A Parameter Estimation Method of Non-instantaneous Diffusion Point Source Based on Finite Rate of Innovation
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摘要: 扩散现象在农业真菌传播、大气污染等现实场景广泛存在，扩散源参数估计也因此在农业、工业等实际应用中具有重要意义。目前针对扩散源参数估计提出的方法大多针对理想的瞬时点源信号，对于非瞬时的实际扩散过程存在模型失配问题，极大地限制了算法的实际应用场景。为了解决模型不匹配的问题，同时有效估计扩散源持续时间参数，该文将扩散源信号模型拓展为脉宽可变信号，并提出相应的非瞬时点源模型的参数估计算法。该算法中，利用无线传感网络采样得到实际测量值，找到一个组合系数将实际测量值线性组合为指数函数，再根据有限新息率(FRI)采样理论对组合后的数据用零化滤波器方法求解扩散源参数。仿真结果表明，在信噪比20 dB，位置参数重构MAE能够达到0.008左右，脉宽参数能够达到0.1左右，持续时间参数能够达到0.05左右，这验证了非瞬时点源参数估计的准确性。同时我们分析了传感器个数等因素对参数恢复性能的影响。Abstract: Many physical phenomena can be described by the diffusion equations, such as the emission of chimney pollutants, chemical substance leakage, etc. Therefore, the estimation of diffusion source parameters is of great significance in practical applications. Currently, most of the proposed methods for estimating parameters of diffusion sources are aimed at instantaneous point source signals. For non-instantaneous actual diffusion processes, there is a problem of model mismatch. In this paper, the diffusion source model is extended to variable pulse-width signals, and the parameter estimation algorithm of corresponding non-instantaneous point sources are proposed. In this algorithm, the actual measurement value is obtained by sampling with the wireless sensor network, a combination coefficient is found to combine linearly the actual measurement value into an exponential function, and then the combined data is analyzed according to the Finite Rate of Innovation (FRI) sampling theory by using the annihilation filter method to solve the diffusion source parameters. The simulation results analyze the performance factors that affect parameter recovery, including noise, the number of sensors, etc., and the accuracy of the non-instantaneous diffusion point source parameter estimation method is validated.
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表 1 非理想时间脉冲扩散点源参数估计过程
输入：传感器位置${{\boldsymbol{x}}_n}$，采样时刻${t_l}$，传感器采样值$ {\varphi _n}({t_l}) $，扩散
系数$\mu $，脉冲个数K，采样时长T输出：位置参数${\boldsymbol{\xi}}$，时延参数，脉宽参数，幅度参数。 (1) 根据式(40)计算当$ r = 0 $时的组合加权系数
$\{ {w_{{\boldsymbol{n}},l} }({\boldsymbol{k} },0)\} _{ {\boldsymbol{k} } = (0,0)}^{({K_1},{K_2})}$。(2) 根据式(28)对采样值进行组合，得到$\hat Q({\boldsymbol{k}},0)$。 (3) 利用2维谱估计算法得到($ {\xi _1},{\xi _2} $)。 (4) 根据式(40)计算当$ {k_1} = 0 $, $ {k_2} = 1 $时的加权系数
$\{ {w_{{\boldsymbol{n}},l} }(k,r)\} _{r = 0}^{2K + 1}$。(5) 根据式(28)对采样值进行组合，得到$\hat Q({\boldsymbol{k}},r)$。 (6) 根据式(26)与式(27)获取傅里叶系数$ H(r) $。 (7) 利用改进的零化滤波器恢复时延，脉宽和幅度参数。 
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表 2 参数设置及恢复结果
实验组 参数 真实值 估计均值 估计值方差 脉宽$ {r_k} $ (1.0000, 1.0000) (0.9171, 1.0102) (0.0018, 0.0028) 第1组 时延$ {t_k} $ (3.0000, 7.0000) (2.9996, 7.1626) (5.9602$ \times $10–4, 9.7505$ \times $10–4) 幅度$ {c_k} $ (5.0000, 5.0000) (6.4175, 4.6885) (0.7995, 0.8620) 脉宽$ {r_k} $ (0.8000, 0.8000) (0.7122, 0.7689) (7.4302$ \times $10–4, 7.1916$ \times $10–4) 第2组 时延$ {t_k} $ (4.0000, 7.0000) (4.0159, 7.0212) (2.8550$ \times $10–4, 3.3058$ \times $10–4) 幅度$ {c_k} $ (4.0000, 4.0000) (4.6386, 4.0244) (0.1670, 0.1475) 脉宽$ {r_k} $ (0.9000, 0.9000) (0.7965, 0.8395) (8.3376$ \times $10–4, 0.0011) 第3组 时延$ {t_k} $ (2.0000, 7.0000) (1.9525, 7.1132) (8.0755$ \times $10–4, 5.3718$ \times $10–4) 幅度$ {c_k} $ (5.0000, 5.0000) (5.2004, 4.7983) (0.0901, 0.1080)
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表 3 参数估计结果
实验组 参数 真实值 瞬时点源模型估计值 瞬时点源估计方差 本文点源模型估计值 本文点源估计方差 第1组 时延 2.0000 1.5497 6.5657$ \times $10–7 2.0055 2.6604$ \times $10–5 幅值 398.8178 560.8293 84.2846 405.4113 31.8795 第2组 时延 7.0000 6.5475 1.7676$ \times $10–6 6.9988 1.5820$ \times $10–5 幅值 159.5271 222.7944 11.8788 158.6228 6.1016
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