Assist on finding the impulse response fom a simple LTI graph

Question

Could you please advise on the impulse response from this $LTI$ graph? I need to plot it and find if it is stable and causal...

LTI system with response to unitary step ($t$):

Answer 1

s(t) is the convolution of h(t) and u(t) (impulse response and unitary step) in time domain, so the corresponding frequency domain relation is S(s)=H(s)U(s). So if you find the inverse Laplace transform of H(s)=S(s)/U(s) it will give you impulse response h(t), and depending upon the location of poles of S(s)/U(s) of the s-plane you can figure out the stability and causality of system.

Note: you can represent the given S(t) as the sum of shifted t*u(t) (causal ramp signal) and make use of Laplace transform properties. ( s(t)=(t+1)u(t+1)-tu(t)+2(t-2)*u(t-2)-2(t-3)*u(t-3) )