Massive multiple-input multiple-output (MIMO) is a critical technique to significantly improve the performance of the fifth generation (5G) cellular network . In massive MIMO, the base station (BS) is equipped with hundreds, or even thousands, of antennas to provide high spectral and power efficiency. However, both cost and power consumption increase dramatically with the number of antennas, partly because each antenna requires a pair of dedicated analog-to-digital converters (ADCs). Fortunately, there are two potential means of alleviating this challenging issue. On one hand, low-precision ADCs can be employed since the power consumption decreases exponentially with the quantization precision -. An overview on channel estimation, signal detector, and transmit precoding for massive MIMO using low-precision ADCs in future networks has been provided in . Specifically in , it has shown that 1-bit ADCs can achieve satisfactory performance in terms of theoretical capacity and symbol error rate (SER) in massive MIMO uplink systems. Furthermore, the spectral efficiencies of a mixed-ADC system under energy constraint has been studied in . The mixed-ADC architecture in frequency-selective channels has been investigated in . It has been demonstrated in  that low-precision, e.g., 2-3 bits, ADCs only cause limited sum rate loss under some mild assumptions for an amplify-and-forward relay uplink network. Studies in  and  have analyzed the performance of low-precision transceivers in multiuser massive MIMO downlinks. On the other hand, radio-frequency (RF) chains can be also constrained to reduce the total number of required converters, which leads to a hybrid transceiver architecture  . A low-complexity hybrid precoding method has been proposed in . The study in  has shown that hybrid beamforming can asymptotically approach the performance of fully digital beamforming for a sufficiently large number of antennas. However, in many scenarios, low-precision ADCs inevitably deteriorate the performance while the architecture with limited RF chains sacrifices the multiplexing gain. In practice, it is interesting to find a cost-efficiency tradeoff when employing low-precision ADCs and a limited number of RF chains  .
Meanwhile, in order to achieve ultra high data rates, the spectrum ranging from 30 GHz to 300 GHz, namely millimeter wave (mmWave), looks attractive in 5G . The ten-fold increase in carrier frequency, compared to the current majority of wireless systems, implies that mmWave signals experience an order-of-magnitude increase in free-space loss  . Fortunately, the decrease in wavelength enables to pack a large number of antenna elements into small form factors. Large antenna arrays in mmWave systems are leveraged to combat severe pathloss through a large beamforming gain . In  and , hybrid beamforming has been investigated in mmWave MIMO networks. A joint beam selection scheme for analog precoding has been proposed in  and a relay hybrid precoding design has been studied in . Different from conventional wireless channels in cellular networks, spatial sparsity emerges as a dominant nature in mmWave propagations . By exploiting the sparsity, a beamforming training algorithm has been proposed in . Then, random beamforming has been studied in  as well as a user scheduling algorithm proposed for beam aggregation. For low-complexity hybrid precoding, an algorithm using generalized orthogonal matching pursuit has been proposed in  when the knowledge of channel sparsity is known.
It is known that the availability of channel state information (CSI) plays a critical role in beamforming design . A two-stage precoding scheme has been proposed in  to reduce the overhead of both channel training and CSI feedback in massive MIMO systems. Furthermore, an interference alignment and soft-space-reuse based cooperative transmission scheme has been proposed in  and a low-cost channel estimator has been designed. For systems with low-precision ADCs, conventional pilot-aided channel estimation can in some scenarios be used to acquire the CSI . However, it can be hardly applied to the multiuser hybrid system because the number of RF chains is much smaller than the antenna number. Therefore, channel estimation using overlapped beam patterns and rate adaptation has been proposed in  and a limited feedback hybrid channel estimation has been studied in . To overcome the drawback of the feedback-based mechanism in these methods, a low-complexity channel estimation method has been proposed in .
In this paper, we investigate a non-cooperative multi-cell mmWave system with a large-scale antenna array, where low-precision ADCs are used at the BS. We assume that each cell serves multiple users and each user is equipped with multiple antennas but driven by a single RF chain. Analog beamforming is therefore conducted at user sides based on the estimated CSI. Most of the existing works, like -, focused on either low-precision quantization or hybrid architecture. In our work, we study both the low-precision ADCs at the BS and analog beamforming at the user side. This setup is of much interest due to its implementational popularity in practice  . To the best of our knowledge, there is few works investigating the performance of the network . Main contributions of this work are summarized as follows:
1) We derive the ergodic achievable rate of the network with imperfect CSI by using an ADC quantization model based on the Bussgang theorem. Although the popular tools, such as the law of large numbers and the central limit theorem, do not apply here due to the sparsity of mmWave channel, we successfully derive a lower bound for the user ergodic rate with the help of stochastic calculations.
2) Based on the derived lower bound, the impacts of various system parameters, including the ADC precision, signal and pilot SNRs, and the numbers of users and antennas, on the system performance have been characterized. A typical scenario of a single-cell network is investigated by retrieving as a special case from our derived results. We find that the received signal-to-interference-quantization-and-noise ratio (SIQNR) can be expressed as a scaling value of the original low SNR.
The rest of this paper is organized as follows. Both ADC quantization and mmWave channel models are described in Section II. In Section III, we introduce a two-step channel estimation method for the multi-cell hybrid system. In Section IV, we analyze the achievable uplink rate with imperfect CSI and low-precision quantization error and derive a lower rate bound. Then based on the bound, we analyze the performance under two special scenarios in Section V. Simulation results are presented in Section VI and conclusions are drawn in Section VII.
Notations: , and represent the transpose, conjugate and conjugate transpose of A, respectively. represents the th column of A. keeps only the diagonal elements of A, while generates a diagonal matrix with entries . is the expectation operator. U
denotes the uniform distribution betweenand . denotes the almost sure convergence.
Ii System Model
We consider a non-cooperative multi-cell system consisting of cells. In each cell, user terminals are served simultaneously and antennas are equipped at the BS. Universal frequency reuse is exploited, and therefore both intra-cell and inter-cell interferences exist.
Ii-a Quantization Model for Low-Precision ADCs
As in Fig. 1, each user equips antennas driven by a single RF chain. The RF chain can access to all the antennas through phase shifters, which allows analog beamforming for both transmitting and receiving. At the BS side, a pair of low-precision ADCs is exploited for each antenna for processing the in-phase and quadrature input signals.
It is in general difficult to accurately analyze the signal quantization error of low-precision ADCs. Fortunately, an approximately linear representation has been widely adopted by using the Bussgang theorem . This quantization model has been verified accurate enough for characterizing commonly used ADCs, especially for popular quantization levels in practice  . It decomposes the ADC quantization into two uncorrelated parts as
where is the quantization operation of ADC,
denotes the vector before quantization,F represents the quantization processing matrix, and denotes the quantization noise. From  , it follows
where represents the distortion factor. The distortion factor depends on the ADC precision, , representing the number of the quantized bits of the ADC.
Ii-B Channel Model with Hybrid Architecture
where denotes the large-scale fading from the th user in cell to BS , and denote the antenna array response vectors of the th user in cell and the BS in cell , respectively. The small-scale fadings are represented by and . Due to the sparsity of mmWave channels, each of and is in general a single line-of-sight (LoS) path depending on the corresponding angle of incidence. In particular, we have
where U and U are the corresponding angles of incidence at the antenna arrays of user in cell and BS , respectively, is the distance between adjacent antennas, and is the wavelength of radio signals at the carrier frequency. Typically, let to minimize the space occupied by the massive antenna array while still achieving the optimal diversity .
Since each user is equipped with a single RF chain even if it has multiple antennas as illustrated in Fig. 1, analog beamforming is conducted at the user sides. Let be the beamforming vector at user in cell , which is determined by the estimated angle of arrival (AoA) to be elaborated in Section III.A. The equivalent uplink channel from users in cell to BS can be expressed as
where represents the beamforming gain of the th user, defined as
In the uplink, the received analog signals at BS are converted by ADCs before detection. After the ADC operation, maximal ratio combining (MRC) is utilized for signal detection. At BS , the received signal, , can be expressed as
where is normalized uplink data from all users in cell , i.e., , is the transmit power of each user, and denotes the additive white Gaussian noise (AWGN) in cell with representing the noise power. Here, denotes the estimate of equivalent channel within cell , which will be discussed later in Section III.B.
Iii Channel Estimation
In the above communication process, a critical procedure includes determining the analog beamforming vector, , at user side and the digital combining matrices, , at the BS. The design relies on the availability of CSI at the corresponding nodes.
Iii-a AoA Estimation
In order to determine the beamforming vectors, we first estimate the angle of incidence at each user from the BS in the same cell. A similar procedure as in  is introduced. In order to avoid inter-cell interference, each cell conducts this step in an orthogonal way. Taking cell for instance, BS broadcasts a frequency tone from an arbitrary antenna to all users. Assuming channel reciprocity, the received signal at user in cell can be expressed as
where is the AWGN at user in cell . The receiving beamforming vector, , is expressed as
where is the phase shift of the receiving antenna array. To estimate AoA, we resort to choosing the optimal to maximize the power of received signal . Since ideal analog phase shifter with continuous phase is less practical, we consider an analog beamformer of limited resolution. The value of is chosen from a codebook:
where and is the number of quantization bits for phases. Then, the estimated AoA of user in cell is chosen as
After obtaining , the beamforming vector for user in cell is accordingly set to:
Given that is determined, beamforming gain from BS to user in cell , i.e., in (8), can be obtained by measuring the received signal power. Note that the above AoA estimation is conducted in the downlink while the obtained analog beamforming vector, , is used for uplink transmission. This is realizable thanks to the assumption of channel reciprocity in time division duplex (TDD) mode.
Iii-B Demodulation Channel Estimation
With in (16) and the obtained beamforming gain , we can estimate the uplink channel by transmitting orthogonal pilots from users to the BS at this step. After analog beamforming, we only need to estimate equivalent channel in (7), instead of the original in (4) with a much larger size. Thus, the number of the required pilots decreases from to and the dimension of matrix computation is greatly reduced. If all cells reuse the same pilot sequences, pilot contamination should also be considered. Since low-precision ADCs are deployed at the BS, the accuracy of channel estimation is also affected by the ADC quantization.
Let user in each cell send pilot vector where is the pilot length, which is orthonormal for different users. Define the pilot matrix, , as
then . The received pilot signal at the th BS before ADCs equals
where is the pilot power and denotes the AWGN with for .
where denotes the quantization noise and
. Applying a popular discrete Fourier transform matrix, by substituting (3), (6), and (7), the quantized noise power equals 
After the ADC quantization, the minimum mean-square-error (MMSE) estimator in  is used. The channel estimate can be expressed as
where is the estimation matrix and is the channel estimation error matrix denoted as . By using (7), we have
where we define that , for , and
Iv Uplink Achievable Rate
In this section, we are ready to analyze the uplink achievable rate with low-precision ADC quantization and the above channel estimation. We also derive a tight lower bound for the achievable rate, which provides more insights.
Iv-a Ergodic Achievable Rate Analysis
where denotes the th diagonal element of and denotes the th element of vector . In (26), represents the equivalent thermal noise, denotes the quantization noise, and represents the received signal at BS from all the users, among which the desired signal term from user in cell equals
Note that the common scaler, , in the left hand side of (26) does not affect the evaluation of the received SIQNR. Therefore, we can drop out and remove the subscript for notational brevity. Using (27) and (6), the desired signal power can be expressed as
From (26), we can get the power of interferences and noises as
where comes from the fact that both channel and quantization noises are uncorrelated with the received signal and utilizes (28). Detailed derivations of the first three terms in (29) are given in Appendix A. Thus far, the SIQNR can be expressed as
By applying the assumption of the worst-case Gaussian interference, the ergodic achievable rate of each user can be evaluated as follows
In most literature on massive MIMO, a concise closed-form expression of can be further achieved by applying the law of large numbers to the expression of . The effectiveness relies on the assumption that the dimension of the channel matrix tends large and all the channel coefficients contain a large amount of independent, and possibly identically, random components. Here as observed in (50), (51), and (52) in Appendix A, the terms involving the channel coefficients do not tend to an asymptotically deterministic value even with large . This is because the mmWave MIMO channel is sparse in general. The sparsity makes the channel matrix, in (4), to have only few terms and the law of large number becomes invalid. In particular, even with an infinitely large antenna number , consistently contains only two terms coming from the random angles and in and , respectively. On the other hand, the AoA estimation error lies on the exponent term in the design of and thus affects the value of in (8) highly nonlinearly. Therefore, the analog beamforming gain, , in (50), (51), and (52) is also hard to express in closed form. Consequently, a direct analysis on (31) is difficult.
Iv-B Lower Rate Bound
Since the expression of the achievable rate in (31) is complicated, especially the expression of in (29), we derive a tight lower bound for the rate. Assuming that long-term uplink power control is conducted to compensate for the large-scale fadings of different users in the same cell, the large-scale fading within each cell can be considered identical. For simplicity, assume that the attenuations between different cells remain the same and we have
where . We have the following theorem on the lower bound for the ergodic achievable rate.
A lower bound for the ergodic uplink rate in (31) is given by
where and represent the inter-user and inter-cell interferences, respectively. is the AWGN and denotes the interference caused by ADC quantization. represents the interference due to channel estimation error. They are expressed as:
See Appendix E. ∎
Due to the effect of pilot contamination, the uplink rate converges to a constant with the antenna number increasing to infinity, i.e., . From (33), we have
where we utilize the facts that
where use the property of the Bessel function that , and for , comes from as indicated in (60), and utilizes (40). Further considering , the result in (42) implies . From (39), the desired signal is interfered by signals from other cells with large-scaling fading due to pilot reuse. is a lower bound for the analog beamforming gain at the user side while represents an upper bound for beamforming gain from other cells. Note that the asymptotic SIQNR is not affected by the ADC distortion factor . It is because that the dominating interference caused by pilot contamination is quantized as well as the desired signal.
V Rate Analysis for Single-Cell Scenario
Pilot contamination suppression has been widely investigated in literature, e.g., in . This section then pays attention to the performance of a single-cell network where pilot contamination is temporarily assumed well suppressed. By setting in Theorem 1, we obtain the lower bound for the achievable data rate in a single-cell network as in (43) at the top of the next page, where and are the uplink data and pilot SNRs, respectively. Obviously, BS antenna number , user antenna number , ADC distortion factor , data SNR , and pilot SNR contribute differently to the achievable rate. In addition, the rate decreases with increasing since more users cause more pronounced multiuser interference. Due to the large frequency bandwidth in mmWave communications and the use of massive MIMO, low SNR is able to provide satisfactory data transmission rate  . In the following, we therefore focus on low SNR scenarios, which is of common interest in mmWave massive MIMO applications.
V-a Imperfect CSI with Low Pilot SNR
First, we consider the case with low data and pilot SNRs, i.e., and . Under this condition, the lower bound in (43) can be further simplified as
where follows by substituting the expression of in Lemma 1 and applying the assumption that and , and comes from the fact that when the analog beamforming interval is small enough. It is obvious that the SIQNR is a scaled value of the data SNR by a factor
where represents the SNR attenuation due to the low-precision ADC quantization to the received data signals, and represents the beamforming gain at both the BS and user sides. Factor represents the SNR attenuation due to the channel estimation error. Specifically, the channel estimation error is mainly caused by AWGN with and the pilot quantization error from low-precision ADCs, while the analog beamforming at user side improves the estimation accuracy by .
From the above discussion, we have the following important remarks:
1) From (44), the achievable rate per user is independent of user number . This is because that the channel estimation error is mainly caused by AWGN under the assumption of , which overwhelms the effect of multiuser pilot interference. When transmitting data with , the inter-user interference is negligibly small compared to the thermal noise and the interference caused by imperfect CSI. In this condition, a large user number hardly degrades the achievable rate of each user.
2) Expression (44) explicitly characterizing the relationship between increasing the antenna number and the reduction in and . In particular, a dB reduction in data or pilot SNR needs doubling the BS antennas, or alternatively increasing user antennas by times, in order to maintain the same rate at a low SNR. Therefore, increasing antenna number at the user side is more efficient than that at the BS. However, in practice, the number of antennas at the user side is more tightly restricted by the size of terminals than that at BS.
3) For fixed , remains the same if in (45) keeps as a constant. More antennas or higher pilot power can compensate for the rate loss caused by low-precision ADCs. According to typical values of , the BS needs times receiving antennas when ADC resolution decreases from to , in order to maintain the same rate. In particular at a low SNR, employing antennas with 5-bit ADCs at the BS achieves the same rate as using antennas with 1-bit ADCs, which is also verified by numerical results in Section VI.B.
V-B Imperfect CSI with ADC Quantization Error
In order to improve the accuracy of channel estimation, the pilot power may be set higher than the data transmit power in applications. Here we assume that to clearly see the impact of the low-precision ADCs on channel estimation. In this case, the lower bound in (43) approximately equals
which shares some similarities as in (45). The factor represents the SNR attenuation caused by low-precision ADCs, and represents the array gain obtained by beamforming. The difference between and is the factor , representing the SNR attenuation due to different channel estimation qualities.
Based on the above result, we have the following remarks:
1) Comparing in (47) with in (45), the difference lies in the last multiplicative term because the dominating factors for the imperfect CSI are different. In channel estimation, multiuser interference exists because the received pilot signals are quantized by low-precision ADCs. This quantization operation, to some extent, breaks the orthogonality among pilots from different users in . Under the assumption of , the channel estimation error due to ADC quantization, instead of AWGN, becomes dominating. On one hand, the channel estimation error decreases with because longer pilot improves the channel estimation accuracy. On the other hand, a larger yields more channel estimation error and consequently leads to a lower rate. For a specific choice of