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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 6, NO. 5, OCTOBER 2017
Spectral Efficiency of DFT-Based Processing Hybrid Architectures in Massive MIMO Weiqiang Tan, Michail Matthaiou, Senior Member, IEEE, Shi Jin, Member, IEEE, and Xiao Li, Member, IEEE
Abstract—This letter investigates the achievable spectral efficiency (SE) of massive multiple-input multiple-output transmission systems with hybrid architectures based on discrete Fourier transform (DFT) processing, where the base station (BS) has perfect channel state information and baseband processing is performed by zero-forcing precoding. We derive tractable upper and lower bounds on the achievable SE. Based on these results, the effects of the number of radio frequency (RF) chains, signalto-noise ratio (SNR), and the number of users are revealed. Compared to hybrid architectures with ideal and quantized phase shifters, simulations indicate that the achievable SE with DFT processing is inferior to the one with ideal phase shifters. In addition, the achievable SE with DFT processing is independent of the number of BS antennas and can be improved by increasing the SNR and the number of RF chains, while there exists an optimal number of users that maximizes the total achievable SE. Index Terms—DFT processing, hybrid architectures, massive MIMO, spectral efficiency, zero-forcing.
I. I NTRODUCTION N CONVENTIONAL MIMO systems, each transceiver requires a dedicated radio frequency (RF) chain, which includes a low-noise amplifier, analog-to-digital converter (ADC) and so on [1]. When the number of antennas is very large, deploying a separate RF chain for each antenna not only consumes high levels of energy but also increases substantially the hardware cost, which is a technical bottleneck for massive MIMO systems. This motivates the use of hybrid analog/digital architectures, which use a small number of RF chains [2], typically much smaller than the number of antennas. Recently Liang et al. [3] proposed a low RF-complexity hybrid architecture and designed a novel low-dimensional
I
Manuscript received May 26, 2017; accepted June 16, 2017. Date of publication June 23, 2017; date of current version October 11, 2017. This work was supported in part by the National Science Foundation for Distinguished Young Scholars of China under Grant 61625106, in part by the National Natural Science Foundation of China under Grant 61531011 and Grant 61571112, in part by the Foundation for the Author of National Excellent Doctoral Dissertation, China under Grant 201446, in part by the Fundamental Research Funds for the Central Universities under Grant 2242015R30006, and in part by the Engineering and Physical Sciences Research Council under Grant EP/P000673/1. The associate editor coordinating the review of this paper and approving it for publication was D. So. (Corresponding author: Shi Jin.) W. Tan is now with the School of Computer Science and Educational Software, Guangzhou University, Guangzhou 510006, China. He was with the National Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail:
[email protected]). S. Jin and X. Li are with the National Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail:
[email protected];
[email protected]). M. Matthaiou is with the Institute of Electronics, Communications and Information Technology, Queen’s University Belfast, Belfast BT3 9DT, U.K. (e-mail: m:
[email protected]). Digital Object Identifier 10.1109/LWC.2017.2719036
baseband zero-forcing (ZF) precoding scheme. The results indicated that there is a negligible rate loss for ideal phase shifters and a considerable rate loss for quantized phase shifters. An alternative approach to reduce the number of RF chains is the antenna subset selection scheme, which assumes that each RF chain is connected to a subset of the antennas [4]. Unfortunately, variable phase shifters require accurate and instantaneous phase information, which is challenging to acquire in practice. Besides, to reduce the number of RF chains, the work of [5] utilized a discrete lens antenna array, which provides variable phase shifting for electromagnetic rays so as to direct the signals toward different points of the focal surface. Amadori and Masouros [6] studied a low RFcomplexity topology with lens antenna arrays. They showed that there is no notable performance degradation for multiuser MIMO systems. However, the works of [5] and [6] are limited to sparse millimeter wave MIMO channels. In [7], a hybrid antenna selection scheme was developed based on DFT for spatially correlated MIMO channels. This can be realized by placing a Butler matrix between the antenna elements and the receiver switch. Moreover, the authors showed that an analog DFT circuit can be easily implemented with low power consumption by using floating-gate transistors on a field-programmable analog array (FPAA) [8]. Motivated by this observation, the analysis of the achievable SE for massive MIMO with DFT processing is of special interest. The main contribution of this letter is to investigate the achievable SE of a novel hybrid architecture based on DFT processing and derive tractable upper and lower bounds by assuming ZF baseband precoding. Based on the derived analytical expressions, the effects of the number of BS antennas, the SNR, and the number of users on the achievable SE are revealed. Compared to the hybrid architectures with ideal and quantized phase shifters, simulations show that the achievable SE with DFT processing is inferior to the one with ideal phase shifters. II. S YSTEM M ODEL Consider a single cell massive MIMO system, where the BS is equipped with Nt transmit antennas, which are connected to Ns RF chains and simultaneously serve K single-antenna users.1 We assume that the number of BS antennas is much larger than the number of users (Nt K) and the number of RF chains is more than the number of active users and less than the number of BS antennas (K ≤ Ns ≤ Nt ). Let x 1 We use diag{·}, E{·}, and det(·) to denote a diagonal matrix, expectation, and matrix determinant, respectively; · represents rounding to the nearest integer; IK denotes an K × K identity matrix; Ak is A with the k-th column removed while [A]n,m is the element at the n-th row and m-th column of A, ψ(·) is the Euler’s digamma function, W0 (·) denotes the Lambert √ function [12, eq. (3.1)], and i = −1 denotes the imaginary unit.
c 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2162-2345 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
TAN et al.: SE OF DFT-BASED PROCESSING HYBRID ARCHITECTURES IN MASSIVE MIMO
be the K × 1 signal vector for a total of K users, satisfying E[xxH ] = IK . If Gaussian inputs are used, the received signal after analog signal processing and digital precoding, can be expressed as P H G FWx + n, y= (1) K where P denotes the SNR, W denotes the Ns × K digital precoding matrix, F represents the Nt × Ns analog processing matrix, n is the complex circular symmetric Gaussian noise with zero mean and unit variance, and G is the Nt × K channel matrix from the BS to the users that is modeled as (2) G = HD1/2 , where H ∈ CNt ×K contains the fast-fading coefficients, whose entries are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and unitvariance, and D is a K × K diagonal matrix with diagonal −γ elements given by [D]k,k = βk . Here, βk = zk rk models both path loss and shadowing, where rk is the distance from the k-th user to the BS, γ is the decay exponent, and zk is a log-normal random variable. In the previous works [3], [4], the analog processing was performed using ideal phase shifters. However, such shifters with continuous phase are not feasible because perfect channel state information (CSI) is hard to acquire in practice. Therefore, we consider a hybrid architecture based on DFT processing. In this way, phase shifters are replaced by an integrated circuit or a software implementation, and analog processing can be done by a Butler matrix integrated [7] or floating-gate transistors on a FPAA [8], which efficiently reduces hardware complexity and power consumption. For such as architecture, all transmitted signals pass through a part of DFT matrix pre-processing before baseband processing. The analog processing matrix F is designed as ⎡ ⎤ 1 1 1 ··· 1 ⎢1 w w2 ··· w(Ns −1) ⎥ ⎥ 1 ⎢ 2 4 ⎢1 w w ··· w2(Ns −1) ⎥ F= √ ⎢ ⎥, (3) ⎥ .. .. .. Nt ⎢ .. .. ⎣. ⎦ . . . . 1 w(Nt −1)
w2(Nt −1)
···
w(Nt −1)(Ns −1)
where w = e−2π i/Nt . We note that all RF chains are always activated as DFT processing is implemented offline. III. S PECTRAL E FFICIENCY A NALYSIS In this section, we derive tractable upper and lower bounds on the achievable SE. Based on the derived analytical expression, the effects of some physical parameters are revealed. A. The Achievable Ergodic Spectral Efficiency Assume the BS has perfect CSI and we adopt ZF precoding scheme in the baseband processing, which is able to completely cancel out inter-user interference. In this case, the equivalent low dimension matrix Geq = GH F comes up as the product of the channel matrix and analog matrix. Therefore, the baseband precoding matrix with ZF precoding is given by [3]
−1 H W = GH , (4) eq Geq Geq where denotes a normalized diagonal matrix, i.e., []k,k = 1/ [WH W]k,k . For the case under consideration, the
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achievable SE of the k-th user in bit/s/Hz can ⎧ ⎛ ⎪ ⎪ ⎪ ⎜ ⎨ P ⎜ Rk = E log2 ⎜1 +
−1 ⎪ ⎝ ⎪ H ⎪ K G G eq eq ⎩
be given by ⎞⎫ ⎪ ⎪ ⎬ ⎟⎪ ⎟ (5) ⎟ . ⎠⎪ ⎪ ⎪ ⎭
k,k
From (5), it is worth noting that the expectation is taken over all channel realizations of Geq and the equivalent channel matrix is needed to be ergodic. An exact evaluation of (5) involves exponential integral functions and therefore offers no physical insights. Alternatively, we seek to derive tractable bounds, which enable us to draw engineering insights into the system performance. We now present the following theorem, which derives upper and lower bounds on the achievable SE. B. Bounds on the Achievable Spectral Efficiency Theorem 1: For a hybrid architecture based on DFT processing, the exact achievable SE of the k-th user with ZF precoding can be bounded as where
RLk ≤ Rk ≤ RU k,
(6)
Pβk ψ(Ns −K) RLk = log2 1 + , e K
(7)
and
RU k
Ns ! Pβk Ns ! + = log2 K (Ns − K)! (Ns − (K − 1))! K−1 1 ψ(Ns − k) . − ln 2
(8)
k=0
Proof: See the Appendix. From Theorem 1, it can be seen that the lower and upper bounds can be easily evaluated. We now draw some additional engineering insights by considering several special scenarios. The following corollary presents the achievable SE limit as the number of RF chains becomes large. Corollary 1: For a hybrid architecture with DFT processing, when the number of RF chains becomes large, the exact analytical expression of the achievable SE is expressed by Pβk ¯ (9) Rk → Rk = log2 1 + (Ns − K + 1) . K Proof: By using the property that ψ(Ns − K) ≈ ln (Ns − K + 1) when Ns becomes large in [11, eq. (6.3.18)], the lower bound in (7) reduces to Pβk (10) RLk = log2 1 + (Ns − K + 1) . K Using the same method, it can be shown that the last term in (8) can be simplified as K−1 1 Ns ! . (11) ψ(Ns − k) = log2 ln 2 (Ns − (K − 1))! k=0
Substituting (11) into (8), the upper bound is calculated as Pβk U (12) Rk = log2 1 + (Ns − K + 1) . K By utilizing the squeeze theorem [11] for (10) and (12), we derive the desired result.
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From Corollary 1, we observe that as the number of RF chains grows large, the lower and upper bounds coincide and the approximated expression is equivalent to the exact analytical expression. In addition, we also see that for a fixed number of users, the achievable SE in (9) becomes a strictly logarithmic increasing function with the number of RF chains and the SNR. This implies that the achievable SE can be boosted by increasing the number of RF chains or the SNR. We now investigate the impact of the number of users on the total achievable SE. Corollary 2: Supposed that every user has the same largescale fading factor, i.e, βk = β, ∀k and a large number of RF chains are deployed at the BS. In this case, there exists an optimal number of users to maximize the total achievable SE, which can be calculated as ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Pβ(Ns + 1) ! ⎦. (13) K opt = ⎣ 1 +1 (Pβ − 1) W (Pβ−1)e −1 ) 0( Proof: For the case βk = β, ∀k, the total achievable SE of the hybrid architecture MIMO system is calculated as Pβ (14) Rsum = Klog2 1 + (Ns − K + 1) . K By differentiating the total achievable SE in (14) with respect to K, the number of users, the first order partial derivative of Rsum (K) can be written as ∂Rsum 1 Pβ (Ns + 1) = ln 1 + . (Ns − K + 1) − K ∂K ln 2 K Pβ + (Ns − K + 1) (15) Since it is challenging to work out whether (15) is strictly positive or negative, the second order partial derivative of Rsum (K) is further checked, which gives
(Ns +1)2 1 2 2 1 − Pβ (Ns + 1) − K ∂Rsum = (16) !2 < 0.
∂K 2 1 K Pβ − 1 + (Ns + 1) This indicates that Rsum (K) is concave with respect to the number of users. Therefore, we can infer that there exists a unique globally optimal number of users maximizing the total achievable SE. In order to find this optimal value, we set the first order partial derivative equal to zero, that is ∂Rsum /∂K = 0, and we have Pβ Pβ K (Ns + 1) ln 1 + . (17) (Ns − K + 1) = Pβ K 1 + K (Ns − K + 1) Denote y = 1 + (Pβ/K)(Ns − K + 1), (17) is simplified as Pβ − 1 Pβ−1 e y . (18) (Pβ − 1)e−1 = y Using the definition of the Lambert function [12], we attain Pβ − 1 " #. y= (19) W0 (Pβ − 1)e−1 Substituting the definition of y into (19) and performing some basic algebraic manipulations yields the desired result. From Corollary 2, we observe that for a fixed number of RF chains and SNR, the total achievable SE of system first increases and then decreases with the number of users. This is because a reduced amount of power is allocated to each user
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 6, NO. 5, OCTOBER 2017
Fig. 1. Total achievable SE versus the SNR for fixed Nt = 128, K = 8 and Ns = 50, 20, 10. Results are shown for full-digital and hybrid architectures with ideal, 1-bit quantized phase shifters, and DFT processing.
and the achievable SE per user decreases with an increase in the number of activated users. Corollary 2 indicates that we can find an optimal number of users to maximize the total achievable SE, which depends on the number of RF chains, large-scale fading factor, and the SNR. IV. N UMERICAL R ESULTS This section provides simulation results to validate the derived analytical expressions. We consider a circular cell with a radius (from center to edge) of 500 meters and a guard zone of 20 meters with a random distribution, the decay exponent is γ = 2.1, the shadow-fading standard deviation is σshad = 4.9 dB, and the number of users is set to K = 8. The users’ large-scale fading coefficients βk , ∀k are randomly generated as follows: {0.19, 3.35, 5.28, 0.15, 11.84, 1.79, 3.03, 0.08}×10−3 . In Fig. 1, the simulated total achievable SE of hybrid architecture with DFT processing, as well as, the derived analytical expressions of Theorem 1 are plotted against the SNR. We see that the theoretical bounds of the total achievable SE remain very tight with the numerical simulation results in the entire SNR regime, which validates the analytical results in Theorem 1. The total achievable SE for the full-digital architecture is also provided, along with the hybrid architecture with ideal phase shifters and 1-bit quantized phase shifters [3]. Compared with a full-digital architecture, the total achievable SE with ideal phase shifters suffers a 16% loss at SNR =20 dB, and the total achievable SE with DFT processing and 1-bit quantized phase shifters is inferior to the one with ideal phase shifters. The performance loss is caused by inaccurate angle phase information. Compared with the ideal phase shifter case, the achievable SE with 1-bit quantized phase shifter suffers a 43% loss at SNR = 20 dB and the achievable SE with DFT processing has 30%, 52% loss at Ns = 50, 20, respectively. For comparison, we illustrate the achievable SE with DFT processing for different number of RF chains Ns = 50, 20, 10, respectively. We find that the total achievable SE can be improved by increasing the number of RF chains since more data streams can be multiplexed, which is in accordance with the result in Corollary 1. Fig. 2 shows the total achievable SE for the proposed hybrid architecture based on DFT processing versus the number of users. Without loss of generality, we assume that every user has the same large-scale fading factor β = 1 × 10−3 . We
TAN et al.: SE OF DFT-BASED PROCESSING HYBRID ARCHITECTURES IN MASSIVE MIMO
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be written as
⎧ ⎛ ⎞⎫ ⎬ ⎨ Pβ k ⎠ . Rk = E log2 ⎝1 + $ %−1 ⎩ K TH T k,k ⎭
(22)
By applying the Jensen’s inequality on log2 (1 + a exp(x)) for a > 0, the lower bound on the achievable SE is expressed as ⎛ ⎧ ⎛ ⎛ ⎞⎫⎞⎞ ⎨ ⎬ 1 Pβk ⎠ ⎠⎠. (23) exp⎝E ln⎝ $ RLk = log2 ⎝1 + % −1 ⎭ ⎩ K TH T k,k
Fig. 2. Total achievable SE with DFT processing versus the number of users for fixed Nt = 128, SNR = 20 dB, and different number of RF chains (Ns = 40, 60, and 80).
see that for a given number of BS antennas, SNR, and the number of RF chains, the total achievable SE first increases and then decreases when the number of users varies from 3 to 30. Regardless of the number of RF chains, there indeed exists an optimal number of users to maximize the total achievable SE, which validates the analytical theoretical in Corollary 2. Furthermore, by substituting the fixed Ns , P, and β into (13), the optimal number of users can be calculated as 7, 9, and 12, respectively, which aligns with the result in Corollary 2. V. C ONCLUSION This letter investigated the achievable SE of massive MIMO systems for a hybrid architecture based on DFT processing and derived tractable upper and lower bounds on the SE. Simulations showed that hybrid architecture based on DFT processing can have comparable performance with quantized phase shifters. Our results also showcased that the achievable SE is independent of the number of BS antennas and can be improved by increasing the number of RF chains and the SNR, while there exists an optimal number of users that maximizes the total achievable SE. We conclude that a hybrid architecture based on DFT processing is a viable low-complexity alternative for future massive MIMO systems. A PPENDIX P ROOF OF T HEOREM 1 Recalling the definition of Geq = GH F with G = HD1/2 , the achievable ergodic SE with ZF precoding is written as ⎧ ⎛ ⎞⎫ ⎬ ⎨ Pβ k ⎠ (20) Rk = E log2 ⎝1 + $ %−1 ⎭, ⎩ K HH AH k,k
where A is defined as an auxiliary matrix that equals to FFH . Since each column vector of the DFT matrix F is orthogonal to each other [9], the eigen-decomposition A can be given by A = FFH = UUH ,
(21)
where = diag{1, . . . , 1, 0, . . . , 0}. With the aid of the uni& '( ) Ns
tary transformation, HH AH reduces in distribution to TH T, where T is a K × Ns dimensional matrix and equals to Ns column of H. Therefore, the achievable ergodic SE in (20) can
$ %−1 With the help of the key matrix property that TH T k,k = # " H # " det TH k Tk /det T T , the lower bound in (23) can be calculated as " * " " ##+ Pβk exp E ln det TH T (24) RLk = log2 (1 + * " K" H ##+# − E ln det Tk Tk ). We move on to study the upper bound. By applying Jensen’s inequality again, the upper bound can be calculated as * " H #+ Pβk * " H #+ U E det Tk Tk Rk = log2 E det T T + K 1 * " " H ##+ E ln det Tk Tk . (25) − ln 2 According to the conclusion in [10, eq. (21)] and [10, eq. (20)], we" obtain closed-form of ## " forH #the expectation " # ## " " expressions " H ln det TH T , ln det TH k Tk , det TT , and det Tk Tk . Substituting the corresponding results into (24) and (25), respectively, and after some simple simplifications, we obtain the desired results. R EFERENCES [1] M. Vu and A. Paulraj, “MIMO wireless linear precoding,” IEEE Signal Process. Mag., vol. 24, no. 5, pp. 86–105, Sep. 2007. [2] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, Jr., “Hybrid precoding for millimeter wave cellular systems with partial channel knowledge,” in Proc. IEEE ITA, San Diego, CA, USA, Feb. 2013, pp. 1–5. [3] L. Liang, W. Xu, and X. Dong, “Low-complexity hybrid precoding in massive multiuser MIMO systems,” IEEE Wireless Commun. Lett., vol. 3, no. 6, pp. 653–656, Dec. 2014. [4] S. Han, C.-L. I, Z. Xu, and C. Rowell, “Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave 5G,” IEEE Commun. Mag., vol. 53, no. 1, pp. 186–194, Jan. 2015. [5] J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for millimeter-wave communications: System architecture, modeling, analysis, and measurements,” IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 3814–3827, Jul. 2013. [6] P. V. Amadori and C. Masouros, “Low RF-complexity millimeter-wave beamspace-MIMO systems by beam selection,” IEEE Trans. Commun., vol. 63, no. 6, pp. 2212–2223, Jun. 2015. [7] A. F. Molisch, X. Zhang, S. Y. Kung, and J. Zhang, “DFT-based hybrid antenna selection schemes for spatially correlated MIMO channels,” in Proc. IEEE PIMRC, Beijing, China, Feb. 2003, pp. 1119–1123. [8] S. Suh, A. Basu, C. Schlottmann, P. E. Hasler, and J. R. Barry, “Low-power discrete Fourier transform for OFDM: A programmable analog approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 2, pp. 290–298, Feb. 2011. [9] 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. Veh. Technol., vol. 66, no. 4, pp. 3170–3184, Apr. 2017. [10] M. Matthaiou, C. Zhong, and T. Ratnarajah, “Novel generic bounds on the sum rate of MIMO ZF receivers,” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4341–4353, Sep. 2011. [11] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York, NY, USA: Dover, 1974. [12] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., vol. 5, no. 4, pp. 329–359, Dec. 1996.
