The input and output of a continuous system are respectively denoted by x(t) and y(t). Which of the following descriptions corresponds to a causal system?

The impulse response h(t) of a linear time-invariant continuous time system is described by $$h\left( t \right) = \,\,\exp \left( {\alpha t} \right)u\left( t \right)\,\,\, + \,\,\exp \left( {\beta t} \right)u\left( { - t} \right),$$ where u(t) denotes the unit step function, and $$\alpha $$ and $$\beta $$ are real constants. This system is stable if

Let x(t) be the input and y(t) be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, R3, R4.

Properties

P1 : Linear but NOT time-invariant
P2: Time-invariant but NOT linear
P3: Linear and time-invariant

Relations

R1: y(t) = $${t^2}$$ x(t)
R2: y(t) = t$$\left| {x(t)} \right|$$
R3: y(t) = $$\left| {x(t)} \right|$$
R4: y(t) = x(t-5)

In the following network (Fig .1), the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz.
The expression and the region of convergence of the z-transform of the sampled signal are