An Integrated Orthogonal Frequency-Division Multiplexing Chirp Waveform Processing Method for Joint Radar and Communication Based on Low-Density Parity-Check Coding and Channel Estimation

Abstract

With the advancement of information technology construction, the integration of radar and communication represents a crucial technological evolution. Driven by the research boom of integrated sensing and communications (ISACs), some scholars have proposed utilizing orthogonal frequency-division multiplexing (OFDM) to separately modulate radar and communication signals. However, the OFDM symbols in this paper incorporate a cyclic prefix (CP) and a virtual carrier (VC) instead of zero padding (ZP). This approach mitigates out-of-band power caused by ZP, in addition to reducing adjacent channel interference (ACI). In addition, we introduce low-density parity-check (LDPC) and use an improved normalized min-sum algorithm (NMSA) in decoding. The enhanced decoding efficiency and minimized system errors render the proposed waveform more suitable for complex environments. In terms of signal processing methods, this paper continues to use radar signals as a priori information to participate in channel estimation. Further, we consider the symbol timing offset (STO) and carrier frequency offset (CFO) issues. In order to obtain more reliable data, we use the minimum mean-square error (MMSE) estimation based on the discrete Fourier transform (DFT) to evaluate the channel. Simulation experiments verify that the system we propose not only realizes the transmission and detection functions but also improves the performance index of the integrated signal, such as the bit error rate (BER) of 7 × 10⁻⁵, the peak side lobe ratio (PSLR) of −13.81 dB, and the integrated side lobe ratio (ISLR) of −8.98 dB at a signal-to-noise ratio (SNR) of 10 dB.

1. Introduction

Facing the arrival of the 6G communication era, the proliferation of wireless communication devices is inevitable, and spectrum resources will become even scarcer. In the design of lightweight systems such as earthquake relief, intelligent transportation, and smart agriculture, the integration of independent radar and communication devices may result in system redundancy [1]. To efficiently utilize spectrum and hardware resources, radar communication spectrum sharing (RCSS) is categorized into two research lines: radar communication spectrum coexistence (RCC) and radar communication integrated system (DFRC) [2]. Regarding joint radar and communication systems, the core issue is the integrated waveform design. Therefore, there is a need to design a new multiplexed waveform that can carry communication information and be used for target detection at the same time.

Integrated signals can be classified into two major categories: multiplexed signals and shared signals [3]. The performance of multiplexed signals depends on their ability to suppress interference between radar and communication through multiplexing techniques. However, under the constraints of simultaneous frequency and spatial sharing, multiplexing techniques limited to code division multiplexing do not meet the orthogonal requirements defined by the radar ambiguity functions. Also, code division multiplexing is more prone to co-frequency interference [4,5,6]. In the case of shared signals, radar-shared signals are used to carry communication information by adjusting the parameters of the radar signal [7,8]. To ensure the detection performance of the radar, this method has a few adjustable parameters. And the communication data rate of the radar-shared signals is low because the communication channel estimation and channel equalization are not considered [9,10,11]. Therefore, with a focus on orthogonal frequency-division multiplexing (OFDM), communication-shared waveforms are currently a research hotspot [12,13,14,15].

An integrated signal design, a processing scheme for radar communication based on OFDM, has been proposed by both Sturm et al. [16] and Garmatyuk et al. [17]. Han et al. summarized the research and progress on integrated waveform design, highlighting that communication-shared signals based on OFDM are the most favorable integrated signal scheme [18]. Yuzhen Zhao et al. achieved the integrated waveform design by separately modulating radar and communication functions using orthogonal frequency-division multiplexing with zero padding (ZP-OFDM) signals [19]. They improved radar detection performance by avoiding false targets introduced during detection through blank guard intervals. Even if the length of the zero padding (ZP) is greater than the maximum delay of the multipath channel, a small symbol timing offset (STO) will cause discontinuities in the OFDM symbols within the effective period inside the FFT window. Additionally, compared to the orthogonal frequency-division multiplexing with cyclic prefix (CP-OFDM) symbols, ZP-OFDM symbols will introduce higher out-of-band power.

Therefore, to enhance the system’s robustness against channel-induced distortion, this paper integrates the cyclic prefix (CP) and virtual carriers (VC) into the composite signal. Furthermore, to reduce communication error rates, we introduce low-density parity-check (LDPC) in this system. On the receiver side, channel estimation performance is improved by the symbol timing offset (STO) estimation, the carrier frequency offset (CFO) estimation, and channel estimation techniques assisted by radar-based a priori information. During radar detection, the joint estimation of radar target distance and velocity is achieved by using matched filtering, pulse compression, and windowing to obtain the delay and Doppler frequency. We reduce the false alarm probability by setting a fixed threshold.

The content arrangement of this article is as follows: Section 1 is an introduction, providing an improvement plan based on the current research status. Section 2 describes the design methods of integrated signals, including the existing integrated signals with OFDM odd–even subcarrier modulation and the integrated signals based on the LDPC encoding in OFDM chirp. Section 3 presents the processing methods for the proposed integrated signals, including the improved normalized minimum-sum algorithm in LDPC, synchronization estimation, channel estimation, and equalization, as well as radar signal processing methods. Section 4 presents the experimental setup parameters and results’ discussion. Section 5 provides conclusions.

2. Design Methodology for Integrated Signal

2.1. OFDM Chirp Signal

To improve bandwidth efficiency, extending the Nyquist criterion from single-carrier to multi-carrier communication leads to the generation of overlapping orthogonal subcarrier signals in the frequency domain, known as OFDM [20]. Since the signals between different subcarriers of OFDM satisfy orthogonality, its odd and even carriers can be utilized to modulate radar and communication signals, respectively [19,21,22]. Figure 1 shows the frequency domain subcarrier assignments (left) and the corresponding time–frequency plots (right) for each orthogonal waveform [22]. Figure 2 represents a schematic diagram of the integrated radar and communication signal waveform, where $\mathrm{x}\_com$ is the communication message modulated by $16-QAM$ and $x\_rad$ is the chirp information sampled in the frequency domain.

Let $X_l[2K+1]$ denote the $\left(2k+1\right)-th$ radar transmission symbol sent on the $l-th$ subcarrier and $X_l[2K]$ represent the $2k-th$ communication transmission symbol sent on the $l-th$ subcarrier, where $l=0,1,2,\cdots ,\infty ,k=0,1,2,\cdots ,N-1$. The transmission time of $N$ symbols is expanded to the duration of a single symbol, i.e., $T_{sym}=N{T}_s$, the integrated signal on the $k-th$ subcarrier is given by the following:

(1)${\Psi }_{l,k}\left(t\right)=\{\begin{array}{l}{e}^{j2\mathsf{\pi }{f}_k\left(t-T_{sym}\right)},0<t⩽T_{sym}\hfill \\ 0,\hspace{1em}\hspace{1em}\hspace{1em} else\hfill \end{array}$

The time-continuous passband radar and communication signals are denoted, respectively [19,20], as follows:

(2)$\{\begin{array}{l}{x}_{rad}\left(t\right)=Re\left\{\frac{1}{T_{sym}}{\displaystyle {\displaystyle \sum }_{l=0}^{\infty }}\left\{{\displaystyle {\displaystyle \sum }_{k=0}^{N-1}}{X}_l\left[2k+1\right]{\Psi }_{l,2k+1}\left(t\right)\right\}\right\}\hfill \\ x_{com}\left(t\right)=Re\left\{\frac{1}{T_{sym}}{\displaystyle {\displaystyle \sum }_{l=0}^{\infty }}\left\{{\displaystyle {\displaystyle \sum }_{k=0}^{N-1}}{X}_l\left[2k\right]{\Psi }_{l,2k}\left(t\right)\right\}\right\}\hfill \end{array}$

The baseband signals for radar and communication are denoted, respectively, as follows:

(3)$\{\begin{array}{l}{x}_{rad}\left(t\right)={\displaystyle {\displaystyle \sum }_{l=0}^{\infty }{\displaystyle \sum }_{k=0}^{N-1}}{X}_l\left[2k+1\right]e^{j2\mathsf{\pi }{f}_{2k+1}\left(t-l{T}_{sym}\right)}\hfill \\ x_{com}\left(t\right)={\displaystyle {\displaystyle \sum }_{l=0}^{\infty }{\displaystyle \sum }_{k=0}^{N-1}}{X}_l\left[2k\right]e^{j2\mathsf{\pi }{f}_{2k}\left(t-l{T}_{sym}\right)}\hfill \end{array}$

Sampling the integration signal $x\left(t\right)=x_{rad}\left(t\right)+x_{com}\left(t\right)$ at $t=l{T}_{sym}+n{T}_s$, $T_s=T_{sym}/N$, $f_k=K/T_{sym}$ yields the corresponding discrete-time integration symbols:

(4)$x_l\left[n\right]={\displaystyle \sum }_{k=0}^{N-1}{X}_l\left[k\right]e^{j2\mathsf{\pi }kn/N},n=0,1,\cdots ,N-1$

Since the multipath channel generates inter-symbol interference (ISI) to this integrated symbol, we insert a guard interval between two consecutive symbols. In this paper, we utilize the CP to achieve cyclic expansion of the integrated symbol. For the second subcarrier signal with a delay of $t_0+T_s$, the following equation is satisfied:

(5)$\frac{1}{T_{sub}}{{\displaystyle \int }}_0^{T_{sub}}{e}^{j2\mathsf{\pi }{f}_k\left(t-t_0\right)}{e}^{-j2\mathsf{\pi }{f}_k\left(t-t_0-T_s\right)}dt=0,k\ne i$

where $T_{sub}=N{T}_s=1/\Delta f$, orthogonality between subcarriers can be maintained when the length of the CP is greater than the maximum delay of the multipath channel.

Additionally, since each subcarrier signal on each symbol is time-limited, the integrated signal will generate out-of-band radiation, leading to adjacent channel interference (ACI). To minimize out-of-band radiation, it is necessary to allocate guard bandwidth on the outermost subcarriers, i.e., using Hanning windows and VC.

Under the condition that the transmit samples and channel samples satisfy the circular convolution, the integrated signal can be obtained at the receiver as follows [20]:

(6)$\begin{array}{ll}{Y}_l\left[k\right]& ={\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{y}_l\left[n\right]e^{-j2\mathsf{\pi }kn/N}\\ & ={\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}\left\{{\displaystyle {\displaystyle \sum }_{m=0}^{\infty }}{h}_l\left[m\right]x_l\left[n-m\right]+z_l\left[m\right]\right\}{e}^{-j2\mathsf{\pi }kn/N}\\ & ={\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}\left\{{\displaystyle {\displaystyle \sum }_{m=0}^{\infty }}{h}_l\left[m\right]\left\{\frac{1}{N}{\displaystyle {\displaystyle \sum }_{N=0}^{N-1}}{X}_l\left[i\right]e^{-j2\mathsf{\pi }i\left(n-m\right)/N}\right\}\right\}{e}^{-j2\mathsf{\pi }kn/N}+Z_l\left[k\right]\\ & =\frac{1}{N}{\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}\left\{\left\{{\displaystyle {\displaystyle \sum }_{m=0}^{\infty }}{h}_l\left[m\right]e^{-j2\mathsf{\pi }im/N}\right\}{X}_l\left[i\right]{\displaystyle {\displaystyle \sum }_{n=0}^{\infty }}{e}^{-j2\mathsf{\pi }\left(k-i\right)n/N}\right\}{e}^{-j2\mathsf{\pi }kn/N}+Z_l\left[k\right]\\ & =H_l\left[k\right]X_l\left[k\right]+Z_l\left[k\right]\end{array}$

where $X_l[k],Y_l[k],H_l[k],Z_l[k]$ are denoted as the transmitted symbols, the received symbols, the frequency response of the channel, and the noise in the frequency domain on the $k-th$ subcarrier of the $l-th$ symbol, respectively.

2.2. OFDM Chirp Signal Based on LDPC Coding

Low-density parity-check (LDPC) codes have become the preferred channel coding scheme for wireless communication standards such as IEEE802.16e and 5G [23] because of their error-correcting performance that approaches the Shannon limit [23]. In this integrated system, some consecutive subcarriers may undergo deep fading [20]. Therefore, this paper introduces LDPC coding to address the random errors caused by deep fading in a multi-carrier scenario. The LDPC code $v$ is a set of linear packet codes $(n,k)$, where $n$ is the code length and $k$ is the length of the message sequence, which can be uniquely determined by a sparse check matrix $H$. Each row of the checksum matrix corresponds to a checksum equation, and each column corresponds to a variable node. The LDPC encoder adds redundancy to the data based on the LDPC checksum matrix to enhance the error correction of the signal [23,24].

Additionally, we chose an LDPC code rate of 1/2 to enhance the error correction performance of the signal. The error correction capability of LDPC codes is closely tied to their code rate. For LDPC codes with a code rate of 1/2, the error correction performance is more robust under the same codeword length. Additionally, the use of a lower code rate can reduce the complexity of decoding, making it suitable for communication systems with constrained resources. Some communication standards and protocols have adopted the design of LDPC codes with a rate of 1/2, such as the Wi-Fi standard, enhancing compatibility between devices and improving the interoperability of communication systems.

Figure 3 represents a block diagram of the OFDM chirp transceiver system design based on LDPC coding. The communication information is LDPC-coded in the form of a bit stream [25,26]. To enhance communication efficiency, every four bits of encoded data are converted to decimal for $16-QAM$ modulation. After inserting VC into the communication signal, it undergoes odd–even subcarrier modulation using OFDM, along with the chirp signal sampled in the frequency domain, to generate multiple integrated symbols. The CP is then inserted between symbols to obtain the integrated signal. At the receiving end, upon receiving the integrated signal carrying channel information, the signal is subjected to orthogonal demodulation. In communication signal processing, the radar signal can be treated as pilot information inserted at a 1:1 ratio for channel estimation. This, along with the parameters obtained from timing synchronization, is used for channel equalization, resulting in more accurate communication data. In radar signal processing, joint super-resolution estimation of radar target range and velocity is achieved using matched filtering and pulse compression.

3. The Processing Method for Integrated Radar–Communication Signals

3.1. Improved Normalized Minimum-Sum Algorithm in LDPC

Some scholars have simulated communication systems with integrated LDPC in Rayleigh fading channels, and the results indicate that the use of the sum–product algorithm (SPA) significantly enhances system performance [27]. The SPA in traditional LDPC involves a large number of multiplications, resulting in high computational complexity during implementation [28]. Therefore, the probability information is represented using likelihood ratios, leading to the log-likelihood ratio belief propagation (LLR BP) algorithm [23]. In this case, the initial messages for the channels are as follows [23]:

(7)$L\left(p_i\right)=\frac{\ln p_i\left(0\right)}{\ln p_i\left(1\right)}=\ln \frac{P\left(v_i=0|y_i\right)}{P\left(v_i=1|y_i\right)}$

Here, $v_i$ represents the $i-th$ variable node, and $p_i\left(b\right)$ represents the posterior probability of codeword $c_i=b$ when $y_i$ is received at the receiving end. The external information passed by the calibration node to the variable node is as follows:

(8)$r_{ji}=\ln \frac{r_{ji}\left(0\right)}{r_{ji}\left(1\right)}=\ln \frac{1+{\displaystyle {\displaystyle \prod }_{i^{\prime }\in V\left(j\right)/i}}\left(1-2q_{i^{\prime }j}\left(1\right)\right)}{1-{\displaystyle {\displaystyle \prod }_{i^{\prime }\in V\left(j\right)/i}}\left(1-2q_{i^{\prime }j}\left(1\right)\right)}$

in which, $r_{ij}$ represents the external information passed from check node $j$ to variable node $i$, and $V(j)/i$ denotes the set of other variable nodes connected to $j-th$ check node, except for the $i-th$ variable node. The simplification of external information yields the following with the introduction of $\tanh (x)=(e^x-e^{-x})/(e^x+e^{-x})$:

(9)$r_{ji}=2{\tanh }^{-1}({\displaystyle \prod }_{i^{\prime }\in V\left(j\right)/i}\tanh \frac{q_{i^{\prime }j}}{2})$

Even with the introduction of $\tanh$ in the logarithmic domain, it still consumes a significant amount of hardware resources. In this paper, the minimum-sum belief propagation (min-sum BP) algorithm is introduced to convert complex function operations into symbol value operations and numerical comparison operations. Here, the minimum value is used to replace the multiplication results:

(10)${\displaystyle \prod }_{i^{\prime }\in V\left(j\right)/i}\tanh \left|\frac{q_{i^{\prime }j}}{2}\right|\approx \underset{i^{\prime }\in V\left(j\right)/i}{\min }(2\tanh \left|\frac{q_{i^{\prime }j}}{2}\right|)$

The check node update value of the min-sum BP algorithm is greater than that obtained by the LLR BP algorithm, i.e.,

(11)$2{\tanh }^{-1}({\displaystyle \prod }_{i^{\prime }\in V\left(j\right)/i}\tanh \frac{q_{i^{\prime }j}}{2})<\underset{i^{\prime }\in V\left(j\right)/i}{\min }(\tanh \left|q_{i^{\prime }j}\right|)$

Therefore, in this paper, we multiply this outer information by the multiplicative scaling factor $\alpha$ to obtain the LDPC-improved normalized min-sum algorithm (NMSA). Hence, the external information transmitted from check nodes to variable nodes can be simplified as follows:

(12)$r_{ji}=\alpha \cdot {\displaystyle \prod }_{i^{\prime }\in V\left(j\right)/i}\mathrm{sgn}(q_{i^{\prime }j})\cdot \underset{i^{\prime }\in V\left(j\right)/i}{\min }(\tanh \left|q_{i^{\prime }j}\right|)$

As depicted in Figure 4, it is discernible that the convergence velocity is maximized when $\alpha =0.8$. Consequently, in the subsequent simulation experiments, we intend to configure the corresponding multiplicative diminishing factor to 0.8. During the decoding decision process, after iterations, the posterior probability of variable nodes is given by the following:

(13)$q_{ij}=\ln \frac{p_i(0){\displaystyle \prod _{j^{\prime }\in C(i)}{r}_{j^{\prime }i}(0)}}{p_i(1){\displaystyle \prod _{j^{\prime }\in C(i)}{r}_{j^{\prime }i}(1)}}=L(p_i)+{\displaystyle \sum _{j^{\prime }\in C(i)}{r}_{j^{\prime }i}}$

If the value is greater than 0, i.e., $P(v_i=0)>P(v_i=1)$, $v_i$ can be adjudicated as 0.

3.2. Symbol Time Offset and Carrier Frequency Offset

In this integrated system, the starting point of the integrated signal front end and the FFT window can be inconsistent due to the symbol timing offset (STO). We perform a fast Fourier transform (FFT) on the received samples ${\{x_l[n+\delta ]\}}_{n=0}^{N-1}$ in the time domain, where $\delta$ is the number of samples. We obtain the received signal in the frequency domain as follows:

(14)$Y_l=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{x}_l[n+\delta ]}{e}^{-j2\pi nk/N}=X_l[k]e^{j2\pi k\delta /N}$

When there is a phase offset in the received signal, it causes the constellation of the signal to rotate around the origin. To ensure accurate sampling of the transmitted signal, we need to consider symbol timing synchronization. Due to the overlap between the CP and the corresponding data, as Figure 5 shows, when the CP falls within the sliding window, the similarity between the two sampling blocks within the window reaches its maximum.

We use the correlation between the two sampling blocks in $W1$ and $W2$ and then estimate the STO by maximizing the log-likelihood function. Let $\rho =SNR/(SNR+1)$ and $L$ represent the actual number of samples averaged within the window. Then, STO is estimated as follows:

(15)${\widehat{\delta }}_{ML}=\underset{\delta }{\arg \max }\{\left|\gamma [\delta ]\right|-\rho \varphi [\delta ]\}$

(16)$\begin{array}{l}\gamma [m]={\displaystyle \sum _{n=m}^{m+L-1}{y}_l[n]}{y}_l{}^{*}[n+N]\\ \varphi [m]=\frac{1}{2}{\displaystyle \sum _{n=m}^{m+L-1}\{{\left|y_l[n]\right|}^2+{\left|y_l[n+N]\right|}^2\}}\end{array}$

The $\gamma (\delta )$ and $\varphi (\delta )$ in (15) are shown in (16). Inevitably, carrier signal distortion occurs at the receiving end of the signal, and we also consider the carrier frequency offset (CFO) caused by the Doppler shift $f_d$. Let $f_{offset}=f_c-f_{c^{\prime }}$, where $f_c$ and $f_{c^{\prime }}$ represent the carrier frequencies of the transmitter and receiver, respectively. The Doppler shift $f_d$ is determined by both the carrier frequency $f_c$ and the speed of the mobile terminal $v$, i.e., $f_d=v\cdot f_c/c$.

Define the normalized CFO as the ratio of the CFO to the subcarrier spacing: $\epsilon =f_{offset}/\Delta f$. For normalized CFO, it can be decomposed into two parts, i.e., $\epsilon ={\epsilon }_i+{\epsilon }_f$, where ${\epsilon }_i$ represents the integer frequency offset (IFO), and ${\epsilon }_f$ represents the fractional frequency offset (FFO). When there is a CFO of size $\epsilon$ between the transmitter and the receiver, the received signal is represented as follows:

(17)${\mathrm{y}}_l[n]=\frac{1}{N}{\displaystyle \sum _{k=0}^{N-1}H[K]X_L[K+\delta ]}{e}^{j2\pi (k+\epsilon )n/N}+z_l[n]$

We transform the two symbols $y_l[n]$ and $y_{l+D}[n]$ into a frequency domain signal ${\{Y_l[k]\}}_{K=0}^{N-1}$ and ${\{Y_{l+D}[k]\}}_{K=0}^{N-1}$ by FFT, which is used to extract the guide frequency [20]. Then, the CFO of the guide frequency estimation is used to compensate for the time-domain received signal. The specific scheme is shown in Figure 6.

We obtain the IFO and FFO estimates, respectively:

(18)$\begin{array}{c}{\widehat{\epsilon }}_i=\frac{1}{2\pi \cdot T_{sub}}\underset{\epsilon }{\max }\left\{\left|{\displaystyle \sum _{j=0}^{L-1}{Y}_{l+D}[p[j],\epsilon ]Y_l{}^{*}[p[j],\epsilon ]}{X}_{l+D}{}^{*}[p[j]]X_l{}^{*}[p[j]]\right|\right\}\\ {\widehat{\epsilon }}_f=\frac{1}{2\pi \cdot T_{sub}\cdot D}\arg \{{\displaystyle \sum _{j=0}^{L-1}{Y}_{l+D}[p[j],{\widehat{\epsilon }}_i]Y_l{}^{*}[p[j],{\widehat{\epsilon }}_i]}{X}_{l+D}{}^{*}[p[j]]X_l{}^{*}[p[j]]\}\end{array}$

Figure 7 shows the performance of CP-based STO estimation using correlation-based maximum and interpolation minimum techniques. The solid line represents the method based on maximum correlation, while the dashed line represents the method based on minimum difference. STO is located at the point where the difference between the CP sampling block and the data sampling block is minimized, or at the point where their correlation is maximized.

Figure 8 shows the CFO estimation using the phase difference between CP and the corresponding OFDM symbols at the end, as well as the CFO estimation based on the phase difference between two repeated preambles. We can observe that the mean squared error (MSE) of the CFO estimation using pilot signals is smaller. As the signal-to-noise ratio (SNR) increases, the MSE of CFO estimation continues to decrease.

3.3. DFT-Based Minimum Mean Square Error Channel Estimation

The integrated signal contains not only unknown transmission symbol data but also known chirp signals. Therefore, we can use them as training symbols for the minimum mean-square error (MMSE) channel estimation. By minimizing the cost function $J(\widehat{H})={\Vert Y-X\widehat{H}\Vert }^2$, we can obtain the least squares (LS) estimate of the channel as follows:

(19)${\widehat{H}}_{LS}={(X^{H}X)}^{-1}{X}^{H}Y=X^{-1}Y$

Considering the LS solution in the equation, let ${\widehat{H}}_{LS}\triangleq \tilde{H}$ and we define MMSE as $\widehat{H}\triangleq W\tilde{H}$ using a weighted matrix [20]. The mean squared error (MSE) of MMSE channel estimation $\tilde{H}$ can be expressed as follows:

(20)$J(\tilde{H})=E\{{\Vert e\Vert }^2\}=E\{{\Vert H-\widehat{H}\Vert }^2\}$

where $e=H-\widehat{H}$ is the error vector, and when the error matrix is orthogonal to $\widehat{H}$, we obtain the minimum mean square error. In this case, the weighting matrix is $W=R_{H{\tilde{H}}^H}{R}_{H\tilde{H}}{}^{-1}$, where $R$ represents cross-correlation operations.

Channel estimation based on the discrete Fourier transform (DFT) can improve the estimation performance of MMSE by eliminating noise beyond the maximum channel delay [22]. Let $\widehat{H}[k]$ represent the channel gain of the $k-th$ subcarrier obtained by the MMSE channel estimation method. We take the inverse discrete Fourier transform (IDFT) of the estimated channel ${\{\widehat{H}[k]\}}_{k=0}^{N-1}$, i.e.,

(21)$\begin{array}{ll}IDFT\{\widehat{H}[k]\}& =h[n]+z[n]\\ & =\widehat{h}[n],n=0,1,\cdots ,N-1\end{array}$

Then, the remaining L channel coefficients are transformed to the frequency domain ${\widehat{H}}_{DFT}[k]=DFT\{{\widehat{h}}_{DFT}[n]\}$. From Figure 9, by comparing the results of various types of channel estimation, it is obvious that the DFT-based channel estimation method improves the estimation performance. The MMSE estimation performance is better than the LS estimation performance.

3.4. Windowing in Signal Processing

The power spectrum of OFDM consists of the superposition of multiple sinc functions, which usually have large out-of-band power and may cause neighbor channel interference. To strike a balance between computational complexity and performance, this paper employs a time-domain shaping function to reduce the out-of-band power of OFDM symbols. Considering the communication information carried by OFDM, where a signal encompasses multiple frequency components and exhibits a complex spectrum, and recognizing that signal processing primarily focuses on frequency points rather than energy magnitude, in this paper, we choose the Hanning window with a wider main flap to reduce the leakage phenomenon. To enhance the mitigation of adjacent channel interference, virtual carriers can be added at both ends of the transmission bandwidth. In this paper, the virtual carrier is combined with the Hanning window in order to play a role in protecting the frequency band.

The design of the radar waveforms directly affects the PSLR and ISLR. In this paper, linear frequency modulation (chirp) signals are used, resulting in a more concentrated main flap and relatively small side flaps. Compared with some conventional waveforms, the chirp signal can have a narrower main flap, and the resolution of the radar system can be improved by using the pulse compression technique.

For the received radar signals, windowing and pulse compression are performed in this paper. The rectangular window is the simplest window function, characterized by a wider transition band and higher-amplitude side lobes in the frequency domain. In this paper, the Blackman window is used to adjust the spectral characteristics of the waveform to optimize the PSLR and ISLR. The Blackman window is a weighted window function that has a more complex shape in the time domain than the Hanning window, better suppression of the side lobe, and smaller ripples in the frequency domain. The Blackman window is a second-order, ascending cosine window, and even though its equivalent noise bandwidth is larger than that of the Hanning window, the fluctuations are small and can realize higher-amplitude recognition accuracy. Therefore, we apply the Blackman window to make the chirp signal more concentrated in the frequency domain, thus reducing the spectral leakage.

4. Integrated Signal Performance Metrics from Simulation Experiments

We verify the effectiveness of the design and processing method of the proposed integrated signal waveform through simulation experiments in MATLAB 2018a. Some of the simulation parameters are shown in Table 1. Figure 10 shows the constellation diagram of the received signals before and after channel compensation for the 16-QAM integrated system, which can be seen to be clearer after channel compensation.

In radar detection, a lower peak-to-side lobe ratio (PSLR) reduces the likelihood of the main lobe of a weak target being overwhelmed by the side lobe of a nearby strong target. Similarly, a lower integral side lobe ratio (ISLR) in the image makes the dark echo regions in the image less susceptible to being affected by nearby strong scattering. In the simulation experiments, the imaging results for five randomly generated target signals after passing through a Gaussian channel are shown in Figure 11. By applying pulse compression to the received echoes, we can obtain the velocity and distance of the five sets of targets.

The performance comparison is obtained through simulation experiments. Figure 12 shows the pulse-compressed echoes with a PSLR of −13.81 dB, which is better than the −13.24 dB reported in [19]. The ISLR is −8.98 dB, which is an improvement compared to the traditional method’s −6.04 dB. Figure 13 shows the communication BER analysis of the transmitted image information. It can be observed that after using LDPC encoding, the bit error rate is lower than that of the OFDM chirp-integrated signal. With an SNR of 10 dB, after timing synchronization and Doppler compensation, the communication BER of the OFDM chirp-integrated signal mentioned in this paper is 4.8 × 10⁻⁴ under the Gaussian channel, and the BER for the integrated signal with LDPC encoding is 7 × 10⁻⁵, both of which are lower than the 7.1 × 10⁻⁴ reported in [19]. Compared to Reference [19], under the signal-to-noise ratio (SNR) range of 0–12 dB, the average BER of OFDM chirp-integrated signals decreased by 55.97%. For the OFDM chirp-integrated signals with LDPC encoding, the average BER decreased by 90.96%. Moreover, within the proposed approach in this paper, the scheme incorporating LDPC coding exhibited a 78.91% reduction in average BER compared to the scheme without LDPC integration.

In practical scenarios, due to system imperfections such as phase noise, carrier leakage, and other defects, the position of the signal on the constellation diagram may deviate from the ideal locations. To comprehensively assess the quality of the modulated signal, we computed the vector differences between the reference signal and the actual transmitted signal. Figure 14 illustrates the discrepancies between the actual signal and the ideal signal according to the proposed approach. As shown in Figure 15, a comparison of the transmitted image quality at different SNRs is shown. We can see that the quality of transmitted data can already be improved significantly at an SNR of 15 dB.

However, the system proposed in this paper still has some limitations. First, the integrated signal uses a cyclic prefix and a virtual carrier. Since they are not used for transmitting data, the spectral efficiency will be reduced. Second, when facing high-speed moving mobile devices, the OFDM system has certain limitations in Doppler resistance, potentially leading to performance degradation due to the frequency offset phenomena. Third, although chirp signals have good autocorrelation performance and can suppress a certain degree of interference, such as non-target scattering and clutter, multi-target interference, time bias, and speed processing, they still need to be paid attention to in practical applications. Therefore, in order to meet different application scenarios and needs, more advanced processing techniques and a more flexible LDPC coding structure [29] are needed. Finally, we will continue to consider the trade-offs [30] between communication and sensing performance in different operating environments, such as urban and rural areas, in future research.

5. Conclusions

In this paper, enhancements are made to the communication aspect of the integrated waveform design framework. To reduce out-of-band radiation between different subcarriers, this paper chooses to add a VC and a CP to the waveform instead of adding ZP before symbols. Moreover, the use of STO estimation and CFO estimation at the receiver ensures temporal and spectral synchronization of the received signal. Consequently, it effectively improves the communication BER of the integrated signal. For radar detection, this paper uses PSLR and ISLR in the frequency domain to characterize their radar detection performance. Chirp signals sampled in the frequency domain are emitted as wide pulses to ensure an ample range of action. A corresponding pulse compression algorithm is used at the time of reception to obtain a narrow pulse to ensure its resolution. MATLAB 2018a is used as the software platform for simulation experiments to verify that the integrated system proposed in this paper can accurately transmit and receive communication data, reach a BER of 7 × 10⁻⁵ and an EVM of 0.2087 at an SNR of 10 dB, and perform speed and distance measurements on radar targets with a PSLR of −13.81 dB.

Author Contributions

Data curation, C.Z.; funding acquisition, P.H. and S.W.; investigation, G.W.; methodology, C.Z., P.H. and G.W.; project administration, P.H. and S.W.; resources, P.H.; software, C.Z.; validation, C.Z. and G.W.; writing—original draft, C.Z.; writing—review and editing, P.H. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1. The subcarrier distribution diagram of the chirp signal and the time–frequency plot of the modulated signal in the time domain.

Figure 2. Schematic diagram of OFDM-Chirp integrated signal.

Figure 3. Steps in integrated signal design and processing.

Figure 4. BER variation with different multiplicative reduction factors [Forumla omitted. See PDF.] under NMSA algorithm.

Figure 5. STO estimation technique using double sliding windows.

Figure 6. CFO synchronization scheme using pilot signals.

Figure 7. STO performance estimation based on correlation maximization and difference minimization.

Figure 8. Mean square estimation of CFO estimation techniques.

Figure 9. Comparison of different channel estimation methods. (a) LS linear interpolation estimation without DFT technique; (b) LS linear interpolation estimation based on the DFT technique; (c) LS spline interpolation estimation without DFT technique; (d) LS spline interpolation estimation based on the DFT technique; (e) MMSE estimation without DFT technique; (f) MMSE estimation based on DFT technique.

Figure 10. Constellation diagram of the received signal before and after channel compensation. (a) Before channel compensation; (b) after channel compensation.

Figure 11. Speed and distance of radar detection. (a) Two-dimensional images of target information; (b) Three-dimensional images of target information.

Figure 12. The target echo after pulse compression. (a) Radar echoes of five targets; (b) radar echo from a single target.

Figure 13. Comparison of communication BER performance under different methods [19].

Figure 14. Comparison of error vector magnitude (EVM) values for different scenarios.

Figure 15. The comparison of the transmitted image quality at different SNR. (a) Image to be transferred; (b) image received at SNR = 5 dB; (c) image received at SNR = 5 dB; (d) image received at SNR = 15 dB.

Table 1

Simulation parameters.

Symbol	Name	Value
$B$	signal bandwidth (MHZ)	5
$F_S$	sample rate (MHZ)	50
$f_c$	carrier frequency (GHZ)	3
$T$	signal bandwidth ($\mu s$)	5
$N$	number of subcarriers	2048
$N_{cp}$	length of cyclic prefix	512
$N_{VC}$	length of virtual carrier	256
$N_{frame}$	number of symbols per frame	5
M	modulation order	16
$PRF$	pulse repetition frequency	2000
$SNR$	signal-to-noise ratio	0–12 dB
${\gamma }_{\mathrm{rate}}$	code rate at LDPC	1/2
$\alpha$	correction factor	0.8

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