Homework: Sampling: A class review

Sampling: A Class Review

A continuous function has an infinite amount of information stored on it - a continuous line has an infinite amount of points on it to document. So, the only way we can manipulate transforms on the computer is to quanticize them using the Fourier series.

$X_{s}(f)\,$	$\ {\mathrm {def} }{=}\ {\mathcal {F}}\left\{x_{s}(t)\right\}=\int _{-\infty }^{\infty }x_{s}(t)e^{-i2\pi ft}\,dt\$
	$={\mathcal {F}}\left\{\sum _{k=-\infty }^{\infty }x(t)\cdot e^{i2\pi kf_{s}t}\right\}\$
	$=\sum _{k=-\infty }^{\infty }{\mathcal {F}}\left\{x(t)\cdot e^{i2\pi kf_{s}t}\right\}\$
	$=\sum _{k=-\infty }^{\infty }X(f-kf_{s})\$

To make our job easier when dealing with discrete time based equations, it is helpful to have a