Consider a eascade connection of a causal LTI system shown in the Fig. $1$ below

$(6$ pts$)$

Fig. $1.$ Block diagram representation of a cascade connection of two systems

a$.$ Determine the transfer function $H_1(s)=W\frac{s}{x}(s)$ and then $h_1(t)$

b$.$ Determine the transfer function $H_2(s)=Y\frac{s}{W}(s)$ and then $h_2(t)$

c$.$ Determine the overall transfer function $H(s),$ the poles and zeros of $H(s)$ and draw the pole$-$zero diagram and indicate the region of convergence.

d$.$ Determine the overall impulse response $h(t)$ and justify whether the system is stable or not stable.

e$.$ Determine the differential equation relating the input $x(t)$ to the output $y(t).$

f$.$ Draw the block diagram of the overall system as a parallel of connection of two systems.