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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:
Image
A periodic discrete-time signal. Note how it repeats every 8 time units.
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 x n N 1 n N 2 undefined else

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 :

Image
Infinite-length signal (only a portion of it is shown)
Image
Finite-length signal
(a) an infinite-length signal x n (only part of it shown) has a portion extracted via windowing to create (b) a finite-length signal 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 n N 2 :

y n 0 N 0 n N 1 x n N 1 n N 2 0 N 2 n N 3

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.

Questions & Answers

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Source:  OpenStax, Discrete-time signals and systems. OpenStax CNX. Oct 07, 2015 Download for free at https://legacy.cnx.org/content/col11868/1.2
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