SIGNALS AND SYSTEMS 402067

TON DUC THANG UNIVERSITY FACULTY OF ELECTRICAL & ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS 402067 Syllabus ACKNOWLEDGEMENT The picture content of this slide is from Charles L Phillips, [2014],Signals, Systems, and Transforms, 5e Pearson 9/22/2016 402067 - SIGNALS AND SYSTEM COURSE OBJECTIVES By the end of this course, student will be able to • understand and use math models to represent basic signals and systems • understand the relationship between time and frequency domain of basic systems’ math model • transform signals and system models from time domain to frequency domain and vice versa • understand the relationship between continuoustime and discrete-time models of system 9/22/2016 402067 - SIGNALS AND SYSTEM COURSE CONTENTS • Introduce the mathematical tools for analysis signals and systems • Provide a basis for applying the above techniques in control and communication engineering 9/22/2016 402067 - SIGNALS AND SYSTEM COURSE CONTENTS This course will cover the following topics: • Continuous time signals and systems • Fourier transform and its applications • Linear time invariant systems • Discrete time signals and systems Simulation software: MATLAB 9/22/2016 402067 - SIGNALS AND SYSTEM OUTCOMES • Remember definitions of mathematical models to represent signals and systems • Understand the mathematical principles of these models to represent signals and systems • Analyze models of continuous time and discrete time signals and systems • Apply these above knowledge in solving and simulating problems related to signals and systems 9/22/2016 402067 - SIGNALS AND SYSTEM REFERENCES Textbook: Charles L Phillips Signals, Systems, and Transforms 5e Pearson Prentice Hall 9/22/2016 402067 - SIGNALS AND SYSTEM REFERENCES Phạm Thị Cư, [2009], Lý thuyết tín hiệu, NXB Giáo dục, Hà Nội Steven T Karris, [2012], Signals and Systems with MATLAB Computing and Simulink modeling, 5e, Orchad Publications Other references please refer to the detail syllabus found in the library 9/22/2016 402067 - SIGNALS AND SYSTEM PREREQUISITE ENGINEERING ANALYSIS - 402064 9/22/2016 402067 - SIGNALS AND SYSTEM GRADES • 10%: writing test • 20%: homework, quizzes, project • 20%: midterm exam (writing test) • 50%: final exam (50% writing test + 50% multiple choices) 9/22/2016 402067 - SIGNALS AND SYSTEM 10 DISCRETE-TIME SIGNALS PROPERTIES • Periodic signals: A discrete-time signal x[n] is periodic with period N if 𝑥[𝑛 + 𝑁] = 𝑥[𝑛] 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 12 DISCRETE-TIME SYSTEMS Denote: • x[n] is the input of a discrete-time system • y[n] is the output of a discrete-time system We have y[n] = T(x[n]) : mathematical model of the system (Recall: T is a transformation) System can be represented by block diagram 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 13 INTERCONNECTING SYSTEMS • Parallel connection 𝑦 𝑛 = 𝑦1 𝑛 + 𝑦2 𝑛 = 𝑇1 𝑥 𝑛 + 𝑇2 𝑥 𝑛 = 𝑇(𝑥[𝑛]) • Series connection 𝑦[𝑛] = 𝑇2(𝑦1[𝑛]) = 𝑇2(𝑇1(𝑥[𝑛])) = 𝑇(𝑥[𝑛]) 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 14 DISCRETE-TIME SYSTEMS PROPERTIES Similar to continuous-time systems, discrete-time systems have the following properties: • Memory • Invertibility • Causality • Stability • Time invariance • Linearity (refer to the textbook for more details) 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 15 DISCRETE-TIME LTI SYSTEMS • Discrete-time LTI system With h[.] is denoted as unit impulse response We have: 𝛿[𝑛] → ℎ[𝑛] Time-invariant system: 𝛿[𝑛 − 𝑘] → ℎ[𝑛 − 𝑘] Linear system: 𝑥[𝑘]𝛿[𝑛 − 𝑘] → 𝑥[𝑘]ℎ[𝑛 − 𝑘] If 𝑥[𝑛] = 9/22/2016 ∞ 𝑘=−∞ 𝑥[𝑘]𝛿[𝑛 − 𝑘] then y[𝑛] = ∞ 𝑘=−∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘] 402067 - Chapter 5: Discrete-Time Signals & Systems 16 CONVOLUTION SUM • The output y[𝑛] = ∞ 𝑘=−∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘] is denoted as convolution sum of x[n] and h[n] y 𝑛 =𝑥 𝑛 ∗ℎ 𝑛 = 9/22/2016 ∞ 𝑘=−∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘] 402067 - Chapter 5: Discrete-Time Signals & Systems 17 IMPULSE & UNIT STEP RESPONSE Denote: Unit step response of system is s[n] Then, s[n] can be calculated from unit impulse response h[n] as follow: ∞ 𝑛 𝑠𝑛 = 𝑢 𝑛−𝑘 ℎ 𝑘 = 𝑘=−∞ 9/22/2016 ℎ𝑘 𝑘=−∞ 402067 - Chapter 5: Discrete-Time Signals & Systems 18 BLOCK DIAGRAMS • If signal 𝑥[𝑛] goes through an ideal delay, then 𝑥[𝑛] → 𝑦[𝑛] = 𝑥[𝑛 − 1] • Given a discrete system described as follow: 𝑦 𝑛 = 𝑎𝑦 𝑛 − + 𝑥[𝑛] (first order system) The above system can be represented as 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 19 BLOCK DIAGRAMS There are standard forms: • Direct form I: series connection of the systems • Direct form II: manipulate Direct form I Example: Consider a second order system given: 𝑎0𝑦 𝑛 + 𝑎1𝑦 𝑛 − + 𝑎2𝑦 𝑛 − = 𝑏0𝑥 𝑛 + 𝑏1𝑥 𝑛 − + 𝑏2𝑥 𝑛 − 𝑎0𝑦[𝑛] + 𝑎1𝑦[𝑛 − 1] + 𝑎2𝑦[𝑛 − 2] = 𝑤[𝑛] 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 20 BLOCK DIAGRAMS Given system: 𝑎0𝑦 𝑛 + 𝑎1𝑦 𝑛 − + 𝑎2𝑦 𝑛 − = 𝑤 𝑛 then 𝑦𝑛 = 𝑎0 (𝑤 𝑛 − 𝑎1𝑦 𝑛 − − 𝑎2𝑦 𝑛 − ) with a0≠0 Direct form I is the combination of block diagrams for w[n] and y[n] 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 21 BLOCK DIAGRAMS • Direct form I 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 22 BLOCK DIAGRAMS • Direct form II: Same signal is delayed by two sets of cascaded delays > eliminate one set of delays 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 23 SUMMARY In this chapter, you have learned: • the details of discrete-time systems • how to find the frequency response of a discretetime LTI system • how to represent system using block diagrams 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 24 MATLAB TUTORIAL MATLAB computing: • Discrete-time systems Pages 9-1 to 9-55 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 25 HOMEWORK [1] page 510 to 518 9.1, 9.2, 9.3, 9.5, 9.6, 9.7, 9.10, 9.14, 9.15, 9.16, 9.18, 9.21, 9.23, 9.29 [1] page 564 to 575 10.1, 10.3, 10.7, 10.8, 10.11, 10.12, 10.15, 10.29, 10.33 9/22/2016 402067 - Chapter 5: Discrete-Time Signals & Systems 26