# 荧光寿命测温多指数数据处理方法Multi-Exponential Data Processing Method in Fluorescence Lifetime Temperature Measurement

DOI: 10.12677/JSTA.2020.83008, PDF, 下载: 249  浏览: 930

Abstract: The key point of optical fiber temperature measurement technology based on fluorescence lifetime is to remove the background interference such as the dc component and the noise signal in the fluorescence signal so as to obtain the fluorescence life τ as accurately as possible. However, in the case of high-precision measurement, when the traditional single exponential model algorithms, such as the least squares fitting, the integral area ratio, the FFT fitting and other methods are used to process the multi-exponential fluorescence signal, the fluorescence life error obtained will be large. LM algorithm and Prony algorithm are introduced into fluorescence temperature measurement. Through simulation analysis, it can be seen that the two algorithms have obvious advantages over the traditional methods in processing multi-exponential fluorescence signals. Under low noise condition, the maximum error of LM algorithm is between ±10−4 ms, and the re-sponse speed is 3 × 10−1 s. It is very suitable for high-precision measurement. Prony algorithm is significantly affected by noise. The maximum error is ±10−3 ms in the low noise environment, and the response speed can reach 2 × 10−3 s. It is suitable for application in the case of fast response.

1. 引言

$I\left(t\right)={I}_{0}\mathrm{exp}\left(-t/\tau \right)+{I}_{d}+{I}_{n}$ (1)

$I\left(t\right)={I}_{0}\mathrm{exp}\left(-t/{\tau }_{1}\right)+{I}_{1}\mathrm{exp}\left(-t/{\tau }_{2}\right)+\cdots +{I}_{N}\mathrm{exp}\left(-t/{\tau }_{J}\right)+{I}_{d}+{I}_{n}$ (2)

LM算法和Prony算法是针对多指数模型的数据处理方法 [9] [10]。本文通过仿真分析，研究将LM算法和Prony算法引入荧光测温的可行性和测量精度的问题。

2. L-M算法

2.1. L-M算法在测温系统中的实现

$I\left(t\right)=\underset{i=1}{\overset{k}{\sum }}{A}_{i}\mathrm{exp}\left(-t/{\tau }_{i}\right)$ (3)

$\begin{array}{c}S\left(\beta ;t\right)=\frac{1}{2}\underset{j=1}{\overset{n}{\sum }}{\left[{I}_{j}\left(\beta ;t\right)-{y}_{j}\right]}^{2}\\ =\frac{1}{2}\underset{j=1}{\overset{n}{\sum }}{f}_{j}{\left(x\right)}^{2}\\ \equiv \frac{1}{2}F{\left(x\right)}^{\text{T}}F\left(x\right)\end{array}$ (4)

LM算法的迭代格式为：

${x}_{k+1}={x}_{k}+{d}_{k}$ (5)

${x}_{k+1}={x}_{k}-{\left({J}^{\text{T}}J+\lambda \text{diag}\left({J}^{\text{T}}J\right)\right)}^{-1}JF\left(x\right)$ (6)

2.2. L-M算法单指数仿真分析

$I\left(t\right)={A}_{1}\mathrm{exp}\left(-t/\tau \right)+{I}_{d}+{I}_{n}$

$y=A\mathrm{exp}\left(-i\Delta t/\tau \right)+{I}_{d}+{I}_{n}$

2.2.1. 直流分量的影响

$A=1$，噪声信号 ${I}_{n}=0$，改变直流分量Id的幅度与荧光信号强度的比，结果如表1所示。可以看出，利用LM算法求取荧光寿命 $\tau$ 时，会受到直流分量的影响，随着直流分量的增大，荧光寿命 $\tau$ 也逐渐偏离理论值。因此若想应用此法得到较高的测量精度需要想办法去掉直流分量，这是此法的一个弊端。

Table 1. The τ value table of the DC component is 0.5% - 2.5% of the fluorescence intensity

2.2.2. 噪声信号的影响

Table 2. The τ value table of the noise amplitude is 0.5% - 2.5% of the fluorescence intensity

2.3. LM算法双指数仿真分析

$y={A}_{1}\mathrm{exp}\left(-i\Delta t/{\tau }_{1}\right)+{A}_{2}\text{exp}\left(-i\Delta t/{\tau }_{2}\right)+{I}_{d}+{I}_{n}$

Table 3. The τ value table of the fitting length of l-m algorithm is 120 points

Table 4. The τ value table of the fitting length of traditional algorithm is 120 points

3. Prony算法

3.1. Prony算法在测温系统中的实现

Prony算法是使用多指数函数的一种线性组合来描述等间距采样数据的数学模型，可以根据采样值直接估算出信号频率、衰减、幅值和初相位的分析方法。

$\begin{array}{c}I\left(t\right)={I}_{1}\mathrm{exp}\left(-t/{\tau }_{1}\right)+{I}_{2}\mathrm{exp}\left(-t/{\tau }_{2}\right)+\cdots +{I}_{n}\mathrm{exp}\left(-t/{\tau }_{n}\right)\\ =\underset{j=1}{\overset{n}{\sum }}{I}_{0j}\mathrm{exp}\left(-t/{\tau }_{j}\right)\end{array}$ (7)

$\left\{\begin{array}{l}{I}_{0}={I}_{01}+{I}_{02}+\cdots +{I}_{0n}\\ {I}_{1}={I}_{01}\mathrm{exp}\left(-\Delta t/{\tau }_{1}\right)+{I}_{02}\mathrm{exp}\left(-\Delta t/{\tau }_{2}\right)+\cdots +{I}_{0n}\mathrm{exp}\left(-\Delta t/{\tau }_{n}\right)\\ {I}_{2}={I}_{01}\mathrm{exp}\left(-2\Delta t/{\tau }_{1}\right)+{I}_{02}\mathrm{exp}\left(-2\Delta t/{\tau }_{2}\right)+\cdots +{I}_{0n}\mathrm{exp}\left(-2\Delta t/{\tau }_{n}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {I}_{2n-1}={I}_{01}\mathrm{exp}\left(-\left(2n-1\right)\Delta t/{\tau }_{1}\right)+{I}_{02}\mathrm{exp}\left(-\left(2n-1\right)\Delta t/{\tau }_{2}\right)+\cdots +{I}_{0n}\mathrm{exp}\left(-\left(2n-1\right)\Delta t/{\tau }_{n}\right)\end{array}$ (8)

${I}_{i}=I\left(i\Delta t\right)=\underset{j=1}{\overset{n}{\sum }}{I}_{0j}\mathrm{exp}\left(-i\Delta t/{\tau }_{j}\right)$ (9)

${\stackrel{^}{I}}_{n}=\underset{i=1}{\overset{p}{\sum }}{b}_{i}{z}_{i}^{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=0,1,\cdots ,N-1\right)$ (10)

${b}_{m}={A}_{m}$ (11)

${z}_{m}=\mathrm{exp}\left({a}_{m}\Delta t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a=-1/\tau$ (12)

$\mathrm{min}\left(\epsilon =\underset{n=0}{\overset{N-1}{\sum }}{|I\left(n\right)-\stackrel{^}{I}\left(n\right)|}^{2}\right)$ (13)

$\phi \left(z\right)=\underset{t=0}{\overset{p}{\prod }}\left(z-{z}_{k}\right)=\underset{i=0}{\overset{p}{\sum }}{a}_{i}{z}^{p-1}$ (14)

$\stackrel{^}{I}\left(n-m\right)=\underset{i=1}{\overset{p}{\sum }}{b}_{i}{z}^{n-m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(0\le n-m\le N-1\right)$ (15)

$\underset{m=0}{\overset{p}{\sum }}{a}_{m}\stackrel{^}{I}\left(n-m\right)=\underset{l=1}{\overset{p}{\sum }}{b}_{l}\underset{m=0}{\overset{p}{\sum }}{a}_{m}{z}_{l}^{n-m},\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\le n\le N-1$ (16)

$\underset{m=0}{\overset{p}{\sum }}{a}_{m}\stackrel{^}{I}\left(n-m\right)=\underset{l=1}{\overset{p}{\sum }}{b}_{l}{z}_{l}^{n-p}\underset{m=0}{\overset{p}{\sum }}{a}_{m}{z}_{l}^{p-m}=0$ (17)

$\varphi \left({z}_{i}\right)=0$。因此可知 $\stackrel{^}{I}\left(n\right)$ 满足递推的差分方程：

$\stackrel{^}{I}\left(n\right)=-\underset{m=1}{\overset{p}{\sum }}\stackrel{^}{I}\left(n-m\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p\le n\le N-1\right)$ (18)

$\begin{array}{c}I\left(n\right)=-\underset{m=1}{\overset{p}{\sum }}{a}_{m}\stackrel{^}{I}\left(n-m\right)+e\left(n\right)\\ =\underset{m=1}{\overset{p}{\sum }}{a}_{m}I\left(n-m\right)+\underset{m=0}{\overset{p}{\sum }}{a}_{m}e\left(n-m\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p\le n\le N-1\right)\end{array}$ (19)

$\epsilon \left(n\right)=\underset{m=0}{\overset{p}{\sum }}{a}_{m}e\left(n-m\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=p,\cdots ,N-1\right)$ (20)

$I\left(n\right)=-\underset{m=1}{\overset{p}{\sum }}{a}_{m}I\left(n-m\right)+\epsilon \left(n\right)$ (21)

$\left[\begin{array}{cccc}x\left(p\right)& I\left(p-1\right)& \cdots & I\left(0\right)\\ x\left(p+1\right)& I\left(p\right)& \cdots & I\left(1\right)\\ ⋮& ⋮& \ddots & ⋮\\ x\left(p\right)& I\left(p\right)& \cdots & I\left(p\right)\end{array}\right]\left[\begin{array}{c}1\\ {a}_{1}\\ ⋮\\ {a}_{p}\end{array}\right]=\left[\begin{array}{c}\epsilon \left(0\right)\\ \epsilon \left(1\right)\\ ⋮\\ \epsilon \left(N-1\right)\end{array}\right]$

$Ia=\epsilon$ (22)

$\underset{m=0}{\overset{p}{\sum }}{a}_{m}\left[\underset{n=p}{\overset{N-1}{\sum }}I\left(n-m\right){I}^{\ast }\left(n-i\right)\right]=0$ (23)

${\epsilon }_{p}=\underset{m=0}{\overset{p}{\sum }}{a}_{m}\left[\underset{n=p}{\overset{N-1}{\sum }}I\left(n-m\right){I}^{\ast }\left(n\right)\right]$ (24)

$r\left(i,j\right)=\underset{n=p}{\overset{N-1}{\sum }}I\left(n-j\right){I}^{\ast }\left(n-i\right)$ (25)

$\left[\begin{array}{cccc}r\left(0,0\right)& r\left(0,1\right)& \cdots & r\left(0,p\right)\\ r\left(1,0\right)& r\left(1,1\right)& \cdots & r\left(1,p\right)\\ ⋮& ⋮& \ddots & ⋮\\ r\left(p,0\right)& r\left(p,1\right)& \cdots & r\left(p,p\right)\end{array}\right]\left[\begin{array}{c}1\\ {a}_{1}\\ ⋮\\ {a}_{p}\end{array}\right]=\left[\begin{array}{c}{\epsilon }_{p}\\ 0\\ ⋮\\ 0\end{array}\right]$ (26)

$1+{a}_{1}{z}^{-1}+\cdots +{a}_{p}{z}^{-p}=0$ (27)

$\left[\begin{array}{cccc}1& 1& \cdots & 1\\ {z}_{1}& {z}_{2}& \cdots & {z}_{p}\\ ⋮& ⋮& \ddots & ⋮\\ {z}_{1}^{N-1}& {z}_{2}^{N-2}& \cdots & {z}_{p}^{N-1}\end{array}\right]\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{p}\end{array}\right]=\left[\begin{array}{c}\stackrel{^}{x}\left(1\right)\\ \stackrel{^}{x}\left(2\right)\\ ⋮\\ \stackrel{^}{x}\left(p\right)\end{array}\right]$ (28)

$\left\{\begin{array}{l}{A}_{i}=|{b}_{i}|\\ {a}_{i}=\mathrm{ln}\left({z}_{i}\right)/\Delta t\end{array}$ (29)

3.2. Prony算法的单指数仿真

$y=A\mathrm{exp}\left(-i\Delta t/\tau \right)+{I}_{d}+{I}_{n}$

Table 5. The τ value table of the DC component is 0.5% - 2.5% of the fluorescence intensity (sampling point 120)

Table 6. The τ value table of the noise amplitude is 0.5% - 2.5% of the fluorescence intensity (sampling point 120)

3.3. Prony算法的双指数仿真

$y={A}_{\text{ }}\mathrm{exp}\left(-i\Delta t/{\tau }_{1}\right)+{A}_{2}\mathrm{exp}\left(-i\Delta t/{\tau }_{2}\right)+{I}_{d}+{I}_{n}$

Table 7. The τ value table of the noise amplitude is 0.5% - 2.5% of the fluorescence intensity (sampling point 120)

4. 结论

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