差值定理在离散数据一阶导数解算中的应用 -- 西北工业大学学报,2016,34(1):166-175
论文:2016,Vol:34,Issue(1):166-175
引用本文:
梁红. 差值定理在离散数据一阶导数解算中的应用[J]. 西北工业大学学报
Liang Hong. Applying Difference Theorem to Calculating First Order Derivatives of Discrete Data[J]. Northwestern polytechnical university

差值定理在离散数据一阶导数解算中的应用
梁红
中国人民解放军91550部队94分队, 辽宁 大连 116023
摘要:
针对差值定理在离散数据一阶导数解算中易受测量误差影响而导致一些测量数据难以获得解算结果的问题,对差值定理的适用条件进行了解析,更正了已有文献中的错误,并对其应用方法进行了分析和推导,提出了基于极值点判别原则下差值定理与最小二乘算法相结合并对三次拟合多项式的一次项系数和二次项系数进行调整的一种新的离散数据一阶导数解算方法,给出了等间隔采样条件下的计算公式。仿真数据和实测数据验证结果表明,新算法能够对测量序列不包括端点在内的所有数据的一阶导数进行有效解算,解算结果不受测量误差限的影响,且解算精度总体上优于不进行多项式系数调整的情况,使差值定理能够更好地进行工程化应用,可显著改善测量序列端点附近和剧烈变化段一阶导数解算精度差的状况。
关键词:    差值定理    微分    一阶导数    数字滤波    截断误差   
Applying Difference Theorem to Calculating First Order Derivatives of Discrete Data
Liang Hong
PLA Unit 91550, Dalian 116023, China
Abstract:
When the Difference Theorem is applied to calculating first order derivatives of discrete data, the measurement error can lead to some of the data having no result. To resolve this problem, application conditions of the Difference Theorem are analyzed, and the inaccuracies about them in some literature are corrected. Through analyzing and deducing the application method, a new algorithm on first order derivatives of discrete data is put forward; it combines the Difference Theorem with least squares algorithm and adjusts item coefficient and binomial coefficient of third-order fit polynomial for extreme point identifying, and the calculation-formula is given on the premise that measurement data are at equal intervals. Verification of the algorithm is made with simulation and measurement; the results indicate that: (1)this new algorithm can calculate effectively first order derivatives of all data of the measurement sequence, exclusive of the endpoints; (2)the measurement error bounds do not affect the results; (3)the calculated precision is better in general than that of the case in which polynomial coefficients are not adjusted. These enable the Difference Theorem to improve significantly calculation precision of the first order derivatives of the points in the interval where the data change abruptly or near the endpoints of measurement sequence.
Key words:    algorithms    calculations    efficiency    errors    functions    Kalman filters    least squares approximations    linear regression    measurements    polynomials    schematic diagrams    statistics    Difference Theorem    differential coefficient    digital filtering    first order derivative    truncation error   
收稿日期: 2015-04-18     修回日期:
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作者简介: 梁红(1971-),女,91550部队高级工程师,主要从事试验数据处理方法研究。
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