Contents
Sampling Theorem
Let $ \omega_m $ be a non-negative number.
Let $ x(t) $ be a signal with $ X(\omega) = 0 $ when $ |\omega| > \omega_m $.
Consider the samples $ x(nT) $ for $ n = 0, +-1, +-2, ... $
If $ T < \frac{1}{2}(\frac{2\pi}{\omega_m}) $ then $ x(t) $ can be uniquely recovered from its samples.
Variable Definitions
$ T $ Sampling Period
$ \frac{2\pi}{T} = \omega_s $ Sampling Frequency
$ T < \frac{1}{2}(\frac{2\pi}{\omega_m}) <==> \omega_s>2\omega_m $
$ \omega_m $ Maximum frequencye for a band limited signal
$ NQ = 2\omega_m $ Nyquist Rate - The frequencye the sampling frequency should be, or greater.
$ \omega_c $ Cut off frequency for a filter
Impulse-Train Sampling
Let $ x(t) $ be a continuous signal
Let $ p(t) = \sum_{n = -\infty}^\infty \delta(t - nT) $
$ x(t)p(t) = x_p(t) $
$ x_p(t) = x(t) \sum_{n = -\infty}^\infty \delta(t - nT) $
$ x_p(t) = \sum_{n = -\infty}^\infty x(t) \delta(t - nT) $
$ x_p(t) = \sum_{n = -\infty}^\infty x(xT) \delta(t - nT) $
If $ \omega_s $ is not greater then $ 2\omega_m $ Aliasing with happen.
Recoving an Impulse-Train
$ x_p(t) -> H(\omega) -> x(t) $
Where $ H(\omega) = T, |\omega| < \omega_c, else 0 $
$ F^{-1}(H(\omega)) = \frac{T\sin(\omega_ct)}{\pi t} $
Pretty much apply a low pass filter to $ x_p(t) $
Note: $ \omega_m < \omega_c < \omega_s - \omega_m $ exmaple $ \omega_c = \frac{\omega_s}{2} $
Honorable Mentions in Sampling
Zero-order hold operation
First-order hold operation
Discrete-Time Processing of Continous-Time Signals
$ x(t) $ --> C/D Conversion --> $ x_d[n] $ --> $ H_d $ --> $ y_d[n] $ --> D/C Conversion --> $ y(t) $
