# 1.2 Signal properties

## Signal classification

Signals can be broadly classified as discrete-time or continuous-time, depending on whether the independent variable is integer-valued or real-valued. Signals may also be either real-valued or complex-valued. We will now consider some of the other ways we can classify signals.

## Signal length: finite/infinite

This classification is just as it sounds. An infinite-length discrete-time signal takes values for all time indices: all integer values $n$ on the number line from $-$ all the way up to  . A finite-length signal is defined only for a certain range of $n$ , from some ${N}_{1}$ to ${N}_{2}$ . The signal is not defined outside of that range.

## Signal periodicity

As the name suggests, periodic signals are those that repeat themselves. Mathematically, this means that there exists some integer value $N$ for which $x(n+N)=x(n)$ , for all values of $n$ . So if we define a fundamental period of this particular signal of length, like $N=8$ , then we will see the same signal values shifted by $8$ time indices, by $16$ , $-8$ , $-16$ , etc. Below is an example of a periodic signal: So periodic signals repeat, and clearly periodic signals are going to be, therefore, infinite in length.It's also important to remember that to be periodic in discrete-time, the period $N$ must be an integer. If there is no such integer-valued $N$ for which $x(n+N)=x(n)$ (for all values of $n$ ), then we classify the signal as being aperiodic .

## Converting between infinite and finite length

In different applications, the need will arise to convert a signal from infinite-length to finite-length, and vice versa. There are many ways this operation can be accomplished, but we will consider the most common.

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.

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