II. PROBLEM 2 1) Find the Zee-transform of the sequence ¤[k] = ul-k). 2) Determine the sequence r[k] whose Zee-transform is given by X(z) log(1+az-1), |z| > |a|

1. If a system function has zeros at the origin of the z-plane then the system function is a properrational function. Why?  2. Can you obtain the z-transform of u[n] at z = 0? If you can, what is its value? If you cannot,why not?  3. What is the preferred method for obtaining the inverse z-transform of rational functions?Describe this method.  4. Explain why a one-sided z-transform is able to determine response to a LCCDE with initialconditions while the two-sided z-transform cannot.  5. Can any system with a rational system function be decomposed into a product of an allpasssystem and a minimum-phase system? If yes, then explain how?

According to Time shifting property of z-transform, if X(z) is the z-transform of x(n) then what is the z-transform of x(n-k)?a) zkX(z)b) z-kX(z)c) X(z-k)d) X(z+k)

If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then what is the z-transform of correlation between the two signals?a) X1(z).X2(z-1)b) X1(z).X2(z-1)c) X1(z).X2(z)d) X1(z).X2(-z)

What is the ROC of z-transform of finite duration anti-causal sequence? a) z=0 b) z=0 c) Entire z-plane, except at z=0 d) Entire z-plane, except at z=0

12 By first resolving Y(z)/z into partial fractions, find Inverse Z- transform (g) 22²-7z (z-1)²(z-3)

show all you work

For each of the following z-domain functions, determine the (causal) sampled time function using the z-transform tables of one-sided z-transform transform pairs and attributes. Assume the one-sided z-transform requires a Region of Convergence of |z| > 1.

QUESTION 4 Z-transform of any điscrete -time signal can be defined using expression below, +00 X(Z) = ) x[n]z-n, x,[n] = a"u[n] x2[n] = -a"u[-n – 1] (b) Instead of having the same X(Z) expression, prove that, both discrete-time signals having different ROC.

Q6