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 samples x (n) (n=0, 1, 2,...........) are given by

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

The z-transform X (z) f a sequence x$$\left[ n \right]$$ is given by = $${{0.5} \over {1 - 2{z^{ - 1}}}}$$ . It is given that the region of convergence of X$$\left[ z \right]$$ includes the unit circle. The value of x$$\left[ 0 \right]$$ is

The z-transform of a signal is given by c(z)=$${1 \over 4}{{{z^{ - 1}}(1 - {z^{ - 4}})} \over {{{(1 - {z^{ - 1}})}^2}}}$$. Its final value is