BER Performance Analysis of Multiuser Diversity with Antenna Selection in MRC MIMO Systems

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1 Performance Analyss of ultuser Dversty wth Antenna Selecton n RC IO Systems ohammad Torab 1, Davd Haccoun 1 and Wessam Ajb 2 1 Department of Electrcal Engneerng, École Polytechnque de ontréal, ontréal, QC, Canada. 2 Department of Computer Scence, Unversté duquébec à ontréal, Canada. Abstract In ths paper, we present a performance analyss for the user schedulng for the multuser RC IO systems explotng the user and antenna dverstes. We consder two schedulng schemes ncludng absolute SNR-based schedulng and normalzed SNR-based schedulng schemes. We propose the utlzaton of an antenna selecton scheme to overcome the drawback of channel hardenng n multuser IO systems and to mprove the system performance. We derve new closedform expressons for the average bt error rate of the presented schemes for two scenaros: Heterogeneous ndependent nondentcally dstrbuted SNRs-.n.d. and Homogeneous ndependent dentcally dstrbuted SNRs-..d. cases. Usng mathematcal analyss and numercal smulatons, we compare the presented schemes. I. INTRODUCTION In multuser wreless communcatons, schedulng technques have been proposed to explot the multuser dversty and hence mprove the performance of the system. Consderng the downlnk, from the base staton to the moble statons users at each tme slot, the user havng the best channel condton s scheduled to access the channel. Whle a proper schedulng may ncrease the system throughput, the farness n resource allocaton among the users s a key parameter and should be taken nto consderaton. The round-robn scheduler s a smple and full-far schedulng scheme, where all users have the same prorty for accessng the channel, but t does not explot the multuser dversty. On the other hand, the bestuser scheduler, the so called opportunstc or greedy scheduler, selects the user wth the hghest throughput. It acheves a hgher throughput than round-robn scheduler does, but ths gan s obtaned at the prce of unfar resource allocaton among the users, as for example for the users that are closer to the base staton, and hence havng more chance to access the channel. The proportonal far scheduler PFS has been proposed for explotng multuser dversty whle mantanng the farness among the users. Instead of selectng the best user wth hghest absolute throughput or absolute sgnal-to-nose-rato SNR, PFS chooses the user wth the hghest normalzed-throughput or normalzed-snr, normalzed each to ts own average [1], [2]. In multple-nput multple-output IO multuser systems, a scheduler can be used to explot the antenna and multuser dverstes at the same tme and to ncrease the system performance [1] [5]. An overvew of user selecton and schedulng algorthms n IO systems has been presented n [6]. It has been observed that due to the channel hardenng n the IO systems, the full advantages of both multuser and antenna dverstes cannot be exploted [6], [7]. In order to reduce the channel hardenng, antenna selecton can be employed [8], [9]. It has been shown that a sgnfcant gan from both multuser and antenna dverstes can be obtaned at the same tme for the..d. user lnks. However, to the best of our knowledge, a complete performance analyss for user schedulng schemes n maxmum rato combnng RC IO systems wth antenna selecton especally for nondentcally dstrbuted.n.d. users s not avalable n the lterature. Furthermore, bt error rate analyss for these systems s of nterest but has not been reported prevously. In ths paper, we present a performance analyss for the average for multuser IO systems employng RC technque at the recever. In order to reduce the effects of IO channel hardenng, we propose utlzng transmt antenna selecton TAS at the base staton. Ths can both mprove the system performance and reduce the hardware complexty. We consder two scenaros: Heterogeneous ndependent non-dentcally dstrbuted SNRs-.n.d. and Homogeneous ndependent dentcally dstrbuted SNRs-..d. cases. We derve the expressons for the probablty densty functon PDF, the cumulatve dstrbuton functon CDF, and the moment generatng functon GF of the sgnal-tonose-rato SNR n a IO channel wth antenna selecton for the consdered multuser schedulng schemes. Ths enable us to establsh a mathematcal analyss and formulaton for the performance of the system under study. The paper s organzed as follows. Secton II presents the system model. The dervatons of PDF, CDF, and GF of the user s channel SNRs for the multuser dversty system are also presented. In Secton III, two multuser schedulng schemes n conjuncton wth antenna selecton are presented. In each case, the analyss of the average for dfferent modulaton schemes s presented. Numercal results are gven an dscussed n Secton IV, and fnally, Secton V concludes the paper. II. SYSTE ODE Fg. 1 shows a block dagram of the multuser IO system model consdered n ths paper for downlnk data transmsson for a cellular system. A base staton employng transmt antennas s servng K users moble statons where each users has N receve antennas. The base staton allocates the resources to users and schedules the data for downlnk transmssons to the users. In conventonal users schedulng system, t s assumed that the base staton uses a feedback channel from the users n order to acqure channel state nformaton CSI knowledge or SNR from each of the users [6]. To do ths, for example usng plot sgnals, the recever

2 at each moble staton user estmates the SNR receved from each transmt antenna of the base staton, and then through a robust feedback channel sends that SNR to the base staton. We assume a perfect channel estmaton and feedback channel to be error free. However, from a practcal pont of vew, the effects of channel estmaton error and feedback delay on the performance s not addressed here, but may be the subject for future work. Here, we assume that the channel between the base staton and user u s a quas-statc Raylegh fadng IO channel and expressed by H u matrx of sze N, wth elements h j, u correspondng to the complex channel gan of the channel response between the -th transmt and j-th receve antennas for the u-th user. The receved sgnal expresson for each user at the output of RC can be wrtten as r u = h,u 2 F s u + η u, u =1, 2,,K 1 j=1 u where r u s the output sgnal of RC, s u s the transmtted symbol, h,u s the -th column of the IO channel matrx H u, and. 2 F s the squared Frobenus norm gven by h,u 2 F = N h j, 2, and where η u s a complex Gaussan nose wth dstrbuton N, N 2 h,u 2 F per dmenson. The nstantaneous SNR per symbol at the recever of the user u may be wrtten as,u =,u h,u 2 F 2 where,u s the average receve SNR per antenna for user u. Snce we assumed that the channels fadng,.e., h j, u are Raylegh flat-fadng, therefore, h j, u 2 for each user wll be a Ch-squared dstrbuted random varable. For a gven u and, snce h,u 2 F s the sum of N..d. random varables whch they may be.n.d. wth respect to u, then h,u 2 F wll be a Ch-squared dstrbuted wth 2N degrees of freedom. Thus, usng a change n varables, we can show that the PDF and CDF of the receved SNR for each user; gven n 2, can be expressed as N 1 f,u = N 1!,u N exp 3,u F,u =1 exp N 1 l 1. 4,u l!,u l= III. UTIUSER SCHEDUING AND ANTENNA SEECTION In ths secton, we explan an antenna selecton scheme and two user schedulng schemes for the system under study. A. Antenna Selecton As stated earler, at the recever sde for each user, the SNR receved from each transmt antenna of the base staton s obtaned usng an estmaton technque. At the recevers sde, based on those SNR values, each user s recever selects ts best transmt antenna b u -th antenna, havng the hghest channel SNR,.e. b u = arg max A {,ut}, u =1,,K, 5 where A = {1, 2,...,},,u t s the nstantaneous SNR, for user u n tme-slot t for the -th transmt antenna. In other Fg. 1. User schedulng n a RC IO system wth antenna selecton. words, consderng a K matrx, Γ = {,u t} wth the elements,u t =1,,, and u =1,,K, n the u-th row of that matrx, the largest element, denoted by bu,u s selected out of elements. Then, for a total of K rows, a vector of those selected SNRs can be consdered, gven by T t b1,1 b, 2 t bk, K t 2, where. T denotes the transpose operaton. To ease the notaton, the tme-slot ndex t wll be gnored. In ths scheme, the recever needs only to feedback the ndex of the selected transmt antenna the ndex b u and the SNR of the users gven by the above vector to the transmtter, nstead of the full CSI of all transmt antennas gven by the K matrx, Γ. Ths scheme s attractve, snce t sgnfcantly reduces the amount of the requred feedback to the transmtter. Usng the order statstcs and assumng that the SNR values correspondng to the transmt antennas are..d. wth respect to, =1,,, we can express the CDF and PDF of the SNR for the best transmt antenna for user u as F bu,u =[F,u ] 6 and f bu,u =f,u [F,u ] 1. 7 where the f,u and F,u are defned n 3 and 4. B. User Schedulng We assume that each user feedbacks ts correspondng SNR; bu,u and the ndex b u {1,,} to the base staton through a relable feedback channel. The goal of scheduler s to select the best user m-th user among all the actve users,.e., to select the hghest SNR among K followng SNRs b1,1 b2, 2 bk, K. We frst consder the homogeneous case where the channel SNR values for the users, bu,u values are..d. wth respect to the user ndex u. Then, we consder the heterogeneous case where the short-term average SNR values of users are.n.d. wth respect to ndex u. We present the schedulng schemes ncludng absolute SNR-based schedulng and normalzed SNR-based schedulng. Homogeneous Case In the homogeneous case, the statstcs of all users are the same, and all the users SNRs are..d. Hence, the absolute- SNR based scheduler selects the best user m-th user wth the maxmum absolute-snr,.e.

3 m = arg max u U { b u,ut}, 8 where U = {1, 2,...,K}, and bu,ut s the nstantaneous SNR of user u wth the best transmt antenna of base staton for that user, b u n the tme-slot t. Then, usng the theory of order statstcs the CDF and PDF of the best user wth SNR bm,m selected from K avalable users can be obtaned from and F bm,m =[F bu,u ] K 9 f bm,m =Kf bu,u [F bu,u ] K 1 1 where f bu,u and F bu,u are defned n 6-7. Substtutng 6 and 7 nto 9 and 1 yelds and F bm,m =[F,u ] K 11 f bm,m =Kf,u [F,u ] K By substtutng 4 nto 11 and usng the bnomal expanson of 1 x Z defned by 1 x Z = Z Z 1 x, and assumng Z = K and,u =, after some manpulatons we obtan [ N 1 ] Z l 1 F bm,m = 1 exp,u l! l=,u Z N 1 Z t = 1 exp a,t 13 where a,t denotes the coeffcent of [ N 1 1 l= l! 1 mnt, N 1 t n=1 t n the expanson l ], defned by: a, =1, a,1 =, a,t = n+1 t n! a,t n, for 2 t<n 1, and a,t =[N 1!], for t = N 1 [5]. Smlarly, by substtutng 3 and 4 nto 12, after some manpulatons, we obtan Z 1 Z 1 f bm,m = 1 N 1! exp +1 N 1 Usng the defnton of the GF gven by bm,m s = a,t t+n e s f bm,m d, 15 we substtute 14 nto 15 and obtan after some manpulatons Z N 1 Z 1 bm,m s = 1 N 1! t + N 1! a,t. 16 t+n +1+s Knowng bm,m s gven by 16, respectvely, we can derve closed-form expressons for the average, as shown below. 1 Average Bt Error Rate: The average bt error rate can be obtaned by ntegratng the nstantaneous bt error probablty at the destnaton termnal denoted bm,m, over the PDF of bm,m and can be wrtten as = x f bm,m x dx. 17 Usng the GF-based analyss approach, the average error rate performance of dgtal communcaton systems over fadng channels can be determned [1]. We derve the average bt error rate of bnary sgnals and mportant -ary modulatons schemes such as -PSK and -QA. Bnary Sgnals: The average, denoted for bnary sgnals s gven by [1] = 1 π/2 g bm,m π sn 2 d 18 where g =1for BPSK and g =.5 for orthogonal BFSK, and the GF bm,m. s gven n 16. Substtutng 16 nto 18, after some manpulatons the average can be expressed as = N 1 Z N 1! t + N 1 a,t +1 t+n Z 1 1 π 1 π/2 sn 2 n sn 2 d 19 + c where c = g +1 and n = t + N. The closed-form expresson for the soluton of above ntegral s gven by [1]eq.5.17 I n c = 1 π/2 sn 2 n π sn 2 d + c [ = 1 n 1 c ] q 2q c q 41+c q= Therefore, expresson 19 can be wrtten as Z N 1 Z 1 = 1 t + N 1 a,t 2N 1! +1 t+n [ t+n 1 g ] q 2q g q 4 +1+g q= -PSK Sgnals: The average for -PSK sgnals can be wrtten as [1] PSK = 1 1π/ gpsk bm,m πβ sn 2 d 22 where g PSK =sn 2 π/ and β s the numbers of bts/symbol β = log 2. Substtutng 16 nto 22 and followng the same steps as n 19-21, after some manpulatons, the average can be wrtten as PSK = Z βn 1! t + N 1! a,t +1 t+n Z 1 1π 1 N 1, g PSK +1 23

4 where assumng n = t + N, the closed-form expresson for K n.,. s gven by [1]eq.5A.24 K n φ, c = 1 φ sn 2 n π sn 2 d, π φ π + c { = φ π B π n 1 π 2 + 2q 1 tan 1 A q c q +sn tan 1 A n 1 where B = 2q q q q=1 p=1 c [ 2q p q p QA = 4 πβ q= T pq [ 1 + c q cos tan 1 A ] } 2q p c sgnφ, A = B cot φ, and T pq = 1 4 p [2q p+1]]. -QA Sgnals: The average for -QA sgnals can be wrtten as [1] 1 1 π/2 gqa bm,m d 4 πβ π/4 sn 2 gqa bm,m sn 2 d 25 where g QA =1.5/ 1. Substtutng 16 nto 25 and followng smlar steps as for -PSK sgnals, the average can be wrtten as QA = 4Z βn 1! t + N 1! a,t +1 t+n N 1 Z 1 1 { 1 1 π 2, g QA π 4, g QA +1 where K n.,. s defned n 24. } 26 2 Access Probablty n the Homogeneous Case: The access probablty can be defned as the probablty that user u s scheduled to access the channel. In the homogeneous case employng absolute-snr based scheduler, t can be expressed by [11] P u =Pr bu,u all other bj,j. 27 Thus usng 1, expresson 27 can be wrtten as [ K 1 P u = f bu,u F bu,u ] d = 1 K f bm,m d = 1 K. 28 As expected, n the homogeneous case, the access probablty of all users are the same,.e. P u =1/K. Heterogeneous Case In the heterogeneous case, as a more practcal scenaro, the statstcs of users are non-dentcally dstrbuted.e. the shortterm average SNR values of the users are.n.d. Absolute- SNR based schedulng and normalzed SNR-based schedulng are agan two possble schedulng schemes. The absolute-snr based scheduler always selects the user wth the hghest SNR. Snce ths scheduler selects for each tme-slot the user havng the best qualty channel,.e. hghest SNR, users sufferng from bad channel condtons may never have access to the channel, yeldng an unfar resource allocaton. To overcome ths stuaton, we now consder a normalzed SNR-based schedulng scheme that s far to all users. Consderng that users are.n.d., a crteron for the user schedulng wth proportonal far schedulng, s defned as the rato of the nstantaneous SNR of each user over ts own average SNR, [2], [11],.e., m = arg max u U { bu,ut bu,ut }, 29 where bu,ut and bu,ut are the nstantaneous and the short-term average SNR values, respectvely for the user u and best transmt antenna of base staton for user u, b u n the tme-slot t. When the average SNR bu,u, of all users are the same,.e. all users are..d. ths scheduler wll be equvalent to the absolute SNR-based scheduler stated earler. In ths schedulng scheme, the base staton selects the users wth the largest normalzed SNR value. Smlar to the method n [11], the PDF and CDF of the best user wth SNR bm,m selected from K avalable users can be respectvely obtaned from K K 1 f bm,m = 3 and u=1 F bm,m = bu,u f bu,u K u=1 bu,u bu,u f bu,u x k=1 k u K k=1 k u F bk,k bu,u F bk x dx 31,k where F bu,u. and f bu,u. are the CDF and the PDF of normalzed SNR bu,u, of each user and can be obtaned bu,u from F bu,u. and f bu,u., gven n 6-7. Then, the PDF, f bm,m can be obtaned from 3 followng smlar steps as for the homogeneous case, as follows: f bm,m = N 1! exp K 1 u=1 +1 u u N 1 Z 1 a,t u 1 t+n Then, after substtutng 32 nto 15 and solvng the ntegral, the GF expresson can be wrtten as K Z 1 bm,m s = 1 N 1! u=1 N 1 t + N 1! a,t. 33 t+n +1+ u s

5 1 Average Bt Error Rate: Smlarly to the homogeneous case, n the followng, we derve the average bt error rate of some modulaton schemes for the heterogeneous case. Bnary Sgnals: The average for bnary sgnals can be obtaned by substtutng the GF expresson gven by 33 nto 18 as follows K N 1 Z 1 = 1 t + N 1 a,t 2N 1! u=1 +1 t+n [ t+n 1 1 u g ] q 2q u g q u g q= -PSK Sgnals: The average for -PSK modulaton s obtaned by substtutng the GF expresson gven by 33 nto 22 as follows: PSK = N 1 βn 1! K u=1 t + N 1! a,t +1 t+n where K n.,. s gven by 24. Z 1 1 1π, u g PSK QA Sgnals: The average for -QA sgnals n ths case s obtaned by substtutng the GF expresson gven by 33 nto 25 as follows 4 QA = βn 1! t + N 1! a,t +1 t+n K u=1 N 1 Z 1 1 { 1 1 π 2, u g QA π 4, u g QA +1 } 36 where the closed-form expresson for K n.,. s gven n Access Probablty n the Heterogeneous Case: Smlarly to the homogeneous case, the access probablty n normalzed SNR-based schedulng can be wrtten as bu,u P u =Pr all other b j,j bu,u bj,j 1 = f bu,u bu,u bu,u K j=1 j u F bj,j bu,u d 37 whch yelds P u = 1 K, mplyng that all the users have the same probablty of access. Ths states that normalzed-snr based schedulng s a completely far schedulng scheme. IV. SIUATION AND NUERICA RESUTS In ths secton, we provde results obtaned from the onte Carlo smulaton and from the mathematcal expressons for the multuser,n IO system employng transmt antennas at the base staton and N receve antennas at each user sde moble staton, wth user schedulng and antenna selecton over IO Raylegh fadng channels. We consder both..d. and.n.d. Raylegh fadng dstrbuted channels. The results are obtaned from the analytcal formulas expressed n ths paper. Smulaton results are also provded to verfy the analyss. The setup n fgures 2-5 s as follows. At every tme slot, we consder..d. or.n.d. varables for the short-term average SNR values gven by,u, for {1,...,}, and {1,...,K}. When the SNR values,,u are..d wth respect to both and u, weset,u = = E s /N, long-term average SNR. And when the SNR values,,u are..d. wth respect to and are.n.d. wth respect to u, we assume bu,u = u. In the.n.d. users, an exponentally power decayng profle PDP gven by u = e δu 1, for u {1,...,K} s consdered n whch δ s the power decayng factor. Settng smply δ =, the user lnks act as..d. lnks. We then normalze them so that ther sum s equal to K. Wth these assumptons we can make a far comparson between..d. and.n.d. cases. For the followng fgures, the homogeneous..d. and the heterogeneous.n.d. cases, respectvely correspond to absolute-snr based schedulng and normalzed-snr based schedulng. Fg.s 2-4 show the average s; versus average long-term SNR, for BPSK, 16-QA, and 8-PSK modulaton schemes, respectvely when the users /antennas SNRs are..d. Dfferent values for the number of actve users are chosen K =1, 2, 4, and 1. As expected the average reduces when K or ncreases. We can also observe that the dversty order of K N s mantaned. For example, for K =4, =1, the result s the same as for K =2, =2, where N =2has been assumed for both cases. Snce the users are assumed to be..d. the results are dentcal for both absolute SNR-based and normalzed SNR-based schedulng schemes. In Fg. 5 the average versus average long-term SNR, for 16-QA and 64-QA modulaton schemes are presented. Several possble cases ncludng..d. and/or.n.d. SNRs are consdered. We observe that n the heterogeneous.n.d. case, the normalzed SNR-based schedulng provdes a worse average than n the homogeneous..d. case, whch s the cost for guarantyng the user farness. We can also observe that the average ncreases as δ, the power decayng factor, ncreases. Ths s expected snce as δ ncreases, the user dversty gan decreases. However, usng antenna selecton can mprove the average and can compensate the performance loss, whch s clearly shown n Fg. 5. For example, n both 16-QA and 64-QA schemes, for the.n.d. case wth δ =.1, when the number of transmt antennas s =1, the performance loss has been compensated by usng antenna selecton,.e., selectng the best antenna out of =2 antennas, whereas the performance n ths case s almost the same as for the..d. case wth =1. Obvously, for a larger value of δ e.g. δ =.135, a larger value of, e.g. =3 s requred to compensate the performance loss. From Fg.s 2-5, we can see that the results obtaned from the closed-form expressons match closely wth those obtaned by the smulaton. Ths verfes the accuracy of the analyss.

6 User Selecton, = 1 User and Antenna Selecton, = 2 Smulaton User Selecton, = 1 User and Antenna Selecton, = 2 Smulaton K, = 1, K, = 1, ,2, 2, ,1 4,1, 2,2 1,2, 2,1 IO RC, N=2, BPSK 1, Average SNR db ,1 1,2 IO RC, N = 2, 8 PSK 4,2 4,1, 2, Average SNR db Fg. 2. Average vs. for BPSK,..d. case, K, and N =2. Fg. 4. Average vs. for 8-PSK, K, and N =2for the..d. case User Selecton, = 1 User and Antenna Selecton, = 2 Smulaton IO RC, TAS N = 2, K =1 1 2 K, = 1, ,1 1,2 IO RC, N=2, 16 QA 4,2 1,2, 2,1 4,1, 2, Average SNR db Fg. 3. Average vs. for 16-QA,..d. case, K, and N =2. Fg QA.n.d., δ=.135, = 1.n.d., δ=.1, = 1..d., δ=., = 1.n.d., δ=.135, = 3.n.d., δ=.1, = 2..d., δ=., = 2 Smulaton 64 QA Average SNR db Average vs. for 16-QA and 64-QA,..d. and.n.d. cases. V. CONCUSION Ths paper presents a performance analyss of the user schedulng schemes for multuser RC IO systems explotng the multuser and antenna dverstes. Absolute SNR-based schedulng and normalzed SNR-based schedulng schemes have been presented. An antenna selecton scheme has been used to overcome the drawbacks of channel hardenng n multuser IO systems, n addton to mprovng the system performance. It was shown that the normalzed SNR-based schedulng guarantees the farness among the users whle the slght performance loss due to the far schedulng can be compensated wth an antenna selecton scheme. REFERENCES [1] P. Vswanath, D. Tse, and R. aroa, Opportunstc beamformng usng dumb antennas, IEEE Trans. Inform. Theory, vol. 48, no. 6, pp , June 22. [2] N. Sharma and. H. Ozarow, A study of opportunsm for multpleantenna systems, IEEE Trans. Inform. Theory, vol. 51, no. 5, pp , ay 25. [3] J. Jang, R.. Buehrer, and W. H. Tranter, Antenna dversty n multuser data networks, IEEE Trans. Commun., vol. 52, no. 3, pp , ar. 24. [4] S. Serbetl and A. Yener, Tme-slotted multuser IO systems: beamformng and schedulng strateges, EURASIP Journal on Wreless Communcatons and Networkng, vol. 2, pp , 24. [5] C. J. Chen and. C. Wang, A unfed capacty analyss for wreless systems wth jont multuser schedulng and antenna dversty n Nakagam fadng channels, IEEE Trans. Commun., vol. 54, no. 3, pp , arch 26. [6] W. Ajb and D. Haccoun, An overvew of schedulng algorthms n IO-based fourth-generaton wreless systems, IEEE Network, vol. 19, no. 5, pp , September/October 25. [7] B.. Hochwald, T.. arzetta, and V. Tarokh, ult-antenna channel hardenng and ts mplcaton for rate feedback and schedulng, IEEE Trans. Inform. Theory, vol. 5, no. 9, pp , Sept. 24. [8] D. Ba, P. tran, S. Ghassemzadeh, R. R. ller, and V. Tarokh, Channel hardenng and the schedulng gan of antenna selecton dversty schemes, n Proc. IEEE Int. Symp. on Inf. Theory, ISIT 27, 27, pp [9] X. Zhang, F. Chen, and W. Wang, Outage probablty study of multuser dversty n IO transmt antenna selecton systems, IEEE Sgnal Proc. etters, vol. 14, no. 3, pp , 27. [1]. K. Smon and. S. Aloun, Dgtal Communcatons over Fadng Channels: A Unfed Approach to Performance Analyss. Wley, 2. [11]. Yang and.-s. Aloun, Performance analyss of multuser selecton dversty, IEEE Trans. Veh. Technol., vol. 55, no. 3, pp , ay 26.