Reference no: EM132396812

Assignment -

Problem 1 - Given is the LCC difference equation that represents some LTI system:

y(n) + ½y(n - 1) = x(n) - 2x(n - 1)

a) Find the system function H(z) of this system.

b) What is Y(z) when x(n) = δ(n+1) + (¼)^n-1u(n-1).

c) When x(n) is as in part b), invert Y(z), i.e. obtain y(n) in time domain using contour integration (method of residues) and using partial-fraction expansion ( note: make sure the rational function you are working with is proper)

Problem 2 - Given the Z-transform

X(z) = (5z²-3z)/(z²-2z-3)

What are the 3 possible regions of convergence for this X(z)?

a) Using contour integration in the region of convergence that has a shape |z|> R (R depends on X(z)), invert X(z) to obtain the time domain signal x(n).

b) Using contour integration in the region of convergence that has a shape of a disk, invert X(z) to obtain the time domain signal x(n).

c) Using contour integration in the ROC that has a shape |z|< R, invert X(z) to obtain the time domain signal x(n).

d) Assuming the same ROC as in part a), invert X(z) using the method of partial-fraction expansion.

Problem 3 - Assume a system function of an LTI system is

H(z) = 5z/z-3

The input to the system x(n) = 3ⁿu(n). Assuming y(n) and Y(z) are the Z-transform pair, and y(n) is the output of the above system:

a) Determine Y(z) assuming ROC of H(z) is |z| > 3.

b) Determine y(n) by inverting Y(z) using the method of residuals (assuming ROC of H(z) is |z|>3).

c) Determine y(n) by inverting Y(z) using the method of partial-fraction-expansion for the same ROC as in parts a) and b).

d) What is the ROC of a stable system given by H(z).