Off-grid Blind Near-Field Integrated Sensing And Communication: Algorithm Design and Lower Bound
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摘要: 超大规模天线阵列场景下用户多处于近场区域，现有通信与感知算法面临离格功率泄露、幅度衰减导致模型失配等问题，且大多依赖导频辅助感知。为此，该文提出一种离格盲近场通信感知一体化(NF-ISAC)算法，可在无导频条件下实现近场用户的高精度坐标感知、信道估计与符号检测。该算法首先构建NF-ISAC系统的因子图模型，提出幅度-相位分离的混合导向矢量建模方法，通过神经网络实现拟合，并将其作为函数节点嵌入因子图；其次结合矩阵分解与消息传递算法，实现因子图中神经网络的穿透计算，完成离格盲NF-ISAC算法的完整推导；同时推导了基于神经网络的近场感知克拉美罗下界，明确了近场位置感知的理论极限。仿真结果表明，所提算法以与主流算法同阶的计算复杂度，在通信误码率与感知精度上均取得显著性能提升，感知精度较现有近场离格算法提升2～3 dB，且性能最接近理论下界。Abstract:
Objective With the widespread deployment of extra-large-scale antenna arrays in 6G networks, user terminals are increasingly located in the near-field region. Existing Near-Field Integrated Sensing And Communication (NF-ISAC) algorithms face key challenges, including off-grid power leakage, severe model mismatch, and strong pilot dependence. These limitations make them unsuitable for low-overhead, high-performance 6G transmission. This paper aims to design an off-grid blind NF-ISAC algorithm and derive the theoretical performance bound for near-field sensing. Methods To overcome the limitations of analytical geometric steering vectors and adapt to more accurate electromagnetic propagation characteristics without closed-form expressions, an amplitude-phase separation method is first proposed. This method decomposes the nonlinear near-field steering vector into amplitude and phase terms, enabling high-precision characterization of the steering vector using a single-hidden-layer neural network. Second, the NF-ISAC problem is formulated as a constrained matrix factorization problem, and a corresponding factor graph model is constructed. The trained neural network is embedded into the factor graph as a function node. Message passing through the embedded neural network is then achieved, enabling joint blind coordinate sensing, channel estimation, and signal detection in a pilot-free manner. Finally, the Cramér-Rao Lower Bound (CRLB) for multi-user near-field joint distance and angle sensing in polar coordinates is derived based on the neural-network-fitted steering vector. Results and Discussions Extensive Monte Carlo simulations are conducted to evaluate the performance of the proposed algorithm. The simulation results show that the proposed algorithm achieves millimeter-level position sensing. Compared with existing mainstream algorithms, it improves both communication Bit Error Rate (BER) and sensing accuracy. The proposed algorithm achieves a 2～3 dB gain in sensing accuracy over the state-of-the-art near-field off-grid algorithm, and its performance is closest to the derived theoretical CRLB. These results indicate that the proposed algorithm effectively mitigates off-grid power leakage and model mismatch. Conclusions The proposed off-grid blind NF-ISAC algorithm overcomes the pilot dependence and model mismatch of existing NF-ISAC schemes. It achieves integrated high-precision sensing and reliable communication for near-field users in a pilot-free manner. The derived CRLB provides a theoretical benchmark for evaluating the sensing performance of NF-ISAC systems. This work provides technical support for the design of 6G NF-ISAC systems. -
表 1 因式分解和函数节点含义
函数 概率 表达式 函数 概率 表达式 函数 概率 表达式 $ f_{\boldsymbol{Y}} $ $ P(\boldsymbol{Y}|\boldsymbol{A},\boldsymbol{X},\beta ) $ $ \mathcal{M}\mathcal{N}\left(\boldsymbol{Y};\boldsymbol{AX},{\boldsymbol{I}}_{R},{\boldsymbol{I}}_{L}\right) $ $ {f}_{\beta } $ $ P(\beta ) $ $ 1/\beta $ $ f_{\boldsymbol{\gamma}} $ $ P(\boldsymbol{\gamma }) $ $ \text{Ga}\left(\boldsymbol{\gamma };\varepsilon ,\eta \right) $ $ f_{\boldsymbol{x}_l} $ $ P({\boldsymbol{x}}_{l}|\boldsymbol{\gamma }) $ $ \text{CN}\left({\boldsymbol{x}}_{l};0,{\boldsymbol{\gamma }}^{-1}\right) $ $ f_{\boldsymbol{\alpha}_j} $ $ P({\boldsymbol{\alpha }}_{j}|{d}_{j},{\theta }_{j}) $ $ \mathcal{N}{\mathcal{N}}_{j} $ $ {f}_{{{d}_{j}}},{f}_{{{\theta }_{j}}} $ $ P({d}_{j}),P({\theta }_{j}) $ $ \text{Unif}() $ 
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1 矩阵分解算法
1 初始化：$ {\boldsymbol{U}}_{\mathbf{A}}={\boldsymbol{I}}_{R} $, $ {\boldsymbol{V}}_{\mathbf{A}}={\boldsymbol{I}}_{J} $, $ \boldsymbol{\hat{A}}={\boldsymbol{A}}_{0} $, $ {\boldsymbol{V}}_{\mathbf{X}}={\boldsymbol{I}}_{L} $, $ {\mathbf{\varXi} }_{\mathbf{X}}={{{\textit{1}}}}_{J\times L} $, $ {\boldsymbol{S}}_{\mathbf{X}}={{{\textit{0}}}}_{J\times L} $, $ {\mathbf{\varXi} }_{\mathbf{A}}={{{\textit{1}}}}_{R\times J} $, $ {\boldsymbol{S}}_{\mathbf{A}}={{{\textit{0}}}}_{R\times J} $ 2 For $ t=1\colon T $ 3 $ {\overline{{\boldsymbol{W}}}}_{\mathbf{X}}={\boldsymbol{\hat{A}}}^{\text{H}}{\mathbf{A}+}R{\boldsymbol{V}}_{\mathbf{A}} $, $ \left[{\boldsymbol{C}}_{\mathbf{X}},{\boldsymbol{D}}_{\mathbf{X}}\right]=\text{eig}({\boldsymbol{\overline{W}}}_{\mathbf{X}}) $, $ {\boldsymbol{R}}_{\mathbf{X}}=\boldsymbol{D}_{\mathbf{X}}^{-1/2}\boldsymbol{C}_{\mathbf{X}}^{\text{H}}{\boldsymbol{\hat{A}}}^{\text{H}}\boldsymbol{Y},\;\;{\boldsymbol{{\varPhi }}}_{\mathbf{X}}=\boldsymbol{D}_{\mathbf{X}}^{-1/2}\boldsymbol{C}_{\mathbf{X}}^{\text{H}} $ 4 $ {\boldsymbol{V}}_{{{\mathbf{P}}_{\mathbf{X}}}}={\left| {\boldsymbol{{\varPhi }}}_{\mathbf{X}}\right| }^{.2}{\mathbf{\varXi} }_{\mathbf{X}},\; {\boldsymbol{\hat{P}}}_{\mathbf{X}}={\boldsymbol{{\varPhi }}}_{\mathbf{X}}\boldsymbol{\hat{X}}-{\boldsymbol{V}}_{{{\mathbf{P}}_{\mathbf{X}}}}\cdot {\boldsymbol{S}}_{\mathbf{X}} $, $ {\boldsymbol{V}}_{{{S}_{X}}}={{\textit{1}}}./\left({\boldsymbol{V}}_{{{P}_{X}}}+{\beta }^{-1}\right),\; \; {\boldsymbol{S}}_{X}={\boldsymbol{V}}_{{{S}_{X}}}\cdot ({\boldsymbol{R}}_{X}-{\boldsymbol{P}}_{X}) $ 5 $ {\boldsymbol{V}}_{{{Q}_{X}}}={{\textit{1}}}./\left({\left| \boldsymbol{{\varPhi }}_{X}^{\text{H}}\right| }^{.2}\cdot \; {\boldsymbol{V}}_{{{S}_{X}}}\right),\; {\boldsymbol{Q}}_{X}=\boldsymbol{\hat{X}}+{\boldsymbol{V}}_{{{Q}_{X}}}\cdot \left(\boldsymbol{{\varPhi }}_{X}^{\text{H}}{\boldsymbol{S}}_{X}\right) $，%计算$ \boldsymbol{X} $的外信息均值和方差 6 $ {\mathbf{\varXi }}_{X}={\boldsymbol{V}}_{{{Q}_{X}}}\cdot \boldsymbol{G}_{X}^{\prime}({\boldsymbol{Q}}_{X},{\boldsymbol{V}}_{{{Q}_{X}}}),\; \boldsymbol{\hat{X}}={\boldsymbol{G}}_{X}({\boldsymbol{Q}}_{X},{\boldsymbol{V}}_{{{Q}_{X}}}) $， %外信息和先验合并，计算$ \boldsymbol{X} $后验 7 $ {\boldsymbol{U}}_{X}=\text{diag}(\text{mean}({\mathbf{\varXi }}_{X},2)) $， %计算调制符号矩阵$ \boldsymbol{X} $的逐元素方差 8 $ {\boldsymbol{\overline{W}}}_{A}=\boldsymbol{X}{\boldsymbol{\hat{X}}}^{\text{H}}\boldsymbol{+}L{\boldsymbol{U}}_{X} $, $ \left[{\boldsymbol{C}}_{A},{\boldsymbol{D}}_{A}\right]={\mathrm{eig}}({\boldsymbol{\overline{W}}}_{A}) $, $ {\boldsymbol{R}}_{A}=\boldsymbol{D}_{A}^{-1/2}\boldsymbol{C}_{A}^{\text{H}}\boldsymbol{\hat{X}}{\boldsymbol{Y}}^{\text{H}},\; \; {\boldsymbol{{\varPhi }}}_{A}=\boldsymbol{D}_{A}^{-1/2}\boldsymbol{C}_{A}^{\text{H}} $ 9 $ {\boldsymbol{V}}_{{{P}_{A}}}={\left| {\boldsymbol{{\varPhi }}}_{A}\right| }^{.2}\varXi _{A}^{\text{H}},\; {\boldsymbol{\hat{P}}}_{A}={\boldsymbol{{\varPhi }}}_{A}{\boldsymbol{\hat{A}}}^{\text{H}}-{\boldsymbol{V}}_{{{P}_{A}}}\cdot {\boldsymbol{S}}_{A} $, $ {\boldsymbol{V}}_{{{S}_{A}}}={{\textit{1}}}./\left({\boldsymbol{V}}_{{{P}_{A}}}+{\beta }^{-1}\right),\; \; {\boldsymbol{S}}_{A}={\boldsymbol{V}}_{{{S}_{A}}}\cdot ({\boldsymbol{R}}_{A}-{\boldsymbol{P}}_{A}) $ 10 $ {\boldsymbol{V}}_{{{Q}_{A}}}={{\textit{1}}}./\left({\left| \boldsymbol{{\varPhi }}_{A}^{\text{H}}\right| }^{.2}\cdot \; {\boldsymbol{V}}_{{{S}_{A}}}\right),\; {\boldsymbol{Q}}_{A}={\boldsymbol{\hat{A}}}^{\text{H}}+{\boldsymbol{V}}_{{{Q}_{A}}}\cdot \left(\boldsymbol{{\varPhi }}_{A}^{\text{H}}{\boldsymbol{S}}_{A}\right) $，%导向矢量$ \boldsymbol{A} $的外信息均值和方差 11 $ {\mathbf{\varXi }}_{A}={\boldsymbol{V}}_{{{Q}_{A}}}\cdot \boldsymbol{G}_{A}^{\prime}({\boldsymbol{Q}}_{A},{\boldsymbol{V}}_{{{Q}_{A}}}),\; \boldsymbol{\hat{A}}={\boldsymbol{G}}_{A}({\boldsymbol{Q}}_{A},{\boldsymbol{V}}_{{{Q}_{A}}}) $ %外信息和先验合并，计算$ \boldsymbol{A} $后验均值和方差 12 $ {\boldsymbol{U}}_{A}=\text{diag}(\text{mean}({\mathbf{\varXi }}_{A},{{\textit{1}}})) $ %计算矢量$ \boldsymbol{A} $的逐元素方差 13 噪声精度$ \beta =ML/C $，其中$ \boldsymbol{C}={\left|\left|\boldsymbol{Y}-\boldsymbol{AX}\right|\right|}^{2}+M\text{Tr}(\boldsymbol{\hat{X}}{\boldsymbol{\hat{X}}}^{\text{H}}{\boldsymbol{V}}_{A})+L\text{Tr}({\boldsymbol{U}}_{X}{\boldsymbol{\hat{A}}}^{\text{H}}\boldsymbol{\hat{A}})+ML\text{Tr}({\boldsymbol{U}}_{X}{\boldsymbol{V}}_{X}) $ 14 EndFor
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2 通信感知一体化算法
1 初始化：网格筛选得到$ (d_{m}^{t-1},\theta _{n}^{t-1})=({d}_{m},{\theta }_{n}),\forall m,n $和表1第1行初始化步骤。 2 For $ t=1\colon T $ 3 由算法1的3～5行得到$ {\boldsymbol{Q}}_{X},{\boldsymbol{V}}_{{{Q}_{X}}} $，分解为向量$ \boldsymbol{q}_{l}^{x},\boldsymbol{v}_{{q}_{l}}^{x},\forall l $； %引用算法1 4 由式(15)计算$ {\boldsymbol{\hat{x}}}_{l},{\boldsymbol{v}}_{{{x}_{l}}},\forall l $，并排列为矩阵$ \boldsymbol{\hat{X}},{\boldsymbol{{\varXi }}}_{X} $，由式(16)计算稀疏贝叶斯超先验$ \boldsymbol{\gamma } $； 5 由算法1的8～10行计算$ {\boldsymbol{Q}}_{A},{\boldsymbol{V}}_{{{Q}_{A}}} $，分解为向量$ \boldsymbol{q}_{j}^{a},\boldsymbol{v}_{{q}_{j}}^{a},\forall j $； %引用算法1 6 利用$ (d_{m}^{t-1},\theta _{n}^{t-1}) $，由式(7)计算矢量$ {\boldsymbol{\alpha }}_{{{j}_{0}}} $偏导数$ \boldsymbol{e}_{0}^{{\theta }_{j}} $和$ \boldsymbol{e}_{0}^{{d}_{j}},\forall j $； 7 $ {\boldsymbol{\xi }}_{{{j}_{0}}}={\boldsymbol{\alpha }}_{{{j}_{0}}}-{d}^{t-1}\boldsymbol{e}_{{j}_{0}}^{d}-{\theta }^{t-1}\boldsymbol{e}_{{j}_{0}}^{\theta },\forall j $，由式(9)计算$ {\boldsymbol{\overrightarrow{d}}}_{j},{\boldsymbol{\overrightarrow{v}}}_{{{d}_{j}}},\forall j $； 8 由式(10)计算$ d_{j}^{} $和$ {v}_{{{d}_{j}}},\forall j $，由(11)计算$ {\boldsymbol{\overleftarrow{v}}}_{{{d}_{j}}} $和$ {\boldsymbol{\overleftarrow{d}}}_{j},\forall j $； 9 同理计算$ {\theta }_{j} $,$ {v}_{{{\theta }_{j}}} $,$ {\boldsymbol{\overleftarrow{v}}}_{{{\theta }_{j}}} $和$ {\boldsymbol{\overleftarrow{\theta }}}_{j},\forall j $； 10 由式(14)计算$ {\boldsymbol{v}}_{{{\boldsymbol{\alpha }}_{j}}} $和$ {\boldsymbol{\hat{\alpha }}}_{j},\forall j $，并排列为矩阵$ \boldsymbol{\hat{A}},{\boldsymbol{{\varXi }}}_{A} $，由算法1的13行计算$ \beta $； 11 EndFor 12 对$ \boldsymbol{\hat{X}} $解差分解调并判决得到信息比特$ {\boldsymbol{\hat{b}}}_{k} $。
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表 2 近场通信感知一体化系统参数
参数 数值 活跃用户个数K 1～5 基站天线个数R 128 信噪比SNR –3～–10 dB 感知距离范围$ {D}_{{\mathrm{Min}}},{D}_{{\mathrm{Max}}} $ $ d\sim \left[5\;{\mathrm{m}},50\;{\mathrm{m}}\right] $ 数据调制方式 QPSK 每帧数据长度L 100～500 路径衰落$ {a}_{k} $ Unif(0.8,1) 感知角度范围$ {\phi }_{{\mathrm{Min}}},{\phi }_{{\mathrm{Max}}} $ $ \theta =\left[30{^{\circ}},150{^{\circ}}\right] $
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