Dual Function Radar and Communication Signal Design with Combined Waveform Selection and Pulse Repetition Interval Agility

Abstract

The traditional probe–pass integration system embeds communication information into a radar waveform, which leads to a high level of waveform autocorrelation sidelobes and a poor false symbol rate at low signal-to-noise ratios. This article proposes a three-dimensional indexed modulation-based design method for probe–pass integration waveforms. This method realises communication information modulation and demodulation by simultaneously indexing orthogonal waveform selection, transmitting pulse PRI changes and carrier frequency changes in three dimensions, and applying compressed perception technology to solve the problems of PRI shortcuts and carrier frequency, resulting in a velocity term in the received waveform that cannot be accumulated by phase reference to realise velocity super-resolution. Finally, the radar detection performance and communication performance are simulated and analysed, and the simulation results reveal that the method proposed in this paper can not only satisfy the radar detection performance requirements but also achieve a lower unsigned rate on the basis of an improved communication rate.

1. Introduction

Radar detection and communication integration systems have the advantages of resource intensification and functional complementarity, which can effectively solve the problems of resource waste and high cost. This approach is one of the important research directions for the comprehensive integration systems of many types of electronic equipment, and detection and communication integration waveform design is one of the key research components of detection and communication integration systems [1,2,3,4]. Integrated waveform design is one of the key research components of integrated systems of probes and passes. At present, integrated waveform research is based on three main categories: integrated waveforms based on typical communication waveforms, integrated waveforms based on radar waveforms, and integrated waveforms based on joint design [5].

Integrated waveform design methods based on radar waveforms commonly embed communication information into the radar waveform. Commonly, the radar waveform is modified, mainly through the linear frequency modulation (LFM) pulse within the frequency modulation or phase modulation, to complete the embedding of communication information [6,7]. In combination with continuous phase modulation (CPM), the embedding of communication information in the LFM pulse is achieved through the modulation of the phase [8]. The above methods increase the size of the integrated waveform. All of the above methods increase the autocorrelation level of the integrated waveform, and the communication transmission rate is generally not high. In recent years, by borrowing index modulation from communication systems [9,10,11], the communication rate has been greatly improved. The communication transmission rate can be improved by indexing the pulse repetition interval (PRI) shortcut to achieve communication information embedding [12] or by indexing the carrier frequency corresponding to the location of the transmitting antenna [13]. Alternatively, multiple-input multiple-output (MIMO) system multicarrier phase modulation can be used by indexing the carrier frequency of the sparse array, and the position of the transmitting antenna can be modulated to further improve the communication transmission rate [14]. Lin et al. proposed an information modulation and demodulation method based on orthogonal waveforms and the phase union of multiple sub-pulses [15], which further enhances the performance of the integrated radar communication system. The method makes each sub-pulse not only have orthogonal waveforms but also different phase states by introducing phase union modulation. Such phase states can be used to carry additional information, which improves the communication rate and efficiency. Amin M.G. et al. [16] made a new breakthrough in the design of phase modulation sequences and frequency-hopping code words. They used a genetic algorithm to design continuous phase modulation CSK; the objective of this method is to obtain a combination of a communication phase modulation sequence and a frequency-hopping code word with the lowest autocorrelation function side-flap level. In addition, researchers have proposed an indexed modulation-based information transmission method for PRI shortcut communication [12], which greatly enhances the communication transmission rate. Yu et al. proposed a multicarrier phase-modulated PM-FMCW architecture for MIMO systems [14], which achieves a significant increase in the communication rate by carefully selecting the carrier frequency of the sparse array, as well as the transmit antenna positions in a joint modulation strategy. Researchers have summarised and compared the proposed communication message-transmission schemes for FH-MIMO systems transmitting orthogonal signals to give a high-rate communication approach with joint phasing and frequency allocation on the antennas [17], which also greatly enhances the communication transmission method. Wang et al. proposed a multiple-index modulation based on the FH-MIMO system [18] which achieves the goal of high-speed communication by jointly optimising phase and frequency allocation on antennas and antenna position selection.

Three-dimensional indexed modulation is a technique that expands on traditional modulation methods by adding a third dimension (e.g., time, frequency, space, etc.) to improve the efficiency of information transmission. Conventional methods (e.g., BPSK, QPSK, etc.) usually rely on a single dimension such as signal amplitude, phase, or frequency to transmit information. In contrast, 3D indexed modulation can transmit more data information with the same bandwidth and time resources by encoding information in three dimensions, e.g., combining different combinations of time, frequency, and space. This extension allows for a more complex representation of each signal, thus increasing the information capacity. Compressed perception is a mathematical theory for signal sampling and reconstruction that presents significantly different ideas in the signal recovery process and has several unique properties compared to traditional signal processing methods. Traditional signal sampling theorems (e.g., the Nyquist Sampling Theorem) require that signals be sampled at a certain sampling rate (usually twice the signal bandwidth). Compressed perception theory, on the other hand, states that as long as the signal is sparse on some transform domain (e.g., Fourier transform or wavelet transform), the signal can be recovered by sampling at a much lower rate than required by traditional methods.

In this paper, we draw on indexed modulation and propose a probe-pass integrated waveform design method based on three-dimensional indexed modulation. An orthogonal waveform set with good autocorrelation and cross-correlation performance is achieved by attaching a perturbation phase to the LFM signal. The three dimensions of indexed orthogonal waveform selection, PRI shortcut, and carrier frequency shortcut are used to achieve communication modulation, and the detection of the orthogonal waveform, PRI shortcut, and carrier frequency shortcut is used to achieve communication demodulation. On the radar-processing side, the compressed perceptual sparse reconstruction algorithm is used to solve the problem of the velocity term not being able to be accumulated by the phase parameter. After this, target information detection is complete.

2. Materials and Methods

2.1. Integrated Waveform Design

2.1.1. Orthogonal Waveform Set Design

In radar detection, assuming that there is a strong target and a weak target in the radar scene, the weak target will be annihilated if the autocorrelation sidelobe level of the waveform is too high, affecting the judgement of the weak target and thus affecting the radar detection of the target. In communication, different orthogonal waveforms are transmitted to map different code elements, and the set of orthogonal waveforms is intercorrelated with the transmitted signals at the receiving end to judge the transmitted orthogonal waveform. If the intercorrelation sidelobe level is too high, it affects the judgement of the real transmitted orthogonal waveform, which in turn affects the demodulation of the communication message.

Therefore, to ensure that the orthogonal waveform in the waveform set has good target detection and information transmission capabilities simultaneously, the following two design guidelines are used:

• Minimise the autocorrelation peak sidelobe level (APSL) to meet the target detection capability of the radar.
• Minimise the cross-correlation peak sidelobe level (CPSL) for demodulation of communication messages.

To satisfy criteria (1) and (2), a total of $M$ orthogonal waveforms $[s_1,s_2,\cdots ,s_M]$ are designed in this paper, and the autocorrelation function of the $m$th orthogonal waveform $s_m(t)$ is as follows:

$$X_m(\tau )={\displaystyle \int s_m(t)}{s}_m^{*}(t-\tau )dt$$

(1)

The autocorrelation peak sidelobe level is expressed as

$$\mathrm{APSL}=\underset{\tau }{\max }\left|{\displaystyle \frac{X(\tau )}{X(0)}}\right|\begin{array}{cc}& \tau \in [{\tau }_{main},T_r]\end{array}$$

(2)

In Equation (2), the value of $\left|X(\tau )\right|$ has a range of $[-T_r, T_r]$ symmetric about the origin. If $[0, {\tau }_{main}]$ is defined in the range of $[0, T_r]$ for the time range of the main flap and then $[{\tau }_{main}, T_r]$ for the time range of the side flap, and if $T_r$ for the pulse repetition period, then the time range of the main flap is defined in the time range of the side flap.

The cross-correlation function between the orthogonal waveforms $s_{m_1}(t)$ and $s_{m_2}(t)$ is as follows:

$$X_{m_1{m}_2}(\tau )={\displaystyle \int s_{m_1}(\tau )}{s}_{m_2}^{\ast }(t-\tau )dt\begin{array}{cc}& m_1\ne \end{array}{m}_2$$

(3)

The intercorrelation peak sidelobe level can be expressed as

$$\begin{array}{cc}\mathrm{CPSL}=\underset{\tau }{\max }\left|{\displaystyle \frac{X_{m_{1}m2}(\tau )}{X_{m_{1}m2}(0)}}\right|& m_1\ne m_2,\tau \in [{\tau }_{main},T_r]\end{array}$$

(4)

If the phase modulation of the waveform is achieved by adding the perturbation phase to the LFM-based signal and then the orthogonal waveform is designed [19], then the $m$th orthogonal waveform is represented as follows:

$$\begin{array}{c}{s}_m(t)={\beta }_m(t)\exp (\mathrm{j}2\pi (f_{c}t+K{t}^2/2))\\ 0\le t\le T_r, m=1,2,\cdots ,M\end{array}$$

(5)

In Equation (5), ${\beta }_m(t)=\exp (\mathrm{j}{\phi }_m(t))$, where ${\phi }_m(t)$ is the phase perturbation function, $K$ is the FM slope, and $f_c$ is the initial carrier frequency.

The phase perturbation function ${\phi }_m(t)$ is defined as

$${\phi }_m(t)={\displaystyle \sum _{n=1}^N{\vartheta }_{mn}\cos ({\omega }_{mn}t+{\theta }_{mn})}$$

(6)

where $N$ is the number of subphase perturbation functions, ${\vartheta }_{mn}$ is the amplitude parameter of the subphase perturbation function, ${\theta }_{mn}$ is the phase parameter of the subphase perturbation function, and ${\omega }_{mn}$ is the frequency parameter of the subphase perturbation function.

According to Equation (6), the phase perturbation function ${\phi }_m(t)$ can be obtained by designing the parameters $N$, ${\vartheta }_{mn}$, ${\theta }_{mn}$, and ${\omega }_{mn}$ to realise the orthogonal waveform set design. Among them, $N\in \left[1,100\right]$ is a positive integer; if sub-band $\left[B_1,B_2\right]\in \left[0,B\right]$ is assumed, the value of ${\omega }_{mn}$ can be set at equal intervals inside sub-band $\left[B_1,B_2\right]$; ${\vartheta }_{mn}$ and ${\theta }_{mn}$ are randomly selected inside $\left[0,2\pi \right)$.

Based on the two design criteria, i.e., satisfying $\mathrm{APSL}\le \xi$ and $\mathrm{CPSL}\le \xi$ (where $\xi$ is the optimisation threshold), the design flow of the orthogonal waveform set on the basis of phase modulation of linear FM signals is as follows:

• Based on the operating parameters of the radar transmitter, the reference LFM signal is determined to be: $s = \exp (\mathrm{j}2\pi (f_{c}t+K{t}^2/2)),0\le t\le T_r$;
• ${\vartheta }_{mn}$ and ${\theta }_{mn}$ are randomly generated and the parameters $N$ and ${\omega }_{mn}$ are determined, where ${\omega }_{mn}$ takes values at equal intervals within the sub-band $\left[B_1,B_2\right]$; then, ${\omega }_{mn}=B_1+(nN-m+n)\Delta B,$$\Delta B=(B_2-B_1)/(MN-1), m=1,2,\cdots ,M$ $n=1,2,\cdots ,N$.
• Creating a waveform set of size $M$, the LFM phase modulation-based waveform is represented as $s(t)={\displaystyle \sum _{m=1}^M}{\beta }_m(t)\exp (\mathrm{j}2\pi (f_{c}t+K{t}^2/2))$, where ${\beta }_m(t)=\exp (\mathrm{j}{\phi }_m(t))$. ${\phi }_m(t)={\displaystyle \sum _{n=1}^N{\vartheta }_{mn}\cos ({\omega }_{mn}t+{\theta }_{mn})}$;
• $\mathrm{APSL}$ is calculated for each waveform and any two signals for $\mathrm{CPSL}$;
• If $\mathrm{APSL}\le \xi$ and $\mathrm{CPSL}\le \xi$ are satisfied, then the waveform set is output; otherwise, the process returns to step 2 until the design is satisfied.

2.1.2. Integrated Signal Model

First, a radar-emission model is constructed as shown in Figure 1, assuming that the radar transmits a carrier frequency shortcut and a PRI shortcut and transmits a total of $M$ pulses in one coherent processing time, with a pulse repetition interval of $T_r$ and a carrier frequency of $f_m$. The total pulse repetition time is $t_m$. Then, the transmitted signal can be expressed as:

$$S(t)={\displaystyle \sum _{m=1}^M\mathrm{rect}(\frac{t-t_m}{T_p})}{\beta }_m(t-t_m)\exp (\mathrm{j}\pi K(t-t_m{)}^2)\exp (\mathrm{j}2\pi f_m\left(t-t_m\right))$$

(7)

where $\mathrm{rect}(\cdot )$ is the unit rectangular window, $T_p$ is the pulse width, ${\beta }_m(t)=\exp (j{\varphi }_m(t))$ is the perturbation term, and ${\varphi }_m(t)$ is the corresponding phase perturbation function. $f_m=f_0+b(m)\Delta f$, where $\Delta f$ is the minimum carrier frequency shortcut interval. $t_m=(m-1)T_r+a(m)\Delta T$, where $\Delta T$ is the minimum PRI shortcut interval. $K=B/T_p$ is the modulation frequency. $a(m)\in \{1,2,3,\dots ,N\}$, where $a(1)=0$ is the pulse modulation code word and $b(m)\in \{1,2,3,\dots ,\zeta \}, b(1)=0$ is the carrier frequency modulation code word.

The different orthogonal waveforms, pulse repetition intervals (PRIs) of the transmitted pulses, and carrier frequencies of the transmitted pulses are used as unique code element information. By indexing these parameters, the modulation of the communication message can be achieved. If a frame of data is to be transmitted within a coherent processing interval (CPI), this frame can be converted into multiple parallel lines of data with a fixed number of bits per line.

Specifically, the first $D_1$ bits of each line of data represent the communication information for different PRI mappings, where $D_1$ depends on the number of PRIs in the system. For example, if there are eight different PRIs, three bits are needed. The subsequent $D_2$ bits reflect the different carrier-mapped communications, with a size of $D_2$ depending on the number of carriers in the system. If there are 8 different carriers, 3 bits are again needed. Finally, the $D_3$ bits represent the communication information of different quadrature waveform mappings, the number of which is determined by the type of quadrature waveform. For example, if there are 4 different quadrature waveforms, 2 bits are required to represent them. Therefore, the total number of bits per line of data is $D^{‴}=D_1+D_2+D_3$.

This section transmits a total of $L$ PRIs, $\zeta$ carrier frequencies and $M$ orthogonal waveforms and then carries the total communication data $D^{‴}$ and the communication rate $R_b^{‴}$.

$$D^{‴}=⌊{\log }_{2}L\ast \zeta *M⌋$$

(8)

$$R_b^{‴}={\displaystyle \frac{⌊{\log }_{2}L\ast \zeta *M⌋}{T_r}}$$

(9)

When communicating with four pulse position modulation codewords, four quadrature waveforms, and eight carrier frequency modulation codewords, we can map the communication data into three parts: carrier frequency shortcut mapping, pulse position mapping, and quadrature waveform mapping.

First, the communication data for the carrier frequency shortcut mapping occupy 3 bits. This is because we have 8 different carrier frequencies to choose from, and 3 bits are just enough to represent all possible carrier frequency states. With these 3 bits of data, we can specify exactly which carrier frequency to use for the transmitted signal.

Second, the communication data for pulse position mapping occupy 2 bits. This is because there are 4 different pulse positions to choose from, and 2 bits can cover all possible pulse position states. These 2 bits of data determine the exact position of the pulse in the signal.

Finally, the communication data for orthogonal waveform mapping also occupy 2 bits. This is because there are 4 different orthogonal waveforms to choose from, and 2 bits of data are enough to represent all possible waveform states. These 2 bits of data determine the particular orthogonal waveform used to modulate the signal.

The communication data transmitted in each line total 7 bits. Among these 7 bits of data, the first 3 bits represent the communication information of carrier frequency shortcut mapping, which determines the carrier frequency in the signal; the middle 2 bits represent the communication information of the PRI shortcut, which determines the specific position of the pulse in the signal; and the last 2 bits represent the communication information of the waveform, which determines the waveform in the signal. The mapping relationships are shown in

Table 1. Mapping of pulse positions to code elements.

PRI	Code Point
$t_1$	00
$t_2$	01
$t_3$	10
$t_4$	11

,

Table 2. PRI shortcut time and code element mapping relationship.

Carrier-Frequency Shortcut Time	Code Point
$f_1$	000
$f_2$	001
$f_3$	010
$f_4$	011
$f_5$	100
$f_6$	101
$f_7$	110
$f_8$	111

and

Table 3. Mapping of waveform to code elements.

Waveform	Code Point
Waveform 1	00
Waveform 2	01
Waveform 3	10
Waveform 4	11

, and the communication information modulation is shown in Figure 2:

2.2. Integrated Signal Processing

2.2.1. Radar Signal Processing

Suppose that a total of $G$ targets are present in the radar’s observation scene. Each target has its own radial distance $r_g$ and radial velocity $v_g$ relative to the radar, where $g\in \{1,2,\dots ,G\}$ denotes the index of the number of targets, which is used to distinguish different targets. When the signal emitted by the radar is reflected by these targets, the echo formed can be expressed as

$$\begin{array}{l}{S}_r(t)={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{m=1}^M\mathrm{rect}(\frac{t-t_m-{\tau }_g}{T_p}){\beta }_m(t-t_m-{\tau }_g)}}\\ \exp (\mathrm{j}\pi K(t-t_m-{\tau }_g{)}^2)\exp (\mathrm{j}2\pi f_m(t-t_m-{\tau }_g)+n(t,t_m)\end{array}$$

(10)

Let ${\varphi }_r^{‴}(t)=\mathrm{rect}({\displaystyle \frac{t-t_m-{\tau }_g}{T_p}}){\beta }_m(t)\exp (\mathrm{j}\pi K{t}^2)$, and then

$$S_r(t)={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{m=1}^M{\varphi }_r^{‴}(t-t_m-{\tau }_g)\exp (\mathrm{j}2\pi f_m(t-t_m-{\tau }_g))}}+n(t,t_m)$$

(11)

where ${\mathrm{\Phi }}_r^{‴}(t-t_m-2r_g/c)={\displaystyle \int {\varphi }_r^{‴}(t-t_m){\varphi }_r^{‴}(t-t_m-2r_g/c)}dt$ denotes the autocorrelation of $\varphi (t-t_m)$ and $A_g$ denotes the compressed amplitude of the pulse of the first $g$ target, which can be obtained by substituting ${\tau }_g=2(r_g-{\nu }_g{t}_m)/c$ into Eq.

$$\begin{array}{ll}\hfill S_r(t)& ={\displaystyle \sum _{g=1}^G{\displaystyle \sum _m^M{A}_g{\mathrm{\Phi }}_r^{‴}(t-t_m-\frac{2r_g}{c})}}\exp (-\mathrm{j}2\pi f_m(2{\displaystyle \frac{(r_g-{\nu }_g{t}_m)}{c}}))+n(t,t_m)\\ & ={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{m=1}^M{A}_g{\mathrm{\Phi }}_r^{‴}}}(t-t_m-{\displaystyle \frac{2r_g}{c}})\exp (-\mathrm{j}{\displaystyle \frac{4\pi f_m{r}_g}{c}})\exp (\mathrm{j}{\displaystyle \frac{4\pi f_m{\nu }_g{t}_m}{c}})+n(t,t_m)\end{array}$$

(12)

Bringing in $f_m=f_0+b(m)\Delta f$, $t_m=(m-1)T_r+a(m)\Delta T$ yields

$$\begin{array}{ll}\hfill S_r(t)& ={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{m=1}^M{A}_g{\mathrm{\Phi }}_r^{‴}}}(t-t_m-{\displaystyle \frac{2r_g}{c}})\exp (-\mathrm{j}{\displaystyle \frac{4\pi f_m{r}_g}{c}})\exp (\mathrm{j}{\displaystyle \frac{4\pi f_m{\nu }_g{t}_m}{c}})+n(t,t_m)\\ & ={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{m=1}^M{A}_g{\mathrm{\Phi }}_r^{‴}}}(t-t_m-{\displaystyle \frac{2r_g}{c}})\exp (-\mathrm{j}{\displaystyle \frac{4\pi (f_0+b(m)\Delta f)r_g}{c}})\\ & \exp (\mathrm{j}{\displaystyle \frac{4\pi (f_0+b(m)\Delta f){\nu }_g((m-1)T_r+a(m)\Delta T)}{c}})+n(t,t_m)\end{array}$$

(13)

Writing and expanding the phase term of Equation (13) separately gives

$$\begin{array}{l}{h}_m(t)=\exp (-\mathrm{j}{\displaystyle \frac{4\pi f_0{r}_g}{c}})\exp (-\mathrm{j}{\displaystyle \frac{4\pi b(m)\Delta f{r}_g}{c}})\\ \exp (\mathrm{j}{\displaystyle \frac{4\pi (f_0+b(m)\Delta f){\nu }_g((m-1)T_r+a(m)\Delta T)}{c}})\end{array}$$

(14)

From Equation (14), the above equation can be obtained by the presence of a fixed-phase term $\exp (-j4\pi f_0{r}_g/c)$, a coupling term $\exp (-j4\pi b(m)\Delta f{r}_g/c)$ between the initial distance and carrier frequency variation term $a(m)$, and a coupling term $\exp (j4\pi (f_0+b(m)\Delta f){\nu }_g((m-1)T_r+a(m)\Delta T)/c)$ between the target velocity and random PRI and random carrier frequency.

The phase term of the carrier frequency shortcuts and PRI shortcuts is

$$x_m(t)=\exp (-\mathrm{j}{\displaystyle \frac{4\pi b(m)\Delta f{r}_g}{c}})\exp (\mathrm{j}{\displaystyle \frac{4\pi (f_0+b(m)\Delta f){\nu }_g((m-1)T_r+a(m)\Delta T)}{c}})$$

(15)

The distance resolution is $\Delta R$, and the speed resolution is $\Delta \nu$.

$$\left\{\begin{array}{c}{p}_d=2\pi \Delta f{\displaystyle \frac{2(r_0+d\Delta R)}{c}},0\le d\le D\\ q_l=2\pi f_m{\displaystyle \frac{2({\nu }_0+l\Delta \nu )t_m}{c}},0\le l\le L\end{array}\right.$$

(16)

The vertical coordinates in Figure 3 indicate that $R_0$ is the initial distance and that $\Delta R$ is the amount of change increasing line by line, dividing the distance into $D$ individual grid points. The horizontal coordinates indicate that $V_0$ is the initial speed, $\Delta V$ is the number of changes that increase line by line, and the speed is divided into $L$ independent grid points.

Substituting Equation (16) into Equation (15) yields the phase term about the PRI, and the carrier frequency shortcut is expressed as

$${\mathit{x}}_m(d,l)=\exp (-\mathrm{j}b(m){\mathit{p}}_d)\exp (\mathrm{j}{\mathit{q}}_l)$$

(17)

The amplitude value of the echo signal is

$${\mathit{z}}_m(d,l)={\mathit{A}}_{\mathrm{g}(d,l)}\mathrm{\Phi }(t-{\displaystyle \frac{2r}{c}})\exp (-\mathrm{j}4\pi f_0{r}_g/c)$$

(18)

The echo of the signal can be re-expressed as

$${\mathit{S}}_m(d,l)={\mathit{z}}_m(d,l){\mathit{x}}_m(d,l)+\mathit{n}(t_m)$$

(19)

Constructing a Perception Matrix,

$$\mathit{E}={[{\mathit{e}}_{1,1},{\mathit{e}}_{1,2},{\mathit{e}}_{1,3},\cdots ,{\mathit{e}}_{D,L}]}_{M\times (D\times L)}$$

(20)

$${\mathit{e}}_{d,l}=[{\mathit{p}}_d(1)\times {\mathit{q}}_l(1),{\mathit{p}}_d(2)\times {\mathit{q}}_l(2),{\mathit{p}}_d(3)\times {\mathit{q}}_l(3),\cdots ,{\mathit{p}}_d(M)\times {\mathit{q}}_l(M)$$

(21)

This is obtained by substituting Equation (21) into Equation (20):

$$E={\left[\begin{array}{ccccccc}{e}_1(0,0)& \dots & e_1(D-1,0)& \dots & e_1(0,L-1)& \dots & e_1(D-1,L-1)\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ e_M(0,0)& \dots & e_M(D-1,0)& \dots & e_M(0,L-1)& \dots & e_M(D-1,L-1)\end{array}\right]}_{M\times DL}$$

(22)

Schematic diagram of the perceptual matrix is shown in Figure 4.

Then, the echo signal can be expressed as

$${\mathit{S}}_m=\mathit{E}{\mathit{\theta }}_m+\mathit{n}(t_m)$$

(23)

After constructing Equation (23), the sparse vector $\theta$ can be obtained by solving the underdetermined equation. The estimate of $\theta$, $\widehat{\theta }$ can be obtained by solving the following ${\mathsf{\ell }}_1$ paradigm optimisation problem:

$$\begin{array}{l}\widehat{\mathit{\theta }}=\arg \min \left({‖\theta ‖}_1\right)\\ \mathit{s.t.}{‖\mathit{S}-\mathit{E}\mathit{\theta }‖}_2\le {‖\mathit{n}‖}_2\end{array}$$

(24)

In summary, the algorithm flow of OMP is shown in

Table 4. OMP algorithm flow.

Inputs: observation signal $\mathit{S}$, observation matrix $\mathbf{E}$, number of iterations $V$

Initialisation: the residuals $\mathit{r}$ are initially $\mathit{S}$$, t=1$$, A_t$ are empty matrices

Start Iteration

Step 1: Calculate the inner product of each column of $\mathit{S}$ and $\mathbf{E}$, find the largest column and record it in ${\mathit{A}}_t$.

Step 2: Find the least squares solution of $\mathit{S}={\mathit{A}}_t\mathit{\theta }$, the $\mathit{\theta }={({\mathit{A}}_t^T{\mathit{A}}_t)}^{-1}{\mathit{A}}_t^T\mathit{S}$.

Step 3: Update the residuals. $\mathit{r}=\mathit{s}-{({\mathit{A}}_t^T{\mathit{A}}_t)}^{-1}{\mathit{A}}_t^T\mathit{S}$.

Step 4: $t=t+1$, if $t>V$ then stop the iteration.

Output: $\mathit{\theta }$

.

2.2.2. Communication Signal Processing

The echo signal received by the communication receiver is as follows:

$$S_c(t,t_m)={\displaystyle \sum _{m=1}^M{\varphi }_{m_c}(t-t_m-{\tau }_c)\exp (\mathrm{j}2\pi f_m(t-t_m-{\tau }_c))}+n(t,t_m)$$

(25)

where ${\varphi }_{m_c}^{‴}(t-t_m)=\mathrm{rect}((t-t_m)/T_p){\beta }_m(t-t_m)\exp (\mathrm{j}\pi K{\left(t-t_m\right)}^2)$ and ${\tau }_c=R/c$ denote the time delay between the transmitter and the communication receiver, and the baseband signal obtained after mixing at the communication receiver is

$$S_c(t,t_m)={\displaystyle \sum _{m=1}^M{\varphi }_{m_c}(t-t_m-{\tau }_c)}\exp (-\mathrm{j}2\pi f_m{\tau }_c)+n(t,t_m)$$

(26)

A signal received at a communication receiver is matched and filtered with a set of matched filters ($h_m(t),m=1,2,3,\cdots ,M*M$), where $h_m(t)=s_m^{\ast }(t)$ and its $M$ carrier frequencies correspond to the matched filters.

$$S_c(t,t_m)={\displaystyle \sum _{m=1}^M{A}_{m_1{m}_2}{\mathrm{\Phi }}_{m_1{m}_2}(t-t_m-{\tau }_c)}\exp (-\mathrm{j}2\pi f_m{\tau }_c)+n(t,t_m)$$

(27)

Because the orthogonal waveform set has good intercorrelation characteristics, each $s_m(t)$ only has greater spikes at the output of its corresponding $h_m(t)$, i.e., when the value of $A_{m_1{m}_2}$ is the largest when $m_1=m_2$ is used, then the largest peak can be selected as the original transmitting orthogonal waveform and its carrier frequency, i.e., we can obtain the orthogonal waveform and its carrier mapping of the communication data.

For the communication information mapped by the PRI shortcut, the corresponding time at the peak, i.e., $t_m+{\tau }_c$, can then be subtracted from ${\tau }_c$ to obtain the time of the PRI shortcut $t_m$; then, the communication data of the PRI shortcut can be demodulated according to the mapping relationship.

The communication data of PRI shortcut demodulation are $D_1$, the communication data of carrier frequency shortcut demodulation are $D_2$, the communication data of waveform demodulation are $D_3$, and the total communication data are $D^{‴}=D_1+D_2+D_3$, which completes the demodulation of the total communication information. The communication demodulation processing flowchart is shown in Figure 5.

3. Results

3.1. System Simulation Analysis

To verify the effectiveness of the integrated waveform design of the probe-through, the simulation parameters in this paper are shown in

Table 5. Simulation parameters.

Parameter Type	Numerical Value	Parameter Type	Numerical Value
$\mathrm{Pulse} \mathrm{width}/\mathrm{\mu }\mathrm{s}$	50	Number of pulse sets/pc	64
$\mathrm{Carrier} \mathrm{frequency}/\mathrm{G}\mathrm{H}\mathrm{z}$	14	Total pulse modulation code words/each	128
$\mathrm{Signal} \mathrm{bandwidth}/\mathrm{M}\mathrm{H}\mathrm{z}$	20	$\mathrm{PRI} \mathrm{shortcut} \mathrm{interval}/\mathrm{\mu }\mathrm{s}$	1
$\mathrm{Sub}-\mathrm{band}/\mathrm{M}\mathrm{H}\mathrm{z}$	(16, 20)	$\mathrm{Pulse} \mathrm{repetition} \mathrm{period}/\mathrm{\mu }\mathrm{s}$	400
$N$/each	90	$\xi$ dB	−20

:

3.1.1. Orthogonal Waveform Analysis

To verify that the orthogonal waveform has good autocorrelation and cross-correlation performance, this paper compares the autocorrelation peak sidelobe level with that of the LFM-BPSK waveform. From Figure 6a, the average values of the autocorrelation peak parallax levels of the LFM-BPSK waveform and the orthogonal waveform are −15.66 dB and −22.69 dB, and the two comparisons of the latter have decreased by 7.03 dB; from Figure 6b, the worst values of the autocorrelation peak parallax levels of the LFM-BPSK waveform and the orthogonal waveform are −13.78 dB and −20.37 dB, and the two comparisons of the latter have decreased by 6.59 dB; from Figure 6c, the best values of the autocorrelation peak parallax levels of the LFM-BPSK waveform and the orthogonal waveform, the best values of the autocorrelation peak sidelobe level are −17.26 dB and −24.78 dB, which have decreased by 7.52 dB in comparison with the latter. Figure 6d shows the CPSL plot of the quadrature waveform. The simulation results show that the APSL of both orthogonal waveforms is smaller than that of LFM-BPSK.

3.1.2. Effect of PRI Shortcuts on Pulse Pressure

To verify that the introduction of PRI shortcuts in this chapter did not affect radar detection, conventional methods of embedding communication information in radar waveforms lose radar detection performance. The PRI shortcut times of the 1st, 37th, 45th, and 52nd waveforms are selected, and the PRI shortcut times are $0 \mathrm{\mu }\mathrm{s}$, $6 \mathrm{\mu }\mathrm{s}$, $4 \mathrm{\mu }\mathrm{s}$, and $8 \mathrm{\mu }\mathrm{s}$, respectively. Figure 7 shows the comparison of the pulse compression results of the PRI shortcut for the echo delay compensation and PRI unchanged at the distance, Figure 8 shows the comparison of the pulse compression results of the PRI unchanged and shortcut compensation, and the simulation test shows that the PRI shortcut has almost no effect on the pulse compression results after the compensation of the echo delay, which indicates that the PRI shortcut has no effect on the radar detection.

3.1.3. Effect of Carrier Frequency Shortcuts on Pulse Pressure

Conventional methods of embedding communication information in the radar waveform lose radar detection performance to verify that the carrier frequency shortcut does not affect radar detection. The target is at 4300 $\mathrm{m}$ under a signal-to-noise ratio of 10 dB. The carrier frequencies of the waveform are 1 GHz and 1.58 GHz. Figure 9 shows the pulse pressure results of the waveform with a carrier frequency of 1 GHz and a carrier frequency of 1 GHz. The simulation results in Figure 9 and Figure 10 show that the different carrier frequency pulse pressures do not affect the distance of the target, indicating that the introduction of the carrier frequency shortcut does not affect radar detection.

3.1.4. Impact of the Carrier Frequency Shortcut on the Waveform Set Performance

To verify that the carrier frequency shortcut has no effect on the performance of the waveform set, it is assumed that waveform 3 in the orthogonal waveform set is selected, in which the carrier frequencies of the waveform are 1 GHz and 1.58 GHz. Figure 11 shows the autocorrelation results for the waveform with a carrier frequency of 1 GHz, and Figure 12 shows the mutual correlation results for the waveform with a carrier frequency of 1 GHz and a carrier frequency of 1.58 GHz. The simulation results in Figure 11 and Figure 12 show that it does not affect the performance of the orthogonal waveform set of the waveform.

3.1.5. Radar Performance Analysis

The detection performance of the three-dimensional index-modulated probe-through integration waveform for single-target distance and velocity information is verified. Under a signal-to-noise ratio of 10 dB, the number of PRI shortcuts is 128 and the PRI shortcut interval is $1 \mathrm{\mu }\mathrm{s}$. The PRI shortcut sequence is shown in Figure 13; the number of carrier frequency shortcuts is 64 and the carrier frequency shortcut interval is 20 MHz. The carrier frequency shortcut sequence is shown in Figure 14. The number of waveforms selected from the waveform set is 64. The perception matrix in the OMP algorithm is constructed with the coupling terms of the initial distance and carrier frequency change term, the target velocity and random PRI, and random carrier frequency coupling terms. Assuming that there is a single target in the radar scene, the number of iterations of the OMP algorithm $V$ is 1. The distance of the target is assumed to be 3080 $\mathrm{m}$, and the speed is 100 m/s. The simulation results are shown in Figure 15, where the horizontal coordinates denote the distance and the vertical coordinates denote the speed. Figure 15 shows that the distance of the target is 3080 $\mathrm{m}$, and the speed is 100 m/s. The results show that the probe-through-integrated waveform with multidimensional index modulation is able to detect information about a single target.

The detection performance of the three-dimensional indexed modulated probe-through integration waveform for multiple targets with distance and velocity information is verified. Under a signal-to-noise ratio of 10 dB, the number of PRI shortcuts is 128 and the PRI shortcut interval is $1 \mathrm{\mu }\mathrm{s}$. The PRI shortcut sequence is shown in Figure 16; the number of carrier frequency shortcuts is 64 and the carrier frequency shortcut interval is 20 MHz. The carrier frequency shortcut sequence is shown in Figure 17. The number of waveforms selected from the waveform set is 64. The perception matrix in the OMP algorithm is constructed with the coupling term of the initial distance and carrier frequency change term, the target speed and random PRI, and the random carrier frequency coupling term. Assuming that there are multiple targets in the radar scene, the number of iterations of the OMP algorithm $V$ is 3. The distance and speed information corresponding to Target 1 is 3080 m and 90 m/s; the distance and speed information corresponding to Target 2 is 3100 m and 50 m/s; and the distance and speed information corresponding to Target 3 is 3140 m and 100 m/s. Figure 18 shows that the distance and speed information of Target 1 is 3080 m and 90 m/s; the distance and speed information of Target 2 is 3100 m and 50 m/s; and the distance and speed information of Target 3 is 3140 m and 100 m/s. The results show that the probing integrated waveform with multidimensional index modulation is able to detect the information of multiple targets. For radar systems, bandwidth directly affects detection resolution and the operating range of the system. Modern radar systems typically employ wider bandwidths (e.g., GHz levels), which help to provide higher resolution and better target identification. However, a relatively narrow bandwidth of 20 MHz is chosen in this paper, the reason being that in some cases, hardware limitations or spectrum licences may make the use of a narrower bandwidth the only viable option. In addition, some radar applications such as short-range detection may not require very high resolution, and narrow bandwidths can be effective in reducing system complexity. In addition, in practical radar systems, it is ensured that the 64 waveforms are highly orthogonal to each other and maintain low levels of autocorrelation and cross-correlation; therefore, to ensure high orthogonality between waveforms, the signals can be assigned to different frequency subcarriers by assigning the signals to different frequency subcarriers. Orthogonal Frequency Division Multiplexing ensures that a plurality of waveforms are orthogonal to each other in the frequency domain, and this method generates waveforms that do not interfere with each other in either the time or frequency domains. In order to reduce the autocorrelation of the waveforms, the frequency of the signals can be changed. FM waveforms such as linear FM or spread spectrum FM have low autocorrelation and can reduce the interference of temporally repetitive signals.

3.1.6. Communication Performance Analysis

The communication performance of the three-dimensional indexed modulated probe–pass integrated waveform is verified. The false symbol rate of the three-dimensional index-modulated tan–pass integrated waveform is calculated via 10,000 Monte Carlo simulations under the conditions of signal-to-noise ratios (SNRs) of −30 dB to −16 dB. The variation curve of the false symbol rate with the SNR is shown in Figure 19 shows that, compared with the false symbol rate of the conventional LFM-BPSK, the design of the probe-through integrated waveform with multidimensional index modulation in this paper is able to achieve a lower false symbol rate with a low signal-to-noise ratio. However, it is worth noting that achieving a signal-to-noise ratio of −30 dB to −16 dB is indeed a considerable challenge for most radar systems, especially in the absence of efficient noise reduction techniques. Therefore, in order to remain robust under such conditions, the signal strength can be increased in the design by gain control and power amplifier optimisation to cope with the low SNR environment, while the use of advanced signal processing techniques, such as adaptive filtering, waveform design, and multiple-input multiple-output MIMO techniques, can help to enhance the detection performance at lower SNR.

In addition, this paper also compares the number of individual pulse transmission bits of different integrated waveforms, and the comparison results are shown in

Table 6. Number of bits transmitted for a single pulse of an integrated waveform.

Integrated Waveform	A Single Pulse Transmits a Number of Binary Bits (Bit)	Time (s)
Combined waveform selection and PRI shortcut integration waveforms	42	2.049
Combined waveform selection and carrier frequency shortcut integration waveforms	36	3.689
Combined carrier-frequency shortcut and PRI shortcut integration waveforms	42	0.327
Methodology of this paper	256	5.169

, from which we can clearly see that the performance of the integrated waveforms of this paper’s method is better than that of other similar integrated waveforms as a whole, and thus further proves the effectiveness and superiority of this paper’s method.

4. Discussion

Research on the integration of radar communication based on index modulation has always been a challenging and practical topic with great value. Despite in-depth research in this field, due to subjective and objective constraints such as personal ability, space limitations, and time constraints, this paper still has deficiencies in terms of the breadth and depth of research. Specifically, this paper may not have fully explored or thoroughly researched certain aspects, and there are still many directions that warrant further exploration and research.

In the future, these deficiencies can be addressed through more in-depth research to promote the further development and application of radar communication integration technology.

First, regarding the research on the performance verification of the radar communication integration system, this paper focuses on the discussion at the level of software simulation and has not yet been empirically verified at the level of hardware equipment. Although software simulation provides an important basis for theoretical verification and performance prediction, verification of actual hardware equipment is crucial to ensuring the performance and reliability of the system in actual application. An important direction for future research is to practically test and verify the two-dimensional index modulation exploration integration waveform design and the multi-dimensional index modulation exploration integration waveform design scheme proposed in this paper on hardware equipment for actual testing and verification.

Second, MIMO technology, as an advanced wireless communication technology, has shown great application potential in the field of radar communication integration. Applying it to index modulation-based integrated waveform design not only significantly enhances the detection performance of the radar, but also provides greater bandwidth for communication, thereby further improving transmission efficiency and communication quality.

Third, compared to radar waveform coding technology’s anti-jamming ability, ordinary radar waveforms tend to be relatively simple and easily recognised by the enemy’s electronic warfare equipment and interference, while the coded waveforms, due to a special time-domain or frequency-domain characteristics, can effectively resist interference. Compared to radar waveform coding technology with higher target identification capabilities, coded waveforms usually provide richer signal information, which helps to more accurately analyse the target characteristics. Compared with ordinary radar waveforms, they can better cope with the complex battlefield environment in the target identification task. Compared to radar waveform coding technology’s better concealment, ordinary radar waveform signal patterns may be relatively easily intercepted by the enemy radar, while the complexity of the coded waveform makes the radar detection more difficult to identify and localise by the enemy.

Fourth, research on the integrated radar-communication signal is mostly based on the assumption of Gaussian white noise. However, the real-world electromagnetic environment is far more complex, especially in fading channel conditions, where signal quality can be severely interfered with and affected. Therefore, it is necessary to explore the actual performance of the integrated radar-communication signal in fading channels.

In addition, the 3D index modulation of waveforms is a technique to improve the efficiency of data transmission in wireless communication systems by adding extra dimensions. However, this technique is also challenging, with potential errors and implementation limitations associated with the hardware. Three-dimensional modulation requires the system to be able to accurately control the signal in multiple dimensions (e.g., time, frequency, space, etc.). Any inaccuracies or errors in the hardware can affect the accurate transmission of the signal, leading to data loss or incorrect decoding. This limitation is reflected in the fact that 3D modulation increases the complexity of the system, especially in terms of encoding, decoding, and signal processing. In order to be successfully implemented, the hardware must have more computational power, which may be beyond the capabilities of many legacy devices.

With the continued in-depth exploration of integrated radar-communication waveforms, this field will continue to produce more high-quality research results. These studies will not only promote the technical progress of radar and wireless communication fusion systems, but will also promote the widespread deployment and application of such systems in many practical application scenarios.

5. Conclusions

This paper proposes an integrated waveform design method for radar detection and communication, which realises an orthogonal waveform design by attaching perturbation phases to LFM signals and indexes the three dimensions of orthogonal waveform selection, the PRI shortcut, and the carrier frequency shortcut to realise the modulation of communication information. The integrated waveform is introduced for radar detection and communication processing flow, and a simulation is used to prove that the integrated waveform designed in this paper can meet radar detection requirements while improving the communication rate and stable communication transmission.