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Analysis of Signals and Systems

Classification of Systems

Dr. David Antonio-Torres

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2Signals and Systems

Objectives

To introduce the concept of system identify the models ofbasic systems and understand the classification of systems

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!Signals and Systems

Contents

Concept of systemsTypes of systems

"epresentation of systems

#nterconnection of systems

$roperties of systems

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%Signals and Systems

Concept of Systems

A system is a an operator or rule that associates an inputsignal &'t( or stimulus to an output signal y't( or response

)iven an input signal a system yields an output signal

y(t) = H[ x(t) ]

System'* Operator(

#nput Signal&'t(

Output Signaly't(

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+Signals and Systems

Types of Systems

"egarding the signals they processA continuous-time system is a system that accepts acontinuous-time signal at the input and yields acontinuous-time signal at the output

A discrete-time system is a system that accepts adiscrete-time signal at the input and yields a discrete-timesignal at the output

A mixed-signal systemhandles both types of signals

"egarding the number of signals at the input and at the output

An SISO system is a system ,ith a single signal at theinput and a single signal at the output

An MIMO systemis a system ,ith multiple signals at theinput and multiple signals at the output

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Signals and Systems

"epresentation of Systems

A mathematical e&pression that characteries a physicalsystem is termed the representation or model of the system

The process of assigning a model to a system is calledmodeling

y

t=

A x1

t+

B x2

t

y t= x t1

y t= cos

x t

4 d

dt

y t+ y t= 2x t

Y s

Xs=

s 1

s 2s+ 1

h t= e

2tu t

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/Signals and Systems

#nterconnection of Systems

0ith the purpose of understanding the behavior of a systemor dealing ,ith its comple&ity a system can be represented asan interconnection of components or sub-systems

Sub-systems can be interconnected

Serially#n parallel

#n a feedbac1 loop

#n a combination of the previous ones

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Signals and Systems

#nterconnection of Systems

Serial connection

$arallel connection

System 3 System 2#nput Output

System 3

System 2

+#nput Output

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4Signals and Systems

#nterconnection of Systems

Serial-parallel connection

Connection in a feedbac1 loop

System 3 System 2

#nput

System !

+ System % Output

System 2

System 3#nput Output+

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35Signals and Systems

$roperties of Systems

Systems can be classified according to the follo,ingproperties

6emory

#nvertibility

CausalityStability

Time invariance

7inearity

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33Signals and Systems

Systems ,ithout 6emory '6emoryless(

A system is said to be memorylessif its output for each valueof the independent variable at a given time is dependent onlyon the input at the same time

y

t=

x

t

y t= K x t

y t= cos

x t

y [ n]=x [n]

y[

n]=

x[

n]

x2[

n]

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32Signals and Systems

Systems ,ith 6emory

A system ,ith memory is a system that retains informationabout input values at times other than the current time

y

t=

x

t

1

y

t=

t

x

d

y [ n]=x [n1 ]

y [ n]=

n

x [ k ]

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3!Signals and Systems

#nvertibility

A system is said to be an invertible system if distinct inputslead to distinct outputs

An invertible system ,hen serially connected to its inversesystem outputs the original input signal

#nvertibleSystem

#nverseSystem

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3%Signals and Systems

Causality

A system is causal if the output at any time depends onvalues of the input at the current time and8or in the past9 asystem is non-causal if the output depends on values of theinput in the future

6emoryless systems are causal systems

:elo, are the e&amples of non-causal systems

y t=x t+1

y[

n]=

1

3 {x[

n+

1]+

x[

n]+

x[

n

1]

}

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3+Signals and Systems

:#:O Stability

A system is said to be stable if ,hen e&cited by a bounded-amplitude input signal yields a bounded-amplitude outputsignal9 an e&ample of a stable system is given belo,

0hen e&cited by a bounded-amplitude input signal anunstable system ,ill yield an unbounded-amplitude outputsignal9 an e&ample of an unstable system is given belo,

y t= x tcos t

y

t=

t x

t

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3Signals and Systems

Time #nvariance

A system is time invariant if its behavior and characteristicsremain fi&ed over time

A system is time invariant if a time shift in the input signalproduces results in an identical time shift in the output signal

Time-#nvariantSystem

&'t( y't(

Time-#nvariantSystem

&'t ; t5( y't ; t5(

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3/Signals and Systems

Time #nvariance

:elo, is the e&ample of a time-invariant system

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3Signals and Systems

7inearity

A linear system is a system that possesses the property ofsuperposition= if an input signal consists of the ,eightedsum of several signals then the output is the ,eighted sum ofthe responses of the system to each of those signals

Superposition is made up of the properties of additivityandscaling 'or homogeneity(9 scaling is graphically representedbelo,

7inearSystem

&3't( y

3't(

7inearSystem

A &3't( A y3't(

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34Signals and Systems

7inearity

Additivityis graphically represented ne&t

7inearSystem

&3't( y

3't(

7inearSystem

&2't( y

2't(

7inearSystem

&3't( > &

2't( y3't( > y2't(

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25Signals and Systems

7inearity

Superpositionis graphically represented ne&t

7inearSystem

&3't( y

3't(

7inearSystem

&2't( y

2't(

7inearSystem

A &3't( > : &

2't( A y3't( > : y2't(

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23Signals and Systems

7inearity