Sum-Rate Maximization of Multicell MISO Networks with Limited Information Exchange

Sum-Rate Maximization of Multicell MISO Networks with Limited Information Exchange

Youjin Kim, , and Hyun Jong Yang, Y. Kim and H. J. Yang (corresponding author) are with the School of Electrical and Computer Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, Republic of Korea (e-mail: {nick0822, hjyang}@unist.ac.kr).
Abstract

Although there have been extensive studies on transmit beamforming in multi-input single-output (MISO) multicell networks, achieving optimal sum-rate with limited channel state information (CSI) is still a challenge even with a single user per cell. A novel cooperative downlink multicell MISO beamforming scheme is proposed with highly limited information exchange among the base stations (BSs) to maximize the sum-rate. In the proposed scheme, each BS can design its beamforming vector with only local CSI based on limited information exchange on CSI. Unlike previous studies, the proposed beamforming design is non-iterative and does not require any vector or matrix feedback but requires only quantized scalar information. The proposed scheme closely achieves the optimal sum-rate bound in almost all signal-to-noise ratio regime based on non-iterative optimization with lower amount of information exchange than existing schemes, which is justified by numerical simulations.

Multi-input single-output (MISO), downlink beamforming, small cells, scalar information exchange, multicell downlink

I Introduction

In dense multicell networks, the signal-to-interference-plus-noise ratio (SINR) cannot grow unless the interference signals are kept weak enough compared to the desired channel gain [1]. If the transmitter is equipped with multiple antennas, intercell interference can be significantly mitigated or even cancelled via spatial transmit beamforming [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The interference alignment framework [12, 13] achieves asymptotically optimal multiplexing gain based on global channel state information (CSI) at the cost of excessive use of frequency- or time-domain signal extension, but with no guarantee of optimal sum-rate achievability. Though massive multi-input multi-output (MIMO) employed at the transmitter provide significant spectral efficiency gain [14, 15], the number of transmit antennas even at base stations (BSs) is often limited by up to 8 in the pervasive conventional mobile networks [16].

In the downlink scenario, if information exchange for global CSI is allowed among the BSs via direct link, coordinated beamforming transmission can be employed [10]. In coordinated beamforming, only the beamforming vectors are jointly optimized, and each user’s data streams are transmitted by a single serving BS. In this paper, the focus is on the coordinated downlink multi-input single-output (MISO) beamforming design with limited direct link capacity. With a wireless direct link, which is put on the highest priority by 3GPP, the capacity is limited by 10-100Mbps typically. In such a case, highly limited information exchange is required, particularly in dense networks. Although the MISO multicell network is a well-studied area, achieving the optimal sum-rate with limited information exchange on CSI is still a major challenge.

I-a Related Works

With global CSI, coordinated beamforming offers optimal multiplexing gain [4, 8, 17], an optimal Pareto rate boundary [7], or a significant sum-rate gain over the conventional distributed beamforming [18, 19]. However, in MISO networks, the amount of CSI information exchange in general increases as the number of transmit antennas grows, which make them difficult to be implemented in systems with limited direct link or backhaul capacity.

Several studies have proposed cooperative beamforming methods with vector quantization to reduce the amount of information exchange [20, 21, 22, 23, 3, 24, 25, 26, 27, 28, 29, 11, 30]. However, with the vector quantization, the number of quantization bits increases linearly with respect to the number of antennas to achieve the same rate.

Distributed beamforming also has been proposed based only on local CSI requiring no information exchange [2, 31, 32, 7, 33, 34, 35, 36]. In [7], the condition of beamforming vector which corresponds to Pareto’s optimal rate boundary is derived for a multicell MISO channel with local CSI. However, no closed-form solution of beamforming vector is derived. In [34], a simple MIMO downlink precoding is proposed in a single cell maximizing each user’s signal-to-leakage-plus-noise ratio (SLNR)111The terminology is also known as signal-to-generating-interference-and-noise ratio (SGINR) [33] or distributed virtual SINR [2]. while decoupling each user’s beamforming vector design. In [35, 33, 31, 36], the SLNR-maximizing beamforming scheme is applied to the multicell MISO channel, and the achievability of Pareto’s optimal rate bound is discussed. The same idea was extended in [2, 32] to the multicell MISO network where each user is served by all the BSs assuming each user’s data being shared by all the BSs, i.e., coordinated multi-point joint transmission. Statistical beamforming design schemes robust to instantaneous CSI have also been proposed based only on the second order statistics of local CSI [2, 21]. However, the sum-rate of these SLNR-maximizing schemes with only local CSI is far below the channel capacity of the multicell MISO channel, especially in high-SNR regime.

Iterative beamforming design approaches, in which the BSs update their beamforming vectors iteratively exchanging interference pricing measures with other BSs or users, have been proposed in pursuit of maximizing the sum-rate of the two-user MIMO interference channel [37] and minimizing transmission power of the multicell MISO channel [38, 39, 40] with the use of limited information exchange. In the scheme proposed in [41], beamforming vectors, receive equalizers, and weight coefficients are designed iteratively between the transmitters and receivers. However, it requires excessive amount of information exchange due to the vector information exchange about the beamforming vectors. Furthermore, iterative optimization can significantly increase the overhead of information exchange for convergence of the solutions.

In [42], the beamforming vectors design based on neural network is proposed. However, the optimal beamforming solution to the sum-rate maximization problem is still unknown.

I-B Contribution

In this paper, we propose a non-iterative cooperative downlink beamforming scheme in multicell MISO networks, each cell of which consists of a BS with multiple antennas and a user with a single antenna, based on local CSI with limited information exchange of scalar values. Our contribution in summary is as follows:

  • We first give inspiration that the sum-rate maximization may be achieved by choosing a proper set of users and making them interference-free. From this inspiration, we propose a novel multicell beamforming design based on the mixture of the maximization of weighted signal-to-leakage-plus-noise ratio (WSLNR) and the minimization of weighted generating-interference (WGI). Unlike previous related studies, where the SLNR or generating-interference (GI) formulation with identical weights was used, we focus on the design of the weights in WSLNR and WGI via choosing a proper set of interference-free users.

  • For each selection on the number of interference-free users, we provide an information exchange protocol with limited direct link capacity, and present an adaptive beamforming design scheme. In the proposed protocol, only scalar information, not vector CSI, is exchanged, and hence the amount of information exchange does not grow for increasing number of antennas.

  • Then, a scalar quantization method for the information to be exchanged is derived, based on which quantitative evaluation of the amount of information exchange is provided compared with existing schemes.

  • We derive conditions of system parameters for which the optimal sum-rate is asymptotically achievable with the proposed scheme. We also confirm by extensive simulations that the proposed scheme closely achieves the optimal sum-rate bound for almost all the SNR regime requiring less information exchange compared to the existing schemes. Although there have been extensive studies on multicell MISO beamforming, to the best of authors’ knowledge, this is the first non-iterative beamforming design that achieves the optimal sum-rate bound even with the lowest information exchange overhead.

Ii System model and Proposed Protocol

It is assumed that each cell is composed of a single BS and user assuming frequency-, code-, or time-division multi-user orthogonal multiplexing222Though our focus is to build a beamforming design framework in case of a single user per cell, the system can be readily extended to multiuser cases, which shall be described in Section V.. Each small cell BS is assumed to have antennas, whereas each user has a single antenna. The number of cells considered is denoted by , and it is assumed that and . The channel vector from the -th BS (referred to as BS henceforth) to the user in the -th cell (referred to as user henceforth) is denoted by . Block fading and time-division duplexing with channel reciprocity are assumed. Resorting to channel reciprocity, each BS is assumed to have local CSI at the transmitter [2], i.e., BS has the information of , .

The beamforming vector at BS is denoted by , where . The received signal at user is written by

(1)

where is the unit-variance transmit symbol at the -th BS, , and is the additive white Gaussian noise (AWGN) at user with zero-mean and variance of . Thus, the corresponding SINR is expressed by

(2)

and the achievable sum-rate is given by

(3)

Iii Optimization of the Beamforming Vector Design

Iii-a Beamforming vector design: Selection of interference-free users

The sum-rate maximization problem should be formulated jointly for all the beamforming vectors as

(4)

which requires global CSI to find the optimal solution. According to [2], the solution of the sum-rate maximization problem can also be obtained by solving the max-WSLNR problem. Specifically, let us denote the weight coefficient for the channel gain from BS to user by , and the set of , , by . Then, the beamforming vector in the max-WSLNR problem for given weights is obtained from

(5)

Here, the weights should be jointly optimized to maximize the sum-rate as

(6)

The problems (5) and (6) are coupled with each other, and thus global CSI is required to solve these problems. To design the beamforming vectors with local CSI, in majority of the previous studies, all the weights are assumed to be identical, i.e., , .

Our aim is to design , , to maximize the sum-rate with local CSI and limited information exchange among the BSs. To gain intuition, we start with the following numerical example introducing the notion of interference-free users. If the received interference at user , i.e., in (2), is significantly small, e.g., smaller than 1/100 of the maximum out of the interference strengths at all the users, then let us denote user by an almost-interference-free user. Figure 1 shows that the optimal per-cell average rate (left y-axis) and the average number of almost-interference-free users (right y-axis) versus SNR for and , where each channel is identically and independently distributed (i.i.d.) according to the complex Gaussian distribution. Here, the beamforming vectors are optimally designed through exhaustive numerical simulations based on global CSI. As shown in the figure, the average number of users with noticeably low interference increases from 0 to as SNR increases. The lesson from Fig. 1 is that choosing a proper number of interference-free users for given channel condition is essential to maximize the sum-rate. Indubitably, choosing a right set of interference-free users, i.e., who shall be interfere-free, is also critical.

Figure 1: Average per-cell sum-rate and the average number of interference-free users versus SNR for and

In what follows, we first propose a beamforming design framework based on the mixture of the WSLNR maximization and the WGI minimization for each possible number of interference-free users. To begin, we define a general WSLNR in pursuit of incorporating the notion of WGI as

(7)

where is the weight coefficient for the channel gain from BS to user . The essence of the proposed beamforming design is to restrict to to work with limited information exchange among the BSs. The set of interference-free users is denoted by , and the number of interference-free users is denoted by , i.e., .

Iii-B Beamforming vector design for

Assuming global CSI, the maximum multiplexing gain without the time or frequency domain dimension extension can be obtained by the interference alignment framework as summarized in the following proposition.

Proposition 1 (Theorem 1 in [43])

With the interference alignment without dimension extension for the case of , the maximum multiplexing gain is .

Proposition 1 implies that there can exist up to users, the effective SINRs of which after proper receive processing incorporate zero inter-user interference, i.e., interference-free users. Since a single antenna at the receiver is assumed, no zero-forcing-like receive processing is possible. Thus, Proposition 1 in fact means that the SINRs of up to users can be interfere-free only via transmit beamforming.

To shed light on obtaining interference-free users with local CSI, we introduce the following lemma.

Lemma 1

For , given that each BS transmits with the equal power constraint , the optimal multiplexing gain of the multicell MISO downlink channel is without time or frequency-domain signal extension.

Proof:

Lemma 1 can be proved by following the similar footsteps of [43]. Note that the number of interference-free users is , and hence the multiplexing gain is . Suppose that user is an interference-free user. Then, the interference-free constraints at the receiver side are given by

(8)

The number of these equalities for the interference-free users is . On the other hand, the number of effective variables in each is considering the unit-norm constraint. For the existence of the solution on of the equalities (8), we need the number of effective variables to be equal to or greater than the number of equalities, i.e., . Therefore, the maximum number of interference-free users is given by

(9)

for , which proves the lemma. \qed

Lemma 1 implies that the multiplexing gain of cannot be obtained with the equal power constraint. Inspired by this fact, we notice that interference-free users can be obtained by employing for some BSs, i.e., no effective transmission. The following lemma discusses the maximum number of interference-free users based on this zero transmission power concept.

Lemma 2

The maximum number of interference-free users in the MISO interference channel with BSs having zero transmission power is given by

(10)

where is the number of BSs with non-zero transmission power.

Proof:

Note that the number of BSs with non-zero transmit power and the number of interference-free users having non-zero strength of the desired signal are denoted as and , respectively. The condition on can be obtained following the analogous footsteps of the proof of Lemma 1 by replacing with as . Therefore, the maximum number of interference-free users is given by

(11)

Thus, choosing , we have . Note that for and for , which proves the lemma. \qed

From Lemma 2, the maximum number of interference-free users, , can be obtained by simply muting BSs. In such a case, the index set of the active BSs with non-zero transmission power should be the same as the index set of the interference-free users, denoted by . Specifically, the beamforming vectors are designed as follows. BS for designs the beamforming vector that maximizes in (7) setting and for , and for as

(12)
(13)

where . Then, the solution for the problem (12) is obtained by choosing the right singular vector of associated with the smallest singular value. Note that since we choose and for , and for , the rank of is ; that is, the smallest singular value is 0, yielding .

For and , we choose

(14)

With this choice, the interference received at user , , becomes zero, and the sum-rate is given by

(15)

It is crucial to design properly to maximize the sum-rate, which shall be obtained in Section III-E.

Remark 1

Turning off a set of base stations in small cell networks is used as one of the sum-rate improving technologies in 3GPP [44]. However, which and how many BSs should be turned off to maximize the sum-rate for given network has been investigated only empirically or heuristically. In this study, we derive which and how many BSs should be turned off in case of to nearly achieve the maximum capacity bound.

Iii-C Beamforming vector design for

From Lemma 2, can be obtained by having . Setting to its maximum value, i.e., , does not harm the sum-rate because more non-zero rates from BS , , are added in the sum-rate than with . Thus, for , we choose to set . For , we consider the following beamforming designs with local CSI.

Iii-C1 Bs for

Note that each beamforming vector of size has null space size of . Thus, to make user , , interference-free, BS , should employ the min-WGI beamforming design in (12) as follows:

(16)

Iii-C2 Bs for

Since BS for only needs to make zero interference to the BSs with the indices in , where , BS can utilize the space of rank one either to improve the desired channel gain or to make zero-interference to user for . Specifically, to make zero-interference to user for , BS for would set for and for and design its beamforming vector maximizing (7) from

(17)

On the other hand, to improve the desired channel gain, BS for would set for and for and design its beamforming vector maximizing (7) as

(18)
(19)

where and . Then, the solution of (18) is given by the eigenvector of associated with the maximum eigenvalue.

To discuss the difference between the aforementioned two strategies in the sense of maximizing the sum-rate, we establish the following theorem.

Theorem 1

For BS , , and , let us denote the sum-rate for the case (referred to as Case 1) where BS employs the max-WSLNR beamforming from (18) as , and the sum-rate for the case (referred to as Case 2) where BS employs the min-WGI beamforming from (17) by . For both the cases, BS , , designs its beamforming vector from (16). Then, we have in low- and high-SNR regime.

Proof:

The sum-rate for Case 1, , can be represented as

(20)

where and .

To compute , suppose that BSs , , make GI to another user , , zero. Then, can be represented as

(21)

i) In low-SNR regime, i.e., is arbitrarily large,

(22)
(23)

Consequently, we have

(24)
(25)

where (25) follows from the fact that the max-WSLNR problem (18) becomes the max-SNR problem for arbitrarily large , yielding . Since for any unit-norm , we have for , which proves the theorem for low-SNR regime.

ii) In high-SNR regime, i.e., is arbitrarily small, the achievable rates of the interference-free users, which have zero interference, are dominant due to the interference terms in the achievable rates of the other users. Thus, we have and , and hence, we again have the same expression as in (24). In Case 1, for is designed to have the direction of the orthogonal projection of onto the null space of , . On the other hand, in Case 2, the beamforming vector is designed to have the direction of the null space of and . That is, the beamforming vector is designed independently of on the null space of , . Therefore, we have

(26)

which proves the theorem for high-SNR regime. \qed

From Theorem 1, we propose to design for , , from the max-WSLNR problem of (18).The sum-rate with such a choice is given by (27).

(27)

Again, the design of shall be provided in Section III-E.

Iii-D Beamforming vector design for

For , all the BSs design their beamforming vectors making zero GI to user , . The number of neighboring users, to which each BS makes GI zero, is for BS , , and for BS , . That is, BS , , designs its beamforming vector maximizing the desired channel gain and making GI zero to user , , and BS , , designs its beamforming vectors maximizing the desired channel gain and making GI zero to user , . Then, the beamforming vectors of BS and the -BS are designed in the null spaces of ranks and , respectively. Then, for , all the beamforming vectors are obtained from the max-WSLNR problem of (18). The sum-rate in such a case is given by

(28)

The examples of the beamforming vector design protocol with and for , , and are illustrated in Figs. 1(a), 1(b), and 1(c), respectively.

(a)
(b)
(c)
Figure 2: Illustration of the proposed multicell beamforming vector design

Iii-E Selection of : Design of

Now, the aim is to determine a proper number of interference-free users, , and the set of interference-free users, , out of all possible cases in pursuit of maximizing the sum-rate with local CSI and limited information exchange. Totally, there exist possible interference-free users selection.

Let us denote the -th interference-free user selection, , by , i.e., users , , have received interference of zero. With this interference-free users selection of , the rate of user is denoted by and the beamforming vector of BS is denoted by . Then, the sum-rate for the -th interference-free user selection can be represented as

(29)

where and . Herein, the first term is the sum of rates of user , , which can be computed with only local CSI by BS . On the other hand, the second term is the sum of rates of user which require global CSI to be computed by BS , .

For , is zero since BS , , has zero transmit power. Therefore, we have and it requires only local CSI to be available at BS , . However, for , is non-zero and requires global CSI to be available at all the BSs. Thus, we propose to consider the upper bound of the average which can be computed at all the BSs with only local CSI. To get the upper bound of for , we establish the following lemma.

Lemma 3

For all and ,

(30)

where is the incomplete gamma function.

Proof:

for , the expectation of can be bounded as follows:

(31)
(32)

i) , : For , is designed independently with , and hence, is a Chi-square random variable with degrees of freedom (DoF) 2. On the other hand, for , lies in the orthogonal projection of onto the null space of , . Let us denote as the -th basis vector of the null space of . The rank of the space composed of these basis vectors is . Then, the desired channel gain can be represented as

(33)

and it is a Chi-square random variable with DoF of which is for . Thus, we get for .

ii) , : For , , , is designed independently with . Thus, is a Chi-square random variable with DoF 2. Then, we get

(34)

where is the gamma function.

From the above two results, the expectation of for can be further bounded as follows:

(35)

which proves the lemma. \qed

From Lemma 3, we propose to select the index set for , which maximizes . Note that for , and hence the cost function can be used for all possible values discussed. At this point, to compromise between the amount of information exchange among BSs and the sum-rate performance, let us assume that the information of , , is collected only for the cases with selected . In this case, let us denote the set of considered and the index set of the considered cases as and , respectively. If the set of considered is for and , we have . Then, the index set optimization problem is formulated as

(36)

where

(37)

Iii-E1 Tightness of the upper bound

The gap of and results only from the Jensen’s inequality in (32). The analysis of Jensen’s gap has been extensively studied in the literature [45, 46]. The gap in the inequality (32) tends to 0 if the random variable is almost surely constant. The bound of the gap in case where is mean-centric is derived in [46]. In addition, the function becomes an affine function for small , resulting in the gap tending to 0. In summary, as received interference at user , , becomes significantly stronger than the desired signal gain, the gap in (30) tends to zero. Furthermore, the more the SINR becomes mean-centric, the tighter upper bound we can get from (30).

Iii-E2 Asymptotic performance of using

In the high SNR regime, i.e., is arbitrarily small, becomes dominant in (29) and we have , for all . In addition, also tends to 0 in the high SNR regime. Therefore, the proposed design is asymptotically optimal as SNR increases. In finite SNR regime, as grows for fixed , in (29) becomes dominant since the number of rate terms in which require global CSI, , decreases. On the other hand, as increases for fixed , the number of interference terms in increases, and the number of rate terms in also increases. It can be readily shown that this global CSI term tends to be bounded by a constant value even in the high-SNR regime, following the analysis in [1]. Hence, for high-SNR regime, where the terms tend to be infinite, or for large compared with , the global CSI terms become negligible compared to the local CSI terms, resulting in also tending to 0.

Iii-E3 Performance of using in finite SNR, , and

Figure 3 shows the per-cell average and versus SNR for and , where each channel is i.i.d. according to the complex Gaussian distribution. As shown in this figure, the gap between and is smaller than 0.04bps/Hz for all possible values, showing that is a good estimator of even with finite parameter values.

Figure 3: Per-cell average and versus SNR for and

Iv Information Exchange Protocol and Quantization

In this section, an information exchange protocol and quantization method are proposed based on the beamforming vector design proposed in Section III.

Iv-a Information exchange

To compute the cost function of the problem (37), , each rate term of , , needs to be computed by BS , , with local CSI and be shared by all the BSs. The term can be computed by any BS without any extra information on instantaneous channels. Let us denote the rate of user for with the -th interference-free user selection by

(38)

An example case is as shown in Table I, where , , and . Here, BS 1 can compute the achievable rates in the white cells of the column of BS1 with only local CSI and does not compute the achievable rates correspond to the dark gray cells in the column of BS1 in Table I, because they require global CSI to computed. Though each BS can compute rate terms with local CSI, BS shares values only for to restrict the amount of information exchange. Then, for given , , is computed by adding all the collected rate terms, i.e., collected rate terms in each row of Table I, the problem (37) can be formulated together with .

Table I: Achievable rates table for all the interference-free user selection cases for and

Iv-B Quantization optimization

In this subsection, the quantization of rate terms that need to be exchanged is analyzed, which is crucial to exchange the information with finite bits. Let us denote the number of nonzero rates to be exchanged by and the number of information exchange bits to be used for quantization of each rate by . BS quantizes rates terms, i.e., , , . Thus, the number of information exchange bits used at each BS is

(39)

For optimal quantization, the probability density function (PDF) of , , , is needed, which is denoted by . To get the PDF , we establish the following lemma.

Lemma 4

The random variable , , , is distributed as a Chi-square random variable with degrees of freedom (DoF) of .

Proof:

i) For , the beamforming vector is designed to only minimize the GI to user , . Thus, is designed independently with the desired channel vector and is distributed as a Chi-square random variable with DoF 2.

ii) For , the beamforming vector is designed to maximize its WSLNR and it has the direction of the orthogonal projection of onto the null space of , where and . Let us denote is the -th basis vector of the null space of . The number of the basis vector is . Then, the desired channel gain can be represented as