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Signals and Systems Lecture 3

Signals and Systems Lecture 3. CT and DT Systems Basic Systems Properties. CT and DT Systems (P28). This subject deals with mathematical method used to describe signals and to analyze and synthesize systems. Signals are variables that carry information

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Signals and Systems Lecture 3

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  1. Signals and SystemsLecture 3 CT and DT Systems Basic Systems Properties

  2. CT and DT Systems(P28) • This subject deals with mathematical method used to describe signals and to analyze and synthesize systems. • Signals are variables that carry information • Systems process input signals to produce output signals. • In this course, systems are described from input/output perspective, that is, input to the system x causes the output y

  3. SYSTEM EXAMPLESA RCL Circuit - A Electric network(P29)

  4. SYSTEM EXAMPLESA shock absorber – a mechanical system

  5. Observation: different systems could bedescribed by the same input/output relations. The mechanical and electrical systems are dynamically analogous Thus, understanding one of these systems gives insights into the other.

  6. SYSTEM EXAMPLESA shock absorber – a mechanical system Signals and Systems version Med. school description

  7. SYSTEM EXAMPLESA robot car – a close-loop system

  8. Observations • A very rich class of systems (but by no means all systems of interest to us) are described by differential and difference equations. • Such an equation, by itself, does not completely describe the input-output behavior of a system: we need auxiliary conditions (initial conditions, boundary conditions). • In some cases the system of interest has time as the natural independent variable and is causal. However, that is not always the case. • Very different physical systems may have very similar mathematical descriptions.

  9. Chapter 1 Signals and Systems Continuous-Time System N-order Linear Constant-coefficient Differential Equation Discrete-Time System N-order Linear Constant-coefficient Difference Equation

  10. Block diagram(P32) • A block diagram using integrators, adders, and gains

  11. Electronic synthesis of block diagram • The op-amp itself is a functional model of a device that we can synthesize with an electronic circuit including a number of transistors.

  12. Conclusion • We have seen several types of descriptions of systems • All four descriptions define a system with the same dynamic properties. We will develop methods that characterize these systems efficiently and that abstract their critical dynamic properties.

  13. Chapter 1 Signals and Systems output input System 1 System 2 System 1 output input System 2 input output System 1 System 2 § 1.5.2 Interconnection of Systems Series interconnection (级联) Parallel interconnection (并联) Feedback interconnection (反馈)

  14. SYSTEM PROPERTIES(Causality, Linearity, Time-invariance, etc.) • Why bother with such general properties? • Important practical/physical implications. We can make many important predictions of the system behaviors without having to do any mathematical derivations. • They allow us to develop powerful tools (transformations, more on this later) for analysis and design.

  15. Memoryless systems(P32) • The output of a memoryless system at some time depends only on its input at the same time . • For example, for the resistive divider network, • Therefore, depends upon the value of and not on .

  16. Systems with Memory(P33) • Note that v(t) depends not just on i(t) at one point in time t .Therefore, the system that relates v to i exhibits memory.

  17. Chapter 1 Signals and Systems ① ② ③ summer Systems with memory ④ delay ⑤ integrate 无记忆系统:在任意时刻(t)的输出仅仅与同时刻(t)的输入有关。 ——memoryless(无记忆) ——identity system ,memoryless

  18. Chapter 1 Signals and Systems System Inverse System § 1.6.2 Invertibility and Inverse Systems 可逆系统与可逆性 可逆系统:不同的输入导致不同的输出(一一对应)。 ——noninvertible systems 不可逆系统

  19. Causal and Noncausal Systems(P35) • For a causal system the output at time depends only on the input for ,i.e., the system cannot anticipate the input.

  20. Causal and Noncausal Systems(P35) • Physical systems with time as the independent variable are causal systems.There are examples of systems that are not causal. • Physical systems for which time is not the independent variable,e.g.,the independent variable is space (x,y,z ) as in an optical system. • Processing of signals where time is the independent variable but the signal has been recorded or generated in a computer.Processing is not in real time.

  21. Causal and Noncausal Systems(P35) • Mathematically (in CT): A system x(t) → y(t) is causal If x1(t) → y1(t) x2(t) → y2(t) And x1(t) = x2(t) for all t ≤ to Then y1(t) = y2(t) for all t ≤ to • More explicit mathematical definition of causality later.

  22. Chapter 1 Signals and Systems Not Causal ① ② Causal systems ④ 不可预测系统 因果系统 物理上可实现 ③ Systems without memory 非因果系统: 适用于非时间自变量信号的处理.

  23. Stable and Unstable Systems(P36) • Stability can be defined in a variety of ways. • Definition 1: a stable system is one for which an incremental input leads to an incremental output. • Definition 2:A system is BIBO stable if every bounded input leads to a bounded output.We will use this definition.

  24. Chapter 1 Signals and Systems § 1.6.4 Stability (稳定性) Stable System —— not stable ① —— stable ②

  25. Time-Invariant Systems(P37)

  26. TIME-INVARIANCE (TI)(P37) • Informally, a system is time-invariant (TI) if its behavior does not depend on the choice of t = 0. Then two identical experiments will yield the same results, regardless the starting time. • Mathematically (in DT): A system x[n] → y[n] is TI if for any input x[n] and any time shift n0, If x[n] → y[n] then x[n - n0] → y[n - n0] . • Similarly for CT time-invariant system, If x(t) → y(t) then x(t - to) → y(t - to) .

  27. TIME-INVARIANT OR TIME-VARYING ?

  28. LINEAR AND NONLINEAR SYSTEMS(P39) • Many systems are nonlinear. Ex. Economic system. Input: fiscal and monetary policies, labor, resources, etc. → Output: GDP, inflation, etc. System behavior is very unpredictable because it is highly nonlinear. • In this course, we will deal with only linear systems, which are good approximations of nonlinear systems in certain ranges. Ex. Small-signal conductance of a nonlinear diode. • Linear systems can be analyzed accurately.

  29. Linear systems(P39)

  30. LINEARITY(P39) • A (CT) system is linear if it has the superposition property: If x1(t) → y1(t) and x2(t) → y2(t) then ax1(t) + bx2(t) → ay1(t) + by2(t) • For linear systems, zero input → zero output "Proof" 0 = 0⋅ x[n]→ 0 ⋅ y[n] = 0 Question: Is the system y = A + x linear?

  31. PROPERTIES OF LINEAR SYSTEMS(P39) • Superposition If xk[n] → yk[n] Then This property seems to be almost trivial now, but it is one of the most important ones used in this course. • A linear system is causal if and only if it satisfies the conditions of initial rest: (both necessary and sufficient) “Proof” x(t) = 0 for t ≤ to→ y(t) = 0 for t ≤ to(*). a) Suppose system is causal. Show that (*) holds. b) Suppose (*) holds. Show that the system is causal

  32. Chapter 1 Signals and Systems Example 1.17 Example 1.18 Example 1.19 Example 1.20

  33. Chapter 1 Signals and Systems 线性 系统 线性系统的三个特性 ① 微分特性 ② 积分特性 • 频率保持性: • 信号通过线性系统不会产生新的频率分量

  34. Linear and time-invariant (LTI)systems • Many man-made and naturally occurring systems can be modeled as LTI systems. • Powerful techniques have been developed to analyze and to characterize LTI systems. • The analysis of LTI systems is an essential precursor to the analysis of more complex systems.

  35. Problem —Multiplication by a time function • A system is defined by the functional description • Is this system linear? • Is this system time-invariant? Y

  36. Summary • What we have learned? • CT and DT Systems and their basic Properties • Unit Impulse and Unit Step Signal • Singular Functions • What was the most important point in the lecture? • What was the muddiest point? • What would you like to hear more about?

  37. Reading List • Signals and Systems : • Chapter 2.1~ 2.2, • P55~74 • Question : • What is the base of convolution?

  38. Problem Set • 1.27(a),(c),(d) • 1.28(b),(d),(f)

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