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==DT LTI System== | ==DT LTI System== | ||
| − | <math>y[n] = | + | <math>y[n] = x[n] + 2x[n-1]</math> |
==h[n]== | ==h[n]== | ||
| − | <math>h[n] = \ | + | <math>h[n] = \delta [n] + 2 \delta [n-1]</math> |
==H(z)== | ==H(z)== | ||
| − | <math>H(z) = \sum_{m=-\infty}^{\infty}h[m] e^{-j \omega m} = \sum_{m=-\infty}^{\infty} \ | + | <math>H(z) = \sum_{m=-\infty}^{\infty}h[m] e^{-j \omega m} = \sum_{m=-\infty}^{\infty} (\delta [m] + 2 \delta [m-1]) e^{-j \omega m} = \sum_{m=-\infty}^{\infty} \delta [m] e^{-j \omega m} + \sum_{m=-\infty}^{\infty} 2 \delta [m-1] e^{-j \omega m} = 1 + 2 e^{-j \omega}</math> |
| − | <math>\ | + | ==Response to x[n]== |
| + | |||
| + | Input <math>x[n]</math> is the following signal: | ||
| + | |||
| + | [[Image:SawDTJP_ECE301Fall2008mboutin.jpg]] | ||
| + | |||
| + | The Fourier series coefficients for <math>x[n]</math> are: | ||
| + | |||
| + | <font size="4.5"> | ||
| + | <math>a_{0} = 1</math> | ||
| + | </font> | ||
| + | |||
| + | <math>a_{1} = -\frac{1}{2}</math> | ||
| + | |||
| + | <font size="4.5"> | ||
| + | <math>a_{2} = 0</math> | ||
| + | </font> | ||
| + | |||
| + | <math>a_{3} = -\frac{1}{2}</math> | ||
| + | |||
| + | <math>y[n] = \sum a_{k} H(e^{jk \omega_{o}}) e^{jk \omega_{o}n} = \sum_{k=0}^{3} a_{k} H(e^{jk \frac{\pi}{2}}) e^{jk \frac{\pi}{2}n} = \sum_{k=0}^{3} a_{k} (1 + 2 e^{-jk \frac{\pi}{2}}) e^{jk \frac{\pi}{2}n} = (1)(3)(1) + (-\frac{1}{2})(1-2j)e^{j \frac{\pi}{2} n} + 0 + (-\frac{1}{2})(1+2j)e^{j \frac{3 \pi}{2}n}</math> | ||
| + | |||
| + | <math>y[n] = 3 + (-\frac{1}{2}+j)e^{j \frac{\pi}{2} n} + (-\frac{1}{2}-j)e^{j \frac{3 \pi}{2}n}</math> | ||
| + | |||
| + | The system response y[n] is shown below: | ||
| + | |||
| + | [[Image:RespSawDTJP_ECE301Fall2008mboutin.jpg]] | ||
Latest revision as of 12:55, 24 September 2008
Contents
DT LTI System
$ y[n] = x[n] + 2x[n-1] $
h[n]
$ h[n] = \delta [n] + 2 \delta [n-1] $
H(z)
$ H(z) = \sum_{m=-\infty}^{\infty}h[m] e^{-j \omega m} = \sum_{m=-\infty}^{\infty} (\delta [m] + 2 \delta [m-1]) e^{-j \omega m} = \sum_{m=-\infty}^{\infty} \delta [m] e^{-j \omega m} + \sum_{m=-\infty}^{\infty} 2 \delta [m-1] e^{-j \omega m} = 1 + 2 e^{-j \omega} $
Response to x[n]
Input $ x[n] $ is the following signal:
The Fourier series coefficients for $ x[n] $ are:
$ a_{0} = 1 $
$ a_{1} = -\frac{1}{2} $
$ a_{2} = 0 $
$ a_{3} = -\frac{1}{2} $
$ y[n] = \sum a_{k} H(e^{jk \omega_{o}}) e^{jk \omega_{o}n} = \sum_{k=0}^{3} a_{k} H(e^{jk \frac{\pi}{2}}) e^{jk \frac{\pi}{2}n} = \sum_{k=0}^{3} a_{k} (1 + 2 e^{-jk \frac{\pi}{2}}) e^{jk \frac{\pi}{2}n} = (1)(3)(1) + (-\frac{1}{2})(1-2j)e^{j \frac{\pi}{2} n} + 0 + (-\frac{1}{2})(1+2j)e^{j \frac{3 \pi}{2}n} $
$ y[n] = 3 + (-\frac{1}{2}+j)e^{j \frac{\pi}{2} n} + (-\frac{1}{2}-j)e^{j \frac{3 \pi}{2}n} $
The system response y[n] is shown below:
