## 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.

## Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras. 