Final answer:
The question seeks the transfer function and causality of a system with impulse response h(t). The system is non-causal since it starts responding before the input signal.
Step-by-step explanation:
The question asks to find the transfer functions for the given system characterized by h(t) = e⁻ₐᵗu(t+2), where a > 0, and to determine if the systems are causal. A system is causal if its output depends only on present and past inputs, not future inputs. The given impulse response, along with the step function u(t+2), suggests that the system starts responding at t = -2, which implies non-causality because it anticipates the input signal. To derive the transfer function in the Laplace domain, the Laplace transform of the given impulse response function h(t) would be computed. This would result in a rational function depending on the complex Laplace variable s and the parameter a.
The transfer function of the system h(t) = e⁻ᵃᵗu(t+2), a>0 can be found by taking the Laplace transform of the given function. The Laplace transform of e⁻ᵃᵗ is 1/(s+a) and the Laplace transform of u(t+2) is e⁻²ˢ/s. Therefore, the transfer function becomes H(s) = 1/(s+a) * e⁻²ˢ/s.
To determine if the system is causal, we need to check if its impulse response h(t) is zero for t<0. In this case, h(t) = e⁻ᵃᵗu(t+2). As long as a>0, the function e⁻ᵃᵗ is non-zero for all t>0, which means the system is causal.