Abstract- In this paper, we present a performance analysis forthe channel capacity and outage probability of the user schedul­ing for multiuser multiple-input multiple-output (MIMO) systemswith maximum ratio combining (MRC) exploiting the user andantenna diversities. We consider two scheduling schemes: abso­lute SNR-based and normalized SNR-based scheduling schemes.We propose the utilization of a transmit antenna selectionscheme in multiuser MIMO systems in order to overcomethe drawback of channel hardening and also to improve thesystem performance. We derive new closed-form expressionsfor the outage probability and average channel capacity of thepresented schemes for two scenarios: Heterogeneous (indepen­dent non-identically distributed SNRs- i.ni.d.) and Homogeneous(independent identically distributed SNRs- i.i.d.) cases. Usingmathematical analysis and numerical simulations, we comparethe performances of presented schemes.

Capacity and Outage Probability Analysis ofMultiuser Diversity in MIMO MRC Systems with

Transmit Antenna SelectionMohammad Torabi1, David Haccoun ' and Wessam Ajib''

1 Department of Electrical Engineering, Ecole Polytechnique de Montreal, Montreal, QC, Canada.2 Department of Computer Science, Universite du Quebec it Montreal, QC, Canada.

in MIMO systems, the full advantages of both multiuserand antenna diversities cannot be exploited [6], [7]. In orderto reduce the channel hardening, antenna selection can beemployed [8]-[10]. It has been shown that a significant gainfrom both multiuser and antenna diversities can be obtainedat the same time for the independent identically distributed(i.i.d.) user links. However, to the best of our knowledge, acomplete performance analysis of user scheduling schemes inmaximum ratio combining (MRC) MIMO systems with an­tenna selection especially for independent and non-identicallydistributed (i.ni.d.) users is not available in the literature.Likewise, channel capacity and outage probability analysesfor these systems are of interest but have not been reportedpreviously.

I. INTRODUCTION In this paper, we present a performance analysis for theIn multiuser wireless communications, scheduling tech- outage probability and average channel capacity for multiuser

niques have been proposed to exploit the multiuser diversity MIMO systems employing MRC technique at the receiver. Inand hence improve the performance of the system. Considering order to reduce the effects of MIMO channel hardening, wethe downlink, from the base station to the mobile stations propose utilizing a transmit antenna selection (TAS) scheme(users) at each time slot, the user having the best channel at the base station for improving the system performance andcondition is scheduled to access the channel. While a proper reducing the hardware complexity. We consider two scenarios:scheduling may increase the system throughput, the fairness Heterogeneous (i.ni.d.) and Homogeneous (i.i.d.) cases. Wein resource allocation among the users is a key parameter and derive the expressions for the probability density functionshould be taken into consideration. The round-robin scheduler (PDF), and the cumulative distribution function (CDF) of theis a simple and full-fair scheduling scheme, where all users signal-to-noise-ratio (SNR) in a MIMO channel with antennahave the same priority for accessing the channel, but it does selection for the considered multiuser scheduling schemes.not exploit the multiuser diversity. On the other hand, the best- This enables us to establish a mathematical analysis anduser scheduler, the so called opportunistic or greedy scheduler, formulation for the performance of the system under study.selects the user with the highest throughput. It achieves a The paper is organized as follows. Section II presents thehigher throughput than the round-robin scheduler but this gain system model. The derivations of PDF and CDF of the user'sis obtained at the cost of unfair resource allocation among the channel SNRs for the multiuser diversity system are alsousers, as for example for the users that are closer to the base presented. In Section III, two multiuser scheduling schemesstation, and hence having more chance to access the channel. in conjunction with antenna selection are presented. In each

The proportional fair scheduler (PFS) exploiting the mul- case, the analyses of the outage probability and averagetiuser diversity while maintaining the fairness among the channel capacity are presented. Numerical results are givenusers. Instead of selecting the best user with the highest and discussed in Section IV, and finally, Section V concludesabsolute throughput or absolute signal-to-noise-ratio (SNR), the paper.PFS chooses the user with the highest normalized-throughput II. SYSTEM MODELor normalized-SNR, (normalized each to its own average) Fig. 1 shows a block diagram of the multiuser MIMO[1], [2]. In multiple-input multiple-output (MIMO) multiuser system model considered in this paper for the downlink datasystems, a scheduler can be used to exploit the antenna and transmission of a cellular system. A base station employingmultiuser diversities at the same time and to increase the L transmit antennas is serving K users (mobile stations)system performance [1]-[5]. An overview of user selection and where each user has N receive antennas. The base stationscheduling algorithms in MIMO systems has been presented allocates the resources to the users and schedules the downlinkin [6]. It has been observed that due to the channel hardening transmission of the date to the users. In conventional users

59

978-1-4244-4561-5/09/$25.00 ©2009 IEEE PACRIM'09

B. User Scheduling

We assume that each user feedbacks its correspondingSNRs; 1'bu,u and the index bu E {I"" , L} to the basestation through a reliable feedback channel. The goal ofscheduler is to select the best user (rn-th user) among allthe active users, i.e., to select the highest SNR among Kfollowing SNRs ( 1'bl,I 1'b2,2 . .. 1'bK ,K l. We first considerthe homogeneous case where the channel SNR values for theusers, 1'bu ,u values are i.i.d. with respect to the user index u.Then, we consider the heterogeneous case where the short­term average SNR values of users are i.ni.d. with respect toindex u. We present the scheduling schemes including absoluteSNR-based scheduling and normalized SNR-based scheduling.

60

(5)

(6)

UserK

User 1

MRCUser 1

MRC rK

User K

N L.-....----'

(b, and Ib" , )

N L.-,---'

. (b. and-Ib,. ,')

1

u= 1" " ,K,

K Mobile Stations.... - - - - - - - - - - - - ...

(b. and I b,,.) • . . . , (bK and -I"" K)

CSI Feedback Channel

Base Station...... -- ---- ------tl>

r/Data for51 1User 1

Data forUser One antenna

User 2 5 2 Scheduler is selectedSelects

One User

~Data for5 K L

UserK

f "YbU'u (')') = L t-: (')') [F"Yi,u (')')]£-1. (7)

where the f "Y (')') and F"Y (')') are defined in (3) and (4).'t ,t./. t ,U

where A = {I ,2, ... ,L}, 1'i ,u(t) is the instantaneous SNR,for user u in time-slot t for the i-th transmit antenna. In otherwords, considering a K x L matrix, r = (')'i ,u(t)} with theelements 1'i ,u(t) (i = 1" " , L, and u = 1" " ,K), in theu-th row of that matrix, the largest element, denoted by 1'bu ,uis selected out of L elements . Then, for a total of K rows,a vector of those selected SNRs can be considered, given by

(1'bl ,l (t) 1'b2,2(t) ... 1'bK ,K(t))T,where(.)Tdenotesthe

transpose operation . To ease the notation, time-slot index (t)will be ignored. In this scheme, the receiver needs only tofeedback the index of the selected transmit antenna (the indexbu ) and the SNR of the users (given by the above vector) tothe transmitter, instead of the full CSI of all transmit antennas(given by the K x L matrix, T), This scheme is attractive, sinceit significantly reduces the amount of the required feedback tothe transmitter. Using the order statistics and assuming that theSNR values corresponding to the transmit antennas are i.i.d.(with respect to i , i = 1" " , L) , we can express the CDFand PDF of the SNR for the best transmit antenna for user uas

Fig, 1. User scheduling in a MIMO MRC system with antenna selection.

and

(I)

(3)

(4)

u = 1,2, ·· ·,K

where r u is the output signal of MRC, SU is the transmittedsymbol, h i,u is the i-th column of the MIMO channel ma­trix H u , and II .II~ is the squared Frobenius norm given by

Ilhi ,ull~ = L:f=1 Ih~ ' i)12, and where 'flu is a complex Gaus­

sian noise with distribution N(O, ~ Ilhi,u II~) per dimension .The instantaneous SNR per symbol at the receiver of the useru may be written as

1'i ,u = 'Yi,u Ilhi,ull~ (2)

scheduling system, it is assumed that the base station uses afeedback channel from the users in order to acquire channelstate information (CSI) knowledge or SNR value from each ofthe users [6]. To do this, for example using pilot signals, thereceiver at each mobile station (user) estimates the SNR valuereceived from each transmit antenna of the base station, andthen sends that SNR value to the base station through a robustfeedback channel. We assume a perfect channel estimationand the feedback channel to be error free. However, from apractical point of view, the effects of channel estimation errorand feedback delay on the performance is not addressed here,but may be the subject of future work.

Here, we assume that the channel between the base stationand user u (u = 1,2,· .. ,K) is a quasi-static Rayleigh fadingMIMO channel and expressed by H ; matrix of size N xL, with elements h~, i) corresponding to the complex channelgain of the channel response between the z-th transmit andj-th receive antennas for the u-th user. The received signalexpression for each user at the output of MRC can be writtenas

where 'Yi u is the average receive SNR per antenna for user

u. Since ~e assumed that the channels fading, i.e., h~,i) are

Rayleigh flat-fading, therefore, Ih~,i) 1

2for each user will be a

Chi-squared distributed random variable . For a given u and i,since Ilhi ,ull~ is the sum of N i.i.d. random variables (whichthey may be i.ni.d. with respect to u), then Ilhi,ull~ will bea Chi-squared distributed with 2N degrees of freedom. Thus,using a change in variables, we can show that the PDF andCDF of the received SNR for each user; given in (2), can beexpressed as

1'N-1 ( 1' )f"Yi,,,(')') = (N-1)! NexP -;:y o

1't,U It,U

F"Yi ,,,(')') = 1- exp ( __1' ) >= ~ (_1' )11't ,U 1=0 l. 1't,U

III. MULTIUSER SCHEDULING AND ANTENNA SELECTION

In this section, we explain an antenna selection scheme andtwo user scheduling schemes for the system under study.

A. Antenna Selection

As stated earlier, at the receiver sides for each user, theSNR received from each transmit antenna of the base stationis obtained using an estimation technique . At the receiversside, based on those SNR values, each user's receiver selectsits best transmit antenna (bu-th antenna), having the highestchannel SNR, i.e.

and

f'Ybm,m (1') = K f'Ybu,u (1') [F'Ybu,u (1')]K-1 (10)

where f'Y (1') and F'Y (1') are defined in (6)-(7). Substi-tuting (6)bua~d (7) into (9)'~nd (10) yields

where U = {I, 2, ... , K}, and 1'bu,u(t) is the instantaneousSNR of user u with the best transmit antenna of base stationfor that user, bu in the time-slot t. Then, using the theory oforder statistics the CDP and PDP of the best user (with SNR1'bm,m) selected from K available users can be obtained from

1) Outage Probability: The outage probability is an impor­tant measure for the performance evaluation of communicationsystems over fading channels. It can be defined as the proba­bility that the capacity of the channel from the best transmitantenna of base station to the best user in the system understudy falls bellow a predefined threshold value Co.

Since, the capacity of the channel between i-th transmitantenna of base station and u-th user can be written as

Oi.« = log2 (1 + 7fi,u Ilhi,u II~) , (15)

then, the achieved capacity of the channel between the besttransmit antenna of base station to the best user in this casecan be expressed as

Cbm ,m = log2 (1 + 1'bm,m) (16)

where 1'bm,m = 7fbm,m Ilhbm,mll~.The outage probability can be expressed as

Pout = Pr (Cbm,m < Co) = Pr (log2(1 + 1'bm,m) <Co)

= Pr (»:» < AD) = F'Ybm,m (AD). (17)

where AD = 20 0 - 1. Substituting (13) into (17) yields

(9)

(8)

Homogeneous Case

In the homogeneous case, the statistics of all users are thesame, and all the users' SNRs are i.i.d. Hence, the absolute­SNR based scheduler selects the best user (m-th user) withthe maximum absolute-SNR, i.e.

(11)

and

Pout = t(:) (-l)i exp (-~AO) (~)~i,t (~)t .(18)i=O l' t=O l'

(12) Clearly, the outage probability decreases as the value of ADbecomes smaller.

By substituting (4) into (11) and using the binomial expansion

of (1- x)Z defined by (1- x)Z = 2::~=o ( :) (-I)i xi, and

assuming Z = K Land 7fi u = 7f, after some manipulationswe obtain '

where ai,t denotes the coefficient of (~) t in the expansion

[2::[:01 n(~Yrdefined by: ai,O = 1, ai,1 = i, ai,t =

1 ~min(t,N-1) n(i+1)-t. f 2 < t < q(N - 1),t L.Jn= 1 n! .a1"t-n, or _ f/

and ai,t = [(N - 1)!]-1" for t = i(N - 1) [5].

Similarly, by substituting (3) and (4) into (12) and using thebinomial expansion, after some manipulations, we obtain

Z (1)Z-1(Z_1) .i-;.: b) = (N-l)! ~ ~ i (_I)t

((i + 1)1') i(N-1) (1')t+N-1

x exp _ L ai,t = .(14)l' t=O l'

Knowing F'Y (1') and f'Y (1') given by (13) and (14),bm,m b"'l,m

respectively, we can derive closed-form expressions for theoutage probability and average capacity as shown below.

2) Average Channel Capacity: The average capacity of thechannel between the best transmit antenna of base station tothe best user can be expressed by

C = (XJ log2(1 + 'Yb""",) !'Yb '" bb""",) d'Yb""",' (19)io m,

U sing the expression of f'Yb ,m (1'bm ,m) for this schedulingscheme, given in (14) and substituting into (19), we obtaina closed-form expression for the average capacity as thefollowing expression, which is proved in the Appendix.

Z-1 i(N-1)C= Zlog2(e) ~ (Z~l) (_I)i " (t+N-l)!ai,t

(N - I)! ~ ~ ~ (" + l)t+N1,=0 t=O ~

{ ( "+1) ("+1)x Pt+N - ~ 'Y E1 ~ 'Y

t+N-1 1 (i+1) ( i+1)}+ {; qP q 'Y Pt+N-q - 'Y .(20)

where Pq ( .) is the Poisson distribution defined by Pq (x) =2::~:6 x: e-x, and E1 (.) is the exponential integral of firstorder, defined by E1 (x) = Ixoo

C 1 e- t dt for x > O.

3) Access Probability in the Homogeneous Case: Theaccess probability can be defined as the probability that user uis scheduled to access the channel. In the homogeneous caseemploying absolute-SNR based scheduler, it can be expressedby [11]

Pu = Pr(1'bu,u 2: all other 1'bj,j). (21)

Thus using (9), expression (21) can be written as61

(22)

(29)

roo [ ]K-1r; = Jo f"lb",,, ('r) F"Ib",,, ('r) d'Y

1 roo 1=KJo f"lb""",('r)d'Y=K'

As expected, in the homogeneous case, the access probabilityof all users are the same, i.e. Pu = 1/K.

Heterogeneous Case

In the heterogeneous case, as a more practical scenario, thestatistics of users are non-identically distributed i.e. the short­term average SNR values of the users are i.ni.d. Absolute­SNR based scheduling and normalized SNR-based schedulingare again two possible scheduling schemes. The absolute-SNRbased scheduler always selects the user with the highest SNR.Since this scheduler selects for each time-slot the user havingthe best quality channel, i.e. highest SNR, users suffering frombad channel conditions may never have access to the channel,yielding an unfair resource allocation. To overcome this situ­ation, we now consider a normalized SNR-based schedulingscheme that is fair to all users. Considering that users arei.ni.d., a criterion for the user scheduling with proportionalfair scheduling, is defined as the ratio of the instantaneousSNR of each user over its own average SNR, [2], [11], i.e.,

m = argmax{~bu'U(t)} (23)uEU ;Ybu,u (t) ,

where ~bu ,u (t ) and ;Yi; ,u ( t ) are the instantaneous and theshort-term average SNR values, respectively for the user uand best transmit antenna of base station for user u, bu inthe time-slot t. When the average SNR ;Ybu,u' of all usersare the same, (i.e. all users are i.i.d.) this scheduler will beequivalent to the absolute SNR-based scheduler stated earlier.In this scheduling scheme, the base station selects the userswith the largest normalized SNR value. Similar to the methodin [11], the PDF and CDF of the best user (with SNR ~brn,rn)

selected from K available users can be respectively obtainedfrom

f"lb""", ('r) = t'Y 1 fb",,, ("1 'Y ) ITFbk,k ("1 'Y ) (24)u=1 bu,u bu,u k=1 bu,u

kfu

andK _'"Y_ K

F"Ib""", ('r) = L l"'b"'" fb",,, (x) II Fbk,k (x) dx (25)u=1 0 k=1

kfu

where Fbu,u (.) and fbu,u (.) are the CDF and the PDF ofthe normalized SNR ~bu,U , of each user and can be obtained

bu,ufrom F'Y (.) and f'Y (.), given in (6)-(7). Then, (25) can

bu,u bu,ube written as

F"Ib""", ('r) = ~ t [Fbk,k ("1 'Y )] Ku=1 bu,u

= ~t [1-exp (-2)~ i (2 )l]~~6)K u=1 "t« l=O l. ~u

where we set ;Yb = ;Yu:u,u

Then, the PDF, f'Y (~) can be obtained from (24) andbrn,rn

following similar steps as for the homogeneous case, asfollows:

L K(1)Z-1(Z_1) .f"lb""", ('r) = (N -I)!~ 'Yu ~ i (-1)~

((i + 1)~) i(N-1) ( ~ )t+N-1

x exp _ L ai,t =- . (27)"t« t=O "t«

1) Outage Probability: In this case, by substituting (26) into(17) we obtain the outage probability expression as follows

1 K [ ( AO) N-1 1 (Ao)l]KLPout = - L 1 - exp - =- L,. =- .(28)

K u=1 "t« l=O l. "t«

2) Average Channel Capacity: Similarly to the homoge­neous case, the average channel capacity in this case can beobtained by substituting (27) into (19), which yields

K Z-1 i(N-1)C = Llog2 (e)" " (Z ~ 1) (-1) i " (t + N- 1)ai,t

(N - 1)!L...--~ i L...-- (i + 1)t+Nu=1 't=O t=O

{ ( ·+1) (·+1)x Pt+N - t 'Yu E1 t 'Yu

t+N-1 1 (i+1) ( i+1)}+ " -p - Pt N- --L...-- q - +q -q=1 q ~u ~u

where Pq ( .) and E1 (.) are defined in the lines after (20).

3) Access Probability in the Heterogeneous Case: Similarlyto the homogeneous case, the access probability in normalizedSNR-based scheduling can be written as

t; = Pr (2b" 'U ::::: all other 2bj

'] )~bu,u ~bj,J

= 100

_ 1 [s;,« (_ ~ ) IT Fbj,j (_ ~ ) d~. (30)o ~bu,u ~bu,u j=1 ~bu,u

jfu

Substituting Fbu,u (.) and [s;,« (.) into (30) yields r; = -k,implying that all the users have the same probability ofaccess. This states that normalized-SNR based scheduling isa "completely fair" scheduling scheme.

IV. SIMULATION AND NUMERICAL RESULTS

In this section, we provide results obtained from the MonteCarlo simulation and from the mathematical expressions forthe multiuser (L,N) MIMO system employing L transmitantennas at the base station and N receive antennas at eachuser side (mobile station), with user scheduling and antennaselection over MIMO Rayleigh fading channels. We considerboth i.i.d. and i.ni.d. Rayleigh fading distributed channels. Theresults are obtained from the analytical formulas expressedin this paper. Simulation results are also provided to verifythe analysis. The setup in figures 2-6 is as follows. At everytime slot, we consider i.i.d. or i.ni.d. variables for the short­term average SNR values given by ;Yi u» for i E {1, ... , L },

62 '

For solving the above integral we use the following result [12]

'4

(32)

\.12

\ .

Li.d .- - - t.nr.o.

o Sim ulation

' 0

\

..

ApPENDIX

V. CONCLUSION

. . . \'~ \ \

I ... • ; .

p, roo In (1 + x ) (p, X).,-1 e: ":" dx(A-l)!Jo

).,-1 1= P).,( -p,) E1(p,) + L - Pq(p,) P).,-q( -p,)

q=1 q

. . . "•. ~. . . . ,, (= \

'0·' " . \ .a :') \ .

Z (1) K-1 (Z) i(N - 1)C = (N _ I)! ~ L ~ 1 (_I)i L a i ,t

t=O t=O

roo ( ) t + N -1 (C + 1) )x Jo log2 (1 + x ) ~ exp - t 'Y x dx . (31)

Fig. 2. Outage Probability vs. "y for the i.i.d, and i.ni.d, cases.

Substituting into (19) the expressions for f 'YTn

('Y) given by(14) , we obtain the corresponding average capacity expressionas follows

In this paper, we have presented a performance analysis ofthe user scheduling schemes for multiuser MIMO MRC sys­tems exploiting the multiuser and antenna diversities. AbsoluteSNR-based scheduling and normalized SNR-based schedulingschemes have been presented for homogeneous and hetero­geneous cases , respectively. An antenna selection scheme hasbeen used to overcome the drawbacks of channel hardening inmultiuser MIMO systems, in addition to improving the systemperformance. It was shown that the normalized SNR-basedscheduling guarantees the fairness among the users at the costslight performance loss .

discussed in the paper due to the user unfairness. On the otherhand, in the normalized-SNR based scheduling in the i.ni.d.case, all users have equal access probability (1/ K) , showingthat the normalized-SNR based scheduler can provide userfairness .

and i E {I, . . . , K}. When the SNR values, 'Yi,u are i.i.dwith respect to both i and u, we set 'Yi ,u = 'Y = Es /No,(long-term average SNR). And when the SNR values, 'Yi,u

are i.i.d. with respect to i and are i.ni.d. with respect to u,we assume 'Yb" ,u = 'Yu' In the i.ni.d , users, an exponentiallypower decaying profile (PDP) given by 'Yu = 'Ye-8(u-1),

for U E {I, . . . , K} is considered in which is is the powerdecaying factor. Setting simply is = 0, the user links act as i.i.d.links . We then normalize them so that their sum is equal toK x;:y. With these assumptions we can make a fair comparisonbetween i.i.d. and i.ni.d. cases. For the following figures ,the homogeneous (i.i.d.) and the heterogeneous (i.ni.d.) cases,respectively correspond to absolute-SNR based scheduling andnormalized-SNR based scheduling.

Fig.s 2-4 show the outage probability, respectively versusaverage long-term SNR, number of users, and number oftransmit antennas. We assumed N = 2, Co = 3 b/sIHz and thehomogeneous (i.i .d.) case as well as a heterogeneous (i.ni.d.)case with is = 0.05. We observe that antenna selection and userselection improve the outage probability. In these Figures, weobserve that as expected, increasing either number of usersor number of available transmit antennas reduces the outageprobability. It can be observed that the analytical results matchclosely those obtained by simulations. However, when theSNR of the users are i.ni.d ., the outage probability increases;this is the cost for using normalized SNR-based scheduling toguaranty the user fairness. As shown, using antenna selectioncan reduce the outage probability and can compensate theperformance loss. We conclude that a normalized SNR-basedscheduler can guaranty the user fairness whereas the utilizationof antenna selection can compensate the performance loss .

Fig. 5 shows the average channel capacity of system understudy in terms of bits/sIHz, versus the number of users, forthe average SNR of'Y = 10 dB, for different user schedulingschemes and with transmit antenna selection (TAS) with L = 4or without transmit antenna selection (no-TAS, i.e, L = 1).It can be observed that, in all cases , increasing the numberof users increases the system capacity. As can be seen, non­identically distributed 'Yi u (t) values cause a loss in channelcapacity for normalized-SNR based scheduling scheme. Thisis expected since a normalized-SNR based scheduling schemeprovides the complete fairness among the users but at theexpense of a slight loss in the system capacity, while thisslight performance loss can be compensated by the utilizationof an antenna selection.

From Fig.s 2-5, we can see that the results obtained fromthe closed-form expressions match closely those obtained bysimulations. This verifies the accuracy of the analysis.

Finally, Fig. 6 shows the access probability of the usersfor both schedulers as obtained from simulation. As expected,in the i.i.d. case , in the long term, the users have the sameaccess probability that is equal to 1/K . We can see that inthe i.ni.d. case with the absolute-SNR based scheduling, theaccess probability of the users are not the same and userswith a higher SNR have more chance to access the channel.Therefore, users suffering from bad channel conditions may where P).,( .) is the Poisson distribution defined by P).,(x) =never transmit their information, yielding an unfair resource I:~:6 :~ e- x , and where E1 (.) is the exponential integral ofallocation among the users . This scheme for i.ni.d. case is not first order, defined by E1 (x) = .t oC 1e- tdt for x > O.

63 x

2219

.... _ .. tc- +-

' 6

B.....=':tr-'1i'"::"lF ..... ..:" - k· ... -» .....

10 13Num ber of Users, K

5.6~ .../f _ '-' ~-:a-:;c;, .. : ~ ~7 ~

6.2~--~--~--~--~-----.---~----,

¥ 5 .4 . rf ·· : :. •~

~

10984 567Number of Users, K

32

f.... - - - i.ni.d.i.i.d.r- -.... 0 Simulation

"~

"\.

1 '"' -.....::a.i-'" ).. -,~ ~

----'---2

........

" ...... .1' .- __3

,~ \."}... ~- --

"•

...... ..; , L ......

4r "~ '\J' l <,

"'". <>I'M =

"N = 2 = .oe "..

,'\. -, .•5

Fig. 3. Outage Probability vs. K for 'Y= 4 dB for the i.i.d. and i.ni.d. cases. .. 0 dls f he i . d d i dFig. 5. Average Capacity vs. K ror 'Y = 1 ror t e 1.1.. an i.m..

105 6User Ind ex

.35 I-E> i.nLd ., Absolute SN R Sc hed ulingK = 10 . : , .:t : . ~ i.d . , Norm alized- SNR SchedulingL =·4 . N = .. l.l.d .

O .~ ;; ~ O. 1 ,. . . . . .. . . .

.2 5 : .. . . . . . .

0 .2

. ' 5

.,,.. .I .. I .. ·.0 5

0

. :.

:984 5 6 7

Number of Transmit Antennas, L

i.i.d.- - - i.ni.d.o Simulation

2 3

Fig. 4. Outage Probability vs. L for 'Y= 4 dB for the i.i.d. and i.ni.d. cases. Fig. 6. Access Probability of users for 'Y= 10 dB, for i.i.d and i.ni.d. users.

Therefore, using (32) and after performing changes of vari­ables together with some simplifications, we obtain the solu­tion of integral in (31) as

roo ()t+N-' (('+1))} 0 logz (1+ x ) ~ exp - z 'Y x dx

= 'Y (t +N - 1)! logz (e) {Pt+N(_ i~l) E, (i~l)(i+1)t+ N "y "y

t+N-'l ('+1) ( '+1)}+ {; qPq Z 'Y Pt+N-q - Z 'Y . (33)

Finally, replacing (33) in (31) yields expression (20), whichconcludes the proof.

REFERENCES

[I] P. Viswanath, D. Tse, and R. Laroia, "Opportunistic beamforruing usingdumb antennas," IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1277­1294, June 2002.

[2] N. Sharma and L. H. Ozarow, "A study of opportunism for multiple­antenna systems," IEEE Trans. Inform. Theory, vol. 51, no. 5, pp. 1804­1814, May 2005.

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