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.....
