Homework 7 Solution, ECE438 Fall 2014, Prof. Boutin

Questions 1

Compute the z-transform of the signal

$ x[n]= \left( \frac{1}{2} \right)^n u[-n] $

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} = \sum_{n=-\infty}^{+\infty} (\frac{1}{2})^n u[-n] z^{-n} = \sum_{n=-\infty}^{+\infty} (2z)^{-n} u[-n] $

Let k=-n, then

$ X(z) = \sum_{k=-\infty}^{+\infty} (2z)^k u[k] $

$ X(z) = \left\{ \begin{array}{l l} (\frac{3}{z})^3 &, if \quad |z| < \frac{1}{2}\\ \text{diverges} &, \quad \text{otherwise} \end{array} \right. $

$ \mathcal{F}(x[n]r^{-n}) = X(3e^{jw}) = \mathcal{X}(w) = \frac{\frac{3}{3e^{jw}}}{1-e^{jw}} $

Questions 2

Compute the z-transform of the signal

$ x[n]= 5^n u[n-3] \ $

Questions 3

Compute the z-transform of the signal

$ x[n]= 5^{-|n|} \ $

Question 4

Compute the z-transform of the signal

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

Question 4

Compute the inverse z-transform of

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

Question 5

Compute the inverse z-transform of

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

Question 6

Compute the inverse z-transform of

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

Question 7

Compute the inverse z-transform of

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

Question 8

Compute the inverse z-transform of

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

Question 9

Compute the inverse z-transform of

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

Back to ECE438, Fall 2014, Prof. Boutin