When we were going over the material for chapter 7, I got really confused about all the omegas. Which is which and what do they mean? So I made a list:
$ \omega_s $: The sampling frequency of the sampling function $ p(t) $.
$ \omega_s = \frac{2\pi}{T} $, where T is the sampling period of $ p(t) $
$ \omega_M $: $ \omega_M $ is the frequency that satisfies $ X(j\omega) = 0 $ for $ |{\omega}| > \omega_M $ where $ X(j\omega) $ is the Fourier Transform of the signal we are sampling. This means that $ X(j\omega) $ can only be non-zero between the frequencies of $ -\omega_M $ and $ \omega_M $.
Why is this important? Well, because if $ \omega_s > 2\omega_M $ then we know that the signal can be reconstructed from it's samples. Why is this so? When you Fourier Transform the sampled signal you will get copies of the Fourier Transform of the original signal separated by $ \omega_s $ from the middle point. So inorder for these copies not to overlap $ \omega_s > 2\omega_M $. If the copies were allowed to overlap they would interfere with eachother and make something new so when Inverse Fourier Transforming the signal you would not get back the original signal.
$ 2\omega_M $ is known as the Nyquist Rate.
$ \omega_c $: Band frequency for the lowpass filter used to recover the original signal from it's samples. Must be greater than $ \omega_M $ but less than $ \omega_s - \omega_M $.
