Multipair Massive MIMO Relaying Systems with OneBit ADCs and DACs
Abstract
This paper considers a multipair amplifyandforward massive MIMO relaying system with onebit ADCs and onebit DACs at the relay. The channel state information is estimated via pilot training, and then utilized by the relay to perform simple maximumratio combining/maximumratio transmission processing. Leveraging on the Bussgang decomposition, an exact achievable rate is derived for the system with correlated quantization noise. Based on this, a closedform asymptotic approximation for the achievable rate is presented, thereby enabling efficient evaluation of the impact of key parameters on the system performance. Furthermore, power scaling laws are characterized to study the potential energy efficiency associated with deploying massive onebit antenna arrays at the relay. In addition, a power allocation strategy is designed to compensate for the rate degradation caused by the coarse quantization. Our results suggest that the quality of the channel estimates depends on the specific orthogonal pilot sequences that are used, contrary to unquantized systems where any set of orthogonal pilot sequences gives the same result. Moreover, the sum rate gap between the doublequantized relay system and an ideal nonquantized system is a moderate factor of in the low power regime.
Index terms— Massive MIMO, relays, onebit quantization, power allocation
I Introduction
Multipair multipleinput multipleoutput (MIMO) relaying networks have recently attracted considerable attention since they can provide a costeffective way of achieving performance gains in wireless systems via coverage extension and maintaining a uniform quality of service. In such a system, multiple sources simultaneously exchange information with multiple destinations via a shared multipleantenna relay in the same timefrequency resource. Hence, multiuser interference is the primary system bottleneck. The deployment of massive antenna arrays at the relay has been proposed to address this issue due to their ability to suppress interference, provide large array and spatial multiplexing gains, and in turn to yield large improvements in spectral and energy efficiency [1, 2, 3, 4, 5].
There has recently been considerable research interest in multipair massive MIMO relaying systems. For example, [6] derived the ergodic rate of the system when maximum ratio combining/maximum ratio transmission (MRC/MRT) beamforming is employed and showed that the energy efficiency gain scales with the number of relay antennas in Rayleigh fading channels. Then, [7] extended the analysis to the Ricean fading case and obtained similar power scaling behavior. For fullduplex systems, [8, 9] analytically compared the performance of MRC/MRT and zeroforcing reception/transmission and characterized the impact of the number of user pairs on the spectral efficiency.
All the aforementioned works are based on the assumption of perfect hardware. However, a large number of antennas at the relay implies a very large deployment cost and significant energy consumption if a separate RF chain is implemented for each antenna in order to maintain full beamforming flexibility. In particular, the fabrication cost, chip area and power consumption of the analogtodigital converters (ADCs) and the digitaltoanalog converters (DACs) grow roughly exponentially with the number of quantization bits [10, 11]. The cumulative cost and power required to implement a relay with a very large array can be prohibitive, and thus it is desirable to investigate the use of cheaper and more energyefficient components, such as lowresolution (e.g., one bit) ADCs and DACs. Fortunately, it has been shown in [12, 13] that large arrays exhibit a certain resilience to RF hardware impairments that could be caused by such lowcost components.
Ia Related Work
Several recent contributions have investigated the impact of lowresolution ADCs on the massive MIMO uplink [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. For example, [16] optimized the training pilot length to maximize the spectral efficiency, while [17] revealed that in terms of overall energy efficiency, the optimal level of quantization is 45 bits. In [18], the Bussgang decomposition [24] was used to reformulate the nonlinear quantization using a secondorder statistically equivalent linear operator, and to derive a linear minimum meansquared error (LMMSE) channel estimator for onebit ADCs. In [19], a nearoptimal low complexity bit allocation scheme was presented for millimeter wave channels exhibiting sparsity. The work of [20] examined the impact of onebit ADCs on wideband channels with frequencyselective fading. Other work has focused on balancing the spectral and energy efficiency, either through the combined use of hybrid architectures with a small number of RF chains and low resolution ADCs, or using mixed ADCs architectures with high and low resolution.
In contrast to the uplink case, there are relatively fewer contributions that consider the massive MIMO downlink with lowresolution DACs. In [25], it was shown that performance approaching the unquantized case can be achieved using DACs with only 34 bits of resolution. The nearly optimal quantized Wiener precoder with lowresolution DACs was studied in [26], and the resulting solution was shown to outperform the conventional Wiener precoder with 46 bits of resolution at high signaltonoise ratio (SNR). For the case of onebit DACs, [27, 28] showed that even simple MRT precoding can achieve reasonable results. In [29], an LMMSE precoder was proposed by taking the quantization nonlinearities into account, and different precoding schemes were compared in terms of uncoded bit error rate.
IB Contributions
All these prior works are for singlehop systems rather than dualhop connections via a relay. Recently, [30] considered a relaybased system that uses mixedresolution ADCs at the base station. Unlike [30], we consider a multipair amplifyandforward (AF) relaying system where the relay uses both onebit ADCs and onebit DACs. The onebit ADCs cause errors in the channel estimation stage and subsequently in the reception of the uplink data; then, after a linear transformation, the onebit DACs produce distortion when the downlink signal is coarsely quantized. In this paper, we present a detailed performance investigation of the achievable rate of such doubly quantized systems. In particular, the main contributions are summarized as follows:

We investigate a multipair AF relaying system that employs onebit ADCs and DACs at the relay and uses MRC/MRT beamforming to process the signals. We take the correlation of the quantization noise into account, and present an exact achievable rate by using the arcsine law. Then, we use asymptotic arguments to provide an approximate closedform expression for the achievable rate. Numerical results demonstrate that the approximate rate is accurate in typical massive MIMO scenarios, even with only a moderate number of users.

We show that the channel estimation accuracy of the quantized system depends on the specific orthogonal pilot matrix that is used, which is in contrast to unquantized systems where any orthogonal pilot sequence yields the same result. We consider the specific case of identity and Hadamard pilot matrices, and we show that the identity training scheme provides better channel estimation performance for users with weaker than average channels, while the Hadamard training sequence is better for users with stronger channels.

We compare the achievable rate of different ADC and DAC configurations, and show that a system with onebit DACs and perfect ADCs outperforms a system with onebit ADCs and perfect DACs. Focusing on the low transmit power regime, we show that the sum rate of the relay system with onebit ADCs and DACs is times that achievable with perfect ADCs and DACs. Also, it is shown that the transmit power of each source or the relay can be reduced inversely proportional to the number of relay antennas, while maintaining a given qualityofservice.

We formulate a power allocation problem to allocate power to each source and the relay, subject to a sum power budget. Locally optimum solutions are obtained by solving a sequence of geometric programming (GP) problems. Our numerical results suggest that the power allocation strategy can efficiently compensate for the rate degradation caused by the coarse quantization.
IC Paper Outline and Notations
The remainder of the paper is organized as follows. Section II introduces the multipair AF relaying system model under consideration. Section III presents an approximate closedform expression for the sum rate, and compares the rate achieved with different ADC and DAC configurations. Section IV formulates a power allocation problem to compensate for the rate loss caused by the coarse quantization. Numerical results are provided in Section V. Finally, Section VI summarizes the key findings.
Notation: We use bold upper case letters to denote matrices, bold lower case letters to denote vectors and lower case letters to denote scalars. The notation , , , and respectively represent the conjugate transpose operator, the conjugate operator, the transpose operator, and the matrix inverse. The Euclidian norm is denoted by , the absolute value by , and represents the th entry of . Also, denote a circularly symmetric complex Gaussian random vector with zero mean and covariance matrix , while is the identity matrix of size . The symbol is the Kronecker product, represents a column vector containing the stacked columns of matrix , denotes a diagonal matrix formed by the diagonal elements of matrix , and stand for the real and imaginary part of , respectively. Finally, the statistical expectation operator is represented by , the variance operator is , and the trace is denoted by .
Ii System Model
Consider a multipair relaying system with onebit quantization, as shown in Fig. 1. There are singleantenna user pairs, denoted as and , , intending to exchange information with each other with the assistance of a shared relay. The relay is equipped with receive antennas with onebit ADCs and transmit antennas with onebit DACs. The onebit ADCs cause errors in the channel estimation stage and subsequently in the reception of the uplink data; then, after a linear transformation, the onebit DACs produce distortion when the downlink signal is coarsely quantized. Thus, the system we study is double quantized. We assume that direct links between and do not exist due to large obstacles or severe shadowing. In addition, we further assume that the relay operates in halfduplex mode, and hence it cannot receive and transmit signals simultaneously. Accordingly, information transmission from to is completed in two phases. In the first phase, the sources transmit independent data symbols to the relay, and in the second phase the relay broadcasts the doublequantized signals to the destinations. The signals at the relay’s receive antennas and at the destinations before quantization are respectively given by
(1)  
(2) 
where is chosen to satisfy a total power constraint at the relay, i.e., , which will be specified shortly. The source symbols are represented by , whose elements are assumed to be Gaussian distributed with zero mean and unit power. is a diagonal matrix that denotes the transmit power of the sources with . The vectors and represent additive white Gaussian noise (AWGN) at the relay and destinations, whose elements are both identically and independently distributed (i.i.d.) . Note that to keep the notation clean and without loss of generality, we take the noise variance to be here, and also in the subsequent sections. With this convention, and also the subsequent transmit powers can be interpreted as the normalized SNR. The matrices and respectively represent the uncorrelated Rayleigh fading channels from the sources to the relay with and the channels from the relay to the destinations with . The terms and model the largescale pathloss, which is assumed to be constant over many coherence intervals and known a priori.
Iia Channel Estimation
We assume training pilots are used to estimate the channel matrices and , as in other massive MIMO AF relaying systems [31]. Therefore, during each coherence interval of length (in symbols), all sources simultaneously transmit their mutually orthogonal pilot sequences satisfying to the relay while the destinations remain silent (). Afterwards, all destinations simultaneously transmit their mutually orthogonal pilot sequences satisfying to the relay while the sources remain silent.
Since the channels and are estimated in the same fashion, we focus only on the first link . The received training signal at the relay is given by
(3) 
where represents the transmit power of each pilot symbol, and denotes the noise at the relay, which has i.i.d. elements. After vectorizing the matrix , we obtain
(4) 
where , , and .
IiA1 Onebit ADCs
After the onebit ADCs, the quantized signal can be expressed as
(5) 
where denotes the onebit quantization operation, which separately processes the real and imaginary parts of the signal. Therefore, the output set of the onebit ADCs is . Using the Bussgang decomposition [24, 32], can be represented by a linear signal component and an uncorrelated quantization noise :
(6) 
where is the linear operator obtained by minimizing the power of the quantization noise :
(7) 
where denotes the crosscorrelation matrix between the received signal and the quantized signal , and represents the autocorrelation matrix of , which is computed as
(8) 
where and is a diagonal matrix whose elements are for .
IiA2 LMMSE estimator
Based on the observation and the training pilots , we use the LMMSE technique to estimate . Hence, the estimated channel is given by
(16) 
As a result, the covariance matrix of the estimated channel is expressed as
(17)  
where .
Remark 1
From (17), we can see that is a nontrivial function of , which indicates that the quality of the channel estimates depends on the specific realization of the pilot sequence, which is contrary to unquantized systems where any set of orthogonal pilot sequences gives the same result.
Remark 2
Although our conclusion in Remark 1 is obtained based on the LMMSE estimator, it also holds for the maximum likelihood estimator [35].
In the following, we study the performance of two specific pilot sequences to show how the pilot matrix affects the channel estimation. Here, we choose , which is the minimum possible length of the pilot sequence.
a) Identity Matrix. In this case, , and hence we have
(18) 
Consequently,
(19)  
(20) 
where is a diagonal matrix with . Substituting (19) and (20) into (17), we obtain
(21) 
where is a diagonal matrix with elements
(22) 
b) Hadamard Matrix. In this case, every element of is or , and hence we have
(23)  
(24) 
where the approximation in (24) holds for low . Substituting (23) and (24) into (17), we obtain
(25) 
where is a diagonal matrix with entries
(26) 
where
(27) 
For both cases, the channels from the sources to the relay can be decomposed as
(28) 
where is the estimation error vector. The elements of and are respectively distributed as and when is an identity matrix, while they are distributed as and when is a Hadamard matrix, where and . In what follows we define and .
Similarly, the channels from the relay to the destinations can be decomposed as
(29) 
where and are the estimated channel and estimation error vectors. The elements of and are distributed as and when is an identity matrix, while they are and when is a Hadamard matrix, where
(30)  
(31) 
with
(32) 
and , . We also define and .
For the channel from the kth source to the relay, the meansquare error (MSE) is given by
(33) 
Based on the above results, we have for the identity matrix and for the Hadamard matrix. The following proposition compares the MSE of the two approaches.
Proposition 1
For estimating the channel , the identity matrix is preferable to the Hadamard matrix for user if , and vice versa.
Proof: The proof is trivial since if .
Proposition 1 reveals that the accuracy of the individual channel estimates depends on the particular choice of the orthogonal training scheme, contrary to the ideal case without quantization. More precisely, the scaled identity matrix is beneficial for any user with higher path loss than the average. This is because a weak user benefits from being the only one transmitting at a given time, without the presence of stronger users that dominate the behavior of the ADC. In the case of Hadamard matrix, all users are transmitting simultaneously, resulting in an average quantization noise level for all users jointly, which is advantageous for users with stronger channels.
The question of optimizing the pilot sequence for a given performance metric is an interesting one, but is beyond the scope of the paper. For simplicity, we will assume the identity matrix approach in which each user’s channel is estimated one at a time.
IiB Data Transmission
IiB1 Quantization with Onebit ADCs
With onebit ADCs at the receiver, the resulting quantized signals can be expressed as
(34) 
where is the linear operator, which is uncorrelated with . By adopting the same technique in the previous subsection, we have
(35)  
(36) 
where
IiB2 Digital Linear Processing
We assume that the relay adopts an AF protocol to process the quantized signals by onebit ADCs , yielding
(37) 
where for MRC/MRT beamforming.
IiB3 Quantization with Onebit DACs
Assuming onebit DACs at the transmitter, the resulting quantized signals to be sent by the relay’s transmit antennas can be expressed as
(38) 
where is the linear operator, and is the quantization noise at the relay’s transmit antennas, which is uncorrelated with . Due to the onebit DACs, we have . Therefore, the normalization factor (c.f. (2)) can be expressed as
(39) 
Following in the same fashion as with the ADCs derivations, we obtain
(40)  
(41) 
where
Iii Achievable Rate Analysis
In this section, we investigate the achievable rate of the considered system. In particular, we first provide an expression for the exact achievable rate, which is applicable to arbitrary system configurations. Then we use asymptotic arguments to derive an approximate rate to provide some key insights.
Iiia Exact Achievable Rate Analysis
We consider the realistic case where the destinations do not have access to the instantaneous CSI, which is a typical assumption in the massive MIMO literature since the dissemination of instantaneous CSI leads to excessively high computational and signaling costs for very large antenna arrays. Hence, uses only statistical CSI to decode the desired signal. Combining (1), (2), (34), (37), (38), and (39) yields the received signal at the kth destination
(42) 
where where , where is the kth element of the noise vector . Noticing that the “desired signal” and the “effective noise” in (42) are uncorrelated, and capitalizing on the fact that the worstcase uncorrelated additive noise is independent Gaussian, we obtain the following achievable rate for the kth destination:
(43)  
where
(44)  
(45)  
(46)  
(47)  
(48)  
(49) 
IiiB Asymptotic Simplifications
As we can see, the matrices , , and all involve arcsine functions, which does not give much insight into how the rate changes with various parameters. To facilitate the analysis, we focus on the asymptotic regime for a large number of users, in which (8) can be approximated by
(50) 
Substituting (50) into (35) and (36), we have
(51)  
(52) 
Similarly, asymptotically we have
(53) 
where
(54)  
Note that the proof of calculating the approximate can be found in the Appendix A.
As a result, the matrices and can be approximated by
(55)  
(56) 
IiiC Approximate Rate Analysis
In this section, we derive a simpler closedform approximation for the achievable rate. Substituting (51), (52), (55), and (56) into (43), the exact achievable rate can be approximated by
(57)  
where
(58)  
(59)  
(60)  
(61)  
(62)  
(63)  
(64) 
With this expression, we can compute by using random matrix theory and present a closedform approximate rate for the kth destination, as formalized in the following theorem.
Theorem 1
With onebit ADCs and DACs at the relay, the approximate achievable rate of the kth destination is given by (57), where
(65)  
(66)  
(67)  
(68)  
(69)  
(70)  
(71)  
with .
Proof: See Appendix A.
From Theorem 1, we can more readily see the impact of key parameters on the achievable rate. For instance, decreases with the number of user pairs . This is expected since a higher number of users increases the amount of interuser interference. In addition, is an increasing function of , which reveals that increasing the number of relay’s antennas always boosts the system performance. As approaches infinity, converges to a constant that is independent of . In this case, the system becomes interferencelimited.
To quantify the impact of the double quantization on system performance, in the following corollaries we compare the achievable rate with several different ADC and DAC configurations.
Corollary 1
With perfect ADCs and onebit DACs, the achievable rate of the kth destination can be expressed as (72) (shown on the top of the next page),