Quantized Precoding for Multi-Antenna Downlink Channels with MAGIQ Andrei Nedelcu∗, Fabian Steiner∗, Markus Staudacher∗, Gerhard Kramer∗, Wolfgang Zirwas†, Rakash Sivasiva Ganesan†, Paolo Baracca‡, Stefan Wesemann‡ ∗Institute for Communications Engineering, Technical University of Munich, Germany †Nokia Bell Labs, Munich, Germany, ‡Nokia Bell Labs, Stuttgart, Germany Abstract—A multi-antenna, greedy, iterative, and quantized Recently, the downlink has received attention [7]–[12]. The (MAGIQ) precoding algorithm is proposed for downlink chan- authors of [7] use the Bussgang approximation to design nels. MAGIQ allows a straightforward integration with orthogo- quantized linear precoders (QLPs). The paper [8] introduces a nal frequency-division multiplexing (OFDM) for frequency selec- tive channels. MAGIQ is compared to three existing algorithms lookup-table based precoder for quadrature phase-shift keying in terms of information rates and complexity: quantized linear (QPSK) that minimizes the uncoded bit error rate (BER) at the precoding (QLP), SQUID, and an ADMM-based algorithm. The UE. The paper [9] introduces a hybrid RF architecture that information rate is measured by using a lower bound for finite combines a constrained and a conventional MIMO array. A modulation sets, and the complexity is measured by the number greedy knapsack-like algorithm is used to minimize the mean of multiplications and comparisons. MAGIQ and ADMM achieve similar information rates for Rayleigh flat-fading channels and square error (MSE) between the desired and constructed UE one-bit quantization per real dimension, and they outperform signal points. QLP and SQUID for higher order modulation formats. The authors of [10] describe nonlinear approaches based on semidefinite relaxation, ` norm relaxation and sphere ∞ I. INTRODUCTION decoding. For a reasonable performance and complexity trade- off, they recommend ` norm relaxation, which is named Massive multiple-input multiple-output (massive MIMO) ∞ SQUID [10, Sec. IV]. Another approach is described in [11], uses large antenna arrays at base stations to serve many where the authors extend the framework of alternating direc- users that each have a small number of antennas [1]. The tion of multipliers (ADMM), and they report slight improve- gains of massive MIMO include improved power and spectral ments over SQUID. The precoding problem for coarsely quan- efficiencies, and simplified signal processing [2]. The gains tized, frequency-selective channels and its integration with are often stated for a large number N of base station antennas OFDM was considered in [12], where the authors use linear and a large number K of user equipments (UEs) when the precoding and a frequency domain approach. An extension ratio N/K is held constant. of SQUID to OFDM and frequency selective channels was A practical implementation for large N and K is chal- presented in [13]. lenging. For example, consider a base station deployment with N radio frequency (RF) chains. It seems impractical to use high-resolution analog-to-digital and digital-to-analog B. Contributions and Organization converters (ADCs/DACs), along with linear but low-efficiency Our work is inspired by the greedy approach of [9]. We power amplifiers. Two potential solutions are as follows. First, introduce a multi-antenna, greedy, iterative, and quantized hybrid-beamforming [3] uses analog beamformers in the RF (MAGIQ) precoding algorithm for downlink channels. The chain of each antenna, and the digital baseband processing algorithm decomposes a high dimensional nonlinear and non- arXiv:1712.08735v2 [cs.IT] 2 Mar 2018 is shared among different RF chains. Second, low-resolution convex problem into low dimensional sub-problems that can ADCs/DACs simplify the transceivers, e.g., one bit quantizers be solved efficiently. We compare MAGIQ’s performance to use simple comparators and they might permit using non-linear QLP, SQUID, and the ADMM-based algorithm [7], [10]– power amplifiers. [12]. We consider both low-order and higher-order modulation formats (4, 16, 64-quadrature amplitude modulation (QAM)). A. Uplink and Downlink As a key performance metric, we compute lower bounds We consider the low-resolution quantizer approach. There on the information rates by using the mismatched decoding are numerous studies on the uplink with either linear detectors, framework [14], [15]. We thus take modulation constraints e.g., matched filter (MF), zero forcing (ZF), and Wiener filter into account, rather than idealized Gaussian signaling. For the (WF), or non-linear detectors such as approximate message frequency-selective case, our approach operates in the time passing (AMP) [4]–[6]. For example, it is known that even domain and avoids switching between domains to enforce the for low-resolution quantization at the base station antennas, the discrete alphabet constraint. A related problem was presented UEs can communicate with higher order modulation formats in the context of precoding for frequency selective channels if N is sufficiently large. with constant envelope continuous modulation in [16]. UE1 CSI b bits / Uˆ1[t] u1[t] x1[t] IFFT / / UE2 Phase . Precoder . LO . PA . shifter . xN [t] UˆK [t] uK [t] IFFT / / / UEK b bits Fig. 1. Multi-user MIMO downlink with a low resolution digitally controlled analog architecture. This paper is organized as follows. In Sec. II, we describe where h [l], l = 0, 1,...,L 1, is the channel impulse kn − the system model, the precoding problem for a K-user down- response from the n-th antenna at the base station to the link channel with coarsely quantized transmit symbols and the k-th UE. We study a Rayleigh fading frequency selective QLP approaches. In Sec. III, we present the MAGIQ precoding channel with a uniform power delay profile, i.e., we choose algorithm. Sec. IV provides a comprehensive complexity com- E h [l] 2 = 1/L, where the h [l] (0, 1/L) are i.i.d. | kn | kn ∼ CN parison of SQUID, ADMM and MAGIQ in terms of arithmetic circularly-symmetric, complex Gaussian random variables. We operations. Sec. V presents simulation results and we conclude further assume a block fading channel model, i.e., the channel in Sec. VI. realization remains constant for the coherence interval, and the instantaneous realizations H[l], l = 0, 1,...,L 1, are known − II. PRELIMINARIES at the transmitter. A. System Model We remark that the alphabet (2) can be interpreted as Consider the downlink of a multi-user MIMO channel permitting per-symbol antenna selection through the zero with N transmit antennas and K UEs that each have a symbol. The idea of joint precoding and antenna selection also single antenna. A discrete time, frequency selective, baseband appeared in [17], but our algorithm selects antennas without channel has a finite impulse response (FIR) filter between each an extra metric that enforces sparsity. pair of transmit and receive antennas. We collect the received B. Flat Fading Channels signals yk[t] of user k, k = 1, 2,...,K, at time t into the K-dimensional column vector For flat fading channels (L = 1) the channel (1) becomes L 1 − y = Hx + z. (4) y[t] = H[l]x[t l] + z[t] (1) − Let u be the noise-free complex symbol that we wish to l=0 k ∈ U X generate at the k-th UE for k = 1,...,K, where is either a U where the noise z[t] is a circularly-symmetric, complex, Gaus- 4-, 16-, or 64-QAM signaling set. Let u be the column vector sian, random, column vector with a scaled identity covariance with the K symbols. Consider the precoding problem matrix, i.e., we have z (0, σ2I). The transmit column ∼ CN T 2 2 2 vector x[t] = [x1[t], x2[t], . , xN [t]] has entries taken from min u αHx + α Kσ x,α k − k2 a discrete alphabet that has 2b+1 values where b bits encode X s.t. x N (5) the phase. More precisely, we choose ∈ X α > 0. = 0, exp(j 2πq/2b) (2) X { } The factor α permits trading off noise enhancement and the with q = 0, 1,..., 2b 1. We choose the transmit power to be − received signal power, the latter being more important at low at most unity, and we define the transmit signal-to-noise ratio SNR and maximized by the MF [?]. For a fixed value of α, the 2 (SNR) as SNR = 1/σ . The channel impulse response matrix problem (5) represents a classic nonlinear integer least-squares is problem [18]. h11[l] h12[l] . h1N [l] h21[l] h22[l] . h2N [l] C. Quantized Linear Precoding H[l] = . (3) . .. QLP approximates the solution of (5) by x = Q(P u), CN K h [l] h [l] . h [l] where P × is a precoding matrix and Q( ) is a K1 K2 KN ∈ · B. Frequency Selective Channels Algorithm 1: MAGIQ for frequency-flat channels For frequency selective channels (L > 1), the precoder 1 Inputs: u, H, = 1,...,N , errmin S { } puts out a string of column vectors x[1],..., x[T ], each with 2 Initialize: x = xinit, α = αinit alphabet N , where T is at most the coherence time of the 3 for iter = 1 : I do X channel. In practice, T is chosen to balance the need for 4 loopIdx = 1 2 accurate channel state information (CSI) and quality-of-service 5 err = u 0 k k (QoS) requirements. 6 while (errloopIdx > errmin) (errloopIdx < ∨ Consider the K-dimensional column vectors u[1],..., u[T ] errloopIdx 1) (loopIdx N) do ? ?− ∨ ≤ that we would like to generate at the K UEs. We state our ? 7 (xn , n ) = argminxn ,n F (x, α) T∈X ∈S ? T optimization problem as follows. 8 (x1, . , xn, . , xN ) = (x1, . , xn? , . , xN ) ? 2 9 n T L 1 S ← S \ { } − 2 2 10 loopIdx loopIdx + 1 min u[t] α H[l]x[t l] + α T Kσ ← 2 x[1],...,x[T ],α − − 2 2 t=1 l=0 11 errloopIdx = u αHx 2 + α Kσ 2 k − k X N X 12 end s.t.