ON PERFORMANCE BOUNDS FOR MIMO OFDM BASED WIRELESS COMMUNICATION SYSTEMS. Stefan Schwarz, Michal Šimko and Markus Rupp

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1 ON PERFORMANCE BOUNDS FOR MIMO OFDM BASED WIRELESS COMMUNICATION SYSTEMS Sefan Schwarz, Michal Šimko and Markus Rupp Insiue of Telecommunicaions, Vienna Universiy of Technology Gusshaussrasse 5/389, A-1 Vienna, Ausria {sschwarz, msimko, ABSTRACT In his paper, we develop several increasingly igh and resricive performance bounds for OFDM based MIMO wireless communicaion sysems. Saring wih channel capaciy as he ulimae upper bound on he hroughpu, sep by sep we develop igher bounds on he pracically achievable hroughpu by incorporaing ypical design consrains. The presened bounds are applied o a Long Term Evoluion sysem, allowing us o idenify dominan sources of performance losses. The invesigaion reveals flaws of wireless communicaion sysems hereby poining a direcions o efficienly improve performance in sysem design. Index Terms Performance bounds, LTE, OFDMA, MIMO 1. INTRODUCTION Since many years he channel capaciy of muli-anenna Gaussian channels is well known [1], bu sill modern wireless communicaion sysems are far from achieving hroughpus ha come close o capaciy. Alhough modern urbo and LDPC codes are performing close o ideal Shannon codes [], here are several oher sources of performance loss ha all ogeher accumulae o a large fracion of he heoreically possible hroughpu. In his work, we sysemaically idenify hese sources and accoun for heir corresponding hroughpu losses. The invesigaion shows ha a large par of he performance degradaion is caused by he sysem overhead required for iming and frequency synchronizaion as well as channel esimaion.there are also oher more fundamenal losses, like he finie code block lengh or uilizaion of Bi Inerleaved Coded Modulaion BICM) [3] insead of Gaussian signaling, which can be quanified heoreically. We do no consider losses due o sub-opimaliies in e.g. he channel esimaion process. We compare our bounds o pracically achievable hroughpus in a Long Term Evoluion LTE) sysem, uilizing he sandard complian Vienna LTE Link Level Simulaor [], [5]. The gap ha is finally lef beween he ighes upper bound on hroughpu and he real achievable performance can be accouned o he channel code iself. This work has been funded by A1 Telekom Ausria AG and he Insiue of Telecommunicaions, Vienna Universiy of Technology. Fig. 1. Sysem archiecure of a modern wireless communicaion sysem. We focus on Muliple Inpu Muliple Oupu MIMO) Orhogonal Frequency Division Muliplexing OFDM) communicaion sysems, because pracically all modern sandardized sysems are based on his archiecure e.g. 3GPP LTE, IEEE Worldwide Ineroperabiliy for Microwave Access WiMAX)). Furhermore, hese sysems employ BICM which separaes he coding and modulaion mapping ino wo independen eniies, hereby slighly sacrificing opimaliy bu gaining in erms of decreased complexiy. Finally, we assume closed loop linearly precoded MIMO sysems ha uilize feedback from he receiver o choose he appropriae precoder from a finie code book. By aking ino accoun a variey of pracical design consrains, we develop increasingly igh bounds for he hroughpu of a single user MIMO OFDM sysem in Secion. Aferwards, in Secion 3, we apply he proposed bounds o an LTysem and evaluae he hroughpu loss caused by he individual consrains. We conclude in Secion.. BOUNDS ON THE ACHIEVABLE THROUGHPUT The amoun of informaion ha any communicaion sysem can ransmi reliably over a given channel is upper bounded by he well known channel capaciy. We consider a MIMO OFDM sysem wih N ransmi and N r receive anennas, which splis he oal sysem bandwidh ino K orhogonal subcarriers. In such a sysem channel capaciy can be achieved by employing Singular Value Decomposiion SVD)-based precoders and receive filers on each subcarrier a he ransmier and receiver side respecively. Furhermore,

2 he oal ransmi power mus be opimally disribued over he resuling parallel SISO channels, which can be achieved uilizing he well known waer filling power allocaion algorihm [], []. We will now consider a pracical wireless communicaion sysem wih an archiecure as shown in Figure 1. Such a sysem ypically firs codes he user daa AMC scheme selecion in he figure), nex he daa is mapped ono he spaial ransmission layers and appropriaely precoded MIMO preprocessing ) and aferwards he ransmi signal is generaed Transmi signal generaion ) by insering he sysem overhead, mapping he daa ono subcarriers and applying an Inverse Fas Fourier Transform IFFT). If he ransmier does no have any channel knowledge, he Muual Informaion MI) for a paricular channel realizaion H k C Nr N,k {1,...,K}, assuming zero mean complex Gaussian disribued ransmi signals and receiver noise, is he appropriae performance measure insead of he channel capaciy [7]: I OL) = log de I Nr + P ) σnn H k H H k. 1) Here, σn denoes he variance of he addiive Gaussian receiver noise and P he ransmi power. Precoding can be applied wihou channel knowledge o increase he diversiy of he ransmission e.g. LTE uilizes cyclic delay diversiy precoding [8]). Then he precoder is already conained in he effecive channel marix H k in Equaion 1). This bound is denoed Open Loop Muual Informaion OLMI). In a Frequency Division Duplex FDD) sysem only by means of receiver feedback, knowledge abou he channel from ransmier o receiver can be obained. To limi he amoun of feedback required, he possible precoders are resriced o a given uniary code book W e.g. [8]). An opimum power allocaion by means of waer filling is no possible in ha case as he effecive channel marix precoder and channel) is no diagonal anymore, leading in mos cases o uniform power allocaion. The appropriae performance measure for such a sysem is he Closed Loop Muual Informaion CLMI) defined as: I CL) = max log de I Nr + P ) W k W σnn H k W k Wk H H H k. To furher reduce he feedback overhead, ofen he same precoder is used for he oal sysem bandwidh which leads o he Wideband Closed Loop Muual Informaion WB- CLMI) bound: I CL,WB) = max ) log de I Nr + P ) σnn H k WW H H H k. 3) Unil now we assumed an ideal Maximum Likelihood ML) deecor [] a he receiver. Because such deecors have exponenial complexiy in he number of anennas hey are seldom used in pracice. Insead, simple linear receive filers Zero Forcing ZF), Minimum Mean Squared Error MMSE)) are used o decouple he individual spaial daa sreams and decode hem independenly. Denoing wih F k he receive equalizer filer for subcarrier k, he pos-equalizaion Signal o Inerference and Noise Raio SINR) for spaial ransmission layer l is given by [9]: P K k [l,l] SINR k,l = P i l K k[l,i] +σn i F k[l,i] ) K k = F k H k W. 5) K k [l,l] denoes he elemen in he lh row and lh column of he marix K k. As we are assuming Gaussian ransmi signals, he channel experienced on each layer afer equalizaion corresponds o an AWGN channel wih Signal o Noise Raio SNR) given by ). Using AWGN channel capaciy, we herefore define he Wideband Closed Loop Muual Informaion wih Linear Receiver WB-CLMI-LR) bound as: I CL,WB,LR) = max l=1 log de1+sinr k,l ). ) Here, denoes he number of parallel spaial daa sreams ransmission rank), which depends on he precoder. The precoder deermines wheher a spaial muliplexing or a beam forming gain or a mixure of boh) is obained by he MIMO sysem. Anoher hroughpu loss is caused by he design of he channel coding and modulaion mapping. To achieve channel capaciy, coding wih a Gaussian code book is required [1]. In pracical sysems, only a finie modulaion alphabe is suppored e.g. /1/ QAM). Depending on wheher he code is joinly designed wih he modulaion mapping or he wo are independenly designed, he scheme is denoed Coded Modulaion CM) or BICM. The capaciy of such sysems is well known, albei no in closed form [11]. A low SNR CM performs beer han BICM. Neverheless, BICM is preferred in pracice because i allows o combine any channel code wih any arbirary modulaion alphabe via a bi inerleaver. For a Single Inpu Single Oupu SISO) AWGN channel he BICM capaciy for arbirary modulaion alphabes can easily be evaluaed by means of Mone Carlo simulaions see [11]). This defines a mapping BSNR) from SNR o specral efficiency in bis per channel use. If muliple modulaion alphabes are defined, he mapping BSNR) chooses he maximum BICM capaciy wih respec o he alphabes. Alhough in case of finie modulaion alphabes he iner-layer inerference experienced afer equalizaion is no Gaussian anymore, we employ BSNR) insead of he AWGN capaciy in Equaion ) o ake ino accoun he effec of uilizing a BICM

3 sysem 1. We herefore define he BICM bound according o: BI CL,WB,LR) = max l=1 BSINR k,l W)). 7) The necessarily finie block lengh of he code also causes a performance degradaion. I prevens an opimal Shannon code a code ha achieves he Shannon capaciy limi for infinie block lengh) from achieving channel capaciy. Consider an ideal Shannon code of given raer, in informaion bis per channel symbol, wih infinie block lengh. The Block Error Probabiliy BLEP) of such a code corresponds o a sep funcion over SNR dropping from one o zero a exacly he SNR ha corresponds o channel capaciy C = R. The finie block lengh N in channel symbols) causes he BLEP o decrease coninuously over SNR. The BLEP P B of he bes code is upper bounded by he Gallager bound given by []: P B < N ER) ; ER) = max max [E ρ,q) ρr] 8) q ρ 1 [ 1+ρ E ρ,q) = log qx)py x) 1+ρ] 1. 9) y In hese equaions, x denoes he channel inpu signal, qx) is he channel inpu disribuion, y is he channel oupu signal, py x) is he condiional probabiliy of he channel oupu given he channel inpu andρis an auxiliary opimizaion variable. ER) is called he Gallager exponen. Similarly, he BLEP of he bes code is lower bounded by he Sphere Packing bound [13] whose formulaion is equal o he Gallager bound excep ρ is jus lower bounded by zero and no upper bounded. If he opimizing ρ lies beween zero and one, he wo bounds coincide and define he BLEP of he bes code. In our calculaions his was always he case. Employing BICM renders he channel inpu signal uniformly disribued over he QAM alphabe. Therefore, he maximizaion wih respec oqx) can be skipped. Assuming a linear receiver, he MIMO OFDM channel decomposes ino parallel SISO AWGN channels wih SNR given by Equaion ). Therefore one needs o evaluae he wo bounds on BLEP for an AWGN channel wih uniform inpu disribuion over a given QAM consellaion e.g. numerically). Because he code block lengh is finie, in general a BLEP of zero canno be obained anymore. A ypical operaing poin for wireless communicaion sysems is a arge Block Error Raio BLER) of 1 1 [9]. Fixing he BLEP o he arge BLER, he code rae of he bes code ha achieves he arge BLER can be compued. This defines a mapping from SNR o code rae or specral efficiency. An example is shown in Figure for a block lengh of N = 1 and QAM. The figure compares he obained specral efficiency o he corresponding BICM capaciy. The obained specral efficiency is 1 For ZF receivers he pos-equalizaion inerference vanishes, validaing he assumpion of Gaussian noise, for MMSE receivers we resor o he Gaussian approximaion of pos-equalizaion inerference [1]. x Specral efficiency [bpcu] Efficiency obained from Gallager bound Bi inerleaved coded modulaion capaciy.35 db /N Fig.. Specral efficiency obained via he Gallager bound compared o BICM capaciy of a QAM sysem. below BICM capaciy a high SNR 1 db). A low SNR he obained efficiency is larger han BICM capaciy. The reason is ha he Gallager and Sphere Packing bounds hold for a CM and no a BICM sysem. CM achieves higher specral efficiency han BICM a low SNR [3]. Neverheless, because i is known ha CM and BICM perform equally good a high SNR [11], one can compue how much he BICM capaciy needs o be shifed o ake ino accoun he finie code block lengh. This shif is obained by maching he BICM capaciy and he obained specral efficiency a high SNR. In Figure he shif equals.35 db. Using his shifed BICM efficiency SSNR) insead of he BICM capaciy BSNR) in 7), we arrive a he Shifed BICM SBICM) bound, denoed SBI CL,WB,LR). To achieve he SBICM bound, he communicaion sysem mus suppor any possible code rae. In pracical sysems, ypically jus a small se of possible Adapive Modulaion and Coding AMC) schemes is employed 15 in LTE/LTE- A). To sill guaranee he desired BLER consrain, he suppored AMC scheme wih larges specral efficiency less han or equal o he SBICM bound has o be uilized. Mahemaically his can be formulaed as quanizing he SBICM bound o he neares AMC scheme wih lower or equal specral efficiency: QSBI CL,WB,LR) = SBI CL,WB,LR) C. 1) The operaor. C means flooring wih respec o he AMC schemes defined in he code book C of he considered sysem. We denoe his bound Quanized SBICM QSBICM) bound. In general he MIMO channel is varying over ime. Assuming ergodiciy we obain ergodic values for he bounds e.g. ergodic capaciy []) by means of Mone Carlo simulaions for a chosen channel model in Secion 3. During hese Mone Carlo simulaions we ake ino accoun he mos obvious source of effecive user hroughpu loss, namely he inserion of sysem overhead e.g. for synchronizaion, channel esimaion), by skipping he appropriae subcarriers, as

4 Specral efficiency [bpcu] /N Fig. 3. Specral efficiency obained via link level simulaions of a SISO LTysem over an AWGN channel. defined by he considered sandard. We refer o he bounds obained in ha way as achievable bounds, e.g. achievable muual informaion [1]. The shifed BICM efficiencyssnr) is useful for predicing he opimal performance of a pracical sysem as demonsraed in Secion 3. This is achieved by compuing he average efficiency over all subcarriers for all combinaions of sandard defined precoders W and modulaion alphabes A: EW,A) = 1 K l=1 S A SINR k,l W)). 11) S A SNR) denoes he shifed BICM efficiency corresponding o he modulaion alphabe A A. The SNR value of an AWGN channel ha achieves he average efficiency EW, A) can be compued via he inverse funcion of S A SNR). This is very similar o Muual Informaion Effecive SNR Mapping MIESM) [15]. Then lookup ables, obained from link level simulaions of a SISO AWGN channel, can be used o find he AMC scheme ha achieves he highes specral efficiency for he given SNR. Figure 3 shows an example of such a mapping for an LTysem. Noe ha for each AMC scheme he corresponding modulaion alphabe dependen SNR value mus be used. Knowing he bes AMC scheme, he hroughpu can be compued by muliplying wih he number of channel uses. 3. SIMULATION RESULTS FOR AN LTE SYSTEM In his secion, we apply he proposed bounds o an LTE complian sysem. We compare he bounds o he opimal sysem performance obained by exhausive search simulaions, uilizing he Vienna LTE Link Level Simulaor [], [5]. This means ha we generae a channel and noise realizaion and simulae daa ransmission over his channel wih all possible combinaions of ransmission ranks, precoders and AMC schemes. The opimum is he maximum hroughpu obained over all feasible combinaions. The resuls are averaged over 1 independen channel and noise realizaions. Channel model VehA [1] Sysem bandwidh 1. MHz Receiver equalizer ZF Anenna configuraion CQI feedback RI feedback PMI feedback Table 1. Seings for he comparison of he proposed hroughpu bounds o he simulaed performance of an LTysem. In our simulaions we consider a Closed Loop Spaial Muliplexing CLSM) LTysem, whose seings are summarized in Table 1. All anennas are assumed o be spaially uncorrelaed. In CLSM ransmission mode, LTE enables he adapaion of he AMC scheme, ransmission rank and spaial precoding marix, by means of User Equipmen UE) feedback, in order o opimize he sysem performance. Thereby, he Channel Qualiy Indicaor CQI) chooses he appropriae AMC scheme o obain a given arge BLER ypically.1), he Rank Indicaor RI) signals he preferred number of parallel spaial daa sreams and he Precoding Marix Indicaor PMI) informs he enodeb abou he currenly bes precoder, chosen from a finie code book [8]. Figure shows he resuls for he bounds and simulaed hroughpu obained for he sysem. Considering an operaing poin of SNR = 1. db he following is observed: Channel capaciy he lefmos curve) equals 1 Mbi/s. An LTysem, employing realisic feedback algorihms o compue he feedback indicaors [9], achieves a hroughpu of 5.9 Mbi/s corresponding o 9% of channel capaciy. The opimal LTE performance, assuming perfec knowledge of he channel and noise realizaion a he ransmier, equals 5.% of channel capaciy. By means of he derived hroughpu bounds we nex analyze he reasons for he 5% hroughpu gap beween he pracical sysem and channel capaciy: The signaling overhead achiev. channel capaciy) causes a loss of Mbi/s 1.7% of channel capaciy). The loss caused by he finie se of precoders and equal power allocaion equals. Mbi/s 3.3%). The loss is raher insignifican because a he considered SNR he achievable muual informaion wihou any precoding) already closes up o he achievable channel capaciy. Wih linear receiver CLMI-LR bound) he hroughpu decreases o 8.7 Mbi/s corresponding o a loss of 7.5%. Uilizing he same precoder for feedback reducion) for he oal sysem bandwidh WB-CLMI-LR bound) leads o a loss of.5 Mbi/s or.%. Noe ha his loss depends on he coherence bandwidh of he considered channel model 1.35 MHz for VehA [17]). The BICM bound, aking ino accoun he modulaion alphabes, lies. Mbi/s or 5% of channel capaciy below he WB-CLMI-LR bound.

5 Throughpu [Mbi/s] TP =.1 Mbi/s SNR = 9. db 1 Channel capaciy Achievable channel capaciy 8 Achievable CLMI bound Achievable CLMI LR bound Achievable WB CLMI LR bound Achievable BICM bound Achievable QSBICM bound Predicion of LTE opimum Simulaed opimum Feedback mehod Achievable muual informaion /N Fig.. Comparison of hroughpu bounds and simulaed sysem performance for a CLSM sysem. The finie code block lengh and finie se of suppored code raes decrease he performance by.7% of channel capaciy o.75 Mbi/s QSBICM) corresponding o 5% of channel capaciy. The mehod o predic he opimal LTE performance, described a he end of Secion, overlaps wih he simulaed opimal performance, obained by link level simulaions. Our ighes hroughpu bound is % off he simulaed opimal performance. This difference is caused by he acual channel code performance. Noe ha we do no include addiional losses caused by guard bands and a like.. CONCLUSION In his paper, we analyze he performance of MIMO OFDM based wireless communicaion sysems by means of hroughpu bounds. Taking ino accoun ypical pracical design consrains like he srucure of spaial precoding, finie ses of suppored AMC schemes, finie code block lengh, sysem overhead and receiver design we arrive a increasingly igh bounds. The loss caused by each of hese consrains is quanified. We compare he obained bounds o he performance of a pracical LTysem by means of link level simulaions. This shows ha LTE, uilizing ZF equalizers a he receivers, achieves around 5% of channel capaciy, assuming perfec channel esimaion and frequency synchronicaion, and ignoring losses caused by guard bands. The invesigaion reveals ha a major loss of user hroughpu is caused by sysem overhead, especially for large anenna configuraions. Also ZF equalizers, o separae he spaial sreams, are no a useful choice. The limied number of suppored AMC schemes 15 in LTE) also causes a srong hroughpu degradaion. Finally, he performance of he channel code iself is far from wha Shannon promises. 5. REFERENCES [1] I. E. Telaar, Capaciy of muli-anenna gaussian channels, European Transacions on Telecommunicaions, vol. 1, no., Nov [] C. Schlegel and L. Perez, On error bounds on urbocodes, IEEE Communicaion leers, vol. 3, no. 7, [3] G. Caire, G. Taricco, and E. Biglieri, Bi-inerleaved coded modulaion, IEEE Transacion on Informaion Theory, vol., no. 3, May [] C. Mehlführer, J. C. Ikuno, M. Simko, S. Schwarz, and M. Rupp, The Vienna LTE Simulaors Enabling Reproducibiliy in Wireless Communicaions Research, 11. under review a EURASIP JASP. [5] [Online]. hp:// [] A. Paulraj, R. Nabar, and D. Gore, Inroducion o space-ime wireless communicaions. Cambridge Universiy Press, 3. [7] Z. Wang and G. Giannakis, Ouage muual informaion of space-ime mimo channels, Informaion Theory, IEEE Transacions on, vol. 5, no., pp. 57,. [8] 3GPP, E-UTRA and E-UTRAN; Physical Channels and Modulaion Release 8), 9. [Online]. hp:// [9] S. Schwarz, C. Mehlführer, and M. Rupp, Calculaion of he Spaial Preprocessing and Link Adapion Feedback for 3GPP UMTS/LTE, in Proc. IEEE Wireless Advanced 1, London, UK), 1. [1] T. Cover and J. Thomas, Elemens of Informaion Theory. Wiley and Sons, [11] G. Caire, G. Taricco, and E. Biglieri, Capaciy of biinerleaved channels, Elecron. Le., vol. 3, issue 1, pp. 1 11, 199. [1] D. Seehaler, G. Maz, and F. Hlawasch, An efficien mmse-based demodulaor for mimo bi-inerleaved coded modulaion, in Global Telecommunicaions Conference,. GLOBECOM. IEEE, vol., pp Vol., Nov.. [13] F. Babich, On he performance of efficien coding echniques over fading channels, IEEE Transacions on wireless communicaions, vol. 3, no. 1, Jan.. [1] M. Rupp, C. Mehlführer, and S. Caban, On achieving he Shannon bound in cellular sysems, in Proc. h Inernaional Conference Radioelekronika 1, Brno, Czech Republik), Apr. 1. [15] L. Wan, S. Tsai, and M. Almgren, A Fading-Insensive Performance Meric for a Unified Link Qualiy Model, in Proc. IEEE Wireless Communicaions & Neworking Conference WCNC,. [1] 3GPP, Technical Specificaion Group Radio Access Neworks; Deploymen aspecs Release 8), Dec. 8. [Online]. hp:// [17] S. Schwarz, M. Wrulich, and M. Rupp, Muual Informaion based Calculaion of he Precoding Marix Indicaor for 3GPP UMTS/LTE, in Proc. IEEE Workshop on Smar Anennas 1, Bremen, Germany), Feb. 1.