# Massive MIMO Systems with Low-Resolution ADCs: Achievable Rates and Allocation of Quantization Bits

Authors: Wence Zhang; Jing Xia; Xu Bao
Journal: Wireless Communications and Mobile Computing
 (2023)

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## Abstract

In massive multiple-input multiple-output (MIMO) systems, the large number of high-resolution analog-to-digital converters (ADCs) lead to high hardware cost and power consumption. In this work, the uplink achievable rates of massive MIMO systems with low-resolution ADCs are studied with consideration of both “Uniform-ADC” that uses ADCs with the same number of quantization bits and “Mixed-ADC” that allows the use of ADCs with different resolutions. By leveraging an additive quantization noise model (AQNM), the asymptotic achievable rates are obtained for maximum ratio combining (MRC), zero-forcing (ZF), and linear minimum mean squared error (LMMSE) receivers in very simple forms. Taking advantages of the theoretical results, we propose two criteria for allocation of quantization bits. It is found that the optimal quantization bits allocation for LMMSE is Mixed-ADCs with number of quantization bits that are polarized, while Uniform-ADC is optimal for MRC and ZF. When there is a constraint on the total ADC power consumption, the proposed quantization-bit allocation scheme for LMMSE becomes Uniform-ADC when the transmit signal-to-noise ratio (SNR) is below a threshold, which is related to the system scale and the ADC power consumption. The theoretical results are verified by Monte-Carlo simulations.


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## Body

## 1. Introduction

Massive multiple-input multiple-output (MIMO) systems that employ a large number of antennas at the base station (BS) are able to provide excess spatial degrees of freedom and significantly improve both the spectral and energy efficiency [1, 2], which makes it a key technology in future wireless communication systems [3]. Despite these appealing advantages, the use of a large number of antennas leads to high hardware cost and power consumption. Therefore, recent works have advocated the use of low-cost and energy-efficient devices [4–6]. Since the main source of power consumption is the power-hungry high-resolution analog-to-digital converters (ADCs) at the receiver, the use of low-resolution ADCs in massive MIMO systems has drawn extensive research interest [7].There are different ways of exploiting the benefits of low-resolution ADCs in massive MIMO systems, i.e., “Uniform-ADC” that uses ADCs with the same resolution [8–20], and “Mixed-ADC” that uses ADCs with different resolutions [21–26]. Regarding Uniform-ADC, the extreme case of using one-bit ADCs in massive MIMO systems was studied in [8–11]. In [8], the optimal detection for receivers with one-bit ADCs was investigated and a near maximum likelihood (ML) detector was proposed. The optimal quantization threshold was designed in [9]. It is shown that using one-bit ADCs can approach the channel estimation error of unquantized systems with an increased by a factor of π/2. In [10, 11], the impact of using one-bit ADCs on the spectral efficiency of massive MIMO systems was studied. It is shown that the high spatial multiplexing gain owing to the use of large antenna array is still achievable with one-bit ADCs. To attain the same performance as massive MIMO systems with high-resolution ADCs, however, much more antennas (over 2–2.5 times) are required. Due to the extremely coarse quantization, receivers with one-bit ADCs suffer from severe quantization noise. Therefore, to relax the ADC resolution from one bit to a few bits may significantly improve the performance. The uplink achievable rates and energy efficiency of Uniform-ADC massive MIMO systems were investigated in [12, 13]. Results show that 4-bit ADCs would approach the performance of perfect quantization for MRC receivers [12], while the most energy-efficient ADC resolution is 4–8 bits and it does not decrease with more BS antennas in most cases [13]. In [14], a Bayes inference-based joint channel and data detection scheme was proposed for Uniform-ADC massive MIMO systems. A novel channel estimation scheme based on Bussgang decomposition was proposed and the approximate achievable rates in the uplink was derived in [15], which reveals that massive MIMO systems with ADCs of only a few bits can approach the performance of unquantized systems. The joint antenna and user selection in Uniform-ADC massive MIMO systems was studied in [16] by solving a cross-entropy optimization problem. In [17], performance analysis for massive MIMO systems with uniform ADCs was carried out under Nakagami-m channels. The implementation of uniform ADCs in millimeter wave (mmWave) was discussed in [18–20]. It is discovered that using ADCs with uniform low resolution can reduce the hardware cost and improve the energy efficiency.Massive MIMO systems with all antennas connected to high-resolution ADCs cause huge power consumption, while using only low-resolution ADCs leads to spectral efficiency degradation. Therefore, Mixed-ADC architecture that uses ADCs with different resolutions was studied in [21–26]. In [21], the authors use one-bit ADCs to partially replace conventional full-precision ADCs and propose that Mixed-ADC scheme is able to achieve a large fraction of the channel capacity obtained by conventional schemes. In [22], a family of detectors were devised through Bayesian inference for uplink massive MIMO systems with mixed ADCs. In [23], the authors investigated channel estimation in Mixed-ADC architecture and analyzed the spectral efficiency. Mixed-ADC massive MIMO systems were studied under Rician fading channel in [24]. In [25], power allocation algorithms for mixed-ADC massive MIMO systems were proposed. In [26], the impact of ADCs was studied in the presence of IQ imbalance.In the design of massive MIMO systems with Uniform-ADC or Mixed-ADC, a key topic is how to choose the optimal resolution bits for ADCs. In [13], it is found that 4–8 bits are energy-efficient. In [27, 28], efficient optimal quantization-bit allocation algorithms are designed for cell-free and mmWave MIMO systems. However, these works are mostly based on simulations and few theoretical results are provided.In this work, we consider a general model for massive MIMO system with low-resolution ADCs that covers both Uniform-ADC and Mixed-ADC architectures. The additive quantization noise model (AQNM) is used to model the quantization process that is widely used in the literature [17, 18]. In contrast to previous work, we aim to provide more theoretical guidance for implementing low-resolution ADCs in massive MIMO systems. The asymptotic achievable rates of linear detectors (i.e., MRC, ZF, and LMMSE) are provided as functions of the ADC quantization bits. Most of the analytical results are presented in very simple forms such that more insights could be discovered. Based on the theoretical results, we formulate the optimization problem for the allocation of ADC bits with a constraint on total power consumption. By leveraging the theory of majorization, an approximate solution is derived in closed form. The main contributions of this work are summarized as follows:
(i)
The asymptotic achievable rates of MRC and ZF detectors are derived in closed form. It is found that the received signal-to-interference-plus-noise ratio (SINR) is closely related to the inverse of quantization coefficients. Moreover, there is an upper bound for the SINR in the high transmit signal-to-noise ratio (SNR) region(ii)
For the LMMSE detection, approximations of the achievable rates are provided for both high and low transmit SNR region in very simple forms(iii)
The optimal ADC quantization-bit allocation is studied with or without a total power constraint. By using majorization theory, we propose simple design criteria for massive MIMO systems with low-resolution ADCs for MRC, ZF, and LMMSE receivers(iv)
Monte-Carlo simulations are carried out. The theoretical analysis and the proposed quantization-bit allocation criteria are verified by numerical resultsThe rest of the paper is organized as follows. The system model is described in Section2. In Section 3, the achievable rates for MRC, ZF, and LMMSE receivers are derived. In Section 4, the optimal allocation of resolution bits is investigated. The numerical results are given in Section 5 and conclusions are drawn in Section 6.


## 2. System Model

As shown in Figure1, we consider the uplink of a single-cell massive MIMO system with a BS which employs an array of M antennas and serves K single-antenna users. The received signal y∈ℂM at the BS is described as
(1)y=puHx+n,where H∈ℂM×K denotes the channel matrix between the BS and the users; x=x1,⋯,xKT represents the transmitted symbol vector of all the users, the entries of which are identically independently distributed (i.i.d.) and follow the distributions of CN0,1, i.e., circularly symmetric complex Gaussian with zero mean and unit variance; pu is the average transmit power of each user; n~CN0,σn2IK represents the additive Gaussian white noise with zero mean and covariance given by σn2IK.Figure 1
System model of the uplink of a massive MIMO system with ADCs with an arbitrary number of bits.The wireless channelH is modeled as
(2)H=GD1/2,where G∈ℂM×K stands for small-scale fading with i.i.d entries that follow the distributions of CN0,1/M; D is a K×K diagonal matrix with diagonal entries βn (n=1,⋯,K), which models both geometric attenuation and shadow fading. βn is assumed to be known at the receiver and is constant across the antenna array since the BS antennas are considered to be colocated.In order to study the optimal allocation of bits of ADCs, we consider that each antenna at the BS is connected to an ADC with arbitrary resolutions. The numbers of ADC resolution bits of all antennas can be expressed asb=b1,b2,⋯,bMT, where bm can take any integer value from 1 to the maximum resolution bits bmax.The received signal at the BS after quantization is obtained using AQNM in [12] as
(3)yq=Ay+nq=puAHx+An+nq,with diagonal matrix A=diagα1,⋯,αM and αm=1−ρm, where ρm is the inverse of the signal-to-quantization-noise ratio. nq is the additive Gaussian quantization noise vector that is uncorrelated with y. According to [12], the values of ρm are listed in Table 1 for bm≤5 and can be approximated by ρm=π3/2·2−2bm for bm>5. The covariance of the quantization noise nq is given by
(4)Rnqnq=EnqnqH=AIM−AEdiagpuHHH+σn2IM.Table 1
Distortion factors for different quantization bits.
bm12345ρm0.36340.11750.034540.0094970.002499Since the entries ofG follow the distribution of CN0,1/M, each of the diagonal elements of EHHH is obtained as L/M with L being the trace of D, i.e. L=TrD. Therefore, the covariance of the quantization noise nq is obtained as
(5)Rnqnq=AIM−ApuL/M+σn2.In this paper, the ADC resolution bits are considered to be arbitrary, thus to cover all cases of Uniform-ADC and Mixed-ADC schemes.


## 3. Achievable Rate Analysis for Linear Receivers

In this section, the uplink achievable rates of each user in massive MIMO systems with low-resolution ADCs and linear receivers (i.e., MRC, ZF, and LMMSE) are derived. An asymptotic analysis is carried out and simplified expressions for the achievable rates are also provided.


### 3.1. Achievable Rates Analysis for MRC Receivers

The receive filter for MRC is given byWMRC=HHA−1, where A−1 is introduced for ADC calibration so as to simplify signal processing, such as channel estimation. Then the estimate of the signal is obtained as
(6)rMRC=puHHA−1AHx+HHA−1An+HHA−1nq.The estimated signal of userk is derived as
(7)rMRC,k=puhkHhkxk+pu∑i=1,i≠kKhkHhixi+hkHn+hkHA−1nq,where hk is the k-th column of H. According to (7), the achievable rates of MRC are derived and summarized in the following proposition.Proposition 1.
WhenM and K are large, the achievable rate of user k for MRC receivers is given by RMRC,k⟶log21+SINRMRC,k, where
(8)SINRMRC,k=M+1puβkpuL−βk+TrE,where E=σn2IM+puL/M+σn2A−1−IM.Proof.
According to (7), the signal power of user k is obtained as
(9)EpuhkHhkhkHhk=puβk2Egk22=M+1puβk2M,where gk denotes the k-th column of G. The noise power is obtained as EhkHnnHhk=βkσn2. Similarly, the interference power is derived as ∑i=1,i≠kKEpuhkHhihiHhk=βkpuL−βk/M. The quantization noise power is given by
(10)EhkHA−1RnqnqA−1hk=EhkH−σn2IM+Ehk=βkMTrE−βkσn2.
Then the SINR is obtained straightforwardly as in (8).As can be seen from Proposition1, the low-resolution ADCs affect SINRMRC,k through E. Denoting the transmit SNR as SNRt=pu/σn2, there is an upper bound of SINRMRC,k in the high SNRt region.Corollary 2.
The asymptotic achievable rates of userk for MRC receivers is upper bounded by
(11)SINRMRC,kUB=MM+1βkML−βk+LTrA−1−I.Proof.
The proof is obtained by lettingpu⟶∞.Corollary2 indicates besides interuser interference, in the high SNRt region the achievable rates are limited by low-resolution ADCs and different choices of A greatly affect the performance of MRC receivers.


### 3.2. Achievable Rates Analysis for ZF Receivers

For ZF receivers, the receive filter is given byWZF=HHH−1HHA−1. Then the estimate of the signal is obtained as
(12)rZF=puHHH−1HHA−1Ax+HHH−1HHA−1n+HHH−1HHA−1nq=pux+HHH−1HHn+HHH−1HHA−1nq.The expression for the achievable rates of ZF receivers are given in the following proposition.Proposition 3.
WhenM and K are large, the achievable rate of user k for ZF receivers is given by RZF,k⟶log21+SINRZF,k, where SINRZF,k is given by
(13)SINRZF,k=M−KpuβkTrE,where E is the same as in (8).Proof.
For ZF receivers, the estimated signal of userk is derived from (12) as
(14)rZF,k=puxk+HHHk−1HHn+HHHk−1HHA−1nq,where HHHk−1 is the k-th row of HHH−1. Then the desired signal power is pu and the interference power is 0. Now we only need to compute the noise and quantization noise power, which is given by EHHHk−1HHnnHHHHHk−1H=EHHHkk−1σn2=aM/M−Kβk−1σn2, where (a) follows the property of central Wishart’s matrix.
The quantization noise power is obtained as(15)EHHHk−1HHA−1nqnqHA−1HHHHk−1H=puL/M+σn21KETrHHH−1HHA−1−IMHHHH−1=puL/M+σn21KETrHHHH−2HHA−1−IM=puL/M+σn21KEHmHHH−2HmHTrA−1−IM=puL/M+σn2TrA−1−IM1K1METrHHHH−2HH=puL/M+σn2TrA−1−IMβk−1M−K,where Hm is the m-th row of H.
By combining the above results, the SINR of userk is easily obtained as (13).Similarly as MRC, ZF receivers also have an upper bound in the highSINRt region.Corollary 4.
The asymptotic achievable rates of userk for ZF receiver with ADC calibration is upper bounded by
(16)SINRZF,kUB=MM−KβkLTrA−1−I.Proof.
The proof is obtained by lettingpu⟶∞.Unlike MRC, the interuser interference diminishes for ZF. However, the achievable rates of ZF receivers are limited by low-resolution ADCs and carefully choosingA may help improve the performance.


### 3.3. Achievable Rates Analysis for LMMSE Receivers

We first derive the LMMSE receive filterWLMMSE by solving the optimization problem WLMMSE=argminWEx−Wyq2, the solution to which is expressed as WLMMSE=RxyqRyqyq−1, where
(17)Rxyq=ExyqH=puHHAH,Ryqyq=EyqyqH=puAHHHAH+σn2AAH+Rnqnq.DenoteQ≜puHHH+σn2IM+puL/M+σn2A−1−IM−1 and then we have
(18)WLMMSE=HHQA−1where pu is omitted since it does not affect the analysis. Note that compared with the traditional LMMSE receive filter, there are additional terms that are related to A in WLMMSE. This makes the achievable rates analysis completely different, especially for Mixed-ADC schemes where the entries of A take different values. Note that the term A−1−IM exists in both high and low SNR region. Therefore, LMMSE behaves differently from ZF and MRC which will become clear in the following sections.The estimate of the signal is given by(19)rLMMSE=puWLMMSEAHx+WLMMSEAn+WLMMSEnq=puHHQHx+HHQn+HHQA−1nq,The estimated signal of userk is thus obtained by simple algebraic manipulations as
(20)rLMMSE,k=fk,kpuxk+∑i=1,i≠kKfk,ipuxi+fnk+fqk,where fk,i=hkHQhi (i=1,⋯,K) is the effective gain for the signal of user i; fnk=hkHQn and fqk=hkHQA−1nq are the effective white noise and quantization noise, respectively.Following a similar procedure as in [29, 30], given any allocation of bits in the ADCs, i.e., A, the SINR of user k can thus be approximated as
(21)SINRLMMSE,k=puEfk,k2puvarfk,k+Epu∑i≠kfk,i2+fnk2+fqk2.Assuming Gaussian signaling, the expression for the achievable rates of thek-th user RLMMSE,k is given by RLMMSE,k=log21+SINRLMMSE,k.An accurate analytical result ofSINRLMMSE,k is derived in our previous work and summarized as Proposition 1 in [30]. This result is derived under the assumption that M⟶∞ and K⟶∞, but it provides accurate approximation for limited values of M and K, though the expression is complicated. In this paper, we provide simplified expressions in Proposition 5.Proposition 5.
For LMMSE receivers, the SINR of userk is approximated in the low SNRt region by
(22)SINRLMMSE,kL≈puβkTrE−12puL−βkTrE−2+MTrE−1,and in the high SNRt region by
(23)SINRLMMSE,kH≈puβkTrE−1M.Proof.
Let us rewriteWLMMSE=IK+puD1/2GHE−1GD1/2−1HHE−1A−1. Since the elements of G, i.e., gmk are i.i.d and follows the distribution of CN0,1/M, according to Lemma 4 in [31], WLMMSE is approximated by
(24)WLMMSE≈IK+puTrE−1MD−1HHE−1A−1,where the off-diagonal elements of GHE−1G is omitted since they have little impact on the the overall matrix-inverse in W when pu is small in the low SNRt region.
Therefore, the received signal can be approximated by(25)r≈pupuTrE−1MD+IK−1HHE−1Hx+n+A−1nq.
Consequently, the received signal of userk is obtained as
(26)rk=pupuTrE−1Mβk+1−1hkHE−1hkxk+∑i≠kKhixi+n+A−1nq.
In (26), the desired signal power is given by
(27)EpupuTrE−1Mβk+1−1hkHE−1hk2≈apupuTrE−1Mβk+1−1βkTrE−1M2.where (a) follows Lemma 4 in [31]. The interference power is obtained as
(28)E∑i=1,i≠kKpupuTrE−1Mβk+1−1hkHE−1hihiHE−1hkpuTrE−1Mβk+1−1pu=pupuTrE−1Mβk+1−2EhkHE−1H−kH−kHE−1hk=pupuTrE−1Mβk+1−2L−βkMEhkHE−1E−1hk≈apupuTrE−1Mβk+1−21M2βkL−βkTrE−2,where (a) is obtained using Lemma 4 in [31]. Similarly, the noise power is obtained as
(29)EpuTrE−1Mβk+1−1hkHE−1nnHE−1hkpuTrE−1Mβk+1−1≈apuTrE−1Mβk+1−21Mβkσn2TrE−2,in which (a) follows Lemma 4 in [31]. The quantization noise power is given by
(30)EpuTrE−1Mβk+1−1hkHE−1A−1nqnqHA−1E−1hkpuTrE−1Mβk+1−1=puTrE−1Mβk+1−2EhkHE−1IM−ApuL/M+σn2A−1E−1hk≈apuTrE−1Mβk+1−2βkpuL/M+σn21MTrE−1IM−AA−1E−1,where (a) follows Lemma 4 in [31].
According to (27), (28), (29), and (30), we obtain the approximate SINR of the user k in the low SNRt region as
(31)SINRkL≈puTr2E−1βkpuL−βkTrE−2+σn2MTrE−2+MpuL/M+σn2TrE−2A−1−IM.
SinceE=σn2IM+puL/M+σn2A−1−IM, we have A−1=1/puL/M+σn2E+puL/MIM. Substituting A−1 into (31) yields (22) in Proposition 5.
Regarding the highSNRt region, WLMMSE is approximated by
(32)WLMMSE=IK+puHHE−1H−1HHE−1A−1≈puHHE−1H−1HHE−1A−1,where IK is omitted since it has little impact when pu is significantly large.
Similarly as in the lowSNRt region, the received signal in the high SNRt region can be expressed as
(33)r=puWLMMSEAHx+WLMMSEAn+WLMMSEnq=pupuHHE−1H−1HHE−1Hx+n+A−1nq≈1pux+puTrE−1MD−1HHE−1n+puTrE−1MD−1HHE−1A−1nq.
Then the received signal of userk can be approximated by
(34)rk≈1puxk+puTrE−1Mβk+1−1hkHE−1n+puTrE−1Mβk+1−1hkHE−1A−1nq.
The desired signal power is given by(35)1puExkxk∗=1pu.
The interference power is omitted and the noise power is given by(36)EpuTrE−1Mβk−1hkHE−1nnHE−1hkpuTrE−1Mβk−1≈apuTrE−1Mβk−21Mβkσn2TrE−2,in which a is obtained using Lemma 4 in [31]. The quantization noise is given by
(37)EpuTrE−1Mβk−1hkHE−1A−1nqnqHA−1E−1hkpuTrE−1Mβk−1=puTrE−1Mβk−2hkHE−1IM−ApuL/M+σn2A−1E−1hk≈apuTrE−1Mβk−2βkpuL/M+σn21MTrE−1IM−AA−1E−1,where a follows Lemma 4 in [31].
Combining (35), (36), and (37), the approximated SINR of user k in high SNRt region is obtained as
(38)SINRLMMSE,kH≈1MpuβkpuTrE−1βk−2σn2TrE−2+puL/M+σn2TrE−2A−1−IM.
From (38), it easy to obtain (23).Due to the impact of low-resolution ADCs, the achievable rates of LMMSE receiver have an upper bound in the highSNRt region.Corollary 6.
The SINR of userk is upper bounded in the high SNRt region as
(39)SINRLMMSE,kH≤βkLTrA−1−IM−1.Proof.
The proof is obtained by lettingpu⟶∞.From Proposition5, it is found that ADCs with higher resolution will lead to smaller diagonal entries of E, thus resulting in better performance. When the channel gain βk is larger, SINRLMMSE,k becomes higher as expected.Corollary 7.
LetSNR¯r≜puL/σn2K be the average received SNR of all users at the BS and denote γ≜K/M. Among all the users, the performance of users that have improved channel gain will become better while that of other users will decrease as SNR¯r increases.Proof.
It is ready to see thatSINRLMMSE,k always increases with βk. In the high SNR region, from (23) we get
(40)SINRkH≈puβkMTrσn2IM+puLM+σn2A−1−IM−1=puβkMTrpuLM+σn2A−1−puLMIM−1,
LetSNRr,k be the received SNR of user k at the BS. By simple algebraic manipulations, we obtain
(41)SINRkH≈puβkσn21MTrγSNR¯r+1A−1−γSNR¯rIM−1=SNRr,k1MTrγSNR¯r+1A−1−γSNR¯rIM−1=SNRr,k1M∑m=1M1γSNR¯r+1αi−1−γSNR¯r.
It is apparent from (41) that for fixed values of A and SNRr,k, the receive SINR will decrease with larger values of SNR¯r.
The proof for the low SNR region is similar by using the fact that∂/∂XiiTrX2>∂/∂XiiTrX2 for any positive definite matrix X with diagonal elements given by Xii.Remarks.
When there are a large number of users,SNR¯r is determined by their topology in the cell. Corollary 7 reveals how users interfere with each other due to the impact of low-resolution ADCs. For some specific user j, if βj becomes larger which indicates a better channel condition, then SINRLMMSE,j will improve. However, the SINR for all the other users will decrease due to the additional quantization noise caused by increased signal power of user j.


## 3.1. Achievable Rates Analysis for MRC Receivers

The receive filter for MRC is given byWMRC=HHA−1, where A−1 is introduced for ADC calibration so as to simplify signal processing, such as channel estimation. Then the estimate of the signal is obtained as
(6)rMRC=puHHA−1AHx+HHA−1An+HHA−1nq.The estimated signal of userk is derived as
(7)rMRC,k=puhkHhkxk+pu∑i=1,i≠kKhkHhixi+hkHn+hkHA−1nq,where hk is the k-th column of H. According to (7), the achievable rates of MRC are derived and summarized in the following proposition.Proposition 1.
WhenM and K are large, the achievable rate of user k for MRC receivers is given by RMRC,k⟶log21+SINRMRC,k, where
(8)SINRMRC,k=M+1puβkpuL−βk+TrE,where E=σn2IM+puL/M+σn2A−1−IM.Proof.
According to (7), the signal power of user k is obtained as
(9)EpuhkHhkhkHhk=puβk2Egk22=M+1puβk2M,where gk denotes the k-th column of G. The noise power is obtained as EhkHnnHhk=βkσn2. Similarly, the interference power is derived as ∑i=1,i≠kKEpuhkHhihiHhk=βkpuL−βk/M. The quantization noise power is given by
(10)EhkHA−1RnqnqA−1hk=EhkH−σn2IM+Ehk=βkMTrE−βkσn2.
Then the SINR is obtained straightforwardly as in (8).As can be seen from Proposition1, the low-resolution ADCs affect SINRMRC,k through E. Denoting the transmit SNR as SNRt=pu/σn2, there is an upper bound of SINRMRC,k in the high SNRt region.Corollary 2.
The asymptotic achievable rates of userk for MRC receivers is upper bounded by
(11)SINRMRC,kUB=MM+1βkML−βk+LTrA−1−I.Proof.
The proof is obtained by lettingpu⟶∞.Corollary2 indicates besides interuser interference, in the high SNRt region the achievable rates are limited by low-resolution ADCs and different choices of A greatly affect the performance of MRC receivers.


## 3.2. Achievable Rates Analysis for ZF Receivers

For ZF receivers, the receive filter is given byWZF=HHH−1HHA−1. Then the estimate of the signal is obtained as
(12)rZF=puHHH−1HHA−1Ax+HHH−1HHA−1n+HHH−1HHA−1nq=pux+HHH−1HHn+HHH−1HHA−1nq.The expression for the achievable rates of ZF receivers are given in the following proposition.Proposition 3.
WhenM and K are large, the achievable rate of user k for ZF receivers is given by RZF,k⟶log21+SINRZF,k, where SINRZF,k is given by
(13)SINRZF,k=M−KpuβkTrE,where E is the same as in (8).Proof.
For ZF receivers, the estimated signal of userk is derived from (12) as
(14)rZF,k=puxk+HHHk−1HHn+HHHk−1HHA−1nq,where HHHk−1 is the k-th row of HHH−1. Then the desired signal power is pu and the interference power is 0. Now we only need to compute the noise and quantization noise power, which is given by EHHHk−1HHnnHHHHHk−1H=EHHHkk−1σn2=aM/M−Kβk−1σn2, where (a) follows the property of central Wishart’s matrix.
The quantization noise power is obtained as(15)EHHHk−1HHA−1nqnqHA−1HHHHk−1H=puL/M+σn21KETrHHH−1HHA−1−IMHHHH−1=puL/M+σn21KETrHHHH−2HHA−1−IM=puL/M+σn21KEHmHHH−2HmHTrA−1−IM=puL/M+σn2TrA−1−IM1K1METrHHHH−2HH=puL/M+σn2TrA−1−IMβk−1M−K,where Hm is the m-th row of H.
By combining the above results, the SINR of userk is easily obtained as (13).Similarly as MRC, ZF receivers also have an upper bound in the highSINRt region.Corollary 4.
The asymptotic achievable rates of userk for ZF receiver with ADC calibration is upper bounded by
(16)SINRZF,kUB=MM−KβkLTrA−1−I.Proof.
The proof is obtained by lettingpu⟶∞.Unlike MRC, the interuser interference diminishes for ZF. However, the achievable rates of ZF receivers are limited by low-resolution ADCs and carefully choosingA may help improve the performance.


## 3.3. Achievable Rates Analysis for LMMSE Receivers

We first derive the LMMSE receive filterWLMMSE by solving the optimization problem WLMMSE=argminWEx−Wyq2, the solution to which is expressed as WLMMSE=RxyqRyqyq−1, where
(17)Rxyq=ExyqH=puHHAH,Ryqyq=EyqyqH=puAHHHAH+σn2AAH+Rnqnq.DenoteQ≜puHHH+σn2IM+puL/M+σn2A−1−IM−1 and then we have
(18)WLMMSE=HHQA−1where pu is omitted since it does not affect the analysis. Note that compared with the traditional LMMSE receive filter, there are additional terms that are related to A in WLMMSE. This makes the achievable rates analysis completely different, especially for Mixed-ADC schemes where the entries of A take different values. Note that the term A−1−IM exists in both high and low SNR region. Therefore, LMMSE behaves differently from ZF and MRC which will become clear in the following sections.The estimate of the signal is given by(19)rLMMSE=puWLMMSEAHx+WLMMSEAn+WLMMSEnq=puHHQHx+HHQn+HHQA−1nq,The estimated signal of userk is thus obtained by simple algebraic manipulations as
(20)rLMMSE,k=fk,kpuxk+∑i=1,i≠kKfk,ipuxi+fnk+fqk,where fk,i=hkHQhi (i=1,⋯,K) is the effective gain for the signal of user i; fnk=hkHQn and fqk=hkHQA−1nq are the effective white noise and quantization noise, respectively.Following a similar procedure as in [29, 30], given any allocation of bits in the ADCs, i.e., A, the SINR of user k can thus be approximated as
(21)SINRLMMSE,k=puEfk,k2puvarfk,k+Epu∑i≠kfk,i2+fnk2+fqk2.Assuming Gaussian signaling, the expression for the achievable rates of thek-th user RLMMSE,k is given by RLMMSE,k=log21+SINRLMMSE,k.An accurate analytical result ofSINRLMMSE,k is derived in our previous work and summarized as Proposition 1 in [30]. This result is derived under the assumption that M⟶∞ and K⟶∞, but it provides accurate approximation for limited values of M and K, though the expression is complicated. In this paper, we provide simplified expressions in Proposition 5.Proposition 5.
For LMMSE receivers, the SINR of userk is approximated in the low SNRt region by
(22)SINRLMMSE,kL≈puβkTrE−12puL−βkTrE−2+MTrE−1,and in the high SNRt region by
(23)SINRLMMSE,kH≈puβkTrE−1M.Proof.
Let us rewriteWLMMSE=IK+puD1/2GHE−1GD1/2−1HHE−1A−1. Since the elements of G, i.e., gmk are i.i.d and follows the distribution of CN0,1/M, according to Lemma 4 in [31], WLMMSE is approximated by
(24)WLMMSE≈IK+puTrE−1MD−1HHE−1A−1,where the off-diagonal elements of GHE−1G is omitted since they have little impact on the the overall matrix-inverse in W when pu is small in the low SNRt region.
Therefore, the received signal can be approximated by(25)r≈pupuTrE−1MD+IK−1HHE−1Hx+n+A−1nq.
Consequently, the received signal of userk is obtained as
(26)rk=pupuTrE−1Mβk+1−1hkHE−1hkxk+∑i≠kKhixi+n+A−1nq.
In (26), the desired signal power is given by
(27)EpupuTrE−1Mβk+1−1hkHE−1hk2≈apupuTrE−1Mβk+1−1βkTrE−1M2.where (a) follows Lemma 4 in [31]. The interference power is obtained as
(28)E∑i=1,i≠kKpupuTrE−1Mβk+1−1hkHE−1hihiHE−1hkpuTrE−1Mβk+1−1pu=pupuTrE−1Mβk+1−2EhkHE−1H−kH−kHE−1hk=pupuTrE−1Mβk+1−2L−βkMEhkHE−1E−1hk≈apupuTrE−1Mβk+1−21M2βkL−βkTrE−2,where (a) is obtained using Lemma 4 in [31]. Similarly, the noise power is obtained as
(29)EpuTrE−1Mβk+1−1hkHE−1nnHE−1hkpuTrE−1Mβk+1−1≈apuTrE−1Mβk+1−21Mβkσn2TrE−2,in which (a) follows Lemma 4 in [31]. The quantization noise power is given by
(30)EpuTrE−1Mβk+1−1hkHE−1A−1nqnqHA−1E−1hkpuTrE−1Mβk+1−1=puTrE−1Mβk+1−2EhkHE−1IM−ApuL/M+σn2A−1E−1hk≈apuTrE−1Mβk+1−2βkpuL/M+σn21MTrE−1IM−AA−1E−1,where (a) follows Lemma 4 in [31].
According to (27), (28), (29), and (30), we obtain the approximate SINR of the user k in the low SNRt region as
(31)SINRkL≈puTr2E−1βkpuL−βkTrE−2+σn2MTrE−2+MpuL/M+σn2TrE−2A−1−IM.
SinceE=σn2IM+puL/M+σn2A−1−IM, we have A−1=1/puL/M+σn2E+puL/MIM. Substituting A−1 into (31) yields (22) in Proposition 5.
Regarding the highSNRt region, WLMMSE is approximated by
(32)WLMMSE=IK+puHHE−1H−1HHE−1A−1≈puHHE−1H−1HHE−1A−1,where IK is omitted since it has little impact when pu is significantly large.
Similarly as in the lowSNRt region, the received signal in the high SNRt region can be expressed as
(33)r=puWLMMSEAHx+WLMMSEAn+WLMMSEnq=pupuHHE−1H−1HHE−1Hx+n+A−1nq≈1pux+puTrE−1MD−1HHE−1n+puTrE−1MD−1HHE−1A−1nq.
Then the received signal of userk can be approximated by
(34)rk≈1puxk+puTrE−1Mβk+1−1hkHE−1n+puTrE−1Mβk+1−1hkHE−1A−1nq.
The desired signal power is given by(35)1puExkxk∗=1pu.
The interference power is omitted and the noise power is given by(36)EpuTrE−1Mβk−1hkHE−1nnHE−1hkpuTrE−1Mβk−1≈apuTrE−1Mβk−21Mβkσn2TrE−2,in which a is obtained using Lemma 4 in [31]. The quantization noise is given by
(37)EpuTrE−1Mβk−1hkHE−1A−1nqnqHA−1E−1hkpuTrE−1Mβk−1=puTrE−1Mβk−2hkHE−1IM−ApuL/M+σn2A−1E−1hk≈apuTrE−1Mβk−2βkpuL/M+σn21MTrE−1IM−AA−1E−1,where a follows Lemma 4 in [31].
Combining (35), (36), and (37), the approximated SINR of user k in high SNRt region is obtained as
(38)SINRLMMSE,kH≈1MpuβkpuTrE−1βk−2σn2TrE−2+puL/M+σn2TrE−2A−1−IM.
From (38), it easy to obtain (23).Due to the impact of low-resolution ADCs, the achievable rates of LMMSE receiver have an upper bound in the highSNRt region.Corollary 6.
The SINR of userk is upper bounded in the high SNRt region as
(39)SINRLMMSE,kH≤βkLTrA−1−IM−1.Proof.
The proof is obtained by lettingpu⟶∞.From Proposition5, it is found that ADCs with higher resolution will lead to smaller diagonal entries of E, thus resulting in better performance. When the channel gain βk is larger, SINRLMMSE,k becomes higher as expected.Corollary 7.
LetSNR¯r≜puL/σn2K be the average received SNR of all users at the BS and denote γ≜K/M. Among all the users, the performance of users that have improved channel gain will become better while that of other users will decrease as SNR¯r increases.Proof.
It is ready to see thatSINRLMMSE,k always increases with βk. In the high SNR region, from (23) we get
(40)SINRkH≈puβkMTrσn2IM+puLM+σn2A−1−IM−1=puβkMTrpuLM+σn2A−1−puLMIM−1,
LetSNRr,k be the received SNR of user k at the BS. By simple algebraic manipulations, we obtain
(41)SINRkH≈puβkσn21MTrγSNR¯r+1A−1−γSNR¯rIM−1=SNRr,k1MTrγSNR¯r+1A−1−γSNR¯rIM−1=SNRr,k1M∑m=1M1γSNR¯r+1αi−1−γSNR¯r.
It is apparent from (41) that for fixed values of A and SNRr,k, the receive SINR will decrease with larger values of SNR¯r.
The proof for the low SNR region is similar by using the fact that∂/∂XiiTrX2>∂/∂XiiTrX2 for any positive definite matrix X with diagonal elements given by Xii.Remarks.
When there are a large number of users,SNR¯r is determined by their topology in the cell. Corollary 7 reveals how users interfere with each other due to the impact of low-resolution ADCs. For some specific user j, if βj becomes larger which indicates a better channel condition, then SINRLMMSE,j will improve. However, the SINR for all the other users will decrease due to the additional quantization noise caused by increased signal power of user j.


## 4. Optimal Allocation of Quantization Bits

In this section, the optimal allocation of quantization bits is studied. Unlike previous work in [13, 27] and [28], where the optimal resolutions for ADCs are either obtained by simulations or numerical algorithms, in this work, we aim to find theoretical criteria for the optimal quantization bits of all ADCs.Conjecture 1.
For any two allocations of bits of ADCsA1 and A2 that satisfy TrA1=TrA2, denote α1 and α2 as the vectors formed by diagonal entries of A1 and A2, respectively. If α1 is majorized by α2 (i.e., α1≺α2), the achievable rates corresponding to A1 is less than that of A2 for LMMSE, but is greater than the latter for MRC and ZF.Let us start with LMMSE in lowSNRt region. After simple scaling and expansion we get
(42)puβkpuL−βk+MTr−1E−1≤SINRLMMSE,kL≤puβkTrE−1puL−βk+M.SinceE is a diagonal matrix, we can rewrite TrE−1 as TrE−1=∑m=1M1/σn2+puL/M+σn2αm−1−1, where the second order derivative of each term in the summation is given by
(43)∂2∂αm2Emm−1=2puL/MpuL/M+σn2puL/M+σn2−αmpuL/M3.Whenαm<1, (43) is larger than 0. Therefore, each term in the summation in TrE−1 is convex over αm, and TrE−1 is a convex function of αm’s. According to the theory of majorization [32], it is obtained that when TrA is fixed, TrE−1α1<TrE−1α2 if α1≺α2. Consequently, both the upper bound and lower bound of SINRkL for α1 are higher, and thus it is reasonable to result in better performance for A1. The analysis for MRC, ZF, and LMMSE in the high SNRt region is similar and thus omitted.Conjecture1 reveals that the optimal architecture for LMMSE is Mixed-ADC with polarized quantization bits. In this case, it tends to use combination of one-bit ADCs and those with the highest resolution. For MRC and ZF, however, completely opposite conclusion is obtained, where the optimal solution is Uniform-ADC that uses ADCs with all the same resolution. Conjecture 1 indicates that Mixed-ADC does not always outperform Uniform-ADC. It depends on the specific detection method used at the receiver.Based on Conjecture1, we propose Majorization-based Allocation of Bits with Fixed Trace of A (MAB-FT) algorithm to find the optimal ADC quantization bits b∗. In Algorithm 1, bvalue=bmax,bmax−1,⋯,1T is the group of ADC quantization bits that can be used. The quantization coefficient αvalue is computed according to Table 1. The proposed algorithm can be considered “optimal” in the sense that it finds b∗ that majorizes the other ADCs resolution profiles and thus provides the maximal achievable rates with fixed TrA.Algorithm 1: Majorization based Allocation of Bits for ADCs with Fixed TrA (MAB-FT).
Input:M, bvalue, TrA;i=1, j=0, αsum=0, I=lengthbvalue;Computeαvalue according to Table 1;Whilei<IdoIfαsum+αvaluei+M−1∗αvalueI≤TrAthenj=j+1;M=M−1;αsum=αsum+αvaluei;b∗i=bvaluei;Elsei=i+1;End ifEnd whileb∗j+1:j+M=bvaluei;Output:b∗;In order to provide insights on the performance of massive MIMO systems with low-resolution ADCs, we need to take the ADC power consumption into account, i.e.,pADCbm=cW2bm, where W is the sampling rate and c is the energy consumption per conversion step. Therefore, the following optimization problem is brought about as
(44)P1:maxbRks.t.∑m=1MpADCbm=C,where C is the maximum power consumption of all ADCs. The objective function can be RLMMSE,k, RMRC,k, or RZF,k. By solving Problem P1, we aim to find the optimal quantization-bit allocation with the ADC power-consumption constraint.Conjecture 2.
Assuming that the amount of ADC power consumption is fixed toC and ρm=c12−2bm. The solution to P1 with MRC and ZF receivers is to use Uniform-ADC. However, the solution for LMMSE receivers is given by
(45)Mixed−ADCifγc1SNR¯r>3·22bmax,Uniform−ADCifγc1SNR¯r<12,where the optimal Mixed-ADC scheme has ADC resolutions that mostly spread out.Let us start with the proof for MRC and ZF. In order to maximize the achievable rates, we need to selectA that minimizes TrE. Denote xm=2bm and we obtain αm=1−c1xm−2. Therefore, P1 is reformulated as
(46)minTrEs.t.∑m=1Mxm=CcW.Thei-th diagonal entry of E is rewritten as Eii=σn2+puL/M+σn2c1fxm, where fx=x2−c1−1, the second order derivative of which is given by d2f/dx2=6x2+2c1/x2−c13. According to the values of c1, in most cases we have xm2>c1. Therefore, d2f/dx2>0 and thus fx is convex over x. Consequently, Eii is convex over xm. Then it is readily shown that TrE is convex over xm’s. According to the theory of majorization, TrE is minimized when xm’s take the same value, which leads to a Uniform-ADC scheme.Regarding LMMSE, the optimization problem can be reformulated as(47)maxTrE−1s.t.∑m=1Mxm=CcW.Thei-th diagonal entry of E−1 is rewritten as Eii−1=1/σn2+puL/M+σn2c1σn−2gxm, where gx=−x2+puLM−1c1σn−2−1, the second order derivative of which is given by d2g/dx2=2puLM−1c1σn−2−6x2/x2+puLM−1c1σn−23. It is now straightforward to prove when γc1SNR¯r>3·2bmax, TrE−1 is convex over xm’s. According to [32], the optimal solution to P1 is Mixed-ADC with polarized quantization bits. Similarly, when γc1SNR¯r<6, TrE−1 is concave over xm’s and the optimal solution becomes Uniform-ADC that uses the same number of resolution bits across the antennas.It can be seen from Conjecture2, the optimal structure for MRC and ZF is Uniform-ADC, while that for LMMSE depends on the value of γ, c1, and SNR¯r. With fixed value of C, the optimal allocation of bits for Mixed-ADC can be computed similarly as Algorithm 1 by replacing the constraint. We refer to it as Majorization-based Allocation of Bits with Fixed Power consumption (MAB-FP). Details of the algorithm is omitted for conciseness.Remark.
Although it would be interesting to see the optimal bit allocation under the constraint of total number of quantization bits, this is no easy task due to the truth that there is no closed-form expression forαm as a function of bm. Even though we can extend the approximation αm=1−ρm=1−3π/2·2−2bm for bm≥5 to arbitrary value of bm, it is still very complicated to obtain useful results. Since the paper is already lengthy, we defer the detailed analysis to future work.


## 5. Numerical Results

In this section, we carry out Monte-Carlo simulations to verify the correctness of the theoretical analysis in previous sections. All users are considered to be randomly distributed in a circular cell with a radius of 1000 meters, except for a central disk of radiusrh=100 meters. The large-scale fading is modeled as βk=zk/rk/rhv, where zk is a log-normal variable with standard deviation σshadow=8 dB, rk is the distance between the user k and the BS, and v=4.8 is the path loss exponent. In all the simulations of low-resolution ADCs, bmax is set to 6. The noise power spectrum density is set to be -174 dBm/Hz and the c=494 fJ for all ADCs. The number of BS antennas and users are set to be M=128 and K=20, respectively, unless otherwise specified.Figure2 illustrates the achievable sum rates with respect to the transmit SNR. The sum rates for LMMSE, ZF, and MRC increase with respect to SNRt and then level out in the high SNRt region, due to extra quantization noise of ADCs caused by high-input signal power. The upper bounds in high SNRt region are also derived in (11), (16), and (39). It can be seen from Figure 2 that the asymptotic analysis for ZF and MRC match well with the simulation results. For LMMSE, (22) provides accurate approximation in low SNRt region and (23) is accurate for modest SNRt and is slightly higher than the simulation results when SNRt is very high. Despite an approximation error within about 10% for M=128, (25) shows the same trend as the simulation results. Therefore, it is still very helpful for analysis.Figure 2
Achievable sum rate versus the transmit SNR, whereM=128 and K=20.Figure3 demonstrates how the increased channel gain of a user affects the performance of both users for LMMSE. As the distance between user 1 and the BS decreases, β1 increases, which leads to a higher achievable rates of user 1. However, as described in Corollary 7, the performance of user 2 will decrease due to the increased quantization noise. Different from systems with perfect quantization, when low-resolution ADCs are employed, in addition to interuser interference, users can affect each other through the quantization noise.Figure 3
Achievable rates of two user whereM=128, r1=500 m and r2=700m for Case 1 and r1=300 m and r2=700 m for Case 2.Figure4 shows the comparison of different allocation schemes of quantization bits for ADCs in terms of achievable sum rates, where “Mixed-ADC Random” randomly allocates a number of bits to each ADC with equal probability. In the simulations, TrA is set to be 120. For LMMSE, the proposed MAB-FT performs the best, while uniform-ADC is the worst. In contrast, Uniform-ADC provides the highest sum rates and outperforms Mixed-ADC schemes for MRC and ZF. The simulation results show that Conjecture 1 is correct.Figure 4
Achievable sum rates comparisons for different quantization-bit allocation withTrA=120.Figure5 shows the comparison for different quantization-bit allocation with fixed total power C=30.4 mW. It can be seen that the “Uniform-ADC” scheme performs the best for MRC and ZF. For LMMSE, according to (45), when SNRt>74 dB, the optimal allocation scheme for LMMSE is the proposed MAB-FP, while “Uniform-ADC” is the best when SNRt<44 dB. Figure 5 demonstrates that Conjecture 2 provides sufficient conditions for the optimal allocation schemes for LMMSE.Figure 5
Achievable sum rates comparisons for different quantization-bit allocation with fixed total power.


## 6. Conclusion

In this paper, the uplink performance of MRC, ZF, and LMMSE receivers have been investigated for massive MIMO systems with low-resolution ADCs. Leveraging AQNM and random matrix theory, the asymptotic achievable rates for MRC, ZF, znd LMMSE receivers have been derived as functions of the ADC quantization bits in closed and simple forms. Based on the theoretical analysis, we have proposed two quantization-bit allocation criteria for all the three detection schemes with or without the ADC power constraint. Results indicate that Uniform-ADC is optimal for MRC and ZF, while for LMMSE it depends onc1 and the ratio of K to M. Numerical results verified the correctness of the theoretical analysis.



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*Source: 4012841-2023-02-17.xml*