零输入响应的两种定义的等效性

The Equivalence Relation between Two Definitions of Zero-Input Response

连续LTI系统的零输入响应可定义为输入的反因果分量与系统冲激响应的卷积后产生的因果分量,也可定义为由初始状态产生的等效零输入激励与系统对应的规范化系统的冲激响应的卷积;两种不同的定义给出了两种不同的计算系统零输入响应的解法。本文利用反因果微分性质严格证明了,这两种不同的定义和解法是等价的,它们会给出同一个计算结果,并用例题证明其正确性。这有助于学生深刻理解系统零输入响应的形成原因和内含的物理意义。

Zero-input response of an LTI continuous-time system Can be defined as the causal component generated by the convolution of system impulse response and input anti-causal component. It can be alsc defined as the convolution of the equivalent zero-input excitation, generated by system initial state, and the impulse response of the corresponding canonical system. Each definition gives out the corresponding algorithm. In this paper, the equivalence relation between the two different definitions of zero-input response is rigorously proved theoretically by means of anti-causal differential theorem. It means that the same computational result must be given out by the two different definitions. Its correctness is demonstrated by the typical examples. The discussion is helpful for students to profoundly understand the cause of formation ot the system＇s zero-input response with its physical meaning.