Generalized Singular Value Decomposition-Based Secure Beam Hybrid Precoding for Millimeter Wave Massive Multiple-Input Multiple-Output Systems

Abstract

The precoder obtained using the traditional singular value decomposition (SVD) method for legitimate user’s channel, while achieving the highest spectral efficiency for the legitimate user, cannot defend against eavesdropping attacks, thus posing a security vulnerability. This paper investigates the millimeter wave (mmWave) secure beam hybrid precoding technology and proposes a generalized singular value decomposition (GSVD)-based secure beam hybrid precoding algorithm, termed GSVD-Sparsity, leveraging the sparsity of the mmWave beamspace channel. The algorithm selects the most powerful paths from the legitimate user’s beamspace channel representation and utilizes their corresponding angle information to construct a radio frequency (RF) precoder. It then constructs a hybrid precoder that closely approximates the optimal digital precoder derived from the GSVD-based scheme in a fully digital system. The simulation results indicate that, compared to the SVD-based scheme that focuses on spectral efficiency, the GSVD-based precoding scheme can form secure beams in a fully digital system. Under the condition that the legitimate user experiences a certain loss in the received signal-to-noise ratio (SNR), the eavesdropper is unable to correctly reconstruct the original constellation diagram, ensuring the scheme has strong anti-eavesdropping capabilities. In a hybrid precoding system, the low-complexity GSVD-Sparsity algorithm can achieve a spectral efficiency close to that of the GSVD-based scheme in a fully digital system while maintaining anti-eavesdropping capabilities.

1. Introduction

Wireless communication systems based on mmWave massive hybrid arrays offer higher network security, as the line-of-sight transmission and high directionality of the mmWave create a wired-like environment, while array antenna systems provide greater flexibility for physical layer network security technologies. Through the appropriate configuration and the use of beamforming technologies, users and intruders can receive entirely different signals. This approach can even achieve physical layer security without any prior knowledge of intruder information. For single-user access, transmission security depends on the magnitude of the sidelobe beams. In the case of multi-user spatial division access, where multiple beams are used simultaneously, the magnitude of the sidelobe beams becomes more challenging to characterize, making the description of the secure capacity extremely complex.

Beamforming is primarily carried out from two aspects: one is the design based on secrecy capacity, which ensures that system communication efficiency is maximized under the condition that eavesdroppers cannot intercept, with the main goal being to maximize the secrecy rate during communication; the other is the design based on SNR, which aims to weaken the eavesdropper’s interception capability as much as possible while ensuring the normal communication efficiency of the main channel. The authors of [1] design a low-complexity beamforming scheme in reconfigurable intelligent surface (RIS)-assisted communication systems to enhance physical layer security. The security performance of quasi-orthogonal space-time block codes with power scaling in MIMO Rayleigh fading eavesdropping channels is investigated in [2]. By employing moment matching approximation technology, a closed-form approximation of the secrecy outage probability is derived.

As a commonly used auxiliary method, artificial noise (AN) is often combined with beamforming to implement secure transmission strategies. The channel covariance matrix is used for beam design in [3]. By performing eigenvalue decomposition on the channel covariance matrix, beam domain information is obtained, which serves as the basis for jointly optimizing beam selection and artificial noise power allocation. The optimization problem of eavesdropping prevention and the main lobe bit error rate (BER) under the constrained AN power is investigated in [4]. The spatial and frequency domains are combined in [5] and redundant AN is introduced to interfere with the eavesdropper. Under the condition where only the statistical channel state information (CSI) of the legitimate user and the eavesdropper is available, an AN-aided transmission strategy can be used to enhance the security of single-cell massive MIMO systems [6]. Moreover, beamforming technology can also be applied to multi-user systems. An RIS-assisted multi-user massive MIMO system is considered in [7], utilizing AN for secure communication. The system’s secrecy rate in spatially correlated channels is derived, and the power allocation of AN and phase shifts of the RIS are further optimized. In [8], a transmission method combining space-time line coding and AN is proposed for the multi-user MIMO downlink, and the lower bound of the secrecy sum rate is derived. For the downlink multi-user generalized spatial modulation system, block diagonalization is first employed to eliminate inter-user interference and then AN is used to prevent eavesdropping [9]. Similarly, in the context of multi-user interference, a more powerful eavesdropper with more antennas than the legitimate users is considered in [10]. User utility is defined as the secrecy rate minus the inter-user interference, and the study focuses on optimizing the total utility of all users.

On the other hand, security schemes based on the GSVD have also garnered widespread attention in communications. Employing a GSVD-based beamforming scheme and allocating corresponding transmission power to spatial streams can optimize the secrecy rate [11,12]. The authors of [13] propose a low-complexity security precoding scheme based on GSVD for the visible light communication wiretap model. Secure transmission in relay communication wiretap models is studied in [14], where the base station uses GSVD to diagonalize the relay and user channels, generating independent parallel channels. In multi-user systems, a MIMO downlink communication scenario involving a base station and two users is considered in [15]. To make full use of spectral resources, a transmission protocol combining GSVD and non-orthogonal multiple access (NOMA) is proposed. In Ref. [16], a highly efficient precoding method leveraging GSVD and vector perturbation is proposed. In Ref. [17], a GSVD-based multi-user MIMO precoding scheme is studied, which transforms the multi-user MIMO channel into precoding weights to enhance the separability of effective channels among users.

Compared to the microwave frequency band, the mmWave can provide a larger bandwidth to meet the communication demands for ultra-high transmission rates. Moreover, the hybrid beamforming (HBF) architecture used in mmWave communication enables a trade-off between system power consumption and spectral efficiency. The authors of [18] study secure transmission in unmanned aerial vehicle (UAV) scenarios and propose an HBF architecture based on generalized eigenvalue decomposition (GEVD) and SVD. When the ideal CSI of users is unknown, a neural network is constructed to learn the statistical features of the wireless channel for beamforming design. Even in the absence of the eavesdropper’s CSI, this method can still achieve a positive secrecy rate. The secure design of intelligent reflecting surface (IRS)-assisted mmWave systems is studied in [19], where the base station simultaneously transmits information signals and AN. The IRS performs passive beamforming (PB) to reconstruct the wireless channel and optimize the secrecy rate. For IRS-assisted mmWave MISO systems, secure transmission in the scenario with multiple distributed eavesdroppers is investigated in [20]. Under the condition of unknown perfect cascaded eavesdropping channels, a robust secure beamforming design is proposed.

It can be seen that, although some studies on the physical layer security of mmWave communication have been conducted, they mostly rely on methods such as beamforming or AN without exploring the inherent characteristics of mmWave channels. In fact, due to high path loss and limited propagation paths, mmWave channels exhibit inherent sparsity, which is perfectly reflected in the beamspace channel matrix. Additionally, wireless channels possess uniqueness, distinguishing legitimate user’s channel from the eavesdropping channel, forming the foundation for secure beam design. On the other hand, unlike traditional microwave communication, mmWave communication relies on large-scale arrays to counteract path loss. To reduce the power consumption and hardware costs associated with large-scale arrays, hybrid precoding architectures are employed, requiring the design of both a baseband precoder in the digital domain and an RF precoder in the analog domain. Inspired by the sparsity of mmWave channels, we selected the array response vectors corresponding to the dominant paths in the beamspace channel representation to construct the RF precoder. The transmit beams generated by these array response vectors exhibit significant power. Furthermore, by linearly combining these array response vectors using a baseband precoder, the security performance of the hybrid precoder can approximate that of the optimal digital precoder obtained through the GSVD-based scheme.

The main contributions of this paper are as follows:

• A mmWave massive MIMO analog–digital hybrid precoding model was constructed and the sparsity characteristic of mmWave channels was investigated. This sparsity was validated in the beamspace channel matrix, which contains only a few dominant propagation paths. Furthermore, we analyzed the eavesdropping channel model and provided its secrecy capacity expression.
• In a fully digital system, we investigated a secure beam precoding scheme based on GSVD, providing the optimal digital precoder and its corresponding combiner design. As a comparison, an SVD-based precoder design aimed at optimizing spectral efficiency was also presented. The GSVD-based scheme, at the cost of sacrificing some transmission efficiency, can effectively prevent eavesdropping even at a low SNR. In contrast, the SVD-based scheme fails to offer any eavesdropping protection.
• In a hybrid precoding system, the GSVD-Sparsity algorithm is proposed to achieve secure beams. Peak search was performed in the beamspace channel matrix to identify several optimal propagation paths, and their array response vectors were used to construct the RF precoder. The baseband precoder was then obtained using a simple minimum mean square error (MMSE) method. The low-complexity GSVD-Sparsity algorithm ensures secure communication while approximating the transmission efficiency of the optimal digital precoder.

The rest of the paper is organized as follows. Section 2 introduces the mmWave massive MIMO system model and channel sparsity, providing a general expression for the secrecy capacity in the eavesdropping channel model. Section 3 explores the GSVD-based secure beam digital precoding scheme and proposes the GSVD-Sparsity mmWave secure beam hybrid precoding algorithm. Simulation evaluations of the security performance and transmission efficiency of the GSVD-based digital precoding scheme and the GSVD-Sparsity algorithm are conducted in Section 4. Section 5 concludes the paper.

2. System Model

2.1. mmWave Massive MIMO System Model

Consider the Massive MIMO analog–digital hybrid precoding system model as shown in Figure 1.

Assume there are $N_s$ data streams in the system, and the base station Alice and the user Bob are equipped with $N_t$ and $N_r$ antennas, respectively. To facilitate multi-stream communication, Alice has $N_{RF}^t$ RF chains, satisfying $N_s\le N_{RF}^t\le N_t$. Under this hardware architecture, a baseband precoder $F_{BB}$ of size $N_{RF}^t\times N_s$ and an RF precoder $F_{RF}$ of size $N_t\times N_{RF}^t$ are adopted. The RF precoder $F_{RF}$ is implemented using phase shifters, where the elements have equal magnitude. By adjusting $F_{BB}$, Alice normalizes the transmit power, ensuring ${‖F_{RF}{F}_{BB}‖}_F^2=N_s$. On the receiving side, a baseband combiner $W_{BB}$ of size $N_{RF}^r\times N_s$ and an RF combiner $W_{RF}$ of size $N_{RF}^r\times N_s$ are used.

In a fully digital communication system, the digital precoder is denoted as $F$. In a hybrid precoding system, where the number of RF chains is smaller than the number of transmit antennas, $F$ needs to be decomposed into an RF precoder $F_{RF}$ and a baseband precoder $F_{BB}$ using algorithms such as orthogonal matching pursuit (OMP). The relationship is expressed as

$$F\approx F_{RF}{F}_{BB}.$$

(1)

2.2. Sparsity of mmWave MIMO Channels

The construction of mmWave channels adopts a clustered ray-based model, assuming that the channel contains $N_{cl}$ clusters, each cluster comprising $N_{ray}$ paths. The physical channel is determined in a nonlinear manner by the angle of departure (AoD) and angle of arrival (AoA) of each path. It can be expressed as

$$H=\sqrt{{\displaystyle \frac{N_t{N}_r}{N_{cl}{N}_{ray}}}}{\displaystyle \sum _{i=1}^{N_{cl}}{\displaystyle \sum _{l=1}^{N_{ray}}{\alpha }_{il}{a}_r({\varphi }_{il}^r)a_t^H({\varphi }_{il}^t)}},$$

(2)

where ${\varphi }_{il}^t$ and ${\varphi }_{il}^r$ represent the AoD and AoA for the $l$-th path in the $i$-th cluster, respectively, and ${\alpha }_{il}$ denotes the path gain. Figure 2 illustrates the propagation of scattering paths in mmWave channels.

The mmWave MIMO channel representation in the beamspace can be expressed as

$$\begin{array}{ll}H& =\sqrt{{\displaystyle \frac{N_t{N}_r}{N_{cl}{N}_{ray}}}}{\displaystyle \sum _{i=1}^{N_r}{\displaystyle \sum _{j=1}^{N_t}{H}_b\left(i,j\right)a_r({\overline{\varphi }}_i^r)a_t^H({\overline{\varphi }}_j^t)}}\\ & =U_r{H}_b{U}_t^H,\end{array}$$

(3)

where $H_b$ is referred to as the beamspace channel matrix. ${\overline{\varphi }}_i^r=\mathrm{arc}\sin \left({\displaystyle \frac{\lambda {\overline{\vartheta }}_i^r}{d}}\right)$ and ${\overline{\varphi }}_j^t=\mathrm{arc}\sin \left({\displaystyle \frac{\lambda {\overline{\vartheta }}_j^t}{d}}\right)$ represent the AoA and AoD in the beamspace channel representation, corresponding to the uniformly spaced spatial angles ${\overline{\vartheta }}_i^r$ and ${\overline{\vartheta }}_j^t$, respectively. $\lambda$ is the mmWave wavelength, and d is the antenna spacing.

$$\begin{array}{l}{\overline{\vartheta }}_i^r={\displaystyle \frac{i-1-\left(N_r-1\right)/2}{N_r}} i\in \{1,2,\dots ,N_r\},\\ {\overline{\vartheta }}_j^t={\displaystyle \frac{j-1-\left(N_t-1\right)/2}{N_t}} j\in \{1,2,\dots ,N_t\},\end{array}$$

(4)

$U_r$ and $U_t$ are unitary discrete Fourier transform (DFT) matrices, whose columns are orthogonal array response vectors.

$$\begin{array}{l}{U}_r={\displaystyle \frac{1}{\sqrt{N_r}}}\left[a({\overline{\varphi }}_1^r),a({\overline{\varphi }}_2^r),\dots ,a({\overline{\varphi }}_{N_r}^r)\right],\\ U_t={\displaystyle \frac{1}{\sqrt{N_t}}}\left[a({\overline{\varphi }}_1^t),a({\overline{\varphi }}_2^t),\dots ,a({\overline{\varphi }}_{N_t}^t)\right].\end{array}$$

(5)

Thus, we have

$$H_b=U_r^{H}H{U}_t.$$

(6)

The beamspace channel matrix $H_b$ is a unitary equivalent representation of the antenna domain channel matrix $H$, meaning that $H_b$ is the projection of $H$ onto the Fourier orthogonal basis. Due to the significant propagation loss and limited number of propagation paths in mmWave transmission, the channel exhibits a low-rank (sparse) characteristic. It is important to note that the channel $H$ itself is not sparse. The sparsity of the mmWave channel becomes evident in $H_b$.

Figure 3a illustrates the response characteristics of the antenna domain channel $H$, which consists of six scattering clusters, each containing eight propagation paths. Figure 3b displays the response characteristics of the beamspace channel $H_b$. The sparsity of $H_b$ is clearly evident.

In fact, the sparse beamspace channel $H_b$ has a property that it is only related to the position of the legitimate user, reflecting the uniqueness of the channel. When an eavesdropper deviates from the legitimate user’s position, he cannot obtain the beamspace channel $H_b$. This property can be leveraged to generate a secure beam hybrid precoder/combiner pair $(F_{RF}{F}_{BB},W_{RF}{W}_{BB})$ for the legitimate user.

2.3. Eavesdropping Channel Model and Secrecy Capacity

Assuming the presence of an eavesdropper in the system, as shown in Figure 4, the eavesdropper Eve is equipped with $N_e$ antennas.

Alice needs to transmit confidential information $x$ to Bob. The received signals at Bob and Eve are, respectively, expressed as

$$\begin{array}{l}{y}_b=Hx+n_b,\\ y_e=Gx+n_e,\end{array}$$

(7)

where $H\in {ℂ}^{N_r\times N_t}$ and $G\in {ℂ}^{N_e\times N_t}$ represent the channel matrices of the legitimate user and the eavesdropper, respectively. $x\in {ℂ}^{N_t\times 1}$ represents the signal transmitted by Alice, with its covariance matrix given by $\mathbb{E}\left\{x{x}^H\right\}=Q_x$. The total transmission power is constrained by $\mathrm{Tr}(Q_x)\le P_t$. $n_b\in {ℂ}^{N_r\times 1}$ and $n_e\in {ℂ}^{N_e\times 1}$ represent the additive complex Gaussian noise vectors for Bob and Eve, respectively, with $n_b\sim \mathcal{C}\mathcal{N}(0,{\sigma }_b^{2}I)$ and $n_e\sim \mathcal{C}\mathcal{N}(0,{\sigma }_e^{2}I)$.

Assuming that Alice has perfect CSI for the channel matrices $H$ and $G$. Under the assumption of a single receive antenna, the spectral efficiencies of the legitimate user’s channel and the eavesdropping channel can be expressed based on Shannon’s theorem as follows:

$$\begin{array}{l}{R}_b={\log }_2\det (I+H{Q}_x{H}^H),\\ R_e={\log }_2\det (I+G{Q}_x{G}^H).\end{array}$$

(8)

The system’s secrecy rate can be expressed as the difference between the spectral efficiency of the legitimate user’s channel and that of the eavesdropping channel. Considering the non-negativity of the secrecy rate, we have

$$R_s=\left\{\begin{array}{ll}{R}_b-R_e,& R_b>R_e,\\ 0,& R_b\le R_e.\end{array}\right.$$

(9)

The secrecy capacity of the eavesdropping channel can be expressed as

$$C_s(P_t)=\underset{Q_x\succcurlyeq 0,\mathrm{Tr}(Q_x)\le P_t}{\max }{R}_s.$$

(10)

3. mmWave Secure Beam Hybrid Precoding

3.1. GSVD-Based Secure Beam Digital Precoding Scheme

We first consider a fully digital system and assume that the input signal $x$ has the following form:

$$x=Fs,$$

(11)

where $F\in {ℂ}^{N_t\times N_s}$ is Alice’s precoder, $W_b\in {ℂ}^{N_r\times N_s}$ is Bob’s combiner, $W_e\in {ℂ}^{N_e\times N_s}$ is Eve’s combiner, and $s$ is the source signal, satisfying $\mathbb{E}\{s{s}^H\}={\displaystyle \frac{1}{N_s}}{I}_{N_s}$.

Then, (7) can be rewritten as

$$\begin{array}{l}{y}_b=W_b^{H}HFs,+W_b^H{n}_b,\\ y_e=W_e^{H}GFs,+W_e^H{n}_e.\end{array}$$

(12)

The secrecy rate is

$$\begin{array}{ll}{R}_s=& {\log }_2\det (I+{\displaystyle \frac{1}{N_s}}{R}_b^{-1}{W}_b^{H}HF{F}^H{H}^H{W}_b^H)-\\ & {\log }_2\det (I+{\displaystyle \frac{1}{N_s}}{R}_e^{-1}{W}_e^{H}GF{F}^H{G}^H{W}_e),\end{array}$$

(13)

where $R_b={\sigma }_b^2{W}_b^H{W}_b$, $R_e={\sigma }_e^2{W}_e^H{W}_e$.

The secure beam can now be implemented through digital precoder $F_{opt}$, i.e., $F=F_{opt}$. $F_{opt}$ is obtained using the GSVD of $(H,G)$ [11]. The detailed procedure is presented in Algorithm 1. By utilizing GSVD and the appropriate power allocation of the initial input data streams, the secrecy capacity of the multiple-input multiple-output multiple-eavesdropper (MIMOME) system can be achieved [21]. To simulate a typical communication system scenario, this paper assumes equal power allocation for the input data streams, i.e., $\mathbb{E}[s{s}^H]={\displaystyle \frac{1}{N_s}}{I}_{N_s}$. Under this condition, the GSVD-based precoding can still achieve a relatively high secrecy rate.

Algorithm 1. GSVD-based secure beam digital precoding.

Input:   legitimate user’s channel $H$;   eavesdropping channel $G$; 1: GSVD   $[U,V,X,C,S]=gsvd(H,G)$; 2: pseudo-inverse   $P=pinv(X^H)$; 3: Extraction   $c=column(P)$;   $F_{opt}=P(:,c:-1:c-N_s+1)$; 4: Normalization $F_{opt}=\sqrt{N_s}{\displaystyle \frac{F_{opt}}{{‖F_{opt}‖}_F}}$ Output:   digital precoder $F_{opt}$

The channel matrices $H\in {ℂ}^{N_r\times N_t}$ and $G\in {ℂ}^{N_e\times N_t}$ have the same number of columns. In the first step, GSVD is performed on $(H,G)$

$$[U,V,X,C,S]=gsvd(H,G),$$

(14)

where $U\in {ℂ}^{N_r\times N_r}$ and $V\in {ℂ}^{N_e\times N_e}$ are unitary matrices, $C\in {ℂ}^{N_r\times q}$ and $S\in {ℂ}^{N_e\times q}$ are non-negative diagonal matrix. We also obtain $X\in {ℂ}^{N_t\times q}$, with $q=\min (N_t,N_r+N_e)$.

The matrices above satisfy the following equations:

$$\begin{array}{l}H=UC{X}^H,\\ G=VS{X}^H.\end{array}$$

(15)

Furthermore, the matrices $C$ and $S$ satisfy

$$C^{T}C+S^{T}S=I.$$

(16)

GSVD is designed to extract the dominant directions of two matrices and align them within a unified coordinate system for comparison. For the legitimate user’s channel $H$ and the eavesdropper’s channel $G$, GSVD is performed on the matrix pair $(H,G)$, simultaneously diagonalizing them. The decomposition yields two unitary matrices $U$ and $V$, a matrix $X$, and two non-negative diagonal matrices $C$ and $S$. GSVD can be viewed as a generalization of SVD. It projects both $H$ and $G$ onto the common basis $X$, allowing them to be represented in a decoupled form. In this basis, matrix $H$ exhibits strong responses along the directions corresponding to large generalized singular values, whereas matrix $G$ exhibits weak responses along these same directions.

In step 2, the pseudo-inverse of matrix $X^H$ is computed, $X\ast pinv(X^H)=I$. In step 3, $F_{opt}$ is composed of the rightmost $N_s$ columns of $\mathrm{pinv}\left({\mathbf{X}}^H\right)$, corresponding to the $N_s$ largest generalized singular values. Here, $c$ denotes the number of columns in matrix $pinv(X^H)$, and $N_s$ represents the number of data streams. Step 4 performs normalization on $F_{opt}$ to satisfy the transmit power constraint.

After obtaining the precoder $F_{opt}$, the MMSE algorithm can be used to compute the receiver combiner matrix $W_{opt}$ [22]. The MMSE-based combiner minimizes the mean square error (MSE) of the signal estimation in the presence of Gaussian noise and interference.

$$\begin{array}{ll}{W}_{opt}& =\mathbb{E}{[y{y}^H]}^{-1}[y{s}^H]\\ & ={(H{F}_{opt}{F}_{opt}^H{H}^H+N_s{\sigma }_b^2{I}_{N_r})}^{-1}H{F}_{opt}.\end{array}$$

(17)

Thus, we obtain the optimal secure beam precoder/combiner pair $(F_{opt},W_{opt})$ based on GSVD.

Additionally, as a comparison to the GSVD-based digital precoding scheme, we describe the SVD-based digital precoding scheme, as shown in Algorithm 2. Assuming the eavesdropping channel is not considered and only the legitimate channel undergoes SVD decomposition, we have

$$[\overline{U},\overline{S},\overline{V}]=svd(H),$$

(18)

where $\overline{U}\in {ℂ}^{N_r\times N_r}$ is the left singular matrix, $\overline{V}\in {ℂ}^{N_t\times N_t}$ is the right singular matrix, and $\overline{S}\in {ℂ}^{N_r\times N_t}$ is the diagonal matrix composed of singular values.

Algorithm 2. SVD-based digital precoding.

Input:   legitimate user’s channel $H$; 1: SVD   $[\overline{U},\overline{S},\overline{V}]=svd(H),$   $H=\overline{U}\overline{S}{\overline{V}}^H$ 2: Extraction   $\overline{F}=\overline{V}(:,1:N_s)$; 3: Normalization   $\overline{F}=\sqrt{N_s}{\displaystyle \frac{\overline{F}}{{‖\overline{F}‖}_F}}$ Output:   digital precoder $\overline{F}$

By selecting the $N_s$ leftmost column vectors from $\overline{V}$, the precoder $\overline{F}$ is formed, i.e., $F=\overline{F}$. The combiner $W=\overline{W}$ is still calculated using the MMSE method. The resulting precoder/combiner pair $(\overline{F},\overline{W})$ maximizes the spectral efficiency for the legitimate user.

$$R={\log }_2(|I_{N_s}+{\displaystyle \frac{1}{N_s}}{\overline{R}}_b^{-1}{\overline{W}}^{H}H\overline{F}{\overline{F}}^H{H}^H\overline{W}|),$$

(19)

where ${\overline{R}}_b={\sigma }_b^2{\overline{W}}^H\overline{W}$ is the covariance matrix of the received noise.

3.2. The Proposed Secure Beam Hybrid Precoding Algorithm: GSVD-Sparsity

In the fully digital architecture, the optimal secure beam can be formed by solving the precoder/combiner pair $(F_{opt},W_{opt})$ based on GSVD. However, in the mmWave hybrid precoding architecture, matrix operations are required to decompose $F_{opt}$ ($W_{opt}$) into $F_{RF}$ and $F_{BB}$ ($W_{RF}$ and $W_{BB}$). However, in general mmWave communication systems, the number of RF chains is not equal to the number of transmit (receive) antennas. Typically, the number of RF chains is much smaller than the number of transmit (receive) antennas. In this case, the decomposition problem becomes an optimization problem, which requires $F_{RF}{F}_{BB}$ ($W_{RF}{W}_{BB}$) to infinitely approach $F_{opt}$ ($W_{opt}$), i.e.,

$$\begin{array}{l}{F}_{RF}{F}_{BB}\to F_{opt},\\ W_{RF}{W}_{BB}\to W_{opt}.\end{array}$$

(20)

Studies have shown that for the mmWave channel model, the antenna array response vectors $a_t({\varphi }_{il}^t),\forall i,l$ can form a finite spanning set for the row space of the channel. Moreover, when $N_{cl}{N}_{ray}\le N_t$, the antenna array response vectors are linearly independent with probability 1. Specifically, when $N_{cl}{N}_{ray}\le \min (N_t,N_r)$, $a_t({\varphi }_{il}^t)$ can form the minimal basis for the row space of the channel [22].

Additionally, the modulus of the elements in $a_t({\varphi }_{il}^t)$ remains constant, with only the phase varying, which perfectly matches the characteristic of analog phase shifters in antenna arrays. Therefore, the base station can select $N_{RF}^t$ elements from the dictionary matrix $A_t=[a_t({\varphi }_{1,1}^t),a_t({\varphi }_{1,2}^t),\dots ,a_t({\varphi }_{N_{cl},N_{ray}}^t)]$, which consists of the transmit array response vectors, to construct the RF precoder $F_{RF}$. Subsequently, the selected array response vectors can be linearly combined through baseband precoder $F_{BB}$, ensuring that $F_{RF}{F}_{BB}$ approximates $F_{opt}$. Similarly, on the user side, $N_{RF}^r$ elements can be selected from the dictionary matrix $A_r=[a_r({\varphi }_{1,1}^r),a_r({\varphi }_{1,2}^r),\dots ,a_r({\varphi }_{N_{cl},N_{ray}}^r)]$, which consists of the receive array response vectors, to construct the RF combiner $W_{RF}$.

In such a hybrid precoding system, the hybrid precoder/combiner pair $(F_{RF}{F}_{BB},W_{RF}{W}_{BB})$ corresponding to the optimal secure beam that maximizes the secrecy rate $R_s$ can be obtained by performing an exhaustive search over all RF precoder/combiner pairs and selecting the pair that achieves the maximum secrecy rate. However, since both the base station and user sides utilize large-scale arrays, the dimensionality of $(F_{RF},W_{RF})$ is high, making it infeasible to bear the substantial computational cost and increased system processing time caused by exhaustive search.

To address this issue, the OMP algorithm is proposed [22]. In this method, $F_{opt}$ is orthogonally projected onto the matrix $A_t$ formed by the transmit array response vectors. The algorithm quickly selects $N_{RF}^t$ elements with the highest correlation to $F_{opt}$ from $A_t$ to construct $F_{RF}$, and then obtains $F_{BB}$ using the least squares method. Similarly, on the user side, $W_{opt}$ is orthogonally projected onto $A_r$, and $N_{RF}^r$ elements with the highest correlation to $W_{opt}$ are selected from $A_r$ to construct $W_{RF}$. Algorithm 3 presents the procedure for solving hybrid precoding using the OMP algorithm.

Algorithm 3. OMP.

Input:   optimal digital precoder $F_{opt}$;   dictionary matrix $A_t$;   number of RF chains $N_{RF}^t$; 1: Initiation   $F_{RF}=\varnothing$;   $F_{res}=F_{opt}$; 2: Iteration   for $i=1:N_{RF}^t$ do   (a) Projection      $G=A_t^H{F}_{res}$;   (b) Support set update      $j=\arg \underset{l=1,\dots ,N_{cl}{N}_{ray}}{\max }{(G{G}^H)}_{l,l}$;      $F_{RF}=[F_{RF}|A_t^{(j)}]$;   (c) Least squares      $F_{BB}={(F_{RF}^H{F}_{RF})}^{-1}{F}_{RF}^H{F}_{opt}$;   (d) Update residual      $F_{res}={\displaystyle \frac{F_{opt}-F_{RF}{F}_{BB}}{{‖F_{opt}-F_{RF}{F}_{BB}‖}_F}}$   end for 3: Normalization   $F_{BB}=\sqrt{N_s}{\displaystyle \frac{F_{BB}}{{‖F_{RF}{F}_{BB}‖}_F}}$; Output:   baseband precoder $F_{BB}$ and RF precoder $F_{RF}$

Inspired by the sparsity of the beamspace channel $H_b$, $F_{RF}$ is not limited to being constructed from the dictionary matrix $A_t$ formed by the array response vectors of the antenna domain channel $H$. It can also be constructed from the dictionary matrix $A_{t,b}=[a_t({\overline{\varphi }}_1^t),a_t({\overline{\varphi }}_2^t),\dots ,a_t({\overline{\varphi }}_{N_t}^t)]$ formed by the array response vectors of the beamspace channel representation. In the beamspace channel, the number of dominant propagation paths is relatively small, and these dominant paths correspond to higher transmission power. Therefore, the array response vectors $a_t({\overline{\varphi }}_j^t)$ corresponding to the AoDs of the $N_{RF}^t$ optimal propagation paths can be selected from $A_{t,b}$ to construct $F_{RF}$, which is known as the peak search method. Each $a_t({\overline{\varphi }}_j^t)$ in $F_{RF}$ can form a transmission beam. The angles of these beams are aligned with the $N_{RF}^t$ optimal propagation paths in the beamspace channel, which exhibit significant transmission power. Thus, when $F_{BB}$ performs a linear combination of these array response vectors, high transmission efficiency can still be achieved. Similarly, by selecting the array response vectors $a_r({\overline{\varphi }}_i^r)$ corresponding to the AoAs of the $N_{RF}^r$ optimal propagation paths from $A_{r,b}=[a_r({\overline{\varphi }}_1^r),a_r({\overline{\varphi }}_2^r),\dots ,a_r({\overline{\varphi }}_{N_r}^r)]$, $W_{RF}$ can be constructed, and $(F_{RF},W_{RF})$ can be determined accordingly.

Given that $(F_{RF},W_{RF})$ has been solved, $(F_{BB},W_{BB})$ can be obtained using the least squares method.

$$\begin{array}{l}{F}_{BB}={(F_{RF}^H{F}_{RF})}^{-1}{F}_{RF}^H{F}_{opt},\\ W_{BB}={(W_{RF}^H{W}_{RF})}^{-1}{W}_{RF}^H{W}_{opt}.\end{array}$$

(21)

Finally, the secure beam precoder/combiner pair $(F_{RF}{F}_{BB},W_{RF}{W}_{BB})$ is obtained. The pseudo-code for computing the hybrid precoder using GSVD-Sparsity is provided in Algorithm 4.

Algorithm 4. GSVD-Sparsity.

Input:   optimal digital precoder $F_{opt}$;   dictionary matrix $A_{t,b}$;   number of RF chains $N_{RF}^t$;   beamspace channel $H_b$; 1: Initiation   $F_{RF}=\varnothing$; 2: Perform quick select on the elements of matrix $H_b$   $quickselect(H_b)\to [{\alpha }_{b,1},{\alpha }_{b,2},\dots ,{\alpha }_{b,N_{RF}^t}]$; 3: Iteration   for $i=1:N_{RF}^t$ do      $j=column\_index({\alpha }_{b,i})$;      $F_{RF}=[F_{RF}|A_{t,b}^{(j)}]$;   end for 4: Least squares   $F_{BB}={(F_{RF}^H{F}_{RF})}^{-1}{F}_{RF}^H{F}_{opt}$; 5: Normalization   $F_{BB}=\sqrt{N_s}{\displaystyle \frac{F_{BB}}{{‖F_{RF}{F}_{BB}‖}_F}}$; Output:   baseband precoder $F_{BB}$ and RF precoder $F_{RF}$

Therefore, to address the non-convex optimization problem in the decomposition of $F_{opt}(W_{opt})$, the GSVD-Sparsity algorithm selects the strongest power paths from the sparse channel matrix of the legitimate user to construct the RF precoder. Based on this, the hybrid precoder at the transmitter and the hybrid combiner at the receiver are then derived. Since the sparse channel matrix is location-dependent, only the legitimate user located at a specific position possesses the corresponding sparse matrix. The eavesdropper, being at a different location, does not have access to such position-specific information. Consequently, the selection of the strongest power paths is inherently location-dependent, and so is the resulting hybrid precoder, thereby enhancing the security of the proposed algorithm.

Precoding, as a physical layer security technique, aims to provide strong signal reception for the legitimate user while suppressing the signal strength at unauthorized users as much as possible. The GSVD-based secure beam precoding enhances security by suppressing the signal received by the eavesdropper, though at the cost of spectral efficiency loss for the legitimate user. The proposed GSVD-Sparsity algorithm exploits the sparsity of the channel space and the strongest propagation paths to maximize energy efficiency while maintaining the required level of security.

4. Numerical Simulations and Analysis

Consider a mmWave MIMO system in the downlink, operating at a frequency of 28 GHz. The base station is equipped with 64 antennas, while the user terminal is equipped with 16 antennas. Both adopt a uniform linear array (ULA) with an antenna spacing of ${\displaystyle \frac{\lambda }{2}}$. The number of RF chains is $N_{RF}^t=N_{RF}^r=4$, and the number of data streams is 1. Assuming that both the base station and the user terminal have perfect knowledge of the channel state information, the mmWave channel $H$ is modeled as consisting of six scattering clusters, with each cluster contributing eight propagation paths. For the $i$-th cluster, the departure (arrival) azimuth angles ${\varphi }_{il}^t$ (${\varphi }_{il}^r$) of the eight paths are assumed to follow a Laplace distribution with a mean of ${\varphi }_i^t$ (${\varphi }_i^r$) and an angular spread of ${\sigma }_{{\varphi }_i^t}$ (${\sigma }_{{\varphi }_i^r}$). ${\varphi }_i^t$ (${\varphi }_i^r$) are uniformly distributed within $[-{\displaystyle \frac{\pi }{3}},{\displaystyle \frac{\pi }{3}}]$, and ${\sigma }_{{\varphi }_i^t}={\sigma }_{{\varphi }_i^r}=5°$. Considering practical scenarios, the departure azimuth angles are constrained within the range $[-{\displaystyle \frac{\pi }{3}},{\displaystyle \frac{\pi }{3}}]$, while the arrival azimuth angles are restricted to the range $[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$.

4.1. Full Digital System

First, we compare the performance of the GSVD-based scheme and the SVD-based scheme in a fully digital system. Figure 5a,b show the constellation diagrams of the legitimate user and the eavesdropper, respectively, formed by $(\overline{F},\overline{W})$ when $\mathrm{SNR}=0 \mathrm{dB}$, where $(\overline{F},\overline{W})$ is obtained through SVD (Equation (18)). Figure 5c,d depict the constellation diagrams formed by $(F_{opt},W_{opt})$, which is obtained through GSVD (Equation (14)). The signal modulation scheme is 16QAM. Figure 6 illustrates the scenario where $\mathrm{SNR}=15 \mathrm{dB}$.

When the GSVD-based scheme is used to design the precoder, a comparison between Figure 5c and Figure 6c shows that the constellation points for the legitimate user cluster together when the SNR is low, making proper demodulation impossible. However, when the SNR is high, proper demodulation becomes achievable. From Figure 5d and Figure 6d, it can be observed that all received signals fall at the $(0,0)$ point, rendering the eavesdropper completely unable to obtain a valid constellation diagram. Therefore, the GSVD-based scheme effectively prevents the eavesdropper from receiving useful signals. This is achieved at the cost of reducing the received SNR for legitimate users, ensuring that the eavesdropper is completely unable to demodulate correctly.

When using an SVD-based scheme for precoder design, a comparison of Figure 6a,b shows that, although the eavesdropper’s received SNR is lower than that of the legitimate user, it does not affect the eavesdropper’s ability to correctly demodulate. Comparing Figure 5b and Figure 6b, it is evident that even at a low SNR level, the eavesdropper can still demodulate the constellation diagram, with only a slight increase in the bit error rate (BER). Therefore, the SVD-based scheme cannot prevent the eavesdropper from receiving useful signals and fails to meet the requirements for secure transmission. Figure 5a and Figure 6a illustrate the constellation diagrams received by the legitimate user, where normal demodulation is achievable.

Figure 7 illustrates the spectral efficiency variations for both the legitimate user and the eavesdropper under the SVD-based scheme and GSVD-based scheme as the SNR changes. The optimal curve in the figure represents the channel capacity of the system, serving as a benchmark. The solid line with $\ast$ indicates the spectral efficiency of the eavesdropper under the GSVD-based scheme, with most values being 0 or close to 0. This is consistent with the constellation diagram results shown in Figure 5d and Figure 6d, where the eavesdropper is unable to correctly demodulate the received signals. Compared to the SVD-based scheme, although the GSVD-based scheme sacrifices part of the transmission efficiency, it ensures transmission security. On the other hand, the SVD-based scheme, while maintaining a higher user rate, completely fails to prevent eavesdropping.

Compared to the SVD-based scheme, the GSVD-based scheme incurs a certain spectral efficiency loss. Specifically, at $\mathrm{SNR}=0 \mathrm{dB}$, the spectral efficiency loss for the legitimate user is approximately 5 bits/s/Hz, and within the SNR range of $[5,50]$, the loss increases to about 6.5 bits/s/Hz. However, in the GSVD-based scheme, the spectral efficiency of the eavesdropper is suppressed to zero, which demonstrates the security capability of the scheme.

Figure 8 compares the secrecy rate performance of the GSVD-based secure beam precoding scheme with that of AN and zero-forcing (ZF) secure precoding in a fully digital system. Both AN and ZF precoding are widely used techniques in physical layer security. The principle of AN is to embed noise into the transmitted signal in such a way that it does not affect the legitimate user’s ability to decode the information, but degrades the signal quality at the eavesdropper [23]. In contrast, the ZF projects the transmitted signal into the null space of the eavesdropper’s channel, thereby ensuring that the eavesdropper receives no useful information [24]. In this simulation, the number of antennas at the eavesdropper is set to 60, since for the ZF secure precoding to be feasible, the system must satisfy certain conditions (e.g., $N_t>\mathrm{rank}(H_e)$) to ensure that a non-trivial null space exists at the eavesdropper’s side, enabling the design of ZF precoding.

As shown in the figure, the secrecy rate achieved by the GSVD-based precoding is lower than that of the AN scheme. This is because the performance of GSVD is highly dependent on the structural differences between the legitimate user’s channel and the eavesdropping channel, such as rank and singular value disparities. When the differences between the two channels are not significant, GSVD is less effective in providing strong secrecy performance. In contrast, even when the legitimate user’s channel and the eavesdropping channel are similar, the AN can still exploit spatial degrees of freedom to enhance secrecy, thus exhibiting greater robustness. It is worth noting that, unlike the AN scheme, the GSVD-based precoding does not allocate any transmit power to artificial noise; instead, the entire transmit power is dedicated to the information signal. Furthermore, to ensure a fair comparison, equal power allocation is applied to the initial data streams in the GSVD-based precoding. When power is properly allocated among the data streams, the GSVD-based precoding is capable of achieving the secrecy capacity [21].

The figure also shows that the GSVD-based precoding outperforms the ZF secure precoding. This is because ZF secure precoding requires the transmitted signal to be completely projected into the null space of the eavesdropping channel. When the eavesdropping channel has a large dimension or significantly overlaps with the legitimate user’s channel, there may be insufficient degrees of freedom, making it impossible to construct a valid null space. In contrast, the GSVD-based approach jointly decomposes the legitimate user’s channel and the eavesdropping channel and projects the signal onto subchannels that are strong for the legitimate user and weak for the eavesdropper. In this way, even when there is partial overlap between the legitimate user’s channel and the eavesdropping channel, the GSVD-based method can still optimize the level of information leakage.

4.2. Hybrid Precoding System

Next, the performance of the proposed algorithm in a hybrid precoding system is discussed. Figure 9 verifies the sparsity of the beamspace channel $H_b$ in the GSVD-Sparsity algorithm. Figure 9a shows a 3D beam pattern, while Figure 9b presents the 2D planar beam pattern. In Figure 9a, only a few prominent peaks can be observed, and in Figure 9b, the red-colored elements indicate beams with relatively high power. It is clearly shown that there are only a few strong paths, demonstrating that the sparse beamspace channel $H_b$ exhibits high directionality.

Figure 10 compares the spectral efficiency of the legitimate user under two secure beam hybrid precoding algorithms, GSVD-Sparsity and OMP, in a hybrid precoding system. In the fully digital system, the optimal precoder/combiner pair obtained is $(F_{opt},W_{opt})$. Both the GSVD-Sparsity and OMP algorithms decompose $F_{opt}$ and $W_{opt}$ to derive the hybrid precoder $(F_{RF}{F}_{BB},W_{RF}{W}_{BB})$. The optimal curve in the figure represents the spectral efficiency of the legitimate user achieved by the GSVD-based scheme in the fully digital system, serving as the benchmark for comparison. In the GSVD-Sparsity algorithm, $F_{RF}$ and $W_{RF}$ are determined by the sparsity of $H_b$. From the figure, it can be observed that the GSVD-Sparsity algorithm closely approximates the performance of the optimal digital precoding scheme based on GSVD. This is because the beams constructed by $F_{RF}$ and $W_{RF}$ are aligned with the paths in the beamspace channel that have the highest power, thereby enhancing the transmission rate. Moreover, the sparsity-based beam selection is location-dependent. Only when the legitimate user is located at a specific position can the corresponding sparse beams at that location be obtained. The power paths are also location-dependent, which ensures the uniqueness of the transmission path.

The OMP algorithm performs a search over the entire RF precoder (combiner) candidate set $A_t(A_r)$, selecting the optimal $N_{RF}^t(N_{RF}^r)$ basis vectors, such that $F_{opt}(W_{opt})$ can be best projected onto these basis vectors. The OMP only selects the column vectors that contribute most to approximating $F_{opt}$, which reflects its greedy nature in numerical approximation. Specifically, $F_{RF}$ and $W_{RF}$ are constructed from the array response vectors that have the highest correlation with $F_{opt}$ and $W_{opt}$, respectively. This algorithm only approximates $F_{RF}{F}_{BB}$ ($W_{RF}{W}_{BB}$) to $F_{opt}$ ($W_{opt}$) as closely as possible from a numerical perspective, without leveraging the inherent sparsity of mmWave channels.

Compared with the OMP algorithm, one of the key advantages of the proposed GSVD-Sparsity approach lies in its lower computational complexity. The following presents a comparative analysis of the computational complexity of GSVD-Sparsity, OMP, and exhaustive search algorithms. To ensure a fair comparison, the number of data streams was set to $N_s=1$. For the OMP algorithm, the complexity of step 2(a) in Algorithm 3, which involves the matrix projection operation, was $\mathcal{O}(N_{cl}{N}_{ray}{N}_t)$; the complexity of step 2(b), which updates the support set, was $\mathcal{O}(N_{cl}{N}_{ray})$; the least squares solution in step 2(c) required a complexity of $\mathcal{O}(N_t{i}^2+i^3)$; and the residual update in step 2(d) had a complexity of $\mathcal{O}(N_{t}i)$. After $N_{RF}^t$ iterations, the total complexity of OMP was $\mathcal{O}(N_{RF}^t{N}_{cl}{N}_{ray}{N}_t+N_t{(N_{RF}^t)}^3+{(N_{RF}^t)}^4)$. For the GSVD-Sparsity algorithm, in step 2, the fast selection algorithm was used to find the top $N_{RF}^t$ largest elements (peak search) in the beamspace channel $H_b$, with a complexity of $\mathcal{O}(N_t{N}_r)$; in step 4, the least-squares solution required $\mathcal{O}(N_t{(N_{RF}^t)}^2+{(N_{RF}^t)}^3)$. Hence, the overall complexity of GSVD-Sparsity was $\mathcal{O}(N_t{N}_r+N_t{(N_{RF}^t)}^2+{(N_{RF}^t)}^3)$. In the exhaustive search algorithm, the analog precoder $F_{RF}$ was constructed by selecting $N_{RF}^t$ columns from the dictionary matrix with $N_{cl}{N}_{ray}$ candidates, resulting in a total of $C_{N_{\mathrm{cl}}{N}_{\mathrm{ray}}}^{N_{\mathrm{RF}}}$ possible combinations. The complexity of solving the least squares problem for each combination was $\mathcal{O}(N_t{(N_{RF}^t)}^2+{(N_{RF}^t)}^3)$, and thus, the overall complexity of the exhaustive search algorithm was $C_{N_{cl}{N}_{ray}}^{N_{RF}^t}\mathcal{O}(N_t{(N_{RF}^t)}^2+{(N_{RF}^t)}^3)$.

Clearly, the proposed GSVD-Sparsity algorithm demonstrates a significant advantage in time complexity. This is because, unlike the OMP algorithm which requires iterative updates to construct $F_{\mathrm{RF}}$ and $F_{\mathrm{BB}}$, GSVD-Sparsity uses a fast selection method to directly identify the top $N_{RF}^t$ strongest beam paths in the beamspace channel, and uses the corresponding array response vectors to construct $F_{\mathrm{RF}}$. This eliminates the need for complex iterative procedures, resulting in a considerable reduction in complexity. Moreover, selecting the strongest paths for transmission also aligns with the physical layer goal of improving transmission efficiency.

Table 1. Complexity comparison.

Algorithm	Computational Complexity
GSVD-Sparsity	$\mathcal{O}(N_t{N}_r+N_t{(N_{RF}^t)}^2+{(N_{RF}^t)}^3)$
OMP	$\mathcal{O}(N_{RF}^t{N}_{cl}{N}_{ray}{N}_t+N_t{(N_{RF}^t)}^3+{(N_{RF}^t)}^4)$
Exhaustive search	$C_{N_{cl}{N}_{ray}}^{N_{RF}^t}\mathcal{O}(N_t{(N_{RF}^t)}^2+{(N_{RF}^t)}^3)$

summarizes the computational complexity of the three algorithms.

Figure 11a,b show the constellation diagrams obtained by the legitimate user in a hybrid precoding system using the GSVD-Sparsity algorithm under $\mathrm{SNR}=0 \mathrm{dB}$ and $\mathrm{SNR}=15 \mathrm{dB}$, respectively. Compared to Figure 5c and Figure 6c, it can be observed that the hybrid precoder obtained using the GSVD-Sparsity algorithm achieved a BER performance comparable to the optimal precoder $(F_{opt},W_{opt})$ in a fully digital system.

To verify the robustness of the proposed GSVD-Sparsity algorithm under different channel conditions, Figure 12 presents the symbol error rate (SER) performance under both additive white Gaussian noise (AWGN) and multipath fading channels, with 16QAM modulation employed at the legitimate user. The figure includes four curves, corresponding to GSVD-OMP with AWGN, GSVD-Sparsity with AWGN, GSVD-OMP with a fading channel, and GSVD-Sparsity with a fading channel. It can be observed that under AWGN channel conditions, the two curves on the left (i.e., GSVD-OMP and GSVD-Sparsity under AWGN) are almost overlapping, indicating that both methods achieved a similar SER performance when the channel was ideal. Under multipath fading conditions, the two curves on the right begin to diverge after 15 dB, showing that in worse channel conditions, GSVD-OMP achieved a slightly lower SER than GSVD-Sparsity. However, as the SNR increases, the SER for both methods drops rapidly to zero. Therefore, GSVD-Sparsity maintained high reliability under high SNR conditions and can be extended to various channel models.

5. Conclusions

This paper investigates a secure beam hybrid precoding algorithm for mmWave communications. Based on Shannon’s theorem, an eavesdropping channel model is established along with its secrecy capacity expression. The masking characterization of the sparse mmWave MIMO channel is studied. The sparse beamspace channel $H_b$ is uniquely determined by the receiver’s location. A secure beam digital precoding scheme based on GSVD is analyzed. Building on GSVD-based scheme, a secure beam hybrid precoding algorithm, GSVD-Sparsity, is proposed, which fully exploits the sparsity of $H_b$. Simulation results validate the security performance and spectral efficiency of digital precoding schemes based on SVD and GSVD, as well as the hybrid precoding algorithm GSVD-Sparsity. The analysis indicates that, in a fully digital system, employing the GSVD-based digital precoding scheme can effectively prevent the eavesdropper from correctly receiving signals at the cost of sacrificing part of the transmission efficiency of the legitimate user, rendering the eavesdropper unable to demodulate the constellation diagram. In a hybrid precoding system, the low-complexity GSVD-Sparsity algorithm can approximate the spectral efficiency of the optimal digital precoding scheme based on GSVD while ensuring transmission security. Massive MIMO and hybrid precoding are key technologies in mmWave communications. Accordingly, the demand for security and privacy protection is also increasing. This work investigates a secure hybrid precoding algorithm based on GSVD and the sparsity of mmWave channels. As a physical layer security technique, it holds promising potential for a wide range of applications.