(New page: Hello, This is my Homework 7 Contribution. I am having some trouble still with the process of doing Fourier Transforms so I thought it would be a good idea to do some examples of how to d...) |
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Lets let : <math>x[n] = a^nu[n], |a| < 1\,</math> | Lets let : <math>x[n] = a^nu[n], |a| < 1\,</math> | ||
| + | |||
Converting to <math>X(e^{j\omega})\,</math> notation we get | Converting to <math>X(e^{j\omega})\,</math> notation we get | ||
| − | <math>X(e^{j\omega}) = \sum^{\infty}_{n = -\infty}\,</math> | + | |
| + | <math>X(e^{j\omega}) = \sum^{\infty}_{n = -\infty} a^n u[n] e^{-j\omega n}\,</math> | ||
| + | |||
| + | Simplifying we get... | ||
| + | |||
| + | <math>X(e^{j\omega}) = \sum^{\infty}_{n = 0} (ae^{-j\omega})^n\,</math> | ||
| + | |||
| + | <math>X(e^{j\omega}) = \frac{1}{1-a e^{-j\omega}}\,</math> | ||
| + | |||
| + | This gives up a magnitude and phase graphs. Something noteworthy is that this function is periodic with an <math>{\omega} = 2\pi\,</math> | ||
| + | |||
| + | ---- | ||
| + | Example 2 | ||
| + | ---- | ||
| + | Let's try something a little more challenging..... | ||
Revision as of 11:32, 24 October 2008
Hello, This is my Homework 7 Contribution.
I am having some trouble still with the process of doing Fourier Transforms so I thought it would be a good idea to do some examples of how to do a Fourier Transform to help clarify the process.
Example 1
Lets take a simple example to start.
Lets let : $ x[n] = a^nu[n], |a| < 1\, $
Converting to $ X(e^{j\omega})\, $ notation we get
$ X(e^{j\omega}) = \sum^{\infty}_{n = -\infty} a^n u[n] e^{-j\omega n}\, $
Simplifying we get...
$ X(e^{j\omega}) = \sum^{\infty}_{n = 0} (ae^{-j\omega})^n\, $
$ X(e^{j\omega}) = \frac{1}{1-a e^{-j\omega}}\, $
This gives up a magnitude and phase graphs. Something noteworthy is that this function is periodic with an $ {\omega} = 2\pi\, $
Example 2
Let's try something a little more challenging.....
