Sampling- A Bridge Between CT and DT
Discrete signals are desirable to work with due to the ability to process them with program such as MATLAB. By use of sampling a continuous signal can be converted to a discrete signal, manipulated via a computer program and then converted back into a continuous time form.
Sampling involves a function known as an impulse train. An impulse train is a series of impulses that are spaced out by a period T, known as the Sampling Period.
Xp(t) is an impulse train that has amplitudes of impulses equal to the samples of X(t) at intervals spaced by T.
$ x_p(t) = x(t)p(t)\! $ where $ p(t) = \sum^{\infty}_{n = -\infty} \delta(t - nT)\! $ and $ x(t)\! $ is the function being sampled.
Since $ x(t) \delta(t - t_0) = x(t_0) \delta(t-t_0)\! $,
$ x_p(t) = \sum^{\infty}_{n = -\infty} x(nT)\delta(t - nT)\! $
Taking the Fourier Transform of this function yields,
$ X_p(j\omega) = \frac{1}{T} \sum^{\infty}_{k = -\infty}X(j(\omega - K*\omega_s)) \! $
which is a scaled and shifted copy of $ X(j\omega)\! $
X(t) can be recovered exactly from Xp(t) by using a low pass filter with gain T and cut off frequency greater than Wm but less than Ws - Wm.
