Signals and Systems Lecture 3

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)