Joint Transmit and Receive Beamforming Design for a Full Duplex UAV Sensing Network

Abstract

Unmanned aerial vehicles (UAVs) are promising and powerful aerial platforms that can execute a variety of complex tasks. However, the increasing complexity of tasks and number of UAV nodes pose significant challenges for UAV sensing networks, such as limiting the spectral resources and increasing device complexity. A potential solution is to implement full-duplex (FD) technology in UAV sensor network transceivers. Although appropriate self-interference (SI) cancellation techniques have been employed in the digital domain, the amplitude of the signal of interest (SoI) is relatively small and can be obscured by SI, especially over longer distances. Moreover, the introduction of phase offsets when filtering measurement signals can lead to signal distortion, resulting in estimation errors in the measurement results. To address these issues, this paper presents a joint transmit (TX) and receive (RX) beamforming algorithm based on the penalty dual decomposition (PDD) algorithm, which considers the constraints of transmission power, reception power, and residual SI power. The simulation analyses demonstrate that with a limited number of antennas, the proposed joint TX-RX beamforming algorithm can effectively suppress SI by up to 140 dB, yielding high-precision measurements in UAV sensor networks without compromising the accuracy of the control signals. Compared with that of the traditional frequency-division duplex (FDD) mode, the measurement accuracy is not decreased; compared with those of the time-division duplex (TDD) mode, the distance and speed measurement accuracies of the UAVs are increased by 10 m and 1.5 m/s, respectively, in the FD mode because there is no interruption of the tracking loop and no continuous retracking in the FD mode.

1. Introduction

Unmanned aerial vehicles (UAVs), which are compact in size and have low power consumption, high integration, flexibility, fast deployment, and low cost, have developed rapidly over the past few years [1,2,3,4,5]. However, the complexity of the tasks and the increase in the number of UAV nodes present difficulties and lead to limited spectrum resources and complex equipment in the UAV sensor network [6,7]. The existing navigation methods for UAVs are mainly based on the global navigation satellite system (GNSS). However, GNSS signals are strongly interfered with when UAVs perform missions in complex environments; for example, there may be electronic interference on a battlefield, signal blockage by mountainous terrain, signal blockage by buildings and trees in a city, etc. In these cases, UAVs can only rely on their own measurement information [8,9,10].

The FDD mode is generally adopted in UAV sensing networks to provide telemetry, track, and command (TT&C) services that support UAV flight monitoring, link monitoring, mission load monitoring, and mission planning functions. In [11], a multifunctional signal that integrates measurement, navigation, and communication was proposed to improve spectrum utilization. In the frequency domain, signals with different functions are orthogonal so that the data transmission signal and measurement signal do not interact. In addition, the TDD mode is adopted to solve the problems of insufficient frequency resources and low utilization caused by FDD, as shown in [12].

To address the shortage of spectrum resources in UAV sensing networks and obtain high-precision real-time measurements, a potential method is to adopt full-duplex (FD) technology, which has received much attention in recent years [13,14,15]. However, one problem for the FD UAV sensing system is that the received TT&C signal is overwhelmed by large SI from the transmitter. Despite employing sufficient SI cancellation techniques before quantization, useful signals may still be drowned out by the SI, as SI can persist at levels exceeding the useful signal by a margin of approximately 80 to 120 dB [16]. A significant challenge in cancelling SI is minimizing the impact of SI cancellation on the phase of the signal of interest (SoI). Phase bias is easily introduced when filtering the TT&C signal. Some applications typically exhibit biases of more than 40° [17].

Many works have studied SI cancellation in FD wireless communication systems, and they have concentrated on active and passive cancellation. In the propagation domain, passive cancellation is primarily performed, including antenna separation (AS), directional antennas, and cross-polarization, to achieve a high degree of isolation between the TX and RX antennas [18,19,20,21]. However, passive cancellation has certain limitations, such as ambient reflections and the resulting increase in the frequency selectivity of the residual SI. Consequently, active cancellation is also necessary to mitigate residual SI [19]. SI cancellation techniques in both the analog and digital domains are employed through adaptive SI cancellation via training sequences. In the analog domain, SI cancellation is performed before the ADC is applied to prevent desensitization of the automatic gain control caused by high-power SI [22,23,24,25,26]. The method of SI cancellation based on intelligent reflecting surfaces (IRSs) and the reconfigurable intelligent surface (RIS) was studied in recent works [27,28,29,30]. However, these approaches impose constraints related to hardware complexity and power consumption, which increase the complexity of hardware design.

In multiantenna systems, an effective approach involves designing hybrid beamformers. A FD transceiver is designed for configuring the analog and digital beamformers of mmWave communication systems in [31,32]. A FD joint hybrid beamforming structure with a multi-tap suppressor is proposed in [33], which achieved approximately 100 dB of self-interference cancellation. Previous studies, such as [34,35], employed null-space projection (NSP) methods for designing TX and RX beamforming in FD integrated sensing and communications (ISAC) systems, aiming to cancel SI and prevent radar signals from being overwhelmed by communication signals. However, the TX and RX beamformers were designed independently in these works, and only the RX beamformer was utilized for SI cancellation. Millimeter-wave FD hybrid beamforming was designed for the ISAC system in [36]. The joint TX and RX beamformer designed in [37] achieved digital-domain SI cancellation up to 60 dB above the noise floor. The measurement methods proposed in these studies were based on a monostatic radar. However, the measurement method based on the echo reflected from the target has weak anti-interference ability, and the transmitting and receiving antennas cannot be separated, which limits the application of this method. Moreover, the literature has focused primarily on extracting measurement signals in cases of communication signal interference. There is a notable lack of studies analyzing how the measurement indicators change when the SI signal originates from the measurement signals transmitted by the sensing system. Therefore, the potential of beamforming and SI cancellation in FD UAV sensing networks warrants further exploration. This paper presents an FD sensing system for UAVs. The signal employs pseudocode modulation techniques. We apply the PDD algorithm [37,38,39] to the UAV network measurement optimization problem in this paper. By introducing auxiliary variables to address the coupling problem between the TX and RX beamformers, the optimization variables in the inner loop are updated in the direction of block coordinate decline, and the subproblems of each block can be solved in a closed form. By adjusting the Lagrange multiplier and penalty parameter, the proposed algorithm, which is based on the PDD algorithm, converges to the stationary solution set of the original optimization problem. The major contributions of this paper are summarized as follows:

• Unlike in previous works [34,35,37], the measurement method does not rely on the echo of the signal, and the UAV sensing system operates in full-duplex mode. The SoI is overwhelmed by the locally transmitted TT&C signal. Because acquisition and tracking rely on autocorrelation characteristics, the local SI and the received signal exhibit strong autocorrelation properties, which interferes with the acquisition and tracking of the SoI. Consequently, it is essential to suppress the SI to an appropriate power level and analyze the effects of the joint TX-RX beamforming algorithm and SI cancellation on the measurement outcomes.
• The signal amplitude is quite small for long-distance measurements, in contrast to the approach in [37], where the SI power is set relative to the noise floor. Therefore, SI is assessed in relation to the signal amplitude in this paper, which implies that the SI power may be high, increasing the difficulty of SI cancellation. In addition, considering the limited size and weight of the UAVs, SI cancellation is performed under near-field conditions.
• The simulation analyses demonstrate that the proposed joint TX-RX beamforming algorithm can effectively suppress up to 140 dB of SI with a limited number of antennas, and it can obtain high-precision measurements in UAV networks without affecting the accuracy of TT&C signals. Compared with that of the traditional FDD mode, the measurement accuracy is not decreased, and compared with those of the TDD mode, the distance and speed measurement accuracies of the UAVs are increased by 10 m and 1.5 m/s, respectively, in FD mode because there is no interruption of the tracking loop and no continuous retracking in FD mode. Futhermore, the proposed algorithm has good dynamic performance.

The rest of the paper is organized as follows. The FD UAV sensing system model is introduced in Section 2. In Section 3, the problem formulation is presented. We then convert the problem into a more optimizable form and propose a joint TX and RX beamforming algorithm design for FD UAV sensing networks. The numerical evaluations are presented in Section 4. Finally, we conclude this work in Section 5.

Notation: $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{C}$ denote the sets of reals, integers, and complex numbers, respectively. $X^T$ and $X^H$ denote the transpose and conjugate transpose of matrix X, respectively. I represents an identity matrix with appropriate dimensions. $\left|X\right|$ and ${||X||}_2$ represent the absolute value and Frobenius norm, respectively.

2. System Model

As shown in Figure 1, we consider a full-duplex measurement system with $N_t\left(n_x\times n_y=N_t\right)$ transmit antennas and $N_r(m_x\times m_y=N_r)$ receive antennas. The adjacent antenna elements are separated by a half-wavelength, and all the arrays are assumed to be uniform arrays. The system can transmit and receive measurement signals simultaneously.

2.1. Signal Model

We let $x_t$ and $x_r$ denote the transmit signals of Nodes 1 and 2, with transmit powers $P_t$ and $P_r$, which are given in [40,41]

$$x_t\left(t\right)=\sqrt{P_t}{d}_t\left(t\right)P{N}_t\left(t\right)c_t\left(t\right)$$

(1)

$$x_r\left(t\right)=\sqrt{P_r}{d}_r\left(t\right)P{N}_r\left(t\right)c_r\left(t\right)$$

(2)

where $d_t\left(t\right)$ and $d_r\left(t\right)$ denote the transmit data, $P{N}_t\left(t\right)$, $P{N}_r\left(t\right)$ are pseudorandom codes, and $c_t\left(t\right)$, $c_r\left(t\right)$ denote the carriers of the TT&C signal. $PN\left(t\right)$, $c_t\left(t\right)$, and $c_r\left(t\right)$ can be expressed as

$$PN\left(t\right)=\sum _{k=-\infty }^{\infty }{\alpha }_k{p}_{T_{code}}(t-k{T}_{code})$$

(3)

$$c_t\left(t\right)=cos(2\pi f_{c}t+{\varphi }_t)$$

(4)

$$c_r\left(t\right)=cos(2\pi f_{c}t+{\varphi }_r)$$

(5)

where ${\alpha }_k=\pm 1$, $T_{code}$ is the width of the pseudorandom code, $p_{T_{code}}(•)$ denotes the pseudo-random sequence, $f_c$ is the carrier frequence, ${\varphi }_t$ and ${\varphi }_r$ denote the initial phases. The transmitting signal, which is filtered by a transmit beamformer $\mathbf{v}\in {\mathbb{C}}^{N_t\times 1}$, is given by

$${\mathbf{y}}_t={\mathbf{v}}^H{x}_t$$

(6)

In an FD TT&C system, the received signal comprises the expected signal, i.e., SoI, as well as residual self-interference. It is assumed that the ADC dynamic range is wide and that the received signal is within the ADC dynamic range; these assumptions are made in numerous methods for self-interference suppression in the analog and spatial domains. Therefore, the received signal at Node 1 can be denoted as

$${\mathbf{y}}_r={\mathbf{H}}_r{x}_r+{\mathbf{H}}_{si}{\mathbf{y}}_t+{\mathbf{n}}_r$$

(7)

where ${\mathbf{H}}_r\in {\mathbb{C}}^{N_r\times 1}$ denotes the downlink channel, ${\mathbf{H}}_{si}\in {\mathbb{C}}^{N_r\times N_t}$ is the SI channel, ${\mathbf{n}}_r\sim CN(0,{\sigma }^{2_r}{\mathbf{I}}_{N_r})\in {\mathbb{C}}^{N_r\times 1}$ is additive complex Gaussian noise. The channel model is given in the next section.

At Node 1, we obtain the estimated signal, which is filtered by the receive beamformer $\mathbf{w}\in {\mathbb{C}}^{N_r\times 1}$ and can be expressed as

$$\begin{array}{cc}\hfill \stackrel{\wedge }{{\mathbf{y}}_r}& ={\mathbf{w}}^H{\mathbf{y}}_r\hfill \\ \hfill & =\underset{SoI}{\underbrace{{\mathbf{w}}^H{\mathbf{H}}_r{x}_r}}+\underset{\mathrm{Re}sidualSI}{\underbrace{{\mathbf{w}}^H{\mathbf{H}}_{si}{\mathbf{y}}_t}}+{\mathbf{w}}^H{\mathbf{n}}_r\hfill \\ \hfill & ={\mathbf{w}}^H{\mathbf{H}}_r{x}_r+{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}{x}_t+{\mathbf{w}}^H{\mathbf{n}}_r\hfill \end{array}$$

(8)

2.2. Channel Model

2.2.1. SI Channel

A challenging problem for FD phased arrays is the amount of SI leaking from the transmit to the receive phased arrays. Many works solve the self-interference suppression problem under far-field conditions [37,42]. Considering the limitations of measurement equipment such as volume/weight and power consumption, the far-field distance condition of the SI channel cannot be satisfied; i.e., $R\ge 2D^2/\lambda$, where $\lambda$ is the wavelength of the carrier, D is the diameter of the antenna aperture, R is the distance between TX and RX phased arrays. For integrated TT&C equipment, the far-field range condition may not hold in general. Therefore, a near-field model with a spherical wavefront should be used for SI channels [43,44].

In the SI channel, in addition to the line-of-sight (LOS) component, there are non-LOS components, including scattering and reflection components; that is, $H_{SI}=H_{LOS}+H_{NLOS}$. For illustration, we consider the placement of FD phased-array antennas, as shown in Figure 2, where the first elements of the TX and RX phased arrays are separated by d and the angle between the TX and RX phased arrays is $\omega$. The spacing between TX/RX phased array elements is $\lambda /2$. The SI channel can be expressed as

$${\mathbf{H}}_{\mathrm{SI}}=\sqrt{{\displaystyle \frac{\kappa }{\kappa +1}}}{\mathbf{H}}_{\mathrm{LOS}}+\sqrt{{\displaystyle \frac{1}{\kappa +1}}}{\mathbf{H}}_{\mathrm{NLOS}},$$

(9)

where k is the Rician factor. $H_{LOS}$ between the nth RX antenna and the mth TX antenna can be denoted as [45]

$$\begin{array}{c}{\left[H_{LOS}\right]}_{nm}=h_{nm}={\displaystyle \frac{\rho }{r_{nm}}}exp(-j2\pi {\displaystyle \frac{r_{nm}}{\lambda }})\end{array}$$

(10)

where $\rho$ is a constant for power normalization and $r_{nm}$ is the distance between the mth TX antenna and the nth RX antenna.

2.2.2. Signal Channel

The angle of departure (AoD) and angle of arrival (AoA) are represented by $({\theta }_t,{\phi }_t)$ and $({\theta }_r,{\phi }_r)$, respectively, at Node 1. We suppose that the transmitting gain of Node 2 is 1 since our focus is on Node 1. Hence, the signal channel can be modeled as follows:

$$\begin{array}{c}{\mathbf{H}}_r\left(t\right)={\beta }_r{e}^{j2\pi f_{d}t}{\mathbf{a}}_r\left({\theta }_r,{\phi }_r\right)\end{array}$$

(11)

where $f_d=v{f}_c/c$ is the doppler frequency, $f_c$ is the carrier frequency, and c denotes the velocity of light; ${\beta }_r$ denotes attenuation factor, and ${\mathbf{a}}_r({\theta }_r,{\phi }_r)$ denotes the steering vector in the direction of $({\theta }_r,{\phi }_r)$, expressed as follows:

$$\begin{array}{c}{\mathbf{a}}_r\left({\theta }_r,{\phi }_r\right)=\left[1,e^{j\pi cos{\theta }_{r}cos{\phi }_r+j\pi cos{\theta }_{r}sin{\phi }_r},\cdots ,e^{j\pi (m_x-1)cos{\theta }_{r}cos{\phi }_r+j\pi (m_y-1)cos{\theta }_{r}sin{\phi }_r}\right]\end{array}$$

(12)

3. Problem Formulation

3.1. Signal-to-Interference-Plus-Noise Ratio (SINR)

For an FD TT&C system, SI has a significant negative effect on the acquisition and tracking of the expected signal, ultimately affecting the accuracy of the measurement results. Generally, the quality of the received signal is assessed by the signal-to-interference-plus-noise ratio (SINR), the expression for which is as follows:

$$\begin{array}{c}{\gamma }_r={\displaystyle \frac{{\left|{\mathbf{w}}^H{\mathbf{H}}_r{x}_r\right|}^2}{{\left|{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}{x}_t\right|}^2+{∥\mathbf{w}∥}^2{{\sigma }_r}^2}}\end{array}$$

(13)

3.2. Beamforming Gain

Similarly to Equation (12), the steering vector in the direction of $({\theta }_t,{\phi }_t)$ is defined as

$$\begin{array}{c}{\mathbf{a}}_t\left({\theta }_t,{\phi }_t\right)=\left[1,e^{j\pi cos{\theta }_{t}cos{\phi }_t+j\pi cos{\theta }_{t}sin{\phi }_t},\cdots ,e^{j\pi (n_x-1)cos{\theta }_{t}cos{\phi }_t+j\pi (n_y-1)cos{\theta }_{t}sin{\phi }_t}\right]\end{array}$$

(14)

The beamforming gain is given by

$$\begin{array}{c}{G}_t={\left|{{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)\mathbf{v}\right|}^2,G_r={\left|{\mathbf{w}}^H{\mathbf{a}}_r({\theta }_r,{\phi }_r)\right|}^2\end{array}$$

(15)

3.3. Objective Function and Constraints

We aim to obtain the optimal transmit and receive beamformers, which maximize the SINR and the receive and transmit gains in the directions of $({\theta }_t,{\phi }_t)$ and $({\theta }_r,{\phi }_r)$, respectively. Moreover, the constraints should also satisfy (1) the power constraints, i.e., ${∥\mathbf{w}∥}^2\le 1$, ${∥\mathbf{v}∥}^2\le 1$, and (2) the SI cancellation constraint, i.e., ${\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}\le P_{nf}$. Therefore, the objective function and constraints can be summarized as follows:

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{\gamma }_r+G_t+G_r\hfill \\ s.t.\hfill & {∥\mathbf{w}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{v}∥}_2^2\le 1\hfill \\ \hfill & {∥{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}∥}_2^2\le P_{nf}\hfill \end{array}$$

(16)

where $P_{nf}$ is the noise floor. We can measure the level of background noise power through practical testing. From (16), $f_d$, obtained by measuring the phase of the expected signal, is part of ${\mathbf{H}}_r$, which is involved in the expression of the SINR. In other words, we cannot take ${\mathbf{H}}_r$ as a known quantity in the objective function. As a result, we need to perform a transformation on the objective function.

Maximizing the SINR involves maximizing the gain of the expected signal and minimizing interference and noise, which aligns with the objectives in the objective function and constraints. Therefore, we can transform (16) into

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{G}_t+G_r\hfill \\ s.t.\hfill & {∥\mathbf{w}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{v}∥}_2^2\le 1\hfill \\ \hfill & {∥{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}∥}_2^2\le P_{nf}\hfill \end{array}$$

(17)

To make (17) convex and obtain the optimal solution, we perform a transformation on $G_t$ and $G_r$. We let $A_t=|{{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t){|}^2$. Then, we have

$$\begin{array}{cc}\hfill G_t& ={\left|{{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)\mathbf{v}\right|}^2={\mathbf{v}}^H{\mathbf{a}}_t({\theta }_t,{\phi }_t){{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)\mathbf{v}\hfill \\ \hfill & ={\mathbf{v}}^H{\mathbf{a}}_t({\theta }_t,{\phi }_t){{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)\mathbf{v}-A_t+A_t\hfill \\ \hfill & \le {\mathbf{v}}^H{\mathbf{a}}_t({\theta }_t,{\phi }_t){{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)\mathbf{v}-{\mathbf{v}}^H{A}_t\mathbf{Iv}+A_t\hfill \\ \hfill & \le {\mathbf{v}}^H({\mathbf{a}}_t({\theta }_t,{\phi }_t){{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)-A_t\mathbf{I})\mathbf{v}+A_t\hfill \end{array}$$

(18)

Equality holds when ${||\mathbf{v}||}^2$. With ${\mathbf{F}}_t({\theta }_t,{\phi }_t)={\mathbf{a}}_t({\theta }_t,{\phi }_t){{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)-A_t\mathbf{I}$, (18) can be reexpressed as

$$\begin{array}{c}{G}_t\le {\mathbf{v}}^H{\mathbf{F}}_t({\theta }_t,{\phi }_t)\mathbf{v}+A_t\end{array}$$

(19)

The expression of the steering vector clearly reveals that ${\mathbf{a}}_t({\theta }_t,{\phi }_t){{\mathbf{a}}_t}^H({\theta }_t,{\phi }_t)$ is a matrix of Rank 1 with eigenvalue ${∥{\mathbf{a}}_t({\theta }_t,{\phi }_t)∥}^2$. Consequently, ${\mathbf{F}}_t({\theta }_t,{\phi }_t)$ is negative semidefinite. Then, the problem of maximizing $G_t$ can be converted into maximizing $\mathbf{F}({\theta }_t,{\phi }_t)$.

Correspondingly, $G_r$ can be denoted as

$$\begin{array}{c}{G}_r\le {\mathbf{w}}^H{\mathbf{F}}_r({\theta }_r,{\phi }_r)\mathbf{w}+A_r\end{array}$$

(20)

where ${\mathbf{F}}_r({\theta }_r,{\phi }_r)={\mathbf{a}}_r({\theta }_r,{\phi }_r){{\mathbf{a}}_r}^H({\theta }_r,{\phi }_r)-A_r\mathbf{I}$, $A_r={∥{{\mathbf{a}}_r}^H({\theta }_r,{\phi }_r)∥}^2$. The equality in (20) holds when ${\left|\right|\mathbf{w}\left|\right|}^2=1$.

Therefore, Problem (17) can be described as

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{\mathbf{v}}^H{\mathbf{F}}_t({\theta }_t,{\phi }_t)\mathbf{v}+A_t+{\mathbf{w}}^H{\mathbf{F}}_r({\theta }_r,{\phi }_r)\mathbf{w}+A_r\hfill \\ s.t.\hfill & {∥\mathbf{w}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{v}∥}_2^2\le 1\hfill \\ \hfill & {∥{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}∥}_2^2\le P_{nf}\hfill \end{array}$$

(21)

Clearly, there is a coupling problem in the constraint ${∥{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}∥}_2^2$ in (21). The presence of a coupling term prevents traditional alternating optimization approaches from obtaining the optimal solution, which creates considerable difficulties for the optimization problem. Inspired by the PDD algorithm, we propose a penalty-based iterative algorithm to address the optimization problem in (21).

To address the coupling problem, we introduce an auxiliary variable $\mathbf{S}$ with the contraints $\mathbf{S}={\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}$. That is,

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{\mathbf{v}}^H{\mathbf{F}}_t({\theta }_t,{\phi }_t)\mathbf{v}+A_t+{\mathbf{w}}^H{\mathbf{F}}_r({\theta }_r,{\phi }_r)\mathbf{w}+A_r\hfill \\ s.t.\hfill & {∥\mathbf{v}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{w}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{S}∥}_2^2\le P_{nf}\hfill \\ \hfill & \mathbf{S}={\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}\hfill \end{array}$$

(22)

We then perform a transformation on the constraint of $\mathbf{S}={\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}$ to convert it into a penalty term with parameter $\alpha$. Therefore, by introducing the Lagrange multiplier $\mathit{\lambda }$, the objective function and constraint can be transformed into

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{\mathbf{v}}^H{\mathbf{F}}_t({\theta }_t,{\phi }_t)\mathbf{v}+A_t+{\mathbf{w}}^H{\mathbf{F}}_r({\theta }_r,{\phi }_r)\mathbf{w}+A_r-{\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2\hfill \\ s.t.\hfill & {∥\mathbf{v}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{w}∥}_2^2\le 1\hfill \\ \hfill & {∥\mathbf{S}∥}_2^2\le P_{nf}\hfill \end{array}$$

(23)

We employ alternating optimization to solve the aforementioned problem by optimizing $\left\{\mathbf{w}\right\}$, $\left\{\mathbf{v}\right\}$, $\left\{\mathbf{S}\right\}$ in turn.

The subproblem of $\left\{\mathbf{w}\right\}$: With fixed $\mathbf{S}$ and $\mathbf{v}$, the optimization problem concerning $\mathbf{w}$ can be simplified to

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{\mathbf{w}}^H{\mathbf{F}}_r({\theta }_r,{\phi }_r)\mathbf{w}+A_r-{\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2\hfill \\ s.t.\hfill & {∥\mathbf{w}∥}_2^2\le 1\hfill \end{array}$$

(24)

The Karush–Kuhn–Tucker (KKT)-condition-based Lagrange multiplier approach is applied since the objective function is convex, with the convex constraint

$$L(\mathbf{w},{\mu }_1)={\mathbf{w}}^H{\mathbf{F}}_r({\theta }_r,{\phi }_r)\mathbf{w}+A_r-{\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2+{\mu }_1({∥\mathbf{w}∥}_2^2-1)$$

(25)

To solve this problem, we calculate the partial derivatives of the Lagrangian function $L(\mathbf{w},{\mu }_1)$ for $\mathbf{w}$ and ${\mu }_1$ and set them to zero; we obtain

$$\mathbf{w}=-{(2{\mathbf{F}}_r({\theta }_r,{\phi }_r)+{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}\mathbf{v}{\mathbf{v}}^H{\mathbf{H}}_{si}^H+2{\mu }_1\mathbf{I})}^{-1}{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}\mathbf{v}({\mathbf{S}}^H+{\left(\alpha \mathit{\lambda }\right)}^H)$$

(26)

and

$${∥\mathbf{w}∥}_2^2=1$$

(27)

With the bisection method, the solution for ${\mu }_1^{\ast }$ can be easily obtained. Furthermore, substituting ${\mu }_1^{\ast }$ into (26) results in $w^{\ast }$; that is,

$${\mathbf{w}}^{\ast }=-{(2{\mathbf{F}}_r({\theta }_r,{\phi }_r)+{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}\mathbf{v}{\mathbf{v}}^H{\mathbf{H}}_{si}^H+2{\mu }_1^{\ast }I)}^{-1}{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}\mathbf{v}({\mathbf{S}}^H+{\left(\alpha \mathit{\lambda }\right)}^H)$$

(28)

The subproblem of $\left\{\mathbf{v}\right\}$: Similar to the subproblem of $\left\{\mathbf{w}\right\}$, the optimization problem with respect to $\mathbf{v}$ can be simplified to

$$\begin{array}{cc}\hfill & \underset{\mathbf{w},\mathbf{v}}{max}{\mathbf{v}}^H{\mathbf{F}}_t({\theta }_t,{\phi }_t)\mathbf{v}+A_t-{\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2\hfill \\ s.t.\hfill & {∥\mathbf{v}∥}_2^2\le 1\hfill \end{array}$$

(29)

By introducing a Lagrange multiplier ${\mu }_2$, we can obtain

$$L(\mathbf{v},{\mu }_2)={\mathbf{v}}^H{\mathbf{F}}_t({\theta }_t,{\phi }_t)\mathbf{w}+A_t-{\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2+{\mu }_2({∥\mathbf{v}∥}_2^2-1)$$

(30)

By performing partial differentiation on the Lagrangian function $L(v,{\mu }_2)$ for v and ${\mu }_2$ and setting them equal to zero, we can obtain

$$\mathbf{v}=-{(2{\mathbf{F}}_t({\theta }_t,{\phi }_t)+{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}^H\mathbf{w}{\mathbf{w}}^H{\mathbf{H}}_{si}+2{\mu }_2\mathbf{I})}^{-1}{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}^H\mathbf{w}(\mathbf{S}+\alpha \mathit{\lambda })$$

(31)

and

$${∥\mathbf{v}∥}_2^2=1$$

(32)

${\mathbf{v}}^{\ast }$ can be calculated by substituting ${\mu }_2^{\ast }$ into (31), which is obtained by the bisection method; i.e.,

$${\mathbf{v}}^{\ast }=-{(2{\mathbf{F}}_t({\theta }_t,{\phi }_t)+{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}^H\mathbf{w}{\mathbf{w}}^H{\mathbf{H}}_{si}+2{\mu }_2^{\ast }\mathbf{I})}^{-1}{\displaystyle \frac{1}{\alpha }}{\mathbf{H}}_{si}^H\mathbf{w}(\mathbf{S}+\alpha \mathit{\lambda })$$

(33)

The subproblem of $\left\{\mathbf{S}\right\}$: The optimization problem with respect to $\mathbf{S}$ can be simplified to

$$\begin{array}{cc}\hfill & \underset{\mathbf{S}}{min}{\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2\hfill \\ s.t.\hfill & {∥\mathbf{S}∥}_2^2\le P_{nf}\hfill \end{array}$$

(34)

By introducing a Lagrange multiplier ${\mu }_3$, we can obtain

$$L(\mathbf{S},{\mu }_3)={\displaystyle \frac{1}{2\alpha }}{∥\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}+\alpha \mathit{\lambda }∥}_2^2+{\mu }_3({∥\mathbf{S}∥}_2^2-P_{nf})$$

(35)

By calculating the partial derivatives of the Lagrangian function $L(\mathbf{S},{\mu }_3)$ for $\mathbf{S}$ and ${\mu }_3$ and setting them to zero, we can obtain

$$\mathbf{S}={({\displaystyle \frac{1}{\alpha }}\mathbf{I}+2{\mu }_3)}^{-1}({\displaystyle \frac{1}{\alpha }}{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}-\mathit{\lambda })$$

(36)

and

$${∥\mathbf{S}∥}_2^2=P_{nf}$$

(37)

By searching for ${\mu }_3^{\ast }$ with the bisection method and substituting it into (36), we obtain the optimal ${\mathbf{S}}^{\ast }$:

$${\mathbf{S}}^{\ast }={({\displaystyle \frac{1}{\alpha }}\mathbf{I}+2{\mu }_3^{\ast })}^{-1}({\displaystyle \frac{1}{\alpha }}{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}-\mathit{\lambda })$$

(38)

We let $F_{w,v}$ denote the objective function (23). We terminate the inner loop when the number of iterations reaches the maximum value or the termination criterion is met. The termination criterion is $\zeta <\delta$, where $\delta$ is a sufficient small number and

$$\zeta \stackrel{\Delta }{=}{\displaystyle \frac{F_{w,v}^n-F_{w,v}^{n-1}}{F_{w,v}^n}}$$

(39)

3.4. Summary of Algorithm

In each iteration of the inner loop of the proposed PDD-based joint transmit and receive beamformer algorithm, we update each variable according to Algorithm 1. Thereafter, we update the dual variables $\mathit{\lambda }$ and penalty parameter $\alpha$ according to Algorithm 2. We define a constraint violation indicator h as

$$\mathbf{h}=\mathbf{S}-{\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}$$

(40)

We define $\mathbf{h}<\epsilon$ as the termination criterion for algorithm iteration, where $\epsilon$ denotes the tolerance of accuracy for the outer loop. Moreover, on the basis of the convergence analysis in [39], we can conclude that the proposed PDD-based joint transmit and receive beamformer algorithm converges to the set of stationary solutions of Problem (17). In addition, in the inner loop, we update ${{\mathbf{w}}_k}^{\ast }$, ${{\mathbf{v}}_k}^{\ast }$, ${{\mathbf{S}}_k}^{\ast }$ through (28), (33), and (38), which have complexities of $\mathcal{O}\left(I_1{N}_r^3\right)$, $\mathcal{O}\left(I_2{N}_t^3\right)$, $\mathcal{O}\left(N_t{N}_r\right)$; $I_1$, $I_2$ denote the numbers of iterations used in searching for ${\mu }_1,{\mu }_2$ with the bisection method. Ultimately, the overall complexity of the proposed PDD-based joint transmit and receive beamformer design is $\mathcal{O}\left(I_3{I}_4(N_t{N}_r+I_1{N}_r^3+I_2{N}_t^3)\right)$, where $I_3,I_4$ are the numbers of iterations of the inner and outer loops, respectively.

Algorithm 1 Proposed PDD-Based Joint Transmit and Receive Beamformer Design in the Inner Loop

Inputs: ${\mathbf{H}}_{si}$, $P_{nf}$, ${\mathbf{a}}_t({\theta }_t,{\phi }_t)$, ${\mathbf{a}}_r({\theta }_r,{\phi }_r)$ Outputs: ${\mathit{w}}^{\ast }$, ${\mathit{v}}^{\ast }$ 1. initialize $\mathbf{w}$, $\mathbf{v}$, $\mathbf{S}={\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}$ 2. reapeat 3. -update $\mathbf{w}$ with (28) 4. -update $\mathbf{v}$ with (33) 5. -update $\mathbf{S}$ with (38) 6. until the termination criterion is met or the number of iterations reaches the maximum.

Algorithm 2 Proposed PDD-Based Joint Transmit and Receive Beamformer Design

Inputs: ${\mathbf{H}}_{si}$, $P_{nf}$, ${\mathbf{a}}_t({\theta }_t,{\phi }_t)$, ${\mathbf{a}}_r({\theta }_r,{\phi }_r)$, $i=1$, $0<c<1$ Outputs: ${\mathbf{w}}^{\ast }$, ${\mathbf{v}}^{\ast }$ 1. initialize $\mathbf{w}$, $\mathbf{v}$, $\mathbf{S}={\mathbf{w}}^H{\mathbf{H}}_{si}\mathbf{v}$ 2. reapeat 3. -optimize (${\mathbf{w}}^i$, ${\mathbf{v}}^i$, ${\mathbf{S}}^i$) according to Algorithm 1 4. -if ${∥{\mathbf{h}}^i∥}_{\infty }\le {\eta }_i$ 5. -${\mathit{\lambda }}_{i+1}={\mathit{\lambda }}_i+{\displaystyle \frac{1}{{\alpha }^i}}{h}^i$ 6. -${\alpha }_{i+1}={\alpha }_i$ 7. -else 8. -${\mathit{\lambda }}_{i+1}={\mathit{\lambda }}_i$ 9. -${\alpha }_{i+1}=c{\alpha }_i$ 10. -endif 11. -until the termination criterion is met or the number of iterations reaches the maximum.

4. Simulation Results

In this section, we conduct several simulations to examine the performance of the proposed PDD-based joint transmit and receive beamformer algorithm. An FD-UAV network scenario with two nodes is considered, where TT&C links exist between each node. The simulation parameters of the links are detailed in

Table 1. Simulation parameters.

Parameter	Value
Modulation type	BPSK
Carrier frequency, $f_c$	32 GHz
Coding frequency, $f_{code}$	10.23 Mcps
Data rate	1000 bps
Intermediate frequency	140 MHz
Sampling frequency, $f_s$	400.23 MHz
FLL bandwidth	20 Hz
PLL bandwidth	10 Hz
DLL bandwith	2 Hz
DLL correlator spacing	0.25 chip
Loop update time	1 ms
Initial velocity, v	1000 m/s
Initial distance, D	10 km
accelerated velocity, a	5 m/s2
Transmit power, $P_t$	30 dBm
Noise power, $P_{noise}$	−119.55 dbm

. To observe the performance of SI cancellation, the gain of SoI, and the measurement accuracy after SI cancellation, we conduct 1000 Monte Carlo simulations. According to the Friis formula in free space [46,47],

$${\displaystyle \frac{P_r}{P_t}}=G_t{G}_r{\left({\displaystyle \frac{\mathit{\lambda }}{4\pi D}}\right)}^2$$

(41)

under the condition set in Table 1, the path loss in free space is $P_{loss}=20lg\left({\displaystyle \frac{\mathit{\lambda }}{4\pi D}}\right)=142.74$ dB. Then, the receiving power at the RX antenna at the FD node can be obtained; that is, $P_r=30-142.74=-112.74$ dBm.

4.1. Convergence Performance

We evaluate the convergence performance of the proposed algorithm in this section. The simulation parameters are shown in

Table 2. Initial value for the simulation.

Parameter	Value
Angle $(({\theta }_t,{\phi }_t)/({\theta }_r,{\phi }_r))$	$({60}^{\circ },{60}^{\circ })/({40}^{\circ },{50}^{\circ })$
The powers of SoI, $P_r$	−112.74 dBm
The relative difference, $\zeta$	${10}^{-7}$
Initial w	Random value
Initial v	Random value
Initial $\alpha$	5
Initial $\mathit{\lambda }$	2
c	0.99

. We define the SI power as $P_{SI}$, which is in the range of −50 dBm∼30 dBm. Figure 3 and Figure 4 show the performance of the objective function and the relative differences and constraint violations of Problem (17). Figure 3 shows that the proposed algorithm can converge well within 140 iterations and that the relative difference $\zeta$ defined in Equation (39) is near zero.

4.2. SI Suppression Performance

The performance of the SI cancellation and gain of the SoI with the proposed algorithm is described in this section; it is measured by the power of the residual SI $P_{SI-res}$ and SoI $P_{SoI}$. We define the SI power as $P_{SI}$, which is in the range of −80 dBm ∼ 30 dBm. We let $G\left(x\right(t\left)\right)$ denote the power spectral density (PSD) of the signal $x\left(t\right)$. The PSDs of the SI, SoI, and noise floor are defined as $PS{D}_{SI}=G(x_{si}\left(t\right)+x_{noise-floor})$, $PS{D}_{SoI}=G(x_{SoI}\left(t\right)+x_{noise-floor})$, and $PS{D}_{noise-floor}=G\left(x_{noise-floor}\left(t\right)\right)$, respectively. We choose a number of groups of angles at random to verify the performance of the algorithm in various directions, as shown in

Table 3. Transmit and receive angle groups.

Angle $(({\mathit{\theta }}_{\mathit{t}},{\mathit{\phi }}_{\mathit{t}})/({\mathit{\theta }}_{\mathit{r}},{\mathit{\phi }}_{\mathit{r}}))$	Value
Group 1	$({60}^{\circ },{60}^{\circ })/({40}^{\circ },{50}^{\circ })$
Group 2	$({60}^{\circ },{60}^{\circ })/({-25}^{\circ },{-35}^{\circ })$
Group 3	$({-45}^{\circ },{-35}^{\circ })/({-25}^{\circ },{-27}^{\circ })$

:

Figure 5 shows the performance of $P_{SI-res}$ and the $SINR$ with varying SI power, different numbers of antennas, and different AoAs and AoDs. Figure 5a shows that SI can be effectively suppressed to an acceptable level with varying numbers of antennas and different AoAs and AoDs. The performance of SI cancellation degrades decreases with increase in the SI power. In addition, the power of the SI is always lesser than the power of the SoI, and it is even lesser than the noise floor in most cases. Figure 5b shows the results for the SINR after SI cancellation, which is defined by (13). As shown in Figure 5b, the SINR is greatly improved after SI cancellation. The PSD of the SI, SoI, transmitting signal, and noise floor when the AoD and AoA are set to $({\theta }_t={60}^{\circ },{\phi }_t={60}^{\circ })$ and $({\theta }_r={40}^{\circ },{\phi }_r={50}^{\circ })$ is shown in Figure 6 for the case in which the SI power is 30 dBm. Figure 6 shows that the SI can be effectively cancelled after the algorithm is applied. Figure 7 shows the results of the Tx and Rx gain in the direction of the AoD and AoA after SI cancellation, which is defined by (15). The results demonstrate that the gain of the SoI and the transmitted signal improves with an increasing number of antennas. Notably, the power of the SoI does not decrease with increasing SI power after the algorithm is applied. Furthermore, we conducted simulations to estimate the signal quality degradation caused by AoA estimation errors. From the expression of SINR defined by (13), only the term of ${\left|{\mathbf{w}}^H{\mathbf{H}}_r{x}_r\right|}^2$ incorporates AoA information. Consequently, we simulated the variation in Rx beamforming gain under different AoA estimation errors. The results shown in Figure 8 demonstrate that the Rx beamforming gain decreases with increasing estimation error, with more pronounced degradation observed in systems employing larger antenna arrays. However, as can be observed from the Figure 8, even when the AoA estimation errors reach 10 degrees, Rx beamforming gain maintains a relatively high level.

4.3. Measurement Performance

Measuring parameters such as angle, distance, speed, and acceleration are an essential feature of measurement systems. Specifically, angle measurement involves carrier phase measurement techniques, whereas distance, speed, and acceleration measurements rely on code phase measurements. Therefore, a key performance indicator for spatial filtering is its effect on measurement signals.

The number of TX and RX antennas of the FD-UAV node is set to $N_t=N_r=16$. We suppose that the FD-UAV node receives a TT&C signal in the direction of $(\theta ={60}^{\circ },\phi ={60}^{\circ })$ at 1 s–4 s and that the angle changes to $(\theta ={40}^{\circ },\phi ={50}^{\circ })$. This change in angle reveals the effect of the algorithm on the measurement results. The FD-UAV node transmits the TT&C signal at an angle of $(\theta ={60}^{\circ },\phi ={60}^{\circ })$. The powers of the noise floor and received signal at the RX antennas are $P_{noise}=-119$ dBm and $P_r=-112.74$ dBm, respectively. Figure 9, Figure 10 and Figure 11 show the measurement results of the FD-UAV system after SI cancellation. The proposed algorithm eliminates the influence of the SI, which has little influence on the measurement results. Moreover, compared with TDD mode, in which the measurement accuracy of the UAVs fluctuates greatly due to interruption of the tracking loop and a lack of continuous retracking, the distance and speed measurement accuracies of UAVs operating in FD mode are increased by 10 m and 1.5 m/s, respectively. The measurement accuracy is comparable to that of UAVs operating in FDD mode.

We conduct a simple UAV sensing network model to further examine the dynamic performance of the proposed algorithm. As shown in Figure 12, the FD node can transmit the measurement signal to Node 5 and receive the measurement signals from Node 2, Node 3, and Node 4 simultaneously. The actual scenario parameters are as in

Table 4. Initial motion state of the UAVs.

Nodes of UAVs	Initial Position [$\mathit{x},\mathit{y},\mathit{z}$] (m)	Initial Velocity [${\mathit{v}}_{\mathit{x}},{\mathit{v}}_{\mathit{y}},{\mathit{v}}_{\mathit{z}}$] (m/s)	Initial Acceleration ([${\mathit{a}}_{\mathit{x}},{\mathit{a}}_{\mathit{y}},{\mathit{a}}_{\mathit{z}}$]) (m/s2)	Motion Type
FD Node	[20, 20, 0]	[3, 3, 3]	[0, 0, 0]	uniform motion
Node 2	[100, 100, 0]	[0, 0, 0]	[0, 0, 0]	complex motion
Node 3	[0, 0, 0]	[0, 0, 0]	[0.04, 0.01, 0.05]	uniformly accelerated motion
Node 4	[50, 150, 70]	[−1, −4, 2]	[0, 0, 0]	uniform motion

.

We focus on the measurement performance of the FD node relative to Node 2, Node 3, and Node 4 in this simulation. The motion trajectories of the four UAVs are shown in Figure 13. The radial distance and radial velocity between the FD node and the other three nodes are shown in Figure 14. Figure 15 shows the measurement results of the FD UAV sensing network after SI cancellation. It can be seen that velocity error is within 0.5 m/s and the range error is within 0.15 m in different motion types.

5. Conclusions

In this paper, the joint TX and RX beamforming design of an FD UAV sensing system is studied. To address this challenging problem, a joint transmit and receive beamforming algorithm based on PDD is proposed, where the TX and RX beamformers are designed to maximize the Tx and Rx beamforming gains and suppress the residual SI. By introducing auxiliary variables to address the coupling problem between the TX and RX beamformers, the subproblems of each block can be solved in a closed form. The proposed joint TX-RX beamforming design algorithm based on the PDD algorithm converges to the stationary solution set of the original optimization problem. Ultimately, the simulation analysis demonstrates that the proposed joint TX-RX beamforming algorithm can effectively suppress up to 140 dB of SI with a limited number of antennas, yielding high-precision measurements in UAV sensor networks. Compared with that of the traditional frequency-division duplex (FDD) mode, the measurement accuracy is not decreased; compared with those of the time-division duplex (TDD) mode, the distance and speed measurement accuracies of the UAVs are increased by 10 m and 1.5 m/s, respectively. Furthermore, the dynamic UAV simulation based on real-world parameters thoroughly evaluates the dynamic performance of the proposed algorithm. For future research, hardware implementation and experimental validation of the proposed algorithm, full domain SI cancellation algorithm in view of more accurate residual SI model and saturation caused by analog SI are worth investigating.