# Power Allocation for Multi-Pair Massive MIMO Two-Way AF Relaying with Linear Processing

## Abstract

In this paper, we consider a multi-pair two-way amplify-and-forward relaying system where multiple sources exchange information via a relay node equipped with large-scale antenna arrays. Given that channel estimation is non-ideal, and that the relay employs either maximum-ratio combining/maximum-ratio transmission (MRC/MRT) or zero-forcing reception/zero-forcing transmission (ZFR/ZFT) beamforming, we derive two corresponding closed-form lower bound expressions for the ergodic achievable rate of each pair sources. The closed-form expressions enable us to design an optimal power allocation (OPA) scheme that maximizes the sum spectral efficiency under certain practical constraints. As the antenna array size tends to infinity and the signal to noise ratios become very large, asymptotically optimal power allocation schemes in simple closed-form are derived. The capacity lower bounds are verified to be accurate predictors of the system performance by simulations, and the proposed OPA outperforms equal power allocation (EPA). It is also found that in the asymptotic regime, when MRC/MRT is used at the relay and the link end-to-end large-scale fading factors among all pairs are equal, the optimal power allocated to a user is inverse to the large-scale fading factor of the channel from the user to the relay, while OPA approaches EPA when ZFR/ZFT is adopted.

## 1Introduction

Massive multiple-input multiple-output (MIMO) transmission, in which a base station is equipped with hundreds of antennas for multiuser operation, is considered as one of the key enabling technologies for 5G [1]. In [2], it was first proposed for multi-cell noncooperative scenarios. Such large antenna arrays can substantially reduce the effects of noise, small-scale fading and inter-user interference, using only simple signal processing techniques with reduced total transmit power, and only inter-cell interference caused by pilot contamination remains [2]. Subsequently, the energy and spectral efficiency of very large multiuser MIMO systems were investigated in the single cell scenarios in [4], which showed that the power radiated by the terminals could be made inversely proportional to the square-root of the number of base station antennas with no reduction in performance when considering imperfect channel state information (CSI), and that the power could be made inversely proportional to the number of antennas if perfect CSI were available.

Currently, massive MIMO combined with cooperative relaying is considered as a strong candidate for the development of future energy-efficient networks and has received increasing attention [5]. In the field of cooperative relaying, two-way relaying technique outperforms one-way relaying in terms of spectral efficiency, since it employs the principle of network coding at the relay in order to mix the signals received simultaneously from two links for subsequent forwarding, and then applies the self-interference cancellation (SIC) at each user to extract the desired information [11]. For the multi-pair two-way relaying with massive MIMO, [7] obtained the asymptotic spectral and energy efficiencies of the system analytically with both maximum-ratio combining/maximum-ratio transmission (MRC/MRT) and zero-forcing reception/zero-forcing transmission (ZFR/ZFT) beamforming, supposing that the number of relay antennas approaches to infinity and the transmit power of all users is equal. However, only asymptotic cases with perfect CSI and perfect SIC were studied and no closed-form expression for the ergodic achievable rate with finite number of relay antennas was derived in [7]. In [9], the ergodic achievable rates were investigated with perfect CSI based MRC/MRT used at the relay, providing a capacity lower bound, the derivation of which involved asymptotic approximations. Neither [7] nor [9] considers imperfect CSI or power allocation (PA) problems.

In the literature, instantaneous power allocation schemes based on instantaneous rate for regular scale MIMO rather than massive MIMO were presented for one way or two way AF wireless relay systems to improve system performance [13]. In massive MIMO systems, ergodic rate is usually used in power allocation because the instantaneous rate approaches the ergodic rate as the number of antennas tends to infinity due to the law of large numbers, and such PA schemes are more practical with lower complexity than instantaneous rate based ones. In [10], an ergodic rate based optimal power allocation (OPA) scheme was proposed for a multi-pair decode-and-forward (DF) one-way relaying with massive arrays. Nevertheless, power allocation has not been addressed in a massive MIMO two-way relaying system. Besides, there is no closed-form ergodic rate expressions derived for massive MIMO two-way relaying with ZFR/ZFT in the literature.

This paper considers a multi-pair two-way AF relaying system where multiple sources exchange information via a relay node equipped with large-scale arrays. Assuming imperfect CSI estimation, the relay station employs the MRC/MRT and ZFR/ZFT beamforming to process the signals, respectively. First, utilizing the technique in [16], we derive for the first time two statistical CSI (SCSI) based closed-form lower bounds for the ergodic achievable rate in the case of arbitrary number of relay antennas (without resorting to asymptotic approximations) with MRC/MRT and ZFR/ZFT processing, respectively, based on the properties of Wishart and inverse Wishart matrices. Having obtained the closed-form expressions, we are able to design an OPA scheme that maximizes the sum spectral efficiency under certain practical constraints. The proposed OPA scheme is based on geometric programming (GP) [18], which can be solved by conventional optimization tools, such as CVX [19]. Considering the massive MIMO properties, an asymptotically OPA is presented for the asymptotic regimes with closed-form solutions. The derived closed-form expressions for the achievable rate are verified to be accurate predictors of the system performance by Monte-Carlo simulations. Furthermore, in order to demonstrate the effectiveness of the developed OPA schemes, simulations of spectral efficiency are conducted under different system configurations, respectively, in comparison to the equal power allocation (EPA) schemes.

The rest of the paper is organized as follows. We briefly describe the system model for the multi-pair two-way AF relaying in Section 2. In Section 3, two closed-form expressions for the achievable rate are derived for MRC/MRT and ZFR/ZFT, respectively, followed by asymptotic analysis. Then, an OPA and an asymptotically OPA are proposed by solving the sum-rate maximization based optimization problem in Section 4. Furthermore, simulation results under different system configurations are given in Section 5 to demonstrate the effectiveness of both derived rate expressions and developed OPAs. Finally, we draw our conclusions in Section 6.

Notations:

For a matrix , we use , , and to denote the trace, the transpose, the Hermitian transpose, and the conjugate, respectively. The symbol indicates the 2-norm of vector and denotes a diagonal matrix with being its diagonal entries. Moreover, denotes the identity matrix, and denotes the expectation and the variance operators, respectively. denotes . Finally, represents a circularly symmetric complex Gaussian vector with zero mean and covariance matrix .

## 2System Model

Figure 1 shows the considered multi-pair two-way AF relaying network, where pairs of users communicate with the help of a common relay station by sharing the same time-frequency resources. In this system, two single-antenna users in the th user pair denoted by or (for ) want to exchange information with each other via the relay equipped with () antennas. Notably, the direct links between the corresponding users are assumed non-existing in the two-way relaying system. Typically, a two-way network is divided into two phases, namely the multiple-access (MA) phase and the broadcast (BC) phase [11]. In the MA phase, information is sent from the user pairs to the relay; while in the BC phase, the relay broadcasts the processed information.

Let () and denote the power transmitted by user and the relay corresponding to the MA and BC phases, respectively. We assume that all the channels between the users and the relay follow independent and identically distributed (i.i.d.) Rayleigh fading and time division duplex (TDD) is adopted in all transceivers. Thus, supposing that is the channel between the th user and the relay, contains the i.i.d. elements, where represents the corresponding large-scale fading coefficient. In this way, we can denote the channel matrix between all the users and the relay accounting for both small-scale fading and large-scale fading by

where includes the i.i.d. small-scale fading coefficients, and is the large-scale fading diagonal matrix with the th diagonal elements denoted by ().

### 2.1Channel Estimation

Practically, the channel matrices in both the MA and BC phases have to be estimated for relay processing. However, due to the large-scale antenna array at the relay, channel estimation at the user side becomes rather impractical. Thus, time division duplex (TDD) is adopted here and channel reciprocity can be utilized, i.e., only channel matrix between all the users and the relay has to be estimated based on the uplink training. The relay then has the estimated CSIs of all uplink and downlink channels. The required channel related information at the user side can be calculated by the relay and fed back to the users, as will be explained later. At the beginning of each coherence interval , all users simultaneously transmit pilot sequences of length symbols. The pilot sequences of all the users are pairwisely orthogonal, i.e., is required. Then the training matrix received at the relay is

where is the transmit power of each pilot symbol, is the additive white Gaussian noise (AWGN) matrix with i.i.d. components following , the training vector transmitted by the th () user is denoted by the th row of , satisfying . Moreover, since the rows of pilot sequence matrices are pairwisely orthogonal, we have ().

In order to estimate the channel matrices , we employ the minimum mean-square-error (MMSE) estimation at the relay. The MMSE channel estimates are given by [20]

where we define and . According to the property of , we conclude that is composed of i.i.d. elements. Then,

where denotes the estimation error matrix which is independent of from the property of MMSE channel estimation [20]. Hence, we have with , and with . The diagonal elements satisfy and with .

### 2.2Data Transmission

Since the relay station estimates all the channels, it employs linear processing MRC/MRT and ZFR/ZFT based on the imperfect CSI. While each user only has the knowledge of its pairwise effective channel coefficient for data detection and self-interference cancellation coefficient for SIC, which are calculated and sent out by the relay. In the MA phase, all the users transmit their signals simultaneously to the relay. That is, the received signal at the relay station is given by

where , with each power satisfying , with the th element representing the transmitted signal by the th user and , , and is the additive white Gaussian noise (AWGN) vector at the relay with zero mean and the variance of .

Then, in the BC phase, the relay multiplies the received signal by a linear receiving and precoding matrix to yield the relay transmitted signal given by , where is the combined beamforming matrix at the relay and its expression will be given in the next subsection. The transmitted signal satisfies the expected transmit power constraint at the relay [21], i.e.,

with a total power constraint ^{1}

where is defined to indicate the th^{2}

Using the estimated CSI, the relay calculates and sends out the SIC coefficient () for each user. Hence, the received signal at the th user after SIC is rewritten as

where the residual self-interference involves , since the SIC coefficient for user is obtained from the estimated CSI. Here, we suppose that there is no error during the SIC coefficients transmission from the relay.

### 2.3MRC/MRT Processing

In this subsection, the simple and widely used MRC/MRT beamforming is adopted. According to [22], the imperfect CSI based MRC/MRT beamforming is given by

where is the block diagonal permutation matrix indicating the user pairing format with , and is a normalization constant, chosen to satisfy the power constraint at the relay station in (Equation 6).

By substituting (Equation 9) into (Equation 6), we have

where

The detailed derivation of the equation is given in Appendix A.

### 2.4ZFR/ZFT Processing

When employing ZFR/ZFT with imperfect CSI, in which the pseudo-inverse of the estimated channels in (Equation 4) are needed for processing, the linear beamforming is given by [22]

where and is the normalization constant, chosen to satisfy the transmit power constraints at the relay. Notably, SIC is not necessary as ZFR/ZFT leads to (). On the basis of (Equation 12) and , we have

where . The detailed derivation of (Equation 13) is given in Appendix B.

## 3Achievable Rate Analysis

In this section, a general form of the ergodic achievable rate of the transmission link for MRC/MRT processing is given first, followed by a rate expression for ZFR/ZFT. In order to obtain a basic and insightful expression that can be used for power allocation optimization, a simplified capacity lower bound is derived utilizing the technique of [16], in which the received signal is rewritten as a known mean times the desired symbol, plus an uncorrelated effective noise. The worst-case uncorrelated effective noise, where each additive term is treated as independent Gaussian noise of the same variance, is employed to derive a lower bound.

From (Equation 8), the ergodic achievable rate of the transmission link is expressed as (Equation 14).

Remark 1:

Here, the ergodic achievable rate is valid based on the assumption that the receiving user knows perfectly in the detection process. To demonstrate the accuracy of the derived lower bounds, we compare the lower bounds with Monte-Carlo realized (Equation 14) in Section 5. The normalization constant for in (Equation 14) is assumed to be calculated based on instantaneous CSI by satisfying .

Further derivation of (Equation 14) is difficult because of the intractability to carry out the ensemble average analytically. Instead, we adopt the technique in [16] to derive a worst-case lower bound of the achievable rate. The first step is to rewrite in (Equation 8) as the sum of and , where the first part is now considered as the “desired signal”. That is, (Equation 8) can be expressed as

where is considered as the effective noise and given by

It is straightforward to show that the first term “desired signal” and the second term “effective noise” in (Equation 15) are uncorrelated. The exact pdf of is not easy to obtain, but we know that the worst-case is to approximate the effective noise as independently Gaussian distributed [16]. Since the relay is equipped with large-scale antenna arrays by assuming , the central limit theorem provides a tight statistical CSI based lower bound for the achievable rate. Then, the statistical CSI based achievable rate lower bound of the transmission link can be obtained as

where , , and denote the residual self-interference after SIC, the inter-pair interference, the amplified noise from relay and the noise at user, respectively, i.e.,

When MRC/MRT beamforming is employed, further mathematical derivation of (Equation 17) leads to the following theorem:

Theorem 1:

With imperfect CSI based MRC/MRT, the ergodic achievable rate of the transmission link , for a finite number of antennas at the relay, is lower bounded by (Equation 18),

where , , , , , and .

Proof:

See Appendix C.

For imperfect CSI based ZFR/ZFT processing, a closed-form expression for the achievable rate in (Equation 17) is derived as follows:

Theorem 2:

With imperfect CSI based ZFR/ZFT beamforming, the achievable rate of the transmission link , for a finite number of antennas at the relay, is lower bounded by (Equation 19), where , , , , , and .

Proof:

See Appendix D.

Theorems 1 and 2 are also valid for conventional MIMO systems, while the bounds become less tight as the antenna scale goes down. The capacity lower bounds for perfect CSI can always be obtained by setting and () in (Equation 18) and (Equation 19). Moreover, it can be observed from (Equation 18) that when the estimation error is severe, the residual SI occupies the major part of the imperfect CSI effect in comparison to other terms. On the other hand, if channel estimation is rather accurate, the residual SI has slight effects in comparison to other terms. While for ZFR/ZFT, both the residual SI and inter-pair interference are determined by the channel estimation accuracy.

### 3.1Asymptotic Analysis with Massive Arrays

Based on the derived closed-form expressions for the achievable rate in (Equation 18) and (Equation 19), this section provides the asymptotic analysis under two different cases when the number of relay antennas approaches to infinity. Suppose that all users have the same transmit power, i.e., .

Proposition 1:

In case I where is fixed, (), , and and are fixed, to achieve non-vanishing user rate as , the user and relay transmit power scaling factor and must satisfy and . When and , the asymptotic achievable rate expressions of the transmission link for imperfect CSI based MRC/MRT and ZFR/ZFT are (Equation 20) and (Equation 21),

respectively, which show that the transmit powers at both users and relay sides can be scaled down by up to to maintain a given rate in case I. When and , the asymptotic achievable rate of each user approaches to infinity as .

In case II where , (), , and and are fixed, to achieve non-vanishing user rate as , the pilot, user and relay transmit power scaling factors , and must satisfy , and . When , and , the asymptotic achievable rate of the transmission link for imperfect CSI based MRC/MRT and ZFR/ZFT are (Equation 22) and (Equation 23), respectively, from which we conclude that the transmit powers of each user and the relay can only be reduced by up to when the pilot transmit power is set as , in order to maintain a given spectral efficiency. Similarly, when and , the asymptotic achievable rate of each user approaches to infinity as .

Remark 2:

When the pilot power scaling factor , which means that the pilot power scales down by , to guarantee user rate there is , which means the relay and user transmit power must stay constant and do not scale down with . The achievable rate in this case can be derived from (Equation 18) and (Equation 19), but not shown here due to space limitation. It is found that channel estimation error induced interference and inter-pair interference cannot be eliminated when is scaled down proportionally to with fixed and in case II.

It can be observed from Theorems 1 and 2 that for fixed (), , and , the achievable rate of each pair-wise user transmission link depends on the user power, i.e., the values of (), and the relay power. Next we propose the optimal power allocation for the studied system.

## 4Power Allocation Schemes

In this section, a power allocation problem is first formulated and solved for multi-pair users in the MA phase transmission and the relay in the BC phase, which maximizes the sum spectral efficiency^{3}

### 4.1Optimal Power Allocation (OPA)

Most power optimization in communications aims to maximize the sum spectral efficiency, which is defined as the sum-rate (in bits) per channel use. Assuming that is the length of the coherent interval (in symbols), in which symbols are used for channel estimation, the sum spectral efficiency^{4}

where in constraint ( ?) is the total power allocated to all the users and relay, and constraints ( ?) specify the peak power limits and for each user and relay, respectively. The objective in ( ?) can be equivalently rewritten as , as is a monotonic increasing function of . We can see that the constraints are posynomial functions. If the objective function is a monomial or posynomial function, the problem ( ?) becomes a GP which can be reformulated as a convex problem, and thus, can be solved efficiently by convex optimization tools, such as CVX [19]. However, the rewritten objective function for ( ?) is still neither a monomial nor posynomial, making solving the problem directly by the convex optimization tools impossible. To solve this problem, an approximation for the objective function can be efficiently found by using the technique in [23]. Specifically, according to [23], we can use a monomial function to approximate near an arbitrary point , where and . Consequently, the objective function can be approximated as , which is a monomial function. In this way, the problem is transformed into a GP problem by the approximation.

Similar to [23], a successive approximation algorithm for the power allocation problem in ( ?) is proposed as Algorithm 1. ( ?) is the relaxed constraint. Notably, the parameter here is utilized to control the desired approximation accuracy. The accuracy is high when is close to 1, but the convergence rate is low, and vice versa. As shown in [23], is an option that introduces a good accuracy trade off in most practical cases.

Algorithm 1: |
---|

Initialization: |

Repeat: Compute and ; Solve the GP: Set , and update , where () are obtained based on the solutions () and of the GP;
Until: Stop if or ; |

Output: |

### 4.2Asymptotically Optimal Power Allocation (AOPA)

Obviously, the optimal power allocation scheme in Algorithm 1 is an iterative numerical solution with no closed-form. However, more tractable expressions can be found for MRC/MRT and ZFR/ZFT, respectively, when we consider the asymptotic regimes with high SNR and .

#### AOPA for MRC/MRT

Suppose that the SNRs at both the relay and user sides are very high, i.e., , and (), and . Then the lower bound can be simplified as given by the following Lemma.

Lemma 1:

When , , (), and , the rate of the transmission link can be approximated as

Proof:

Firstly, we have for due to . Then, we divide both the denominator and numerator of the SINR in (Equation 18) by . Each item with in the denominator is able to be ignored based on . Then, according to and for , (Equation 24) can be obtained.

In order to obtain a closed-form solution for asymptotically optimal power allocation, we set the fixed link condition for . Since the approximated rate expressions involve no , we do not need to find the optimal solution for