1.2 Signal properties

Signal classification

Signal length: finite/infinite

Signal periodicity

Converting between infinite and finite length

The most straightforward way to create a finite-length signal from an infinite-length one is through the process of windowing . A windowing operation extracts a contiguous portion of an infinite-length signal, that portion becoming the new finite-length signal. Sometimes a window will also scale the smaller portion in a particular way. Below is a mathematical expression of windowing (without any scaling):

$y(n)=\begin{cases}x(n) & \text{if $N_1\le n\le N_2$}\\ \text{undefined} & \text{if $\text{else}$}\end{cases}$

Below is a signal $x(n)$ (assume it is infinite-length, with only a part of it shown), with a portion of it extracted to create $y(n)$ :

There are two ways a signal can be converted from a finite-length to infinite-length. The first is referred to as zero-padding . It is easy to take a finite-length signal and then make a larger finite-length signal out of it: just extend the time axis. We have to decide what values to put in the new time locations, and simply putting $0$ at all the new locations is a common approach. Here is how it looks, mathematically, to create a longer signal $y(n)$ from a shorter signal $x(n)$ defined only on $N_1\le n\le N_2$ :

$y(n)=\begin{cases}0 & \text{if $N_0\le n< N_1$}\\ x(n) & \text{if $N_1\le n\le N_2$}\\ 0 & \text{if $N_2< n\le N_3$}\end{cases}$

Here, obviously $N_0< N_1< N_2< N_3$ , and if we extend $N_0$ and $N_3$ to negative and positive infinity, respectively, then $y(n)$ will end up being infinite-length.