Multi-Resolution Codebook Design for Two-Stage Precoding in FDD Massive MIMO Networks

Transcription

1 Multi-Resolution Codeboo Design for Two-Stage Precoding in FDD Massive MIMO Networs Deli Qiao, Haifeng Qian, and Geoffrey Ye Li School of Information Science and Technology, East China Normal University, Shanghai, China School of Computer Science and Software Engineering, East China Normal University, Shanghai, China School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia Abstract 1 In this paper, we develop a new codeboo design method for multi-resolution codeboo based two-stage precoding scheme in massive MIMO networs in frequency division duplex (FDD) mode. A narrow-beam generated by each codeword is intended to provide coverage over certain beam range with strict constraints on the average and the variation in the radiated power. Each narrow-beam or codeword is designed based on a new performance metric, the ratio of the expected average power within the coverage beam range to that outside the beam range, i.e., beam-to-leaage ratio (BLR). An optimization problem for narrow-beam design is formulated to maximize the BLR. The codeboo is generated via solving the optimization problem. From computer evaluation, our novel codeboo design based transmission scheme can improve the overall networ throughput by 54% compared with the DFT-based one and 33% compared with covariance based method when SNR = 1 db. I. INTRODUCTION The research and development on 5G have attracted the interest of both the academia and the industry recently. It is expected that 5G could address the massive capacity and massive connectivity challenges brought by the exponentially growing mobile traffic and machine type applications [1]. Massive MIMO systems [2] are equipped with a large number of transmit antennas at base stations and serve a large number of users at the same frequency band simultaneously. It has been identified as a promising technique to address the challenges in 5G networs [3], [4]. To serve multiple users in a massive MIMO networ, precoding, which exploits downlin channel state information at the base station, is necessary. Most of existing wors on massive MIMO have presumed the time-division duplex (TDD) systems [5]-[7], where the uplin and the downlin wor in the same band and downlin channels can be obtained through the channel reciprocity. However, many existing cellular networs wor in the frequency-division duplex (FDD) mode, where the uplin and the downlin wor in different bands such that channel reciprocity no longer exists and downlin channel estimation turns out to be a challenging issue. Therefore, transmission schemes applicable to FDD massive MIMO systems still need further investigations [8]. Several wors have proposed transmission schemes potentially 1 This wor has been supported in part by the National Natural Science Foundation of China ( , , ) and the Shanghai Sailing Program (16YF1426). suitable to massive MIMO in FDD networs. The two-stage transmission scheme in [9], joint spatial division and multiplexing (JSDM), taes advantage of the channel covariance matrix to effectively reduce the dimension of the channel, which as a result can reduce the cost for channel estimation in FDD networs. It has been shown in [9] that a DFT submatrix is approximately optimal as the number of antennas goes to infinity. Also, different spatial domain based transmission schemes have been proposed. For instance, angle-of-arrival (AOA), which is derived through the uplin channel, based beamforming method has been proposed in [1] for FDD massive MIMO. A multi-resolution codeboo based adaptive beamforming scheme has been proposed in [11] for millimeter wave systems, where the codewords generate beams with designed spatial beamwidth, and later a new codeboo based on phase-shifted DFT has been designed to minimize the variance of the beam gain within the desired beam range [12]. The training overhead of these schemes generally scales with the number of users. In this paper, we apply the multi-resolution codeboo based transmission scheme to the FDD massive MIMO system. We integrate the codeboo with the two-stage precoding strategy. Note that DFT matrix has been shown to have leaage power, i.e., interference, in the spatial domain [13]. Instead of the DFT-based codeboo, we develop a new method for designing the codeboo that intends to maximize the beam-to-leaage ratio (BLR) of each codeword, defined as the ratio between the average radiated power of the beam within the intended coverage and that outside the range. Note that the codewords are designed to provide almost constant beam gain within the beam range. We also evaluate the proposed codeboo in beam pattern and achievable sum rate in the multi-user massive MIMO systems. The organization of this paper is as follows. Section II introduces the multi-resolution codeboo and the two-stage precoding scheme. In Section III, the codeboo design will be discussed in detail. In Section IV, the numerical results will be provided. Finally, Section V concludes this paper. II. MULTI-RESOLUTION CODEBOOK BASED TWO-STAGE PRECODING In this section, we consider a base station (BS) with uniform linear array (ULA). We first introduce the structure of the

2 Fig. 1. Illustration of the Beam pattern of the codeboo. multi-resolution codeboo, and then the downlin training and data transmission procedures for implementing the codeboo integrated two-stage precoding method in FDD massive MI- MO networs. Consider a BS with M transmit antennas placed in a ULA. The steering vector towards the spatial frequency θ = sin(ϕ) ( 1, 1) with ϕ ( π/2, π/2) representing the angle of departure (AOD) has the form [19] a(θ) = [1 a 2 (θ) a M (θ)] T, (1) where [ ] T denotes the transpose of a matrix, and a m (θ) represents phase shift of the signal at the mth antenna in relative to the first antenna with a 1 (θ) = 1. a m (θ) depends (m 1) j2π on the antenna structure. For ULA, a m (θ) = e λ θ, where is the antenna spacing and λ is the wavelength of the carrier. Let v = [v,..., v ] T C M 1 denote the beamforming vector. Then, the corresponding beam pattern generated is given by f(θ) = v H a(θ)a H (θ)v, (2) where [ ] H denotes the Hermitian transpose of a matrix. The multi-resolution codeboo is designed to have N levels, with each codeword defined { by its intended} beam range. The n-th level codeboo C n c (n) 1,..., c(n) V n intends to cover the cell uniformly, e.g., the spatial range Θ = ( 1, 1). Denote C n as the cardinality of C n. The multi-resolution codeboo is designed such that C 1 < < C N, implying narrower beams at higher levels of the codeboo with spatial beamwidth ϑ n = 2 C n. Let Θ(n) denote the spatial interval covered by c (n). Then, without loss of generality, we assume Θ (n) = ( 1 + 2( 1), C n C n ), = 1,..., C n. (3) Ideally, the codeword c (n) should provide an approximately constant beamforming gain over Θ (n).let C 1 = J and C n+1 = J C n with integer J 2, i.e., J codewords in the (n+1)-th level codeboo provide the same coverage as a single codeword in the n-th level, and the number of codewords in the (n + 1)-th level codeboo will be C n+1 = J n+1. Fig. 1 showns an example for J = 2. With the above multi-level codeboo, training can be achieved via differential sounding the N levels of codeboo following a divide-and-conquer manner. In general, at the n- th training phase, the receiver chooses and feeds bac the index with the largest received signal power among the J codewords decided in (n 1)-th training phase. Repeating the above training phases until the signal power is no longer increased, at most JN training symbols are required in total. If the transmitter serves K, K J N, single antenna users simultaneously, then the total training overhead is KJN, which is proportional to K. In this way, the training overhead can be significantly reduced since it scales with the number of users instead of the antenna number. In [9], a two-stage transmission scheme with dimension reduced channel state information has been proposed. Suppose that the channel covariance matrix R = E{h H h }, where h C 1 M denotes the channel between the BS and user, is available at the transmitter, the first-stage precoding is given by 2 V1 = V(:, 1 : Q), (4) where V comes from the singular value decomposition of R = UDV H, V(:, 1 : Q) are the first Q K columns of V, and Q is the dimension of the equivalent channel. Then, the transmitter sends training symbols to estimate the equivalent channel state information of the users, i.e., h = h V 1, which are fed bac by the users. Note that for the multi-resolution codeboos, the users can feedbac the effective channels together with the codewords, i.e. h P with the codewords P = [p 1,..., p K ] decided in the training phase. With the instantaneous equivalent channel state information H = [ h H 1,..., h H K ]H, the transmitter performs the secondstage precoding with zero-forcing (ZF), that is, V 2 = H = H H ( H H H ) 1. (5) Then, the received signal at user is given by y = h V 1 V 2 x + w, (6) where x denotes the input signals intended for the K users. We assume equal power allocation to the users. Denote the total signal-to-noise ratio at the transmitter as SNR. Then, the received signal-to-interference-and-noise ratio (SINR) of user is given by SNR h K V 1 v 2, 2 SINR = 1 + SNR K K j=1,j h V 1 v 2,j 2 (7) where v 2, C Q 1 denotes the second-stage precoding vector of user. The ergodic sum rate of the system can be obtained as K R sum = E{log 2 (1 + SINR )}, bps/hz. (8) III. =1 NARROW-BEAM DESIGN In this section, we first discuss the DFT-based codeboo briefly, and then present our method for designing the narrowbeam codeboo in details. 2 We assume non-parametric channel models in this paper, so the covariance matrices of the users are the same.

3 A. DFT-based In [12], a codeboo using DFT beams has been proposed to minimize the variance of the beam pattern within the intended beam range. Specifically, the codeword c (n) covering Θ (n) is given by c (n) = 1 N n N n p=( 1)N n+1 a(θ p )e jω np where N n = M/ V n, θ p = 1 + 2p 1 M, p = 1, 2,..., M, and ω n [ π/m, π(1 1/M)] is decided by (9) min var (f(θ s)) (1) ω n [ π/m,π(1 1/M)] where var( ) denotes the variance of a vector, and f(θ s ) denotes the beam pattern at uniformly quantized spatial frequencies θ s Θ s. The radiation power in the intended beam range may fluctuate significantly with this design method, and there is beam leaage which may interfere other users in multiuser transmission. B. Narrow-Beam In this part, we tae into account the potential interference caused by the beam, and propose a method for generating a codeboo with the strongest beam-to-leaage ratio (BLR) defined as the ratio of the expected average power within the coverage beam range to that outside the beam range. Note that with the codeboo discussed in Section II, we can see that the radiation power for a user with a codeword v covering θ Θ s is given by v H a(θ)a H (θ)v. We now that the average radiated power for the users lying in the spatial interval, Θ s, as SP = 1 ϑ s θ Θ s v H a(θ)a(θ) H vdθ, (11) where ϑ s denotes the beamwidth of Θ s. At the same time, the average power leaed by v covering Θ s can be expressed as LP = 1 2 ϑ s θ Θ\Θ s v H a(θ)a(θ) H vdθ. (12) Similar to the discussions of signal-to-leaage-and-noise ratio (SLNR) [14], we can formulate an optimization problem that maximizes the BLR ϖ as follows: max ϖ = SP (13a) v LP s.t. t v H a H (θ)a(θ)v 1 ξ/1 t, θ Θ s, (13b) v H v = 1. (13c) where ξ indicates the strength of the ripples within the beam range in db and t is a constant that satisfies v H v = 1. Revisiting (2), we can rewrite the beam pattern as f(θ) = = i= = m= () v i e j2π λ iθ v e j2π λ θ ν m α m(θ) = α H ν, (14) where ν = [ν (),..., ν,..., ν ] T, α(θ) = [α () (θ),..., 1,..., α (θ)] T, and ν m = m i= v i v i+m, (15) ν m = ν m, (16) α m (θ) = e j2π λ mθ. (17) With the above definition, we can rewrite (11) and (12) as SP = γ H ν, LP = β H ν, (18) where γ = 1 ϑ s θ Θ s α(θ)dθ, and β = 1 2 ϑ s θ Θ\Θ s α(θ)dθ. Similar to [17, eq. (17)], we can convert the problem in (13) to the following linear programming problem: where (µ, τ ) = arg max (µ,τ) F ϵ γ H µ (19) F ξ = {(µ, τ) : τ α H (θ)µ τ1 ξ/1, θ Θ s ; α H (θ)µ, θ Θ \ Θ s ; τ > ; β H µ = 1}, (2) with µ = (β H ν) 1 ν and τ = t(β H ν) 1. The problem in (19) can be approximately solved via selecting a set of values 1 θ 1, θ 2, θ L, 1, and replacing F ξ with F ξ = {(µ, τ) : τ α H (θ i, )µ τ1 ξ/1, θ i, Θ s ; α H (θ i, )µ, θ i, Θ \ Θ s ; τ > ; β H µ = 1}. (21) Then, the linear programming problem with constraint F ξ can be solved efficiently in practice, e.g., the LINPROG command in Matlab. From ν, we can obtain v similar to the discussions in [15], [16]. Specifically, expanding (15), we have v v = ν (), (22) v vm 2 + v 1v = ν (M 2), (23). v v 1 + v 1v v M 2v = ν 1, (24) v v 2 = ν. (25) Then, the solutions to the above equation set can form the following equation g(x) = (v + v 1x + + v x ) ( v + v1x vx ()), (26) = ν () x () + ν (M 2) x (M 2) + + ν + ν 1 x + + ν x. (27) From the structure of g(x) in (26), if x 1, x 2,..., x are solutions to g(x) =, then 1 x, x,..., 2 x are too. From the solution set of g(x) =, we can form at most 2 solutions of the problem in (19) by ϕ(x) = (x α m ) = v + v 1 x + + v x, (28) m=1 1 where α m = x m or x. m

4 ( o ) 4 db Sum rate (bps/hz) JSDM JSDM NB JSDM DFT Sum rate (bps/hz) JSDM JSDM NB JSDM DFT SNR (db) SNR (db) 1 ( 9 o ) 1 (9 o ) (a) Correlated channels. (b) IID channels. Fig. 2. Comparison of the Beam pattern. M = 32. Solid and dot-dashed lines represent the beam pattern of the 4th-level of the proposed codeboo and the DFT-based codeboo, respectively. Theorem 1. Given a codeword c 1 providing coverage over spatial interval Θ 1 = ( ϑ/2, ϑ/2) ( 1, 1) with beamwidth ϑ, the codeword c 2 providing coverage over spatial interval Θ 2 = ( ϑ/2 + σ, ϑ/2 + σ) ( 1, 1) is given by c 2 = c 1 a (σ) (29) where denotes the element-wise multiplication, and a (σ) is the conjugate of a(σ). Note that in practice, we can generate a codeword c (n) providing coverage centered at o with beamwidth ϑ n, i.e., Θ s = ( ϑ n /2, ϑ n /2). Then, we can generate all other codewords c (n) in the same level of the codeboo with coverage specified by (3) through rotating the codeword c (n) by spatial frequency σ = C n. This method is generic and can apply to other spatial domain based codeboos as well, e.g., the DFT based codeboo in [12]. IV. NUMERICAL RESULTS Consider a ULA antenna array with M = 32 antennas. For the codeboo construction, we assume ξ = 3 db, J = 2, N = 4. So there will be a total of J N = 16 codewords in the N-th level of the codeboo. In Fig. 2, we compare the beam pattern of the proposed method with that of DFT based codeboo. We can find that the proposed method achieves much less variations in radiated power in the intended beam range. We assume that the BS serves K = 16 single-antenna users simultaneously. We assume that the dimension of the equivalent channel for JSDM is Q = K = 16. We consider a non-parametric model and assume that h is given by [18] h = g iid R 1 2, (3) where R C M M represents the correlation matrix at the BS, and g iid C 1 M CN (, I), whose entries are independently identically distributed circularly symmetric complex Gaussian CN (, 1). For the ULA antenna structure with omnidirectional antennas, R is Toeplitz, i.e., R = T(J) with [18] ( ) ( )] 2πλ 2π J = [J (), J,..., J λ (M 1), (31) where J (x) is the Bessel function of order. We average over channel realizations for each simulation. Fig. 3. Sum rate in SNR. M = 32. In Fig. 3(a), we plot the achievable ergodic sum rate versus SNR for different transmission schemes. In the figure, JSDM, JSDM-NB, and JSDM-DFT represent the twostage schemes with the first-stage precoding matrix decided by the covariance matrix, the codewords of the users selected from the proposed codeboo, and the DFT-based codeboo, respectively. As can be seen from the figure, the proposed codeboo achieves better performance than the DFT-based codeboo in all cases since the designed codeboo maximizes the BLR among the codewords in the spatial domain. It is interesting that the proposed scheme also achieves higher sum rate than the JSDM scheme with covariance matrix information in the low-snr regime. This is because in the nonparametric channel model, covariance matrix based dimension reduction results in significant loss in the useful signal power, which will deteriorate the system performance in the low- SNR regime, while the spatial domain codeboos based JSDM schemes still reserve most of the power. At SNR = 1 db, the performance improvement of the proposed method is 54% with respect to the DFT based codeboo, and 33% with respect to the covariance based method. In the high-snr regime, JSDM with covariance matrix information can exploit the orthogonality of the first stage precoding matrix to reduce the interference of different users to achieve better performance. In Fig. 3(b), we also plot the achievable sum rate versus SNR in independent and identically distributed (IID) fading channels with entries of h given by IID circularly symmetric complex Gaussian random variables with zero-mean and unit variance. Again, we can see performance improvement similar to the above statement. V. CONCLUSIONS In this paper, we have considered the codeboo design for multi-resolution codeboo based two-stage precoding scheme in massive MIMO systems. We have defined a new performance metric, beam-to-leaage ratio, and formulated an optimization problem that maximizes the BLR with little variation in the radiated power within the beam range. By solving the optimization problem, we have developed a method for generating the codeboo. Through numerical results, we have shown that our proposed codeboo based two-stage precoding scheme performs better than the DFT-based codeboo one, and can achieve better performance than the covariance based one in the low-snr regime.

5 REFERENCES [1] Huawei Technologies, 5G: A Technology Vision, White Paper, 213. Available: [2] T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. on Wireless Commun., vol. 9, no. 11, pp , Nov. 21. [3] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashihmin, and R. Zhang, An overview of massive MIMO: Benefits and challenges, IEEE J. Sel. Topics Sig. Proc., vol. 8, no. 5, pp , May 214. [4] F. Boccardi, R.W. Heath, Jr., A. Lozano, T. Marzetta, and P. Popovsi, Five disruptive technology directions for 5G, IEEE Commun. Mag., vol. 52, no. 2, pp. 74-8, Feb [5] F. Ruse, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, Scaling up MIMO: Opportunities and challenges with very large arrays, IEEE Signal Proces. Mag., vol. 3, no. 1, pp. 4-46, Jan [6] H. Huh, G. Caire, H.C. Papadopoulos, and S.A. Ramprashad, Achieving Massive MIMO spectral efficiency with a not-so-large number of antennas, IEEE Trans. Wireless Commun., vol. 11, no. 9, pp , Sep [7] J. Hoydis, S. Brin, and M. Debbah, Massive MIMO in the UL/DL of cellular networs: how many antennas do we need? IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp , Feb [8] E. Bjornson, E. G. Larsson, and T. L. Marzetta, Massive MIMO: ten myths and one critical question, IEEE Comm. Mag., vol. 54, no. 2, pp , Feb [9] A. Adhiary, J. Nam, J.-Y Ahn, and G. Caire, Joint spatial division and multiplexing-the large-scale array regime, IEEE Trans. Inform. Theory, vol. 59, no. 1, pp , Oct [1] X. Zhang, J. Tadrous, E. Everett, F. Xue, and A. Sabharwal, Angleof-arrival based beamforming for FDD massive MIMO, in th Asilomar Conferences on Signals, Systems and Computers. [11] A. Alhateeb, O. E. Ayach, G. Leus, and R. W. Heath, Channel estimation and hybrid precoding for millimeter wave cellular systems, IEEE J. Sel. T. Sig. Proc., vol. 8, no. 5, pp , Oct [12] S. Noh, M. D. Zoltowsi, and D. J. Love, Multi-resolution codeboo and adaptive beamforming sequence design for millimeter wave beam alignment, (215). Dept. of ECE Tech. Rep., paper [13] H. Xie, F. Gao, S. Zhang, and S. Jin, A unified transmission strategy for TDD/FDD massive MIMO systems with spatial basis expansion model, IEEE Trans. Vehi. Technol., vol. pp, no. 99, pp. 1-1, 216. [14] M. Sade, A. Tarighat, and A. H. Sayed, A leaage-based precoding scheme for downlin multi-user MIMO channels, IEEE Trans. Wireless Commun., vol. 6, no. 5, pp , May 27. [15] A. Khabbazibasmenj, S. A. Vorobyov, and A. Hassanien, Transmit radiation pattern invariance in MIMO radar with application to DOA estimation, IEEE Sig. Proc. Letters, vol. 22, no. 1, pp , Oct [16] D. Qiao, H. Qian, and G. Y. Li, Broadbeam for massive MIMO systems, IEEE Trans. Sig. Proc., vol. 64, no. 9, pp , May 216. [17] A. Sabharwal, D. Avidor, and L. Potter, Sector beam synthesis for cellular systems using phased antenna arrays, IEEE Trans. Vehi. Technol., vol. 49, no. 5, pp , Sep. 2. [18] Q.-U.-A. Nadeem, A. Kammoun, M. Debbah, and M.-S. Alouini, A generalized spatial correlation model for 3D MIMO channels based on Fourier coefficients of power spectrums, IEEE Trans. Sig. Proc., vol. 63, no. 14, pp , July 215. [19] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New Yor: Wiley, 25.