system classifications and properties

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System Classifications and PropertiesModule by:Melissa Selik, Richard Baraniuk, Stephen Kruzick.

Summary: Describes various classifications of systems.

Introduction

In this module some o f the basic c lassifications of systems will be briefly introduced and the mostimportant properties of these systems are ex plained. As c an be seen, the properties of a system prov ide

an easy way to distinguish one sy stem from another. Understanding these basic differences between

systems, and their properties, will be a fundamental concept used in all signal and system c ourses. Once a

set of systems c an be identified as sharing particular properties, o ne no longer has to repro v e a certain

character istic of a system each time, but it can simply be known due to the the system c lassification.

Classification of Systems

Continuous vs. Discrete

One of the mo st important distinctions to understand is the difference between discre te time and

continuous time systems. A sy stem in which the input signal and output signal both hav e

continuous domains is said to be a co ntinuous sy stem. One in which the input signal and output

signal both have discrete domains is said to be a c ontinuous system. Of co urse, it is possible to

conc eive of signals that belong to neither category , such as sy stems in which sampling of a

continuous time signal or reconstruction from a discrete time signal take place.

Linear vs. Nonlinear


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A linear sy stem is any system that obey s the properties of scaling (first order homo geneity) and

superposition (additivity ) further described b elow. A nonlinear system is any sy stem that does

not have at least one of these properties.

To show that a system obey s the scaling property is to show that

Figure 1:A block diagra m demonstratin g th e scaling property of l inearity

To demonstrate that a system obey s the superposition property of linearity is to show that

Figure 2:A block diagr am dem onstrat ing t he superposition property of linearity

It is possible to chec k a system for linearity in a single (though larger) step. To do this, simply

combine the first two steps to get

Time Invariant vs. Time Varying

A sy stem is said to be time invariant if it commutes with the parameter shift oper ator defined by

for all , which is to say

for all real . Intuitively , that means that for any input function that produces some output

function, any time shift of that input function will produce an output function identical in ev ery

way ex cept that it is shifted by the same amount. Any system that does not have this property is

said to be time varying.

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Figure 3: This block diagram shows wh at th e condition for t im e inv arian ce. The

output is the sam e wheth er the delay is put on the input or th e output.

Causal vs. Noncausal

A causal sy stem is one in which the o utput depends only on curr ent or past inputs, but not future

inputs. Similarly, an anticausal system is one in which the output depends only o n current o r

future inputs, but not past inputs. Finally, a noncausal sy stem is one in which the output depends

on both past and future inputs. All "realtime" systems must be c ausal, since they can not hav e

future inputs av ailable to them.

One may think the idea of future inputs does not seem to make much phy sical sense; however , we

have only been dealing with time as our dependent v ariable so far, which is not always the case.

Imagine rather that we wanted to do image processing. Then the dependent v ariable might

represent pixel positions to the left and right (the "future") of the current position on the image,

and we would not necessarily hav e a causal system.

(a) For a ty pical sy stem to be causal. . .

(b) . . . the output at t im e , , can only depend on the portion of the input signa l


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Figure 4

before .

Stable vs. Unstable

There are sev eral definitions of stability , but the one that will be used mo st frequently in this

course will be bounded input, bounded output (BIBO) stability. In this context, a stable sy stem is

one in which the output is bo unded if the input is also bo unded. Similarly , an unstable system is

one in which at least one bo unded input produces an unbounded output.

Representing this mathematically, a stable sy stem must have the following propert y, where

is the input and is the output. The output must satisfy the co ndition

whenev er we hav e an input to the system that satisfies

and both represent a set of finite positive numbers and these relationships hold for all

of . Otherwise, the system is unstable.

System Classifications Summary

This module desc ribes just some of the many ways in which systems can be c lassified. Systems can be

continuous time, discrete time, or neither. They can be linear or nonlinear, time invariant or time

vary ing, and stable or unstable. We can also div ide them based on their causality properties. There are

other ways to classify systems, such as use of memory , that are not discussed here but will be desc ribed in

subsequent modules.

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This w ork is licensed byMelissa S elik, Richa rd Bara niuk, an d Stephen Kruzickun der a Creat iv e Com m ons

Attribu tion License (CC-BY 1 .0) , and is an Open Educa tional Resour ce.

Last edited byRich ard Baraniuk on Jun 21, 2010 10:26 am GMT-5.

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