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摘要: 针对现有联合设计的通感一体化波形对运动目标探测性能不足的问题，该文提出了一种具有多普勒容忍性的通感一体化波形联合设计方案。首先，基于脉冲串模糊函数，推导了构造多普勒容忍波形等价于波形在相关区内具有极低的积分旁瓣电平。基于此，构建了以最小化一体化波形的加权积分旁瓣电平为优化准则，以发射波形的能量、峰均功率比以及与通信波形之间的相位差为约束条件的优化问题，从而实现具有多普勒容忍性的通感一体化波形的构造。由于该优化问题的非凸性，该文提出一种基于优化最小化的迭代优化算法对其进行求解。数值仿真实验表明，相比传统一体化波形，该文提出的一体化波形具有更高的多普勒容忍性和更低的误符号率，在保证通信质量的前提下显著提升了通感一体化系统对运动目标的探测性能。Abstract: Because Doppler resilience is limited in the existing joint design of Integrated Sensing And Communication (ISAC) waveforms, a new Doppler resilient ISAC waveform design is proposed based on a joint design. First, with the pulse train ambiguity function, a construction of the Doppler resilient pulse train is deduced, which is equivalent to designing a waveform with a very low integral sidelobe level in a correlation zone. Accordingly, to construct the Doppler resilient ISAC pulse train, an optimization problem is proposed that takes minimizing the weighted integral sidelobe level of the ISAC waveform as the objective function and takes the energy of the transmitted waveform, the peak-to-average power ratio, and the phase difference between the transmitted ISAC waveform and the communication data modulated waveform as constraints. Because the optimization problem is nonconvex, an iterative optimization algorithm based on the Majorization-Minimization (MM) framework is proposed to solve it. Numerical simulation experiments show that compared with the traditional ISAC waveform design method, the ISAC waveform proposed in this paper has higher Doppler resilience and a lower symbol error rate, and the detection performance of the ISAC system for moving targets is considerably improved without loss of communication quality.
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图 3 基于MM的一体化波形在不同参数下的SER比较
Figure 3. SER comparison under various parameters based on MM-based ISAC waveform
图 4 基于MM和基于LFM的一体化波形SER性能比较
Figure 4. SER comparison between MM-based and LFM-based ISAC waveform
表 1 基于FFT/IFFT快速计算
${\boldsymbol{a}}_n^{(t)}$ [30]Table 1. Compute
${\boldsymbol{a}}_n^{(t)}$ based on FFT/IFFT[30]1. 输入：$ {\boldsymbol{x}}_n^{(t)} $, $ \{ {\omega _l}\} _{l = - L + 1}^{L - 1} $，执行： 2. ${\boldsymbol{fx} } = {\rm{fft} }({[{\boldsymbol{x} }_n^{(t)},{ {\bf{0} }_{1 \times L} }]^{\rm{T} } })$ 3. ${\boldsymbol{r} } = {\rm{ifft} }({\left| { {\boldsymbol{fx} } } \right|^2})$ 4. $ {\boldsymbol{c}} = {\boldsymbol{r}} \circ {[0,{\omega _1}, \cdots ,{\omega _{L - 1}},0,{\omega _{L - 1}}, \cdots ,{\omega _1}]^{\text{T}}} $ 5. ${\boldsymbol{\mu}} = {\rm{fft} }({\boldsymbol{c} })$ 6. ${\bf{tmp} } = {\rm{ifft} }\left( {\mu \circ ({\boldsymbol{fx} })} \right)$ 7. ${\boldsymbol{Rx} }_n^{(t)} = {\bf{tm} }{ {\bf{p} }_{1:L} }$ 8. ${\lambda }_{J}=\mathop {\max }\limits_{k}\{ {\omega }_{k}(L-k):k=1,2，\cdots ,L-1\}$ 9. ${\lambda _u} = \dfrac{1}{2}\left( { { \mathop {\max }\limits_{1 \le i \le L} }{\mu _{2i} } + { \mathop {\max }\limits_{1 \le i \le L} }{\mu _{2i - 1} } } \right)$ 10. $ {\boldsymbol{a}}_n^{(t)} = - {\boldsymbol{R}}{{\boldsymbol{x}}^{(t)}} + ({\lambda _J}L + {\lambda _u}){{\boldsymbol{x}}^{(t)}} $ 11. 输出：$ {\boldsymbol{a}}_n^{(t)} $ 
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表 2 二分法求
$ {\boldsymbol{\delta}} $ Table 2. Bisection method for
${\boldsymbol{ \delta }}$ 1. 初始化：设置搜索区间$ ({\delta _L},{\delta _U}) $。设置$ {\delta _L} = 0 $以及
$ {\delta _U} = \dfrac{{{\alpha _U}}}{{\min \{ |{a_{n,l}}| \ne 0,l = 1,2, \cdots ,L\} }} $, 令$ \delta \in ({\delta _L},{\delta _U}) $2. While $ |{\delta _U} - {\delta _L}| > {\text{eps}} \cdot |{\delta _U}| $ 3. $ \delta = ({\delta _L} + {\delta _U})/2 $ 4. If $ {\text{sign}}({f_L}(\delta )) = {\text{sign}}({f_L}({\delta _U})) $ 5. $ {\delta _U} = \delta $ 6. or 7. $ {\delta _L} = \delta $ 8. End if 9. End while，输出：$ \delta $
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表 3 基于MM算法的一体化波形设计
Table 3. MM-based algorithm for ISAC waveform design
1. 输入：$ \alpha $, $\epsilon$, $ {\omega _l} $, $ {{\boldsymbol{e}}_n} $，以及$ {\boldsymbol{x}}_n^{(0)} = {{\boldsymbol{e}}_n} $ 2. for $ t = 0,1,2, \cdots $执行 3. 计算 $ {\boldsymbol{a}}_n^{(t)} $(根据表1执行) 4. 计算 $\bar {\boldsymbol{x} } _n^{(t + 1)}$(根据式(40)) 5. 计算 ${\boldsymbol{x} }_n^{(t + 1)} = \Pi (\bar {\boldsymbol{x} } _n^{(t + 1)})$(根据式(41)) 6. end for (当收敛时) 7. 输出：$ {\boldsymbol{x}}_n^{(t + 1)} $
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