| Line 74: | Line 74: | ||
<math>x[n]= 2^{n}u[n]+ 3^{n}u[-n+1] \ </math> | <math>x[n]= 2^{n}u[n]+ 3^{n}u[-n+1] \ </math> | ||
| + | |||
| + | <math>X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} = \sum_{n=-\infty}^{\infty} (2^{n}u[n]+ 3^{n}u[-n+1]) z^{-n} = \sum_{n=-\infty}^{\infty} 2^{n}u[n] z^{-n} + \sum_{m=-\infty}^{\infty} 3^{m}u[-m+1] z^{-m} </math> | ||
| + | |||
| + | <math>X(z) = \sum_{n=-\infty}^{\infty} (\frac{2}{z})^n u[n] + \sum_{m=-\infty}^{\infty} (\frac{3}{z})^{m}u[-m+1] </math> | ||
| + | |||
| + | Let k = -m+1, then | ||
| + | |||
| + | <math>X(z) = \sum_{n=-\infty}^{\infty} (\frac{2}{z})^n u[n] + \sum_{k=-\infty}^{\infty} (\frac{z}{3})^{k-1}u[k] = \sum_{n=0}^{\infty} (\frac{2}{z})^n + \frac{3}{z} \sum_{k=0}^{\infty} (\frac{z}{3})^{k} </math> | ||
| + | |||
| + | <math>X(z) = \frac{1}{1-\frac{2}{z}} + \frac{3}{z} \frac{1}{1-\frac{z}{3}} = \frac{z}{z-2} + \frac{3}{z} \frac{3}{3-z} , if \quad 2 < |z| < 3 </math> | ||
| + | |||
| + | <math> | ||
| + | X(z) = \left\{ | ||
| + | \begin{array}{l l} | ||
| + | \frac{z}{z-2} + \frac{3}{z} \frac{3}{3-z} &, if \quad 2 < |z| < 3 \\ | ||
| + | \text{diverges} &, \quad \text{otherwise} | ||
| + | \end{array} \right. | ||
| + | </math> | ||
| + | |||
== Question 4 == | == Question 4 == | ||
Revision as of 20:41, 2 November 2014
Contents
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. $
Question 4
Compute the z-transform of the signal
$ x[n]= 2^{n}u[n]+ 3^{n}u[-n+1] \ $
$ X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} = \sum_{n=-\infty}^{\infty} (2^{n}u[n]+ 3^{n}u[-n+1]) z^{-n} = \sum_{n=-\infty}^{\infty} 2^{n}u[n] z^{-n} + \sum_{m=-\infty}^{\infty} 3^{m}u[-m+1] z^{-m} $
$ X(z) = \sum_{n=-\infty}^{\infty} (\frac{2}{z})^n u[n] + \sum_{m=-\infty}^{\infty} (\frac{3}{z})^{m}u[-m+1] $
Let k = -m+1, then
$ X(z) = \sum_{n=-\infty}^{\infty} (\frac{2}{z})^n u[n] + \sum_{k=-\infty}^{\infty} (\frac{z}{3})^{k-1}u[k] = \sum_{n=0}^{\infty} (\frac{2}{z})^n + \frac{3}{z} \sum_{k=0}^{\infty} (\frac{z}{3})^{k} $
$ X(z) = \frac{1}{1-\frac{2}{z}} + \frac{3}{z} \frac{1}{1-\frac{z}{3}} = \frac{z}{z-2} + \frac{3}{z} \frac{3}{3-z} , if \quad 2 < |z| < 3 $
$ X(z) = \left\{ \begin{array}{l l} \frac{z}{z-2} + \frac{3}{z} \frac{3}{3-z} &, if \quad 2 < |z| < 3 \\ \text{diverges} &, \quad \text{otherwise} \end{array} \right. $
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 $
