Difference between revisions of "Josh Long: Sampling (Chapter 7) ECE301Fall2008mboutin" - Rhea

Revision as of 12:02, 10 November 2008

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