Question 1 The Pole-zero plot of a transfer function H(z) for a discrete-time LTI system is shown below

Question 1

The Pole-zero plot of a transfer function H(z) for a discrete-time LTI system is shown below.

                                                                Im(z)

Z=(1)

                                                                                                                                                                           Re(z)                                                                                                                                                                                                                                          

Can a system with this pole-zero plot be both stable and casual? Write “X”, if the pole-zero plot information is insufficient to determine the answer. Explain to get credit.

Question 2

Let N be a positive integer. Let y[n] = x [n/N], when n is divisible by N, and y[n]= 0, otherwise. Find a closed from expression (i.e., without summation, integration, etc.) for Y(z) in terms of X(z).

Question 3

The input to a casual linear time-invariant (LTI) system is

X[n] = u [- n – 1] + (½)ⁿu[n]

Where u[n] is a unit step function. The z-transform of the output of this system is

Y(z) = -1/2z^-1/(1-½z^-1) (1+z^-1)

Determine the z transform h(z) of the system impulse response. Be sure to specify the region of convergense.

Hint: Recall that the z-transform of aⁿu[n] is equal to 1/1-az-1 ad its ROC is defined as IzI > IaI.

On the other hand, the z-transform of aⁿu[n] is equal to 1/1-az-1 while its ROC is defined as IzI < IaI.