Analysis on the Number of Users in MRT and MRC Based Full-Duplex Massive MIMO Systems

Ⅰ. Introduction

To increase data rate, full-duplex has received great attention as one of the representative technologies in 6G wireless communication systems^[1]. Full-duplex is a duplex mode in which both data transmission and reception are allowed at the same time-frequency resources^[2]. Therefore, assuming perfect self-interference cancellation at the receiver, the achievable data rates of both downlink and uplink can be obtained simultaneously, providing the doubled total sum-rate compared to the half-duplex system. Notably, advancements in self-interference cancellation techniques have significantly enhanced the practicality of full-duplex systems^[3,4].

Meanwhile, integration of multi-user multiple-input multiple-output (MIMO) into full-duplex systems further improves the data rate by enjoying both spatial multiplexing gain and doubled data rate. In particular, massive MIMO systems, which utilize a large number of antenna arrays to support a vast number of users, have been considered a key technology for enhancing the performance of full-duplex systems^[5]. Therefore, many researches have focused on analyzing the performance gain of full-duplex massive MIMO systems with various system operation scenarios in [6-10]. Authors in [6] analyzed an antenna ratio to maximize sum-rate of the full-duplex massive MIMO system with half-duplex multiantenna users. Moreover, the sum-rates of the full-duplex massive MIMO system with bidirectional users were analyzed in [7], in which both Zero-forcing, maximum ratio transmission (MRT) and maximum ratio combining (MRC) are utilized at the base station. In [8] and [9], the full-duplex was also incorporated with wireless backhaul links and cell-free massive MIMO systems,

and their theoretical spectral efficiencies were analyzed. The authors in [10] further considered multi-stream transmission with full-duplex masive MIMO systems considering channel estimation error. However, the prior works in [6-10] did not consider the impact of the number of users to maximize the sum-rate. Remarkably, it was already proven in [11] and [12] that the number of users should be carefully chosen to maximize the sum-rate of the half-duplex massive MIMO systems. Therefore, it is essential to investigate full-duplex massive MIMO systems with regard to the number of users, which encourages us to analyze the sum-rate of full-duplex massive MIMO systems with a focus on the impact of the number of users.

Compared to previous works in [6-10], our study considers MRT and MRC in full-duplex massive MIMO to derive the sum-rate in closed form, reducing the need for extensive simulations and enabling an efficient assessment of sum-rate improvement in terms of the number of users. In addition, we here focus on analyzing the number of users based on MRT and MRC. Compared to Zero-forcing which requires high computational complexity due to the matrix inversion, MRT and MRC are attractive in terms of simple conjugate operation in obtaining transmitter and receiver^[13-15], along with asymptotic signal-tointerference- plus-noise ratio performance with a large number of antennas^[16,17].

In this work, we study the number of users to maximize the sum-rate of the full-duplex massive MIMO system. The contributions of this work can be summarized as follows:

· The downlink and uplink sum-rates of the full-duplex massive MIMO system based on maximum ratio transmission (MRT) and maximum ratio combining (MRC) are derived in a closed-form expression, which is an approximated result based on a large number of antennas.

· Based on the analysis, we derive the numbers of downlink and uplink users to maximize the total sum-rate of the full-duplex massive MIMO system based on MRT and MRC. Since the number of uplink users to maximize the total sum-rate is intractable, we obtain the numbers of downlink and uplink users to maximize the downlink and uplink sum-rates, respectively. Then, we propose to select the number of downlink and uplink users considering the target uplink rate.

The remaining paper is organized as follows. Section II describes a system model, the number of users to maximize the sum-rate is analyze in Section III, simulation results are provided in Section IV, and Section V concludes the paper.

Notation: In this paper, [TeX:] $$(\cdot)^H$$ denotes the Hermitian transpose, [TeX:] $$\mathbb{E}\{\cdot\}$$ denotes the expectation operation, Var· denotes the variance operation, [TeX:] $$\|\cdot\|$$ denotes the Frobenius norm, and [TeX:] $$I_N$$ is an N × N identity matrix.

Ⅱ. System Model

As shown in Fig. 1, we consider a multi-user MIMO system where the BS and users are equipped with M antennas and single antenna, respectively. Each antenna at the BS simultaneously transmits and receives data in the same time-frequency resource^[18]. Also, [TeX:] $$K_d$$ downlink and [TeX:] $$K_u$$ uplink users receive and transmit data, respectively. Let [TeX:] $$s_k \text{ and } x_k$$ denotes the transmission symbol for downlink user k with [TeX:] $$E\left\{\left|s_k\right|^2\right\}=1$$ and the transmission symbol of uplink user k with [TeX:] $$E\left\{\left|x_k\right|^2\right\}=1,$$ respectively. Also, let define [TeX:] $$n_k^d \text { and } \mathbf{n}^u$$ as the downlink and uplink additive white Gaussian noise with zero mean and unit variance, respectively. Then, the downlink received signal at downlink user k is given by

where [TeX:] $$P_d \text { and } P_u$$ are the transmission signal-to-noise ratios (SNRs) at the BS and uplink user, respectively, and [TeX:] $$\mathbf{h}_k^d \in \mathbb{C}^{1 \times M}, \quad \mathbf{g}_k \in \mathbb{C}^{1 \times M}, \quad h_{U_m D_k}$$ denote channel between the BS and downlink user k, precoding vector for downlink user k, and channel between uplink user m and downlink user k, respectively. Moreover, the received signal for uplink user k at the BS is given by

where [TeX:] $$\mathbf{h}_k^u \in \mathbb{C}^{M \times 1}, \mathbf{w}_k \in \mathbb{C}^{M \times 1}, \mathbf{H}^s \in \mathbb{C}^{M \times M}, \text { and } \sigma_{\mathrm{SI}}^2$$ denote channel between uplink user k and the BS, receive combining filter vector for uplink user k, selfinterference channel matrix at the BS, and residual self-interference. It is assumed that the elements of [TeX:] $$\mathbf{h}_k^d, \mathbf{h}_k^u \text { and } \mathbf{H}^s$$ are modeled as independent complex Gaussian random variables with zero mean and unit variances as in [6]. To maximize the received signal powers of downlink and uplink data, we consider MRT as a transmission precoding vector and MRC as a receive combining filter for downlink and uplink, respectively, which are expressed as

and

Ⅲ. The Number of Users to Maximize Sum-rate of Full-duplex Massive MIMO

In this section, we analyze the number of users to maximize the sum-rate of the full-duplex massive MIMO system based on MRT and MRC. We first derive the ergodic downlink and uplink sum-rates of the full-duplex massive MIMO system. To gain insights into the analysis, closed-form expressions for the sum-rate are derived using the method outlined in [19]. Based on the derivation, we analyze the numbers of downlink and uplink users to maximize the sum-rate of the full-duplex massive MIMO systems.

3.1 Ergodic Downlink and Uplink Sum-Rates of Full-Duplex Massive MIMO Based on MRT and MRC

The ergodic sum-rate of full-duplex massive MIMO system can be expressed as in [6-10] by

In (5), from (1) and (3), the ergodic achievable rate of downlink user k based on MRT, [TeX:] $$\mathbb{E}\left\{R_d^k\right\},$$ can be obtained as in [12] by

where [TeX:] $$I_{\mathrm{m}, k}=P_d \sum_{l \neq k}^{K_d} \frac{\left|\mathbf{h}_k^d\left(\mathbf{h}_l^d\right)^H\right|^2}{K_d| | \mathbf{h}_l^d \|^2} \text { and } I_{\mathrm{u}, k}=P_u \sum_{m=1}^{K_u}\left|h_{U_m D_k}\right|^2 .$$ Using [TeX:] $$\mathbb{E}\left\{\log _2\left(1+\frac{S}{N}\right)\right\} \approx \log _2\left(1+\frac{\mathbb{E}\{S\}}{\mathbb{E}\{N\}}\right)$$ as in [19]. (6) can be approximated as

In (7), it was proven in [19] that [TeX:] $$\mathbb{E}\left\{\left|\mathbf{h}_k^d\right|^2\right\}=M .$$ Moreover, [TeX:] $$\mathbb{E}\left\{I_{\mathrm{m}, k}\right\}$$ can be written as

According to Lemma 4 of [19], assuming high/low SNR, i.e. [TeX:] $$P_d \operatorname{Var}\left\{\left|\mathbf{h}_k^d\left(\mathbf{h}_l^d\right)^H\right|^2\right\} \rightarrow 0,$$ we have

Substituting (9) into (8), [TeX:] $$\mathbb{E}\left\{I_{\mathrm{m}, k}\right\}$$ can be approximated as

In (10), based on Lemma 4 of [19] with large M, [TeX:] $$\mathbb{E}\left\{\frac{1}{\left\|\mathbf{h}_l^d\right\|^2}\right\}$$ can be approximated as

Putting (9)-(11) into (8), we can further derive [TeX:] $$\mathbb{E}\left\{I_{\mathrm{m}, k}\right\}$$ as

where (a ) is obtained by Appendix B1 of [19]. Furthermore, [TeX:] $$\mathbb{E}\left\{I_{\mathrm{u}, k}\right\}$$ can be derived as

Putting [TeX:] $$\mathbb{E}\left\{\left|\mathbf{h}_k^d\right|^2\right\}=M,$$ (12), and (13) into (7), [TeX:] $$\mathbb{E}\left\{R_k^d\right\}$$ can be approximated as

Therefore, the ergodic downlink sum-rate of the full-duplex massive MIMO with MRT can be approximated as

It can be shown from (15) that the ergodic downlink sum-rate increases as M increases, and decreases as [TeX:] $$K_u$$ increases.

In what follows, we derive the achievable rate of uplink user k, [TeX:] $$\mathbb{E}\left\{R_u^k\right\},$$ which is given by

where [TeX:] $$I_{\mathrm{m}, k}^u=P_u \sum_{l \neq k}^{K_u}\left|\left(\mathbf{h}_k^u\right)^H \mathbf{h}_l^u\right|^2$$ and [TeX:] $$I_{\mathrm{s}, k}=\sigma_{\mathrm{SI}}^2 \sum_{m=1}^{K_d}\left|\left(\mathbf{h}_k^u\right)^H \mathbf{H}^s\left(\mathbf{h}_m^d\right)^H\right|^2 .$$ Then, similar to (7), we can approximate [TeX:] $$\mathbb{E}\left\{R_u^k\right\}$$ as

In (16), [TeX:] $$\mathbb{E}\left\{\left\|\mathbf{h}_k^u\right\|^2\right\}=M .$$ Moreover,

where (b) is obtained by Appendix B1 of [19], and

In (19),

Then, substituting (20) into (19), we have

In (21), it was proven in [20] that

Substituting (22) into (21), we have

Combining [TeX:] $$\mathbb{E}\left\{P_u\left|\mathbf{h}_k^u\right|^2\right\}=M,$$ (18), and (23) into (17), we obtain

Therefore, the ergodic uplink sum-rate of the fullduplex massive MIMO with MRC can be approximated by

It can be seen from (25) that the uplink sum-rate increases as M increases, thanks to the combining gain by a large number of antennas.

3.2 Analysis on the Number of Users to Maximize Sum-Rate

In this subsection, we analyze the number of users to maximize the ergodic sum-rate of the MRT and MRC based full-duplex massive MIMO. We utilize the approximated ergodic downlink and uplink sum-rates derived in (15) and (25), respectively. Then, we can formulate an optimization problem to find the numbers of downlink and uplink users that maximize Rsum such that

However, [TeX:] $$K_d^* \text { and } K_u^*$$ are intractable, due to non-convexity of [TeX:] $$R_{\mathrm{sum}}$$ in terms of [TeX:] $$K_u^*.$$ Therefore, we divide (26) into

and

where [TeX:] $$\bar{R}_d=\sum_{k=1}^{K_d} \mathbb{E}\left\{R_d^k\right\} \text { and } \bar{R}_u=\sum_{k=1}^{K_u} \mathbb{E}\left\{R_u^k\right\} .$$ Ku (27) and (28) are derived based on an insight that [TeX:] $$R_{\mathrm{sum}}$$ can be maximized by maximizing [TeX:] $$\bar{R}_d \text { and } \bar{R}_u$$ respectively.

First, we investigate the existence of the solution of (27). To this end. (27) has to satisfy the second-order necessary condition (SONC)^[21]. The second derivative of [TeX:] $$\bar{R}_d$$ can be expressed as

where [TeX:] $$\Gamma_d=\frac{P_d M}{P_d\left(K_d-1\right)+P_u K_u K_d+K_d} \text { and } \gamma_d=P_d+P_u K_u+1.$$ To satisfy SONC, (29) should be less then 0, which leads to

We can rewrite (30) into

According to the definition of [TeX:] $$\Gamma_d, \Gamma_d$$ is always larger than 0. When [TeX:] $$\Gamma_d \geq 2,$$ (31) is always satisfied. Whereas, when [TeX:] $$0 \leq \Gamma_d\lt 2,$$ the minimum of the left hand side of (31) is [TeX:] $$2 P_d M-\gamma_d,$$ which indicates that SONC in [TeX:] $$0 \leq \Gamma_d\lt 2$$ is satisfied when [TeX:] $$2 P_d M-\gamma_d \gt 0.$$ Therefore, we can conclude that SONC is strictly satisfied if [TeX:] $$\Gamma_d \geq 2,$$ which indicates that [TeX:] $$K_d^*$$ exists over [TeX:] $$0\lt K_d \leq M \text { under } \Gamma_d \geq 2 .$$

To obtain [TeX:] $$K_d^*,$$ we derive the first derivative of [TeX:] $$\bar{R}_d$$ given by

where [TeX:] $$\alpha_d=P_d+P_u K_u+1 .$$ Applying [TeX:] $$\ln x \geq 1-\frac{1}{x}$$ into (32) and dividing ln 2 into left-and right-hand sides of (32), we have

It is shown in (33) that [TeX:] $$\frac{\partial \bar{R}_d}{\partial K_d}$$ is always positive over [TeX:] $$K_d\gt 0,$$ which indicates that [TeX:] $$\bar{R}_d$$ monotonically increases over [TeX:] $$K_d,$$ and therefore [TeX:] $$K_d^*=M .$$

Next, we analyze [TeX:] $$K_u^*$$ to maximize [TeX:] $$\bar{R}_u .$$ To investigate SONC of [TeX:] $$\bar{R}_u,$$ we derive the second derivative of [TeX:] $$\bar{R}_u$$ as

where [TeX:] $$\Gamma_u=\frac{P_u M}{P_u\left(K_u-1\right)+\sigma_{\mathrm{SI}}^2+1} .$$ To satisfy SONC, [TeX:] $$2-\frac{K_u\left(2 \Gamma_u-\Gamma_u^2\right)}{M\left(1+\Gamma_u\right)}$$ should be larger than 0, which leads to

For [TeX:] $$\Gamma_u\gt 0,$$ the left-hand side of (35) is always larger than zero. Therefore, we can conclude that [TeX:] $$\bar{R}_u$$ satisfies SONC.

To analyze [TeX:] $$K_u^*$$, we derive the first derivative of [TeX:] $$\bar{R}_u$$ given by

where [TeX:] $$\varepsilon_u=-P_u+\sigma_{\mathrm{SI}}^2+1$$ Applying [TeX:] $$\ln x \geq 1-\frac{1}{x}$$ into (36) and dividing ln 2 into left-and right-hand sides of (36), we have

It can be shown from (37) that [TeX:] $$\frac{\partial \bar{R}_u}{\partial K_u}$$ is always larger than 0, which indicates that [TeX:] $$\bar{R}_u$$ is an increasing function over [TeX:] $$K_u \geq 1,$$ therefore [TeX:] $$K_u^*=M .$$

3.3 Selection of the Numbers of Downlink and Uplink Users

In the previous subsection, it was proven that the numbers of downlink and uplink users to maximize the MRT-based downlink and MRC-based uplink sumrates are equal to the number of antennas, which implies that a larger number of users helps to enhance the sum-rate of the full-duplex massive MIMO system when MRT and MRC are utilized as the precoder and receiver. However, it can be seen in (6) that the downlink sum-rate decreases as the number of uplink users increases. Therefore, to achieve maximum total sum-rate in (5), we need to choose proper number of uplink users. It is known in general that the required downlink data rate is larger than the required uplink data rate^[22]. Therefore, we consider choosing the number of uplink users that satisfies the target uplink data rate.

Let us define [TeX:] $$R_u^t$$ as the target uplink data rate per user. Then, the ergodic uplink data rate per user should satisfy

With some manipulation, (38) can be rewritten in terms of [TeX:] $$K_u$$ given by

It is shown in (39) that maximum allowed number of uplink users increases as [TeX:] $$R_u^t$$ decreases. Because larger target rate requires lower signal-to-interference plus noise ratio (SINR), large [TeX:] $$K_u$$ can prohibit to achieve [TeX:] $$R_u^t$$ due to small SINR.

Based on the observation from Subsection 3.2 and (39), we introduce a strategy of choosing the numbers of downlink and uplink users to maximize the sum-rates as follows:

· [TeX:] $$K_d$$ is selected as M, to maximize the downlink sum-rate.

· Using (39), [TeX:] $$K_u$$ is selected such that

where [TeX:] $$\lceil a\rceil$$ is the largest integer smaller than a .

Ⅳ. Simulation Results

In this section, we provide numerical results to show the accuracy of the analysis. Unless otherwise stated, we set [TeX:] $$P_d=P_u=10 \mathrm{~dB}$$ and M = 64.

Fig. 2 shows the downlink sum-rate as a function of the number of downlink users. Here, we set [TeX:] $$K_u=\{8,16,24\} .$$ As shown in the figure, the downlink sum-rate increases as [TeX:] $$K_d$$ increases, which result is aligned with the analysis in Section 3.2. The increase of the downlink sum-rate for [TeX:] $$K_d$$ is because the SINR gain by a large number of antennas dominates the multi-user interference. Also, it is observed that the sum-rate decreases as [TeX:] $$K_u$$ increases, because the inter-user interference from the uplink user increases as [TeX:] $$K_u$$ increases.

Fig. 3 shows the uplink sum-rate as a function of the the number of uplink users. Here, we set [TeX:] $$K_d=32 \text { and } \sigma_{\mathrm{SI}}^2=\{0,10,20\} \mathrm{dB} .$$ Similar to the downlink, the uplink sum-rate increases as [TeX:] $$K_u$$ increases, which result is aligned with the analysis in Section 3.2. Further, we can observe that the sum-rate decreases as the residual self-interference increases, which highlights the importance of ensuring effective self-interference cancellation to achieve a sufficient uplink sum-rate.

Fig. 4 shows the total sum-rate as a function of the number of antennas when [TeX:] $$K_d=K_u=\{8,16,32\} .$$ In the simulation, we set [TeX:] $$P_u=3 \mathrm{~dB} \text { and } \sigma_{\mathrm{SI}}^2=10 \mathrm{~dB} .$$ As seen in the figure, the total sum-rate increase as M increases, because of the SINR gain from increasing number of antennas. Moreover, the total sum-rate increases as [TeX:] $$K_d \text { and } K_u$$ increase, because the gains from Fig. 2 and 3 are combined. Therefore, we can conclude that the total sum-rate of the full-duplex massive MIMO system based on MRT and MRC can be improved by increased numbers of downlink and uplink users.

Fig. 5 compares the total sum-rates of full-duplex and half-duplex as a function of the residual self-interference when [TeX:] $$K_d=K_u=\{8,16,32\} .$$ In the simulation, we set [TeX:] $$P_u=3.$$ As shown in the figure, the total sum-rate of the full-duplex massive MIMO system decreases as [TeX:] $$\sigma_{\mathrm{SI}}^2$$ increases, because the uplink sum-rate decreases by residual self-interference. On the other hand, the total sum-rate of half-duplex is constant over [TeX:] $$\sigma_{\mathrm{SI}}^2$$. Therefore, the half-duplex outperforms the full-duplex when [TeX:] $$\sigma_{\mathrm{SI}}^2\gt 10 \mathrm{~dB}$$ with [TeX:] $$K_d=K_u=8.$$ It is worth mentioning that the region of residual self-interference where the half-duplex outperforms the full-duplex decreases as [TeX:] $$K_d \text{ and } K_u$$ increase. Fig. 6 illustrates the downlink sum-rate when the constraint of uplink sumrate is given as [TeX:] $$R_t=\{1,2,3\}$$ bps/Hz. Here, we set [TeX:] $$M=K_d=64 \text { and } P_u=3$$ dB. It is shown in Fig. 6 that for a given M, the downlink sum-rate increases as [TeX:] $$R_t$$ increases. This is because larger [TeX:] $$R_t$$ requires a smaller number of uplink users, resulting in the increase of downlink SINR and sum-rate. In addition, Fig. 7 plots [TeX:] $$K_u$$ as a function of M with [TeX:] $$R_t=\{1,2,3\} .$$ As shown in the figure, [TeX:] $$K_u$$ increases as M increases for all [TeX:] $$R_t .$$ This is because more number of uplink users are allowed for a large M with given [TeX:] $$R_t .$$

Ⅴ. Conclusions

In this paper, we have analyzed the number of users to maximize the ergodic sum-rate of full-duplex massive MIMO system. We have derived the approximated ergodic downlink and uplink sum-rates based on MRT and MRC, respectively. From the analysis, we have shown that the numbers of downlink and uplink users to maximize the sum-rate are equal to the number of antennas. Simulation results have shown that the analyzed ergodic downlink and uplink sum-rates are close to simulation results. Simulation results have also shown that the sum-rates with the analyzed number of users improves the total sum-rate.