Joint Power and Bandwidth Allocation in Collocated MIMO Radar Based on the Quality of Service Framework

Abstract

The simultaneous multi-beam working mode of the collocated multiple-input and multiple-output (MIMO) radar enables the radar to track multiple targets simultaneously. A joint power and bandwidth allocation algorithm in a collocated MIMO radar based on the quality of service (QoS) framework is proposed for the multi-target tracking problem with different threat levels. Firstly, a posterior Cramer–Rao lower bound (PCRLB) concerning the power and bandwidth is derived. In addition, the optimal objective functions of power and bandwidth are designed based on the QoS framework, and the problem is solved using the convex relaxation technique and the cyclical minimization algorithm. The numerical results show that the proposed algorithm has better tracking accuracy and achieves more reasonable resource allocation compared to strategies such as average allocation.

1. Introduction

1.1. Background and Related Studies

The multiple-input multiple-output (MIMO) radar has attracted increasing attention from the academic community [1,2,3,4]. The MIMO radar is a new system radar with the advantages of waveform diversity gain, low interception performance, and high Doppler resolution compared to the phased array radar [5]. The MIMO radar can be divided into two types: collocated and radar, respectively. Among them, the collocated MIMO radar, whose array spacing is small, can transmit orthogonal signals between different arrays at the same time, forming multiple equivalent beams in space, thus having the ability to track multiple targets simultaneously [6]. To achieve a better use of radar resources, a reasonable allocation of radar resources is needed. Therefore, radar resource management has achieved more and more attention.

By introducing the idea of a cognitive radar [7] into radar resource allocation, the closed-loop feedback in the tracking process can be realized. The common optimization principles are mainly divided into two kinds: (1) Maximize the target tracking accuracy with limited resources [8,9,10,11,12,13,14,15]. (2) Achieve the minimum consumption of resources in the condition of meeting performance requirements [16,17,18]. Reference [8] establishes a joint power and bandwidth allocation strategy and demonstrates that the joint allocation strategy can effectively improve the target tracking accuracy. Reference [9] proposes two different allocation schemes, decomposing the bivariate optimization problem into two univariate optimization problems, and the resource utilization rate shows an improvement compared with the conventional allocation strategy. Reference [10] proposes a joint power and bandwidth allocation algorithm, which effectively improves multi-target tracking performance. The paper uses posterior Cramer–Rao lower bound (PCRLB) as the performance index. For the multi-target tracking problem of the collocated MIMO radar, Reference [11] uses the weighted a posteriori Cramer–Rao lower bound as the performance index, which demonstrates better performance in the condition of limited power and bandwidth resources. Reference [12] introduces a joint power and beam allocation algorithm with the min–max principle and takes improving the worst tracking accuracy for multiple targets as the criterion, which effectively improves the worst tracking accuracy of the targets. Reference [13] proposes a resource allocation model based on the quality of service (QoS) framework. The model makes each target satisfy the constraints under the minimum accuracy demand to achieve maximum system efficiency. Reference [14] proposes a new resource allocation strategy based on the QoS framework for a multi-target tracking scenario. It realizes more reasonable resource allocation by making different targets converge to different tracking accuracies. A joint beam and power allocation method is developed to solve the multi-target tracking problem in [15], and a position-accuracy-based objective function is established to quantify the global tracking performance while considering the priority and threat level of the targets. A joint optimization model for transmit resource management and waveform selection is introduced in [16], which effectively improves the tracking accuracy of the grouped radars while reducing the consumption of resources and improving the radar stealth performance. A joint transmitter selection and resource management strategy is developed in [17], which improves the low intercept performance by optimizing parameters such as revisit interval, transmitter selection, and dwell time. A power and bandwidth allocation strategy based on low intercept performance is proposed in [18]. The predicted Bayesian Cramer–Rao lower bound (PCRLB) is used to evaluate the multi-target tracking performance, which effectively reduces power consumption. In Reference [19], the paper introduces a joint target assignment and resource optimization strategy for target tracking. Based on Reference [19], joint aperture and transmit resource allocation is proposed in [20], which improves stealth performance by allocating the aperture, power, and effective bandwidth. To better deal with a jamming situation, the strategy of transmitting resource allocation under jamming is studied in References [21,22].

1.2. Motivation and Main Contributions

The above works provide a foundation for the study of radar resource allocation techniques, but there are some shortcomings. The reference on joint power and bandwidth allocation mostly applies the min–max principle to establish the objective function, while the optimization model under this criterion will make the tracking accuracy of different targets gradually become identical. In the complex battlefield environment, the threat levels of different types of targets [23] are often different, thus putting higher requirements on radar resource allocation. Therefore, to make the joint power and bandwidth allocation strategy more reasonable, this paper proposes an algorithm based on the QoS framework. The main contributions of this paper are summarized as follows.

• A joint optimization model based on the QoS framework is established in this paper. Based on the closed-loop structure of cognitive tracking that is shown in Figure 1, the objective function is improved using the QOS framework. The model overcomes the shortcomings of the min–max principle and has better performance with different threat targets.
• To deal with different threat targets, a target threat assessment model is proposed to evaluate the threat level. This paper provides a more reasonable target threat level given method, which is more suitable for the actual battlefield scenario.
• An algorithm with convex relaxation and cyclical minimization techniques is proposed. Compared with the particle swarm optimization algorithm, the proposed algorithm has a shorter operation time and higher precision.
• A closed-loop structure of cognitive tracking is developed. As is shown in Figure 1, the target state is updated via a square root cubature Kalman filter (SCKF) [24], and then the cost function is composed of the PCRLB based on the QOS framework. Finally, the radar resource allocation results are fed back to the radar.

The rest of the paper is organized as follows. Section 2 introduces the system model. In Section 3, the optimization model is established. Section 4 proposes the improved algorithm. The simulation results and corresponding analysis are presented in Section 5. Finally, Section 6 concludes this paper.

2. System Model

For the subsequent analysis and simulation, the following assumptions were made for the radar operation mode and the target motion state:

• The radar was assumed to operate in the simultaneous multi-beam working mode, which can transmit multiple orthogonal wide beams and perform digital beam formation on the receiver side, and then extract measurement information;
• Each beam transmitted by the radar can only track one target, and the transmitted beam power and bandwidth can be controlled;
• There are Q targets in space, and each target is separated from the other.

2.1. Signal Model

Suppose that the signal transmitted [25] to the target q at time index k is as follows:

$$s_k^q(t)=\sqrt{P_k^q}{\tilde{s}}_k^q(t)e^{-j2\pi f_{c}t}$$

(1)

where $P_k^q$ is the transmit power and $f_c$ is the carrier frequency. ${\tilde{s}}_k^q\left(t\right)$ is the normalized complex envelope of the transmit signal. The effective bandwidth of the transmitted signal satisfies the following conditions:

$$\left|{\beta }_k^q\right|=\frac{{\displaystyle \int f^2{\left|{\tilde{s}}_k^q(f)\right|}^{2}df}}{{\displaystyle \int {\left|{\tilde{s}}_k^q(f)\right|}^{2}df}}$$

(2)

and the effective time duration is of the following form:

$$\left|T_k^q\right|=\frac{{\displaystyle \int t^2{\left|{\tilde{s}}_k^q(t)\right|}^{2}dt}}{{\displaystyle \int {\left|{\tilde{s}}_k^q(t)\right|}^{2}dt}}$$

(3)

where f and t denote the frequency and time domain, respectively.

The received signal of the radar can be expressed as follows:

$$u_k^q(t)={\gamma }_k^q\sqrt{{\alpha }_k^q{P}_k^q}{\tilde{s}}_k^q(t-{\tau }_k^q)e^{-j2\pi f_k^{q}t}+w_k^q(t)$$

(4)

where ${\gamma }_k^q$ is the target reflectivity. ${\alpha }_k^q\propto 1/r_k^q$ is the attenuation coefficient due to path loss. ${\tau }_k^q$ is the signal time delay. $f_k^q$ is the Doppler frequency shift. $w_k^q(t)$ is the zero-mean complex Gaussian white noise. The derivation of the Doppler frequency is given by the following:

$$f_k^q=-\frac{2}{{\lambda }_q}[{\dot{x}}_k^q(x_k^q-x)+{\dot{y}}_k^q(y_k^q-y)]/R_k^q$$

(5)

where ${\lambda }_q$ is the wavelength to target q. ${\dot{x}}_k^q \mathrm{and} {\dot{y}}_k^q$ indicate the velocity in the x-axis direction and the velocity in the y-axis direction, respectively. $x_k^q \mathrm{and} y_k^q$ indicate the position of the target in the x-axis direction and y-axis direction. $R_k^q$ indicates the distance between the target and the radar.

2.2. Target Motion Model

In this paper, we suppose that targets move in a two-dimensional plane with uniform acceleration and establish the equation of the state about the target. Therefore, the model [26,27] of the target is described as follows:

$${\mathit{X}}_k^q={\mathit{F}}_q{\mathit{X}}_{k-1}^q+{\mathit{w}}_k^q(t)$$

(6)

where state vector ${\mathit{X}}_k^q$ has the following form:

$${\mathit{X}}_k^q={[x_k^q,{\dot{x}}_k^q,{\ddot{x}}_k^q,y_k^q,{\dot{y}}_k^q,{\ddot{y}}_k^q]}^T$$

(7)

where each component represents the position, velocity, and acceleration of the target q in the x-axis direction and y-axis direction. ${\mathit{F}}_q \mathrm{and} {\mathit{w}}_k^q(t)$ can be expressed as follows:

$${\mathit{F}}_q={\mathit{I}}_2\otimes \left[\begin{array}{ccc}1& T_s& T_s{}^2/2\\ 0& 1& T_s\\ 0& 0& 1\end{array}\right]$$

(8)

where $\otimes$ indicates the Kronecker operator. $T_s$ is the time interval for sampling; ${\mathit{I}}_2$ denotes the 2 × 2 identity matrix. ${\mathit{w}}_k^q\sim {\rm N}(0,{\mathit{Q}}_q)$, which denotes the Gaussian white noise with zero-mean and covariance matrix ${\mathit{Q}}_q$:

$${\mathit{Q}}_q={\xi }_f{\mathit{I}}_2\otimes \left[\begin{array}{ccc}{T}_s{}^5/20& T_s{}^4/8& T_s{}^3/6\\ T_s{}^4/8& T_s{}^3/3& T_s{}^2/2\\ T_s{}^3/6& T_s{}^2/2& T_s\end{array}\right]$$

(9)

where ${\xi }_f$ denotes the process noise intensity.

2.3. Measurement Model

By processing the received signal, the measurement information can be extracted. The nonlinear measurement model [28] can be expressed as follows:

$${\mathit{Z}}_k^q=h({\mathit{X}}_k^q)+{\mathit{\upsilon }}_k^q$$

(10)

The nonlinear transformation $h({\mathit{X}}_k^q)$ is as follows:

$$h({\mathit{X}}_k^q)=[r_k^q,{\dot{r}}_k^q,{\theta }_k^q]$$

(11)

where

$$\left\{\begin{array}{l}{r}_k^q=\sqrt{x_k^q{}^2+y_k^q{}^2}\\ {\dot{r}}_k^q=({\dot{x}}_k^q{x}_k^q+{\dot{y}}_k^q{y}_k^q)/r_k^q\\ {\theta }_k^q=\mathrm{arc}\tan (y_k^q/x_k^q)\end{array}\right.$$

(12)

and where $r_k^q$ denotes the distance between target q and the radar. ${\dot{r}}_k^q$ indicates the radial velocity of the target. ${\theta }_k^q$ implies the azimuthal angle in two-dimensional space. The measurement error ${\mathit{\upsilon }}_k^q\sim \mathrm{N}(0,{\mathit{R}}_k^q)$ is in the form of the following:

$${\mathit{R}}_k^q=\mathrm{diag}({\sigma }_{r_k^q}^2,{\sigma }_{{\dot{r}}_k^q}^2,{\sigma }_{{\theta }_k^q}^2)$$

(13)

where ${\sigma }_{r_k^q}^2,{\sigma }_{{\dot{r}}_k^q}^2,{\sigma }_{{\theta }_k^q}^2$ represent the distance, velocity, and bearing measurement components, respectively [29], which are of the following form:

$$\left\{\begin{array}{l}{\sigma }_{r_k^q}^2\propto ({\alpha }_k^q{P}_k^q{\left|{\gamma }_k^q\right|}^2{\left|{\beta }_k^q\right|}^2{)}^{-1}\\ {\sigma }_{{\dot{r}}_k^q}^2\propto ({\alpha }_k^q{P}_k^q{\left|{\gamma }_k^q\right|}^2{\left|T_k^q\right|}^2{)}^{-1}\\ {\sigma }_{{\theta }_k^q}^2\propto ({\alpha }_k^q{P}_k^q{\left|{\gamma }_k^q\right|}^2/{\beta }_W{)}^{-1}\end{array}\right.$$

(14)

where $\propto$ is the direct proportion symbol and ${\beta }_W$ is the null-to-null beam width.

Equation (14) indicates that the measurement error of the distance is inversely proportional to the transmit power $P_k^q$ and the signal bandwidth ${\beta }_k^q$. Therefore, the tracking accuracy can be adjusted by adjusting the power and the bandwidth to achieve the desired tracking accuracy.

2.4. Derivation of PCRLB

The PCRLB provides a tight and lower bound for unbiased estimation [30], which is close to the tracking accuracy in actual application; so, it can be used as a measure of tracking accuracy. The PCRLB can predict the estimation error at the next moment, which can be used as a basis for resource allocation at the next moment, expressed as follows:

$$\mathit{J}({\mathit{X}}_k^q)={\mathit{J}}_P({\mathit{X}}_k^q)+{\mathit{J}}_D({\mathit{X}}_k^q)$$

(15)

where $\mathit{J}({\mathit{X}}_k^q)$ is the inverse of PCRLB and ${\mathit{J}}_P({\mathit{X}}_k^q)$ is the priori Fisher information matrix (FIM). ${\mathit{J}}_D({\mathit{X}}_k^q)$ is the data FIM, which can be expressed as follows:

$$\left\{\begin{array}{l}{\mathit{J}}_P({\mathit{X}}_k^q)=({\mathit{Q}}_q+{\mathit{F}}_q{J}^{-1}({\mathit{X}}_{k-1}^q){\mathit{F}}_q{}^T{)}^{-1}\\ {\mathit{J}}_D({\mathit{X}}_k^q)=({\mathit{H}}_k^q{)}^T({\mathit{R}}_k^q{)}^{-1}({\mathit{H}}_k^q)\end{array}\right.$$

(16)

In the case of low process noise, this can be approximated as follows:

$${\mathit{J}}_D({\mathit{X}}_k^q)={({\widehat{\mathit{H}}}_k^q)}^T{({\widehat{\mathit{R}}}_k^q)}^{-1}({\widehat{\mathit{H}}}_k^q)$$

(17)

where ${\widehat{\mathit{H}}}_k^q$ and ${\widehat{\mathit{R}}}_k^q$ represent the estimated Jacobian matrix and measurement covariance matrix, respectively. The inverse of the PCRLB is of the following form [31]:

$$\mathit{J}({\mathit{X}}_k^q)={({\mathit{Q}}_q+{\mathit{F}}_q{\mathit{J}}^{-1}({\mathit{X}}_{k-1}^q){\mathit{F}}_q{}^T)}^{-1}+{({\widehat{\mathit{H}}}_k^q)}^T{({\widehat{\mathit{R}}}_k^q(P_k^q,{\beta }_k^q))}^{-1}({\widehat{\mathit{H}}}_k^q)$$

(18)

Since the position error plays a major role in the state error, the PCRLB of the position error is used as the main basis for resource allocation. The PCRLB of the position error of the target $q$ can be expressed as follows:

$$M_k^q=\sqrt{\mathrm{tr}(J^{-1}({\mathit{X}}_k^q,P_k^q,{\beta }_k^q\left)\right)}$$

(19)

where tr(.) is the trace operation.

3. Optimization Model

3.1. Target Threat Assessment Model

Due to the different threat levels of targets in the battlefield environment, a new resource allocation strategy needs to be proposed to adapt to the real environment. The first step is to reasonably quantify the threat level of the target [32]. This section mainly considers the target distance, radial velocity, target type, and target heading angle. Then, we used the utility function theory and the idea of normalization to establish the threat level function and determine the threat level of each target.

3.1.1. Target Distance

The closer the target is, the shorter the time it takes to reach the radar. The threat level will be higher and the threat utility function is given by the following:

$${\psi }_{k,r}^q=\left\{\begin{array}{l}1,0\le r_k^q\le r_1\\ \exp [-l_r\times (r_k^q-r_1)],r_1\le r_k^q\le r_2\\ 0,r_k^q\ge r_2\end{array}\right.$$

(20)

where $r_1=50 \mathrm{km},r_2=80 \mathrm{km}$. Without considering the target’s height, $r_1$ is the minimum target detection distance in the plane and $r_2$ is the maximum target detection distance in the plane. $l_r={10}^{-2}$, which is an adjustable parameter.

3.1.2. Radial Velocity

The greater the radial velocity, the more difficult it is for the radar to track the target. The threat level will be higher and the threat utility function is given by the following:

$${\psi }_{k,v}^q=1-\exp [-l_v\times {\dot{r}}_k^q]$$

(21)

where $l_v={10}^{-3}$, which is an adjustable parameter.

3.1.3. Target Type

Different types of military targets have different combat missions, and the threat level to the radar is also different. The normalized function of target types is shown in

Table 1. The utility value of the target type.

Target Type	${\mathit{\psi }}_{\mathit{k},\mathit{t}}^{\mathit{q}}$
Bomber Group	0.8
Fighter Group	0.7
Bomber	0.6
Unknown Type	0.5
Fighter	0.4
Unmanned Aerial Vehicle (UAV)	0.3
Jammer	0.2
Early Warning Aircraft	0.1

[33].

3.1.4. Target Heading Angle

The target’s heading angle refers to the angle between the target’s velocity direction and the line between the target and the radar. The threat utility function is as follows:

$${\psi }_{k,{\theta }_j}^q=\exp [-l_{{\theta }_j}\times {\theta }_j{}^2]$$

(22)

where $l_{{\theta }_j}={10}^{-3}$, which is an adjustable parameter.

It is assumed that the assessment factors are independent of each other and do not influence each other. After calculating the threat utility function for each influencing factor, a vector of assessment metrics for each target is denoted as $({\psi }_{k,r}^q,{\psi }_{k,v}^q,{\psi }_{k,t}^q,{\psi }_{k,{\theta }_j}^q)$. A combined utility value for each objective is obtained as follows:

$$E_k^q={\rho }_r{\psi }_{k,r}^q+{\rho }_v{\psi }_{k,v}^q+{\rho }_t{\psi }_{k,t}^q+{\rho }_{{\theta }_j}{\psi }_{k,{\theta }_j}^q$$

(23)

where ${\rho }_r+{\rho }_v+{\rho }_t+{\rho }_{{\theta }_j}=1$. ${\rho }_r,{\rho }_v,{\rho }_t,{\rho }_{{\theta }_j}$ are the weights of each parameter and are acquired via expert experience. The threat level of each target is obtained via normalization:

$${\omega }_k^q=\frac{E_k^q}{{\displaystyle \sum _{j=1}^Q{E}_k^j}}$$

(24)

The target’s threat level vector is as follows:

$$W_k^q=[{\omega }_k^1,{\omega }_k^2\cdot \cdot \cdot {\omega }_k^Q]$$

(25)

In practical application scenarios, the threat assessment of multiple targets can be performed based on the above model. The threat level vector is determined using the utility function as well as the idea of normalization, which lays the foundation for the next step of resource allocation.

3.2. Objective Function

The cost function is often used to describe the cost that the target brings to the system, which can be rewarded and penalized by judging whether the tracking accuracy of the target meets the requirements. Based on the QoS framework, the corresponding optimal objective function is established under the condition that the threat level of the target is determined. The cost function of a single objective [14] is established as follows:

$${\phi }_k^q=\exp (\frac{M_k^q}{\eta }-1)-1$$

(26)

where $\eta$ is the preset tracking accuracy requirement. $M_k^q$ is the position error PCRLB of the target q. Assuming that the tracking accuracy varies over different distance ranges, the value $\eta$ is as follows:

$$\eta =\left\{\begin{array}{l}15,\hspace{1em}{r}_k^q\le 50 \mathrm{km}\\ 50,\hspace{1em}50 \mathrm{km}\le r_k^q\le 80 \mathrm{km}\\ 190,\hspace{1em}{r}_k^q\ge 80 \mathrm{km}\end{array}\right.$$

(27)

Based on the cost function of individual objectives and the established threat level, the global cost function can be obtained, i.e., the objective function is as follows:

$$F(P_k,{\beta }_k)={\displaystyle \sum _{q=1}^Q{\omega }_k^q{\phi }_k^q}={\displaystyle \sum _{q=1}^Q{\omega }_k^q[\exp (\frac{M_k^q}{{\eta }_q}-1)-1}]$$

(28)

where ${\omega }_k^q$ is the normalized weight of the threat level at the time k. $P_k \mathrm{and} {\beta }_k$ are the set of power and bandwidth at the time k, respectively.

4. Solution Algorithm

Our aim is to minimize the global cost function by optimizing power and bandwidth resources. Based on the above derivation of the objective function, the following optimization model can be developed as follows:

$$\begin{array}{l}\underset{P_k,{\beta }_k}{\min }F(P_k,{\beta }_k)\\ \mathrm{s}.\mathrm{t}. P_{\min }\le P_k^q\le P_{\max }\\ \hspace{1em}\hspace{1em}{\beta }_{\min }\le {\beta }_k^q\le {\beta }_{\max }\\ \hspace{1em}\hspace{1em}{1}_Q^T{P}_k=P_{\mathrm{total}}\\ \hspace{1em}\hspace{1em}{1}_Q^T{\beta }_k={\beta }_{\mathrm{total}}\end{array}$$

(29)

where $P_k=[P_k^1,P_k^2\cdot \cdot \cdot P_k^Q]$ is the power set. ${\beta }_k=[{\beta }_k^1,{\beta }_k^2\cdot \cdot \cdot {\beta }_k^Q]$ is the bandwidth set. $P_{\min } \mathrm{and} P_{\max }$ are the lower and upper limits of signal power, respectively. ${\beta }_{\min } \mathrm{and} {\beta }_{\max }$ are the lower and upper limits of the signal bandwidth, respectively. $P_{\mathrm{total}} \mathrm{and} {\beta }_{\mathrm{total}}$ are the total power and the total bandwidth of the radar, respectively.

Equation (30) shows that the optimization function has two variables, power and bandwidth. Since the bandwidth exists as a quadratic term in PCRLB, it is a nonconvex optimization problem. Using the convex relaxation technique, the problem is converted into a two-stage convex optimization problem. The algorithm processing flow chart is shown in Figure 2.

For the optimization model, this paper uses the convex relaxation and the cyclical minimization techniques for solving [9]. The processing flow is as follows:

Firstly, three temporary optimization values are set, which are $P_{k,\mathrm{tem}}$, ${\beta }_{k,\mathrm{tem}1}$, and ${\beta }_{k,\mathrm{tem}2}$, respectively. $P_{k,\mathrm{opt}} \mathrm{and} {\beta }_{k,\mathrm{opt}}$ are the final output optimization values.

Step 1: Initialization, assuming equal allocation, where ${\beta }_{k,\mathrm{tem}1}={\beta }_{average}$.

Step 2: Based on the assumptions in the first step, the optimization model for this problem can be redescribed as follows:

$$\begin{array}{l}\underset{P_k}{\min }F(P_k)\\ \mathrm{s}.\mathrm{t}. P_{\min }\le P_k^q\le P_{\max }\\ \hspace{1em}\hspace{1em}{1}_Q^T{P}_k=P_{\mathrm{total}}\end{array}$$

(30)

In this step, there is only a variable set because the bandwidth allocation is equal. The problem is converted into a minimization problem of the power resource. It is known from Reference [26] that the problem is a convex optimization problem. The interior point method was used to tackle the problem. Finally, the transient power distribution vector $P_{k,\mathrm{tem}}$ is obtained.

Step 3: Based on the transient power distribution vector $P_{k,\mathrm{tem}}$ obtained in the second step, ${\beta }_{k,\mathrm{tem}2}$ is obtained. Because the bandwidth exists as a quadratic term in the PCRLB, the convex relaxation technique is required. A new vector $L_k=[l_k^1,l_k^2\cdot \cdot \cdot l_k^Q]=[{\left|{\beta }_k^1\right|}^2,{\left|{\beta }_k^2\right|}^2\cdot \cdot \cdot {\left|{\beta }_k^Q\right|}^2]$ and factor ${\lambda }_1,{\lambda }_2$ are introduced to the process of constraints. Under the condition that the power variable is set as $P_{k,\mathrm{tem}}$, the problem can be converted into a minimization problem of vector $L_k$. The model can be redescribed as follows:

$$\begin{array}{l}\underset{L_k}{\min }F(L_k)\\ \mathrm{s}.\mathrm{t}. {\left|{\beta }_{\min }\right|}^2\le l_k^q\le {\left|{\beta }_{\max }\right|}^2\\ \hspace{1em}\hspace{1em}{1}_Q^T{L}_k={\lambda }_1{\beta }_{\mathrm{total}}\\ \hspace{1em}\hspace{1em}{1}_Q^T(L_k{)}^{1/2}\ge {\lambda }_2{\beta }_{\mathrm{total}}\end{array}$$

(31)

where ${\lambda }_1,{\lambda }_2$ are adjustable parameters in actual application. The specific constraints are as follows:

$$\left\{\begin{array}{l}-2Q{\beta }_{\max }^2+{\beta }_{\mathrm{total}}\le {\lambda }_1{\beta }_{\mathrm{total}}^2\le -2Q{\beta }_{\min }^2+{\beta }_{\mathrm{total}}\\ Q{\beta }_{\min }\le {\lambda }_2{\beta }_{\mathrm{total}}\le Q{\beta }_{\max }\end{array}\right.$$

(32)

The problem is converted into a convex optimization problem after convex relaxation and can be continued using the interior point method to obtain the transient bandwidth allocation vector ${\beta }_{k,\mathrm{tem}2}$ and make ${\beta }_{k,\mathrm{tem}1}={\beta }_{k,\mathrm{tem}2}$.

Step 4: Determine whether the objective function value satisfies the condition of iteration termination and skip to step 2 if it does not. Loop until the condition is satisfied and output the optimal power and bandwidth allocation results $P_{k,\mathrm{opt}}=P_{k,\mathrm{tem}}\mathrm{and} {\beta }_{k,\mathrm{opt}}={\beta }_{k,\mathrm{tem}1}$. To better understand the algorithm, the proposed algorithm is summarized in Algorithm 1.

Algorithm 1: Pseudo code of the proposed algorithm

Initialization: Fix ${\beta }_{k,\mathrm{tem}1}$ as average allocation, ${\beta }_{k,\mathrm{tem}1}={\beta }_{average}.P{A}_2=\mathrm{Inf}, P{A}_1=0$.

while $\left|P{A}_2-P{A}_1\right|>\epsilon$

(1) Optimize $P_k$ by the convex optimization tools on the condition of ${\beta }_k={\beta }_{k,\mathrm{tem}1}$, obtain the temporary optimization value $P_{k,\mathrm{tem}}$.

$P{A}_1=F(P_{k,\mathrm{tem}},{\beta }_{k,\mathrm{tem}1}).$

(2) Convert the problem into a minimization problem of vector $L_k$ by the convex relaxation technique $L_k=[l_k^1,l_k^2\cdot \cdot \cdot l_k^Q]=[{\left|{\beta }_k^1\right|}^2,{\left|{\beta }_k^2\right|}^2\cdot \cdot \cdot {\left|{\beta }_k^Q\right|}^2].$

(3) Optimize $L_k$ by the convex optimization tools on the condition of $P_k=P_{k,\mathrm{tem}}$, obtain the temporary optimization value ${\beta }_{k,\mathrm{tem}2}.$

$P{A}_2=F(P_{k,\mathrm{tem}},{\beta }_{k,\mathrm{tem}2}), {\beta }_{k,\mathrm{tem}1}={\beta }_{k,\mathrm{tem}2}.$

end while

Output: $P_{k,\mathrm{opt}}=P_{k,\mathrm{tem}}, {\beta }_{k,\mathrm{opt}}={\beta }_{k,\mathrm{tem}1}.$

5. Results

To prove the effectiveness of the proposed algorithm and model, corresponding simulation experiments are designed in this paper. The specific motion parameters of the target are shown in

Table 2. Parameters of target motion.

Target	Location (km)	Velocity (m/s)	Acceleration (m2/s)
Target 1	(10, 40)	(−100, −10)	(0, −5)
Target 2	(30, 50)	(100, 200)	(−10, 0)
Target 3	(50, 80)	(−100, 0)	(−5, 2)

. The relative positions of the target and the radar in space are shown in Figure 3. The impact of the target threat level on the algorithm performance is mainly studied here. This simulation assumes two operational scenarios, i.e., two cases with the same and different target threat levels. In the first scenario, the threat level is the same, the PCRLB values under different allocation strategies are compared, and the resource allocation under the QoS framework is analyzed. The second scenario is a scenario with different threat levels of the targets, and the effect on resource allocation is observed. The simulation parameters are set as follows, assuming the effective time width $T_k^q=0.01 \mathrm{ms}$, the total radar transmitting power $P_{\mathrm{total}}=10 \mathrm{kW}$, and the total effective bandwidth ${\beta }_{\mathrm{total}}=100 \mathrm{MHz}$, where $P_{\min }=0.1 P_{\mathrm{total}}$ and $P_{\mathrm{m}\mathrm{ax}}=0.8 P_{\max }$, ${\beta }_{\min }=0.1 {\beta }_{\mathrm{total}}$ and ${\beta }_{\max }=0.8 {\beta }_{\mathrm{total}}$, ${\sigma }_{r_k^q}=100 \mathrm{m}$, ${\sigma }_{{\dot{r}}_k^q}=10 \mathrm{m}/\mathrm{s}$, and ${\sigma }_{{\theta }_k^q}=0.1 \mathrm{rad}$. The sampling time $T_s=1 \mathrm{s}$. The number of Monte Carlo experimental simulations was 100, and the target tracking accuracy is defined as follows:

$${\mathrm{RMSE}}_q^k=\sqrt{\frac{1}{\mathrm{N}}{\displaystyle \sum _{i=1}^N[{(x_q^k-{\widehat{x}}_{q,i}^k)}^2+{(y_q^k-{\widehat{y}}_{q,i}^k)}^2]}}$$

(33)

where ($x_q^k$,$y_q^k$) and (${\widehat{x}}_{q,i}^k$,${\widehat{y}}_{q,i}^k$) are the true values of the target position and the estimated value of the target position in the i-th Monte Carlo experimental simulation, respectively, and N is the number of Monte Carlo experiments.

5.1. Same Target Threat Level

Assuming the threat level is the same for each target, the threat level vector is $W_k^q=[1/3,1/3,1/3]$, and the average allocation strategy, the min–max strategy, and the QoS-framework-based allocation strategy are simulated, respectively. Figure 4 shows the PCRLB and RMSE of each target under different allocation strategies, and the QoS-framework-based allocation strategy has the best performance compared with the average allocation strategy, which can effectively improve the tracking accuracy of each target. Figure 4a shows that the predetermined tracking accuracy of the three targets is not satisfied under the average allocation strategy. Figure 4b shows that the tracking accuracy of different targets under the min–max strategy tends to be consistent, and the tracking accuracies of target 2 and target 3 eventually tend to be consistent; target 1 maintains high tracking accuracy with minimum consumption due to its proximity to the target. However, only the tracking accuracy of target 3 meets the requirement under this strategy, and the tracking accuracy of the rest of the targets does not meet the requirement, which cannot realize the differentiated allocation of radar resources. From Figure 4c, under the QoS-based allocation strategy, the tracking accuracy of each target can meet the preset tracking accuracy requirement, which makes the resources achieve a more reasonable allocation.

Figure 5 shows the results of the allocation under the min–max strategy, where the resources are mainly allocated to target 3 because target 3 has the lowest tracking accuracy, thus making the tracking accuracies of target 2 and target 3 consistent in the end. Since target 1 is closer to the radar, it still has higher tracking accuracy with the minimum power and bandwidth resources allocated, but still does not meet the preset tracking accuracy requirement. Figure 6 represents the resource allocation results based on the QoS framework policy. The resource allocation result is not only determined by the distance between the target and the radar, but also depends on the tracking accuracy and the target threat level. Since the bandwidth exists in the PCRLB expression in the form of a quadratic term, the effect of bandwidth on resource allocation is greater than that of power. Additionally, from the power resource allocation results, the gap of allocated power resources among the targets is small, and the resource allocation gap is mainly reflected in the bandwidth resource allocation. From Figure 6a, more resources are allocated to target 1 because the tracking accuracy requirement of target 1 is relatively high. Additionally, the resources allocated to target 1 are gradually less because of the movement of target 1 toward the radar. As target 2 moves away from the radar, the power and the bandwidth resources allocated to target 2 gradually increase in pairs to meet the preset tracking accuracy requirements. Since the accuracy requirement of target 3 is lower and gradually closer to the radar, the tracking accuracy requirement can still be met with relatively fewer resources allocated to target 3.

5.2. Different Target Threat Levels

The simulation results of the average allocation policy and the min–max allocation policy are consistent for the same and different target threat levels. Therefore, this section only performs the simulation based on the QoS framework allocation policy and the simulation results in the two cases, firstly assuming that the threat level vector $W_k^q=[0.2,0.2,0.6]$.

Combined with Figure 7, target 3 achieves the preset tracking accuracy requirement for 15 frames. Additionally, in Figure 4c, target 3 achieves the preset accuracy requirement for 25 frames. This indicates that when the threat level of target 3 increases, the resources allocated to target 3 correspondingly increase and the convergence speed improves. Comparing the resource allocation results in Figure 6, the resources allocated to target 3 are significantly increased in Figure 8. The purpose of the rational allocation of resources according to the different threat levels of targets is achieved, and the rationality of resource allocation is improved.

5.3. Algorithm Comparison and Analysis

To verify the algorithm performance, the optimization model was tackled by the proposed algorithm and the particle swarm optimization algorithm. To better compare the tracking accuracy of the two algorithms, the sum of the RMSE is proposed, which is as follows:

$${\mathrm{RMSE}}_{sum}^k={\displaystyle \sum _{q=1}^Q\sqrt{\frac{1}{\mathrm{N}}{\displaystyle \sum _{i=1}^N[{(x_q^k-{\widehat{x}}_{q,i}^k)}^2+{(y_q^k-{\widehat{y}}_{q,i}^k)}^2]}}}$$

(34)

where ($x_q^k$,$y_q^k$) and (${\widehat{x}}_{q,i}^k$,${\widehat{y}}_{q,i}^k$) are the true values of the target position and the estimated value of the target position in the i-th Monte Carlo experimental simulation, respectively, and N is the number of Monte Carlo experiments.

The results were obtained on the condition of the same threat level and the QOS framework. Figure 9a shows that the sum of the RMSE of the proposed algorithm is less than the particle swarm optimization algorithm, which means it has higher precision. Figure 9b indicates that the proposed algorithm has a shorter runtime, and the algorithm is suitable to be implemented for practical applications.

As for the proposed algorithm, the interior point method is adopted, and its computational complexity is $𝒪$$(Q^{3.5}Log(1/\epsilon ))$. At each iteration, the interior point method is used twice, and the number of iterations for the cyclical minimization algorithm is assumed to be $i$. Thus, we can conclude that the total computational complexity of the proposed tracking system is $𝒪$($2i{Q}^{3.5}Log(1/\epsilon )$).

6. Conclusions

To address the multi-target tracking problem, a joint power and bandwidth allocation algorithm fixing the QoS framework is proposed. After deriving the PCRLB, the objective function is established based on the QoS framework and solved using the convex relaxation technique and the cyclical minimization algorithm.

The results show that the allocation strategy based on the QoS framework can effectively improve the radar tracking accuracy compared with the average allocation strategy and the min–max strategy. Moreover, it can allocate resources according to the target threat level, which effectively enhances the radar’s operational effectiveness to cope with the increasingly complex battlefield environment. It also overcomes the drawback that the tracking accuracy eventually tends to be consistent in the min–max strategy and achieves more flexible resource allocation.