Secure downlink transmission with finite resolution analog beamforming in massive multiple-input multiple-output system

Abstract

To reduce the number of radio-frequency chains of base station, the use of finite resolution analog beamforming is desirbale in massive multiple-input multiple-output system. This article investigates the secure downlink massive multiple-input multiple-output data transmission with artificial noise at base station in the presence of a multi-antenna passive eavesdropper. The achievable user’s ergodic information rate and ergodic capacity of the eavesdropper are analyzed in detail, respectively. With maximum ratio transmission or maximum ratio combining, we derive closed-form expressions for a tight lower bound on ergodic secrecy rate and tight upper bound for secrecy outage probability. Based on these analytical expressions, the effects of various system parameters on secrecy performance, such as power allocation factor, number of eavesdropper’s antennas, number of the user terminals, total transmission power, and finite resolution analog beamforming parameters, are investigated in detail. Also, the optimal power allocation scheme between data and artificial noise signals is achieved in closed form to maximize the ergodic secrecy rate. In addition, we derive the conditions that the secure massive multiple-input multiple-output system need to meet to obtain a positive secrecy rate. Finally, numerical simulation results validate the system’s secrecy performance and verify all the theoretical analytical results.

Introduction

Massive multiple-input multiple-output (MIMO) using a few hundred or more antennas for multiuser service is an attractive emerging technology for 5G,^1,2 which enables extremely high spectral efficiency and energy efficiency.^3,4 Because of this distinctive feature, massive MIMO can reduce the burden of user terminals (UTs), that is, low hardware cost, limited power, and low storage capacity, which can boost the promotion of the forthcoming green communications and future networks, along with Internet-of-Things (IoT) and wireless sensor network (WSN).^5,6 Moreover, to obtain the array gain, each antenna needs a complete radio-frequency (RF) chain.^7,8 In fact, the large number of antenna elements will bring an extremely high hardware cost in massive MIMO systems. Hence, some new architectures for massive MIMO with low-complexity RF hardware and small number of RF chains should be studied to tackle these issues.

By certain iterative algorithms, phase-only precoding can obtain full diversity gain and near-optimal beamforming performance.^7,9 In Liang et al.’s ¹⁰ article, the RF phase-only analog precoder was designed by extracting the phases of the conjugate transpose of the transmission channel in massive MIMO system. In addition, using the finite resolution analog beamforming (FRAB) can effectively reduce hardware cost. ¹¹ FRAB is cost-effectively applied to high-dimensional antenna array implementing some variable phase shifters with current circuitry technology, which only modifies signals’ phases and does not change amplitudes as digital beamforming. Finite-precision analog-to-digital conversion (ADC) and quantized phase (QP) control have been taken into account in previous works.^8,12 And the finite resolution constraint is meaningful due to the fact that a practical circuit is finite. A new combining architecture for massive MIMO receivers was proposed consisting of RF antenna switches and finite constant phase shifters in Alkhateeb et al. ¹¹ and Zhu et al. ¹² This phase-only combining architecture can reduce the number of required RF chains and power consumption. Ding et al. ¹³ utilized non-orthogonal multiple access (NOMA) to mitigate the reduced degrees of freedom brought by FRAB, which can obtain the diversity gain to shed light on the loss caused by FRAB.

Security takes an increasingly critical role in wireless communication. Physical layer security is an essentially information-theoretic security. ¹⁴ Secure communication in MIMO systems has been studied extensively and has obtained fruitful research results.^15,16 Lin et al. ¹⁷ studied the impact of quantized channel direction information (CDI) on the achievable secrecy rate with artificial noise (AN) generation, which shows that system’s achievable secrecy rate will has a significant loss under quantized CDI. Also, the article derives a scaling law between feedback bits and transmission power to maintain a constant secrecy rate loss. For large multiple-input single-output (MISO) broadcast channels, Geraci et al. ¹⁸ obtained a deterministic approximation for the achievable secrecy sum rate as the number of transmit antennas $N_T$ and the number of users K grow to infinity but with a fixed ratio $K/N_T$ . Based on these results, this article provides an optimal regularized channel inversion (RCI) precoding scheme for this large MISO system. Using large system analysis and AN generation, Li et al. ¹⁹ derived a closed-form expression for the ergodic secrecy sum rate in large multiuser downlink secure communication system with a passive eavesdropper. Meanwhile, the article studies the optimal power allocation between information signals and the AN. There is a common perspective that massive MIMO is merely an extensive of MIMO and all the ways are propitious to the security of massive MIMO system. In fact, massive MIMO has unique qualities, and we cannot address many standard approaches directly.^20,21 A large number of antennas will average the interference, channel estimation errors, and hardware impairments.^4,5 These random impairments will vanish when the number of antennas grows extremely large, leaving pilot contamination only to limit the secure performance. Unperfect channel state information (CSI) may be a major threat to the security in massive MIMO systems. Therefore, physical layer security of massive MIMO provides a new research avenue, which needs a separate treatment of its own.

The existing literatures about combination of physical layer security and massive MIMO are not sufficient. Kapetanovic et al. ²¹ studied the physical-layer security of downlink massive MIMO communication in detail, which revealed that massive MIMO system was naturally immune to passive eavesdropping, but harmed by training-phase jamming. Some works have presented many insightful researches on passive eavesdropping attacks and active attacks on training phase of massive MIMO system.^22–24 A single-cell massive MIMO system in the presence of an adversary capable of jamming and eavesdropping has been presented in Wu et al. ²² Al-nahari ²³ and Zhang et al. ²⁴ achieved the closed-form achievable secrecy rate for passive and active attacking, which also provided active attacking detection schemes, as realizing the presence of active eavesdropper is crucial to the system’s security. Secure transmission in multi-cell multiuser massive MIMO systems with maximum ratio transmission (MRT) or zero forcing (ZF) precoding and AN generation has been investigated by Zhu et al.^25–27 Large antenna array can get excellent directional gain, so beamforming can be exploited to enhance the security performance.^28,29 In IoT scenario, secure uplink and downlink transmissions for massive MIMO system in the presence of a massive MIMO eavesdropper have been studied in Chen et al.^30,31 Due to the excessive components of massive MIMO system, using cheap and power-efficient hardware is desirable to reduce the overall capital expenditures. The secrecy performance of massive MIMO system with non-ideal hardware has received much attention by some researchers and obtained some research productions.^32,33 However, there are no sufficient considerations of how the FRAB affects system security, and how to improve security under some practical constraints in massive MIMO systems.

Motivated by the aforementioned description, to reduce hardware cost, this article focuses on the physical layer security in massive MIMO exploiting FRAB and AN. To the best of our knowledge, this problem is still an open topic. The contributions of this article are summarized as follows:

• The achievable ergodic information rate and eavesdropper’s ergodic capacity are analyzed in detail. Under some assumptions, closed-form expressions of one tight lower bound on the ergodic secrecy rate and a tight upper bound for the secrecy outage probability are derived in this article.

• Based on the achieved ergodic secrecy rate, we derive closed-form expression of the optimal power allocation schemes for the data signals and the AN signals. Meanwhile, the design scheme of the reliable secure massive MIMO system is revealed to achieve positive secrecy rate.

• The impacts of various system parameters on the ergodic secrecy rate are investigated in detail, such as power allocation factor, number of eavesdropper’s antennas, number of the UTs, total transmission power, and FRAB parameters. The results of these analyses can provide significant insight for secure massive MIMO system design.

The remainder of this article is organized as follows. Section “System model” describes the system model and also introduces the scheme of the downlink data transmission and the architecture of the FRAB. In section “Secrecy performance analysis,” we obtain the UT’s achievable ergodic information rate and eavesdropper’s capacity in detail. Also, we provide a tight lower bound for ergodic secrecy rate and a tight upper bound on secrecy outage probability in closed form. In section “Impact of various parameters on secrecy performance,” we give the details of the impacts by various system parameters on the ergodic secrecy rate, which can provide meaningful insight for secure massive MIMO system design. The numerical and simulation results are presented in section “Numerical results.” Finally, we provide the concluding remarks in section “Conclusion.”

Notations

Bold-face letters denote vectors, for example, $\mathbf{h}$ . And $(\mathbf{h}{)}^{*}$ , $(\mathbf{h}{)}^T$ , $(\mathbf{h}{)}^H$ , and $\Vert \mathbf{h}\Vert$ denote complex conjugate, transpose, Hermitian transpose, and Euclidean norm of the vector $\mathbf{h}$ , respectively. j represents the imaginary unit, that is, $j^2=-1$ . $\mathbb{E}[·]$ and $\mathrm{var}[·]$ denote expectation and variance. The notation $\mathbf{x}~\mathcal{C}\mathcal{N}(\mathbf{m},\mathbf{A})$ stands for a circularly symmetric complex Gaussian random vector x with mean m and covariance matrix A. ${\mathbb{C}}^{M\times N}$ represents $M\times N$ dimensional space with complex-valued elements. $\underset{¯}{R}$ denotes the lower bound of statistic R. In addition, i.i.d indicates the abbreviation of independent and identically distributed. Finally, $[x{]}^{+}=max\{0,x\}$ .

System model

In this article, we consider a typical massive MIMO cellular communication system, where base station (BS) equipped with a massive antenna array (antenna number $N_T>>1$ ) serves K single-antenna UTs. Specifically, downlink data transmission is considered in this article. The considered system model is depicted in Figure 1. Alice is the massive MIMO BS. Without loss of generality, we assume Alice attempts to transmit a confidential signal $s_k$ to the kth UT (Bob) in the cellular cell. In the cell, there is an eavesdropper (Eve) with $N_E$ antennas who seeks to recover the confidential message $s_k$ . The channels between BS and these terminals experience large-scale path loss as well as small-scale fading, which can be denoted by $\mathbf{G}={[{\mathbf{g}}_1,{\mathbf{g}}_2,\dots ,{\mathbf{g}}_K]}^T\in {\mathbb{C}}^{K\times N_T}$ and ${\mathbf{G}}_E\in {\mathbb{C}}^{N_E\times N_T}$ , where ${\mathbf{g}}_i=\sqrt{{\alpha }_i}{\mathbf{h}}_i,i=1,2,\dots ,K$ and ${\mathbf{G}}_E=\sqrt{{\alpha }_E}{\mathbf{H}}_E$ . ${\alpha }_i$ and ${\alpha }_E$ are the non-negative large-scale path fading including path loss and shadow fading from Alice to ith UT and Eve. ${\mathbf{h}}_i\in {\mathbb{C}}^{N_T\times 1}$ and ${\mathbf{H}}_E\in {\mathbb{C}}^{N_E\times N_T}$ represent the corresponding small-scale fading components, respectively. In addition, we assume the transmission experiences Rayleigh fading, hence the elements of the two vectors are modeled as i.i.d complex Gaussian random variables with zero mean and unit variance. We assume the BS has accurate CSI of all UTs.

Figure 1.

Considered massive MIMO secure downlink transmission model, including one massive antenna BS with FRAB precoder and AN generation, K single-antenna UTs, and a multi-antenna Eve.

Downlink data transmission

During the downlink data transmission phase, Alice (BS) sends the data signals to UTs. In addition, Alice simultaneously allocates some power to emit AN signals to degrade Eve’s ability to eavesdrop the intended data to Bob. We assume the transmitter employ $N_{\mathrm{RF}}⩾K$ RF chains. For the sake of simplification, we assume that BS only uses K RF chains serving a multiplicity of K autonomous terminals. Usually, precoding is utilized to reinforce the decoding performance of desired legitimate users constructively and degrade the wiretapping ability of the present eavesdroppers. In fact, compared to the conventional MIMO system, some linear precoding schemes (MRT, ZF, and MMSE) have been exploited extensively in massive MIMO system to obtain large gains in power/bandwidth efficiency.^1,4 However, utilizing these precoding schemes, the massive MIMO system should allocate one complete RF chain to per antenna, which will be too expensive for practical implementation. To reduce power consumption and the complexity of the system structure, we exploit FRAB^11,13 in this article, which employs finite phase shifters. The RF precoding matrix $\mathbf{W}$ can be made by some constant phase shifters. We can rewrite $\mathbf{W}$ as a set of column vectors, that is, $\mathbf{W}=[{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_K]\in {\mathbb{C}}^{N_T\times K}$ . All columns of the precoding matrix $\mathbf{W}$ are normalized with unit norm, that is, $\Vert {\mathbf{w}}_i\Vert =1,\forall i$ . As mentioned in the previous analysis, to degrade decoding ability of the existing eavesdropper, Alice simultaneously transmit both data signals and AN signals to Bob. The AN shaping matrix is denoted by $\mathbf{V}=[{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_{N_T-K}]\in {\mathbb{C}}^{N_T\times (N_T-K)}$ . Similar to the conventional MIMO (non-massive) systems, AN signals are chosen to lie in the null space of the UTs’ transmission channel matrix, so these AN signals only degrade Eve’s received signal-to-interference-plus-noise ratio (SINR) performance, but take little effect to legitimate UTs,^25,26 that is, ${\mathbf{h}}_i^T\mathbf{V}=0,\forall i$ . In this article, to obtain some fundamental insights, for simplification, the uniform power allocation scheme is used across the UTs and AN signals, respectively. We assume each UT has same power p and each one-dimensional AN signal’s power is q. The transmission signal vector of the BS can be given by

$$\mathbf{x}=\sqrt{p}\mathbf{Ws}+\sqrt{q}\mathbf{Vz}$$ (1)

where $p=\frac{\varphi P_T}{K}$ ; $q=\frac{(1-\varphi )P_T}{N_T-K}$ ; s and z represent the data vector and AN vector, respectively, whose elements obey independent and standardized Gaussian distribution: $\mathbf{s}\in \mathcal{C}\mathcal{N}(0,{\mathbf{I}}_K)$ and $\mathbf{z}\in \mathcal{C}\mathcal{N}(0,{\mathbf{I}}_{N_T-K})$ ; $P_T$ is the total transmit power at BS; and $\varphi \in (0,1]$ is the power allocation factor to strike a balance between the information signal and AN signals.

Accordingly, Bob’s received signals in the cell can be denoted by

$$y_k={\mathbf{g}}_k^T\mathbf{x}+n_B$$ (2)

Similarly, the received signals at Eve are represented by

$$y_E={\mathbf{r}}_E^T{\mathbf{G}}_E\mathbf{x}+n_E$$ (3)

In the above expressions, ${\mathbf{r}}_E\in C^{N_E\times 1}$ is the receiving filter used by Eve, and $n_B~\mathcal{C}\mathcal{N}(0,{\delta }_B^2)$ and $n_E~\mathcal{C}\mathcal{N}(0,{\delta }_E^2)$ are the received additive Gauss thermal noise components. With no loss of generality, we set ${\delta }_B^2=1$ to simplify the analysis in this article.

FRAB

In this section, we discuss the design and implementation of the phase-only RF precoding with low complexity. Due to the power constraints, we can set $\left|w_{\mathrm{mn}}\right|=\frac{1}{\sqrt{N_T}}$ , where $w_{\mathrm{mn}}$ represents the (m, n)th element of $\mathbf{W}$ . To gain the highest array gain of the large-scale antennas, the optimal phase-only RF design with continuous phase (CP) can be given by

$$w_{\mathrm{mn}}^{\mathrm{CP}}=\frac{1}{\sqrt{N_T}}{e}^{-j\angle h_{\mathrm{nm}}}=\frac{1}{\sqrt{N_T}}{e}^{-j{\phi }_{\mathrm{nm}}}$$ (4)

where $h_{\mathrm{nm}}$ is the mth element of vector ${\mathbf{h}}_n$ (the channel vector between nth UT and BS) and variable ${\phi }_{\mathrm{nm}}$ denotes the phase of $h_{\mathrm{nm}}$ , ${\phi }_{\mathrm{nm}}=\angle h_{\mathrm{nm}}$ .

As the practical implementation always exploits QP shifters instead of CP values, according to some previous works,^8,10,11 the matrix $\mathbf{W}$ can be quantized to B bits of precision. In this architecture, each element is quantized to its nearest phase value. The beamforming vector ${\mathbf{w}}_i$ can be denoted by

$${\mathbf{w}}_i^{\mathrm{QP}}=\frac{1}{\sqrt{N_T}}{\mathbf{S}}_i{\mathbf{p}}^{*}$$ (5)

where $p={\left[1,e^{j2\pi {/2}^B}\mathrm{,}\dots \mathrm{,}{e}^{j2\pi (2^B-1){/2}^B}\right]}^T$ and ${\mathbf{S}}_i$ is the $N_T\times 2^B$ dimensional binary phase switching matrix. $s_{i,\mathrm{mn}}$ is the (m, n)th element of matrix ${\mathbf{S}}_i$ , which is subjected to these constraints $s_{i,\mathrm{mn}}\in \{0,1\}$ ; $\sum _{n=1}^{2^B}{s}_{i,\mathrm{mn}}=1$ , that is, for ${\mathbf{S}}_i$ , only one element of each row vector is 1, $s_{i,\mathrm{mk}}=1$ , where $k=\arg \underset{n}{min}\left(|p_n-h_{\mathrm{im}}/\Vert h_{\mathrm{im}}\Vert |\right)$ and $p_n$ is the nth element of vector p. In fact, the operation ${\mathbf{S}}_i\mathbf{p}$ is to find the QP value which is nearest to channel parameter phase ${\phi }_{\mathrm{im}}$ .

Secrecy performance analysis

In this section, the achievable intended information rate and Eve’s ergodic capacity will be analyzed under the perfect CSI assumption. Subsequently, exploiting these results, a simple tight lower bound on the ergodic secrecy rate and a closed-form upper bound for secrecy outage probability are derived. Following, we analyze the achievable rate of Bob, Eve’s ergodic capacity, achievable ergodic secrecy rate, and secrecy outage probability in following subsections.

Achievable ergodic information rate of Bob

Accordingly, Bob’s received SINR ( $\mathrm{SIN}{R}_B$ ) is denoted by

$$\mathrm{SIN}{R}_B=\frac{p{\left|{\mathbf{g}}_k^T{\mathbf{w}}_k\right|}^2}{p\sum _{i=1,i\ne k}^K{\left|{\mathbf{g}}_k^T{\mathbf{w}}_i\right|}^2+1}$$ (6)

Obviously, calculating the accurate results of the $\mathrm{SIN}{R}_B$ is very sophisticated. According to some previous works,^25–27 we can achieve a tight lower bound of the achievable rate of Bob’s MISO noise channel as formula (2). This resulting lower bound can be represented by

$$R_k>{\underset{¯}{R}}_k={\log }_2(1+{\lambda }_k)$$ (7)

where ${\lambda }_k$ denotes the lower bound of the $\mathrm{SIN}{R}_B$ . This lower bound for achievable UT’s ergodic rate is achieved assuming legitimate UT only has the statistic information of its own effective channel, that is, $\mathbb{E}\left[{\mathbf{g}}_k^T{\mathbf{w}}_k\right]$ . The assumption on the legitimate users is reasonable and has been widely used in existing papers.^25,34,35 ${\lambda }_k$ is given by

$${\lambda }_k=\frac{{\left|\mathbb{E}\left[\sqrt{p}{\mathbf{g}}_k^T{\mathbf{w}}_k\right]\right|}^2}{\mathrm{var}\left[\sqrt{p}{\mathbf{g}}_k^T{\mathbf{w}}_k\right]+\sum _{i=1,i\ne k}^K\mathbb{E}\left({\left|\sqrt{p}{\mathbf{g}}_k^T{\mathbf{w}}_i\right|}^2\right)+1}$$ (8)

In the above expression, the expection and variance operator are taken with respect to all channel realizations. We can first analyze the optimal phase-only RF beaforming SINR performance. As described before, each element of the small-scale channels is i.i.d complex Gaussian random variable with zero mean and unit variance. Thus, we can conclude that $\left|h_{i,j}\right|$ follows Rayleigh distribution, whose mean and variance are $\sqrt{\pi }/2$ and $1-(\pi /4)$ , respectively. According to the Lindeberg–Lévy central limit theorem,^4,10 the desired signal term can be expressed as

$${\mathbf{g}}_k^T{\mathbf{w}}_k~\mathcal{N}(\frac{\sqrt{\pi N_T{\alpha }_k}}{2},(1-\frac{\pi }{4}){\alpha }_k)$$ (9)

According to Lemma 1 in the literature, ¹⁰ the interference term ${\mathbf{w}}_k^T{\mathbf{g}}_i$ from other UTs in the denominator of equation (8) has its distribution as

$${\mathbf{g}}_k^T{\mathbf{w}}_i~\mathcal{C}\mathcal{N}(0,{\alpha }_k)$$ (10)

So, $\left|{\mathbf{g}}_k^T{\mathbf{w}}_i\right|$ also follows Rayleigh distribution with mean $\sqrt{\pi {\alpha }_k}/2$ and variance $(1-(\pi /4)){\alpha }_k$ . Clearly, plugging the numerator and denominator into formula (8), the lower bound of the achievable rate of Bob with CP analog precoder can be given by

$${\underset{¯}{R}}_k^{\mathrm{CP}}={\log }_2(1+\frac{N_T\pi {\alpha }_k}{4K({\alpha }_k+\frac{1}{\varphi P_T})-\pi {\alpha }_k})$$ (11)

According to these results, we conclude that the upper limit SINR gain of the phase-only RF combining is $\pi /4$ of the gain achieved by fully digital transmission (MRT).^23,24

Then, we consider the FRAB with QP shifters. The received intended signal power component can be expressed as

$${\mathbb{E}}^2\left[{\mathbf{g}}_k^T{\mathbf{w}}_k\right]=\frac{{\alpha }_k}{N_T}{\left|\mathbb{E}\left[\sum _{i=1}^{N_T}\left|h_{\mathrm{ik}}\right|e^{j{\tilde{\phi }}_{\mathrm{ik}}}\right]\right|}^2$$ (12)

where ${\tilde{\phi }}_{\mathrm{ik}}=\angle h_{\mathrm{ik}}+\angle w_{\mathrm{ik}}$ . These phases lie in $[-\frac{\pi }{2^B},\frac{\pi }{2^B}]$ with uniform distribution. Considering the independence between channel amplitude and the phase error, it is known that $\left|h_{\mathrm{ik}}\right|$ and ${\tilde{\phi }}_{\mathrm{ik}}$ obey Rayleigh distribution and uniform distribution, respectively. Furthermore, $\left|h_{\mathrm{ik}}\right|$ and ${\tilde{\phi }}_{\mathrm{ik}}$ are mutually independent.^10,11 So, we can obtain that

$$\begin{array}{cc}\multicolumn{1}{c}{\mathbb{E}\left[\left|h_{\mathrm{ik}}\right|e^{j{\tilde{\phi }}_{\mathrm{ik}}}\right]}& =\mathbb{E}\left[\left|h_{\mathrm{ik}}\right|\right]\mathbb{E}\left[e^{j{\tilde{\phi }}_{\mathrm{ik}}}\right]\hfill \\ \multicolumn{1}{c}{}& =\frac{2^B}{2\sqrt{\pi }}\sin \left(\frac{\pi }{2^B}\right)\hfill \end{array}$$ (13)

Thus, we know that

$${\mathbb{E}}^2\left[\left|{\mathbf{g}}_k^T{\mathbf{w}}_k\right|\right]=\frac{2^{2B}{N}_T{\alpha }_k}{4\pi }\stackrel{2}{\sin }\left(\frac{\pi }{2^B}\right)$$ (14)

Moreover, $\mathbb{E}\left[{\left|{\mathbf{g}}_k^T{\mathbf{w}}_k\right|}^2\right]$ can be expressed as equation (15), shown at the top of the next page

$$\begin{array}{c}\mathbb{E}\left[{\left|{\mathbf{g}}_k^T{\mathbf{w}}_k\right|}^2\right]\hfill \\ =\frac{{\alpha }_k}{N_T}\mathbb{E}[\left(\sum _{i=1}^{N_T}\left|h_{\mathrm{ik}}\right|e^{j{\tilde{\phi }}_{\mathrm{ik}}}\right)·\left(\sum _{m=1}^{N_T}\left|h_{\mathrm{mk}}\right|e^{-j{\tilde{\phi }}_{\mathrm{mk}}}\right)]\hfill \\ =\frac{{\alpha }_k}{N_T}\mathbb{E}[\sum _{i=1}^{N_T}{\left|h_{\mathrm{ik}}\right|}^2+\sum _{i\ne m}^{N_T}\left|h_{\mathrm{ik}}\right|\left|h_{\mathrm{mk}}\right|e^{j{\tilde{\phi }}_{\mathrm{ik}}-j{\tilde{\phi }}_{\mathrm{mk}}}]\hfill \\ ={\alpha }_k+(N_T-1)\frac{2^{2B}{\alpha }_k}{4\pi }\stackrel{2}{\sin }\left(\frac{\pi }{2^B}\right)\hfill \end{array}$$ (15)

Hence, the variance of ${\mathbf{g}}_k^T{\mathbf{w}}_k$ can be gained by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{var}\left[{\mathbf{g}}_k^T{\mathbf{w}}_k\right]=\mathbb{E}\left[{\left|{\mathbf{g}}_k^T{\mathbf{w}}_k\right|}^2\right]-{\mathbb{E}}^2\left[\left|{\mathbf{g}}_k^T{\mathbf{w}}_k\right|\right]}\\ \multicolumn{1}{c}{={\alpha }_k-\frac{2^{2B}{\alpha }_k}{4\pi }\stackrel{2}{\sin }\left(\frac{\pi }{2^B}\right)}\end{array}$$ (16)

Similarly, the interference component ${\mathbf{w}}_k^T{\mathbf{g}}_i$ from other UTs have the following distribution as

$${\mathbf{g}}_k^T{\mathbf{w}}_i~\mathcal{C}\mathcal{N}(0,{\alpha }_k)$$ (17)

Finally, we can get the ${\underset{¯}{R}}_k$ exploiting FRAB, which is given by

$${\underset{¯}{R}}_k^{\mathrm{FRAB}}={\log }_2(1+\frac{2^{2B}{N}_T{\alpha }_k{\sin }^2\left(\frac{\pi }{2^B}\right)}{4\pi K({\alpha }_k+\frac{1}{\varphi P_T})-2^{2B}{\alpha }_k{\sin }^2\left(\frac{\pi }{2^B}\right)})$$ (18)

This result reveals that ${\underset{¯}{R}}_k^{\mathrm{FRAB}}$ is increasing in quantitative precision B. As $B\to \infty$ , ${\underset{¯}{R}}_k^{\mathrm{FRAB}}$ has an upper bound as same as ${\underset{¯}{R}}_k^{\mathrm{CP}}$ . This is in agreement with our expectation.

Ergodic capacity of Eve

Since Eve is assumed to be totally passive, its CSI and noise power level are not acknowledged by BS. Unlike UT’s ergodic rate, we cannot obtain a lower bound for Eve’s rate since, otherwise, we cannot get a lower bound on the ergodic secrecy rate. Therefore, we attempt to analyze an upper bound for Eve’s ergodic capacity. It is assumed that Eve can acquire perfect knowledge of ${\mathbf{G}}_E$ to cancel the received signals from all UTs except the intended signal. Actually, it is scarcely possible to gain the perfect CSI if the Eve is pure passive in the cell. So, this assuming is extremely pessimistic. Based on the effective CSI, Eve is able to obtain the array gain by performing maximum ratio combining (MRC). In addition, since BS has no knowledge of the Eve’s noise level, we consider the worst case that the additive thermal noise at Eve is negligible, that is, ${\delta }_E^2\to 0$ . Hence, we can denote the SINR of Eve as

$${\lambda }_E=\frac{p{\left|{\mathbf{r}}_E^T{\mathbf{G}}_E{\mathbf{w}}_k\right|}^2}{p\sum _{m=1,m\ne k}^K{\left|{\mathbf{r}}_E^T{\mathbf{G}}_E{\mathbf{w}}_m\right|}^2+q\sum _{m=1}^{N_T-K}{\left|{\mathbf{r}}_E^T{\mathbf{G}}_E{\mathbf{v}}_m\right|}^2}$$ (19)

MRC is usually preferred in massive MIMO system, which can achieve an approximately optimal performance with linear operations. So, the receive filter ${\mathbf{r}}_E$ can be designed as

$${\mathbf{r}}_E=\frac{{\mathbf{H}}_E^{*}{\mathbf{w}}_k^{*}}{\Vert {\mathbf{H}}_E{\mathbf{w}}_k\Vert }$$ (20)

For the convenience of analysis, we rewrite the Eve’s SINR expression as

$${\lambda }_E=\frac{{\left|{\mathbf{r}}_E^T{\mathbf{H}}_E{\mathbf{w}}_k\right|}^2}{\sum _{m=1,m\ne k}^K{\left|{\mathbf{r}}_E^T{\mathbf{H}}_E{\mathbf{w}}_m\right|}^2+\gamma \sum _{m=1}^{N_T-K}{\left|{\mathbf{r}}_E^T{\mathbf{H}}_E{\mathbf{v}}_m\right|}^2}=\frac{U}{T}$$ (21)

where $\gamma =q/p$ , and U and T denote the numerator and denominator of ${\lambda }_E$ , respectively.

So, the ergodic capacity of Eve is given by

$$C_E=\mathbb{E}\left[{\log }_2(1+{\lambda }_E)\right]$$ (22)

The received intended signal power at the eavesdropper can be given by

$$\mathbb{E}\left[U\right]=\mathbb{E}\left[{\left|{\mathbf{r}}_E^T{\mathbf{H}}_E{\mathbf{w}}_k\right|}^2\right]=\mathbb{E}\left[z^2\right]$$ (23)

where $z^2=\sum _{i=1}^{N_E}{\left|u_i\right|}^2$ , in which $u_i~\mathcal{C}\mathcal{N}(0,1)$ , and ${\left|u_i\right|}^2$ and $z^2$ obey scaled chi-square distribution with 2 and $2N_E$ degrees of freedom, respectively. Hence, we can get that

$$\mathbb{E}\left[U\right]=\mathbb{E}\left[z^2\right]=N_E$$ (24)

And the probability density function (PDF) of U can be represented by

$$f_U\left(x\right)=\frac{x^{N_E-1}{e}^{-x}}{\Gamma \left(N_E\right)},x>0$$ (25)

where $\Gamma (x)$ denotes gamma function. Then, we consider the term T. From the statistical meaning, variable T is equivalent to a weighted sum of two scaled chi-square distributed variables, which can be expressed as^25,27

$$\begin{array}{cc}\multicolumn{1}{c}{T}& =\sum _{m=1,m\ne k}^K{\left|{\mathbf{r}}_E^T{\mathbf{H}}_E{\mathbf{w}}_m\right|}^2+\gamma \sum _{m=1}^{N_T-K}{\left|{\mathbf{r}}_E^T{\mathbf{H}}_E{\mathbf{v}}_m\right|}^2\hfill \\ =T_1+\gamma T_2\hfill \end{array}$$ (26)

where $T_1~{\chi }_{2(K-1)}^2$ and $T_2~{\chi }_{2(N_T-K)}^2$ denote scaled chi-square distributed variables with $2(K-1)$ and $2(N_T-K)$ degrees of freedom, respectively.

By exploiting Laplace transform and inverse Laplace transform, the PDF of T can be given by

$$f_T\left(x\right)=\frac{1}{{\gamma }^{N_T-K}}(\sum _{i=1}^{K-1}\frac{a_i{x}^{i-1}{e}^{-x}}{\Gamma \left(i\right)}+\sum _{i=1}^{N_T-K}\frac{b_i{x}^{i-1}{e}^{-\frac{x}{\gamma }}}{\Gamma \left(i\right)})$$ (27)

where the individual fractions of the numerators are given by

$$a_i=\frac{{(-1)}^{K-i-1}(N_T-i-2)!}{(N_T-K-1)!(K-i-1)!{(\frac{1}{\gamma }-1)}^{N_T-i-1}}$$ (28)

and

$$b_i=\frac{{(-1)}^{N_T-K-i}(N_T-i-2)!}{(N_T-K-i)!(K-2)!{(1-\frac{1}{\gamma })}^{N_T-i-1}}$$ (29)

Proof

For proof, see Appendix 1.

As U and T are mutually independent, we can obtain the cumulative density function (CDF) of Eve’s SINR ${\lambda }_E$ exploiting the definition $F_{{\lambda }_E}(x)=Pr(\frac{U}{T}⩽x)$ .

Thus, we can obtain the CDF expression of ${\lambda }_E$ and Eve’s ergodic capacity as equations (30) and (31), where $I(\mu ,k)={\int }_0^{\infty }\frac{1}{(x+1){(x+\mu )}^k}\mathrm{dx}$

$$F_{{\lambda }_E}\left(x\right)=1-\frac{1}{{\gamma }^{N_T-K}}(\sum _{i=1}^{K-1}\frac{a_i}{\Gamma \left(i\right)}\sum _{j=1}^{N_E-1}\frac{(i+j-1)!x^j}{j!{(x+1)}^{i+j-2}}+\sum _{i=1}^{N_T-K}\frac{b_i}{\Gamma \left(i\right)}\sum _{j=1}^{N_E-1}\frac{(i+j-1)!x^j}{j!{(x+\frac{1}{\gamma })}^{i+j-2}})$$ (30)

$$\begin{array}{cc}\multicolumn{1}{c}{C_E}& =\mathbb{E}\left[{\log }_2(1+{\lambda }_E)\right]=\frac{1}{\ln 2}\underset{0}{\overset{+\infty }{\int }}\frac{1-F_{{\lambda }_E}\left(x\right)}{1+x}\mathrm{dx}\hfill \\ \multicolumn{1}{c}{}& =\frac{1}{\ln 2{\gamma }^{N_T-K}}(\sum _{i=1}^{K-1}\frac{a_i}{\Gamma \left(i\right)}\sum _{j=1}^{N_E-1}\frac{(i+j-1)!}{j!}I(1,i+j-2)\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=1}^{N_T-K}\frac{b_i}{\Gamma \left(i\right)}\sum _{j=1}^{N_E-1}\frac{(i+j-1)!}{j!}I(\frac{1}{\gamma },i+j-2))\hfill \end{array}$$ (31)

Note that, the exact expression of the ergodic capacity of Eve is not difficult to achieve. However, the expressions are complicated and difficult for theoretical calculation and analysis, which are not easy to provide some insightful results. In the following, we analyze a simple tight upper bound on the ergodic capacity of Eve.

Using Jensen’s inequality, a tight upper bound on Eve’s ergodic capacity is obtained as

$$\begin{array}{c}\multicolumn{1}{c}{C_E⩽{\overline{C}}_E=\underset{2}{\log }(1+\mathbb{E}\left[{\lambda }_E\right])}\\ \multicolumn{1}{c}{=\underset{2}{\log }(1+\mathbb{E}\left[U\right]\mathbb{E}\left[\frac{1}{T}\right])}\end{array}$$ (32)

The above equality is based on independence between U and T. Specifically, recalling some existing results in Zhu et al.,^25,27 the variable T can be approximatively regarded as one single-scaled chi-square distributed variable $T\approx \epsilon T_{\phi }$ , where $T_{\phi }~{\chi }_{\phi }^2$ . And the parameters $\epsilon$ and $\phi$ should be selected to make the first two moments of T and $\epsilon T_{\phi }$ equal. Equating the two moments of the matrices’ traces leads to

$$\epsilon \phi =(K-1)+\gamma (N_T-K)$$ (33)

and

$${\epsilon }^2\phi =(K-1)+{\gamma }^2(N_T-K)$$ (34)

Utilizing the inverse Wishart matrix’s expectation expression, we can obtain that

$$\mathbb{E}\left[\frac{1}{T}\right]=\frac{1}{\epsilon (\phi -1)}$$ (35)

Plugging these results into equation (32), the upper bound of Eve’s ergodic capacity can be obtained by

$$C_E<{\overline{C}}_E=\underset{2}{\log }(1+\frac{{\beta }_E}{{\beta }_K+\gamma (1-{\beta }_K)})$$ (36)

where ${\beta }_K$ represents the ratio of the number of UTs in the cell to the number of BS antennas ( ${\beta }_K=K/N_T$ ), ${\beta }_E$ denotes the ratio of Eve’s antenna number to the number of BS antennas ( ${\beta }_E=N_E/N_T$ ).

As Eve’s noise component is considered negligible in our analysis, we note that ${\overline{C}}_E$ is independent of the path loss ${\alpha }_{\mathrm{AE}}$ .

Achievable ergodic secrecy rate

If the system can afford delays and operate coding over many independent channel realizations, the ergodic secrecy rate is one suitable secrecy performance measure. Based on the previous analysis, a simple lower bound for the ergodic secrecy rate with FRAB can be given by equation (37), where $\mu =2^{2B}{\sin }^2(\pi /2^B)/4\pi$

$$C_{SEC}\mathrm{>}{\underset{\_}{C}}_{SEC}={\left[{\underset{\_}{R}}_k^{FRAB}-{\overline{C}}_E\right]}^{+}={\left[{\log }_2\left(\frac{P_T{\alpha }_k\left(\mu +{\beta }_K\right)\varphi +{\beta }_K}{P_T{\alpha }_k{\beta }_E{\varphi }^2+\left(P_T{\alpha }_k{\beta }_K+{\beta }_E\right)\varphi +{\beta }_K}\right)\right]}^{+}$$ (37)

The large-scale path fading always changes on a very slow time scale, as these path-loss values generally depend on the geographic location and transmission environment. To obtain more insights into various parameters’ impact, we can simplify that ${\alpha }_i=1,i=1,2,\dots ,K$ . So the lower bound of the ergodic secrecy rate can be further simplified as

$${\underset{¯}{C}}_{\mathrm{SEC}}={\left[{\log }_2\left(\frac{P_T(\mu +{\beta }_K)\varphi +{\beta }_K}{P_T{\beta }_E{\varphi }^2+(P_T{\beta }_K+{\beta }_E)\varphi +{\beta }_K}\right)\right]}^{+}$$ (38)

Secrecy outage probability

However, secrecy outages will be unavoidable when the system only afford limited delays, as BS have no information of Eve’s channel and decoding ability. Secrecy outage probability is an appropriate metric to evaluate the system’s secrecy performance, which is defined as the probability that the actual achieved instantaneous secrecy rate is less than the target secrecy rate $R_0$ . For massive MIMO system, we know that the desired information rate is deterministic as $N_T\to \infty$ , but Eve’s achievable capacity is a random variable. Hence, a secrecy outage occurs when Eve’s capacity exceeds the system’s security threshold. According to equations (22) and (30), the secrecy outage probability can be given by

$$\begin{array}{cc}\multicolumn{1}{c}{P_{\mathrm{out}}\left(R_0\right)}& =\mathrm{Pr}(R_k-C_E⩽R_0)\hfill \\ \multicolumn{1}{c}{}& =\mathrm{Pr}({\lambda }_E⩾2^{R_k-R_0}-1)\hfill \\ \multicolumn{1}{c}{}& =1-F_{{\lambda }_E}(2^{R_k-R_0}-1)\hfill \end{array}$$ (39)

where $R_0$ denotes system’s desirable target secrecy rate. Then, we can derive an upper bound for the secrecy outage probability by substituting ${\underset{¯}{R}}_k^{\mathrm{FRAB}}$ for $R_k$ . We can obtain the closed-form expression of secrecy outage probability as equation (40), where $\epsilon =2^{{\underset{¯}{R}}_k^{\mathrm{FRAB}}-R_0}-1$

$$P_{\mathrm{out}}\left(R_0\right)=\frac{1}{{\gamma }^{N_T-K}}(\sum _{i=1}^{K-1}\frac{a_i}{\Gamma \left(i\right)}\sum _{j=1}^{N_E-1}\frac{(i+j-1)!{\epsilon }^j}{j!{(\epsilon +1)}^{i+j-2}}+\sum _{i=1}^{N_T-K}\frac{b_i}{\Gamma \left(i\right)}\sum _{j=1}^{N_E-1}\frac{(i+j-1)!{\epsilon }^j}{j!{(\epsilon +\frac{1}{\gamma })}^{i+j-2}})$$ (40)

Impact of various parameters on secrecy performance

In this section, we analyze the impacts of the various system parameters on secrecy performance in detail. To gain some insightful and instructive results, we exploit the achieved lower bound on the ergodic secrecy rate to make the following analysis.

Power allocation factor $\varphi$

First, we can consider one special case, that is, normalized number of UTs ${\beta }_K$ and the passive eavesdropper’s antenna number $N_E$ are fixed as $N_T\to \infty$ . Under this setting, we are reminded that ${\beta }_E$ is negligible, that is, ${\beta }_E\to 0$ . Thus, we can obtain the ${\underset{¯}{C}}_{\mathrm{SEC}}$ as

$${\underset{¯}{C}}_{\mathrm{SEC}}={\log }_2(1+\frac{\mu }{{\beta }_K}-\frac{\mu }{{\beta }_K(P_T\varphi +1)})$$ (41)

We note that the secrecy performance is restricted by $\varphi$ and ${\beta }_K$ (multiuser interference) and the secrecy rate increases significantly with the increase in $\varphi$ . We realize the information leakage is almost negligible if Eve only exploits finite antennas in massive MIMO system. This is consistent with some previous researches.^22–24 Hence, if the massive MIMO system informs the Eve only have small number of antennas in advance, and BS can allocate total power for data transmission, that is, $\varphi =1$ . And the desired UT can obtain the maximal secrecy information rate, that is, ${\underset{¯}{C}}_{\mathrm{SEC}}=\underset{2}{\log }(1+\frac{\mu P_T}{{\beta }_K(P_T+1)})$ .

For the general situation, first of all, we can reveal the conditions of zero secrecy rate, that is, ${\underset{¯}{C}}_{\mathrm{SEC}}=0$ . It is easy to see that the solutions of the equation are $\varphi =0$ and $\varphi ={\varphi }_1$ . Specifically, ${\varphi }_1$ can be obtained by

$${\varphi }_1=\frac{\mu }{{\beta }_E}-\frac{1}{P_T}$$ (42)

From the expression, we can conclude that this massive MIMO system can obtain positive secrecy rate as ${\beta }_E<\frac{P_T\mu }{1+P_T}$ or $\mu >{\beta }_E(1+\frac{1}{P_T})$ . Otherwise, in order to achieve secrecy rate, the power allocation factor should meet the condition that $\varphi \in (0,{\varphi }_1)$ .

Easily, we can get that $\frac{{\partial }^2{\underset{¯}{C}}_{\mathrm{SEC}}}{\partial {\varphi }^2}<0$ which can prove that the function is a concave function with respect to $\varphi$ at $(0,{\varphi }_1)$ . Hence, we can obtain the optimal $\varphi$ that maximizes the ergodic secrecy rate. The optimal $\varphi$ can be gained by derivative operation from equation (38) as

$${\varphi }_{\mathrm{opt}}={\left[\frac{\sqrt{P_T{\mu }^2{\beta }_K{\beta }_E+P_T\mu {\beta }_K^2{\beta }_E-\mu {\beta }_K{\beta }_E^2}-{\beta }_K{\beta }_E}{P_T{\beta }_E(\mu +{\beta }_K)}\right]}_0^1$$ (43)

where $[x{]}_0^1=x$ if $0<x<1$ , $[x{]}_0^1=0$ if $x⩽0$ , and $[x{]}_0^1=1$ if $x⩾1$ .

Numbers of Eve’s antennas and the UTs

In this section, we use the normalized ratios ${\beta }_K$ and ${\beta }_E$ to evaluate the influence of K and $N_E$ .

To investigate the influence of ${\beta }_K$ , the expression of ${\underset{¯}{C}}_{\mathrm{SEC}}$ can be rewritten as

$${\underset{¯}{C}}_{\mathrm{SEC}}=\underset{2}{\log }(1+\frac{P_T\mu \varphi -P_T{\beta }_E{\varphi }^2-{\beta }_E\varphi }{(P_T\varphi +1){\beta }_K+P_T{\beta }_E{\varphi }^2+{\beta }_E\varphi })$$ (44)

According to relational expression equation (42), it is easily known that

$$P_T\mu \varphi -P_T{\beta }_E{\varphi }^2-{\beta }_E\varphi >0$$ (45)

Thus, the ergodic secrecy rate is a monotonically decreasing function of ${\beta }_K$ . We can explain that as the number of UTs increases, the power allocated to each UT decreases and the multiuser interference term increases. To overcome this effect, more power should be allocated for data transmission. Moreover, from equations (18) and (36), it is shown that the changes in ${\beta }_K$ have a greater effect on the achievable information rate than Eve’s ergodic capacity. Similarly, the secrecy rate decreases as ${\beta }_E$ increases. This is easily explained that Eve can achieve higher capacity with more antennas, while more Eve’s antennas has no effect on Bob’s achievable rate. Hence, if Eve get the knowledge of the power allocation scheme and the precoding architecture of BS, Eve can fully eavesdrop the intended information by equipping adequate antennas, which can be represented as

$${\beta }_E⩾\frac{P_T\mu }{P_T\varphi +1}$$ (46)

Total transmit power P_T

It is noted that the exact expression of the Eve’s achievable capacity is complicated and not easy to evaluate. So, we can utilize the upper bound for Eve’s ergodic capacity as equation (36) in the following analysis. For $N_T\to \infty$ , it is easy to find that ${\overline{C}}_E$ has nothing to do with $P_T$ .

Then, we focus on the UT’s achievable ergodic information rate. From equation (18), we observe that $P_T$ only affects component $1/\varphi P_T$ in the denominator of ${\lambda }_k$ . Hence, the UT’s ergodic information rate monotonically increases in $P_T$ . Based on equations (18) and (36), we can conclude that the ergodic secrecy rate is an increasing function of $P_T$ . As $P_T\to \infty$ , the achievable secrecy rate becomes deterministic, which can be upper bounded by

$${\underset{¯}{\tilde{C}}}_{\mathrm{SEC}}={\left[{\log }_2\left(\frac{\mu +{\beta }_K}{{\beta }_E\varphi +{\beta }_K}\right)\right]}^{+}$$ (47)

FRAB parameter B

Now, we analyze the impact of the FRAB parameter B. In this article, we choose the phase shifters with uniform phases. So, B is the only parameter that determines the precision of FRAB. Physically, larger B means FRAB is closer to CP precoder and can get a larger array gain. Moreover, as we know, Eve’s channel ${\mathbf{G}}_E$ is independent with the column vectors of precoding matrix $\mathbf{W}$ . Hence, parameter B has no effect on Eve’s ergodic capacity, which can be proved by formula (36).

Then, we consider the impact of parameter B on Bob’s ergodic information rate. From equations (8) and (18), we know that

$${\lambda }_k=\frac{2^{2B}{N}_T{\alpha }_k{\sin }^2\left(\frac{\pi }{2^B}\right)}{4\pi K({\alpha }_k+\frac{1}{\varphi P_T})-2^{2B}{\alpha }_k{\sin }^2\left(\frac{\pi }{2^B}\right)}$$ (48)

As ${\underset{¯}{R}}_k^{\mathrm{FRAB}}$ is a logarithmic function of variable ${\lambda }_k$ , ${\underset{¯}{R}}_k^{\mathrm{FRAB}}$ and ${\lambda }_k$ have the same monotonicity of B. By introducing a new function $f(x)=2^x\sin (\pi /2^x)$ , we first consider the monotonicity of $f(x)$ by calculating the first-order derivative with respect to $x>1$ as

$$\frac{\partial f\left(x\right)}{\partial x}=\ln 2·2^x·\cos \left(\frac{\pi }{2^x}\right)·[\tan \left(\frac{\pi }{2^x}\right)-\frac{\pi }{2^x}]$$ (49)

With $x>1$ , it is easily observed that $\frac{\partial f(x)}{\partial x}>0$ . Thereby $f(B)$ monotonically increases in $B>1$ . Then ${\lambda }_k$ can be rewritten as

$${\lambda }_k=N_T{\alpha }_k·{(\frac{4\pi K({\alpha }_k+\frac{1}{\varphi P_T})}{f^2\left(B\right)}-{\alpha }_k)}^{-1}$$ (50)

Hence, ${\lambda }_k$ is also a monotonically increasing function of B. We can conclude that Bob’s ergodic information rate and the ergodic secrecy rate are both monotonically increasing functions in quantization precision B. Meanwhile, for $B\to \infty$ , the upper bound of the ergodic secrecy rate by FRAB precoder actually is the same as the secrecy rate by exploiting CP beamforming. This can be expressed as

$$\underset{B\to \infty }{lim}{\underset{¯}{C}}_k^{\mathrm{FRAB}}={\underset{¯}{C}}_k^{\mathrm{CP}}$$ (51)

Numerical results

In this section, the secrecy performance of the considered massive MIMO system is evaluated by Monte Carlo simulations. Without loss of generality, we set ${\delta }_B^2=1$ and ${\alpha }_i={\alpha }_E=1$ , $i=1,2,\dots ,K$ . We assume all UTs have the identical receiving and processing abilities, so all UTs have the same achievable secrecy rate. The total secrecy rate of the cell can be counted by multiplying the single UT’s secrecy rate by UTs’ number K. It is a meaningful topic to evaluate the system-wide secure performance. As this issue is outside the scope of this article, we will pay attention to this secrecy indicator in the future works.

We first illustrate the impact of the power allocation factor $\varphi$ on the ergodic secrecy rate. Figure 2 shows the achievable ergodic secrecy rates as functions of $\varphi$ with different values ${\beta }_K$ and ${\beta }_E$ . In the simulations, we set $P_T=10\mathrm{dBW}$ and $B=2\mathrm{bits}$ . As expected, the secrecy rate increases first and then decreases with the power allocation factor $\varphi$ increasing from $\varphi =0$ . For any scenario, to maximize the ergodic secrecy rate, there will be an optimal power allocation scheme that are denoted by small circles in the curves. The values of the optimal $\varphi$ are in good agreement with the obtained functional expression in the previous section as equation (43). In addition, we note that the optimal $\varphi$ decreases when ${\beta }_E$ increases, that is, more power needs to be used for AN generation when Eve equips more antennas. Under these simulation conditions, we note that $\frac{P_T\mu }{1+P_T}\approx 0.58$ . For ${\beta }_E⩽0.58$ , the non-negative secrecy rate can be gained regardless of the value of power allocation factor $\varphi \in (0,1]$ . However, for ${\beta }_E>0.58$ , $\varphi$ should meet the condition that $\varphi \in (0,\frac{\mu }{{\beta }_E}-\frac{1}{P_T})$ to achieve a positive secrecy rate. We can observe that the limit of $\varphi$ to gain a positive secrecy rate monotonically decreases in ${\beta }_E$ , that is, more Eve’s antennas will significantly damage the system’s secrecy performance. In addition, we can observe that increasing ${\beta }_K$ or ${\beta }_E$ brings about a negative effect on the ergodic secrecy rate, whereas as expected.

Figure 2.

Ergodic secrecy rate versus power allocation factor $\varphi$ for a system with total transmit power $P_T=10\mathrm{dB}$ , FRAB parameter $B=2\mathrm{bits}$ , and under different system parameters ${\beta }_K=0.1,0.3$ and ${\beta }_E=0.3,0.58,0.9$ .

In Figure 3, the ergodic secrecy rate curves are depicted as functions of ${\beta }_K$ and ${\beta }_E$ , respectively. The values of the ergodic secrecy rate are calculated exploiting the optimal power allocation scheme ${\varphi }_{\mathrm{opt}}$ which was gained in equation (43) or fixed allocation factor $\varphi =0.3$ under different ${\beta }_E=0.1,0.58,0.9$ and ${\beta }_K=0.1,0.3,0.6$ , respectively. From the two simulation diagrams, it is easy to see that the ergodic secrecy rate monotonically decreases with ${\beta }_K$ or ${\beta }_E$ . We can explain this situation that each UT’s power $\varphi P_T/K$ decreases as ${\beta }_K$ increases, which means that the UT’s achievable ergodic secrecy rate decreases. Moreover, increasing Eve’s antenna number can improve its eavesdropping ability, that is, increasing the ergodic capacity of Eve. Hence, the ergodic secrecy rate decreases in ${\beta }_K$ or ${\beta }_E$ , which is in the agreement with the analysis results in section “Numbers of Eve’s antennas and the UTs.” Furthermore, there is a performance gap between schemes with optimal power allocation and one fixed random allocation scheme. In each curve, there is only one coincident point where the power allocation value is just equal with the optimal value $\varphi ={\varphi }_{\mathrm{opt}}$ .

Figure 3.

Ergodic secrecy rate versus the normalized number of UTs ${\beta }_K$ and the normalized number of eavesdropper antennas ${\beta }_E$ , with total transmit power $P_T=10\mathrm{dBW}$ , FRAB parameter $B=2\mathrm{bits}$ , and different system parameters ${\beta }_E=0.1,0.58,0.9$ and ${\beta }_K=0.1,0.3,0.6$ , utilizing the optimal power allocation scheme ${\varphi }_{\mathrm{opt}}$ or fixed allocation factor $\varphi =0.3$ .

Figure 4 depicts the ergodic secrecy rate versus FRAB precision parameter B under different total transmit power $P_T$ . Also, the upper bounds of Bob’s achievable ergodic rate with CP RF precoder or full-digital MRT precoder (DP) are achieved and shown in the figure. First, we can observe that there is a fix performance difference between the CP RF precoder with unbounded $P_T$ and the full-digital MRT, that is, the SINR of the CP RF precoder is $\pi /4$ of the full-digital MRT. This result accords with the conclusion in Alkhateeb et al. ¹¹ From Figure 4, it is obvious that both $P_T$ and B have an important impact on the system secrecy performance. The achieved ergodic secrecy rate monotonically increases with $P_T$ and B. Moreover, the figure indicates that the performance of the FRAB architecture is close to the upper bound with a high phase precision ( $B>3$ ) thanks to the large antenna array gain. However, CP RF precoder or full-digital MRT needs full-complexity $N_T$ RF chains, which is practically costly and infeasible. Our quantized FRAB scheme suffers measurably negligible degradation compared to CP phase beamforming solution, only exploiting finite phase shifters and limited power consumption. Due to the potential advantages to reduce the hardware cost and improve power efficiency, the FRAB architecture is extremely suitable for the massive MIMO system that can keep the total capital expenditures manageable. Figure 4 also considers the ergodic secrecy rate with different transmit power level. As expected from the previous analysis, the ergodic secrecy rate increases with increasing $P_T$ . In addition, simulation results indicate that upper bounds of the ergodic secrcy rate converge to finite limit as $P_T\to \infty$ . This means we cannot get an unbound capacity even with infinite power. We should set an appropriate transmit power in practical system design.

Figure 4.

Achievable ergodic secrecy rate with transmit power $P_T=5,10,15,20(\mathrm{dBW})$ converges to the derived analytical limit exploiting CP precoder (FRAB with parameter $B\to \infty$ ). Total system secrecy rate converges to an upper bound as $P_T\to \infty$ .

Figures 5 and 6 depict the secrecy outage probabilities as functions of the target secrecy rate $R_0$ , where the curves are gained at different FRAB precisions $B=1,2,3,\infty (\mathrm{bits})$ and BS antenna numbers $N_T=100,150,200$ . We know that the achievable ergodic rate of legitimate UT monotonically increases at B, so secrecy outage probability decreases with increasing B, which is consistent with the results in Figure 5. And the curve is close to the lower bound by CP beamforming as $B>3$ , which is in accordance with the previous analysis of Figure 4. In addition, we can observe that B only affects the parameter $\epsilon$ in equation (40). And $\epsilon$ will converge to an upper bound value as $B\to \infty$ . Hence, we can obtain a limit lower bound of the secrecy outage probabilities, which is shown as the solid line in Figure 5. However, Figure 6 shows that the performance of secrecy outage probability can also be improved by enlarging the scale size of the BS’s antenna array $N_T$ . In conclusion, we can improve the performance of the secrecy outage probability by exploiting more phase shifters or increasing the size of the antenna array as expected.

Figure 5.

Secrecy outage probability versus target secrecy rate $R_0$ , for a system with total power $P_T=10\mathrm{dBW}$ , $N_T=100$ , $K=10$ , $N_E=10$ , and under different FRAB parameters $B=1,2,3,\infty (\mathrm{bits})$ .

Figure 6.

Secrecy outage probability versus target secrecy rate $R_0$ , for a system with total power $P_T=10\mathrm{dBW}$ , $B=2\mathrm{bits}$ , $K=10$ , $N_E=10$ , and under different BS antenna numbers $N_T=100,150,200$ .

Conclusion

In this article, we investigated a secure massive MIMO system exploiting simple FRAB and AN with a multi-antenna eavesdropper. Tight closed-form lower bound and upper bound for the achievable ergodic information rate and the eavesdropper’s ergodic capacity were analyzed in detail. Meanwhile, to evaluate the secrecy performance, a tight lower bound of the ergodic secrecy rate and an upper bound on secrecy outage probability were achieved. Based on these results, we achieved optimal power allocation scheme and the conditions of non-zero secrecy rate. Also, some significant insights have been provided about the impacts of various system parameters on secrecy performance. The proposed FRAB architecture could obtain comparable secrecy rate to CP scheme only utilizing inexpensive ADC and limited baseband processing power, which makes the application of low-cost devices desirable in massive MIMO systems. Moreover, this article only studies the system’s performance with perfect CSI. We will further consider the imperfect CSI scene in future.