commath

1. commath Lecture 03 –Systems

2. Outline • Definition and Block Diagrams • Properties of CT Systems • Linear Time Invariant Systems • Impulse Response • Step Response • Convolution

3. Ice Breaker… • Can you give me an example of a system in your body? • Brief description • Input/Output • Process • Body adjustments

4. Definitions • A signal is a function which represents the time variation of a physical variable. • A systemgenerates a response, or output signal, for a given input signal. • If the input signal and output signal of a system are continuous-time signals (discrete-time signals), then the system is called a continuous-time system (discrete-time system).

5. Definitions • If the system has one input signal and one output signal, we call that system a single-input- single-output (SISO) system. If the system has more than one input and/or output signal, then the system is called a multiple-input-multiple-output (MIMO) system.

6. Definitions • An explicit mathematical expression for a system is called a system representationor model. • The process of deriving a system representation is called modeling. • The development of a system model from measured input and output signals is called system identification.

7. f y x Block Diagrams • A block in a block diagram represents a system, a function on signal spaces. • x and y are variables representing signals in some function space, and f is a system that relates these variables by y = f(x) • Example

8. Block Diagrams* • Series Connected *http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

9. Block Diagrams* • Parallel Connected *http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

10. Block Diagrams* • Feedback System *http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

11. Example

12. Properties of CT Systems • Recall, If the input signal and output signal of a system are continuous-time signals then the system is called a continuous-time system. • Stability Memory • Invertibility Time Invariance • Linearity Feedback • Causality

13. Stability • A stable system is defined as a system with a bounded (limited) system response. That is, if the system is subjected to a bounded input or disturbance and the response is bounded in magnitude, the system is said to be stable.

14. Stability • http://ljs.academicdirect.org/A15/083_098.htm

15. Memory • A system whose output at time t only depends on input at time t and not any other time. • A static system is memoryless (instantaneous).

16. Memory • A system whose current output (at time t) depends on past inputs • Dynamics systems have MEMORY • Most engineering systems are dynamic!

17. Invertibility • If we can obtain the input f(t) back from output y(t) by some operations, the system is said to be invertible. • Essential that distinct inputs result in distinct outputs so there is a one-to-one mapping between an input and corresponding output • For noninvertible systems, different inputs can result in the same output and it is impossible to determine the input for a given output.

18. Examples of Invertibility • Rectification – noninvertible system • Differentiator – noninvertible system unless the initial condition is given. • Y(t) = af(t) + b – invertible system

19. Time Invariance • System is time invariant if a time-shift in the input produces a corresponding time-shift in the output. • Essentially time invariance means that an experiment performed at a certain time will give exactly the same results as that performed at any other time.

20. Time Invariance Example 1

21. Time Invariance Example 2

22. Linearity • A system G is said to be linear if the following properties hold: • Superposition principle – response of a system to the sum of two inputs is identical to sum of the responses of the system, had those inputs been applied alone.

23. Linearity • Homogeneity property • A scaling of the system’s input results in a corresponding scaling of the system’s output. • Most of our work will concentrate on linear systems.

24. Example of Linearity • Note: • In superposition, the output must have the same form as the given. • In homogeneity, the scaling value must be equal.

25. Feedback • Can either be positive or negative • Decreased sensitivity of the system to variations in the parameters of the process. • Improved rejection of the disturbances. • Improved measurement noise attenuation. • Improved reduction of the steady-state error of the system. • Easy control and adjustment of the transient response of the system.

26. Causality • A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values of the input up to that time. • All real-time physical systems are causal, because time only moves forward. Effect occurs after cause. (Imagine if you own a noncausal system whose output depends on tomorrow’s stock price.)

27. Causality • Causality does not apply to spatially varying signals. (We can move both left and right, up and down.) • Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast.

28. Examples

29. Linear Time-Invariant Systems • They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply. • A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs. • Powerful tools associated with LTI System

30. Properties of LTI Systems • Linear – superposition + homogeneity • Time-Invariant – delayed input results to a delayed output • Causal – dependent on current and past signal values • Stable – bounded input, bounded output (BIBO)

31. Impulse Response of LTI Systems • Is the output of a LTI system due to an impulse input at t = 0. • Completely characterizes the behavior of any LTI system. • Often determined from knowledge of the system configuration and dynamics or can be measured by applying an approximate impulse to the system input.

32. Impulse Response of LTI Systems • A true impulse • Zero width • Infinite amplitude • Cannot actually be generated • Physically approximated as a pulse of large amplitude and narrow width. • Impulse Response • System behavior in response to a high-amplitude, extremely short-duration input.

33. Impulse Response of LTI Systems • If the input to a linear system is expressed as a weighted superposition of time-shifted impulses, then the output is a weighted superposition of the system response to each time-shifted impulse. • If the system is also time-invariant, then the system response to a time-shifted impulse is a time-shifted version of the system response to an impulse. • Therefore, the output of the LTI system is given by a weighted superposition of time-shifted impulse response.

34. In simplest terms analogy… • # hammer blow to a structure • # hand clap or gun blast in a room • # air gun blast under water http://cnx.org/content/m12191/latest/

35. Solving the impulse response h(t)

36. Example 1

37. Example 1

38. Examples

39. Step Response • System response to a step input • f(t) = 1 • Solution is just like how the impulse response is derived however, there is now a particular solution

40. Example 1

42. Definition • The convolution of two signals x1(t) and x2(t) is: • If both signals are one-sided

43. Steps of Convolution • Flip: x2(-) • Shift: x2(t-) • Multiply: x1() x2(t-) • Integrate

44. Example

45. h(-) h(t)

48. Multiply and Integrate

49. Multiply and Integrate

50. Multiply and Integrate