含参变量t的广义函数

Generalized Functions Containing Parameter

本文依据广义函数的概念研究含参量t的跃闭函数及门函数,文中论证了它们既满足广义函数的定义,又具有广义函数的性质,从而导出了广义函数的一个新分支——含参变量t的广义函数。

For a linear time-invariant system, suppose that its input and impulse response are respectively x(t) =x0(t)u(t-t0) and h(t) =h0(t)u(t-ts). Then the zero state response of the system will bewhere φ(t,τ) = x0(τ)h0(t -τ) , g(t,τ) =Gtk[(t-ts) - t0] Δ=u(τ- t0)u(t- ts -τ).A gate function containing parameter t based on the distribution theory can be defined asnamely,Similarly, a unit step function u(t-τ) containing parameter t can also be defin ed on the distribution theory. Then the conclusion can be drawn that the singularity function containing parameters is a new branch of generalized functions.Let the branch of generalized function be g(t,τ), and the testing function in space D be φ(t,τ), then the theory of distribution used originally to define the generalized function g(t) will still be valid and the properties of generalized function g(t) will also be true in general.