On the achievable rates of multiple antenna broadcast channels with feedback-link capacity constraint

Transcription

1 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 RESEARCH Open Access On the achevable rates of multple antenna broadcast channels wth feedback-lnk capacty constrant Xang Chen *, We Mao, Yunzhou L, Shdong Zhou and Jng Wang Abstract In ths paper, we study a MIMO fadng broadcast channel where each recever has perfect channel state nformaton whle the channel state nformaton at the transmtter s acqured by explct channel feedback from each recever through capacty-constraned feedback lnks. Two feedback schemes are consdered,.e., the analog and dgtal feedback. We analyze the achevable ergodc rates of zero-forcng drty-paper codng ZF-DC), whch s a nonlnear precodng scheme nherently superor to lnear ZF beamformng. Closed-form lower and upper bounds on the achevable ergodc rates of ZF-DC wth Gaussan nputs and unform power allocaton are derved. Based on the closed-form rate bounds, suffcent and necessary condtons on the feedback channels to ensure nonzero and full downlnk multplexng gan are obtaned. Specfcally, for analog feedback n both AWGN and Raylegh fadng feedback channels, t s suffcent and necessary to scale the average feedback lnk SNR lnearly wth the downlnk SNR n order to acheve the full multplexng gan. Whle for the random vector quantzaton-based dgtal feedback wth angle dstorton measure n an error-free feedback lnk, t s suffcent and necessary to scale the number of feedback bts B peruserasb =M )log 2 where M s the number of transmt antennas and s the average downlnk SNR. Keywords: Feedback-lnk capacty constrant, lmted feedback, multple antenna broadcast channel, multplexng gan, multuser MIMO, zero-forcng drty-paper codng ZF-DC) Introducton The multple antenna broadcast channels, also called multple-nput multple-output MIMO) downlnk channels, have attracted great research nterest for a number of years because of ther spectral effcency mprovement and potental for commercal applcaton n wreless systems. Intal research n ths feld has manly focused on the nformaton-theoretc aspect ncludng capacty and downlnk-uplnk dualty [-4] and transmt precodng schemes [5-9]. These results are based on a common assumpton that the transmtter n the downlnk has access to perfect channel state nformaton CSI). It s well known that the multplexng gan of a pont-to-pont MIMO channel s the mnmum of the number of transmt and receve antennas even wthout * Correspondence: chenxang98@mals.tsnghua.edu.cn Research Insttute of Informaton Technology, Tsnghua Natonal Laboratory for Informaton Scence and TechnologyTNLst), Tsnghua Unversty, Bejng, Chna CSIT []. On the other hand, n a MIMO downlnk wth sngle-antenna recevers and..d. channel fadng statstcs, n the case of no CSIT, user multplexng s generally not possble and the multplexng gan s reduced to unty []. As a result, the role of the CSI at the transmtter CSIT) s much more crtcal n MIMO downlnk channels than that n pont-to-pont MIMO channels. The acquston of the CSI at the transmtter s an nterestng and mportant ssue. For tme-dvson duplex TDD) systems, we usually assume that the channel recprocty between the downlnk and uplnk can be exploted and the transmtter n the downlnk utlzes the plot symbols transmtted n the uplnk to estmate the downlnk channel [2]. The mpact of the channel estmaton error and plot desgn on the performance of the MIMO downlnk n TDD systems has been studed n [3-8]. For frequency-dvson duplex FDD) systems, no channel recprocty can be exploted, 2 Chen et al; lcensee Sprnger. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

2 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 2 of 6 and thus t s necessary to ntroduce feedback lnks to convey the CSI acqured at the recevers n the downlnk back to the transmtter. There are generally two knds of CSI feedback schemes appled for MIMO downlnk channels n the lterature. The frst scheme s called the unquantzed and uncoded CSI feedback or analog feedback AF) n short, where each user estmates ts downlnk channel coeffcents and transmts them explctly on the feedback lnk usng unquantzed quadrature-ampltude modulaton [2,9-2]. The performance of the downlnk lnear zero-forcng beamformng ZF-BF) scheme wth AF was evaluated through smulatons n [9], and analytcal results were gven later n [2] and [2]. The second feedback scheme s called the vector quantzed CSI feedback or dgtal feedback DF) n short, where each user quantzes ts downlnk channel coeffcents usng some predetermned quantzaton codebooks and feeds back the bts representng the quantzaton ndex [2-27]. The MIMO broadcast channel wth DF has been consdered n [2,2,24-27]. In [24], a lnear ZF- BF-based multple-nput sngle-output MISO) system s frstly consdered wth random vector quantzaton RVQ) lmted feedback lnk, n whch the closed-form expressons for expected SNR, outage probablty, and bt error probablty were derved. Then the vector quantzaton scheme based on the dstorton measure of the angle between the codevector and the downlnk channel vector was adopted n [2,2,25], and a closedform expresson of the lower and upper bound on the achevable rate of ZF-BF was derved. The results there also showed that the number of feedback bts per user must ncrease lnearly wth the logarthm of the downlnk SNR to mantan the full multplexng gan. Further, the authors n [26] ponted out that n the scenaro where the number of users s larger than that of the transmt antennas, wth smple user selecton, havng more users reduces feedback load per user for a target performance. However, the aforementoned lteratures [2,2,25] both focus on the lnear ZF-BF scheme, whch s not asymptotcally optmal compared wth nonlnear schemes, such as zero-forcng drty-paper codng ZF- DC) []. So, t s necessary to nvestgate the lmtng performance for MIMO downlnk channels wth lmted dgtal feedback lnk. In [27], the authors analyzed both the lnear ZF-BF and nonlnear zero-forcng drty-paper codng ZF-DC) and derved loose upper bounds of the achevable rates wth lmted feedback. But dfferent from the dstorton measure of the angle n [2,2,25], another vector quantzaton approach based on the dstorton measure of mean-square error MSE) between the codevector and the downlnk channel vector was adopted n [27]. Smultaneously, the exact lower bounds of the achevable rates wth lmted feedback for ZF-DC are not gven n [27]. In ths paper, we consder both analog and dgtal feedback schemes and study the achevable rates of a MIMO broadcast channel wth these two feedback schemes, respectvely. Dfferent from [2,25] focusng on the ZF-BF, the ZF-DC s analyzed n our work whch s nherently superor to the ZF-BF due to ts nonlnear nterference precancelaton characterstc and s asymptotcally optmal [] as [27]. Specally, for DF, we adopt the vector quantzaton dstorton measure of the angle between the codevector and the downlnk channel vector, and perform RVQ [2,2,25] for analytcal convenence. Our man contrbutons and key fndngs n ths paper are as follows: A comprehensve analyss of the achevable rates of ZF-DC wth ether analog or dgtal feedback s presented, and closed-form lower and upper bounds on the achevable rates are derved. For fxed feedback-lnk capacty constrant, the downlnk achevable rates of ZF-DC are bounded as the downlnk SNR tends to nfnty, whch ndcates that the downlnk multplexng gan wth fxed feedback-lnk capacty constrant s zero. In order to acheve full downlnk multplexng gan, t s suffcent and necessary to scale the average feedback lnk SNR lnearly wth the downlnk SNR for AF n both AWGN and Raylegh fadng feedback channels. Whle for DF n an error-free feedback lnk, t s suffcent and necessary to scale the feedback bts per user as B =M )log 2 where M s the number of transmt antennas and s the average downlnk SNR. We note that although the ZF-DC wth DF has been consdered n [27], our work also dffers from t n several aspects. Frst, a dfferent dstorton measure for channel vector quantzaton s appled n our work compared to that n [27] as stated earler. Actually, for RVQ-based DF, the angle dstorton measure n [2,2,25] seems more reasonable than the MSE dstorton measure n [27], whch wll be dscussed n ths paper. Second, a more thorough analyss about the downlnk achevable rates ncludng upper and lower bounds) and multplexng gan s presented n ths paper than that n [27] only upper bounds are gven), coverng both AF and DF. The remander of ths paper s organzed as follows. We gve a bref ntroducton to the ZF-DC wth perfect CSIT n Secton 2. Comprehensve analyss of achevable rates and multplexng gan for both AF and DF are

3 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 3 of 6 presented n Sectons 3 and 4, respectvely. A rough comparson of AF and DF s also gven n Secton 4. Fnally, conclusons and dscussons for future work are gven n Secton 5. Throughout the paper, the symbols ) T, )*and ) H represent matrx transposton, complex conjugate and Hermtan, respectvely. [ ] m, n denotes the element n the mth row and the nth column of a matrx. represents the Eucldean norm of a vector. and ) denote the magntude and the phase angle of a complex number, respectvely. E{ } represents expectaton operator. Var ) s the varance of a random varable. CNa, b) denotes a crcularly symmetrc complex Gaussan random varable wth mean of a and varance of b. Zero-forcng drty-paper codng wth perfect CSIT Consder a multple antenna broadcast channel composed of one base staton BS) wth M transmt antennas and K users each wth a sngle receve antenna. Assumng the channel s at and..d. block fadng, the receved sgnal at user n a gven block s y = h x + v, ) where h Î C M s the complex channel gan vector between the BS and user, x Î C M s the transmtted sgnal wth a total transmt power constrant,. e., E{x H x} =, andv s the complex whte Gaussan nose wth varance. For analytcal convenence, we assume spatally ndependent Raylegh fadng channels between the BS and the users,.e., the entres of h are..d. CN, ), andh, =,..., K are mutually ndependent. Under the assumpton of..d. block fadng, h s constant n the duraton of one block and ndependent from block to block. By stackng the receved sgnals of all the users nto y =[y... y K ] T,thesgnalmodels compactly expressed as y = Hx + v, 2) where H =[h T ht 2 ht K ]T and v =[v v 2... v k ] T. In ths paper, we focus on the case K = M. IfK<M, there wll be a loss of multplexng gan. The case K> M wll ntroduce mult-user dversty gan and we wll leave t for future work. We frst gve a bref ntroducton of ZF-DC under perfect CSIT n ths secton. In the ZF-DC scheme, the BS performs a QR-type decomposton to the overall channel matrx H denoted as H = GQ, whereg s an M Mlower trangular matrx and Q s an M Muntary matrx. We let x = Q H d and the components of d are generated by successve drty-paper encodng wth Gaussan codebooks [], then the resultng sgnal model wth the precoded transmt sgnal can be wrtten as: y = Gd + v. 3) From Equaton 3 the receved sgnal at user s gven by y = g d + g j d j + v, 4) j< where g j =[G], j and d,theth entry of d, s the output of drty-paper codng for user treatng the term as the j < g j d j noncausally known nterference sgnal. From the total transmt power constrant E{x H x} =, we have E{d H d} =. Ifthetransmtpowersunformly allocated to each user,.e., d CN, /M), then for.. d. Raylegh flat fadng channel, the closed-form expresson of the achevable ergodc sum rate usng the ZF- DC s gven by [,27]: R CSIT and sum = M R CSIT = R CSIT 5) = E {log 2 + g 2 = e M )} M M + ) MN log 2 e E j, j= where E n x) e xt t n dt s the exponental ntegral functon of order n [28]. The multplexng gan [] of ZF-DC under perfect CSIT s M,.e., R CSIT sum lm = M, log 2 whch s the full multplexng gan of the downlnk [,25]. Achevable rates of ZF-DC under analog feedback In ths secton, we consder the analog feedback AF) scheme, where each user estmates ts downlnk channel coeffcents and transmts them explctly on the feedback lnk wthout any quantzaton or codng. In order to focus on the mpact of feedback lnk capacty constrant, we assume perfect CSI at each user s recever CSIR), and no feedback delay,.e., the downlnk CSI s fed back nstantaneously n the same block as the subsequent downlnk data transmsson. For ease of analyss, 6) 7)

4 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 4 of 6 we also mpose two restrctons on the transmsson strategy: ) the total transmt power s equally allocated to the users and 2) ndependent Gaussan encodng s appled for each user at the transmtter sde. In order to compare the mpact of dfferent feedback channels for AF scheme, we frst consder the AWGN feedback channels from Sectons 3. to 3.4, then extend the analyss to the Raylegh fadng channels n Secton 3.5. Analog feedback n AWGN feedback channels The M users estmate and feed back ther complex channel coeffcents usng orthogonal feedback channels. A smplfyng assumpton of our work s frstly to consder the AWGN feedback channels,.e., no fadng n the feedback lnks. Each user takes b fb M b fb and b fb M s an nteger) channel uses to feed back ts M complex channel coeffcents by modulatng them wth a group of orthonormal spreadng sequences {s m } M m= where s m s a b fb M vector and s m s H m =, m =,..., M, s m s H n = m n [2]. Then the receved sgnals of the feedback channel from user over b fb M channel uses can be wrtten n a compact form: y fb = β fb SNR fb M m= s m h,m + w fb, 8) where h,m CN, ) denotes the downlnk channel gan from the mth transmt antenna of the BS to user, the b fb M vector w fb wth..d. entres each dstrbuted as CN, ) denotes the addtve whte Gaussan nose on the feedback channel and SNR fb represents the average transmt power and also the average SNR n the feedback channel). After despreadng, the suffcent statstc for estmatng h, m s obtaned as wrtten below: r,m = β fb SNR fb h,m + n,m, 9) where n, m s the equvalent nose dstrbuted as CN, ). MMSE estmaton s performed to estmate h, m. WedenotetheMMSEestmateofh, m as ĥ,m and the correspondng estmaton error h,m ĥ,m as δ, m. Snce h,m CN, ), ĥ,m and δ, m are also crcularly symmetrc complex Gaussan random varables wth zero mean, and ther varances are: Varĥ,m) = Varδ,m )= +β fb SNR fb D, ) +β fb SNR fb D. ) Moreover, ĥ,m and δ, m are ndependent from each other. The vector quantzaton scheme usng the dstorton measure of MSE n [27] leads to the same statstcs of the channel error as the AF scheme ntroduced above, so t s equvalent to the AF scheme. Therefore, the followng analyss framework developed for AF can be readly appled to the case studed n [27]. Lower bound on the achevable rate of ZF-DC wth AF n AWGN feedback channels The BS collects the channel estmates ĥ,m, m =,..., M) to form the estmated channel matrx Ĥ, then smply we have the followng relatonshp between H and Ĥ: H = Ĥ +, 2) where [Ĥ],m = ĥ,m and [Δ], m = δ, m. Obvously, Ĥ and Δ are mutually ndependent. The BS performs ZF-DC treatng the estmated channel matrx Ĥ as the true one. The QR decomposton of Ĥ can be wrtten as Ĥ = Ĝ ˆQ, whereĝ s a lower trangular matrx and ˆQ s a untary matrx. The receved sgnal s modeled as: y = H ˆQ H d + v =Ĥ + ) ˆQ H d + v = Ĝd + ˆQ H d + v. 3) From the above equaton, we can extract the receved sgnal at user as lsted below: y = ĝ d + j< ĝ j d j + ˆQ H d + v, 4) where ĝ j =[Ĝ],j and Δ s the th row of Δ. We have the followng theorem that gves a lower bound on the achevable ergodc rate of ZF-DC under AF. Theorem. If the downlnk channel s..d. Raylegh fadng and the feedback channels are AWGN channels, then the achevable ergodc rate of ZF-DC wth AF s lower bounded as: where M + e β log 2 e E j β ), =,2,..., M, 5) j= D + N β = and D =. +β D ) fb SNR fb M

5 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 5 of 6 roof. We frst consder the lower bound on the achevable rate under gven Ĥ. Recall Equaton 4 and ntroduce three notatons: x = ĝ d, s = j< ĝjd j,and n = ˆQ H d + v. Then we have the followng sgnal model: y = x + s + n. 6) Wth unform power allocaton among the M users and ndependent Gaussan encodng d CN, M ), d and d j j) are ndependent of each other. So x and s are mutually ndependent, but n s no longer Gaussan and s not ndependent of x, so we cannot drectly apply the result of drty-paper codng n [29] to derve the capacty of ths channel. As s s stll known at the transmtter, from [3], we know that the achevable rateofthskndofchannel can be formulated n the form of mutual nformaton as shown below: Ĥ) =Iu ; y ) Iu ; s ) = hu ) hu y ) hu )+hu s ) = hu s ) hu y ), 7) where u s an auxlary random varable. Let u = x + as where a s called the nflaton factor, then Ĥ) =hu αs s ) hu αy y ) = hx s ) h α)x αn y ) = hx ) h α)x αn y ) hx ) h α)x αn ) )) hx ) log 2 πe Var α)x αn, 8) where the frst follows from the fact that the entropy s larger than the condtonal entropy, and the second follows from the fact that a Gaussan random varable has the largest dfferental entropy when the mean and varance of a random varable are gven. Snce d CN, M ),wehavevarx )= ĝ 2 M and h x ) = log 2 πe varx )). As E{ } =, E{ α)x αn } = and E{x n } = Then we can get Var α)x αn ) = α) 2 Varx )+α 2 Varn ), 9) Varn )= M E{ H } + = D +. 2) Substtutng Equaton 9 nto Equaton 8, we have Ĥ) log 2 Varx ) α) 2 Varx ) + α 2 Varn ). 2) Varx ) Choosng α = α opt maxmzes the Varx ) +Varn ) rght-hand sde RHS) of the nequalty n Equaton 2, and thus, we get Ĥ) log 2 + Varx ) ) ĝ 2 =log Varn ) 2 + M. 22) D + The above nequalty shows the lower bound on the achevable rate of user under gven Ĥ. In the followng paragraph, we derve closed-form expresson for the lower bound on the achevable ergodc rate under fadng downlnk channel. Snce ĥ,m CN, D ), Ĥ can be decomposed as Ĥ = ϒĤ where the entres of H are..d. CN, ) and ϒ dag{ D,..., D M } s a dagonal matrx. Denote the QR decomposton of H as H = G Q, then Ĥ = ϒ G Q. Therefore, G = ϒ G and ĝ = D g. where g = [ G], From Lemma 2 n [] we know that g 2 χ2m +) 2 where χ2k 2 denotes the central ch-square dstrbuton wth 2k degrees of freedom, whose pdf s fz) =z k- e -z /k -)! Then by takng the means of both sdes of the nequalty n Equaton 22, the achevable ergodc rate of user s lower bounded as follows: where = EĤ{ g 2 D ) Ĥ)} E log 2 + M D + 23) = e β log 2 e M + j= E j β ), D + N β, 24) D ) M and E j x) s the exponental ntegral functon of order j. The closed-form expresson of the expectaton n Equaton 23 follows from the results n [3]. Thus, we have completed the proof. Upper bound on the achevable rate of ZF-DC wth AF n AWGN feedback channels An upper bound of the achevable rate s derved by assumng a gene who can provde the encoders at the BS and the decoders at the users wth some extra nformaton. Ths upper bound s referred to as the geneaded upper-bound.

6 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 6 of 6 Recall Equaton 4 and rewrte t as follows: y =ĝ + ˆq )d + = x + s + n, j< ĝ j d j + m ˆq m d m + v 25) where ˆq s the th column of ˆQ H, x =ĝ + ˆq )d, s = j< ĝjd j,andn = ˆq m m d m + v. Assume there s a gene who knows the values of ˆq and ˆq m m ) and tells these values to the encoder and decoder for user, then wth..d. channel nputs d m CN, M )m =,..., M), n s Gaussan dstrbuted wth zero mean and varance Varn )= m ˆq m 2 /M + and s ndependent of x. Hence the channel for user n Equaton 25 wll be recognzed as a standard drty-paper channel and ts capacty s log 2 + Varx )/Varn )) [29]. Fnally the downlnk achevable ergodc rate can be upper bounded by the gene-aded upper bound as gven n the followng theorem. Theorem 2. If the downlnk channel s..d. Raylegh fadng and the feedback channels are AWGN channels, the achevable ergodc rate of ZF-DC s bounded by a gene-aded upper-bound as follows: ĝ + ˆq 2 E log 2 + M m ˆq m 2 M +, =,2,..., M. 26) It s dffcult to derve a closed-form expresson for the rght-hand sde RHS) n Equaton 26, so we use Monte Carlo smulatons to obtan ths upper bound. We plot the lower and upper bounds on the achevable ergodc sum rates obtaned n Theorems and 2 wth fxed feedback-lnk capacty constrant n Fgure. We set M =4,b fb =andsnr fb =, 5, 2 db. Achevable rate of ZF-DC wth perfect CSIT s also plotted. An mportant observaton from Fgure s that there s a celng effect on the achevable rate of ZF- DC under AF f the feedback-lnk capacty constrant s fxed,.e., the achevable rate s bounded as the downlnk SNR tends to nfnty. Ths can be explaned ntutvely that the power of the nterference caused by mperfect CSIT always scales lnearly wth the sgnal power. A more rgd explanaton s gven n the followng corollary: Corollary. The achevable ergodc rate of ZF-DC wth AF and fxed feedback-lnk capacty s upper bounded for arbtrary downlnk SNR: ) M + log γ log D 2 e, =,2,..., M, 27) Achevable rates bts/s/hz) Achevable rate wth perfect CSIT Lower bound on the achevable rate Upper bound on the achevable rate SNR fb =2dB 5 SNR fb =5dB SNR fb =db / db) Fgure Lower and upper bounds on the achevable ergodc sum rate of ZF-DC wth AF n AWGN feedback channels. where g s the Euler-Mascheron constant [32] and D =. +β fb SNR fb The proof of the corollary s n Appendx. Although ths upper bound s qute loose, t does predct the celng effect on the achevable rate wth fxed feedback-lnk capacty. Achevable downlnk multplexng gan wth AF n AWGN feedback channels From Corollary, t s obvous that the downlnk multplexng gan wth fxed feedback-lnk capacty s zero. In order to mantan a nonzero multplexng gan, the feedback channel qualty should mprove at some rate as the downlnk SNR ncreases, whch s gven n detal n the followng theorem: Theorem 3. For AF and AWGN feedback channels, ) b and b fb SNR fb scales as a a, b > ), then a suffcent and necessary condton for achevng the multplexng gan of M <b <)sthatb = b ;asuffcent and necessary condton for achevng the full multplexng gan of M s that b. Moreover, for b >,the asymptotc rate gap between the achevable rate of ZF- DC wth perfect CSIT and that under AF s zero as the downlnk SNR goes to nfnty. The proof of the theorem s n Appendx 2. Fgure 2 llustrates the conclusons n Theorem 3. We set M =4, b fb =,a =.5 and b =.5, and.5. The curves concde wth the analytcal results n Theorem 3. Note that ncreasng the value of a can further reduce the rate gap between the perfect CSIT case and the AF case.

7 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 7 of 6 Achevable rates bts/s/hz) Achevable rates and multplexng gan wth AF n Raylegh fadng feedback channels In ths subsecton, we wll further consder the effects of Raylegh fadng feedback channels to the achevable rates and multplexng gan wth AF. From Equaton, we notce that D s the functon of the feedback channel h fb. If the feedback channel s a fadng channel, then D wll become a random varable and thus the lower bound we have obtaned n Equaton 5 s also random. So we need to take the mean of the RHS of the nequalty n Equaton 5 wth respect to h fb to get the new lower bound for the fadng feedback channel case. Frst, we ntroduce a lemma to help us derve the lower bound. Lemma. fx) =e x E n x) n ) s a convex and monotoncally decreasng functon. The proof of ths lemma s n Appendx 3. Then we have the followng closed-form lower bound on the downlnk achevable ergodc rate n the Raylegh fadng feedback channels. Theorem 4. If both the downlnk channel and the feedback channels are..d. Raylegh fadng, then the achevable ergodc rate of ZF-DC wth AF and unform power allocaton s lower bounded as: where γ M + e γ log e E j γ ), 28) M M Achevable rate wth perfect CSIT Lower bound on the achevable rate Upper bound on the achevable rate b= j= / db) + + β fb SNR fb b=.5 b=. Fgure 2 Illustraton of the achevable downlnk multplexng gan of ZF-DC wth AF n AWGN feedback channels. M. 29) roof: Takng the mean of the RHS of the nequalty n Equaton 5 wth respect to h fb, we get the lower bound for fadng feedback channel: M + E fb h eβ log e E j β ) j= 3) M + e γ log e E j γ ), j= where γ E fb h {β }. The second n Equaton 3 follows from Lemma and the Jensen nequalty for convex functons. Substtutng Equaton nto Equaton 24, we have the followng expresson for b : β = + β fb SNR fb M h fb 2 + M. 3) Gven that the entres of h fb are..d. CN, ), we have h fb 2 χ2m 2. Then g can be calculated n a closed form: γ = E fb h {β } + = = M M β fb SNR fb M + + β fb SNR fb x + M M. x M e x M )! dx 32) Ths fnshes the proof. The upper bound of the achevable ergodc rate wth fadng feedback channels can also be derved from Equaton 26 as the followng corollary, and smulatons are stll needed to calculate the upper bound: Corollary 2. The achevable ergodc rate of ZF-DC wth AF and Raylegh fadng feedback channels s upper bounded for arbtrary downlnk SNR: log 2 M +)M βfb SNR fb + M ) + γ log 2 e, =,2,..., M, 33) where g s the Euler-Mascheron constant. The proof s smlar to that of Corollary 2 and thus omtted due to the page lmt. From ths corollary, we also have the observaton for the fadng feedback channel that when the downlnk SNR goes to nfnty whle keepng the parameters of the feedback channel constant, there s also a celng effect on the achevable ergodc rate of ZF-DC.

8 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 8 of 6 Achevable rates bts/s/hz) Achevable rate wth perfect CSIT Lower bound on the achevable rate Upper bound on the achevable rate B=6 B=2 B=2 Achevable rates bts/s/hz) Achevable rate wth perfect CSIT Lower bound on the achevable rate Upper bound on the achevable rate AF = β fb DF AF = 2 β fb / db) Fgure 3 Lower and upper bounds on the achevable ergodc sum rate of ZF-DC wth AF n Raylegh fadng feedback channels / db) Fgure 5 Achevable rate comparson between ZF-DC and ZF-BF wth AF n Raylegh fadng feedback channels-ii: fxed / =2dB. Fgure 3 llustrates the lower and upper bounds on the achevable ergodc sum rates of ZF-DC wth AF n Raylegh fadng feedback channels. We set M =4,SNR fb = 5,, 5 db. The curves verfy the analytcal results n Theorem 4 and Corollary 2. Fgures 4 and 5 compare the achevable ergodc sum rates between ZF-DC and ZF-BF schemes, n whch we set M =4,b fb =. In Fgure 4, the achevable rates under fxed SNR fb =5dBandSNR fb = 5 db are compared for ZF-DC and ZF-BF schemes over dfferent downlnk SNR /N, respectvely. Here, the achevable ergodc sum rates of ZF-BF are obtaned by Monte Carlo smulatons Achevable rates bts/s/hz) Achevable rate wth perfect CSIT Lower bound on the achevable rate Upper bound on the achevable rate α = / db) α = α =.5 Fgure 4 Achevable rate comparson between ZF-DC and ZF- BF wth AF n Raylegh fadng feedback channels-i: fxed SNR fb. as n [9]. From Fgure 4 t can be seen that the ZF-DC can outperforms the ZF-BF n terms of achevable rates at the same settngs of feedback channels. Fgure 5 shows the achevable rates comparson under fxed downlnk SNR / = 2 db, from whch the same concluson can be drawn as Fgure 4 shows. From Corollary 2 we can see that the upper bound also tends to a constant. So the multplexng gan s zero, whch s the same as the AWGN feedback channel case. In order to mantan a multplexng gan of M, the SNR of the feedback channel should scale wth the downlnk SNR, as shown n the followng corollary: Corollary 3. For AF and..d. Raylegh fadng feedback ) b channel, let b fb SNR fb scales as a, a, b >, then f b, the multplexng gan of the downlnk wll mantan as M; f b <, the multplexng gan of at least bm can be acheved. Moreover, for b >, the asymptotc rate gap between the achevable rate of ZF-DC wth perfect CSIT and that under AF s zero as the downlnk SNR goes to nfnty. The proof s qute smlar to that of Theorem 3 and thus omtted here for brevty. We also notce that the results are the same as those for AWGN feedback channels, so no more smulaton results are gven here. Achevable rates of ZF-DC under dgtal feedback We now consder dgtal feedback DF), where the downlnk CSI are estmated and quantzed nto several bts usng a vector quantzaton codebook at each user

9 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 9 of 6 and the quantzaton bts are fed back to the BS. The feedback channel s assumed to be capacty-constraned and error-free,.e., as long as the number of feedback bts does not exceed the feedback-lnk capacty n terms of the maxmum feedback bts per fadng block, the feedback transmsson wll be error-free [23]. We also assume perfect CSIR and no feedback delay as n Secton 3. Moreover, the same restrctons are mposed on the transmsson strategy as n Secton 3. Dgtal feedback The downlnk channel vector h of user can be expressed as h = λ h, where λ h s the ampltude of h and h h / h s the drecton of h.underthe assumpton that the entres of h are..d. CN, ), we have λ 2 χ2m 2 and h s unformly dstrbuted on the M dmensonal complex unt sphere [24]. Moreover, l and h are ndependent of each other [24]. The Random Vector Quantzaton RVQ) [24,25] s adopted n our analyss due to ts analytcal tractablty and close performance to the optmal quantzaton. The quantzaton codebook s randomly generated for each quantzaton process, and we analyze performance averaged over all such choces of random codebooks, n addton to averagng over the fadng dstrbuton. At the recever end of user, h s quantzed usng RVQ. Frst, a random vector codebook C = {c,,..., c,n } s generated for user by selectng each of the N vectors ndependently from the unform dstrbuton on the M dmensonal complex unt sphere,.e., the same dstrbuton as h. The codebooks for dfferent users are also ndependently generated to avod the case that multple users quantze ther channel drectons to the same quantzaton vector. The BS s assumed to know the codebooks generated each tme by the users. Then the code vector that has the largest absolute square nner product wth h s pcked up as the quantzaton result, mathematcally formulated as follows: ĥ =argmax c h H 2. 34) c W Then the B =log 2 N quantzaton bts are fed back to the BS. We note that Equaton 34 s actually based on the dstorton measure of the angle between the codevector and the downlnk channel vector, whch s equvalent to 2) n [25] and 5) n [2]. It s obvously dfferent from the dstorton measure of MSE adopted n [27]. We also fnd out that the MSE dstorton measure n [27] s smlar to the dstorton measure Equaton 2) used n AF n our work; therefore, the analyss based on MSE dstorton measure n [27] can be easly ncorporated nto our AF analyss framework. ) Defne ν ĥ h H 2 and θ ĥ hh, then we ntroduce two lemmas that are useful for further dscusson. Lemma 2. [24]: The cumulatve dstrbuton functon of ν s F ν ν) = ν) M ) N, ν [, ]. 35) Lemma 3. [33]: θ s unformly dstrbuted n the nterval -π, π] and ndependent from ν. In the next subsecton, we wll fnd that the nformaton of θ s necessary for phase compensaton at user s recever. Therefore, we need to store the value of θ at user s recever. Notce that the norm nformaton of the channel vectors s not conveyed to the BS. Lower bound on the achevable rate of ZF-DC wth DF Under the assumpton that the feedback channel s error free, the B bts conveyed by each user can be receved by the BS correctly. The BS reconstructs the quantzed channel vector ĥ usng the B bts fed back from user and treats ĥ asthetruechannelvector.thenthebs performs ZF-DC usng the reconstructed channel [ matrx H T ĥt M] ĥt as dd n Secton 3.2. The QR decomposton of H can be wrtten as H = G Q,where G s a lower trangular matrx and Q s a untary matrx. The receved sgnal s modeled as: H y = HQ d + v H = H Q d + v, 36) where dag{λ,..., λ M } s a dagonal matrx, and [ T. H ht... h M] T At each user s recever, a phase compensaton operaton s carred out by multplyng e jθ to the receved sgnal of user, wrtten n a compact form as follows: r = y H = H Q d + v H = H Q d + w, 37) where dag{e jθ,..., e jθ M} s a dagonal matrx, w v has the same statstcs as v. Denote e jθ h ĥ,thenwecanrewrtetna compact form,.e., H = H +, where [ T T M]... T. Equaton 37 can be rewrtten as: H r = H + ) Q d + w H = Gd + Q d + w, 38)

10 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age of 6 From the above equaton we can extract the receved sgnal at user as lsted below: r = λ ĝ d + ĝ j d j + Q Hd + w. 39) j< We frst gve three lemmas useful for dervng the lower bound of the achevable rate of ZF-DC under DF. Lemma 4. λ ĝ 2 χ2m +). 2 Lemma 5. E{ H } =2 E{ ν } ) n whch E{ ν } = N k= ) N ) k [2kM )]!! k [2kM ) + ]!!,4) where N = 2 B, [2k]!! 2 4 2k 2) 2k and [2k +]!! 3 2k ) 2k +). Lemma 6. fx) =e x E n x) n ) s a monotoncally decreasng functon. The proofs of these three lemmas are n Appendces 4-6, respectvely. Then we have the followng theorem on the lower bound of the achevable ergodc rate of ZF-DC under DF. Theorem 5. If the downlnk channel s..d. Raylegh fadng and the feedback channels are error-free, then the achevable ergodc rate of ZF-DC wth DF s lower bounded as: ) M M R DF log 2 e ψm +)+log 2 e E{ H } M log2 e E j M E{ j= H } ), 4) where ψx) s the Euler ps functon [28] and E{ H } s gven n Lemma 5. roof: Snce l s known by the recever of user, the sgnal model n Equaton 39 can be transformed nto: r = r / λ = ĝ d + / ĝ j d j + Q Hd + w λ j< = x + s + n, 42) where x = ĝ d, s = j<ĝ j d j and / n = Q Hd + w λ. Usng the same methodology as n Secton 3.2, we arrve at the followng nequalty for the downlnk achevable rate of user under fxed H and Λ: R DF H, ) hx ) log 2 πe Var α)x αn )), 43) Wth Gaussan nputs and unform power allocaton, d CN, M ), then hx )=log 2 πe ĝ 2 M ). In the dgtal feedback scheme, the channel norm nformaton s not conveyed back to the BS,.e., l s not known at the BS, so we are not able to adjust a accordng to Varx ) and Varn ). We just smply choose a =, then R DF H, ) s lower bounded by: R DF H, ) hx ) log 2 πe Var n )). 44) Snce E{ n } = E{ Q H} E{d} E{w } / λ =, then Var n )=E{n n } = M E h ĥ { H } + / λ 2.45) Substtutng Equaton 45 nto Equaton 44 we fnally get the followng lower bound under fxed H and Λ: λ R DF H, ĝ 2 MN ) log 2 +λ 2 M E h { H ĥ }. 46) Based on the above results, we can derve the lower bound for the achevable ergodc rate n the Raylegh fadng downlnk channel. Takng the mean of both sdes of the nequalty n Equaton 46, we have { } R DF = E R DF H, ) E { log 2 λĝ 2)} ) )} +log 2 E λ, {log MN ĥ 2 +λ 2 E h MN ĥ{h } E { ) )} log 2 λĝ 2)} +log 2 {log Eλ 2 +λ 2 E{ H }, MN MN 47) where the second follows from the Jensen nequalty of the concave functon. From Lemma 4, we can calculate the closed-form expresson for E { log 2 λ ĝ 2)} : E { log 2 λ ĝ 2)} =log 2 e lnx M )! xm e x dx = log 2 e ƔM +)ψm +) ln ) M )! =log 2 e ψm +), 48) where ψx) s the Euler ps functon [28]. Snce λ 2 χ2m 2, the closed-form expresson for the thrd term n Equaton 47 can be calculated as shown below: { )} M E λ log 2 +λ 2 M E{ H } = e E{ H } M log2 e E j M E{ j= H } ), 49) where the closed-form expresson of E{ H } has been obtaned n Lemma 5. Substtutng Equatons 48 and 49 nto Equaton 47, we fnally get the concluson. Remark: From the above theorem and the monotony of e x E n x) shownnlemma6,wecanseethatdecreasng E{ H } wll rase the lower bound on the achevable rate. Now we gve an explanaton on the necessty of the phase compensaton operaton at each recever.

11 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age of 6 In the absence of the phase compensaton, the channel error vector would be h ĥ. Then { )}) E{ H } =2 E R ĥ hh =2 E{ ν cos θ }). 5) From Lemma 3, θ s ndependent from ν and E{cos θ } =, then E{ H } =2 E{ ν } E{cos θ })=2, 5) and thus, the same lower bound remans no matter how many bts are used to quantze h. Therefore, the nformaton of θ and phase compensaton plays an mportant role n the DF scheme, whch s dfferent from the case n [25] where no phase compensaton s needed. Upper bound on the achevable rate of ZF-DC wth DF TheupperboundontheachevablerateofZF-DC wth DF can be obtaned n a smlar way as n Secton 3.3. Recall Equaton 42 and rewrte t as follows: H Q r =ĝ + ˆq )d + = x + s + n, j< ĝ j d j + m / ˆq m d m + w λ 52) where ˆq s the th column of, x =ĝ + ˆq )d, s = j< ĝjd j, and n = m / ˆq m d m + w λ. We also assume there s a gene who knows the values of l, ˆq and ˆq m m ), then followng the methodology n Secton 3.3, we can see that the channel for user s also recognzed as a standard drty-paper channel. Therefore, the downlnk achevable ergodc rate can be upper bounded by the gene-aded upper bound as shown below: Theorem 6. The achevable ergodc rate of ZF-DC wth DF s bounded by a gene-aded upper-bound,.e., λ R DF E {log ĝ + ˆq 2 )} MN m λ2 ˆq m 2 +, =,2,..., M. 53) MN Smulatons are also necessary to calculate the upper bound gven n Theorem 6. The lower and upper bounds on the achevable ergodc sum rate obtaned n Theorems 5 and 6 wth fxed feedback-lnk capacty constrant are plotted n Fgure 6. We set M = 4 and calculate three groups of curves where the number of feedback bts per user,.e., B, s 2, 6 and 2, respectvely. Achevable rate of ZF-DC wth perfect CSIT s also plotted. The curves n Fgure 6 reveal the celng effect on the achevable rate whch s just the same as the AF case. From Theorem 6, we also derve a closed-form upper bound for the achevable rate wth DF as shown below. Corollary 4. The achevable ergodc rate of ZF-DC wth DF and a fxed number of feedback bts per user B s upper bounded for arbtrary downlnk SNR: R DF Fgure 6 Lower and upper bounds on the achevable ergodc sum rate of ZF-DC wth DF n error-free feedback channels. B +log 2 e M +log 2 M ) + log 2 3e), =,2,..., M. 54) The proof s smlar to those of Corollares and 2, and thus omtted due to the page lmt. Although ths upper bound s qute loose, t does also predct the celng effect on the achevable rate wth fxed feedback bts per user. Achevable downlnk multplexng gan wth DF The multplexng gan of the downlnk wth DF and fxed feedback bts per user s zero due to the celng effect. In order to mantan nonzero multplexng gan, the feedback bts per user should scale wth the downlnk SNR. Wth Theorem 5 and Corollary 4, we can derve the followng suffcent and necessary condtons on the scalng to ensure nonzero and full multplexng gan: Theorem 7. For DF and error-free feedback channels, assume that the number of feedback bts per user scales accordng to: B = αm )log 2, α>, 55) then we have the followng conclusons: ) A suffcent and necessary condton for achevng the downlnk multplexng gan of a M <a < ) s that a = a.

12 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 2 of 6 2) A suffcent and necessary condton for achevng the full downlnk multplexng gan of M s that a. 3) If a >, then lm R CSIT N R DF ) =, =,2,..., M. 56) The proof of Theorem 7 s smlar to that of Theorem3andthusomttedduetothepagelmthere. Note that the same concluson has been drawn for ZF-BF n [25]. Fgure 7 llustrates the conclusons n Theorem 7. We set M =4anda =.5,,.5. The curves concde wth the analytcal results n Theorem 7. Comparson of AF and DF In order to compare DF wth AF, we need to relate the number of feedback bts B peruserwthsnr fb and the number of channel uses n the feedback lnk, or equvalently b fb. In ths paper, we make an dealstc assumpton that the AWGN feedback lnk can operate error-free at ts capacty,.e., t can relably transmt log2 + SNR fb ) bts per channel use. Ths assumpton descrbes the maxmum possble number of bts that can ever be conveyed correctly through the AWGN feedback channel. In [2], the authors have ponted out that for far comparson, b fb M feedback channel uses n AF should correspond to b fb M - ) feedback channel uses n DF, snce no channel norm nformaton s fed back n DF and a system usng DF could use one feedback channel symbol to transmt the norm nformaton. Thus, the number of feedback bts per user n the AWGN feedback channel s B = b fb M - ) log 2 + SNR fb ). Nowwecanmakeacomparsonabouttheperformance of AF and DF. Assume for DF, the feedback bts peruserscaleasb =M )log 2,thenfromTheorem 7, we know that the multplexng gan of M can be acheved. Usng the relatonshp between B and SNR fb obtaned above, we can derve the connecton of SNR fb to the downlnk SNR: SNR fb = ) β fb. 57) Then from Theorem 3, we can see that as long as b fb >, only a multplexng gan of less than M can be acheved for AF. Ths means the DF scheme s asymptotcally superor to the AF scheme when b fb >. Fgure 8 compares the achevable rates under AF and DF for b fb =and2.asanalyzedabove,the asymptotc performance of DF s superor to AF when b fb = 2. The concluson s, however, somewhat optmstc snce we assume the AWGN feedback channel can operate error-free at ts capacty. The true performance of DF may degrade when some specfc codng scheme s used. A thorough comparson when usng practcal codng schemes for DF, such as uncoded QAM modulaton dscussed n [2], s for future work. Concluson and future work We have nvestgated the performance of ZF-DC n the multuser MIMO downlnk of a FDD system where the CSIT s obtaned through capacty-constraned feedback channels. Two CSI feedback schemes,.e., the analog α α α Fgure 7 Illustraton of the achevable downlnk multplexng gan of ZF-DC wth DF n error-free feedback channels. = β fb = 2 β fb Fgure 8 Comparson of AF and DF wth b fb = and 2.

13 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 3 of 6 and dgtal feedback schemes, are consdered n our work. Closed-form expressons for lower and upper bounds on the achevable ergodc rates of ZF-DC wth Gaussan nputs and unform power allocaton are derved. Based on the closed-form rate bounds, suffcent and necessary condtons on the feedback channels to ensure nonzero and full downlnk multplexng gan are obtaned. Our prmary results show that for AF n both AWGN and Raylegh fadng feedback channels, t s both suffcent and necessary to scale the average feedback lnk SNR lnearly wth the downlnk SNR n order to acheve full downlnk multplexng gan. Whle for the RVQ-based DF wth angle dstorton measure n an error-free feedback lnk, t s both suffcent and necessary to scale the feedback bts per user as B =M )log 2 where M s the number of transmt antennas and s the average downlnk SNR. We also menton that there are several ssues not consdered n our current work. In ths paper, we have assumed perfect CSI at the users recevers. In a practcal system, however, there are always channel estmaton errors due to fnte number of tranng symbols, whch wll further degrade the performance of ZF- DC. The mpact of the feedback delay of the downlnk CSI on the achevable rates s also not consdered, whch could be sgnfcant when the downlnk channel s fast fadng. For DF scheme, we apply the RVQ for quantzaton of the channel vector n order to make the analyss easer. Generalzaton to arbtrary vector quantzaton codebooks s an nterestng ssue and we expect the same conclusons could be drawn. In the analyss of DF scheme, we made an optmstc assumpton that the AWGN feedback lnk can operate errorfree at ts capacty. Ths assumpton can be removed by consderng practcal feedback transmsson schemes, such as the uncoded QAM modulaton dscussed n [2]. Throughout the paper, we have assumed that the number of users s equal to the number of transmt antennas. We conjecture that when the number of users s larger than that of transmt antennas and we properly desgn the user selecton scheme, the feedback lnk qualty average feedback SNR for AF and the number of feedback bts for DF) per user could be less strngent whle keepng the same performance. Fnally, the analyss of the achevable ergodc rates are carred out wth the restrctons of Gaussan nputs and unform power allocaton. Determnng whether Gaussan nput s optmal and the optmal power allocaton scheme under mperfect CSIT s a challengng problem. Appendx : roof of Corollary From Theorem 2 we have: ĝ + ˆq E {log 2 )} MN 2 + m ˆq m 2 + MN E {log 2 + ĝ + ˆq )} 2 m ˆq m 2 { ĝ 2 + ˆq 2 + m ˆq 2 m + ĝ ˆq H H + ĝ = E log ˆq )} 2 m ˆq 2 m { ĝ ĝ ˆq H H + ĝ E log ˆq )} 2 ˆq m 2 )} = E {log 2 ĝ ĝ ˆq H H + ĝ ˆq E { log 2 ˆq m 2)} log 2 E { ĝ 2} + E { 2} { } + E ĝ ˆq H H + E { ĝ ˆq } ) E { log 2 ˆq m 2)} A ) From the proof of Theorem, we know that E{ ĝ 2 } = D )M +) and E{ 2 } = MDA. 2) Snce Δ s ndependent of Ĥ, Δ s also ndependent of ĝ and ˆq,so E{ĝ ˆq } = E { } E{ĝ ˆq } =. A 3) As for the term E {log 2 ˆq m 2 )}, because the entres of Δ are..d. as CN, D ) and ˆq m =, we have ˆq m CN, D ) and thus ˆq m 2 = D Y where Y χ2 2. Then we have: E{log 2 ˆq m 2 )} =log 2 D +log 2 e lnx e x dx =log 2 D γ log 2 e, A 4) where g s the Euler-Mascheron constant [32]. Substtutng Equatons A-2, A-3 and A-4 nto Equaton A-, we arrve at the concluson. Appendx 2: roof of Theorem 3 Suffcent Condton Denote the RHS of the nequaltes n Theorem and Corollary as R low lm N R low log 2 / ) and R upp lm N respectvely, then we have: log 2 / ) lm N R upp log 2 / ). B ) Let β fb SNR fb = a ) b, a, b >. Then the expresson of b s expanded as: β = D + D ) M = ++β fb SNR fb = M β fb SNR fb ++a a M ) b ) b+ B. 2) Then we can fnd out that β O /) b ) f<b < and b ~ O /) fb. We now ntroduce two results about the exponental ntegral functons n [34]: E n x), x, forn >, B 3) n

14 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 4 of 6 ) n x n E x) = γ ln x, B 4) n!n n= where g s the Euler-Mascheron constant. From Theorem and Equatons B-3 and B-4, the lower bound of R AF wll have the followng asymptotc behavor: lm N R low log 2 / N = ) lm e β log 2 e M + j= E jβ ) log 2 / ) N ) log 2 + o) β = lm log 2 / ) N Therefore, R low { lm N log 2 / ) = b,f< b <,, f b. From Corollary we have: lm N R upp log 2 / N) = log 2 M +) +a lm N = lm N M +) ab M +) +a N N log 2 N N ) b ) ) ) b + + γ log 2 e ) b ) + = b.. B 5) B 6) B 7) Substtute Equatons B-6 and B-7 nto Equaton B- and notce that the downlnk multplexng gan cannot exceed M, then the followng holds: lm N sum log 2 / N = M ) lm N log 2 / N = ) For the case b >, the followng holds: lm β M / = lm { bm, f< b <, M, f b. B 8) ) b ++a a ) b =. B 9) Therefore, the asymptotc rate gap,.e., the lmt of R AF s: R CSIT lm R CSIT lm log 2 log MN 2 = lm log β 2 N N N Necessary Condton R If lm AF / log 2 / ) = b where <b, then R low lm N log 2 / ) b lm R upp β M / =. B ) log 2 / ). B ) ) Let β fb SNR fb = a b, N a, b >.FromEquatonsB-6, B-7 and B-, we have mnb, ) b b. Soforthe case <b <,wehaveb b b and thus b = b ; whle for the case b =, we have b. Appendx 3: roof of Lemma Usng the fact that E n x) = E n x) n )[28], we can calculate the frst-order dervatve as shown below: f x) = e x E n x) E n x)). C ) Usng the defnton of E n x), we can expand the term E n x) -E n - x) as: E n x) E n x) = = <. e xt t n t n+ )dt e xt t n t)dt C 2) And thus f x) < whch ndcates that fx) =e x E n x) s a monotoncally decreasng functon. Further, the second-order dervatve s calculated as follows: f x) =e x E n x) E n x)) + e x E n x)+e n 2 x)) = e x E n x) 2E n x)+e n 2 x)) = e x = e x >. e xt t n 2t n+ + t n+2 )dt e xt t n t ) 2 dt C 3) Snce f x) >, we can conclude that fx) s convex. Appendx 4: roof of Lemma 4 SnceweuseRVQ,ĥ has the same dstrbuton as h,. e., ĥ h.therefore, H H = H,.e.,theentres of H are..d. CN, ). Notethat H = G Q s the QR decomposton of H where G s a lower trangular matrx and Q s a untary matrx. Snce λ ĝ s the th dagonal element of G,thenfrom[]wecanconclude that λ ĝ 2 χ 2 2M +). Appendx 5: roof of Lemma 5 From the defnton of Δ, we have H = ) ) e jθ h ĥ e jθ hh ĥh = h 2 e jθ h ĥ H e jθ ĥ hh + ĥ E 2 ) )) =2 R e jθ ĥ hh,

15 Chen et al. EURASI Journal on Wreless Communcatons and Networkng 2, 2:2 age 5 of 6 where R ) stands for the real part of a complex number. We also have ) R e jθ ĥ hh = R e jθ ν e jθ ) = ν. E 2) Therefore, the followng holds: E{ H } =2 E{ ν }). E 3) Usng Lemma 2, we can derve the closed-form expressons for E{ ν } as follows: E{ ν } = ν dfν ν) = F ν ) F ν ) = 2 Let x =-ν, then ν) M ) N ν ν) M ) N dν = ν F ν ν) 2 ν dν dν. E 4) x M ) N x dx. E 5) - x M - ) N can be expanded as x M ) N = N Nk ) k= x M )k. Moreover, we have the followng ntegral [28]: x m 2m)!! dx =2 for nteger m. E 6) x 2m +)!! Substtutng the above results nto Equaton E-4, we fnally get E{ ν } = N k= ) N ) k [2kM )]!!. E 7) k [2kM ) + ]!! Thus, we have completed the proof. Appendx 6: roof of Lemma 6 Usng the fact that E n x) = E n x) n )[28], we can calculate the frst-order dervatve as shown below: f x) = e x E n x) E n x)). F ) Usng the defnton of E n x), we can expand the term E n x) -E n - x) as: E n x) E n x) = e xt t n t)dt <. F 2) So f x) < whch ndcates that fx) =e x E n x) sa monotoncally decreasng functon. Acknowledgements Ths work was supported by Chna s 863 roject-no. 29AA5, Natonal Natural Scence Foundaton of Chna-No , Chna s Natonal S&T Major roject-no. 28ZX33-4, rogram for Changjang Scholars and Innovatve Research Team n Unversty CSIRT), Natonal Scence and Technology llar rogram No. 28BAH3B9 and Natonal Basc Research rogram of Chna No. 27CB368. Competng nterests The authors declare that they have no competng nterests. Receved: 26 October 2 Accepted: 23 June 2 ublshed: 23 June 2 References. G Care, S Shama Shtz), On the achevable throughput of a multantenna Gaussan broadcast channel. IEEE Trans. Inf. Theory 497), ) 2. S Vshwanath, N Jndal, A Goldsmth, Dualty, achevable rates, and sum-rate capacty of Gaussan MIMO broadcast channels. IEEE Trans Inf Theory 49), ) 3. W Yu, JM Coff, Sum capacty of Gaussan vector broadcast channels. IEEE Trans Inf Theory 59), ) 4. H Wengarten, Y Stenberg, S Shama Shtz), The capacty regon of the Gaussan multple-nput multple-output broadcast channel. IEEE Trans Inf Theory 529), ) 5. A Wesel, YC Eldar, S Shama, Lnear precodng va conc optmzaton for fxed MIMO recevers. IEEE Trans Sgnal rocess. 54), ) 6. M Codreanu, A Toll, M Juntt, M Latva-aho, Jont desgn of tx-rx beamformers n MIMO downlnk channel. IEEE Trans Sgnal rocess. 559), ) 7. C Wndpassnger, RFH Fscher, T Vencel, JB Huber, recodng n multantenna and multuser communcatons. IEEE Trans Wrel Commun. 34), ) 8. BM Hochwald, CB eel, AL Swndlehurst, A vector-perturbaton technque for near-capacty multantenna multuser communcaton. art II: perturbaton. IEEE Trans Commun. 533), ) 9. S Ln, H Su, ractcal vector drty paper codng for MIMO Gaussan broadcast channels. IEEE J Sel Areas Commun. 257), ). L Zheng, D Tse, Dversty and multplexng: a fundamental tradeoff n multple-antenna channels. IEEE Trans Inf Theory. 495), ). SA Jafar, A Goldsmth, Isotropc fadng vector broadcast channels: the scalar upper bound and loss n degrees of freedom. IEEE Trans Inf Theory. 53), ) 2. T Marzetta, B Hochwald, Fast transfer of channel state nformaton n wreless systems. IEEE Trans Sgnal rocess. 544), ) 3. antanda, Duhamel, Achevable rates for the fadng MIMO broadcast channel wth mperfect channel estmaton. In roceedngs of Allerton Conference on Communcaton, Control, and Computng, Sept AF Dana, M Sharf, B Hassb, On the capacty regon of mult-antenna Gaussan broadcast channels wth estmaton error. In ISIT 26, Seattle, USA, 85 85, 59 4 July 26) 5. T Marzetta, How much tranng s requred for multuser MIMO? In 4th Aslomar Conference on Sgnals, Systems and Computers, 26 ACSSC 6), ) 6. Dng, D Love, M Zoltowsk, On the sum rate of mult-antenna broadcast channels wth channel estmaton error. In 39th Aslomar Conference on Sgnals, Systems and Computers, ) 7. J Sh, M Ho, MIMO broadcast channels wth channel estmaton. In roceedngs of IEEE Internatonal Conference on Communcaton ICC), 27, ) 8. D Samardzja, N Mandayam, Impact of plot desgn on achevable date rates n multple antenna multuser TDD systems. IEEE J Sel Areas Commun. 257), ) 9. D Samardzja, N Mandayam, Unquantzed and uncoded channel state nformaton feedback n multple-antenna multuser systems. IEEE Trans Commun. 547), ) 2. G Care, N Jndal, M Kobayash, N Ravndran, Quantzed vs. analog feedback for the MIMO downlnk: a comparson between zero-forcng based achevable rates. In IEEE Internatonal Symposum on Informaton Theory, 27 ISIT 27), June 27