On Multiple-Antenna Techniques for Physical-Layer Range Security in the Terahertz Band

With an increasing demand for high-speed, reliable, and private information exchange, a new spectrum and advanced security technologies are awaited to be developed for next-generation wireless networks. The Terahertz (THz) band, with frequencies ranging from 0.1 to 10 THz, is envisioned to realize multi-Gbps or even Tbps data rates owing to the abundant bandwidth resource [1, 2, 3]. In addition to the bandwidth merit, the potential of THz communications to improve information security is envisioned in [4, 5, 6]. As an essential technique in addition to the conventional encryption-based methods, physical layer security (PLS) ensures information privacy at the physical layer of the wireless networks by exploiting transceiver and channel properties, including beamforming, fading, noise, and interference [7]. Compared with the encryption-based methods, the PLS has the advantage of owning reduced overhead for secret key distribution. Moreover, PLS can ensure information secrecy even with an eavesdropper having infinite computational power.

With the equipment of ultra-massive multiple-input multiple-output (UM-MIMO) arrays for THz communications, the narrow and directional beams powered by multiple-antenna techniques bring substantial benefits to PLS [5, 8, 9]. Typically, a THz beam is directed towards the legitimate user through the beam training and tracking processes. With an eavesdropper being located outside or inside the beam, the THz PLS can be divided into two categories, namely, angle security and range security, as illustrated in Fig. 1. On the one hand, for the angle security, the aim is to prevent the eavesdropper from being located inside the beam sector, which improves the PLS in the angular domain. On the other hand, the range security aims at enhancing the secrecy rate when the legitimate user and the eavesdropper are both located inside the beam sector, i.e. when their propagation angles are the same while their ranges differ. Although the traditional multiple-antenna techniques of THz communications can enhance angle security, it fails to provide range security. Fortunately, there is a high probability that we only need to address the angle security problem since the THz beam width is narrow, e.g., approximately $1^{\circ}$ for a $32\times 32$ array,.

Despite the promising confidentiality of THz angle security, solutions to the THz range security problem are still awaited to be explored for the following challenges [5]. First, the signals received by the eavesdropper inside the beam benefit from the same antenna gain as legitimate users, which suggests that the THz directivity and traditional multiple-antenna techniques are ineffective in enhancing the range security. Second, due to the sparsity feature of the THz channel, the PLS technique using multi-paths to aid the security proposed in [10] might not be helpful. More critically, when Eve inside the beam sector is located nearer than the legitimate receiver, i.e., in close proximity, secure communication is significantly jeopardized since the secret codewords decodable by Bob are even easier decodable by the eavesdropper who has an even higher received signal-to-noise ratio (SNR) [9]. This research gap motivates this work aiming at addressing the range security for THz communications.

The fundamental limit of a wiretap channel model has been analyzed in [11] including a transmitter Alice (A), a legitimate receiver Bob (B), and an eavesdropper Eve (E). The key metric is characterized by the secrecy rate, defined as the maximum data rate that can be transmitted reliably and confidentially. Traditional multiple-antenna technologies can only improve the secrecy rate in the angular domain. For example, the secrecy capacity with the multiple antennas is calculated in [12], while the multi-antenna-based secure beamforming techniques are proposed in [13], which controls the beamforming pattern to enlarge Bob’s channel capacity while mitigating Eve’s capacity. In [14, 15, 16, 17], as an extension to secure beamforming, the authors develop the artificial noise (AN)-aided beamforming technique to combat eavesdroppers in close proximity, where the AN signals are signals without carrying any information. In particular, AN signals are designed to be injected into the transmitted confidential signal to jam Eve’s channel and enhance the secrecy rate.

Range security technologies, however, are rarely investigated for micro-wave band communications. Since the micro-wave channel contains rich scattering, angle security techniques can utilize the spatial degree of freedoms (SDoF) to mitigate eavesdropping attacks. However, the feature of limited SDoFs in THz communications makes the range security problem essential to be addressed. One useful range security technique in the literature is the receiver AN technique, where the receiver side sends the jamming signals while receiving the confidential signals [18, 19, 20]. On the downside, this technique requires a high-complexity self-interference cancellation (SIC) to transmit the AN signal and receive the information signal simultaneously. Another range security technique is the distance-adaptive molecular absorption modulation (DA-APM) scheme, which hides the signal under the molecular absorption peaks in the THz band [8]. This technique does not work when Eve is farther than Bob from Alice.

In recent years, frequency diverse array (FDA) has been recognized as a good candidate to address the range security problem [21]. By introducing a small frequency offset among different antennas in the antenna array, the FDA shows a periodical range-, angle-, and time-varying beam pattern. FDA has been widely used in radar, navigation, and security applications by taking advantage of the time-varying property. For example, The FDA beamforming approach is proposed in[21] to safeguard wireless transmission for proximal Bob and Eve, where the FDA is used to distinguish two proximal receivers to increase the secrecy rate. For the range security problem, FDA is recently envisioned as a potential technique to enhance information security thanks to its range-varying beam pattern. The effectiveness of the FDA-based techniques in providing THz range security requires investigation.

The goals of this paper are two-fold. First, we lay out a theoretical analysis of the multiple-antenna techniques under the range security condition and stress that multiple-antenna techniques operating in the far-field regions cannot enhance range security. We revisit the FDA communication model as one particular design of multiple-antenna techniques and prove that previous FDA models in the literature [14, 15, 16, 17] draw a wrong conclusion on its effectiveness on range security by missing the synchronization between array factor and symbol. Second, motivated by the failure of multiple-antenna techniques operating in far-field regions, we propose a THz widely-spaced array (WSA) and a hybrid beamforming design as a near-field THz range security solution. Specifically, we present a multiple-antenna-assisted THz range security system model and develop the secrecy capacity, based on which the FDA system model is revisited and revised. Moreover, we propose a THz WSA and beamforming design to enhance the THz range security. An ultra-massive uniform planar array (UM-UPA) with widely-spaced antennas, instead of densely-packed antennas, is applied to provide THz range security. The trade-off analysis between the antenna size and the secrecy rate demonstrates that the THz WSA phased array can achieve an acceptable secrecy performance with a reasonable size thanks to the sub-millimeter wavelengths of THz waves. The contributions of this paper are summarized as follows:

• •

  We perform theoretical analysis on multiple-antenna techniques for THz range security. The secrecy capacity is derived based on which multiple-antenna techniques operating in the far-field regions are proven ineffective in providing range security. As a traditional candidacy for range security, the FDA communication model is analyzed and revised.

• •

  We propose a novel THz WSA and hybrid beamforming scheme to enhance the THz range security. A non-constrained optimum approaching (NCOA) algorithm is developed to solve the non-convex secrecy rate maximization problem, which determines the optimal hybrid beamformer design. Moreover, the trade-off between array size and secrecy rate enhancement of WSA is investigated.

• •

  We evaluate our proposed scheme against the existing PLS techniques. Monte-Carlo simulation demonstrates the convergence of the proposed NCOA algorithm. Extensive numerical results show that for a propagation distance at $10~{}\textrm{m}$ and transmit power of $10~{}\textrm{dBm}$, the proposed method can achieve a secrecy rate of $6~{}\textrm{bps/Hz}$, while the conventional antenna array fails to guarantee range security.

The remainder of the paper is organized as follows. Sec. II presents multiple-antenna-assisted THz secure communication system model. Sec. III conducts a theoretical analysis on multiple-antenna techniques for THz range security, where the FDA technique is proven ineffective in providing range security. This conclusion is extended to the general multiple-antenna techniques operating in the far-field region. The proposed WSA communication and hybrid beamforming scheme are proposed in Sec. IV, where the NCOA hybrid beamforming algorithm is elaborated, and the trade-off between array size and maximum secrecy rate is investigated. Numerical results are described in Sec. V to verify the convergence of the proposed algorithm and the secrecy rate enhancement compared with existing methods. The paper is concluded in Sec. VI.

Notations: matrices and vectors are denoted by boldface upper and lower case letters, respectively. $[\cdot]^{+}\triangleq\max(0,\cdot)$. $\langle\cdot,\cdot\rangle$ denotes inner product. $\ast$ is the convolution operation with respect to time $t$. $\mathbf{A}^{T}$, and $\mathbf{A}^{\dagger}$ are the transpose, and conjugate transpose of the matrix $\mathbf{A}$, respectively.

We first theoretically derive the secrecy capacity of the multiple-antenna secure communication system under the range security scenario. Instead of assuming any specific precoding or up-converting method at the transmitter or receiver side, we focus on the original multiple-antenna secure channel. The original channel block diagram is shown in (a) of Fig. 3, where nodes $\{\textbf{A}_{i}\},i\in[1,N_{t}]$ represent the antennas in the transmit antenna array. Node B and E represent the single-antenna Bob and Eve, respectively. The channel impulse response from $\textbf{A}_{i}$ to $\textbf{U}\in\{\textbf{B},\textbf{E}\}$ is expressed by $a(r_{\textrm{U}})\delta(t-\frac{r_{\textrm{U},i}}{c})$. By substituting $r_{\textrm{U},i}=r_{\textrm{U}}+\Delta r_{\textrm{U},i}$, the system block diagram in Fig. 3(a) can be simplified to the block diagram in Fig. 3(b), due to the following relationship

where $x_{i}(t)$ represents the transmitted signal at node $\textbf{A}_{i}$, and $y_{\textrm{U}}(t)$ denotes the received signal at node U. Two intermediate nodes representing the superimposed signals at Bob and Eve in Fig. 3(b) are defined as $\textbf{A}_{\textbf{B}}$ and $\textbf{A}_{\textbf{E}}$, respectively. As the basic assumption for the range security problem, the propagation angles of Bob and Eve are equal, i.e., $\theta_{\textrm{B}}=\theta_{\textrm{E}}$. Additionally, as described in Sec.II, we assume that the receiving nodes $\mathbf{B}$ and $\mathbf{E}$ are in the far-field region of the antenna array. Therefore, $\Delta r_{\textrm{B},i}=\Delta r_{\textrm{E},i}$, and the signals at node $\textbf{A}_{\textbf{B}}$ and $\textbf{A}_{\textbf{E}}$ are equal. We can superimpose the two nodes and their corresponding overlapped parts in (b), as depicted in Fig. 3(c). Fig. 3(c) shows a cascaded wiretap channel which exhibits as two Markov chains, e.g., $\{\textbf{A}_{i}\}\to\textbf{A}\to\textbf{B}$ and $\{\textbf{A}_{i}\}\to\textbf{A}\to\textbf{E}$.

Based on the system transformation from Fig. 3(a) to Fig. 3(c), the secrecy capacity $C_{s}$ of the multiple-antenna secure communication system can be derived according to the property of Markov chains, as

where $P_{\textrm{A}}$ denotes the maximum signal power of the combined signals at node A. (10) implies that the secrecy rate from node $[\textbf{A}_{i}]$ is upper-bounded by the secrecy rate from node A and

whose secrecy capacity is equal to that of a fully-digital beamforming scheme. This result proves that any signal processing and multiple antenna techniques operating in the far-field regions cannot provide THz range security since the traditional fully-digital beamforming technique can reach the secrecy capacity.

Further explanation from the electromagnetic wave propagation perspective is provided as follows. In Fig. 3(c), the intermediate node A can be interpreted as the wavefront of the transmitted beam. The THz LoS signal propagation from the source to far-field regions can be divided into two phases, as depicted in Fig. 4. In the first phase, the transmitted signal propagates from each antenna to the plane perpendicular to the direction of propagation in the near-field region, i.e., the wavefront $\mathbf{A}$. In the second phase, the combined signal further propagates and enters the far-field regions. According to the Markov chain theory, for all receiving nodes located in the far-field regions with the same propagation angle, the secrecy capacity of a MIMO channel is upper bounded by the SISO channel from the wavefront to the receiving node. This wavefront acts like a bottleneck of the wiretap channel, preventing multiple-antenna techniques from distinguishing Bob and Eve in the far-field region. Therefore, any multiple-antenna schemes operating in the far-field region cannot enhance THz range security.

FDA introduces different small frequency offsets on each antenna to obtain a range-, angle-, and time-dependent array factor, which can steer the beam automatically [27]. The range-dependency of the beam can simultaneously enhance the received SNR at Bob while mitigating the received SNR at Eve since the antenna gain of FDA periodically changes with the distance [21]. Based on this phenomenon, researchers explore FDA-based multiple-antenna techniques to achieve range security, and they assume that Bob and Eve are in the far-field regions of the FDA. However, this idea contradicts our secrecy capacity analysis in Sec. III-A since their derived data rate exceeds our derived secrecy capacity. Therefore, we rigorously revisit the traditional FDA range security model and revise the model to demonstrate that the FDA cannot provide THz range security.

The THz WSA model is illustrated in Fig. 6, where the spacing between antennas in the antenna array is widened such that the Alice-Bob distance and the Alice-Eve is comparable to the antenna size. The goal is that Bob and Eve can be located in the near-field region of the transmitter antenna array, and the closed-form expression for the residual propagation distance term in (7) can be expressed as

where $i_{\textrm{row}}=(i-1)\mod N_{y}+1$ and $i_{\textrm{col}}=(i-i_{\textrm{row}})/N_{y}+1$. Remarkably, (18b) is the ground-truth model, since it considers the spherical-wave propagation nature for EM waves.

For the beamformer design at the transmitter side, since traditional fully-digital beamformer requires the same number of radio frequency (RF) chains as antennas [33], which brings tremendous hardware cost for THz communications, hybrid beamforming technology is therefore proposed as a promising yet lower-complexity structure for UM-UPA [34]. Specifically, the fully-connected hybrid beamforming structure is composed of a digital beamformer represented by $\mathbf{P}_{\textrm{D}}\in\mathbb{C}^{N_{\textrm{RF}}\times 1}$ and an analog beamformer expressed as $\mathbf{P}_{\textrm{A}}\in\mathbb{C}^{N_{t}\times N_{\textrm{RF}}}$. Each element of the matrices $\mathbf{P}_{\textrm{D}}$ and $\mathbf{P}_{\textrm{A}}$ represents the complex gain from the baseband signal to the RF chain, and from the RF chain to the antenna via phase shifters, respectively. $N_{\textrm{RF}}$ denotes the number of RF chains. The analog beamformer matrix is expressed in the form of

where $\phi_{pq}\in[0,2\pi),p\in\{1,\cdots,N_{t}\},q\in\{1,\cdots,N_{\textrm{RF}}\}$ denotes the phase output of the phase shifter which connects the $p^{\textrm{th}}$ RF chain with the $q^{\textrm{th}}$ antenna. We assume that the hybrid beamforming does not affect the transmit power, i.e., $\|\mathbf{P}_{\textrm{A}}\mathbf{P}_{\textrm{D}}\|^{2}=1$. The overall beamforming vector is denoted by

We assume that the carrier frequency of each antenna in the WSA is uniform and denoted as $f_{0}$. The received signal at the receiving node $U$ after down-converting and synchronization of symbol $n_{0}$ can be expressed as

or equivalently we can express the received signal in matrix form as

where the channel combining matrix $\mathbf{H}_{\textrm{U}}$ is expressed as

The secrecy rate, defined as the capacity difference between Bob’s and Eve’s channels, is used as the performance metric of our THz system model. As the focus of our THz secure communication system, the aim is to design transmit power $P$, hybrid beamformer, i.e., $\mathbf{P}_{\textrm{A}}$ and $\mathbf{P}_{\textrm{D}}$, to maximize the secrecy rate. Thus, we formulate the hybrid secure beamforming design problem as a secrecy rate maximization (SRM) optimization problem $\textrm{Q}_{1}$ as

where (24a) denotes the secrecy rate of the THz secure communication system, and the operation $[x]^{+}\triangleq\max(x,0)$. (24b) states the transmit power constraint with the maximum power $P_{\textrm{Tx}}$. $\eqref{eq:Rsmax:PAPD}$ describes the normalization constraint of a hybrid beamformer. The maximal secrecy rate is denoted by $R_{s}^{\textrm{opt}}$ and the optimal transmit beamformer is represented by $\mathbf{w}^{\textrm{opt}}=[\mathbf{P}_{\textrm{A}}^{\textrm{opt}}\mathbf{P}_{\textrm{D}}^{\textrm{opt}}]$. Solving the problem $\textbf{Q}_{1}$ yields the optimal hybrid beamforming strategy achieving maximum secrecy. Due to the non-convexity of (24a) with respect to $\mathbf{w}$, the optimization problem $\textbf{Q}_{1}$ is non-convex and cannot be tackled directly. Remarkably, according to (23), for the range security problem where $\theta_{\textrm{B}}=\theta_{\textrm{E}}$, if the far-field approximation $\mathcal{L}(i,\theta_{\textrm{U}},r_{\textrm{U}})\approx-(i_{\textrm{row}}-1)d\sin\theta_{\textrm{U}}$ is applied, we have $\mathcal{L}(i,\theta_{\textrm{B}},r_{\textrm{B}})=\mathcal{L}(i,\theta_{\textrm{E}},r_{\textrm{E}})$. In such case, Bob’s and Eve’s channels are perfectly correlated, i.e., $\mathbf{H}_{\textrm{B}}=\mathbf{H}_{\textrm{E}}$. As a result, their beamforming gains satisfy $\left|\mathbf{w}^{\dagger}\mathbf{H}_{\textrm{B}}\right|^{2}=\left|\mathbf{w}^{\dagger}\mathbf{H}_{\textrm{E}}\right|^{2}$. Therefore, by considering that $\left|\mathbf{w}^{\dagger}\mathbf{H}_{\textrm{U}}\right|^{2}\in[0,N_{t}]$, the maximum secrecy rate is upper bounded by $\left[\log\left(1+\frac{P_{\textrm{Tx}}N_{t}\alpha(r_{\textrm{B}})^{2}}{\sigma^{2}}\right)-\log\left(1+\frac{P_{\textrm{Tx}}N_{t}\alpha(r_{\textrm{E}})^{2}}{\sigma^{2}}\right)\right]^{+}$, which is exactly the maximum secrecy rate without adopting any security technique.

We design a hybrid beamformer to maximize the secrecy rate in (24a) for our WSA transmission scheme. Given the transmission distances $r_{\textrm{B}}$ and $r_{\textrm{E}}$, solving the SRM problem $\textbf{Q}_{1}$ yields our hybrid beamformer design. We propose an NCOA algorithm to solve this non-convex problem. First, we temporarily relax the non-convex constraint to transform $\textbf{Q}_{1}$ into a convex optimization problem $\textbf{Q}_{2}$. A globally optimal solution $\tilde{\mathbf{w}}$ to $\textbf{Q}_{2}$ is then derived, referred to as the non-constrained optimum. Finally, by reconsidering the non-convex constraint, an epsilon-convergent sub-optimal solution is achieved by approaching $\tilde{\mathbf{w}}$.

For a traditional antenna array where the antennas are not widely spaced, the far-field approximation is satisfied, which leads that the secrecy rate is upper-bounded by the secrecy capacity derived in (10). Our WSA design widens the antenna spacing to ensure near-field communications. However, an ensuing drawback is that the array size becomes larger as the spacing between neighboring antennas increases, which makes the WSA design less attractive for practical applications.

Thanks to the millimeter down to sub-millimeter THz wavelength, we clarify that THz communications can maintain a reasonable array size even when the antenna spacing is wide enough to enhance the secrecy rate. A sample comparison between mmWave and THz WSA design is provided as follows. For a $4\times 4$ micro-wave communication system where the wavelength is on the order of sub-meters, e.g., $0.3~{}\textrm{m}$ for the $1~{}\textrm{GHz}$ system, the near-field expression in (18b) is valid when the side length of the array $3d\approx 4.0~{}\textrm{m}$, which is an impractically large antenna array. However, for THz communications, with a wavelength on the order of millimeter, e.g., $1~{}\textrm{mm}$ for a $300~{}\textrm{GHz}$ system, the near-field expression is valid when $3d\approx 0.08~{}\textrm{m}$. Therefore, a $0.08~{}\textrm{m}\times 0.08~{}\textrm{m}$ THz antenna array can well accommodate the near-field communication requirement.