1. Numerical Methods Fourier Transform Pair Part: Frequency and Time Domain http://numericalmethods.eng.usf.edu

2. For more details on this topic Go to http://numericalmethods.eng.usf.eduhttp://numericalmethods.eng.usf.edu Click on keyword Click on Fourier Transform Pair

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5. Chapter 11.03: Fourier Transform Pair: Frequency and Time Domain Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates http://numericalmethods.eng.usf.edu 56/2/2014 Lecture # 5

6. http://numericalmethods.eng.usf.edu6 Example 1

7. http://numericalmethods.eng.usf.edu7 Frequency and Time Domain The amplitude (vertical axis) of a given periodic function can be plotted versus time (horizontal axis), but it can also be plotted in the frequency domain as shown in Figure 2. Figure 2 Periodic function (see Example 1 in Chapter 11.02 Continuous Fourier Series) in frequency domain.

8. Frequency and Time Domain cont. Figures 2(a) and 2(b) can be described with the following equations from chapter 11.02, (39, repeated) where (41, repeated) http://numericalmethods.eng.usf.edu8

9. 9 For the periodic function shown in Example 1 of Chapter 11.02 (Figure 1), one has: Frequency and Time Domain cont.

10. Define: or http://numericalmethods.eng.usf.edu10

11. http://numericalmethods.eng.usf.edu11 Also, Frequency and Time Domain cont.

12. Thus: Using the following Euler identities http://numericalmethods.eng.usf.edu12

13. http://numericalmethods.eng.usf.edu13 Noting that for any integer 1 Frequency and Time Domain cont.

14. Also, Thus, http://numericalmethods.eng.usf.edu14

15. http://numericalmethods.eng.usf.edu15 From Equation (36, Ch. 11.02), one has (36, repeated) Hence; upon comparing the previous 2 equations, one concludes: Frequency and Time Domain cont.

16. For the values forand (based on the previous 2 formulas) are exactly identical as the ones presented earlier in Example 1 of Chapter 11.02. http://numericalmethods.eng.usf.edu16

17. Frequency and Time Domain cont. Thus: http://numericalmethods.eng.usf.edu17

18. http://numericalmethods.eng.usf.edu18 Frequency and Time Domain cont.

19. http://numericalmethods.eng.usf.edu19

20. http://numericalmethods.eng.usf.edu20 In general, one has Frequency and Time Domain cont.

21. THE END http://numericalmethods.eng.usf.edu

22. This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate Acknowledgement

23. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

24. The End - Really

25. Numerical Methods Fourier Transform Pair Part: Complex Number in Polar Coordinates http://numericalmethods.eng.usf.edu

26. For more details on this topic Go to http://numericalmethods.eng.usf.eduhttp://numericalmethods.eng.usf.edu Click on keyword Click on Fourier Transform Pair

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29. Chapter 11.03: Complex number in polar coordinates (Contd.) In Cartesian (Rectangular) Coordinates, a complex number can be expressed as: In Polar Coordinates, a complex number can be expressed as: http://numericalmethods.eng.usf.edu29 Lecture # 6

30. Complex number in polar coordinates cont. Thus, one obtains the following relations between the Cartesian and polar coordinate systems: This is represented graphically in Figure 3. Figure 3. Graphical representation of the complex number system in polar coordinates. http://numericalmethods.eng.usf.edu30

31. Complex number in polar coordinates cont. Hence and http://numericalmethods.eng.usf.edu31

32. http://numericalmethods.eng.usf.edu32 Based on the above 3 formulas, the complex numbers can be expressed as: Complex number in polar coordinates cont.

33. http://numericalmethods.eng.usf.edu33 Notes: (a)The amplitude and angle are 0.59 and 2.14 respectively (also see Figures 2a, and 2b in chapter 11.03). (b) The angle (in radian) obtained from will be 2.138 radians (=122.48 o ). However based on Then = 1.004 radians (=57.52 o ). Complex number in polar coordinates cont.

34. http://numericalmethods.eng.usf.edu34 Since the Real and Imaginary components of are negative and positive, respectively, the proper selection for should be 2.1377 radians. Complex number in polar coordinates cont.

35. http://numericalmethods.eng.usf.edu35 Complex number in polar coordinates cont.

36. http://numericalmethods.eng.usf.edu36 Complex number in polar coordinates cont.

37. THE END http://numericalmethods.eng.usf.edu

38. This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate Acknowledgement

39. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

40. The End - Really

41. Numerical Methods Fourier Transform Pair Part: Non-Periodic Functions http://numericalmethods.eng.usf.edu

42. For more details on this topic Go to http://numericalmethods.eng.usf.eduhttp://numericalmethods.eng.usf.edu Click on keyword Click on Fourier Transform Pair

43. You are free to Share – to copy, distribute, display and perform the work to Remix – to make derivative works

44. Under the following conditions Attribution You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial You may not use this work for commercial purposes. Share Alike If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

45. Chapter 11. 03: Non-Periodic Functions (Contd.) Recall (39, repeated) (41, repeated) Define http://numericalmethods.eng.usf.edu45 Lecture # 7 (1)

46. http://numericalmethods.eng.usf.edu46 Then, Equation (41) can be written as And Equation (39) becomes From above equation or Non-Periodic Functions

47. Non-Periodic Functions cont. Figure 4. Frequency are discretized. From Figure 4, http://numericalmethods.eng.usf.edu47

48. Non-Periodic Functions cont. Multiplying and dividing the right-hand-side of the equation by, one obtains ; inverse Fourier transform Also, using the definition stated in Equation (1), one gets ; Fourier transform http://numericalmethods.eng.usf.edu48

49. THE END http://numericalmethods.eng.usf.edu

50. This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate Acknowledgement

51. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

52. The End - Really