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

$X(z) = \sum_{k=-\infty}^{\infty} (2z)^k u[k] = \sum_{k=0}^{\infty} (2z)^k$

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

Questions 2

Compute the z-transform of the signal

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

$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} = \sum_{n=-\infty}^{\infty} 5^n u[n-3] z^{-n} = \sum_{n=-\infty}^{\infty} (\frac{5}{z})^{n} u[n-3]$

$X(z) = \sum_{n=3}^{\infty} (\frac{5}{z})^{n} = \frac{(\frac{5}{z})^3}{1-\frac{5}{z}} = (\frac{5}{z})^3 \frac{z}{z-5} , if \quad |z| > 5$

$X(z) = \left\{ \begin{array}{l l} (\frac{5}{z})^3 \frac{z}{z-5} &, if \quad |z| > 5\\ \text{diverges} &, \quad \text{otherwise} \end{array} \right.$

Questions 3

Compute the z-transform of the signal

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

$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} = \sum_{n=-\infty}^{\infty} 5^{-|n|} z^{-n} = \sum_{n=0}^{\infty} (\frac{1}{5z})^n + \sum_{m=-\infty}^{-1} (\frac{5}{z})^m$

Let k=-m, then

$X(z) = \sum_{n=0}^{\infty} (\frac{1}{5z})^n + \sum_{k=1}^{\infty} (\frac{5}{z})^k$

$X(z) = \frac{1}{1-\frac{1}{5z}} + \frac{ \frac{z}{5}}{1-\frac{z}{5}} = \frac{z}{z-\frac{1}{5}} + \frac{z}{5-z} , if \quad \frac{1}{5} < |z| < 5$

$X(z) = \left\{ \begin{array}{l l} \frac{z}{z-\frac{1}{5}} + \frac{z}{5-z} &, if \quad \frac{1}{5} < |z| < 5$ \\

   \text{diverges} &, \quad \text{otherwise}
 \end{array} \right.

</math>