Homework 8, ECE438, Fall 2016, Prof. Boutin

Hard copy due in class, Wednesday November 9, 2016.

Question 1

Below we describe the ROAC of the transfer function of an LTI system. For each ROAC, determine which each of these system properties apply. (Just list the letters of the properties that apply.)

a) the system is causal;

b) the system is BIBO stable;

c) the system has a well defined and finite frequency response function;

d) the system is FIR;

e) The system is IIR;

f) the unit impulse response of the system is right-sided;

g) the unit impulse response of the system is left-sided;

1.1 ROAC= all finite complex numbers, but not infinity.

1.2 ROAC= all complex numbers, including infinity.

1.3 ROAC= all complex numbers z with |z|>0.5, including infinity.

1.4 ROAC= all finite complex numbers z with |z|>0.5, but not infinity.

1.5 ROAC= all complex numbers z with |z|>3, including infinity.

1.6 ROAC= all finite complex numbers z with |z|>3, but not infinity.

1.7 ROAC= all complex numbers z with |z|<0.5.

1.8 ROAC= all complex numbers z with 0<|z|<0.5.

1.9 ROAC= all complex numbers z with |z|<3.

1.10 ROAC= all complex numbers z with 0<|z|<3.

1.11 ROAC= all complex numbers z with 2<|z|<3.

1.12 ROAC= all complex numbers z with 0.5<|z|<2.

Question 2

Compute the z-transform of the signal

$ x[n]= 6^n u[n-1] \ $

Questions 3

Compute the z-transform of the signal

$ x[n]= \left( \frac{1}{5} \right)^n u[-n] $

Questions 4

Compute the z-transform of the signal

$ x[n]= 3^{-|n+1|} \ $

Question 5

Compute the z-transform of the signal

$ x[n]= 2^{n}u[n]- 3^{n}u[-n+1] \ $

Question 6

Compute the inverse z-transform of

$ X(z)=\frac{7}{1+z}, \text{ ROC } |z|<1 $

Question 7

Compute the inverse z-transform of

$ X(z)=\frac{1}{1-3 z}, \text{ ROC } |z|> \frac{1}{3} $

Question 8

Compute the inverse z-transform of

$ X(z)=\frac{1}{1+z^2}, \text{ ROC } |z|< 1 $

Question 9

Compute the inverse z-transform of

$ X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC } |z|<1 $

Question 10

Compute the inverse z-transform of

$ X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC } |z|>3 $

Question 11

Compute the inverse z-transform of

$ X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC } 1< |z|<3 $

Hand in a hard copy of your solutions. Pay attention to rigor!

Presentation Guidelines

• Write only on one side of the paper.
• Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
• Staple the pages together.
• Include a cover page.
• Do not let your dog play with your homework.

Discussion

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