Multipair Massive MIMO Relaying Systems with One-Bit ADCs and DACs
This paper considers a multipair amplify-and-forward massive MIMO relaying system with one-bit ADCs and one-bit DACs at the relay. The channel state information is estimated via pilot training, and then utilized by the relay to perform simple maximum-ratio combining/maximum-ratio transmission processing. Leveraging on the Bussgang decomposition, an exact achievable rate is derived for the system with correlated quantization noise. Based on this, a closed-form 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 one-bit 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 double-quantized relay system and an ideal non-quantized system is a moderate factor of in the low power regime.
Index terms— Massive MIMO, relays, one-bit quantization, power allocation
Multipair multiple-input multiple-output (MIMO) relaying networks have recently attracted considerable attention since they can provide a cost-effective 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 multiple-antenna relay in the same time-frequency resource. Hence, multi-user 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,  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,  extended the analysis to the Ricean fading case and obtained similar power scaling behavior. For full-duplex systems, [8, 9] analytically compared the performance of MRC/MRT and zero-forcing 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 analog-to-digital converters (ADCs) and the digital-to-analog 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 energy-efficient components, such as low-resolution (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 low-cost components.
I-a Related Work
Several recent contributions have investigated the impact of low-resolution ADCs on the massive MIMO uplink [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. For example,  optimized the training pilot length to maximize the spectral efficiency, while  revealed that in terms of overall energy efficiency, the optimal level of quantization is 4-5 bits. In , the Bussgang decomposition  was used to reformulate the nonlinear quantization using a second-order statistically equivalent linear operator, and to derive a linear minimum mean-squared error (LMMSE) channel estimator for one-bit ADCs. In , a near-optimal low complexity bit allocation scheme was presented for millimeter wave channels exhibiting sparsity. The work of  examined the impact of one-bit ADCs on wideband channels with frequency-selective 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 low-resolution DACs. In , it was shown that performance approaching the unquantized case can be achieved using DACs with only 3-4 bits of resolution. The nearly optimal quantized Wiener precoder with low-resolution DACs was studied in , and the resulting solution was shown to outperform the conventional Wiener precoder with 4-6 bits of resolution at high signal-to-noise ratio (SNR). For the case of one-bit DACs, [27, 28] showed that even simple MRT precoding can achieve reasonable results. In , an LMMSE precoder was proposed by taking the quantization non-linearities into account, and different precoding schemes were compared in terms of uncoded bit error rate.
All these prior works are for single-hop systems rather than dual-hop connections via a relay. Recently,  considered a relay-based system that uses mixed-resolution ADCs at the base station. Unlike , we consider a multipair amplify-and-forward (AF) relaying system where the relay uses both one-bit ADCs and one-bit DACs. The one-bit ADCs cause errors in the channel estimation stage and subsequently in the reception of the uplink data; then, after a linear transformation, the one-bit 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 one-bit 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 closed-form 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 one-bit DACs and perfect ADCs outperforms a system with one-bit ADCs and perfect DACs. Focusing on the low transmit power regime, we show that the sum rate of the relay system with one-bit 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 quality-of-service.
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.
I-C 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 closed-form 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 one-bit quantization, as shown in Fig. 1. There are single-antenna 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 one-bit ADCs and transmit antennas with one-bit DACs. The one-bit ADCs cause errors in the channel estimation stage and subsequently in the reception of the uplink data; then, after a linear transformation, the one-bit 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 half-duplex 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 double-quantized signals to the destinations. The signals at the relay’s receive antennas and at the destinations before quantization are respectively given by
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 large-scale path-loss, which is assumed to be constant over many coherence intervals and known a priori.
Ii-a Channel Estimation
We assume training pilots are used to estimate the channel matrices and , as in other massive MIMO AF relaying systems . 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
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
where , , and .
Ii-A1 One-bit ADCs
After the one-bit ADCs, the quantized signal can be expressed as
where denotes the one-bit quantization operation, which separately processes the real and imaginary parts of the signal. Therefore, the output set of the one-bit ADCs is . Using the Bussgang decomposition [24, 32], can be represented by a linear signal component and an uncorrelated quantization noise :
where is the linear operator obtained by minimizing the power of the quantization noise :
where denotes the cross-correlation matrix between the received signal and the quantized signal , and represents the auto-correlation matrix of , which is computed as
where and is a diagonal matrix whose elements are for .
Ii-A2 LMMSE estimator
Based on the observation and the training pilots , we use the LMMSE technique to estimate . Hence, the estimated channel is given by
As a result, the covariance matrix of the estimated channel is expressed as
From (17), we can see that is a non-trivial 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.
Although our conclusion in Remark 1 is obtained based on the LMMSE estimator, it also holds for the maximum likelihood estimator .
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
where is a diagonal matrix with elements
b) Hadamard Matrix. In this case, every element of is or , and hence we have
where is a diagonal matrix with entries
For both cases, the channels from the sources to the relay can be decomposed as
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
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
and , . We also define and .
For the channel from the k-th source to the relay, the mean-square error (MSE) is given by
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.
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.
Ii-B Data Transmission
Ii-B1 Quantization with One-bit ADCs
With one-bit ADCs at the receiver, the resulting quantized signals can be expressed as
where is the linear operator, which is uncorrelated with . By adopting the same technique in the previous subsection, we have
Ii-B2 Digital Linear Processing
We assume that the relay adopts an AF protocol to process the quantized signals by one-bit ADCs , yielding
where for MRC/MRT beamforming.
Ii-B3 Quantization with One-bit DACs
Assuming one-bit DACs at the transmitter, the resulting quantized signals to be sent by the relay’s transmit antennas can be expressed as
where is the linear operator, and is the quantization noise at the relay’s transmit antennas, which is uncorrelated with . Due to the one-bit DACs, we have . Therefore, the normalization factor (c.f. (2)) can be expressed as
Following in the same fashion as with the ADCs derivations, we obtain
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.
Iii-a 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 k-th destination
where where , where is the k-th 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 worst-case uncorrelated additive noise is independent Gaussian, we obtain the following achievable rate for the k-th destination:
Iii-B 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
Similarly, asymptotically we have
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
Iii-C Approximate Rate Analysis
With this expression, we can compute by using random matrix theory and present a closed-form approximate rate for the k-th destination, as formalized in the following theorem.
With one-bit ADCs and DACs at the relay, the approximate achievable rate of the k-th destination is given by (57), where
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 inter-user 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 interference-limited.
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.
With perfect ADCs and one-bit DACs, the achievable rate of the k-th destination can be expressed as (72) (shown on the top of the next page),