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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&les 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&le 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&le 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&le 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