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\$

$={\mathcal {F}}\left\{\sum _{k=-\infty }^{\infty }x(t)\cdot e^{i2\pi kf_{s}t}\right\}$

But this Fourier Series is still infinite! What can we do to fix this, so we can store a finite number of values in our computer that can reproduce the waveform we are trying to save?