On the achievable rates of multiple antenna

Chen et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:21 http://jwcn.eurasipjournals.com/content/2011/1/21

RESEARCH

Open Access

On the achievable rates of multiple antenna broadcast channels with feedback-link capacity constraint Xiang Chen*, Wei Miao, Yunzhou Li, Shidong Zhou and Jing Wang

Abstract In this paper, we study a MIMO fading broadcast channel where each receiver has perfect channel state information while the channel state information at the transmitter is acquired by explicit channel feedback from each receiver through capacity-constrained feedback links. Two feedback schemes are considered, i.e., the analog and digital feedback. We analyze the achievable ergodic rates of zero-forcing dirty-paper coding (ZF-DPC), which is a nonlinear precoding scheme inherently superior to linear ZF beamforming. Closed-form lower and upper bounds on the achievable ergodic rates of ZF-DPC with Gaussian inputs and uniform power allocation are derived. Based on the closed-form rate bounds, sufficient and necessary conditions on the feedback channels to ensure nonzero and full downlink multiplexing gain are obtained. Specifically, for analog feedback in both AWGN and Rayleigh fading feedback channels, it is sufficient and necessary to scale the average feedback link SNR linearly with the downlink SNR in order to achieve the full multiplexing gain. While for the random vector quantization-based digital feedback with angle distortion measure in an error-free feedback link, it is sufficient and necessary to scale the number of feedback bits B per user as B = (M − 1)log2

P P where M is the number of transmit antennas and is the average downlink SNR. N0 N0

Keywords: Feedback-link capacity constraint, limited feedback, multiple antenna broadcast channel, multiplexing gain, multiuser MIMO, zero-forcing dirty-paper coding (ZF-DPC)

Introduction The multiple antenna broadcast channels, also called multiple-input multiple-output (MIMO) downlink channels, have attracted great research interest for a number of years because of their spectral efficiency improvement and potential for commercial application in wireless systems. Initial research in this field has mainly focused on the information-theoretic aspect including capacity and downlink-uplink duality [1-4] and transmit precoding schemes [5-9]. These results are based on a common assumption that the transmitter in the downlink has access to perfect channel state information (CSI). It is well known that the multiplexing gain of a point-to-point MIMO channel is the minimum of the number of transmit and receive antennas even without * Correspondence: [email protected] Research Institute of Information Technology, Tsinghua National Laboratory for Information Science and Technology(TNList), Tsinghua University, Beijing, China

CSIT [10]. On the other hand, in a MIMO downlink with single-antenna receivers and i.i.d. channel fading statistics, in the case of no CSIT, user multiplexing is generally not possible and the multiplexing gain is reduced to unity [11]. As a result, the role of the CSI at the transmitter (CSIT) is much more critical in MIMO downlink channels than that in point-to-point MIMO channels. The acquisition of the CSI at the transmitter is an interesting and important issue. For time-division duplex (TDD) systems, we usually assume that the channel reciprocity between the downlink and uplink can be exploited and the transmitter in the downlink utilizes the pilot symbols transmitted in the uplink to estimate the downlink channel [12]. The impact of the channel estimation error and pilot design on the performance of the MIMO downlink in TDD systems has been studied in [13-18]. For frequency-division duplex (FDD) systems, no channel reciprocity can be exploited,

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and thus it is necessary to introduce feedback links to convey the CSI acquired at the receivers in the downlink back to the transmitter. There are generally two kinds of CSI feedback schemes applied for MIMO downlink channels in the literature. The first scheme is called the unquantized and uncoded CSI feedback or analog feedback (AF) in short, where each user estimates its downlink channel coefficients and transmits them explicitly on the feedback link using unquantized quadrature-amplitude modulation [12,19-21]. The performance of the downlink linear zero-forcing beamforming (ZF-BF) scheme with AF was evaluated through simulations in [19], and analytical results were given later in [20] and [21]. The second feedback scheme is called the vector quantized CSI feedback or digital feedback (DF) in short, where each user quantizes its downlink channel coefficients using some predetermined quantization codebooks and feeds back the bits representing the quantization index [20-27]. The MIMO broadcast channel with DF has been considered in [20,21,24-27]. In [24], a linear ZFBF-based multiple-input single-output (MISO) system is firstly considered with random vector quantization (RVQ) limited feedback link, in which the closed-form expressions for expected SNR, outage probability, and bit error probability were derived. Then the vector quantization scheme based on the distortion measure of the angle between the codevector and the downlink channel vector was adopted in [20,21,25], and a closedform expression of the lower and upper bound on the achievable rate of ZF-BF was derived. The results there also showed that the number of feedback bits per user must increase linearly with the logarithm of the downlink SNR to maintain the full multiplexing gain. Further, the authors in [26] pointed out that in the scenario where the number of users is larger than that of the transmit antennas, with simple user selection, having more users reduces feedback load per user for a target performance. However, the aforementioned literatures [20,21,25] both focus on the linear ZF-BF scheme, which is not asymptotically optimal compared with nonlinear schemes, such as zero-forcing dirty-paper coding (ZFDPC) [1]. So, it is necessary to investigate the limiting performance for MIMO downlink channels with limited digital feedback link. In [27], the authors analyzed both the linear ZF-BF and nonlinear zero-forcing dirty-paper coding (ZF-DPC) and derived loose upper bounds of the achievable rates with limited feedback. But different from the distortion measure of the angle in [20,21,25], another vector quantization approach based on the distortion measure of mean-square error (MSE) between the codevector and the downlink channel vector was adopted in [27]. Simultaneously, the exact lower bounds

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of the achievable rates with limited feedback for ZF-DPC are not given in [27]. In this paper, we consider both analog and digital feedback schemes and study the achievable rates of a MIMO broadcast channel with these two feedback schemes, respectively. Different from [21,25] focusing on the ZF-BF, the ZF-DPC is analyzed in our work which is inherently superior to the ZF-BF due to its nonlinear interference precancelation characteristic and is asymptotically optimal [1] as [27]. Specially, for DF, we adopt the vector quantization distortion measure of the angle between the codevector and the downlink channel vector, and perform RVQ [20,21,25] for analytical convenience. Our main contributions and key findings in this paper are as follows: • A comprehensive analysis of the achievable rates of ZF-DPC with either analog or digital feedback is presented, and closed-form lower and upper bounds on the achievable rates are derived. For fixed feedback-link capacity constraint, the downlink achievable rates of ZF-DPC are bounded as the downlink SNR tends to infinity, which indicates that the downlink multiplexing gain with fixed feedback-link capacity constraint is zero. • In order to achieve full downlink multiplexing gain, it is sufficient and necessary to scale the average feedback link SNR linearly with the downlink SNR for AF in both AWGN and Rayleigh fading feedback channels. While for DF in an error-free feedback link, it is sufficient and necessary to scale P the feedback bits per user as B = (M − 1)log2 N0 where M is the number of transmit antennas and P is the average downlink SNR. N0 We note that although the ZF-DPC with DF has been considered in [27], our work also differs from it in several aspects. First, a different distortion measure for channel vector quantization is applied in our work compared to that in [27] as stated earlier. Actually, for RVQ-based DF, the angle distortion measure in [20,21,25] seems more reasonable than the MSE distortion measure in [27], which will be discussed in this paper. Second, a more thorough analysis about the downlink achievable rates (including upper and lower bounds) and multiplexing gain is presented in this paper than that in [27] (only upper bounds are given), covering both AF and DF. The remainder of this paper is organized as follows. We give a brief introduction to the ZF-DPC with perfect CSIT in Section 2. Comprehensive analysis of achievable rates and multiplexing gain for both AF and DF are


presented in Sections 3 and 4, respectively. A rough comparison of AF and DF is also given in Section 4. Finally, conclusions and discussions for future work are given in Section 5. Throughout the paper, the symbols (·)T, (·)* and (·)H represent matrix transposition, complex conjugate and Hermitian, respectively. [·]m, n denotes the element in the mth row and the nth column of a matrix. ||·|| represents the Euclidean norm of a vector. |·| and ∠(·) denote the magnitude and the phase angle of a complex number, respectively. E{·} represents expectation operator. Var(·) is the variance of a random variable. CN (a, b) denotes a circularly symmetric complex Gaussian random variable with mean of a and variance of b.

Zero-forcing dirty-paper coding with perfect CSIT Consider a multiple antenna broadcast channel composed of one base station (BS) with M transmit antennas and K users each with a single receive antenna. Assuming the channel is at and i.i.d. block fading, the received signal at user i in a given block is yi = hi x + vi ,

(1)

where hi Î ℂ1 × M is the complex channel gain vector between the BS and user i, x Î ℂ M × 1 is the transmitted signal with a total transmit power constraint P, i. e., E{xH x} = P, and v i is the complex white Gaussian noise with variance N0. For analytical convenience, we assume spatially independent Rayleigh fading channels between the BS and the users, i.e., the entries of hi are i. i.d. CN (0, 1), and hi, i = 1, ..., K are mutually independent. Under the assumption of i.i.d. block fading, hi is constant in the duration of one block and independent from block to block. By stacking the received signals of all the users into y = [y 1 ... y K ] T , the signal model is compactly expressed as y = Hx + v,

(2)

where H = [hT1 hT2 · · · hTK ]T and v = [v1 v2 ... vk]T. In this paper, we focus on the case K = M. If K < M, there will be a loss of multiplexing gain. The case K > M will introduce multi-user diversity gain and we will leave it for future work. We first give a brief introduction of ZF-DPC under perfect CSIT in this section. In the ZF-DPC scheme, the BS performs a QR-type decomposition to the overall channel matrix H denoted as H = GQ, where G is an M × M lower triangular matrix and Q is an M × M unitary matrix. We let x = QHd and the components of d are generated by successive dirty-paper encoding with Gaussian codebooks [1],

Page 3 of 16

then the resulting signal model with the precoded transmit signal can be written as: y = Gd + v.

(3)

From Equation 3 the received signal at user i is given by yi = gii di + gij dj + vi , (4) j 0. Then the expression

of bi is expanded as: βi =

P N0 Di

+1

P (1 − Di ) MN 0

=

P N0

+ 1 + βfb SNRfb P MN0 βfb SNRfb

=

P N0

b0 + 1 + a NP0 b0 +1 (B. a P · M N0

2)

Then we can find out that βi ∼ O((N0 /P)b0 ) if 0 1, n−1

(B

3)


E1 (x) = −γ − ln x −

∞ (−1)n xn

n!n

n=1

,

(B

4)

where g is the Euler-Mascheroni constant. From Theorem 1 and Equations B-3 and B-4, the lower bound of RiAF will have the following asymptotic behavior: lim

P N0 →∞

eβi log2 e

Rlow i

" = lim P log2 (P N0 ) →∞

M−i+1 j=1

Ej (βi )

" log2 (P N0 )

N0

= lim

1 βi

+ o(1) " . log2 (P N0 )

log2

P N0 →∞

(B

5)

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Appendix 3: Proof of Lemma 1 Using the fact that E n (x) = −En−1 (x) (n ≥ 0)[28], we can calculate the first-order derivative as shown below: f (x) = ex (En (x) − En−1 (x)).

Rlow i " lim = P →∞ log2 (P N0 )

N0

∞

upp

N0

(B

∞

6)

=

log2 NP0

(B

b0 (M − i + 1) · ab0 NP0 = b0 . b0 P P +i−1 N0 →∞ (M − i + 1) 1 + a N 0

7)

Substitute Equations B-6 and B-7 into Equation B-1 and notice that the downlink multiplexing gain cannot exceed M, then the following holds: RAF RAF sum i " " = M · lim = P →∞ log2 (P N0 ) →∞ log2 (P N0 )

lim

N0

b0 M, if 0 < b0 < 1, M, if b0 ≥ 1.

(B

8)

βi " = lim P →∞ MN0 P →∞

lim

P N0

N0

lim RCSIT − RAF i i ≤ lim log2 P N0 →∞

P N0

+1+a b a NP0

b

RiAF log (P/N0 ) 2 P/N0 →∞

lim

= 1. (B

9)

N0

(B

10)

(B

11)

= b0 where 0 0. From Equations B-6,

B-7 and B-11, we have min(b, 1) ≤ b 0 ≤ b. So for the case 0 0, we can conclude that f(x) is convex. ■

Appendix 4: Proof of Lemma 4 Since we use RVQ, ˆhi has the same distribution as ¯hi, i. e., ˆhi ∼ ¯hi. Therefore, H ∼ H ¯ = H, i.e., the entries of H are i.i.d. CN (0, 1). Note that H = GQ is the QR decomposition of H where G is a lower triangu lar matrix and Q is a unitary matrix. Since λi gˆ ii is the ith diagonal element of G, then from [1] we can con2 clude that | λi gˆ ii |2 ∼ χ2(M−i+1) . Appendix 5: Proof of Lemma 5 From the definition of Δi, we have jθi ¯ ¯H− h ˆ ˆH i H e−jθi h i i = e hi − hi i H H (E2 = || ¯hi ||2 − ejθi ¯hi ˆhi − e−jθi ˆhi ¯hi + || ˆhi || H = 2 1 − e−jθi ˆhi ¯hi ,

1)


where ℜ(·) stands for the real part of a complex number. We also have √

√ H

e−jθi ˆhi ¯hi = e−jθi νi ejθi = νi . (E 2) Therefore, the following holds: √ E{i H i } = 2(1 − E{ νi }).

(E

3)

Using Lemma 2, we can derive the closed-form √ expressions for E{ νi } as follows: √ E{ νi } =

1

√ ν dFνi (ν)

0

=

√

√ 1 0 · Fνi (0) − Fνi (ν) √ dν (E 2 ν 1

1 · Fνi (1) −

4)

0

1 =1− 2

1 0

(1 − (1 − ν)M−1 ) √ ν

N

dν.

Let x = 1 - ν, then 1 0

(1 − (1 − ν)M−1 ) √ ν

1

N

dν = 0

(1 − xM−1 ) √ 1−x

N

dx.

(E

5)

1 N ) can be expanded as

M−1 k N N M−1 N . Moreover, we have (1 − x ) = k=0 k −x the following integral [28]:

(1

1 0

-

√

xM

xm (2m)!! for integer m ≥ 0. dx = 2 · (2m + 1)!! 1−x

(E

6)

Substituting the above results into Equation E-4, we finally get N √ [2k(M − 1)]!! N E{ νi } = 1 − . (−1)k · k [2k(M − 1) + 1]!!

(E

7)

k=0

Thus, we have completed the proof. ■

Appendix 6: Proof of Lemma 6 Using the fact that E n (x) = −En−1 (x) (n ≥ 0)[28], we can calculate the first-order derivative as shown below: f (x) = ex (En (x) − En−1 (x)).

(F

1)

Using the definition of En(x), we can expand the term En(x) - En - 1(x) as: ∞ En (x) − En−1 (x) =

e−xt t−n (1 − t)dt < 0.

(F

2)

1

So f’(x) < 0 which indicates that f(x) = e x E n (x) is a monotonically decreasing function.

Page 15 of 16

Acknowledgements This work was supported by China’s 863 Project-No. 2009AA011501, National Natural Science Foundation of China-No. 60832008, China’s National S&T Major Project-No. 2008ZX03003-004, Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT), National Science and Technology Pillar Program No. 2008BAH30B09 and National Basic Research Program of China No. 2007CB310608. Competing interests The authors declare that they have no competing interests. Received: 26 October 2010 Accepted: 23 June 2011 Published: 23 June 2011 References 1. G Caire, S Shamai (Shitz), On the achievable throughput of a multiantenna Gaussian broadcast channel. IEEE Trans. Inf. Theory 49(7), 1691–1706 (2003) 2. S Vishwanath, N Jindal, A Goldsmith, Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels. IEEE Trans Inf Theory 49(10), 2658–1668 (2003) 3. W Yu, JM Cioffi, Sum capacity of Gaussian vector broadcast channels. IEEE Trans Inf Theory 50(9), 1875–1892 (2004) 4. H Weingarten, Y Steinberg, S Shamai (Shitz), The capacity region of the Gaussian multiple-input multiple-output broadcast channel. IEEE Trans Inf Theory 52(9), 3936–3964 (2006) 5. A Wiesel, YC Eldar, S Shamai, Linear precoding via conic optimization for fixed MIMO receivers. IEEE Trans Signal Process. 54(1), 161–176 (2006) 6. M Codreanu, A Tolli, M Juntti, M Latva-aho, Joint design of tx-rx beamformers in MIMO downlink channel. IEEE Trans Signal Process. 55(9), 4639–4655 (2007) 7. C Windpassinger, RFH Fischer, T Vencel, JB Huber, Precoding in multiantenna and multiuser communications. IEEE Trans Wirel Commun. 3(4), 1305–1316 (2004) 8. BM Hochwald, CB Peel, AL Swindlehurst, A vector-perturbation technique for near-capacity multiantenna multiuser communication. Part II: perturbation. IEEE Trans Commun. 53(3), 537–544 (2005) 9. S Lin, H Su, Practical vector dirty paper coding for MIMO Gaussian broadcast channels. IEEE J Sel Areas Commun. 25(7), 1345–1357 (2007) 10. L Zheng, D Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels. IEEE Trans Inf Theory. 49(5), 1073–1096 (2003) 11. SA Jafar, A Goldsmith, Isotropic fading vector broadcast channels: the scalar upper bound and loss in degrees of freedom. IEEE Trans Inf Theory. 51(3), 848–857 (2005) 12. T Marzetta, B Hochwald, Fast transfer of channel state information in wireless systems. IEEE Trans Signal Process. 54(4), 1268–1278 (2006) 13. P Piantanida, P Duhamel, Achievable rates for the fading MIMO broadcast channel with imperfect channel estimation. In Proceedings of Allerton Conference on Communication, Control, and Computing, 27–29 Sept. 2006 14. AF Dana, M Sharif, B Hassibi, On the capacity region of multi-antenna Gaussian broadcast channels with estimation error. In ISIT 2006, Seattle, USA, 1851–185, 59–14 July (2006) 15. T Marzetta, How much training is required for multiuser MIMO? In 40th Asilomar Conference on Signals, Systems and Computers, 2006 (ACSSC’06), 359–363 (2006) 16. P Ding, D Love, M Zoltowski, On the sum rate of multi-antenna broadcast channels with channel estimation error. In 39th Asilomar Conference on Signals, Systems and Computers, 1524–1528 (2005) 17. J Shi, M Ho, MIMO broadcast channels with channel estimation. In Proceedings of IEEE International Conference on Communication (ICC), 2007, 1042–1047 (2007) 18. D Samardzija, N Mandayam, Impact of pilot design on achievable date rates in multiple antenna multiuser TDD systems. IEEE J Sel Areas Commun. 25(7), 1370–1379 (2007) 19. D Samardzija, N Mandayam, Unquantized and uncoded channel state information feedback in multiple-antenna multiuser systems. IEEE Trans Commun. 54(7), 1335–1344 (2006) 20. G Caire, N Jindal, M Kobayashi, N Ravindran, Quantized vs. analog feedback for the MIMO downlink: a comparison between zero-forcing based achievable rates. In IEEE International Symposium on Information Theory, 2007 (ISIT 2007), 2046–2050 June 2007


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