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?

The answer lies in the Dirac Delta function. The Dirac delta function is a function on the real line which is zero everywhere except at the origin, where it is infinite,

$\delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}$

and which is also constrained to satisfy the identity

$\int _{-\infty }^{\infty }\delta (x)\,dx=1.$

So the dirac delta function only works once at the origin, so far. We can change the function so it's effect occurs periodically, like this:

$\sum _{n=-\infty }^{\infty }x(t)\cdot \delta (t-nT)\$
Don't forget the Fourier transform! It will help you look at the frequency response of your new dirac function.
${\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }x(t)\cdot \delta (t-nT)\}\right\}$

Presto! You have a new tool for getting discrete values that can reproduce a continuous waveform. You need to sample the waveform at least twice as fast as the highest frequency that you are recording, or you might permanently lose the information from that frequency due to aliasing. Once these values are recorded, more processing can be done to make sure the signal coming out is just as smooth a wave as the original, or at least to a degree where the loss from sampling is undetected.