Capacity of Beamforming with Limited Training and Feedback

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1 Capacity of Beamforming with Limited Training and Feedback Wiroonsak Santipach and Michael L. Honig Department of Electrical Engineering and Compter Science Northwestern University Evanston, IL USA Abstract We examine the capacity of beamforming over a Mlti-Inpt/Single-Otpt block Rayleigh fading channel with finite training for channel estimation and limited feedback. A fixed-length packet is assmed, which is spanned by training symbols, feedback bits, and the data symbols. The training symbols are sed to obtain a Minimm Mean Sqared Error (MMSE) estimate of the channel vector. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing i.i.d. random vectors, and relays the corresponding bits back to the transmitter. We derive bonds on the capacity and show that for a large nmber of transmit antennas, the optimal and, which maximize the bonds, are approximately eqal and both increase as. We conclde that with limited training and feedback, the optimal nmber of antennas to activate also increases as. I. INTRODUCTION The capacity of a mlti-antenna system with independent Rayleigh fading and perfect channel knowledge at the transmitter and receiver increases with nmber of antennas [1], []. In practice, the channel estimate at the receiver will not be perfect, and frthermore, this estimate mst be qantized before it is relayed back to the transmitter. This has motivated recent work on the performance of feedback schemes with imperfect channel knowledge [3] [5], and the design and performance of limited feedback schemes for Mlti-Inpt/Mlti-Otpt (MIMO) and Mlti-Inpt/Single-Otpt (MISO) channels [6] [14]. All of the preceding work on limited feedback assmes perfect channel knowledge at the receiver. Here we consider the performance of beamforming for a MISO channel with both an imperfect channel estimate at the receiver and limited feedback. We consider an i.i.d. block Rayleigh fading channel in which the channel parameters are stationary within each block, and are independent from block to block. The block size is assmed be constant, and the transmitted codewords span many blocks, so that the maximm achievable rate is the ergodic capacity. Each coherence block contains training symbols and data symbols. Frthermore, we assme that after transmission of the training symbols, the transmitter waits for the receiver to relay bits over a feedback channel, which specify a particlar beamforming vector. This delay, This work was spported by the U.S. Army Research Office nder grant DAAD and the National Science Fondation nder grant CCR in addition to the training symbols, is conted as part of the packet overhead. We assme that the receiver comptes a Minimm Mean Sqared Error (MMSE) estimate of the channel, based on the training symbols, and ses the noisy channel estimate to choose a transmit beamforming vector. A Random ector Qantization (RQ) scheme is assmed [15] in which the beamformer is selected from a codebook consisting of random vectors, which are independent and isotropically distribted, and known a priori at the transmitter and receiver. The associated codebook index is relayed sing bits via a noiseless feedback channel to the transmitter. The capacity of this scheme with perfect channel estimation is analyzed in [9], [1]. It is shown in [9], [1] that the RQ codebook is optimal (i.e., maximizes the capacity) in the large system limit in which nmber of transmit antennas and tend to infinity with fixed ratio! "#. RQ has been observed to give excellent performance for systems with small [16]. Frthermore, for the MISO channel considered, the performance averaged over the random codebooks can be explicitly compted [14]. The capacity with MMSE channel estimates at the receiver (with or withot limited feedback) is nknown. We derive pper and lower bonds on the capacity with RQ and limited feedback, which are fnctions of the nmber of training symbols and feedback bits. Given a fixed block size, or packet length $, we then optimize the capacity bonds over and. Namely, small leads to a poor channel estimate, which decreases capacity, as large leads to an accrate channel estimate, bt leaves few symbols in the packet for transmitting the message. This tradeoff has been stdied in [17], [18] for MIMO channels withot feedback. Here there is also an optimal amont of feedback, which increases with the training interval. That is, more feedback is needed to qantize more accrate channel estimates. As the packet length $&%(' with fixed show that the optimal $)*$+,, we -. " and /01,, which maximize the bonds on capacity, both tend to zero at the rate Conseqently, increases as :465879, and we observe that the associated capacity can be achieved by activating only :46587; antennas. Eqivalently, for this pilot-based scheme with limited feedback, the optimal nmber of (active) transmit antennas increases as $+46587$.

2 II. SYSTEM MODEL We consider a point-to-point MISO i.i.d. block fading channel with # transmit antennas. We assme a rich scattering environment in which the channel gains across transmit antennas are independent and Rayleigh distribted. The < th received symbol of a particlar block is given by =?> <A@; > <A@KJML > <A@ for ONP<9NQ (1) B is an #RS channel vector whose elements are independent, complex Gassian random variables with zero mean and nit variance, F is an TRO nit-norm beamforming vector, H is the transmitted symbol with nit variance, and L is additive white Gassian noise with variance U W. In prior work [9], [1], we have analyzed the channel capacity with perfect channel knowledge at the receiver, bt with limited channel knowledge at the transmitter. Specifically, a qantized beamforming vector is relayed from the receiver to the transmitter, given by FYX [Z]\^7`_aZcb dfe"g]hi > jjlkm B D F?n Bqp o () kr&ccu W, and st F3v wxwxwxvif zy is the qantization codebook, which is known at both the transmitter and receiver a priori. The (ncoded) index corresponding to the best beamforming vector is relayed to the transmitter via an errorfree feedback link. The capacity depends on the beamforming codebook s and. As {%/', the F X that maximizes the capacity is the normalized channel vector B } B. We have shown in [9], [1] that RQ, in which the codebook vectors are independent and isotropically distribted, is optimal (i.e. maximizes capacity) in the large system limit in which > ' with fixed normalized feedback -1,. The reslting capacity was shown to grow as > Althogh, strictly speaking, RQ is sboptimal for a finite-size system, nmerical reslts show that it gives excellent performance [16]. In addition to limited channel information at the transmitter, here we also accont for channel estimation error at the receiver. Letting B be the estimated channel vector, we have B B JMƒ (3) ƒ is the error vector whose elements are i.i.d. with zero mean and variance U. Here we assme that the receiver comptes the MMSE estimate of B. As a reslt, B and ƒ are independent and B has zero mean and covariance > + The receiver then selects F9ˆ B is the actal channel, i.e., F9ˆ X tz\g7`_!zb d e g]h X, assming that > jjlkmš B D F n o oo B (4) The qality of the channel estimate depends on the nmber of training symbols, and so does the capacity. In what follows, we assme that the forward and feedback links are time-division mltiplexed, and each block consists of training symbols, feedback bits, and data symbols. Given that the size of each block is $ symbols, we have the constraint $MtlJMŒŽJ (5) Œ is a conversion factor, which relates bits to symbols. Determining the ergodic capacity of RQ with channel estimation appears to be intractable, so instead we derive pper and lower bonds, which are fnctions of,, and. We wold like to optimize both bonds over v v, sbject to (5). III. CAPACITY BOUNDS The ergodic capacity with channel estimation and qantized beamforming is the maximm mtal information between the received and transmitted symbols, and is given by t _azb > = H]mŠ B`v^F9ˆ š (6) œž is the probability density fnction (pdf) for H and B and F ˆ the expectation is over X. Conditioning on the actal channel vector, instead of the estimate, gives the pper bond NŸ r _azcb > = HmC B`v^F Xˆ š [ 4 5]7 > jjmk >ŠF š (7) N4 5]7 > +JMk >ŠF XDˆ (8) we se the fact that the maximizing pdf is Gassian, and apply Jensen s ineqality (8). Sbstitting (3) into the expectation in (8), and simplifying gives Since we have > F Dˆ >ŠF XDˆ štu JM > F XDˆ š w (9) B, and >ŠF XDˆ B } } B x@a are independent [10], [14], X št } B š r Tš > Tš (10).(_aZcb " }n, zy n >ŠF nd B } With RQ the n s are i.i.d. with pdf given in [8]. The pdf for and associated mean can be explicitly compted [14]. The mean is given by r Tš q l {ªY v # (11) «ŸY the beta fnction >Š v LK@;t 1 A±² > W ³ ² and L Ÿµ. We can bond Tš as follows. Lemma 1: For ) Qµ and #; [, š N + M ²¹ J jj > º Ÿ3@I ²¹»JŸ ²¹ «¼«½ Ÿ š + M ²¹ for (1) (13) º is Eler s nmber. The proof is based on the ineqality derived in [19]. We note that Tš«%/ M ²¹ as j%¾'. Sbstitting (9)-(1) into (8) gives an pper bond on capacity. To derive a lower bond on capacity, we sbstitte (3) into (1) and obtain =?> <A@; > F Dˆ > <A@KJ > F Dˆ w (14) X X > <À@«JlL > Á Â"à <À@ Ä Å Æ6Ç È

3 N B N Å Þ y Ø Ù Å ù ï ³ ñ ð è è È v v Œ ê Since ƒ and are independent, it can be shown that É > <A@^H > <A@ š:êµ. It is shown in [17], [0] that replacing É > <A@ with a zero-mean Gassian random variable minimizes the mtal information > = Hm B`v^F9ˆ and therefore gives a lower bond on the capacity with channel estimation and qantized beamforming. The lower bond is maximized when H > <A@ has a Gassian pdf, i.e., Ÿ _azcb _1Ë Ì xí > = HmC B`v^F X ˆ > št ÏÎ Ð:;J D F X U Ñ¹Ò (15) œ Å and U Å denote the pdf and variance for É, respectively. We derive the following lower bond on by applying the ineqality in [1]. Lemma : U *Ó ª jj > BEDGF9ˆ U qô ª q UžÕ ] šg 4 5]7 ª jj U Å Õ denotes the variance of. > BEDGF9ˆ š (16) Exact evalation of U Õ appears to be intractable; however, we are able to derive the pper bond U Õ jj ]Ú ] Tš N Ö Ø Ù ¼ ½ ² jj ]Ú IÛ ÛÝÜ ¼ ½ ² tá (17) ½ÀßTà l jj Ú ¼«½ Ø Ù ¼ ½ ² is the gamma fnction. We note that á ¼:½ %0µ as %0' Ø. To obtain a lower bond on capacity, we sbstitte U Å U JPU W, (13), and (16)-(17) into (15). The capacity bonds are smmarized as follows. Theorem 1: The capacity with channel estimation variance satisfies U and normalized feedback ; + ; for ) Qµ and #; [ (18) > q åá ¼ 4 5]7 ª jjmk q åu > jjmktu q l v (19) + t46587 ª jjmktu JMk > RæªK` l ²¹ J jj > º Pf@I ²¹»JŸ ²¹ K¼ ½ w (0) : Ÿ The gap between the two bonds tends to zero as U or k tend to zero. With fixed and U both bonds (and the capacity) grow as ç > 4 5]7 > #À@^@ as E%-'. I. OPTIMIZED TRAINING AND FEEDBACK LENGTH A. Channel Estimation Error We first evalate the channel estimation error in terms of the training length and feedback. We assme that the transmitter transmits training symbols H"è > f@ v âxâ âv H"è > é@, and that the training symbol H"è > <À@ modlates the corresponding beamforming vector F è > <A@. The vector of received samples from (1) is given by ê Së è;ì D B Jlí (1) ërèî Ë Z7 H"è > <A@z is a -RÊ matrix, ì«èî F è > f@ â âxâ^f è > é@ š, and í0/ L > f@ â âxâ L > é@ š è. The {R# linear MMSE channel estimation filter is given by ê ê Z]\^79_!Ë Ì B 0ò ï D š () > ì è D ì«è JMU W ž@ ² ë è;ì è D (3) î ï trace DGó ï c, ó D š ë è ì è D ì è ë è D J~U W is the received covariance and the MSE U matrix. Note that the matrix of beamforming vectors dring training, ì«è, is known to the transmitter and receiver, and can be chosen a priori. It is shown in [17] that the set of (nitnorm) beamforming vectors, which achieve the Welch bond with eqality, minimizes the Mean Sqared Error (MSE). We therefore have that [] ì:è9ì è D - if æ Ÿ v (4) ì è D ì«è»s if ænÿ w (5) Applying (3)-(5), we obtain the variance of the estimation error U õô B. Asymptotic Behavior q IÛö è IÛö ßTà æ S æ S (6) We now stdy the behavior of the optimal v and, and the capacity as ø%('. With transmitted symbols in an $ -symbol packet the effective capacity ù > ú $;@ {[ " and $MS$+,. The associated bonds are > : $j@ + and ù > : $;@ ;. From Theorem 1 and (6), we can write ù and ù as fnctions of v v û and optimize, i.e., for the lower bond we wish to v è«ü _azcb ü ý ù (7) sbject to JMŒú SJþ ÿ $ w (8) v denote the optimal vales of,, and, Let respectively, and let ù denote the maximized lower bond on capacity. Similarly, maximizing the pper bond gives the optimal parameters v v and the corresponding bond ù. These soltions can be easily compted nmerically, and also allow s to characterize the asymptotic behavior of the actal capacity. Theorem : Let v v c étz]\^79_azb èkü ü ý ù sbject to (8). As %0', 4 5]7 > A@;% $ (9) > #I@9% $ (30) q Æ ¼ ½ % $ (31)

4 5.5 5 Capacity bonds; B/ = 1; σ = 0.15; ρ = 5 db w L/ D o l /L B o l /L T o l /L Bits / Channel se Lower bond eq.(16) (SIM) Lower bond (Thm. 1) Upper bond eq.(8) (SIM) Upper bond (Thm. 1) Fig. 1. The capacity bonds in Theorem 1 (bits/channel se) verss nmber of transmit antennas. 0.1 Fig ! "# $ &% verss nmber of transmit antennas 'E. and the capacity satisfies ù > l4 5]7 > 4 5]7 (3) ` > +Jlk?@+NaN (33) and [46587 > $j > jjlk?@ 4 5]7 > ]@^@ > Œk?@ l. According to the theorem, as becomes large, to maximize the achievable rate the fraction of $ devoted to training and feedback tends to zero, in which case the rate increases as 4 5]7 > M4 5]7 > 4 5]7 Recall that the achievable rate with RQ and perfect channel estimation grows as > kt#à@. Hence the loss of 4 5]7 > > A@I@ is de to imperfect channel estimation. Theorem also implies that Œ!3Ž%/, i.e., the fraction of the packet devoted to feedback is asymptotically the same as that for training. We observe that the preceding analysis applies if the beamforming vectors dring training are chosen to be nit vectors. Namely, the matrix ì«è can be taken to be diagonal, which corresponds to transmitting the seqence of training symbols over the transmit antennas sccessively one at a time. Hence the fact that the optimal increases as :4 5]7; implies that only :4 5]7j antennas are activated. Eqivalently, we conclde that as the packet size $ increases, the optimal nmber of transmit antennas shold increase as $+ :4 5]79$.. NUMERICAL RESULTS In Fig. 1, we compare the analytical bonds in Theorem 1 with the tighter bonds in (7) and (15). The tighter bonds, which are analytically intractable, are evalated by Monte Carlo simlation and shown as s and R s in the figre. The plots show that the bonds in Theorem 1 are close to (7) and (15) even for small. Since RQ reqires an exhastive search, and the nmber of entries in the codebook grows exponentially with the nmber of antennas, simlation reslts are not shown for + f. As expected, both the pper and lower bonds grow at the same rate as # increases. Fig. shows the set of optimal vales : $ v $ v ( : $q, which maximizes ù, verss $ ) and Œ. As with normalized block length expected from Theorem, the optimal training and feedback lengths decrease to zero. The associated rate with this set of parameters is shown in Fig. 3 with a solid line. The dots correspond to simlation reslts with the same set of parameters as in Fig.. The nmerical reslts for the bond in (15) nearly match the analytical lower bond ù, even for #+*. We also compare this performance with optimized parameters to that with {Ï, *, and 0-,, which may be a reasonable heristic choice of parameters. The rate loss at. µ is abot fµ.. Both rates are sbstantially less than the rate with perfect channel information at the transmitter and receiver, which is displayed by the dashed line. The dash-dot crve is the capacity with perfect channel estimation, is taken to be the optimized vale corresponding to the solid line. Here we see a sbstantial gain relative to the solid line, since with perfect channel knowledge the receiver does not reqire training overhead. Fig. 4 shows the capacity lower bond verss total overhead > JOŒ 1@^ : $ with Œ. The capacity is zero when OJæ ~Sµ, since the estimate is ncorrelated with the channel, and when J $, since 0µ. The solid line corresponds to optimized parameters with $æ0/, 0/, ŒŸ, and kû / db. Different crves correspond to different ratios between and. With eqal amonts of training and feedback, the rate is essentially eqal to that with optimized parameters. The peak is achieved when > ŸJ- 1@^ : $Qæµ w *1/. The performance degrades when deviates significantly from. Also shown are the simlation reslts for (15) when [t. The analytical bond is qite close to the bond in (15) for > øj~ 1@^ $Ÿ Qµ w /. I. CONCLUSIONS We have presented bonds on the capacity of a MISO block Rayleigh fading channel with beamforming, assming

5 Achievable Rates (bits / channel se) Tx and Rx know channel L/ Rx knows channel; finite B o l Lower bond (SIM) w/ { T o l, Bo } l Lower bond w/ { T o l, Bo } l Lower bond w/ { T/ =, B/ = } Fig. 3. Achievable rate verss nmber of transmit antennas 'E with different assmptions abot channel knowledge at the receiver and transmitter Capacity lower bond (bits / channel se) Fig Capacity lower bond; = 5; L/ B o l /, To l / T = B T = B (SIM) T = 3*B T = 0.5*B (T + µb)/l 87 Lower bond on capacity verss normalized training and feedback. limited training and feedback. For a large nmber of transmit antennas, we have characterized the optimal amont of training and feedback as a fraction of the packet dration, assming linear MMSE estimation of the channel, and an RQ codebook for qantizing the beamforming vector. Or reslts show that when optimized, the fraction of the packet devoted to training is asymptotically the same as the fraction of the packet devoted to feedback. Frthermore, the optimal training length increases as #I :46587 > A@, which can be interpreted as the optimal nmber of transmit antennas to activate. Althogh the pilot-based scheme considered is practical, it is most likely sboptimal. Namely, in the absence of feedback sch a pilot-based scheme is strictly sboptimal, althogh it is nearly optimal at high SNRs [17]. With feedback the capacity of the block fading MISO channel considered (i.e., no channel knowledge at the receiver and transmitter) is nknown. Extensions of the model presented here, which we intend to stdy, inclde allocating different powers for the training and data portions, and beamforming for a MIMO block fading channel. REFERENCES [1] İ. E. Telatar, Capacity of mlti-antenna Gassian channels, Eropean Trans. on Telecommn., vol. 10, pp , Nov [] G. J. Foschini and M. J. 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