11/15/2006 Ch 7 System Consideration- Paul Lin 1 ECET 307 Analog Networks Signal Processing Ch 7...

11/15/200611/15/2006 Ch 7 System Consideration- Paul LinCh 7 System Consideration- Paul Lin 11

ECET 307ECET 307

Analog Networks Signal ProcessingAnalog Networks Signal Processing

Ch 7 System ConsiderationsCh 7 System Considerations1 of 31 of 3

Fall 2006Fall 2006

http://www.etcs.ipfw.edu/~linhttp://www.etcs.ipfw.edu/~lin


Ch 7: System ConsiderationsCh 7: System Considerations

Definition of SystemDefinition of System Subsystems or componentsSubsystems or components System Theory System Theory System EngineeringSystem Engineering


Type of SystemsType of Systems

Continuous SystemsContinuous Systems Discrete Systems – discrete Discrete Systems – discrete Digital SystemsDigital Systems Linear SystemsLinear Systems Non-linear SystemsNon-linear Systems Dynamic SystemsDynamic Systems


SignalsSignals

Continuous signals x(t)Continuous signals x(t) Discrete signals Discrete signals

• x[n] = x(nT)x[n] = x(nT)• x(nT) - x(t) is sampled by a Sample-x(nT) - x(t) is sampled by a Sample-

Hold circuit at nT clock pulseHold circuit at nT clock pulse• T – sampling interval or sampling T – sampling interval or sampling

timetime Complex signalsComplex signals


SignalsSignals

Complex signals Complex signals eejωt jωt = cosωt + j sinωt= cosωt + j sinωt

)sin(cos

cossin)sincos(

)(

tjtAj

tAjtAtjAtAdt

d

eAjdt

deAAe

dt

d tjtj

tj


Definition of Linear SystemDefinition of Linear System

Definition of Linear SystemDefinition of Linear System A system is said to be linear with A system is said to be linear with

respect to an excitation x(t) and respect to an excitation x(t) and a response y(t) if the following a response y(t) if the following two properties are satisfied:two properties are satisfied:• Property 1 (amplitude linearity) Property 1 (amplitude linearity)

Property 2 (superposition principle)Property 2 (superposition principle)



Property 1 (amplitude linearity). Property 1 (amplitude linearity). If an excitation x(t) produces a If an excitation x(t) produces a response y(t), them an excitation response y(t), them an excitation Kx(t) should produce a response Kx(t) should produce a response Ky(t) for any value of K, where K Ky(t) for any value of K, where K is a constantis a constant



Example: Property 1 (amplitude Example: Property 1 (amplitude linearity). linearity).

MultiplicationMultiplication y(t) = Kx(t)y(t) = Kx(t)

• y(t) = 2 amperes, x(t) = 10 voltsy(t) = 2 amperes, x(t) = 10 volts• y(t) = 5 amperes, x(t) = 25 voltsy(t) = 5 amperes, x(t) = 25 volts



Property 2 (superposition Property 2 (superposition principle). principle).

If an excitation x1(t) produces a If an excitation x1(t) produces a response y1(t), and an excitation response y1(t), and an excitation x2(t) produces a response y2(t), x2(t) produces a response y2(t), then an excitation x1(t) + x2(t) then an excitation x1(t) + x2(t) should produce a response y1(t) should produce a response y1(t) + y2(t) for arbitrary waveforms + y2(t) for arbitrary waveforms x1(t) and x2(t)x1(t) and x2(t)



AdditionAddition x(t) = x1(t) + x2(t) = 10u(t) + 5 x(t) = x1(t) + x2(t) = 10u(t) + 5

cos(t)cos(t)• x1(t) = 10 u(t)x1(t) = 10 u(t)• x2(t) = 5 cos(t)x2(t) = 5 cos(t)

y(t) = y1(t) + y2(t) = 20 t + 10 y(t) = y1(t) + y2(t) = 20 t + 10 sin(t)sin(t)• y1(t) = 20t y1(t) = 20t • y2(t) = 10 sin(t)y2(t) = 10 sin(t)


Operations of Linear SystemsOperations of Linear Systems

Continuous SystemsContinuous Systems• Laplace TransformsLaplace Transforms• Convolution integral (time)Convolution integral (time)• Correlation integral (time)Correlation integral (time)

Discrete SystemsDiscrete Systems• Z transformZ transform• Convolution sum (time)Convolution sum (time)• Correlation sum (time)Correlation sum (time)


Example: A Circuit SystemExample: A Circuit System

X(s) = X(s) = LL [x(t)] [x(t)] - System Input- System Input Y(s) = Y(s) = LL [y(t)] [y(t)] - System - System

OutputOutput

Y(s) = G(s) X(s)Y(s) = G(s) X(s) G(s) = Y(s)/X(s)G(s) = Y(s)/X(s) - System’s - System’s

Transfer FunctionTransfer Function


Example: A Circuit System (cont.)Example: A Circuit System (cont.)

G(s) is defined by the natural or G(s) is defined by the natural or physical property (R, L, C, etc) of the physical property (R, L, C, etc) of the system, and is not depend on the system, and is not depend on the type of excitationtype of excitation

The poles and zeros of G(s) are due The poles and zeros of G(s) are due only to the circuit or systemonly to the circuit or system

The order of the system = the order The order of the system = the order of denominator polynomial of G(s) of denominator polynomial of G(s)

A circuit containing m non-redundant A circuit containing m non-redundant polynomial is of order mpolynomial is of order m


Example: Unit Impulse ResponseExample: Unit Impulse Response

Input/OutputInput/Output• x(t) = δ(t)x(t) = δ(t)• X(s) = X(s) = LL [x(t)] = 1 [x(t)] = 1• Y(s) = G(s)X(s) = G(s) x 1 = Y(s) = G(s)X(s) = G(s) x 1 =

G(s)G(s) Time domain impulse responseTime domain impulse response

y(t) = y(t) = LL-1-1[Y(s)][Y(s)]


Example: Unit Impulse Response Example: Unit Impulse Response (cont.)(cont.)

The impulse response of the systemThe impulse response of the system- the inverse transform of the - the inverse transform of the

transfer transfer

g(t) = g(t) = LL-1-1[G(s)][G(s)] Frequency Domain (multiplication)Frequency Domain (multiplication) Y(s) = G(s) X(s)Y(s) = G(s) X(s) Time Domain (Convolution)Time Domain (Convolution) y(t) = g(t) * x(t)y(t) = g(t) * x(t)


System Classification by System Classification by ApplicationsApplications

Electrical Systems – a combination Electrical Systems – a combination of electrical componentsof electrical components

Electrical Power SystemsElectrical Power Systems Analog SystemsAnalog Systems Digital Signal Processing SystemsDigital Signal Processing Systems Control SystemsControl Systems Computer-based Controlled Space Computer-based Controlled Space

Shuttle SystemShuttle System


System Classification by System Classification by Applications Applications (cont.)(cont.)

Robotics SystemRobotics System Communication SystemsCommunication Systems Computer SystemsComputer Systems Network SystemsNetwork Systems Wireless Sensor Networks – Wireless Sensor Networks –

Monitoring Monitoring


Examples of Circuit Systems Examples of Circuit Systems (cont.)(cont.)

FiltersFilters• x(t) = s(t) + n(t)x(t) = s(t) + n(t)• y(t) = s(t)y(t) = s(t)



Equalizers Equalizers • s(t) - transmitted through a channels(t) - transmitted through a channel• x(t) - received signal is a distorted x(t) - received signal is a distorted

version of s(t)version of s(t)• A system that its response to x(t) A system that its response to x(t)

equals s(t)equals s(t)• y(t) = s(t)y(t) = s(t)



FeedbackFeedback• A control plant PA control plant P• Input function – x(t) = U(t), unit stepInput function – x(t) = U(t), unit step• Output y(t) = U(t - t0), a delay version Output y(t) = U(t - t0), a delay version

of the input x(t)of the input x(t)



DC Power SupplyDC Power Supply• A full-wave rectifier system RA full-wave rectifier system R• x(t) = Acosωt, inputx(t) = Acosωt, input• y(t) = K, constant response to |Acosωt|y(t) = K, constant response to |Acosωt|


Modeling (representation) of a Modeling (representation) of a Systems Systems

Differential equations – continuous Differential equations – continuous systemsystem

Difference equations – discrete Difference equations – discrete systemsystem

Signal Flow Graph/Block DiagramsSignal Flow Graph/Block Diagrams Transfer functionTransfer function


Models for a Continuous Systems Models for a Continuous Systems

Differential Equations, DEsDifferential Equations, DEs Transfer Functions, G(s), H(s)Transfer Functions, G(s), H(s) Frequency Response, G(jω), H(jω)Frequency Response, G(jω), H(jω) State Differential EquationsState Differential Equations Unit Impulse (or Impulse Response), Unit Impulse (or Impulse Response),

h(t)h(t) Signal Flow Graph or Block Diagrams Signal Flow Graph or Block Diagrams


Models for a Discrete Systems Models for a Discrete Systems

Difference Equations, DEsDifference Equations, DEs Transfer Functions, G(z), H(z)Transfer Functions, G(z), H(z) Frequency Response, G(ejθ), H(ejθ)Frequency Response, G(ejθ), H(ejθ) State Difference EquationsState Difference Equations Unit Impulse (or Impulse Response), Unit Impulse (or Impulse Response),

h(n)h(n) Signal Flow Graph or Block Diagrams Signal Flow Graph or Block Diagrams


Example: Difference Equations Example: Difference Equations

Express the samples y[n] = y(nT) of Express the samples y[n] = y(nT) of the differential equation the differential equation

of a signal x(t) in terms of samples of a signal x(t) in terms of samples x[n] = x[nT] of x(t). x[n] = x[nT] of x(t).

If T is sufficiently small,If T is sufficiently small,

dt

tdxtxty

)()()( '

T

Ttxtxtx

)()()('


Example: Difference Equations Example: Difference Equations (cont.)(cont.)

With t = nT, we can represent y[n] With t = nT, we can represent y[n] in terms of first difference equationin terms of first difference equation

][1

][

]1[][][

]}1[][{1]1[][

][

nxT

ny

nxnxnx

nxnxTT

nxnxny


Example: Difference Equations Example: Difference Equations (cont.)(cont.)

C-Program ImplementationC-Program Implementationvoid main(){ float T = 0.01; const int N = 10 float x[N]; float dx; float y[N] = {0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ..};

…//read 10 items from ADC and save them in //the x[N] array

for(k =1; k < 10; k++)dx = x[k] – x[k-1];

y[k] = dt/T; …. }


Example: Integral/SummationExample: Integral/Summation

Express sample y[n] = y(nT) of the Express sample y[n] = y(nT) of the integralintegral

Approximating the integral by a Approximating the integral by a sum. sum.

t

dxty0

)()(

n

k

nT

kTxny

nTxTxTxTxTdxnTy

nTt

1

0

][][

)}(...)3()2()({)()(


Example: Integral/Summation Example: Integral/Summation (cont.)(cont.)

C-Program ImplementationC-Program Implementationvoid main(){ float T = 0.01; const int N = 10 float x[N]; float y = 0;

…//read 10 items from ADC and save //them in the x[N] array

for(k =0; k < 10; k++)y = y + x[k]; // y += x[k]

y = T*y; …. }


System-Based Problem Solving System-Based Problem Solving

Problem StatementProblem Statement System AnalysisSystem Analysis System RequirementsSystem Requirements System DesignSystem Design

• Modeling and SimulationModeling and Simulation• Performance and AnalysisPerformance and Analysis


System-Based Problem Solving System-Based Problem Solving (cont.)(cont.)

System Components: Software, System Components: Software, Hardware, Documentation, etcHardware, Documentation, etc

System PrototypingSystem Prototyping System TestingSystem Testing System IntegrationSystem Integration Lesson LearnedLesson Learned

11/15/2006 Ch 7 System Consideration- Paul Lin 1 ECET 307 Analog Networks Signal Processing Ch 7...

Documents

Transcript of 11/15/2006 Ch 7 System Consideration- Paul Lin 1 ECET 307 Analog Networks Signal Processing Ch 7...