# Why the formula below are critically-damped second-order linear systems?

This is part formula of command-response model (Fujisaki Model)：

Analysis and synthesis of fundamental frequency contours of Standard Chinese using the command–response model


http://www.sciencedirect.com/science/article/pii/S0167639305001706 Why they are critically-damped second-order linear systems? • i have no idea what $G_\mathrm{t}(t)$ is. it is not the impulse response of an LTI system with that $\min[.]$ function in there. the reason the $G_\mathrm{p}(t)$ is critically damped is that the expression with form $h(t) \ = \ t \ e^{-\alpha t}$ is the inverse Laplace Transform of something like $$H(s) \ = \ \frac{1}{(s+\alpha)^2}$$ anything less damped than that will have a sinusoid in the impulse response (it "rings"). anything more damped than that has two decaying exponential functions (with different $\alpha$). – robert bristow-johnson Nov 4 '14 at 3:19
• What do you mean by "s" in the denominator of your formula? – Ping Tang Nov 4 '14 at 3:28
• Laplace transform – robert bristow-johnson Nov 4 '14 at 4:40