复合函数形式的冲激函数相关运算探讨

Discussion on the correlation operation of compound impulse function

作为信号与系统分析中重要的基本数学模型,单位冲激函数的引入使得函数间断点处导数存在成为可能,是描述和分析诸多物理现象和过程不可缺少的工具.冲激函数的物理含义和性质前人研究颇多,而复合函数形式的冲激函数及其相关运算却很少涉及和讨论.基于上述原因,采用常规函数的计算方法,从广义函数定义出发,对复合函数形式的冲激函数及其微分、积分和卷积积分运算进行了分析和推导,最后通过验证得出一般性的结论.

As an important and basic mathematic model for the analysis of signals and systems,the unit impulse function makes it possible that the discontinuous point has a derivative.It is an indispensable tool for people to describe and analyze some physical phenomena and processes.There have been many previous studies about physical meaning and properties of impulse function.However,the composite function form and correlation operation are rarely involved and discussed.Based on above reasons,the computing methods for a convenience function are used in this paper,proceed from definition of generalized functions.The differential and integral of compound function and its convolution integrals are analyzed and derived.The conclusions are drawn by verifying at last.