Channel Quantization and Feedback Optimization in Multiuser MIMO ...

Channel Quantization and Feedback Optimization in Multiuser MIMO-OFDM Downlink Systems Matteo Trivellato, Stefano Tomasin, and Nevio Benvenuto Department of Information Engineering University of Padova via Gradenigo 6/B, 35131 Padova, Italy e-mail: {name.surname}@dei.unipd.it

Abstract— We consider a multiuser MIMO-OFDM downlink system with single antenna mobile terminals (MTs) where channel state information at the base station is provided through limited uplink feedback (FB). In order to reduce the FB rate and signal processing complexity, the available bandwidth is divided into resource blocks (RBs) whose number of subcarriers reflects the coherence bandwidth of the channel. This approach is very common in the standardization of 4th generation wireless communication systems and justifies an independent channel quantization per RB. The paper has two main contributions: firstly we show conditions on the coherence bandwidth of the channel and the FB rate per RB that allow for a simpler characterization of the RB channel matrix by a space vector, causing negligible performance loss. This is accomplished after deriving a new performance metric for RB channel quantization that exploits spatial and frequency correlation. As a second contribution we investigate the trade-off between accurate channel knowledge and frequency/multiuser diversity. It is seen that even for a moderate number of MTs in the network, concentrating all the available FB bits in characterizing only one RB provides a significant gain in system throughput over a more classical distributed approach and this result is validated both analytically and by simulations.

I. I NTRODUCTION The demand for higher speed communications in future wireless cellular networks motivated an intensive study of multiple antenna (MIMO) techniques with increasing attention to multiuser (MU) configurations, where a transmitter simultaneously serves multiple users over spatially multiplexed channels [1]. Linear beamforming is a simple transmission scheme that, although suboptimum, has been shown to achieve a large part of the capacity achieving dirty paper coding (DPC) throughput, in case of perfect channel state information (CSI) at the transmitter [2]. Under frequency-division duplexing (FDD), the base station (BS) must rely on uplink feedback (FB) from mobile terminals (MTs) to obtain CSI. In this context an optimization of the FB signalling is essential to limit its impact on the system achievable throughput, motivating a vast research activity especially in single carrier flat fading MIMO transmissions, [3], [4], [5]. Anyway FB optimization might become even more relevant in broadband systems where the channel is dispersive [6]. In this context MU MIMO with orthogonal frequency division multiplexing (OFDM) is considered a good candidate as modulation scheme for the (Long Term Evolution) LTE of 3G cellular systems [7].

In this work we consider a MIMO-OFDM downlink system with beamforming and, differently from most contributions in literature [6], [8], we propose solutions for channel quantization and FB optimization explicitly designed for a multiuser environment. In order to reduce control overhead and signal processing complexity, the available bandwidth is divided into resource blocks (RBs) each comprising adjacent subcarriers and user selection is performed on a RB-basis, [7]. This approach is vastly adopted in the standardization of 4th generation wireless communication systems (e.g. LTE [7]). Moreover the spectral width of a RB is chosen so that almost independent fading realizations are experienced in adjacent RBs, motivating an independent channel quantization per RB. The paper has two main contributions: firstly, we propose a technique for generating a codebook for channel quantization in each RB, based on the LBG algorithm [9], that extends the approach followed in [4] for flat fading MIMO transmissions. We derive a novel performance metric for RB channel quantization that exploits spatial and frequency correlation and is related to the system achievable throughput. Moreover we show conditions on the channel coherence bandwidth and the FB rate per RB that allow for an approximation of the MIMO channel as constant within the RB, yielding to a simpler characterization of the channel matrix as a space vector. This reduces the complexity of i) the quantizer at MT and ii) beamformer design and user selection at BS. As a second contribution we study whether it’s better to i) concentrate all the available FB bits in quantizing only one selected RB for each MT or ii) distribute the available FB bits among more RBs. We show analytically that when the number of MTs is large, the first approach is preferable and validate this finding through simulations even for a practical number of MTs in the system. II. S YSTEM MODEL We consider the downlink of a cellular OFDM system with NC subcarriers. The BS has M transmit antennas and K MTs have one antenna each. The available bandwidth is divided into NR RBs each comprising L adjacent subcarriers, and both FB signalling and user selection are performed on a RB-basis. The channel is assumed frequency selective and the 1×M channel vector of MT k relative to subcarrier of RB n is denoted with

978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.

hk,n (). The L×M channel matrix relative to RB n is denoted with Hk,n = [hk,n (0)T , . . . , hk,n (L − 1)T ]T . We assume that the channel is quasi static, i.e., it can be considered invariant for the duration of one OFDM symbol and vectors hk,n () are assumed to be uncorrelated across MTs. We consider an FDD system where in each OFDM symbol (slot) MTs feed back a partial CSI, which is used by the BS to schedule downlink transmissions and perform beamforming. (1) (|U |) Let Un = {un , . . . , un n } be the set of scheduled MTs receiving data on RB n and xn () the 1 × M transmitted symbol vector on subcarrier = 0, . . . , L − 1, containing the information symbols for MTs in Un . The signal received by MT k ∈ Un on subcarrier can be written as yk,n () = hk,n ()xTn () + nk,n () ,

(1)

where nk,n () is the additive complex Gaussian noise with zero mean and unit variance. The transmit symbol vector xn () is related to information symbols {sj,n ()}, j ∈ Un via linear beamforming, i.e. xn () = j∈Un gj,n ()sj,n (), with {gj,n ()} 1 × M zero-forcing beamforming vectors computed with respect to the quantized channel vectors, [3]. Moreover the transmit signal is subject to the average sum power constraint P across all subcarriers and antennas. We assume equal power allocation across active RBs and the corresponding subcarriers. Moreover the available power on subcarrier is uniformly distributed across the MTs selected on that subcarrier. Assuming that the channel has unitary average gain, the average system signal-to-noise ratio per subcarrier is defined as SN R = P/(LNR ). A. Finite rate feedback (FB) strategies We assume that MT k perfectly estimates its channel ¯ k = [HT , . . . , HT ]T and feeds back frequency response H k,1 k,NR in each slot a quantized version utilized at BS for both user selection and beamformer design. In this paper we study two different FB techniques denoted as distributed feedback (DFB) and best RB feedback (BFB). In DFB a set of D RBs is randomly selected by the BS for each MT, irrespective of channel conditions, with the aim of assuring that CSI relative to at least one MT is available for each RB. Then MT k feeds back two information for each assigned RB: 1) the channel direction ˆ k,n = information (CDI) given by a quantized version H T T T ˆ ˆ [hk,n (0) , . . . , hk,n (L−1) ] of the normalized channel ma˜ k,n (0)T , . . . , h ˜ k,n (L − 1)T ]T , with h ˜ k,n () = ˜ k,n = [h trix H hk,n ()/||hk,n ()|| and 2) an analog channel quality indicator (CQI) related to the estimated achievable throughput on the RB. The total BCDI FB bits for CDI quantization are uniformly distributed among the D RBs leading to bCDI = BCDI /D FB bits per RB. Similarly to DFB with D = 1, BFB still uses all the available FB bits for characterizing only the channel matrix of one RB, but differs from the aforementioned strategy in the choice of the RB. In this case, selection is done by the MT, which feeds back information about the RB providing the highest estimated achievable throughput. Note that with BFB

the MT needs also to feed back the index of the selected RB along with CDI and CQI. Indeed log2 (NR ) bits are reserved as RB index and the remaining BCDI − log2 (NR ) bits are used for channel quantization. In the following, we optimistically assume that by exploiting the time correlation of the MIMO channel, each CQI can be updated with an incremental approach using only one FB bit per slot, similarly to power control schemes in cellular systems [10]. As a consequence, DFB needs at least D FB bits whereas BFB requires only one bit per slot. Adopting this model, if B bits are available, DFB keeps BCQI = D bits to update the CQIs and BCDI = B − D bits for CDI quantization. Differently, BFB uses only one bit for CQI FB and BCDI = B − 1 bits for CDI quantization. B. CDI and CQI feedback models We consider two CDI quantization methods: matrix quantization (MQ) and vector quantization (VQ). In MQ each RB is represented by L frequency components per transmit antenna and we make use of a codebook CM Q of 2bCDI , L × M complex matrices {Ci }. In VQ we approximate the L frequency components per transmit antenna as equal and use a codebook CV Q of 2bCDI , 1 × M complex vectors {ci }. In this paper system performance is evaluated in terms of achievable throughput. We recall that user k cannot have a priori knowledge of the CDIs of other selected users, therefore it cannot know what the beamforming vectors gj,n or the interference will be. Nevertheless, generalizing the approach in [3], if we assume M orthogonal selected users and equal power allocation, the achievable throughput on the generic RB n is given by, ˆ k,n ) = R(Hk,n , H

L−1

log2 (1 + γk,n ()) ,

(2)

=0

where γk,n () =

˜ k,n () h ˆ k,n ()H |2 ρ||hk,n ()||2 |h ˜ k,n () h ˆ k,n ()H |2 ) 1 + ρ||hk,n ()||2 (1 − |h

(3)

is the signal-to-interference plus noise ratio (SINR) of MT k on subcarrier of RB n and ρ = P/(M LNR ) is the power allocated to each user. Interestingly, (3) can be shown to provide a tight approximation of the expected SINR even in case of nearly orthogonal selected MTs. We adopt (2) as quantization metric for MQ, therefore the best codematrix ˆ k,n that maximizes the estimated achievable throughput is H given by, ˆ k,n = arg max R(Hk,n , Ci ) . (4) H Ci ∈CM Q

H ci (0)H , ci (1)H , . . . , ci (L − 1)H with where Ci = ||ci ()|| = 1, = 0, . . . , L − 1, is the generic codematrix. Notice that VQ can be considered a special case with ci () = ci , for = 0, . . . , L − 1. For both MQ and VQ the codebook is designed off-line and known to both BS and MTs a priori. Based on (4) and extending the approach of [3] to OFDM, the equivalent CQI feedback for RB n is given by γk,n = ˆ 2R(Hk,n ,Hk,n )/L − 1.


C. User selection The BS performs user selection and linear beamforming based on the CDIs and CQIs received from the MTs and supports up to M MTs on each RB. We propose a simple generalization of the semi-orthogonal user selection (SUS) scheme [3] that operates separately on each RB. Define the (0) initial user set An containing all the indexes of MTs that fed (1) back CDI and CQI for RB n. The first selected MT is un = 1/2 (j) L−1 ˆ ˆ H(j) ()|2 arg max γk,n . Let d = 1 |hk,n () h , k,n

(0)

k∈An

=0

L

un ,n

after selecting i MTs, the (i + 1)th MT is chosen within the set (j) (i−1) A(i) : dk,n ≤ , j = 1, . . . , i (5) n = k ∈ An as

= arg max γk,n , u(i+1) n

(6)

(i)

k∈An

where is a design parameter setting the maximum correlation allowed between the quantized channel matrices of the selected MTs. It’s easy to verify that for VQ the proposed algorithm simplifies to the SUS algorithm in [3]. III. M ATRIX QUANTIZATION VS VECTOR QUANTIZATION VQ does not characterize the channel variability across the L subcarriers forming a RB and the codebook can be simply designed using the algorithm proposed in [4] for flat MIMO channels. Differently, MQ is performed jointly for the L subcarriers forming a RB, whose correlation strictly depends on the power delay profile of the MIMO channel. For codebook design we still resort to the LBG algorithm [9]. A direct optimization of the quantization codebook based on (2) could be done by numerical methods, however the complexity is very high and there is no guarantee of convergence. It can be shown that a suboptimum performance metric derived from applications of Jensen’s inequality on (2) in both low and high-SNR regimes, is given by R (H, Ci ) =

L−1

˜ |h() ci ()H |2 ,

(7)

=0

where for ease of notation we dropped the indexes k and n of both user and RB. In MQ we adopt (7) as performance metric for codebook design, which leads to optimizing separately each row of Ci in each iteration of the LBG algorithm. A. Analytical comparison of VQ and MQ In this section we compare analytically VQ and MQ using, as metric of comparison, an equivalent time domain representation of (7). Analytic results are then validated using numerical simulation which compare the two quantization schemes in terms of (2) using (4) as quantization rule. In details, applying the discrete inverse Fourier transform to the RB channel matrix H we get p(i) =

L−1 =0

i

h()e+j2π L , i = 0, . . . , L − 1 ,

(8)

where p(i) is a space vector in the time domain. Assuming that vectors {p(i)} are uncorrelated, we can quantize each of them independently with a noteworthy simplification of the ˆ (i) be the quantized version of quantizer. In particular, let p ˜ (i) = p(i)/||p(i)||. With this approach in MQ the available p FB bits bCDI are distributed among the L vectors and VQ becomes a special case where bCDI bits are used to quantize only a space vector. We derive now the optimum bit distribution among vectors {p(i)}. Let’s introduce β = [β1 , . . . , βL ] where βi ≥ 0, i = L−1 0, . . . , L−1, i=0 βi = 1, and say bCDI βi the number of bits assigned to the ith space vector. The quantities βi are chosen to maximize the expectation of the performance metric (7) with respect to both channel and codebook realizations. Using the inverse of (8) and the independence between the norm of a space vector and its direction, after some algebra, the expectation of (7) can be equivalently expressed as L−1

ˆ = z(i) E |˜ p(i)ˆ p(i)|2 + C , E R (H, H)

(9)

i=0

L−1 1 where z(i) = E ||p(i)||4 =0 ||h()|| and C is a constant 4 not involved in the optimization. From (9) we can write the following constrained optimization problem max β

L−1

z(i) E |˜ p(i)ˆ p(i)|2

i=0

βi ≥ 0 , i = 0, . . . , L − 1 ,

L−1 i=0

βi ≤ 1 ,

(10a) (10b)

whose solution provides the best distribution of the available FB bits among the different space vectors. We focus on the special case in which the channel vectors have i.i.d. complex Gaussian entries with zero mean and unit variance and adopt the quantization upper bound (QUB), [11]. Solving the KKT conditions associated to (10) it can be shown that quantizing the entire RB channel matrix as a single vector as in VQ, with βi = 1 and βj = 0, j = i, becomes optimum according to (7) when

E[||p(i)||4 ] , ∀ j = i . (11) bCDI (M − 1) log2 E[||p(j)||4 ] As an example we consider a system with M = 4 transmit antennas, NC = 256 subcarriers and L = 12 adjacent subchannels per RB. We use a frequency selective Rayleigh fading MIMO channel (FSC) with an exponential power delay profile and independent channel taps. VQ and MQ are compared in Fig. 1 for different values of the root mean square delay spread normalized with respect to the sampling period, τrms . In the simulations for MQ, a different codebook has been generated for each value of τrms using the suboptimum performance metric (7). For VQ the same codebook is used for all channels. We observe that VQ is optimal for channels with low τrms , while MQ is preferable for a smaller coherence bandwidth or a higher FB rate. It’s worth underlining that the theoretic bound (11) gives even a pessimistic estimation of the optimality of VQ. Moreover, due to high complexity of MQ with high FB


A. Asymptotic analysis of DFB and BFB

50 theoretic bound (12) simulated bound VQ outperforms MQ MQ outperforms VQ

45 40 35

bCDI

30 25 20

VQ optimal 15 10 5 0

0

5

10

15

normalized τ

rms

Fig. 1.

VQ vs MQ in different channel conditions. M = 4 and FSC model

rate, the validity of (11) could be verified only for bCDI ≤ 11. IV. DFB VS BFB In this section we compare DFB with BFB generalizing the analysis in [3] to multiuser MIMO-OFDM. We use an ideal i.i.d. MIMO channel model (IC) where the frequency response is constant within a RB and independent across different RBs and resort to the QUB approximation for channel quantization. Under the IC the user selection algorithm (6) simplifies to the SUS algorithm while the quantization rule (4) becomes equivalent to the minimum chordal distance [3]. Moreover the CQI for MT k in all subcarriers of RB n ˆ k,n () = h ˆ k,n , simplifies to γk,n () = γk,n computed with h = 0, . . . , L − 1. We claim as in [3] that even if SINR for MT k after SUS is unknown at the BS, γk,n provides a close approximation for small values of in (5). To find an expression of the achievable throughput, we need the distribution of γk,n for a selected MT which depends on both the FB scheme and the user selection algorithm. We recall that according to the SUS algorithm, the (i + 1)th selected (i) MT on RB n has the highest γk,n among |An | MTs with independent channels and the same average SNR. Moreover in DFB the statistics of γk,n is different from BFB due to the maximization performed across NR i.i.d RB channels. Let γ¯i:U,n be the ith largest order statistics among U i.i.d random variables {γk,n }. The user selection rule (6) can be seen as the selection of the (i + 1)th largest order statistics in a set ¯n(i) = |A(i) (n)| + i in DFB and N ¯n(i) = ¯n(i) elements (N with N (i) NR |A (n)| + i in BFB) all having the same statistics. An approximated expression for the achievable throughput per RB is given by M . (12) log2 1 + γ¯i:N¯ (i−1) ,n E[R] E L i=1

n

In this section we derive an approximation of (12) in case (i) of many MTs K. First we note that |An | is in general a random variable depending on the selection of the RB channels (0) to be fed back by each MT. Focusing on An , for DFB (0) MT k belongs to An only if RB n has been selected for transmission. Since each RB has the same probability of being selected for FB, the probability of feeding back CSI on RB n is D/NR and applying the law of large numbers, when K is large, we approximate the cardinality of the initial user set (0) as |An | (KD)/NR . Differently in BFB, MT k belongs to (0) An only if γk,n is the maximum among the NR i.i.d. RB channels. A further application of the law of large numbers (0) yields |An | K/NR . Let αi be the probability that a MT (i) (i) belongs to An . An approximation of |An | can be derived (0) from |An | by applying the law of large numbers on K and we finally get

KD KD ¯n(i) − i αi + i = αi + O(1) , (13) N NR NR ¯n(i) (K − iNR )αi + i = Kαi + O(1) , (14) N for DFB and BFB respectively. From [3, Theorem 1] and (13), (14) an approximation of (12) in case of many MTs is E[R]

L

2bCDI U αi−1 , (15) log2 1 + ρ log ρM −1 i=1

M

where U = (KD)/NR for DFB and U = K for BFB. bCDI The logarithmic term Δ = log 2 ρMU−1αi−1 in (15) can be interpreted as the SNR variation, which includes the effects of both quantization error and frequency, multiuser diversity. It’s easy to see that (15) is maximized for DFB when D = 1 and this behaviour does not change if more bits are used for CQI updating. Interestingly for a given SNR variation Δ that assures a constant gap from ZF beamforming with perfect CSIT, B and K should scale with P and NR for DFB and BFB as B + log2 (KD) = (M − 1) log2 P + log2 NR + c , (16a) D B + log2 K = (M − 1) log2 P + log2 NR + c , (16b) where c depends on Δ. It is worth noticing that DFB with D = 1 gives the same scaling law as BFB. Since in the latter a portion log2 (NR ) of the available FB bits is used to index the selected RB, this suggests that the same performance of DFB with D = 1 can be achieved exploiting frequency diversity and using a smaller codebook, hence requiring less memory. Finally, DFB with D = 1 requires almost D times FB bits to achieve the same throughput of BFB. It is important to observe that (15)-(16) are valid only in a large user regime as K → ∞. For finite K, NR and B, if P is too large, the system enters the interference-limited regime. Following similar arguments applied in [3], an approximated


16 14 achievable throughput [bit/s/Hz]

expression for the average sum rate per RB becomes M L i=1 log2 αi−1 LM (bCDI + log2 U ) + . (17) E[R] M −1 M −1 Again DFB with D = 1 and BFB achieve the highest achievable throughput with BFB requiring less memory. Finally for finite K and NR , if either B is too large or P is too small, the system falls in the high resolution or noise-limited region and the achievable throughput may be approximated as M E[R] L i=1 log2 (1 + ρ log2 U αi−1 ). In this case, the best choice for DFB becomes D = NR , because higher multiuser diversity per RB becomes more important than quantization accuracy. Still, BFB achieves the highest sum rate, as DFB with D = NR , because the loss in multiuser diversity is compensated by the gain in frequency diversity.

12 10 8 6 4

IC FSC BFB DFB, D=1 DFB, D=2

2 0 −5

0

5

10 15 SNR [dB]

20

25

30

V. S IMULATION R ESULTS In this section we present numerical results to validate the asymptotic analysis of Section IV-A. We assume a BS equipped with M = 4 antennas and = 0.35 is the correlation parameter in (5). The OFDM system has NC = 256 subcarriers and NR = 8 RBs with L = 12 subcarriers each. The channel is assumed spatially uncorrelated and we adopt two different channel models: i) the IC introduced in Section IV and ii) the FSC with τrms = 2.5 introduced in Section III-A. In the IC we use QUB approximation while in the FSC we perform VQ using an LBG-based codebook and using (4) as quantization rule. In Fig. 2 we compare DFB and BFB in terms of the achievable throughput as a function of the average SN R. The total amount of FB bits per MT is B = 12 and K = 200 MTs are present in the system. We observe how BFB and DFB with D = 1 have very close performance in the high SNR region while BFB is preferable at low-SNR thanks to frequency diversity exploitation as predicted in Section IV-A. Eventually, for increasing SN R the system becomes interference limited as observed in (17). Moreover both BFB and DFB with D = 1 outperform DFB with D > 1 when a sufficient number of users is available in the system, thanks to a better policy in channel quantization and feedback signalling. Interestingly the mutual relationships between the different FB strategies are similar in both IC and FSC. Only a small degradation in the achievable throughput is observed in FSC, mainly due to frequency variability of the channel within a RB. We underline that DFB with D = 1 and BFB have shown to be preferable even for practical values of K > NR . Moreover the validity of (16a) and (16b) could be verified for both IC and FSC for practical feedback rates. VI. C ONCLUSIONS The paper considers the problem of channel quantization and FB optimization in multiuser MIMO-OFDM downlink systems. For RB channel matrix quantization we derived a new performance metric for generating the codebook, that exploits spatial and frequency correlation of the MIMO channel. Moreover we showed that quantizing the RB channel matrix with a

Fig. 2. DFB vs BFB in terms of achievable throughput vs. average SNR. M = 4, K = 200 and B = 12. Two spatially uncorrelated channels: i) IC and ii) FSC with τrms = 2.5.

single space vector becomes optimum for practical values of the FB rate and in common channel environments. The asymptotic analysis derived for a large number of active MTs revealed that exploiting all the available FB bits to quantize only the channel relative to one RB provides a significant gain in achievable throughput over a more classic distributed FB approach and simulations validated these considerations even for a moderate number of MTs in the network. R EFERENCES [1] D. Gesbert, M. Kountouris, R. Heath, C.-B. Chae, and T. Salzer, “From single user to multiuser communications: Shifting the MIMO paradigm,” IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 36–46, Oct. 2008. [2] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE Journ. Sel. Areas in Commun., vol. 24, no. 3, pp. 528 – 541, Mar. 2006. [3] T. Yoo, N. Jindal and A. Goldsmith, “Multi-antenna downlink channels with limited feedback and user selection,” IEEE Journ. Sel. Areas in Commun., vol. 25, no. 7, pp. 1478–1491, Sep 2007. [4] N. Benvenuto, E. Conte, S. Tomasin and M. Trivellato, “Joint low-rate feedback and channel quantization for the MIMO broadcast channel,” in Proc. IEEE Africon’07, Windhoek, Namibia, Sept. 2007. [5] D. J. Love, R. W. Heath Jr., and Thomas Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Info. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003. [6] S. Zhou, B. Li, and P. Willett, “Recursive and trellis-based feedback reduction for MIMO-OFDM with transmit beamforming,” in GLOBECOM ’05, Nov. 2005. [7] H. Ekstrom, A. Furuskar, J. Karlsson, M. Meyer, and S. Parkvall, “Technical solutions for the 3G long-term evolution,” IEEE Commun. Mag., pp. 38–45, Mar 2006. [8] J. Choi and J. R. W. Heath, “Interpolation based transmit beamforming for MIMO-OFDM with limited feedback,” IEEE Trans. Signal Process., vol. 53, pp. 4125–4135, Aug. 2005. [9] N. Benvenuto and G. Cherubini, Algorithms for Communications Systems and their Applications, Wiley, 2002. [10] 3GPP, “3rd generation partnership project; technical specification group radio access network; physical layer procedures (FDD),” Tech. Rep., 3GPP TS 25.214, 2002-03. [11] S. Zhou, Z. Wang, and G. B. Giannakis, “Quantifying the power loss when transmit beamforming relies on finite-rate feedback,” IEEE Tran. Wireless Commun., vol. 4, no. 4, pp. 1948–1957, July 2005.
