Mode Switching for MIMO Broadcast Channel Based on Delay and Channel Quantization

1 Mode Switching for MIMO Broadca st Channel Based on Delay and Channel Quantiz ation Jun Zhang, Robert W . Heath Jr ., Marios K ountouris, and Jef frey G. Andrews Abstract Imperf ect channel state in formatio n degrades the perfor mance of multip le-input multiple-o utput (MIMO) commun ications; its effect on single-user (SU) and multi-u ser (M U) MI MO tran smissions are quite different. In particular, MU-MIMO suffers from residual inter-user interfer ence du e to imperfect channel state informa tion while SU-MIMO only suffers from a p ower loss. This paper compares the throug hput loss of bo th SU and M U MIMO on the d ownlink due to delay and ch annel quantization. Accurate closed -form app roximatio ns are derived for the achievable rates for bo th SU and MU MIMO. It is sho wn that SU-M IMO is relati vely robust to delaye d and qu antized chann el infor mation, while MU- MIMO with zero-forcin g pre coding loses s patial multiplexing gain with a fixed delay or fixed codeboo k size. Based o n d erived achie vable rates, a mode switchin g alg orithm is pro posed tha t switches between SU and MU MIMO mo des to improve the sp ectral efficienc y , based on the av erage signal-to-no ise r atio (SNR), the normalized Doppler f requen cy , an d the c hannel quantization codebook size. The o perating regions fo r SU and MU mod es with d ifferent delay s an d c odebo ok sizes are deter mined, which can be used to select the preferr ed mo de. It is shown that the MU mo de is acti ve only when the no rmalized Doppler fr equency is very sm all an d the c odebo ok size is large. Index T erms Multi-user MIMO, ad aptive transmission, mo de switching, impe rfect channel state inf ormation at the tr ansmitter (CSIT ), zero -forcin g preco ding. The authors are with the W ireless Networking and Communications Group, Department of Electrical and Computer Engineering, The Uni v ersity of T exas at Austin, 1 Un iv ersity Station C080 3, Austin, TX 78712–02 40. Email: { jzhang2, rheath, mkou ntouris, jandre ws } @ece.utex as.edu. This work has been supported in part by A T&T Labs, Inc. 2 I . I N T RO D U C T I O N Over the last decade, the point-to-poi nt multi ple-input mul tiple-output (M IMO) l ink (SU- MIMO) has bee n extensi vely researched and has t ransited from a theoretical concept to a practical technique [1], [2]. Due to space and complexity constraints, ho we ver , current m obile terminals only have one or two antennas, wh ich limits the performance of the SU-MIMO link. Multi-us er MIMO (MU-MIMO) provides the opportuni ty to ov ercome such a limitati on b y communicatin g with mul tiple mobiles simultaneously . It eff ectiv ely increases the number of equiv alent spat ial channels and provides spatial multiplexing gain proportional to the number of t ransmit antennas at the base station e ven with single-antenna mobiles. In addition, M U-MIMO has higher immunity to propagati on limit ations faced by SU-MIMO, s uch as channel rank lo ss and antenna correlation [3]. There are many techni cal challenges that mu st be overcome to exploit the full benefits of MU- MIMO. A major one is the requi rement of channel state information at t he transmitter (CSIT), which is difficult to get especially for the downlink/broadcast channel. For t he MIMO downlink with N t transmit antenn as and N r recei ve antennas, with full CSIT t he sum throughput can grow linearly with N t e ven when N r = 1 , b ut without CSIT the spat ial mu ltiplexing gain is the same as for SU-MIMO, i.e. the throughput grows linearly with min( N t , N r ) at hi gh SNR [4]. Limited feedback is an efficient way to provide partial CSIT , which feeds back the quantized channel information to the transmit ter via a low-ra te feedback channel [5], [6]. Howe ver , such imperfect C SIT will gre atly degrade the throughput gai n pro vided by MU-MIMO [7], [8]. Besides quantization, there are other im perfections in th e av ail able CSIT , such as esti mation error and feedback delay . W ith im perfect CSIT , it is not clear whether–or m ore to the point, when– MU- MIMO can outperform SU-MIMO. In t his paper , we compare SU and MU-MIMO transmi ssions in the M IMO downlink wit h CSI delay and channel quantization, and propo se to swi tch between SU and MU MIMO modes based o n the achiev able rate of each techni que with practical receiv er assumption s. A. Related W ork For the MIMO downlink, CSIT is required to separate the spatial channels for d iffe rent users. T o obtain th e full s patial multiplexing gain for t he MU-MIMO sys tem employing zero-forcing (ZF) or block-diagonali zation (BD) precoding, it was shown in [7], [9] that the quantization 3 codebook size for limited feedback needs to increase linearly with SNR (in dB) and the number of t ransmit antennas. Zero-forcing d irty-paper coding and channel i n version sy stems with limited feedback were in vestigated in [8], where a sum rate ceiling due to a fixed codebook size was deriv ed for both s chemes. In [10], it was shown that to exploit mul tiuser div ersity for ZF , bot h channel direction and information about s ignal-to-interference-plus-noise ratio (SINR) must be fed back. More recently , a comprehensive stud y of the MIMO downlink wit h ZF precoding was done in [11], which considered downlink traini ng and explicit channel feedback and concluded that si gnificant downlink through put is achiev able with effic ient CSI feedback. For a compound MIMO broadcast channel, the information th eoretic analysis in [12] sho wed t hat scaling the CSIT quality such that the CSIT error is dominated by the in verse of the SNR is both necessary and sufficient to achiev e the full s patial mult iplexing gain. Although previous studies show that the spatial mul tiplexing gain of MU-MIMO can be achie ved with limited feedback, it requires the codebook s ize to increase with SNR and the number of t ransmit antennas. Ev en if such a requi rement is satisfied, there is an ine vitable rate loss due to quantization error , plus other CSIT imperfections such as est imation error and delay . In addition , most of prior work focused o n the achiev able sp atial multip lexing gain, mainly based on t he analysis of the rate loss due to imperfect CSIT , whi ch is usuall y a loose bound [7], [9], [12]. Such analys is cannot accura tely characterize the throughput loss, and no comparison with SU-MIMO has been made. In this paper , we deriv e good approximations for the achiev able throughput for both SU and MU MIMO s ystems with fixed channel information accurac y , i .e. with a fixed delay and a fixed quantization codebo ok si ze. W e are interested in the following question: W ith imperfect CSIT , i ncluding delay and channel quantiza tion, when can MU-MIMO actually deliver a thr oughput gain over SU-MIMO? Based on this, we can select the one with the hig her throughput as the transmission technique. B. Contributions In this paper , we in vestigate SU and M U-MIMO in the broadcast channel wit h CSI delay and limited feedback. The m ain contributions of thi s paper are as follows. • SU vs. MU Analy sis . W e in vestigate the impact of imperfect CSIT due to delay and channel quantization. W e sho w that th e SU m ode is more robust t o im perfect CSIT as it only suffers a constant rate loss, whi le MU-MIMO suffers more seve rely from th e residual int er -user 4 interference. W e characterize the residual interference due to delay and channel quantization, which s hows these two effec ts are equivalent. Based o n an independence approx imation of the int erference terms and the sig nal term, accurate closed-form app roximations are deriv ed for the er go dic rates for both SU and MU MIMO modes. • Mode Switching Algorithm. A SU/MU m ode switching algorit hm is proposed based o n the er godic sum rate as a function of th e aver age SNR, n ormalized Do ppler frequency , and the quantization codebook size. Thi s transmissio n techniqu e o nly requires a s mall number of users to feed back inst antaneous channel i nformation. The m ode swit ching points can be calculated from the p re viously derive d approximatio ns for ergodic rates. • Operating Regions. The operating r e gions for SU and M U modes are determined, from whi ch we can determine the activ e mode and find the condition th at activ ates each mode. W ith a fixed delay and codebook size, i f the MU mode is possible at all, there are two mode switching points, with the SU m ode preferred at both low and high SNRs. The MU m ode will only be activ ated when the normalized Doppler frequency is very small and the codebook size is lar ge. From the numerical results, the minimum feedback bits per user to get the MU mode activ ated gro w app roximately linearly with the number of t ransmit antennas. The rest of the paper is or ganized as follows. The system mod el and some assumptions are presented in Section II. The transmis sion techniques for both SU and MU M IMO modes are described in Section III. The rate analysis for both SU and MU modes and the mode switchin g are do ne in Section IV. Num erical result s and conclusions are in Section V and VI, respectiv ely . I I . S Y S T E M M O D E L W e consider a M IMO downlink, where the transm itter (the base station) has N t antennas and each mobile user has a singl e antenna. The system parameters are list ed in T abl e I. During each transmissio n period, which is less than the channel coherence time and t he channel is assumed to be constant, the base s tation transmits to one (SU-MIMO m ode) or multi ple (M U-MIMO mode) users. The discrete-time com plex b aseband recei ved signal at the u -th user at ti me n is 5 giv en as 1 y u [ n ] = h ∗ u [ n ] U X u ′ =1 f u ′ [ n ] x u ′ [ n ] + z u [ n ] , (1) where h u [ n ] is the N t × 1 channel vector from the transm itter to t he u -th user , and z u [ n ] is the normalized com plex Gaussian noise vector , i.e. z u [ n ] ∼ C N (0 , 1) . x u [ n ] and f u [ n ] are t he transmit signal and N t × 1 precoding vector for the u -th user , respectiv ely . The transmit power constraint is E { x ∗ [ n ] x [ n ] } = P , where x [ n ] = [ x ∗ 1 , x ∗ 2 , · · · , x ∗ U ] ∗ . As th e no ise is normalized, P is also the average transmit SNR. T o assist the analy sis, w e assume that the channel h u [ n ] is well modeled as a spati ally white Gaussian channel, with entries h i,j [ n ] ∼ C N (0 , 1) , and t he chann els are i.i.d. over dif ferent users. The results will be diff erent for diffe rent channel models. For example, a limited feedback system with line of s ight MIMO channel requires fewer feedback bi ts compared to the Rayleigh channel [13]. The in vestigation of other channel models is l eft to fut ure work. W e con sider two of the main sources of the CSIT imperfection–delay and q uantization error 2 , specified as follo ws. A. CSI D elay Model W e consider a stationary ergodic Gaus s-Markov block fading process [14, Sec. 16–1], where the channel stays constant for a symb ol duration and changes from symbol to symbol according to h [ n ] = ρ h [ n − 1] + e [ n ] , (2) where e [ n ] is the channel error vector , w ith i.i.d. ent ries e i [ n ] ∼ C N (0 , ǫ 2 e ) , and it is uncorrelated with h [ n − 1] . W e assume the CSI delay is of one symbol. It is straightforward to extend the results to the scenario with a d elay of multiple symbols. For the numerical analysis, the classi cal Clark e’ s isotropic scatterin g model wi ll b e used as an example, for which the correlation coef ficient is 1 In this paper , we use uppercase boldface letters for matrices ( X ) and lo wercase boldface for vectors ( x ). E [ · ] is the expe ctation operator . The conjugate transpo se of a matrix X (v ector x ) is X ∗ ( x ∗ ). S imilarly , X † denotes the pseudo-in verse, ˜ x denotes the normalized vector of x , i .e. ˜ x = x k x k , and ˆ x denotes the quantized vector of ˜ x . 2 For a practical system, the feedb ack bits for each user is usually fixed, and there will ine vitably be delay in the av ailable CSI, both of which are difficult or e ven impossib le to adjust. Other ef fects such as channel estimation error can be made small such as by increasing the transmit power or the number of pilot symbols. 6 ρ = J 0 (2 π f d T s ) with Doppler spread f d [15], where J 0 ( · ) is the zero-th order Bessel function of the first kind. The va riance of the error vector is ǫ 2 e = 1 − ρ 2 . Therefore, both ρ and ǫ e are determined by the n ormalized Doppler frequenc y f d T s . The channel in (2) is wi dely-used to mo del the time-varying channel. For example, it is used t o in ve stigate the im pact of feedback delay on the performance of closed-loop transmit di versity in [16] and the system capacity and bit error rate of p oint-to-point MIMO li nk i n [17 ]. It simpl ifies the analysis , and the resul ts can be easily extended t o other scenarios. Essentially , this m odel i s of the form h [ n ] = g [ n ] + e [ n ] , (3) where g [ n ] is the av ailable CSI at time n with an uncorrelated error vector e [ n ] , g [ n ] ∼ C N ( 0 , (1 − ǫ 2 e ) I ) , and e [ n ] ∼ C N ( 0 , ǫ 2 e I ) . It can be u sed to consider the ef fect of other imperfect CSIT , such as estimation error and analog feedback. The difference is in e [ n ] , w hich has di ff erent var iance ǫ 2 e for different s cenarios. Some examples are gi ven as follo ws. a) Estim ation Err or: I f the receiv er obtains th e CSI th rough MMSE estimation from τ p pilot sym bols, the error v ariance is ǫ 2 e = 1 1+ τ p γ p , where γ p is the SNR of the pil ot symbo l [18]. b) Analog F eedback: F or analog feedback, the error variance is ǫ 2 e = 1 1+ τ ul γ ul , where τ ul is the number of channel uses per channel coef ficient and γ ul is th e S NR on the uplink feedback channel [19]. c) Analog F eedback with Pr ediction: As shown in [20], for analog feedback with a d - step M MSE predictor and the Gauss -Markov m odel, the error va riance is ǫ 2 e = ρ 2 d ǫ 0 + (1 − ρ 2 ) P d − 1 l =0 ρ 2 l , where ρ is the same as in (2) and ǫ 0 is the Kalman filt ering mean-square error . Therefore, the resul ts in th is paper can be easily extended to t hese systems. In the follo wing parts, we focus on the effect of CSI d elay . B. Channel Qu antizatio n Model W e con sider frequency-division duplexing (FDD) systems, where l imited feedback techniq ues provide partial CSIT th rough a dedicated feedback channel from the receiver to the transmitt er . The channel direction i nformation for the precoder design is fed back using a quantization codebook known at both the transm itter and receive r . The quantization i s chosen from a codebook of unit norm vec tors of size L = 2 B . W e assume each user uses a different codebook to av oid the s ame quantization vector . The cod ebook for 7 user u is C u = { c u, 1 , c u, 2 , · · · , c u,L } . Each user quantizes its channel to t he closest codew ord, where clos eness is measured by the inner product. Therefore, the index of channel for user u is I u = arg max 1 ≤ ℓ ≤ L | ˜ h ∗ u c u,ℓ | . (4) Each user needs to feed back B bits to denote this index, and the t ransmitter has the quantized channel inform ation ˆ h u = c u,I u . As the optimal v ector quantizer for this prob lem is not known in general, random vector quant ization (R VQ) [21] is us ed, where each quant ization vector is independently chosen from the isotropi c distribution on the N t -dimensional unit sphere. It has been shown in [7] t hat R VQ can facilitate th e analysis and provide performance close to t he optimal quantization. In this paper , we analyze the achiev able r ate ave raged over both R VQ-based random codebooks and fading d istributions. An im portant m etric for the limited feedback sy stem is the squared angular distortion, defined as sin 2 ( θ u ) = 1 − | ˜ h ∗ u ˆ h u | 2 , where θ u = ∠  ˜ h u , ˆ h u  . W ith R VQ, it was sho wn in [7], [22] that the expectation in i .i.d. Rayleigh fading is gi ven by E θ  sin 2 ( θ u )  = 2 B · β  2 B , N t N t − 1  , (5) where β ( · ) is the beta function. It can be ti ghtly bounded as [7] N t − 1 N t 2 − B N t − 1 ≤ E  sin 2 ( θ u )  ≤ 2 − B N t − 1 . (6) I I I . T R A N S M I S S I O N T E C H N I Q U E S In this section, w e describe the transmission techniques for bot h SU and MU MIMO systems with perfect CSIT , whi ch will be u sed in the s ubsequent sections for imperfect CS IT systems. By doing thi s, we focus on the im pacts of imperfect CSIT on the con ve ntional t ransmission techniques. Designing imp erfect CSIT -aware precoders is left to fut ure work. Throughout t his paper , we use the achiev able er godic rate as the performance metric for bot h SU and MU-MIMO systems. The base station transmits to a singl e user ( U = 1 ) for the SU-MIM O system and to N t users ( U = N t ) for the MU-MIMO system. The SU/MU mode s witching algorithm is also described. 8 A. SU-MIMO System W it h perfect CS IT , it is op timal for the SU-MIMO system to t ransmit alo ng the channel direction [1], i.e. selecting th e beamforming (BF) vector as f [ n ] = ˜ h [ n ] , denoted as eigen- beamforming in this paper . The ergodic capacity of thi s s ystem is the same as that of a maximal ratio combinin g diver sity system, given by [23] R B F ( P ) = E h  log 2  1 + P k h [ n ] k 2  = log 2 ( e ) e 1 /P N t − 1 X k =0 Γ( − k , 1 /P ) P k , (7) where Γ( · , · ) is the c omplementary incomplete gamma f unction defi ned as Γ( α, x ) = R ∞ x t α − 1 e − t dt . B. MU-MIMO System For MIMO broadcast channels, although dirty-paper coding (DPC) [24] is opti mal [25]–[29], it is diffic ult to implement in practice. As in [7], [11], ZF precoding is used in this paper , which is a linear precoding techniqu e that precancels i nter-user interference at the transmitter . There are several reasons for us to use t his sim ple transmi ssion technique. Firstly , du e to its simple structu re, it is possi ble to deri ve closed-form results, which can provide helpful insight s. Second, the ZF precoding is able to provide full spati al multipl exing gain and only has a power off set compared to the optimal DPC system [30]. In addition, it was shown in [30] that the ZF precoding i s optimal among t he set of all linear precoders at as ymptoticall y high SNR. In Section V, we will show that our results for th e ZF system also apply for the regularized ZF precoding [31], which provides a high er throughput t han the ZF precoding at lo w to moderate SNRs. W it h precoding vec tors f u [ n ] , u = 1 , 2 , · · · , U, assuming equal power all ocation 3 , the recei ved SINR for the u -th user i s given as γ Z F,u = P U | h ∗ u [ n ] f u [ n ] | 2 1 + P U P u ′ 6 = u | h ∗ u [ n ] f u ′ [ n ] | 2 . This is true for a g eneral li near precoding MU-MIMO s ystem. W ith perfect CSIT , this quantity can be calculated at the transmitter , while with imperfect CSIT , it can be estimat ed at the receiv er and fed back to the transm itter giv en knowledge of f u [ n ] . 3 At high SNR, this performs closely to the system employing optimal w ater-filling, as po wer allocation mainly benefits at lo w SNR. 9 Denote ˜ H [ n ] = [ ˜ h 1 [ n ] , ˜ h 2 [ n ] , · · · , ˜ h U [ n ]] ∗ . W ith perfect CSIT , the ZF precoding vectors are determined from the pseudo-in verse of ˜ H [ n ] , as F [ n ] = ˜ H † [ n ] = ˜ H ∗ [ n ]( ˜ H [ n ] ˜ H ∗ [ n ]) − 1 . The precoding vector for the u -th us er is obtained by normalizing the u -th column of F [ n ] . Therefore, h ∗ u [ n ] f u ′ [ n ] = 0 , ∀ u 6 = u ′ , i.e. there is no inter-user interference. The receiv ed SINR for t he u -th user becomes γ Z F,u = P U | h ∗ u [ n ] f u [ n ] | 2 . (8) As f u [ n ] is independent of h u [ n ] , and k f u [ n ] k 2 = 1 , t he effecti ve channel for the u -th user is a single-input sing le-output (SISO) Rayleigh fading channel. Therefore, th e achiev able sum rate for the ZF system is given by R Z F ( P ) = U X u =1 E γ [log 2 (1 + γ Z F,u )] . (9) Each term on the right hand side of (9) is the ergodic capacity of a SISO system in Rayleigh fading, giv en in [23] as R Z F,u = E γ [log 2 (1 + γ Z F,u )] = log 2 ( e ) e U /P E 1 ( U /P ) , (10) where E 1 ( · ) is the exponenti al-integral functi on of the fir st o rder , E 1 ( x ) = R ∞ 1 e − xt t dt . C. SU/MU Mode Switching Imperfect CSI T will degrade the performance of the MIMO communication. In thi s case, it is unclear wh ether and when the M U-MIMO s ystem ca n actually provide a throughput gain over the SU-MIMO system. Based on the analysis of the achie v able ergodic rates in this paper , we propose to switch between SU and MU mo des and select the one with the higher achie vable rate. The channel correlation coef ficient ρ , which captures the CSI delay effect, usually v aries slowly . The quanti zation codebook size is normally fixed for a given sy stem. Therefore, it is reasonable to assume that t he transm itter has knowledge of both delay and chann el quantization, and can estimate the achie v able er godic rates of both SU and MU MIMO modes. Then it ca n determine the activ e mode and select one (SU mode) or N t (MU mode) users to serve. This is a low-complexity transmission strategy , and can be combined with ra ndom user selection, round- robin schedu ling, or schedul ing b ased on q ueue l ength rather than channel status. It only requi res the selected users to feed back instantaneous channel information. Therefore, it is suit able for a 10 system that h as a constraint on t he total feedback bits and only allows a sm all number of users to send feedback, or a syst em with a strict delay constraint that cannot employ opportunisti c scheduling based on instantaneous channel in formation. T o determine the transmiss ion rate, the transmitter sends pi lot sym bols, from which the activ e users es timate th e receiv ed SINRs and feed back them to the transmitter . In this paper , we assume the transmitter kn ows perfectly the actual receiv ed SINR at each activ e u ser . In practice, there will ine vitably be errors in such information due to estimation error and feedback delay , which will result in rate mismatch, i .e. the transmission rate b ased on the estimated SINR does not match the actual SINR on the channel, so t here will be outage events. How to deal with such rate m ismatch is of practical importance, and we mention se veral possible approaches as follows. The full in vestigation of this issue requires furth er research and is out of scope of this paper . Considering the outage events, the transmissi on s trategy can be design ed based on the actual information symb ols successfully delivere d to the receiver , denoted as goodput in [32], [33]. W i th the estimated SINR, anot her approach is to back o ff on t he transmiss ion rate based on the variance of the estimation error , as did in [34], [35] for the s ingle-antenna opportunist ic scheduling system and i n [36] for th e multiple-antenna opportu nistic beamform ing system. Combined wi th u ser selection, the transmissi on rate can al so be determined based on some lower bound of the actual SINR to make sure that n o outage occurs, as di d in [37] for the limited feedback system. I V . P E R F O R M A N C E A N A L Y S I S A N D M O D E S W I T C H I N G In t his section, we in vestigate the achiev able er godic rates for bot h SU and MU MIMO m odes. W e first analyze the a ver age recei ve d SNR for the BF system and the a verage residual interfer ence for the ZF system, which p rovide insights on t he impact of imperfect CSIT . T o select the activ e mode, accurate closed-form app roximations for both SU and M U modes are then deri ved. A. SU Mod e–Eigen-Beamforming First, if t here is no del ay and only channel quantization, the BF vector is based on the quantized feedback, f ( Q ) [ n ] = ˆ h [ n ] . The a verage receive d SNR i s SNR ( Q ) B F = E h , C [ P | h ∗ [ n ] ˆ h [ n ] | 2 ] 11 = E h , C [ P k h [ n ] k 2 | ˜ h ∗ [ n ] ˆ h [ n ] | 2 ] ( a ) ≤ P N t  1 − N t − 1 N t 2 − B N t − 1  , (11) where (a) foll ows th e i ndependence between k h [ n ] k 2 and | ˜ h ∗ [ n ] ˆ h [ n ] | 2 , together with the result in (6). W it h both delay and channel quantization, the BF vector is based on the quant ized channel direction with delay , i.e. f ( QD ) [ n ] = ˆ h [ n − 1] . The ins tantaneous receiv ed SNR for t he BF syst em SNR ( QD ) B F = P    h ∗ [ n ] f ( QD ) [ n ]    2 . (12) Based on (11), we get t he fol lowing theorem on the a verage receiv ed SNR for the SU mod e. Theor em 1: The av erage recei ved SNR for a BF system with channel q uantization and CSI delay is SNR ( QD ) B F ≤ P N t  ρ 2 ∆ ( Q ) B F + ∆ ( D ) B F  , (13) where ∆ ( Q ) B F and ∆ ( D ) B F show the impact of channel q uantization and feedback delay , respectiv ely , giv en b y ∆ ( Q ) B F = 1 − N t − 1 N t 2 − B N t − 1 , ∆ ( D ) B F = ǫ 2 e N t . Pr o of: See Appendix B. From Jensen’ s inequ ality , an upper bound of the achiev able rate for the BF sy stem wi th both quantization and d elay is g iv en b y R ( QD ) B F = E h , C h log 2  1 + SNR ( QD ) B F i ≤ log 2 h 1 + SNR ( QD ) B F i ≤ log 2 h 1 + P N t  ρ 2 ∆ ( Q ) B F + ∆ ( D ) B F i . (14) Remark 1: Note that ρ 2 = 1 − ǫ 2 e , so the av erage SNR decreases with ǫ 2 e . W i th a fixed B and fix ed delay , the SNR degradation is a cons tant factor independent of P . At high SNR, the imperfect CSIT in troduces a cons tant rate loss lo g 2  ρ 2 ∆ ( Q ) B F + ∆ ( D ) B F  . The upper bound provided by Jensen’ s inequality is not tight. T o get a better approximation for the achie v able rate, we first make th e follo wing approximation on th e instantaneous received 12 SNR SNR ( QD ) B F = P | h ∗ [ n ] ˆ h [ n − 1] | 2 = P | ( ρ h [ n − 1] + e [ n ]) ∗ ˆ h [ n − 1] | 2 ≈ P ρ 2 | h ∗ [ n − 1] ˆ h [ n − 1] | 2 , (15) i.e. we remove the term with e [ n ] as it is normally insignificant compared to ρ h [ n − 1] . This will be verified later by simul ation. In this way , the system is approxi mated as the one wi th limi ted feedback and wi th equiv alent SNR ρ 2 P . From [22], th e achiev able rate of the l imited feedback BF system is gi ven by R ( Q ) B F ( P ) = log 2 ( e ) e 1 /P N t − 1 X k =0 E k +1  1 P  − Z 1 0  1 − (1 − x ) N t − 1  2 B N t x e 1 /P x E N t +1  1 P x  dx  , (16) where E n ( x ) = R ∞ 1 e − xt x − n dt is the n -th order exponenti al integral. So R ( QD ) B F can be app roxi- mated as R ( QD ) B F ( P ) ≈ R ( Q ) B F ( ρ 2 P ) . (17) As a special case, consi dering a system with d elay only , e.g. the time-division duplexing (TDD) system which can estim ate the CSI from the uplink with channel reciprocity but with propagation and processing delay , the BF v ector is based on th e delayed channel direction, i.e. f ( D ) [ n ] = ˜ h [ n − 1] . W e provide a good approximatio n for the achiev able rate for s uch a system as follows. The instantaneous re ceiv ed SNR is given as SNR ( D ) B F = P | h ∗ [ n ] f ( D ) [ n ] | 2 = P | ( ρ h [ n − 1] + e [ n ]) ∗ ˜ h [ n − 1] | 2 ( a ) ≈ P ρ 2 k h [ n − 1] k 2 + P | e ∗ [ n ] ˜ h [ n − 1] | 2 . (18) In s tep (a) we elimi nate the cross terms since e [ n ] is normall y small . As e [ n ] is independent of ˜ h [ n − 1] , e [ n ] ∼ C N ( 0 , ǫ 2 e I ) and k ˜ h [ n − 1] k 2 = 1 , we ha ve | e ∗ [ n ] ˜ h [ n − 1] | 2 ∼ χ 2 2 , where χ 2 M denotes chi-square distribution wit h M degrees of freedom. In additi on, k h [ n − 1] k 2 ∼ χ 2 2 N t , and it is i ndependent of | e ∗ [ n ] ˜ h [ n − 1] | 2 . Then th e following theorem can be deri ved. 13 Theor em 2: The achie vable er godic rate of the BF system with delay can be approximated as R ( D ) B F ≈ log 2 ( e ) a 0 N t e 1 /η 2 E 1  1 η 2  − log 2 ( e )(1 − a 0 ) N t − 1 X i =0 i X l =0 a N t − 1 − i 0 ( i − l )! η − ( i − l ) 1 I 1 (1 /η 1 , 1 , i − l ) , (19) where η 1 = P ρ 2 , η 2 = P ǫ 2 e , a 0 = η 2 η 2 − η 1 , and I 1 ( · , · , · ) is gi ven in (36 ) in Appendix A. Pr o of: See Appendix C. B. Zer o-F or cing 1) A verage Residual Interfer ence: If there is no delay and only channel quant ization, the precoding vectors for the ZF system are desi gned based on ˆ h 1 [ n ] , ˆ h 2 [ n ] , · · · , ˆ h U [ n ] to achieve ˆ h ∗ u [ n ] f ( Q ) u ′ [ n ] = 0 , ∀ u 6 = u ′ . W i th random vector quant ization, it is shown in [7] that the average noise plus i nterference for eac h user is ∆ ( Q ) Z F,u = E h , C " 1 + P U X u ′ 6 = u | h ∗ u [ n ] f ( Q ) u ′ [ n ] | 2 # = 1 + 2 − B N t − 1 P . (20) W it h both channel quant ization and CSI delay , p recoding vectors are desig ned based on ˆ h 1 [ n − 1] , ˆ h 2 [ n − 1 ] , · · · , ˆ h U [ n − 1 ] and achie ve ˆ h ∗ u [ n − 1 ] f ( QD ) u ′ [ n ] = 0 , ∀ u 6 = u ′ . The recei ved SINR for the u -t h user is gi ven as γ ( QD ) Z F,u = P U | h ∗ u [ n ] f ( QD ) u [ n ] | 2 1 + P U P u ′ 6 = u | h ∗ u [ n ] f ( QD ) u ′ [ n ] | 2 . (21) As f ( QD ) u [ n ] is i n the nulls pace o f ˆ h u ′ [ n − 1] ∀ u ′ 6 = u , it is isotropically distributed in C N t and independent of ˜ h u [ n − 1] as well as ˜ h u [ n ] , so | h ∗ u [ n ] f ( QD ) u [ n ] | 2 ∼ χ 2 2 . The ave rage noise plus interference is gi ven in t he following t heorem. Theor em 3: The ave rage noise plus interference for th e u -th user of the ZF system with bot h channel quanti zation and CSI delay i s ∆ ( QD ) Z F,u = 1 + ( U − 1) P U  ρ 2 u ∆ ( Q ) Z F,u + ∆ ( D ) Z F,u  , (22) where ∆ ( Q ) Z F,u and ∆ ( D ) Z F,u are the de gradations brought by channel quantization and feedback delay , respecti vely , giv en by ∆ ( Q ) Z F,u = U U − 1 2 − B N t − 1 , ∆ ( D ) Z F,u = ǫ 2 e,u . 14 Pr o of: The proof is si milar to th e one for Theor em 1 in appendix B. Remark 2: From Theor em 3 we see that the av erage residual interference for a given user consists of t hree parts: (i) The num ber of i nterfer ers , U − 1 . The more users the system supp orts, the h igher the m utual interference. (ii) The transmi t po wer of the other acti ve users , P U . As t he t ransmit power increases, the system becomes in terference-limited. It is possible to improve p erformance through power allocation, which i s left to future work. (iii) The CSIT accuracy fo r thi s user , which is reflected from ρ 2 u ∆ ( Q ) Z F,u + ∆ ( D ) Z F,u . The user wit h a larger delay or a smaller codebook size suf fers a higher residual int erference. From this remark, the int erference term, P U ( U − 1) ǫ 2 e,u , equiv alently comes from U − 1 virtual interfering users , each with equivalent SNR as P U  ρ 2 u ∆ ( Q ) Z F,u + ∆ ( D ) Z F,u  . W ith a high P and a fixed ǫ e,u or B , the system i s interference-limited and cannot achieve full spatial mul tiplexing gain. Therefore, to keep a constant rate los s, i.e. to sustain t he spati al multi plexing gain , the channel error due t o both quantization and del ay needs to be reduced as SNR increases. Similar to t he result for the limit ed feedback system in [7], for the ZF system with both delay and channel quantization, we can get th e following corollary for the cond ition to achiev e the full spatial mult iplexing gain . Cor ol lary 1: T o k eep a constant rate l oss of log 2 δ 0 bps/Hz for each user , the codebo ok si ze and CSI delay n eed to satisfy the following cond ition ρ 2 u ∆ ( Q ) Z F,u + ∆ ( D ) Z F,u = U U − 1 · δ 0 − 1 P . (23) Pr o of: As sho wn i n [7], [11 ], the rate lo ss for each user due to imperfect CSIT is upper bounded by ∆ R u ≤ log 2 ∆ ( QD ) Z F,u . The coroll ary follows from solving log 2 ∆ ( QD ) Z F,u = log 2 δ 0 . Equiv alently , this means that for a giv en ρ 2 , the feedback bits per us er needs to scale as B = ( N t − 1) log 2  δ 0 − 1 ρ 2 u P − U − 1 U ·  1 ρ 2 u − 1  − 1 . (24) As ρ 2 u → 1 , i.e. there is no CSI delay , the conditi on b ecomes B = ( N t − 1) lo g 2 P δ 0 − 1 , which agrees with t he result i n [7] with limited feedback only . 15 2) Achievable Rate: For the ZF system with i mperfect CSI, the genie-aided upper bound for the ergodic achie vable rate 4 is given by [11 ] R ( QD ) Z F ≤ U X u =1 E γ h log 2  1 + γ ( QD ) Z F,u i = R ( QD ) Z F,ub . (25) W e assu me the mobile users can perf ectly estimate the noise and interference and feed back it to the transmitter , s o the upper bound is chosen as the performance m etric, i.e. R ( QD ) Z F = R ( QD ) Z F,ub , as in [7], [8], [10]. The following l ower bound based on the rate loss analysis is us ed in [7], [11] R ( QD ) Z F ≥ R Z F − U X u =1 log 2 ∆ ( QD ) Z F,u , (26) where R Z F is th e achiev able rate with p erfect CSIT , given in (9). Howe ver , this lower bou nd is very loose. In the following, we will derive a more accurate approxim ation for the achiev able rate for t he ZF system. T o get a go od approximati on for the achieva ble rate for the ZF system, we first approxi mate the ins tantaneous SINR as γ ( QD ) Z F,u = P U | h ∗ u [ n ] f ( QD ) u [ n ] | 2 1 + P U P u ′ 6 = u | ( ρ u h u [ n − 1] + e u [ n ]) ∗ f ( QD ) u ′ [ n ] | 2 ≈ P U | h ∗ u [ n ] f ( QD ) u [ n ] | 2 1 + P U  P u ′ 6 = u ρ 2 u | h ∗ u [ n − 1] f ( QD ) u ′ [ n ] | 2 + P u ′ 6 = u | e ∗ u [ n ] f ( QD ) u ′ [ n ] | 2  , (27) i.e. elim inating the in terference terms whi ch have both h u [ n − 1] and e u [ n ] as e u [ n ] i s normally very small, so we g et two separate interference sums due to delay and quantization, respectiv ely . For the interference term du e to delay , | e ∗ u [ n ] f ( QD ) u ′ [ n ] | 2 ∼ χ 2 2 , as e [ n ] is independent of f ( QD ) u ′ [ n ] and k f ( QD ) u ′ [ n ] k 2 = 1 . For the i nterference term due to quantizatio n, it was shown in [7] that | ˜ h ∗ u [ n − 1] f ( QD ) u ′ [ n ] | 2 is equivalent t o the prod uct of the quantizatio n error sin 2 θ u and an independent β (1 , N t − 2) random v ariable. Therefore, we ha ve | h ∗ u [ n − 1] f ( QD ) u ′ [ n ] = k h u [ n − 1] k 2 (sin 2 θ u ) · β (1 , N t − 2) . (28) 4 This upper bound is achiev able only when a genie provides users with perfect kno wledge of all interference and the transmitter kno ws perfectly the received SINR at each user . 16 In [10], with a quantization cell app roximation 5 [38], [39], it w as shown that k h u [ n − 1] k 2 (sin 2 θ u ) has a Gamma di stribution with p arameters ( N t − 1 , δ ) , where δ = 2 − B N t − 1 . As shown in [10] the analysis based on the quantization cell approximation is clo se to the performance o f random vector q uantization, so we use this approach to deriv e the achie vable rate. The following l emma gives the d istribution of the interference term due to quantization. Lemma 1: Based on the quantization cell approximation , the interference term due to quan- tization in (27), | h u [ n − 1] f ( QD ) u ′ [ n ] | 2 , is an exponential random variable with mean δ , i.e. its probability distri bution function (pdf) is p ( x ) = 1 δ e − x/δ , x ≥ 0 . (29) Pr o of: See Appendix D. Remark 3: From this lemm a, we s ee that t he residual interference terms due to both delay and quantization are exponenti al random variables, which means the d elay and q uantization error hav e equi valent effects, only with different m eans. By comparing the means of these two terms, i.e. comp aring ǫ 2 e and 2 − B N t − 1 , we can find the do minant one. In addi tion, with this result, we can approx imate th e achiev able rate of the ZF limit ed feedback system, which will be p rovided later in t his section. Based on the distribution of the interference terms, the approximati on for th e achie vable rate for the MU mode is g iv en i n the fol lowing theorem. Theor em 4: The er godic achie v able rate for the u -th user in the MU mode wi th both delay and channel quant ization can be approximated as R ( QD ) Z F,u ≈ log 2 ( e ) M − 1 X i =0 2 X j =1 " a ( j ) i i !  α β  i +1 · I 3  1 α , α β δ j , i + 1  # , (30) where α = β = P U , δ 1 = ρ 2 u δ , δ 2 = ǫ 2 e,u , M = N t − 1 , a (1) i and a (2) i are gi ven in (44) and (45), and I 3 ( · , · , · ) is gi ven in (38) in Appendix A. Pr o of: See Appendix E. The ergodic sum t hroughput is R ( QD ) Z F = U X u =1 R ( QD ) Z F,u . (31) 5 The quantization cell approximation is based on the ideal assumption that each quantization cell is a V oronoi reg ion on a spherical cap with the surface area 2 − B of the total area of t he unit sphere for a B bit s codebook. The detail can be found in [10], [38], [39]. 17 As a special case, for a ZF system with delay only , we can get the following approx imation for the er go dic achiev able rate. Cor ol lary 2: The ergodic achiev able rate for t he u -t h user i n the ZF system with delay i s approximated as R ( D ) Z F,u ≈ log 2 ( e )  α β  − ( M − 1) · I 3  1 α , α β , M − 1  , (32) where α = P U , β = ǫ 2 e,u P U , M = N t − 1 , and I 3 ( · , · , · ) is given in (38) in Appendix A. Pr o of: F ollowing the same s teps in Appendix E with δ 1 = 0 . Remark 4: As s hown in Lemma 1 , the effec ts of delay and channel quantization are equiv alent , so th e approximation in (32) als o appli es for the limited feedback sys tem. This is verifie d by simulatio n in Fig. 1, which sho ws that thi s approximation is very accurate and can be used to analyze the limited feedback system. C. Mode S witching W e first verify the approximation (30) in Fig . 2, which compares the approxim ation with simulatio n results and the lower bound (26), with B = 10 , v = 20 km /hr , f c = 2 GHz, and T s = 1 m sec. W e see that the lo wer bound is very loose, while the approxim ation is accurate especially for N t = 2 . In fact, the approximati on tu rns out to be a lower bound. Note that due to the imperfect CSIT , the sum rate reduces with N t . In Fig. 3, we com pare the BF and ZF systems, with B = 18 , f c = 2 GHz, v = 10 km /hr , and T s = 1 msec. W e see that the approximation for t he BF system almost m atches the sim ulation exactly . The approximation for th e ZF system is accurate at low to medium SNRs, and becomes a lower bound at high SNR, which is approxim ately 0 . 7 bps/Hz i n total, or 0 . 17 5 bps/Hz per user , lower than the simulati on. The throughp ut of the ZF system is limited by t he residual inter-user interference at high SNR, where it is lower than the BF syst em. This motiv ates to switch between the SU and M U MIMO modes. The approximations (17 ) and (30) will be us ed to calculate th e m ode switching points. Th ere may be two s witching points for t he system wi th delay , as the SU mode will be selected at both low and high SNR. These two points can b e calculated by providing dif ferent initi al v alues to the nonlinear equation solver , such as fsol ve in MA TLAB. 18 V . N U M E RI C A L R E S U L T S In this section, num erical resul ts are presented. First , the operating regions for different modes are pl otted, which show the i mpact of differe nt p arameters, including the normalized Doppler frequency , the codebook size, and the number of transmit antennas. Then the extension of our results for the ZF precoding to the MMSE precoding is d emonstrated. A. Operating Re gions As shown in Section IV -C, finding m ode swi tching points requires solving a nonlinear equation, which does not have a clos ed-form solutio n and gi ves li ttle insight. Howe ver , it is easy to e va luate numerically for different p arameters, from which insi ghts can be drawn. In this section, wit h the calculated mode switching p oints for different parameters, we plot the operating regions for both SU and MU modes. The activ e mode for the given parameter and th e cond ition to activate each mode can be found from such plots. In F ig. 4, the operating regions for both SU and MU modes are plotted, for different no rmalized Doppler frequencies and di ff erent number of feedback bits in Fig. 4(a) and Fig. 4(b), respectively , and with U = N t = 4 . There are analogies between two plots. Some key observations are as follows: (i) For the delay plot Fig. 4(a), comparing the two curves for B = 16 and B = 20 , we see that th e smaller the codebook size, the smal ler the operating re gion for the Z F mode. For the ZF mode to be active, f d T s needs to be small, specifically we need f d T s < 0 . 055 and f d T s < 0 . 04 6 for B = 20 and B = 16 , respectively . These condit ions are not easily satisfied in practical s ystems. For example, with carrier frequency f c = 2 GHz, mo bility v = 20 km/hr , the Doppler frequency is 37 Hz, and then t o satisfy f d T s < 0 . 055 the delay should be less than 1 . 5 m sec. (ii) For the codebook size plot Fig. 4(b), comparing the two curves with v = 10 km/hr and v = 20 km/hr , as f d T s increases ( v increases), the ZF op erating region shrinks. For the ZF mode to be acti ve, we should ha ve B ≥ 12 and B ≥ 14 for v = 10 km/hr and v = 20 km/hr , respectively , w hich means a large codebook s ize. Note that for BF we only need a small codebook size to get th e near-optimal performance [5]. (iii) For a gi ven f d T s and B , the SU mod e will b e active at both low and high SNRs, which is due to i ts array gain and the robustness to im perfect CSIT , respectively . 19 The operating regions for dif ferent N t are sho wn in Fig. 5. W e see that as N t increases, the operating re gion for the M U mode shrin ks. Specific ally , we need B > 12 for N t = 4 , B > 1 9 for N t = 6 , and B > 26 for N t = 8 to get t he MU mode activ ated. Not e that the minimum required feedback bi ts per u ser for the MU mode grow approximately lin early with N t . B. ZF vs. MMSE Pr ecoding It is shown in [31] that the regularized ZF precoding, denoted as MMSE pr ecoding in this paper , can s ignificantly increase the throughput at lo w SNR . In this section, we show th at our results on m ode swit ching with ZF p recoding can also be applied to MMSE precoding. Denote ˆ H [ n ] = h ˆ h 1 [ n ] , ˆ h 2 [ n ] , · · · , ˆ h U [ n ] i ∗ . Then th e M MSE precoding vectors are cho sen to be the normalized columns of the matrix [31] ˆ H ∗ [ n ]  ˆ H [ n ] ˆ H ∗ [ n ] + U P I  − 1 . (33) From this, we see that the MM SE precoders con verge to ZF precoders at high SNR. Therefore, our deriv ations for the ZF sys tem also apply to the MMSE system at high SNR. In Fig. 6, we compare the performance of ZF and MMSE precoding systems with delay 6 . W e see t hat the MMSE precoding outperforms ZF at low to medium SNRs, and con ver g es to ZF at high SNR while con ver ges to BF at low SNR. In add ition, it has the s ame rate ceilin g as t he ZF system, and crosses the BF curv e roughly at the same point , after which we need to swi tch to the SU mode. Based on th is, we can use the s econd predicted mode swi tching point (the one at higher SNR) of t he ZF sy stem for the MMSE system. W e com pare t he si mulation results and calculation results by (19) and (32) for the mode switching point s in T able II. For the ZF system, it is the second switching point; for the MMSE sy stem, it is the only switching poi nt. W e see th at the s witching poi nts for MMSE and ZF syst ems are very close, and the calculated ones are roughly 2 . 5 ∼ 3 dB lo wer . V I . C O N C L U S I O N S In this paper , we com pare the SU and M U MIMO transmissions in the broadcast channel w ith delayed and q uantized CSIT , where the amount of delay and t he num ber of feedback bit s per 6 This can also be done in the system with both delay and quantization , which is more time-consuming. As shown in Lemma 1 , the effec ts of delay and quantization are equiv alent, so the conclusion will be the same. 20 user are fixed. The throughput of MU-MIMO saturates at high SNR due to residu al inter -user interference, for which a SU/MU mode swit ching al gorithm is prop osed. W e deriv e accurate closed-form approximations for the er godic rates for bot h SU and MU modes, which are then used to calculate the m ode swi tching poin ts. It is shown that the MU mode is only possi ble to be acti ve in the medium SNR regime, with a small normalized Doppler frequenc y and a lar ge codebook size. For future work, the M U-MIMO mode studied in this paper is d esigned with zero-forcing criterion, which is s hown t o be sensitive to CSI i mperfections, so robust precoding design is needed and the impact of t he imperfect CSIT on non-l inear precoding should be in vestigated. As power control is an eff ectiv e way to combat interference, it is interestin g to consi der the ef ficient power control algorith m rather than equal power allo cation to i mprove the performance, especially in the heterogeneous scenario. It is also of practical importance to in vestigate po ssible approaches to improve the quality of the av ailable CSIT wi th a fixed codeboo k size, e.g. through channel predictio n. A P P E N D I X A. Useful Results fo r Rate Analysis In t his Appendix, we present s ome useful results that are used for rate analy sis in this paper . The following l emma will b e used frequently in the deriv ation of the achiev able rate. Lemma 2: For a random var iable x with probabil ity dis tribution function (pdf) f X ( x ) and cumulative distribution funct ion (cdf) F X ( x ) , we ha ve E X [ln(1 + X )] = Z ∞ 0 1 − F X ( x ) 1 + x dx. (34) Pr o of: The proof follows the integration by parts. E X [ln(1 + X )] = Z ∞ 0 ln(1 + x ) f X ( x ) dx = − Z ∞ 0 ln(1 + x ) [1 − F X ( x )] ′ dx ( a ) = Z ∞ 0 1 − F X ( x ) 1 + x dx, (35) where g ′ is the deri vativ e of the functi on g , and step (a) foll ows the integra tion by parts. The follo wing lemma pro vides s ome us eful integrals for rate analysis, which can be d eriv ed from the resul ts in [40]. 21 Lemma 3: I 1 ( a, b, m ) = Z ∞ 0 x m e − ax x + b dx = m X k =1 ( k − 1)!( − b ) m − k a − k − ( − 1) m − 1 b m e ab E 1 ( ab ) (36) I 2 ( a, b, m ) = Z ∞ 0 e − ax ( x + b ) m dx =    e ab E 1 ( ab ) m = 1 P m − 1 k =1 ( k − 1)! ( m − 1)! ( − a ) m − k − 1 b k + ( − a ) m − 1 ( m − 1)! e ab E 1 ( ab ) m ≥ 2 (37) I 3 ( a, b, m ) = Z ∞ 0 e − ax ( x + b ) m ( x + 1) dx = m X i =1 ( − 1) i − 1 (1 − b ) − i · I 2 ( a, b, m − i + 1) + ( b − 1) − m · I 2 ( a, 1 , 1) , (38) where E 1 ( x ) is t he exponential-integral funct ion of the first order . B. Pr oof of Theor em 1 The av erage SNR is SNR ( QD ) B F = E  P    h ∗ [ n ] f ( QD ) [ n ]    2  = P E     ( ρ h [ n − 1] + e [ n ]) ∗ ˆ h [ n − 1]    2  ( a ) = P | ρ h ∗ [ n − 1] ˆ h [ n − 1] | 2 + P | e ∗ [ n ] ˆ h [ n − 1] | 2 ( b ) ≤ P N t ρ 2  1 − 2 − B N t − 1  + P E h | ˆ h ∗ [ n − 1] · [ e [ n ] e ∗ [ n ]] · ˆ h [ n − 1] | i ( c ) = P N t ρ 2  1 − 2 − B N t − 1  + P ǫ 2 e , As e [ n ] is independent of h [ n − 1] , it i s also independent of ˆ h [ n − 1] , which gives (a). Step (b) follows (11). Step (c) is from the fact e [ n ] ∼ C N ( 0 , ǫ 2 e I N t ) and | ˆ h [ n − 1] | 2 = 1 . C. Pr oof of Theor em 2 Denote y 1 = k h [ n − 1] k 2 and y 2 = 1 ǫ 2 e | e ∗ [ n ] ˜ h [ n − 1] | 2 , then y 1 ∼ χ 2 2 N t , y 2 ∼ χ 2 2 , and they are independent. The recei ved SNR can be written as x = η 1 y 1 + η 2 y 2 , where η 1 = P ρ 2 and 22 η 2 = P ǫ 2 e . The cdf of x is gi ven as [41] F X ( x ) = 1 −  η 2 η 2 − η 1  N t e − x/η 2 + e − x/η 1  η 1 η 2 − η 1  · N t − 1 X i =0 i X l =0 1 ( i − l )!  η 2 η 2 − η 1  N t − 1 − i  x η 1  i − l . (39) Denote a 0 = η 2 η 2 − η 1 and following Lemma 2 we ha ve E X [ln(1 + X )] = Z ∞ 0 1 − F X ( x ) 1 + x dx = a N t 0 Z ∞ 0 e x/η 2 1 + x dx − ( 1 − a 0 ) N t − 1 X i =0 i X l =0 a N t − 1 − i 0 ( i − l )!  1 η 1  i − l Z ∞ 0 x i − l e − x/η 1 1 + x dx = a N t 0 I 2 (1 /η 2 , 1 , 1) − (1 − a 0 ) N t − 1 X i =0 i X l =0 a N t − 1 − i 0 ( i − l )!  1 η 1  i − l I 1 (1 /η 1 , 1 , i − l ) . (40) where I 1 ( · , · , · ) and I 2 ( · , · , · ) are gi ven in (36) and (37), respectively . D. Pr oof of Lemma 1 Let x = k h u [ n − D ] k 2 sin 2 θ ∼ Γ( M − 1 , δ ) , y ∼ β (1 , M − 2) , and x is independent of y . Then the i nterference term du e to qu antization is z = xy . T he cdf of z is P Z ( z ) = P ( xy ≤ z ) = Z ∞ 0 F Y | X  z x  f X ( x ) dx = Z z 0 f X ( x ) dx + Z ∞ z  1 −  1 − z x  M − 2  f X ( x ) dx = Z ∞ 0 f X ( x ) dx − Z ∞ z  1 − z x  M − 2 x M − 2 e − x/δ ( M − 2)! δ M − 1 dx = 1 − e − z /δ Z ∞ z ( x − z ) M − 2 e − ( x − z ) /δ ( M − 2)! δ M − 1 dx ( a ) = 1 − e − z /δ , (41) where step (a) follows the equal ity R ∞ 0 y M e − αy = M ! α − ( M +1) . 23 E. Pr oof of Theor em 4 Assuming each interference term in (27) is independent o f each other and independent of the signal po wer term, denot e P u ′ 6 = u ρ 2 u | h ∗ u [ n − 1] f ( QD ) u ′ [ n ] | 2 = ρ 2 u δ y 1 and P u ′ 6 = u | e ∗ u [ n ] f ( QD ) u ′ [ n ] | 2 = ǫ 2 e,u y 2 , then from Lemma 1 we ha ve y 1 ∼ χ 2 2( N t − 1) , and y 2 ∼ χ 2 2( N t − 1) as e u [ n ] is complex Gaussian with var iance ǫ 2 e,u and independent of the normali zed vector f ( QD ) u ′ [ n ] . In additi on, t he signal po wer | h ∗ u [ n ] f ( QD ) u [ n ] | 2 ∼ χ 2 2 . Then the recei ved SINR for t he u -th user is approxi mated as γ ( QD ) Z F,u ≈ αz 1 + β ( δ 1 y 1 + δ 2 y 2 ) , x, (42) where α = β = P U , δ 1 = ρ 2 u δ , δ 2 = ǫ 2 e,u , y 1 ∼ χ 2 2 M , y 1 ∼ χ 2 2 M , M = N t − 1 , z ∼ χ 2 2 , and y 1 , y 2 , z are independent of ea ch other . Let y = δ 1 y 1 + δ 2 y 2 , then t he pdf o f y , which is the sum of two i ndependent chi -square random var iables, is given as [41] p Y ( y ) = e − y /δ 1 M − 1 X i =0 a (1) i y i + e − y /δ 2 M − 1 X i =0 a (2) i y i = 2 X j =1 M − 1 X i =0 e − y /δ j a ( j ) i y i , (43) where a (1) i = 1 δ i +1 1 ( M − 1)!  δ 1 δ 1 − δ 2  M (2( M − 1) − i )! i !( M − 1 − i )!  δ 2 δ 2 − δ 1  M − 1 − i (44) a (2) i = 1 δ i +1 2 ( M − 1)!  δ 2 δ 2 − δ 1  M (2( M − 1) − i )! i !( M − 1 − i )!  δ 1 δ 1 − δ 2  M − 1 − i . (45) The cdf of x is F X ( x ) = P  αz 1 + β y ≤ x  = Z ∞ 0 F Z | Y  x α (1 + β y )  p Y ( y ) dy = Z ∞ 0  1 − e − x α (1+ β y )  p Y ( y ) dy = 1 − e − x/α Z ∞ 0 e − β xy /α p Y ( y ) dy = 1 − e − x/α Z ∞ 0 ( 2 X j =1 M − 1 X i =0 exp  −  β α x + 1 δ j  y  a ( j ) i y i ) dy 24 ( a ) = 1 − e − x/α 2 X j =1 M − 1 X i =0    a ( j ) i i !  β α x + 1 δ j  i +1    , (46) where step (a) follows the equal ity R ∞ 0 y M e − αy = M ! α − ( M +1) . Then the ergodic achiev able rate for th e u -th user is approximated as R ( QD ) Z F,u = E γ h log 2  1 + γ ( QD ) Z F,u i ≈ log 2 ( e ) E X [ln(1 + X )] ( a ) = log 2 ( e ) Z ∞ 0 1 − F X ( x ) x + 1 dx = log 2 ( e ) Z ∞ 0 M − 1 X i =0 2 X j =1    a ( j ) i i !  α β  e − x/α  x + α β δ j  i +1 ( x + 1)    ( b ) = log 2 ( e ) M − 1 X i =0 2 X j =1 " a ( j ) i i !  α β  i +1 I 3  1 α , α β δ j , i + 1  # , (47) where step (a) follows from Lemma 2 , step (b) follo ws t he expression of I 3 ( · , · , · ) in (38). 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Approximations and simulations for the ZF system with limit ed feedback, N t = U = 4 . 0 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 10 11 SNR (dB) Rate (bps/Hz) ZF (simulation) ZF (approximation) ZF (lower bound) N t =U=2 N t =U=4 N t =U=6 Fig. 2. Comparison of app roximation in (30), the lo wer bound in (26), a nd the simulation results for the ZF sy stem with both delay and channel quantization. B = 10 , f c = 2 GHz, v = 20 km/hr, and T s = 1 msec. 28 0 5 10 15 20 25 30 35 40 45 0 2 4 6 8 10 12 14 16 18 SNR (dB) Rate (bps/Hz) BF (Simulation) BF (Approximation) ZF (Simulation) ZF (Approximation) ZF Region BF Region BF Region Fig. 3. Mode switching between BF and ZF modes with both CS I delay and channel quantization, B = 18 , N t = 4 , f c = 2 GHz, T s = 1 msec, v = 10 km/hr . 10 −2 10 −1 5 10 15 20 25 30 35 40 45 50 Normalized Doppler frequency, f d T s SNR (dB) B=20 ZF Region BF Region ZF Region BF Region B=16 (a) Dif ferent f d T s . 10 15 20 25 30 5 10 15 20 25 30 35 40 45 50 Codebook size, B SNR (dB) v = 10 km/hr v = 20 km/hr BF Region BF Region ZF Region ZF Region (b) Di fferen t B , f c = 2 GHz, T s = 1 msec. Fig. 4. Operating regions for BF and ZF with both CSI delay and quantization, N t = 4 . 29 10 15 20 25 30 5 10 15 20 25 30 35 40 45 50 Codebook size, B SNR (dB) N t =U=4 N t =U=6 N t =U=8 BF Region ZF Region BF Region ZF Region BF Region ZF Region Fig. 5. Operating regions for BF and ZF with different N t , f c = 2 GHz, v = 10 km/hr, T s = 1 msec. −20 −10 0 10 20 30 40 0 2 4 6 8 10 12 14 16 18 SNR (dB) Rate (bps/Hz) MMSE ZF BF Fig. 6. Simulation results for BF , Z F and MMSE systems wit h delay , N t = U = 4 , f d T s = 0 . 04 .