Robust Multiplexing Waveform Design for Integrated OFDM Radar and Communication via Complex Weight Optimization

Abstract

Compared to traditional single orthogonal frequency division multiplexing (OFDM) radar and OFDM communication systems, integrated OFDM radar and communication systems have the advantages of improved sharing of scarce spectrum resources, a simple hardware structure and a reduced interference between signals. In this paper, a constraint relaxation-based robust OFDM multiplexing waveform (ROW-CR) design method for integrated radar and communication systems is proposed. Considering the influence of correlated clutter and jamming signals, transmission and reception models of different system platforms are established by allocating subcarrier complex weights of different antenna signals so as to determine radar conditional mutual information and communication channel capacity parameters. Meanwhile, a restricted closed model is introduced through the limit range of the target frequency response. Then, a robust OFDM waveform optimization problem for integrated radar and communication systems is constructed by the minimax criterion, and the closed-form solution is obtained by adopting an improved method based on trace function properties and constraint relaxation, resulting in a better radar and communication performance trade-off. In addition, a parameter hierarchical optimization-based robust OFDM waveform (ROW-PHO) design method is further explored to reduce the computational complexity, and this method can also ensure a low system performance loss. Finally, the numerical simulation results verify the effectiveness of the proposed methods.

1. Introduction

Radar and communication are two typical functions widely used in modern electronic equipment systems, which shoulder the tasks of target detection and information transmission between devices, respectively [1,2]. However, with the increasing demand for higher bandwidth and data rates and the rapid embracement of wireless devices in the Internet of Things (IoT), the scarcity of the spectrum has become a key issue. Joint communication and radar sensing is a new research field which integrates the two functions into one system by sharing most hardware and signal processing modules [3,4]. The integrated system can adopt various optimization schemes (such as waveform optimization and dynamic spectrum sensing) under the same hardware platform to realize spectrum sharing, so that radar and communication can work normally in the same frequency band, thus improving the spectrum efficiency, energy efficiency and hardware efficiency of the system [5,6]. Furthermore, the performance of radar target detection and communication information transmission can be improved by mutual assistance and mutual gain [7,8]. Therefore, the design of integrated radar and communication systems has become a research hotspot in recent years. They are widely used in transportation systems [9], indoor positioning based on Wi-Fi [10] and unmanned aerial vehicle (UAV) monitoring networks [11].

For radar and communication spectrum sharing, multi-carrier waveforms are considered as one of the best candidates for the integrated system [12,13], which can bring many advantages compared to single-carrier waveforms in radar systems [14,15]. Orthogonal frequency division multiplexing (OFDM) is one of the most popular multi-carrier transmission technologies, and has been widely used in wireless communication and radar fields in recent years [16,17]. The existing research shows that OFDM has high spectral efficiency and it can effectively combat inter-symbol interference by converting frequency-selective channels into multiple parallel flat fading channels. Therefore, the waveform optimization design and received signal processing based on integrated OFDM radar and communication systems have been deeply studied [18,19,20]. In order to improve the communication channel capacity and radar performance, Xu et al. [21] studied conditional mutual information (MI) between the impulse response of random targets and echo signals, but it did not optimize the integrated waveform. In [22,23], the detection probability and MI were maximized by optimizing the OFDM radar waveform under the minimum capacity constraint of the communication system. However, these algorithms do not focus on the power allocation of radar and communication dual-function systems and ignore the influence of signal-related clutter. Paper [24] puts forward an improved method to suppress the amplitude of pilot symbols. This method sets and allocates the guard interval in an improved way, which effectively suppresses the influence of OFDM cyclic prefixes and pilot sidelobes, but its detection performance is poor due to the large signal energy loss.

In order to better allocate the limited transmission power in integrated radar and communication systems based on OFDM, several waveform design schemes have been proposed in the literature [25,26]. The method in [27] implements an optimized waveform design through OFDM subcarrier assignment and power allocation, but the robustness of the system is not considered. Paper [28] introduces an adaptive subcarrier transmission power allocation method based on an OFDM integrated waveform, which can improve the radar detection performance and communication channel capacity of the integrated system, but its channel frequency response is directly defined as a fixed value and thus its applicability is not wide. In order to improve the performance of radar target parameter estimation and the communication transmission rate at the same time, subcarrier power allocation based on multi-target parameter design is proposed in [29,30], but the influence of related clutter is not considered. In addition, Chiriyath et al. [31,32] realize two applications by separating the frequency range within the specified bandwidth. This scheme uses different frequency bands to realize different transmission power allocations and explores the balanced performance of radar and communication systems based on the joint optimization design of the target estimation rate and the communication information rate, but its frequency band utilization rate is low. The method in [33] achieves a robust OFDM shared waveform design through power allocation without considering the effect of clutter, and the optimization problem solving method is also relatively traditional. In fact, a good design scheme should combine theory with practical applications and consider the factors that affect the system performance as comprehensively as possible [34,35].

In this paper, in order to achieve better system practicability and improve the comprehensive performance of radar and communication, we comprehensively consider the influence of uncertain channel frequency responses and signal clutter on the integrated system and introduce a robust OFDM multiplexing waveform design scheme for integrated radar and communication systems by constructing a joint optimization model. In addition, a waveform design method based on parameter hierarchical optimization is used to reduce the computational complexity. The main contributions of this paper are summarized as follows:

(1) Considering the influence of clutter and jamming signals, we establish the joint optimization problem of radar and communication through the radar conditional mutual information and communication channel capacity under the integrated system. Furthermore, we construct restricted closed sets (RCS) of impulse response and introduce them into the joint optimization problem to realize a robust OFDM multiplexing waveform design. Compared to existing methods, it can achieve a better radar and communication performance trade-off and has better practical environmental adaptability.

(2) According to the characteristics of objective function in the joint optimization problem, we introduce the concept of a trace function. Combining the properties of the trace function and the characteristics of the objective function, an improved method based on model transformation and constraint relaxation is adopted to obtain the closed-form solution of the optimization problem. This method is a novel approach for solving non-convex joint optimization problems, solving the problem of the low practicability of existing methods and achieving better radar and communication performance.

(3) We further introduce a waveform design method based on parameter hierarchical optimization. According to the characteristics of the reconstructed joint optimization problem, the subcarrier amplitude vector of the communication signal is designed first, then the subcarrier weight vector of the radar signal is designed. The numerical results show that this method greatly reduces the computational complexity of waveform design with a slight performance loss.

The remainder of this paper is organized as follows. In Section 2, the system model and problem description are introduced. In Section 3, the robust OFDM multiplexing waveform design schemes are proposed. In Section 4, several numerical simulations are provided. Finally, conclusions are drawn in Section 5.

2. System Model and Problem Description

Consider the scene model of an OFDM integrated radar and communication system as shown in Figure 1, which includes an integrated platform, a target and a communication user. Among them, the integrated platform can carry out radar target detection and communication information transmission simultaneously, and the communication user can receive and transmit communication signals to the integrated platform. In order to improve the spectrum efficiency, we consider that radar and communication systems coexist in the same frequency band. In the considered scenario, the transmitter of the integrated platform transmits OFDM radar waveforms to hostile targets and OFDM communication waveforms to communication users [36].

Assume that the integrated system has N transmitting antennas, and its radar waveform and communication waveform use OFDM multicarrier signals with $N_c$ subcarriers. In terms of OFDM radar signals, it can be quickly generated by the inverse discrete Fourier transform (IDFT) matrix of the frequency domain subcarrier complex-valued weight sequence [37]. Then, the discrete matrix formula of an OFDM radar signal can be expressed as

$${\mathit{s}}_r={\mathit{F}}_{N_c}^{-1}{\mathit{W}}_r$$

(1)

where ${\mathit{F}}_{N_c}^{-1}$ is an IDFT matrix, which is represented as

$${\mathit{F}}_{N_c}^{-1}=\frac{1}{\sqrt{N_c}}{\left[e^{j2\pi \frac{\left(i-1\right)\left(j-1\right)}{N_c}}\right]}_{N_c\times N_c},i,j=1,2,\cdots ,N_c$$

(2)

Furthermore, ${\mathit{W}}_r$ represents a matrix composed of complex weight vectors corresponding to all subcarriers, which can be written as

$${\mathit{W}}_r={\left[\begin{array}{cccc}{w}_r^1\left(0\right)& w_r^1\left(1\right)& \cdots & w_r^1\left(N_c-1\right)\\ \\ w_r^2\left(0\right)& w_r^2\left(1\right)& \cdots & w_r^2\left(N_c-1\right)\\ ⋮& ⋮& \ddots & ⋮\\ w_r^N\left(0\right)& w_r^N\left(1\right)& \cdots & w_r^N\left(N_c-1\right)\end{array}\right]}^T$$

(3)

where $w_r^n\left(m\right)$ is the complex-valued weight corresponding to the $m\mathrm{th}$ subcarrier of the $n\mathrm{th}$ antenna. Similarly, the matrix formula in discrete form of the OFDM communication signal is expressed as

$${\mathit{s}}_c={\mathit{F}}_{N_c}^{-1}{\mathit{A}}_c$$

(4)

where ${\mathit{A}}_c$ represents a matrix composed of complex amplitude value vectors corresponding to all subcarriers, which can be represented as

$${\mathit{A}}_c={\left[\begin{array}{cccc}{a}_c^1\left(0\right)& a_c^1\left(1\right)& \cdots & a_c^1\left(N_c-1\right)\\ \\ a_c^2\left(0\right)& a_c^2\left(1\right)& \cdots & a_c^2\left(N_c-1\right)\\ ⋮& ⋮& \ddots & ⋮\\ a_c^N\left(0\right)& a_c^N\left(1\right)& \cdots & a_c^N\left(N_c-1\right)\end{array}\right]}^T$$

(5)

where $a_c^n\left(m\right)$ denotes the amplitude value of the $m\mathrm{th}$ subcarrier corresponding to the $n\mathrm{th}$ antenna. It is worth noting that in order to eliminate inter-symbol interference (ISI), a guard interval (GI) should be inserted before all OFDM signal blocks. The traditional method is to add a cyclic prefix (CP) directly, but the introduction of a CP in radar signal processing will produce high CP sidelobes and affect the target detection performance [38]. Therefore, this paper adopts the improved method in reference [38] to obtain the GI data corresponding to all OFDM symbols. This method can improve the detection performance of radar and achieve stable communication synchronization. After up-conversion, the time-domain transmission signal of each antenna can be obtained. Based on this, the radar baseband signal transmitted on the antenna can be expressed as

$$s_r^n\left(t\right)=\frac{1}{\sqrt{N_c}}\sum _{m=0}^{N_c-1}{w}_r^n\left(m\right)e^{j2\pi m\Delta ft}\mathrm{rect}\left(\frac{t}{T_s}\right)$$

(6)

where $T_s$ denotes the symbol period of OFDM, $\Delta f=1/T_s$ denotes the frequency interval between two adjacent subcarriers and $\mathrm{rect}(\ast )$ denotes the rectangular window function. Likewise, the communication baseband signal transmitted on the $n\mathrm{th}$ antenna can be written as

$$s_c^n\left(t\right)=\frac{1}{\sqrt{N_c}}\sum _{m=0}^{N_c-1}{a}_c^n\left(m\right)e^{j2\pi m\Delta ft}$$

(7)

According to the system scenario in Figure 1, the time domain signal of the $n\mathrm{th}$ antenna received by the integrated platform receiver can be described as

$$\begin{array}{cc}\hfill y_{rc}^n\left(t\right)=& s_r^n\left(t\right)\ast h_r\left(t\right)+s_c^n\left(t\right)\ast h_r\left(t\right)\hfill \\ \hfill & +s_r^n\left(t\right)\ast h_{cr}\left(t\right)+s_c^n\left(t\right)\ast h_{cc}\left(t\right)+n_{rc}^n\left(t\right)\hfill \end{array}$$

(8)

where $h_r$, $h_{cr}$ and $h_{cc}$, respectively, represent the impulse responses from the integrated system platform to target and then to the integrated system platform (marked as the surveillance channel), the clutter response caused by radar signals and the clutter response caused by communication signals; ∗ denotes the convolution operator; and $n_{rc}^n\left(t\right)$ is complex additive white Gaussian noise (AWGN) with zero mean.

Suppose that $H_r$, $H_{cr}$ and $H_{cc}$, respectively, represent the frequency response of the surveillance channel, the clutter frequency response caused by radar signals and the clutter frequency response caused by communication signals. It is worth mentioning that $H_{cr}$ and $H_{cc}$ are zero-mean Gaussian random processes, which can be obtained by the power spectral density of clutter caused by different signals. Then, according to the description in [39], the matrix formula of the discrete form of the received signal in (8) can be formulated as

$$\begin{array}{cc}\hfill {\mathit{y}}_{rc}=& {\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_r\right)\odot diag\left({\mathit{l}}_{rt}^{1/2}\right)\right]{\mathit{W}}_r\hfill \\ \hfill & +{\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_r\right)\odot diag\left({\mathit{l}}_{rt}^{1/2}\right)\right]{\mathit{A}}_c\hfill \\ \hfill & +{\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_{cr}\right)\odot diag\left({\mathit{l}}_{rt}^{1/2}\right)\right]{\mathit{W}}_r\hfill \\ \hfill & +{\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_{cc}\right)\odot diag\left({\mathit{l}}_{rt}^{1/2}\right)\right]{\mathit{A}}_c+{\mathit{n}}_{rc}\hfill \end{array}$$

(9)

where ${\mathit{H}}_r$, ${\mathit{H}}_{cr}$ and ${\mathit{H}}_{cc}$, respectively, represent an $N_c\times 1$ vector composed of the $H_r$, $H_{cr}$ and $H_{cc}$ of all subcarriers, $\mathrm{diag}(\ast )$ represents a diagonal matrix, ⊙ represents the Hadamard product, ${\mathit{n}}_{rc}$ represents a vector composed of the power spectral density of $n_{rc}$ and ${\mathit{l}}_{rt}$ represents a vector consisting of the propagation losses of surveillance channels corresponding to all subcarriers [40]. Then, the propagation loss of the surveillance channel corresponding to the mth subcarrier can be expressed as

$${\mathit{l}}_{rt}\left(m\right)=\frac{G_t{G}_r{\left[\lambda \left(m\right)\right]}^2}{{\left(4\pi \right)}^3{d_1}^4}$$

(10)

where $G_t$ is the transmit antenna gain of the integrated system platform, $G_r$ is the receiving antenna gain of the integrated system platform, $\lambda \left(n\right)$ is the wavelength in the nth subcarrier and $d_1$ represents the distance between the integrated system platform and the detection target. In addition, the time domain signal of the nth antenna received by a single communication receiver can be described as

$$\begin{array}{cc}\hfill y_c^n\left(t\right)=& s_r^n\left(t\right)\ast h_{dp}\left(t\right)+s_c^n\left(t\right)\ast h_{dp}\left(t\right)\hfill \\ \hfill & +s_r^n\left(t\right)\ast h_c\left(t\right)+s_c^n\left(t\right)\ast h_c\left(t\right)+n_c^n\left(t\right)\hfill \end{array}$$

(11)

where $h_{dp}$ represents the response from the integrated system platform to the communication receiver (marked as the communication channel), $h_c$ represents the impulse response from the integrated system platform to the target and then to the communication receiver (marked as the hybrid channel) and $n_c\left(t\right)$ is complex additive white Gaussian noise (AWGN) with zero mean. Similarly, the matrix formula of the discrete signal of (11) can be formulated as

$$\begin{array}{cc}\hfill {\mathit{y}}_c=& {\mathit{F}}_{N_c}^{-1}{\mathit{l}}_{rc}^{1/2}{\mathit{W}}_r+{\mathit{F}}_{N_c}^{-1}{\mathit{l}}_{rc}^{1/2}{\mathit{A}}_c\hfill \\ \hfill & +{\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_c\right)\odot diag\left({\mathit{l}}_c^{1/2}\right)\right]{\mathit{W}}_r\hfill \\ \hfill & +{\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_c\right)\odot diag\left({\mathit{l}}_c^{1/2}\right)\right]{\mathit{A}}_c+{\mathit{n}}_c\hfill \end{array}$$

(12)

where ${\mathit{H}}_c$ represents a vector consisting of the frequency response of the hybrid channel corresponding to all subcarriers and ${\mathit{l}}_{rc}$ and ${\mathit{l}}_c$, respectively, represent vectors consisting of the propagation losses of the communication channel and the hybrid channel corresponding to all subcarriers. The propagation loss of the communication channel and hybrid channel corresponding to the mth subcarrier can be represented as

$${\mathit{l}}_{rc}\left(m\right)=\frac{G_t{G}_c{\left[\lambda \left(m\right)\right]}^2}{{\left(4\pi \right)}^2{d_2}^2}$$

(13)

$${\mathit{l}}_c\left(m\right)=\frac{G_t{G}_c{\left[\lambda \left(m\right)\right]}^2}{{\left(4\pi \right)}^3{d_1}^2{d_3}^2}$$

(14)

where $G_c$ is the receiving antenna gain of the communication receiver, $d_2$ is the distance between the integrated system platform and the communication receiver and $d_3$ is the distance between the target and the communication receiver.

Radar waveform design based on information theory was first proposed by Bell [39], in which conditional mutual information was used as the performance index of target estimation. Similarly, for the proposed integrated system, given the transmitted signal ${\mathit{s}}_r$, we use the mutual information between ${\mathit{y}}_{rc}$ and ${\mathit{H}}_c$ to describe the amount of information obtained from the radar received signal, thus evaluating the performance of target estimation. The conditional mutual information can be expressed as

$$\begin{array}{cc}\hfill I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right)& =\sum _{m=0}^{N_c-1}{log}_2\left[1+{\left|{\mathit{H}}_r\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2/\phantom{1+{\left|{\mathit{H}}_r\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2\left({\left|{\mathit{H}}_r\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2\right.}\left({\left|{\mathit{H}}_r\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2\right.\right.\hfill \\ \hfill & \left.\left.+{\left|{\mathit{H}}_{cr}\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2+{\left|{\mathit{H}}_{cc}\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2\right)\right]\hfill \end{array}$$

(15)

where $\left|\right|*\left|\right|$ represents the norm of a vector, ${\mathit{W}}_{r,m}$ represents the weight vector corresponding to the mth subcarrier of radar signal, that is, the mth row of matrix ${\mathit{W}}_r$, and ${\mathit{A}}_{c,m}$ represents the amplitude vector corresponding to the mth subcarrier of the communication signal, that is, the mth row of matrix ${\mathit{A}}_c$.

In terms of communication, the communication channel capacity is an important performance metric. In the frequency selective fading channel, the communication channel capacity can be improved by reasonably allocating limited signal transmission power [41]. Therefore, we adopt the communication channel capacity to characterize the performance of communication transmission efficiency in proposed system, and the communication channel capacity can be written as

$$\begin{array}{cc}\hfill C\left({\mathit{A}}_c,{\mathit{W}}_r\right)& =\sum _{m=0}^{N_c-1}\Delta f{log}_2\left[1+{\left|{\mathit{l}}_{rc}^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2/\phantom{{\left|{\mathit{l}}_{rc}^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2\left({\left|{\mathit{l}}_{rc}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2\right.}\left({\left|{\mathit{l}}_{rc}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2\right.\right.\hfill \\ \hfill & \left.\left.+{\left|{\mathit{H}}_c\left(m\right)\right|}^2{\left|{\mathit{l}}_c^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2+{\left|{\mathit{H}}_c\left(m\right)\right|}^2{\left|{\mathit{l}}_c^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_c\left(m\right)\right|}^2\right)\right]\hfill \end{array}$$

(16)

3. Robust OFDM Multiplexing Waveform Design

In this section, two OFDM multiplexing waveform design schemes under the proposed integrated system platform are explored, and the implementation flow is shown in Figure 2. For Scheme 1, we propose a robust OFDM multiplexing waveform design method based on constraint relaxation (ROW-CR). According to the restricted closed model and optimization parameters, the joint waveform optimization problem is constructed and its closed-form solution is realized by combining the properties of the trace function and partial constraint relaxation. Finally, the radar and communication weight parameters are restored. For Scheme 2, we first reconstruct the joint optimization problem and then propose a robust OFDM waveform design method based on parameter hierarchical optimization (ROW-PHO), which greatly reduces the computational complexity with a lower performance loss.

3.1. Constraint Relaxation-Based Robust OFDM Waveform (ROW-CR) Design

In order to obtain the conditional mutual information in (15) and (16), it is necessary to calculate the values of ${\mathit{H}}_r$ and ${\mathit{H}}_c$ effectively. However, ${\mathit{H}}_r$ and ${\mathit{H}}_c$ are uncertain in practical applications. In order to make the designed waveform more adaptable, we construct restricted closed sets (RCSs) of frequency responses of different channels according to the limit range of different parameters, which can be obtained from field measurements and propagation modeling [42,43]. Then, the constructed RCSs of ${\mathit{H}}_r$ and ${\mathit{H}}_c$ can be formulated as

$${\prod }_r=\left\{{\mathit{H}}_r:0<{\mathit{V}}_r\le {\mathit{H}}_r\le {\mathit{U}}_r\right\}$$

(17)

$${\prod }_c=\left\{{\mathit{H}}_c:0<{\mathit{V}}_c\le {\mathit{H}}_c\le {\mathit{U}}_c\right\}$$

(18)

where ${\mathit{V}}_r$ and ${\mathit{U}}_r$ represent the lower bound and upper bound of ${\mathit{H}}_r$ and ${\mathit{V}}_c$ and ${\mathit{U}}_c$ represent the lower bound and upper bound of ${\mathit{H}}_c$, their sizes are $N_c\times 1$. In order to make the proposed radar and communication multiplexing waveform design more robust, we need to consider the system performance under the worst frequency response. In other words, it is very important to optimize the worst conditional mutual information and communication channel capacity caused by frequency response. Based on the above analysis, we construct the joint optimization problem of radar and communication, which can be expressed as

$$\begin{array}{c}\underset{{\mathit{A}}_c,{\mathit{W}}_r}{max}\left[\underset{{\mathit{H}}_r\in {\prod }_r}{min}I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right)\right]\hfill \\ \\ s.t.\left\{\begin{array}{c}\underset{{\mathit{H}}_c\in {\prod }_c}{min}C\left({\mathit{A}}_c,{\mathit{W}}_r\right)\ge C_1\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left({\mathit{W}}_{r,m}^H{\mathit{W}}_{r,m}+{\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}\right)\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(19)

where $C_1$ denotes a preset threshold of channel capacity and $P_1$ denotes the maximum transmit power of the integrated system. According to (15) and (16), it is not difficult to observe that $I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right)$ increases with an increase in ${\mathit{H}}_r$, and $C\left({\mathit{A}}_c,{\mathit{W}}_r\right)$ decreases with an increase in ${\mathit{H}}_c$. Therefore, according to the monotonicity of conditional MI, we can obtain the following inequality:

$$\left\{\begin{array}{c}I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right){|}_{{\mathit{H}}_r={\mathit{V}}_r}\le I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right),\forall {\mathit{H}}_r\in \prod _r\le I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right){|}_{{\mathit{H}}_r={\mathit{U}}_r}\hfill \\ \\ C\left({\mathit{A}}_c,{\mathit{W}}_r\right){|}_{{\mathit{H}}_c={\mathit{U}}_c}\le C\left({\mathit{A}}_c,{\mathit{W}}_r\right),\forall {\mathit{H}}_c\in \prod _c\le C\left({\mathit{A}}_c,{\mathit{W}}_r\right){|}_{{\mathit{H}}_c={\mathit{V}}_c}\hfill \end{array}\right.$$

(20)

Then, the worst conditional MI and communication channel capacity caused by frequency response can be expressed as

$$\left\{\begin{array}{c}\underset{{\mathit{H}}_r\in {\prod }_r}{min}I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right)=I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right){|}_{{\mathit{H}}_r={\mathit{V}}_r}\hfill \\ \\ \underset{{\mathit{H}}_c\in {\prod }_c}{min}C\left({\mathit{A}}_c,{\mathit{W}}_r\right)=C\left({\mathit{A}}_c,{\mathit{W}}_r\right){|}_{{\mathit{H}}_c={\mathit{U}}_c}\hfill \end{array}\right.$$

(21)

Exploiting (21) in (19), the joint optimization problem can be rewritten as

$$\begin{array}{c}\underset{{\mathit{A}}_c,{\mathit{W}}_r}{max}I\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right){|}_{{\mathit{H}}_r={\mathit{V}}_r}\hfill \\ \\ s.t.\left\{\begin{array}{c}C\left({\mathit{A}}_c,{\mathit{W}}_r\right){|}_{{\mathit{H}}_c={\mathit{U}}_c}\ge C_1\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left({\mathit{W}}_{r,m}^H{\mathit{W}}_{r,m}+{\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}\right)\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(22)

In addition, applying (15) and (16) to (22), the optimization problem in (22) can be reformulated as follows:

$$\begin{array}{c}\underset{{\mathit{A}}_c,{\mathit{W}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2}{{\beta }_1{∥{\mathit{W}}_{r,m}∥}^2+{\beta }_2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{∥{\mathit{A}}_{c,m}∥}^2}{{\gamma }_1{∥{\mathit{W}}_{r,m}∥}^2+{\gamma }_2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left({\mathit{W}}_{r,m}{\mathit{W}}_{r,m}^H+{\mathit{A}}_{c,m}{\mathit{A}}_{c,m}^H\right)\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(23)

where ${\beta }_1={\left|{\mathit{H}}_{cr}\left(m\right)\right|}^2\left|{\mathit{l}}_{rt}\left(m\right)\right|$, ${\beta }_2=\left({\left|{\mathit{V}}_r\left(m\right)\right|}^2+{\left|{\mathit{H}}_{cc}\left(m\right)\right|}^2\right)\left|{\mathit{l}}_{rt}\left(m\right)\right|$, ${\gamma }_2={\left|{\mathit{U}}_c\left(m\right)\right|}^2\left|{\mathit{l}}_c\left(m\right)\right|$ and ${\gamma }_1=\left|{\mathit{l}}_{rc}\left(m\right)\right|+{\left|{\mathit{U}}_c\left(m\right)\right|}^2\left|{\mathit{l}}_c\left(m\right)\right|$. It can be seen that the constraint condition of the above optimization problem is not a convex set, so the optimization problem is not a convex problem. It is difficult to solve this non-convex problem directly, and the calculation complexity is high. Based on this, we first transform the optimization problem in (23) as follows:

$$\begin{array}{c}\underset{{\mathit{A}}_c,{\mathit{W}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H}{{\beta }_1{\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H+{\beta }_2{\mathit{A}}_{c,m}\mathbf{I}{\mathit{A}}_{c,m}^H+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\mathit{A}}_{c,m}\mathbf{I}{\mathit{A}}_{c,m}^H}{{\gamma }_1{\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H+{\gamma }_2{\mathit{A}}_{c,m}\mathbf{I}{\mathit{A}}_{c,m}^H+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left({\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H+{\mathit{A}}_{c,m}\mathbf{I}{\mathit{A}}_{c,m}^H\right)\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(24)

where $\mathbf{I}$ represents an $N\times N$ unit diagonal matrix. According to the feature that the objective function in (24) is scalar, we introduce the concept of a trace function here [44]. Combining the properties of the trace function with the function characteristics in (24), we can observe that

$${\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H=tr\left({\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H\right)=tr\left(\mathbf{I}{\mathit{W}}_{r,m}^H{\mathit{W}}_{r,m}\right)$$

(25)

$${\mathit{A}}_{c,m}\mathbf{I}{\mathit{A}}_{c,m}^H=tr\left({\mathit{A}}_{c,m}\mathbf{I}{\mathit{A}}_{c,m}^H\right)=tr\left(\mathbf{I}{\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}\right)$$

(26)

where tr(*) represents the trace of the matrix. According to the properties in (25) and (26), we further introduce a set of new variables, ${\widehat{\mathit{W}}}_{r,m}={\mathit{W}}_{r,m}^H{\mathit{W}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}={\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}$, and we can observe that both ${\widehat{\mathit{W}}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}$ are positive semi-definite matrices with rank 1, so the optimization problem in (24) can be further transformed as follows

$$\begin{array}{c}\underset{{\widehat{\mathit{A}}}_c,{\widehat{\mathit{W}}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2\times tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)}{{\beta }_1\times tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)+{\beta }_2\times tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|\times tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)}{{\gamma }_1\times tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)+{\gamma }_2\times tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \\ \\ \begin{array}{c}0<{\displaystyle \sum _{m=0}^{N_c-1}}\left(tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)+tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)\right)\le P_1\hfill \\ \\ {\widehat{\mathit{W}}}_{r,m}\ge 0,\mathrm{rank}\left({\widehat{\mathit{W}}}_{r,m}\right)=1,m=0,1,\cdots ,N_c-1\hfill \\ \\ {\widehat{\mathit{A}}}_{c,m}\ge 0,\mathrm{rank}\left({\widehat{\mathit{A}}}_{c,m}\right)=1,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \end{array}\right.\hfill \end{array}$$

(27)

where rank(*) represents the rank of the matrix. It can be seen that the introduction of ${\widehat{\mathit{W}}}_{r,m}={\mathit{W}}_{r,m}^H{\mathit{W}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}={\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}$ adds two sets of constraints to the optimization problem. Since the trace function is an affine transformation, it is a convex constraint. In addition, the sum of two positive semi-definite matrices must be a positive semi-definite matrix, and the feasible region of the positive semi-definite constraint is a convex set, so the positive semi-definite constraint is a convex constraint. Based on the above analysis, the objective function and other constraints in (27) are convex except for $\mathrm{rank}\left({\widehat{\mathit{W}}}_{r,m}\right)=1$ and $\mathrm{rank}\left({\widehat{\mathit{A}}}_{c,m}\right)=1,m=0,1,\cdots ,N_c-1$. In order to effectively solve the optimization problem in (26), we partially relax the constraints, so (27) can be rewritten as

$$\begin{array}{c}\underset{{\widehat{\mathit{A}}}_c,{\widehat{\mathit{W}}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2\times tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)}{{\beta }_1\times tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)+{\beta }_2\times tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|\times tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)}{{\gamma }_1\times tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)+{\gamma }_2\times tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \\ \\ \begin{array}{c}0<{\displaystyle \sum _{m=0}^{N_c-1}}\left(tr\left(\mathbf{I}{\widehat{\mathit{W}}}_{r,m}\right)+tr\left(\mathbf{I}{\widehat{\mathit{A}}}_{c,m}\right)\right)\le P_1\hfill \\ \\ {\widehat{\mathit{W}}}_{r,m}\ge 0,m=0,1,\cdots ,N_c-1\hfill \\ \\ {\widehat{\mathit{A}}}_{c,m}\ge 0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \end{array}\right.\hfill \end{array}$$

(28)

Up to now, we have found that the joint optimization problem in (28) is a conventional convex optimization problem, and we can use the classical Lagrange multiplier method to solve it [45]. The optimized ${\widehat{\mathit{W}}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}$ can be obtained by combining the KKT condition and the bisection search method. If the estimated ${\widehat{\mathit{W}}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}$ satisfy $\mathrm{rank}\left({\widehat{\mathit{W}}}_{r,m}\right)=1$ and $\mathrm{rank}\left({\widehat{\mathit{A}}}_{c,m}\right)=1,m=0,1,\cdots ,N_c-1$, then ${\mathit{W}}_{r,m}$ and ${\mathit{A}}_{c,m}$ can be directly solved by singular value decomposition or eigenvalue decomposition. However, when $\mathrm{rank}\left({\widehat{\mathit{W}}}_{r,m}\right)\ne 1$ or $\mathrm{rank}\left({\widehat{\mathit{A}}}_{c,m}\right)\ne 1,m=0,1,\cdots ,N_c-1$, a feasible solution cannot be obtained directly by using existing methods, so we can only obtain the closest solution by constructing a classical optimization problem. First, we construct the optimization problem group as follows

$$\left\{\begin{array}{c}\underset{{\mathit{W}}_{r,m}}{min}{∥{\widehat{\mathit{W}}}_{r,m}-{\mathit{W}}_{r,m}^H{\mathit{W}}_{r,m}∥}_{\mathrm{F}}^2\\ \\ \underset{{\mathit{A}}_{c,m}}{min}{∥{\widehat{\mathit{A}}}_{c,m}-{\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}∥}_{\mathrm{F}}^2\end{array}\right.,m=0,1,\cdots ,N_c-1$$

(29)

where ${\left|\right|*\left|\right|}_{\mathrm{F}}$ denotes the Frobenius norm of matrix. The problem in (29) is a classical unconstrained optimization problem, and the closed-form solution of this problem group is the maximum eigenvector of ${\widehat{\mathit{W}}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}$ multiplied by the square root of the maximum eigenvalue [46]. Suppose that the eigenvalue decomposition of ${\widehat{\mathit{W}}}_{r,m}$ and ${\widehat{\mathit{A}}}_{c,m}$ is expressed as

$$\left\{\begin{array}{c}{\widehat{\mathit{W}}}_{r,m}={\displaystyle \sum _{i=1}^{\mathrm{rank}\left({\widehat{\mathit{W}}}_{r,m}\right)}}{\lambda }_{i,m}{\mathit{x}}_{i,m}{\mathit{x}}_{i,m}^H\\ \\ {\widehat{\mathit{A}}}_{c,m}={\displaystyle \sum _{j=1}^{\mathrm{rank}\left({\widehat{\mathit{A}}}_{c,m}\right)}}{\eta }_{j,m}{\mathit{y}}_{j,m}{\mathit{y}}_{j,m}^H\end{array}\right.,m=0,1,\cdots ,N_c-1$$

(30)

where ${\lambda }_{i,m}$ and ${\mathit{x}}_{i,m}$ represent the ith eigenvalues and eigenvectors of the weight vector corresponding to the mth subcarrier of ${\widehat{\mathit{W}}}_{r,m}$, where its eigenvalues are arranged in descending order. Similarly, ${\eta }_{j,m}$ and ${\mathit{y}}_{j,m}$ represent the jth eigenvalues and eigenvectors of the amplitude vector corresponding to the mth subcarrier of ${\widehat{\mathit{A}}}_{c,m}$, and their eigenvalues are also arranged in descending order. Then, the optimal solution of the optimization problem group in (30) can be written as

$$\left\{\begin{array}{c}{\mathit{W}}_{r,m}=\sqrt{{\lambda }_{1,m}}{\mathit{x}}_{1,m}\hfill \\ \\ {\mathit{A}}_{c,m}=\sqrt{{\eta }_{1,m}}{\mathit{y}}_{1,m}\hfill \end{array},m=0,1,\cdots ,N_c-1\right.$$

(31)

After all the subcarriers of the designed multiplexing waveform are processed according to the above solution method, the complex weight vector of each subcarrier in the radar signal and the corresponding amplitude vector of each subcarrier in the communication signal can be obtained.

3.2. Parameter Hierarchical Optimization-Based Robust OFDM Waveform (ROW-PHO) Design

According to the waveform design idea expounded in the previous section, we can find that the radar parameters and communication parameters are optimized simultaneously in the process of solving the optimization problem, which unfortunately results in a high computational complexity. Based on this, we further propose a low-complexity waveform design scheme based on parameter hierarchical optimization.

In order to realize the hierarchical optimization of OFDM waveform designs, the core idea is to realize the subcarrier amplitude vector design of the communication signal first, then the subcarrier weight vector design of the radar signal, thus reducing the computational complexity of the algorithm through dimensionality reduction. On the one hand, we limit the corresponding radar signal in the communication receiver and use the constraint conditions of radar waveform optimization to reduce the influence of radar signals on communication signals. On the other hand, we also consider the influence of interference information in the integrated system and regard the communication signal and the clutter caused by the communication signal in the integrated receiver as a constraint condition for communication waveform optimization. Firstly, according to the received signal model of the integrated platform in (9), we can obtain the following formula:

$$\left\{\begin{array}{c}{\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_r\right)\odot diag\left({\mathit{l}}_{rt}^{1/2}\right)\right]{\mathit{A}}_c=0\\ \\ {\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_{cc}\right)\odot diag\left({\mathit{l}}_{rt}^{1/2}\right)\right]{\mathit{A}}_c=0\end{array}\right.$$

(32)

Similarly, according to the communication received signal model in (12), we can eliminate the interference by the following formula:

$$\left\{\begin{array}{c}{\mathit{F}}_{N_c}^{-1}{\mathit{l}}_{rc}^{1/2}{\mathit{W}}_r=0\hfill \\ \\ {\mathit{F}}_{N_c}^{-1}\left[diag\left({\mathit{H}}_c\right)\odot diag\left({\mathit{l}}_c^{1/2}\right)\right]{\mathit{W}}_r=0\hfill \end{array}\right.$$

(33)

Then, the conditional mutual information and communication channel capacity in (15) and (16) can be rewritten as

$$I^{\prime }\left({\mathit{y}}_{rc};{\mathit{H}}_r|{\mathit{s}}_r\right)=\sum _{m=0}^{N_c-1}{log}_2\left(1+\frac{{\left|{\mathit{H}}_r\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2}{{\left|{\mathit{H}}_{cr}\left(m\right)\right|}^2{\left|{\mathit{l}}_{rt}^{1/2}\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)$$

(34)

$$C^{\prime }\left({\mathit{A}}_c,{\mathit{W}}_r\right)=\sum _{m=0}^{N_c-1}\Delta f{log}_2\left(1+\frac{{\left|{\mathit{l}}_{rc}^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2}{{\left|{\mathit{H}}_c\left(m\right)\right|}^2{\left|{\mathit{l}}_c^{1/2}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)$$

(35)

Next, the joint optimization problem is reconstructed by using the above new radar and communication performance parameters. Combining the results of (32)–(35), the problem of (23) can be reformulated as follows:

$$\begin{array}{c}\underset{{\mathit{A}}_c,{\mathit{W}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2}{{\beta }_1{∥{\mathit{W}}_{r,m}∥}^2+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}\begin{array}{c}\left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2=0\hfill \\ \\ \left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{H}}_{cc}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2=0\hfill \\ \\ \left|{\mathit{l}}_{rc}\left(m\right)\right|{∥{\mathit{W}}_{r,m}∥}^2=0\hfill \\ \\ {\left|{\mathit{U}}_c\left(m\right)\right|}^2\left|{\mathit{l}}_c\left(m\right)\right|{∥{\mathit{W}}_{r,m}∥}^2=0,m=0,1,\cdots ,N_c-1\hfill \\ \\ {\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{∥{\mathit{A}}_{c,m}∥}^2}{{\gamma }_2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \end{array}\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left({\mathit{W}}_{r,m}{\mathit{W}}_{r,m}^H+{\mathit{A}}_{c,m}{\mathit{A}}_{c,m}^H\right)\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(36)

According to (36), it can be found that the unknown parameters in the objective function only contain the subcarrier weight vector of the radar signal, while the amplitude vector corresponding to the subcarrier of the communication signal only exists in the constraint condition, which has no influence on the objective function value. Therefore, we can realize waveform design based on parameter hierarchical optimization. Firstly, we temporarily extract the amplitude vector information of communication subcarriers from this problem and construct a new optimization problem to realize the optimal design of communication parameters. Then, we bring the designed results back to (36) and realize the optimal design of radar parameters. Based on this, the first level optimization problem based on communication parameters can be represented as

$$\begin{array}{c}\underset{{\mathit{A}}_c}{min}{∥{\mathit{A}}_{c,m}∥}^2,m=0,1,\cdots ,N_c-1\hfill \\ \\ s.t.\left\{\begin{array}{c}\begin{array}{c}\left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2=0\hfill \\ \\ \left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{H}}_{cc}\left(m\right)\right|}^2{∥{\mathit{A}}_{c,m}∥}^2=0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \\ \\ {\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{∥{\mathit{A}}_{c,m}∥}^2}{{\gamma }_2{∥{\mathit{A}}_{c,m}∥}^2+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \end{array}\right.\hfill \end{array}$$

(37)

Similarly, we predefine ${\tilde{\mathit{A}}}_{c,m}={\mathit{A}}_{c,m}^H{\mathit{A}}_{c,m}$ according to the properties of (26). Then, the optimization problem of (37) can be rewritten as

$$\begin{array}{c}\underset{{\tilde{\mathit{A}}}_c}{min}tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right),m=0,1,\cdots ,N_c-1\hfill \\ \\ s.t.\left\{\begin{array}{c}\begin{array}{c}\left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)=0\hfill \\ \\ \left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{H}}_{cc}\left(m\right)\right|}^2\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)=0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \\ \\ \begin{array}{c}{\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|\times Tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)}{{\gamma }_2\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \\ \\ {\tilde{\mathit{A}}}_{c,m}\ge 0,\mathrm{rank}\left({\tilde{\mathit{A}}}_{c,m}\right)=1,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \end{array}\right.\hfill \end{array}$$

(38)

It can be observed that only the constraint condition of $\mathrm{rank}\left({\tilde{\mathit{A}}}_{c,m}\right)=1$ in (38) is non-convex. Therefore, in order to solve this problem quickly and effectively, we relax the constraint condition, and (38) can be reformulated as

$$\begin{array}{c}\underset{{\tilde{\mathit{A}}}_c}{min}tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right),m=0,1,\cdots ,N_c-1\hfill \\ \\ s.t.\left\{\begin{array}{c}\begin{array}{c}\left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)=0\hfill \\ \\ \left|{\mathit{l}}_{rt}\left(m\right)\right|{\left|{\mathit{H}}_{cc}\left(m\right)\right|}^2\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)=0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \\ \\ \begin{array}{c}{\displaystyle \sum _{m=0}^{N_c-1}}\Delta f{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)}{{\gamma }_2\times tr\left(\mathbf{I}{\tilde{\mathit{A}}}_{c,m}\right)+{\left|{\mathit{n}}_c\left(m\right)\right|}^2}\right)\ge C_1\hfill \\ \\ {\tilde{\mathit{A}}}_{c,m}\ge 0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \end{array}\right.\hfill \end{array}$$

(39)

If the estimated ${\tilde{\mathit{A}}}_{c,m}$ satisfies $\mathrm{rank}\left({\tilde{\mathit{A}}}_{c,m}\right)=1,m=0,1,\cdots ,N_c-1$, then ${\mathit{A}}_{c,m}$ can be directly solved by singular value decomposition or eigenvalue decomposition. However, when $\mathrm{rank}\left({\tilde{\mathit{A}}}_{c,m}\right)\ne 1,m=0,1,\cdots ,N_c-1$, we can use a processing method similar to (29)–(31) to obtain the optimal complex weight vector of communication signal, which is recorded as ${\mathit{A}}_{c,m}^{*},m=0,1,\cdots ,N_c-1$. Next, the estimated ${\mathit{A}}_{c,m}^{*}$ is brought into the optimization problem of (36). Then, the second level optimization problem based on radar parameters can be simplified as follows:

$$\begin{array}{c}\underset{{\mathit{W}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{∥{\mathit{W}}_{r,m}∥}^2}{{\beta }_1{∥{\mathit{W}}_{r,m}∥}^2+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}\begin{array}{c}\left|{\mathit{l}}_{rc}\left(m\right)\right|{∥{\mathit{W}}_{r,m}∥}^2=0\hfill \\ \\ {\left|{\mathit{U}}_c\left(m\right)\right|}^2\left|{\mathit{l}}_c\left(m\right)\right|{∥{\mathit{W}}_{r,m}∥}^2=0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left[{\mathit{W}}_{r,m}{\mathit{W}}_{r,m}^H+{\mathit{A}}_{c,m}^{*}{\left({\mathit{A}}_{c,m}^{*}\right)}^H\right]\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(40)

The optimization problem of (40) can be transformed as follows

$$\begin{array}{c}\underset{{\mathit{W}}_r}{max}{\displaystyle \sum _{m=0}^{N_c-1}}{log}_2\left(1+\frac{\left|{\mathit{l}}_{rc}\left(m\right)\right|{\left|{\mathit{V}}_r\left(m\right)\right|}^2{\mathit{W}}_{r,m}^H\mathbf{I}{\mathit{W}}_{r,m}}{{\beta }_1{\mathit{W}}_{r,m}^H\mathbf{I}{\mathit{W}}_{r,m}+{\left|{\mathit{n}}_{rc}\left(m\right)\right|}^2}\right)\hfill \\ \\ s.t.\left\{\begin{array}{c}\begin{array}{c}\left|{\mathit{l}}_{rc}\left(m\right)\right|{\mathit{W}}_{r,m}^H\mathbf{I}{\mathit{W}}_{r,m}=0\hfill \\ \\ {\left|{\mathit{U}}_c\left(m\right)\right|}^2\left|{\mathit{l}}_c\left(m\right)\right|{\mathit{W}}_{r,m}^H\mathbf{I}{\mathit{W}}_{r,m}=0,m=0,1,\cdots ,N_c-1\hfill \end{array}\hfill \\ \\ 0<{\displaystyle \sum _{m=0}^{N_c-1}}\left[{\mathit{W}}_{r,m}\mathbf{I}{\mathit{W}}_{r,m}^H+{\mathit{A}}_{c,m}^{*}\mathbf{I}{\left({\mathit{A}}_{c,m}^{*}\right)}^H\right]\le P_1\hfill \end{array}\right.\hfill \end{array}$$

(41)

It can be observed that the objective function of (41) is an affine function, the inequality constraint condition is a convex feasible set on ${\mathit{W}}_{r,m}$ and the equality constraint function is also an affine function. Therefore, the optimization problem in (41) is a standard convex optimization problem, which can be solved by an interior point method or aa convex optimization toolbox [47,48]. Finally, we can obtain the optimized complex weight vector parameters ${\mathit{W}}_{r,m}^{*},m=0,1,\cdots ,N_c-1$ of the radar signal.

4. Numerical Simulation

In this section, simulation results are provided to validate the accuracy of the theoretical derivations as well as to demonstrate the improvement in the radar and communication performance brought on by our proposed waveform design schemes. First, the necessary parameters in the simulation are listed in

Table 1. The parameters of the simulation.

Parameters	Value
Subcarrier spacing	0.25 MHz
Carrier frequency	5 GHz
Number of subcarriers	64
Number of antennas	8
$G_t$/$G_r$	25 dB
$G_c$	2 dB
$d_1$/$d_2$/$d_3$	10 km

.

Next, in order to verify the robustness of the designed OFDM multiplexing waveform, we need to obtain the upper and lower bounds of RCS. Generally speaking, the target impulse response and channel response obey a normal distribution, and we can construct a propagation model by transforming the normal distribution function to obtain the frequency response of different channels [42,43]. Therefore, the limit range of the corresponding magnitudes of frequency responses of different channels is shown in Figure 3. In addition, we estimate the power spectral density (PSD) of related clutter and noise according to the received signal of the integrated platform. The PSDs of clutter caused by radar signals, clutter caused by communication signals and Gaussian white noise are shown in Figure 4.

4.1. Radar Performance

In this subsection, we examine the radar performance with different waveform design methods. In the simulation, Monte Carlo experiments with ${10}^4$ repetitions were carried out under different signal-to-noise ratio (SNR) settings. Under the parameter conditions in Table 1, the four waveform design methods for simulation and comparison were as follows: (1) optimized OFDM waveform design in [27]; (2) robust OFDM shared waveform design in [33]; (3) the proposed robust OFDM waveform design based on constraint relaxation (ROW-CR); and (4) the proposed low-complexity OFDM waveform design based on parameter hierarchical optimization (ROW-PHO). It is worth mentioning that the methods in [27,33] were used for a performance comparison, mainly because the methods in these two references have the highest correlation with the methods proposed in this paper and they are relatively recent references. The variation in the conditional mutual information of different waveforms with the SNR is shown in Figure 5. Among them, Figure 5a is the result obtained after the actual frequency response (AFR) is randomly selected within the range of RCS, which is recorded as the random AFR. Figure 5b is the result obtained after the AFR takes the value that leads to the worst system performance, which is recorded as the worst-case AFR. It can be seen that the conditional mutual information of all waveforms increases almost linearly with the increase in the SNR, and the performance of the robust OFDM waveform is better than that of the non-robust OFDM waveform, especially for the worst-case AFR (see Figure 5b). In particular, the performance of the proposed ROW-CR is obviously superior to the other three waveforms. It is worth mentioning that although the proposed ROW-PHO has a slight performance loss compared to ROW-CR, its radar performance is better than the other two waveforms and its computational complexity is obviously lower than ROW-CR, which will be detailed in Section 4.3 of the paper.

In addition, in order to further determine the influence of the width of RCS on the conditional mutual information, the variation in conditional mutual information of different waveforms with the width of RCS is depicted in Figure 6. Figure 6a,b shows the results obtained by adjusting the set width under the condition that the lower bound and the upper bound are fixed, respectively. It can be seen from Figure 6a that the conditional mutual information of all waveforms in the worst-case AFR does not change with the width of RCS, because the worst-case AFR is equal to a fixed value. When the actual frequency response is randomly selected, the performance of all waveforms increases with the increase in the width of the RCS, but the proposed ROW-CR is slightly better than the other two schemes. It can be observed from Figure 6b that the performance of all waveforms deteriorates with the increase in the width of the RCS, since with the increase in the width of the RCS, the value of the actual frequency response gradually approaches the worst-case AFR. In this case, the proposed ROW-CR also has better performance. As expected, the worst-case radar performance is greatly improved by the proposed schemes.

4.2. Communication Performance

In this section, the communication performances of the two proposed OFDM waveform design methods are evaluated in a ${10}^4$ Monte Carlo simulation experiment. The waveforms used for performance comparison are the same as those in the previous section. In Figure 7, the variation in the communication channel capacity with the SNR is described. Similar to radar performance, the communication channel capacity increases with the increase in the SNR, and the robust waveform has more advantages than the non-robust waveform. Moreover, the communication performances of the proposed ROW-CR and ROW-PHO are better than the other waveforms under both random AFR and worst-case AFR values.

Figure 8 shows the dependence of the communication channel capacity on the width of RCS, which is consistent with the fixed upper and lower bound parameters in Figure 6. As can be seen from Figure 8a, since the fixed parameters and the worst-case AFR are corresponding, with the increase in the set width, the communication performance of all waveforms remains unchanged. On the contrary, Figure 8b fixes the parameters under the optimal system performance, which shows that with an increase in the width of RCS, the system gradually approaches the worst-case AFR, resulting in the gradual deterioration in communication performance. Overall, in both cases shown in Figure 8, the proposed schemes show better communication performance.

4.3. Comparison of Computational Complexity

In order to comprehensively evaluate the performance of the waveform design, under the same system parameters and hardware conditions, the running times of algorithms corresponding to different waveform designs are shown in

Table 2. Algorithm complexity comparison of different waveform design methods.

Methods	Running Time under the Same Hardware Conditions (s)
	${\mathit{N}}_{\mathit{c}}=\mathbf{16}$	${\mathit{N}}_{\mathit{c}}=\mathbf{32}$	${\mathit{N}}_{\mathit{c}}=\mathbf{64}$	${\mathit{N}}_{\mathit{c}}=\mathbf{128}$
Optimized waveform design in [27]	0.91	1.63	2.93	7.74
Robust waveform design in [33]	1.98	2.81	5.66	15.21
ROW-CR (Proposed scheme 1)	2.04	2.75	5.52	14.34
ROW-PHO (Proposed scheme 2)	1.05	1.56	2.74	6.95

. Obviously, the optimized waveform design in [27] has a relatively low computational complexity, but its radar and communication performance is obviously lower than the other methods (see the previous two subsections). The proposed ROW-CR and the robust waveform design in [33] have a higher computational complexity. In addition, the proposed ROW-PHO has a lower computational complexity than ROW-CR, while ensuring a low performance loss, and its overall performance is better than the other two methods.

4.4. Radar–Communication Trade-Off

In the proposed integrated system, in order to evaluate the trade-off between the radar performance and communication performance of different waveforms, we constructed a trade-off curve between radar conditional mutual information and communication channel capacity under parametric conditions through simulation. In the simulation, ${10}^4$ Monte Carlo experiments were carried out. Figure 9 shows the radar–communication trade-off curves of different waveforms under different SNRs. It is not difficult to see that high SNRs have a better radar–communication trade-off curve. With the increase in conditional mutual information, the communication channel capacity of all waveforms decreases gradually; in other words, the radar performance and communication performance are mutually constrained. In addition, by comparing Figure 9a–c, it can be seen that the proposed schemes have higher communication channel capacities under the same conditional mutual information, especially at low SNRs. It can also be found that the decreasing rate of the communication channel capacity increases with the increase in conditional mutual information. Therefore, according to the different requirements of each practical application scenario, the appropriate parameters can be adjusted to realize a trade-off between radar and communication performance.

5. Conclusions

In this paper, a constraint relaxation-based robust OFDM multiplexing waveform (ROW-CR) design scheme for integrated radar and communications systems is proposed, which can simultaneously improve the performance of the integrated OFDM signal in terms of the accuracy of target estimation for radar and the information transmission efficiency for communication. Considering the influence of clutter and jamming signals, the joint optimization problem of radar and communication is constructed, and a novel method based on trace function characteristics is adopted to obtain an optimized complex weight vector. In addition, a robust OFDM waveform design method based on parameter hierarchical optimization (ROW-PHO) is introduced. Numerical experiments show that the proposed two schemes have good robustness, ROW-CR can achieve a better tradeoff between radar and communication performance, and ROW-PHO can greatly reduce the computational complexity of the waveform design with a low performance loss. In future work, we will investigate the waveform design of PAPR reduction in OFDM integrated systems and an integrated waveform design and signal processing scheme based on MIMO-OFDM.