1. Numerical Methods Discrete Fourier Transform Part: Discrete Fourier Transform 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 Discrete Fourier Transform

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5. Chapter 11.04 : Discrete Fourier Transform (DFT) Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates 8/27/2015 http://numericalmethods.eng.usf.edu 5 Lecture # 8

6. Discrete Fourier Transform Recalled the exponential form of Fourier series (see Eqs. 39, 41 in Ch. 11.02), one gets: (39, repeated) (41, repeated) http://numericalmethods.eng.usf.edu6

7. 7 then Eq. (39) becomes: (1) If time “ ” is discretized at Discrete Fourier Transform

8. Discrete Fourier Transform cont. To simplify the notation, define: (2) Then, Eq. (1) can be written as: (3) Multiplying both sides of Eq. (3) by, and performing the summation on “ ”, one obtains (note: l = integer number) http://numericalmethods.eng.usf.edu8

9. 9 (4) (5) Discrete Fourier Transform cont.

10. Switching the order of summations on the right-hand-side of Eq.(5), one obtains: (6) Define: (7) There are 2 possibilities for to be considered in Eq. (7) http://numericalmethods.eng.usf.edu10

11. Discrete Fourier Transform—Case 1 Case(1): is a multiple integer of N, such as: ; or where Thus, Eq. (7) becomes: (8) Hence: (9) http://numericalmethods.eng.usf.edu11

12. Discrete Fourier Transform—Case 2 Case(2): is NOT a multiple integer of In this case, from Eq. (7) one has: (10) Define: (11) http://numericalmethods.eng.usf.edu12

13. http://numericalmethods.eng.usf.edu13 because is “NOT” a multiple integer of Then, Eq. (10) can be expressed as: (12) Discrete Fourier Transform—Case 2

14. From mathematical handbooks, the right side of Eq. (12) represents the “geometric series”, and can be expressed as: if (13) if (14) http://numericalmethods.eng.usf.edu14

15. http://numericalmethods.eng.usf.edu15 Because of Eq. (11), hence Eq. (14) should be used to compute. Thus: (See Eq. (10)) (15) (16) Discrete Fourier Transform—Case 2

16. Substituting Eq. (16) into Eq. (15), one gets (17) Thus, combining the results of case 1 and case 2, we get (18) http://numericalmethods.eng.usf.edu16

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

18. 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

19. 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.

20. The End - Really

21. Numerical Methods Discrete Fourier Transform Part: Discrete Fourier Transform http://numericalmethods.eng.usf.edu

22. For more details on this topic  Go to http://numericalmethods.eng.usf.eduhttp://numericalmethods.eng.usf.edu  Click on Keyword  Click on Discrete Fourier Transform

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25. http://numericalmethods.eng.usf.edu25 Substituting Eq.(18) into Eq.(7), and then referring to Eq.(6), one gets: (18A) Recall (where are integer numbers), And since must be in the range becomes Chapter 11.04: Discrete Fourier Transform (DFT) Lecture # 9 Thus:

26. Discrete Fourier Transform—Case 2 Eq. (18A) can, therefore, be simplified to (18B) Thus: (19) where and (1, repeated) http://numericalmethods.eng.usf.edu26

27. Discrete Fourier Transform cont. Equations (19) and (1) can be rewritten as (20) (21) http://numericalmethods.eng.usf.edu27

28. http://numericalmethods.eng.usf.edu28 To avoid computation with “complex numbers”, Equation (20) can be expressed as (20A) where Discrete Fourier Transform cont.

29. (20B) The above “complex number” equation is equivalent to the following 2 “real number” equations: (20C) (20D) http://numericalmethods.eng.usf.edu29

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

31. 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

32. 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.

33. The End - Really

34. Numerical Methods Discrete Fourier Transform Part: Aliasing Phenomenon Nyquist Samples, Nyquist rate http://numericalmethods.eng.usf.edu

35. For more details on this topic  Go to http://numericalmethods.eng.usf.eduhttp://numericalmethods.eng.usf.edu  Click on Keyword  Click on Discrete Fourier Transform

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

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38. Chapter 11.04: Aliasing Phenomenon, Nyquist samples, Nyquist rate (Contd.) When a functionwhich may represent the signals from some real-life phenomenon (shown in Figure 1), is sampled, it basically converts that function into a sequence at discrete locations of http://numericalmethods.eng.usf.edu38 Lecture # 10 Figure 1: Function to be sampled and “Aliased” sample problem.

39. Aliasing Phenomenon, Nyquist samples, Nyquist rate cont. represents the value ofThus, whereis the location of the first sample In Figure 1, the samples have been taken with a fairly large Thus, these sequence of discrete data will not be able to recover the original signal function http://numericalmethods.eng.usf.edu39

40. Aliasing Phenomenon, Nyquist samples, Nyquist rate cont. These piecewise linear interpolation (or other interpolation schemes) will NOT produce a curve which closely resembles the original function. This is the case where the data has been “ALIASED”. http://numericalmethods.eng.usf.edu40 For example, if all discrete values of were connected by piecewise linear fashion, then a nearly horizontal straight line will occur between through and through respectively (See Figure 1).

41. “Windowing” phenomenon Another potential difficulty in sampling the function is called “windowing” problem. As indicated in Figure 2, while is small enough so that a piecewise linear interpolation for connecting these discrete values will adequately resemble the original function, however, only a portion of the function has been sampled (from through ) rather than the entire one. In other words, one has placed a “window” over the function. http://numericalmethods.eng.usf.edu41

42. “Windowing” phenomenon cont. Figure 2. Function to be sampled and “windowing” sample problem. http://numericalmethods.eng.usf.edu42

43. “Nyquist samples, Nyquist rate” Figure 3. Frequency of sampling rate versus maximum frequency content In order to satisfy the frequency ( ) should be between points A and B of Figure 3. http://numericalmethods.eng.usf.edu43

44. Hence: which implies: Physically, the above equation states that one must have at least 2 samples per cycle of the highest frequency component present (Nyquist samples, Nyquist rate). http://numericalmethods.eng.usf.edu44 “Nyquist samples, Nyquist rate”

45. Figure 4. Correctly reconstructed signal. http://numericalmethods.eng.usf.edu45 “Nyquist samples, Nyquist rate”

46. In Figure 4, a sinusoidal signal is sampled at the rate of 6 samples per 1 cycle (or ). Since this sampling rate does satisfy the sampling theorem requirement of, the reconstructed signal does correctly represent the original signal. http://numericalmethods.eng.usf.edu46 “Nyquist samples, Nyquist rate”

47. Figure 5. Wrongly reconstructed signal. In Figure 5 a sinusoidal signal is sampled at the rate of 6 samples per 4 cycles Since this sampling rate does NOT satisfy the requirement the reconstructed signal was wrongly represent the original signal! http://numericalmethods.eng.usf.edu47 “Nyquist samples, Nyquist rate”

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

49. 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

50. 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.

51. The End - Really