Study of Joint Automatic Gain Control and MMSE Receiver Design Techniques for Quantized Multiuser Multiple-Antenna Systems T. E. B. Cunha1, R. C. de Lamare1, T. N. Ferreira2, T. Hälsig3 1Centre for Telecommunications Studies (CETUC), Pontifical Catholic University of Rio de Janeiro, Brazil 2Engineering School, Fluminense Federal University (UFF), Niterói, RJ, Brazil 3Institute for Communications Engineering, Universität der Bundeswehr München, Germany Email: {thiagoelias, delamare}@cetuc.puc-rio.br, [email protected], [email protected] Abstract—This paper presents the development of a joint design becomes especially important for low resolution ADCs. optimization of an automatic gain control (AGC) algorithm and Although there are many articles on quantization in MIMO a linear minimum mean square error (MMSE) receiver for multi- systems in the literature, few address the design of AGCs. user multiple input multiple output (MU-MIMO) systems with coarsely quantized signals. The optimization of the AGC is based In [1], the authors presented a modified MMSE receiver that on the minimization of the mean square error (MSE) and the takes into account the quantization effects in a MIMO system proposed receive filter takes into account the presence of the but they do not take into account the presence of an AGC. AGC and the effects due to quantization. Moreover, we provide The effects of an AGC on a quantized MIMO system with a lower bound on the capacity of the MU-MIMO system by a standard Zero-Forcing filter at the receiver were examined deriving an expression for the achievable rate. The performance of the proposed Low-Resolution Aware MMSE (LRA-MMSE) in [6]. However, the authors have not optimized the AGC receiver and AGC algorithm is evaluated by simulations, and algorithm nor used a detector that considers the quantization compared with the conventional MMSE receive filter and Zero- effects. Forcing (ZF) receiver using quantization resolution of 2, 3, 4 and This work presents a framework for jointly designing the 5 bits. AGC and a linear receive filter according to the MMSE Index Terms—Coarse Quantization, AGC, MU-MIMO detec- tion, MMSE receiver criterion for a large-scale MU-MIMO system operating with coarsely quantized signals. The procedure consists of com- I. INTRODUCTION puting the modified MMSE receiver presented in [1] and, In 5G celular systems, high data rates, reliable links, low after that, computing the derivative of the cost function that cost and power consumption are key requirements. Multiple- takes into account the presence of the AGC in order to input multiple-output (MIMO) systems in wireless commu- obtain the optimal AGC coefficients. Then, a Low-Resolution nications provide significant improvements in wireless link Aware MMSE (LRA-MMSE) receiver that considers both reliability and achievable rates. However, as the number quantization effects and the AGC is derived. A lower bound on of antennas scales up, the energy consumption and circuit the capacity of this system is investigated and an expression complexity increases accordingly [1], [2], [3], [4], [5]. For to compute the achievable rates is developed. example, the energy consumption of an analog to digital Notation: Vectors and matrices are denoted by lower and upper case italic bold letters. The operators ( )T , ( )H and tr( ) arXiv:1810.11584v1 [cs.IT] 27 Oct 2018 converter (ADC) grows exponentially as a function of the · · · quantization resolution. To reduce circuit complexity and save stand for transpose, Hermitian transpose and trace of a matrix, energy, novel transmission approaches employ low resolution respectively. 1 denotes a column vector of ones and I denotes ADCs, which generate significant nonlinear distortion and, an identity matrix. The operator E[ ] stands for expectation with respect to the random variables· and the operator cor- thus, require new signal processing techniques to provide ⊙ reliable data transmission. Several studies present detection responds to the Hadamard product. Finally, diag(A) denotes methods that deal with signals quantized with few bits of a diagonal matrix containing only the diagonal elements of A and nondiag(A)= A diag(A). resolution [1], [6], [2], [4]. − Automatic gain control (AGC) is the process of adjusting II. SYSTEM DESCRIPTION the analog signal level to the dynamic range of the analog to digital converter (ADC) in order to minimize the signal A large-scale uplink MU-MIMO system [12], [14] con- distortion due to the quantization [7]. The use of an AGC sisting of a base station (BS) with NR receive antennas, is important in applications where the received power varies and K users equipped with NT transmit antennas each is over time, as is the case in mobile scenarios. Proper AGC considered. At each time instant i, each user transmits NT symbols which are organized into a NT 1 vector xk[i] = T × The authors would like to thank the CAPES Brazilian agency for funding. [x1[i], x2[i], ..., xNT [i]] . Each entry of the vector xk[i] is a symbol taken from the modulation alphabet A. The symbol undergo nearly the same distortion factor ρq. It was shown in vector is then transmitted through flat fading channels and [8] for the uniform quantizer case, that optimal quantization b corrupted by additive white Gaussian noise (AWGN). The step ∆ for a Gaussian source decreases as √b2− and that ρq 2 received signal collected by the receive antennas at the BS ∆ is asymptotically well approximated by 12 . is given by the following equation: III. PROPOSED JOINT AGC AND LINEAR MMSE K RECEIVER DESIGN y[i]= Hkxk[i]+ n[i]= Hx[i]+ n[i], (1) The procedure for joint optimization of the AGC algorithm k=1 X and the LRA-MMSE receiver carries out alternating compu- NR 1 NR NT where y[i] C × , and Hk C × is a matrix that ∈ ∈ tations between the AGC and the LRA-MMSE receiver. The contains the complex channel gains from the NT transmit first step consists of computing an LRA-MMSE receive filter antennas of user k to the NR receive antennas of the BS. The that considers the quantization effects. Other approaches to NR 1 vector n[i] is a zero-mean complex circular symmetric computing linear MMSE or related filters [18], [19], [20], [21] × H Gaussian noise vector with covariance matrix E[n[i]n [i]] = can also be considered. After that, we compute the derivative 2 σ I. H is a NR KNT matrix that contains the coefficients n × of the cost function to obtain the optimal AGC coefficients. of the flat fading channels between the transmit antennas of Then, an updated LRA-MMSE receiver is computed. the K users and the receive antennas of the BS. The symbol T vector x[i] = [x1[i], ..., xk[i], ..., xK [i]] contains all symbols A. Linear LRA-MMSE Receive Filter Design that are transmitted by the users at time instant i. The channel In this first step we do not consider the presence of the AGC state information (CSI) is assumed to be unknown to the users in the system. Thus, the received signal after the quantizer is at the transmit side. Therefore, we assume the same symbol expressed, with the Bussgang decomposition [9], as a linear H 2 energy per user and transmit antenna, i.e. E[xx ]= σxI. model r = y + q. To develop the linear receive filter W that minimizes the MSE we use the Wiener-Hopf equations: 1 W = RxrRrr− , (3) where the auto-correlation matrix Rrr is given by H H Rrr = E[rr ]= Ryy + Ryq + Ryq + Rqq, (4) and the cross-correlation matrix Rxr can be expressed as H Rxr = E[xr ]= Rxy + Rxq (5) Fig. 1. An uplink quantized MU-MIMO system We get the auto-correlation matrix Ryy and the cross- As depicted in Fig.1, before to be quantized, y is jointly pre- correlation matrix Rxy directly from the MIMO model as multiplied by the clipping level factor α and the AGC matrix H H G to minimizes the granular and the overload distortions. After Ryy = E[yy ]= HRxxH + Rnn (6) that, the product αGy is quantized and the estimation of the and, transmitted symbol vector x is performed by the LRA-MMSE H H receiver, represented by L. The estimated symbol vector is Rxy = E[xy ]= RxxH (7) represented by xˆ. The real, yi,R, and imaginary, yi,I , parts To compute (4) and (5) we need to obtain the covariance of the complex received signal at each antenna are quantized matrices Ryq, Rqq and Rxq as a function of the channel separately by uniform b-bit resolution ADCs. Therefore, the parameters and the distortion factor ρq. The procedure of how resulting quantized signals read as to obtain these matrices was developed in [1] and we will ri,l = Q(yi,l)= yi,l + qi,l, l R, I , 1 i NR, (2) use some of these results in this work. The cross-correlation ∈{ } ≤ ≤ between the received signal vector and the quantization error where Q( ) denotes the quantization operation and qi,l is the is approximated by resulting quantization· error. Ryq ρqRyy (8) The distortion factor indicates the relative amount of quan- ≈ − tization noise generated by the quantizer, and is given by The covariance matrix of the quantization error is deduced (i,l) 2 2 , where 2 is the variance of the quantizer ρq = σqi,l /σyi,l σqi,l from 2 error and, σ is the variance of the input yi,l. This factor yi,l 2 Rqq ρq diag(Ryy)+ ρ nondiag(Ryy) depends on the number of quantization bits b, the quantizer ≈ q type, and the probability density function of yi,l [1]. In this = ρqRyy (1 ρq)ρq nondiag(Ryy), (9) work the scalar uniform quantizer processes the real and − − and the cross-correlation matrix between the desired signal imaginary parts of the input signal y in a range √b .