr( )(+ γ3tr( 3) / ) (+ γtr( ) / ) RR R R + + ( ) + γ γtr( R R)(+ γtr( R) / ) (+ γ tr( R ) / ) γγγ 3tr( RRR 3)(+ γ4tr( R4) / ) (+ γtr( R) / ) + 3 (!) (0!) 3 + γ γ γtr( R R R)(+ γtr( R) / ) (+ γ 3tr( R 3) / ) Ct () log + ( ) γγγγ 3 4tr( RR )tr( RR 3 4)(+ γ5tr( R5) / ). 4 (!) (0!) (+ γ tr( ) / ) R γ 3γ γ γtr( R 3R)tr( R R )(+ γtr( R) / ) (+ γ 4tr( R 4) / ) + + (( )!) + ( ) γγ γ{tr( )} RR R APPENDIX II Lettig γ A = + γ + ad DERIVATION OF (5) B = α γ, (0) ca be rewritte as π π EC { } = log ( A Bcos( θ θ )) dθ dθ 4π r 0 0 π π A = log 0 0 B l cos( θ θ ) dθ dθ. 4π + l B (38) Sice k x l( a x) = l a, (38) ca be rewritte as k ka k = EC { } log B l cos ( ) d d k π π A B k r = 0 0 θ θ θ θ 4π + l B k = k A π π B k = log 0 0 A cos ( θ θ ) dθ dθ 4π l k = k A π π B k = log A cos ( θ θ ) dθ dθ 4π l 0 0 k = k A 3/ B k + π Γ A = log A. 4π l k = k + kγ k k k (39) k π ( k )!! From [9], we have Γ =. Therefore, (39) ca be rewritte as ( )/ k

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