11/15/2006 Ch 7 System Consideration- Paul Lin 1 ECET 307 Analog Networks Signal Processing Ch 7 System Considerations 1 of 3 Fall

11/15/2006 Ch 7 System Consideration- Paul Lin 2 Ch 7: System Considerations Definition of System Definition of System Subsystems or components Subsystems or components System Theory System Theory System Engineering System Engineering

11/15/2006 Ch 7 System Consideration- Paul Lin 3 Type of Systems Continuous Systems Continuous Systems Discrete Systems – discrete Discrete Systems – discrete Digital Systems Digital Systems Linear Systems Linear Systems Non-linear Systems Non-linear Systems Dynamic Systems Dynamic Systems

11/15/2006 Ch 7 System Consideration- Paul Lin 4 Signals 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-Hold circuit at nT clock pulsex(nT) - x(t) is sampled by a Sample-Hold circuit at nT clock pulse T – sampling interval or sampling timeT – sampling interval or sampling time Complex signals Complex signals

11/15/2006 Ch 7 System Consideration- Paul Lin 5 Signals Complex signals Complex signals e jωt = cosωt + j sinωt e jωt = cosωt + j sinωt

11/15/2006 Ch 7 System Consideration- Paul Lin 6 Definition of Linear System Definition of Linear System Definition of Linear System A system is said to be linear with respect to an excitation x(t) and a response y(t) if the following two properties are satisfied: A system is said to be linear with respect to an excitation x(t) and a response y(t) if the following two properties are satisfied: Property 1 (amplitude linearity) Property 2 (superposition principle)Property 1 (amplitude linearity) Property 2 (superposition principle)

11/15/2006 Ch 7 System Consideration- Paul Lin 7 Definition of Linear System Property 1 (amplitude linearity). If an excitation x(t) produces a response y(t), them an excitation Kx(t) should produce a response Ky(t) for any value of K, where K is a constant Property 1 (amplitude linearity). If an excitation x(t) produces a response y(t), them an excitation Kx(t) should produce a response Ky(t) for any value of K, where K is a constant

11/15/2006 Ch 7 System Consideration- Paul Lin 8 Definition of Linear System Example: Property 1 (amplitude linearity). Example: Property 1 (amplitude linearity). Multiplication Multiplication 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

11/15/2006 Ch 7 System Consideration- Paul Lin 9 Definition of Linear System Property 2 (superposition principle). Property 2 (superposition principle). If an excitation x1(t) produces a response y1(t), and an excitation x2(t) produces a response y2(t), then an excitation x1(t) + x2(t) should produce a response y1(t) + y2(t) for arbitrary waveforms x1(t) and x2(t)

11/15/2006 Ch 7 System Consideration- Paul Lin 10 Definition of Linear System Addition Addition x(t) = x1(t) + x2(t) = 10u(t) + 5 cos(t) x(t) = x1(t) + x2(t) = 10u(t) + 5 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 sin(t) y(t) = y1(t) + y2(t) = 20 t + 10 sin(t) y1(t) = 20ty1(t) = 20t y2(t) = 10 sin(t)y2(t) = 10 sin(t)

11/15/2006 Ch 7 System Consideration- Paul Lin 11 Operations of Linear Systems Continuous Systems Continuous Systems Laplace TransformsLaplace Transforms Convolution integral (time)Convolution integral (time) Correlation integral (time)Correlation integral (time) Discrete Systems Discrete Systems Z transformZ transform Convolution sum (time)Convolution sum (time) Correlation sum (time)Correlation sum (time)

11/15/2006 Ch 7 System Consideration- Paul Lin 12 Example: A Circuit System X(s) = L [x(t)]- System Input X(s) = L [x(t)]- System Input Y(s) = L [y(t)]- System Output Y(s) = L [y(t)]- System Output Y(s) = G(s) X(s) Y(s) = G(s) X(s) G(s) = Y(s)/X(s)- System’s Transfer Function G(s) = Y(s)/X(s)- System’s Transfer Function

11/15/2006 Ch 7 System Consideration- Paul Lin 13 Example: A Circuit System (cont.) G(s) is defined by the natural or physical property (R, L, C, etc) of the system, and is not depend on the type of excitation G(s) is defined by the natural or physical property (R, L, C, etc) of the system, and is not depend on the type of excitation The poles and zeros of G(s) are due only to the circuit or system The poles and zeros of G(s) are due only to the circuit or system The order of the system = the order of denominator polynomial of G(s) The order of the system = the order of denominator polynomial of G(s) A circuit containing m non-redundant polynomial is of order m A circuit containing m non-redundant polynomial is of order m

11/15/2006 Ch 7 System Consideration- Paul Lin 14 Example: Unit Impulse Response Input/Output Input/Output x(t) = δ(t)x(t) = δ(t) X(s) = L [x(t)] = 1X(s) = L [x(t)] = 1 Y(s) = G(s)X(s) = G(s) x 1 = G(s)Y(s) = G(s)X(s) = G(s) x 1 = G(s) Time domain impulse response Time domain impulse response y(t) = L -1 [Y(s)] y(t) = L -1 [Y(s)]

11/15/2006 Ch 7 System Consideration- Paul Lin 15 Example: Unit Impulse Response (cont.) The impulse response of the system - the inverse transform of the transfer The impulse response of the system - the inverse transform of the transfer g(t) = L -1 [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)

11/15/2006 Ch 7 System Consideration- Paul Lin 16 System Classification by Applications Electrical Systems – a combination of electrical components Electrical Systems – a combination of electrical components Electrical Power Systems Electrical Power Systems Analog Systems Analog Systems Digital Signal Processing Systems Digital Signal Processing Systems Control Systems Control Systems Computer-based Controlled Space Shuttle System Computer-based Controlled Space Shuttle System

11/15/2006 Ch 7 System Consideration- Paul Lin 17 System Classification by Applications (cont.) Robotics System Robotics System Communication Systems Communication Systems Computer Systems Computer Systems Network Systems Network Systems Wireless Sensor Networks – Monitoring Wireless Sensor Networks – Monitoring

11/15/2006 Ch 7 System Consideration- Paul Lin 18 Examples of Circuit Systems (cont.) Filters Filters x(t) = s(t) + n(t)x(t) = s(t) + n(t) y(t) = s(t)y(t) = s(t)

11/15/2006 Ch 7 System Consideration- Paul Lin 19 Examples of Circuit Systems (cont.) Equalizers Equalizers s(t) - transmitted through a channels(t) - transmitted through a channel x(t) - received signal is a distorted version of s(t)x(t) - received signal is a distorted version of s(t) A system that its response to x(t) equals s(t)A system that its response to x(t) equals s(t) y(t) = s(t)y(t) = s(t)

11/15/2006 Ch 7 System Consideration- Paul Lin 20 Examples of Circuit Systems (cont.) Feedback Feedback 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 of the input x(t)Output y(t) = U(t - t0), a delay version of the input x(t)

11/15/2006 Ch 7 System Consideration- Paul Lin 21 Examples of Circuit Systems (cont.) DC Power Supply DC 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|

11/15/2006 Ch 7 System Consideration- Paul Lin 22 Modeling (representation) of a Systems Differential equations – continuous system Differential equations – continuous system Difference equations – discrete system Difference equations – discrete system Signal Flow Graph/Block Diagrams Signal Flow Graph/Block Diagrams Transfer function Transfer function

11/15/2006 Ch 7 System Consideration- Paul Lin 23 Models for a Continuous Systems Differential Equations, DEs Differential 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 Equations State Differential Equations Unit Impulse (or Impulse Response), h(t) Unit Impulse (or Impulse Response), h(t) Signal Flow Graph or Block Diagrams Signal Flow Graph or Block Diagrams

11/15/2006 Ch 7 System Consideration- Paul Lin 24 Models for a Discrete Systems Difference Equations, DEs Difference 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 Equations State Difference Equations Unit Impulse (or Impulse Response), h(n) Unit Impulse (or Impulse Response), h(n) Signal Flow Graph or Block Diagrams Signal Flow Graph or Block Diagrams

11/15/2006 Ch 7 System Consideration- Paul Lin 25 Example: Difference Equations Express the samples y[n] = y(nT) of the differential equation Express the samples y[n] = y(nT) of the differential equation of a signal x(t) in terms of samples x[n] = x[nT] of x(t). If T is sufficiently small, If T is sufficiently small,

11/15/2006 Ch 7 System Consideration- Paul Lin 26 Example: Difference Equations (cont.) With t = nT, we can represent y[n] in terms of first difference equation With t = nT, we can represent y[n] in terms of first difference equation

11/15/2006 Ch 7 System Consideration- Paul Lin 27 Example: Difference Equations (cont.) C-Program Implementation C-Program Implementation void 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; …. }

11/15/2006 Ch 7 System Consideration- Paul Lin 28 Example: Integral/Summation Express sample y[n] = y(nT) of the integral Express sample y[n] = y(nT) of the integral Approximating the integral by a sum. Approximating the integral by a sum.

11/15/2006 Ch 7 System Consideration- Paul Lin 29 Example: Integral/Summation (cont.) C-Program Implementation C-Program Implementation void 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; …. }

11/15/2006 Ch 7 System Consideration- Paul Lin 30 System-Based Problem Solving Problem Statement Problem Statement System Analysis System Analysis System Requirements System Requirements System Design System Design Modeling and SimulationModeling and Simulation Performance and AnalysisPerformance and Analysis

11/15/2006 Ch 7 System Consideration- Paul Lin 31 System-Based Problem Solving (cont.) System Components: Software, Hardware, Documentation, etc System Components: Software, Hardware, Documentation, etc System Prototyping System Prototyping System Testing System Testing System Integration System Integration Lesson Learned Lesson Learned