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在原始光谱中,级联光栅的反射峰为低频成分,而噪声等干扰信号为高频成分。因此,通过结合EMD和Chebyshev滤波,提出一种高效的信号优化算法。算法流程图如图3所示,算法详细过程介绍如下:
Step 1:原始反射光谱通过EMD分解为一系列本征模函数(Intrinsic Mode Function,IMF)。
$$ {x_o}(\lambda ) = \sum\limits_{i = 1}^{M + 1} {{f_i}} (\lambda ) $$ (1) 式中:
$ {f_{M + 1}}(\lambda ) $ 为分解余量。低阶IMF主要包含了高频噪声等干扰成分,随着IMF阶次增加,高频噪声等干扰成分逐渐减少,而低频的反射峰成分逐渐增多。Step 2:使用Chebyshev波滤器进行自适应低通滤波,提取IMF中的反射峰成分。将包含反射峰成分的IMF阶数定义为“反射光谱阶”,并用
$ {Q_{ref}} $ 表示。将分解余量$ {f_{M + 1}}(\lambda ) $ 作为最后一阶IMF,从$ {f_{M + 1}}(\lambda ) $ 开始,使用Chebyshev低通滤波器$ {c_i}(\lambda )\; $ 对所有“反射光谱阶”IMF进行滤波。滤波器输出即每阶IMF中的反射峰成分。$$ spe{c_i}(\lambda ) = {c_i}(\lambda ) * {f_i}(\lambda ),\; \; i = M + 2 - {Q_{ref}},\cdots ,M + 1 $$ (2) 式中:
$ * $ 表示卷积。随着IMF阶数的降低,IMF中的反射峰成分越来越少,而高频噪声等干扰成分越来越多。因此,将$ {c_i}(\lambda )\; $ 的截止频率规定为:设最后一阶IMF的滤波器$ {c_{M + 1}}(\lambda )\; \; $ 截止频率为$ {\omega _1} $ ,则第$ k $ 个低通滤波器的截止频率为$\dfrac{{{\omega _1}}}{{{P^{{{k}} - 1}}}}$ ,其中$ P > 1 $ 为频率折叠数。滤波器输出$ spe{c_i}(\lambda ) $ 表示从每阶IMF中提取的反射峰成分,将其用于确定“反射光谱阶”$ {Q_{ref}} $ 。每个$ spe{c_i}(\lambda ) $ 的方差定义为:$$ {{{var}}} \left\{ {spe{c_i}(\lambda )} \right\} = \frac{1}{{L - 1}}\sum\limits_0^{L - 1} {{{\left[ {spe{c_i}(\lambda ) - {\mu _{spe{c_i}}}} \right]}^2}} $$ (3) 式中:
$ {\mu _{spe{c_i}}} $ 为$ spe{c_i}(\lambda ) $ 的平均值;$ L $ 为原始反射光谱的采样点数量。从最后一阶IMF开始计算,$ {Q_{ref}} $ 的确定方法为:${{var}} \left\{ {spe{c_{{Q_{ref}} - 1}}(\lambda )} \right\} \geqslant \zeta$ 且${{var}} \left\{ {spe{c_{{Q_{ref}}}}(\lambda )} \right\} < \zeta$ ,其中${{var}} \left\{ {spe{c_i}(\lambda )} \right\}$ 为$ spe{c_i}(\lambda ) $ 的方差,$ \zeta $ 为给定阈值。Step 3:将滤波器输出进行重构,得到优化后的光谱。除去“反射光谱阶”
$ {Q_{ref}} $ 对应的IMF,其余IMF只包含高频噪声等干扰成分而不含反射峰成分,因此直接舍弃。将“反射光谱阶”$ {Q_{ref}} $ 对应IMF的Chebyshev滤波器输出进行重构,即得到原始光谱中的反射峰成分,即优化后的级联光栅反射光谱。$$ \overline {spec} (\lambda ) = \sum\limits_{i = {Q_{ref}}}^{M + 1} {spe{c_i}(\lambda )} $$ (4)
Spectrum optimization algorithm of cascaded grating micro-vibration sensor based on EMD-CF
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摘要: 基于级联光栅的微振动传感器是一种典型的微振动信号测量方案,然而由于光信号在级联光栅中经过多次透射和反射,导致光谱信噪比差、成分复杂等问题。基于此,文中提出一种结合经验模态分解和切比雪夫滤波技术的光谱信号优化算法。首先,将传感器原始光谱通过经验模态分解得到一系列本征模函数;其次,利用所提出的自适应滤波方法,确定包含反射峰成分的本征模函数阶数,并对其进行切比雪夫低通滤波;最后,将滤波器输出进行重构,即得到优化后的传感器光谱。使用振幅为±8 mV、频率为500 Hz的微振动激励信号进行实验验证。结果表明:文中所提出算法可以较好地还原激励源发出的微振动信号,相比传统方法精度提高87.5%以上。Abstract: The micro-vibration sensor based on cascaded grating is a typical micro-vibration signal measurement scheme. However, due to the multiple transmission and reflection of optical signal in the cascaded grating, the sensor is subject to the poor spectral signal-to-noise ratio and complex components. Based on this, a spectrum signal optimization algorithm combined with empirical mode decomposition and chebyshev filter was proposed in this paper. Firstly, the original spectrum of the sensor was decomposed into a series of intrinsic eigenmode functions by empirical mode decomposition; Secondly, the order of the intrinsic mode functions including the reflection peak component was determined by using the proposed adaptive filtering method, and the chebyshev low-pass filtering was performed on these orders; Finally, the optimized sensor spectrum was obtained by reconstructing the output of the filter. A micro-vibration excitation signal with an amplitude of ±8 mV and frequency of 500 Hz was used for experimental verification. The results show that the proposed algorithm can effectively restore the micro-vibration signal from the excitation source, and the accuracy is improved by more than 87.5% compared with the traditional methods.
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Key words:
- fiber optic sensors /
- micro-vibration /
- algorithm /
- spectrum
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