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Numerical Methods Fast Fourier Transform Part: Informal Development of Fast Fourier Transform http://numericalmethods.eng.usf.edu. For more details on this topic Go to http://numericalmethods.eng.usf.edu Click on Keyword Click on Fast Fourier Transform. You are free.

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  1. Numerical MethodsFast Fourier Transform Part: Informal Development of Fast Fourier Transform http://numericalmethods.eng.usf.edu

  2. For more details on this topic • Go to http://numericalmethods.eng.usf.edu • Click on Keyword • Click on Fast Fourier Transform

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

  4. 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.

  5. Lecture # 11 Chapter 11.05: Informal Development of Fast Fourier Transform Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates 11/12/2014 http://numericalmethods.eng.usf.edu 5

  6. Recall the DFT pairs of Equations (20) and (21) of Chapter 11.04 and swapping the indexes , one obtains where Let Informal Development of Fast Fourier Transform (1) (2) (3) (4) 6 http://numericalmethods.eng.usf.edu

  7. Assuming , then Informal Development cont. Then Eq. (1) and Eq. (2) become (5) (5A) 7 http://numericalmethods.eng.usf.edu

  8. To obtain the above unknown vector for a given vector , the coefficient matrix can be easily converted as Informal Development cont. 8 http://numericalmethods.eng.usf.edu

  9. Hence, the unknown vector can be computed as with matrix vector operations, as following Informal Development cont. (6) http://numericalmethods.eng.usf.edu

  10. For and , then: Informal Development cont. (7) 10 http://numericalmethods.eng.usf.edu

  11. where Thus, in general (for ) Informal Development cont. (8) http://numericalmethods.eng.usf.edu

  12. Remarks: Matrix times vector, shown in Eq. (7), will require 16 (or ) complex multiplications and 12 (or ) complex additions. b) Use of Eq. (8) will help to reduce the number of operation counts, as explained in the next section. Informal Development cont. 12 http://numericalmethods.eng.usf.edu

  13. The End http://numericalmethods.eng.usf.edu

  14. Acknowledgement 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

  15. 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.

  16. The End - Really

  17. Numerical MethodsFast Fourier Transform Part: Factorized Matrix and Further Operation Counthttp://numericalmethods.eng.usf.edu

  18. For more details on this topic • Go to http://numericalmethods.eng.usf.edu • Click on Keyword • Click on Fast Fourier Transform

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

  20. 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.

  21. Lecture # 12 Chapter 11.05: Factorized Matrix and Further Operation Count (Contd.) Equation (7) can be factorized as (9) Let’s define the following “inner – product” (10) 21 http://numericalmethods.eng.usf.edu

  22. with Factorized Matrix cont. From Eq. (9) and (10) we obtain (11A) (11B) (11C) (11D) 22 http://numericalmethods.eng.usf.edu

  23. Factorized Matrix cont. Equations(11A through 11D) for the “inner” matrix times vector requires 2 complex multiplications and 4 complex additions. http://numericalmethods.eng.usf.edu

  24. Factorized Matrix cont. Finally, performing the “outer” product (matrix times vector) on the RHS of Equation(9), one obtains (12) 24 http://numericalmethods.eng.usf.edu

  25. Factorized Matrix cont. (13A) (13B) (13C) (13D) http://numericalmethods.eng.usf.edu

  26. Again, Eqs (13A-13D) requires 2 complex multiplications And 4 complex additions. Thus, the complete RHS of Eq. (9) Can be computed by only 4 complex multiplications (or ) and 8 complex additions (or ), where Since computational time is mainly controlled by the number of multiplications, implementing Eq. (9) will significantly reduce the number of multiplication operations, as compared to a direct matrix times vector operations. (as shown in Eq. (7)). Factorized Matrix cont. 26 http://numericalmethods.eng.usf.edu

  27. Equation (14) gives: For Factorized Matrix cont. For a large number of data points, (14) This implies that the number of complex multiplications involved in Eq. (9) is about 372 times less than the one involved in Eq. (7). 27 http://numericalmethods.eng.usf.edu

  28. Computational Vectors Vector 2 (l=2=r) Vector 1 (l=1) Initial data Vector Graphical Flow of Eq. 9 Consider the case Figure 1. Graphical form of FFT (Eq. 9) for the case 28 http://numericalmethods.eng.usf.edu

  29. Computation Arrays Data Array L=1 L=2 L=3 L=4 Figure 2. Graphical Form of FFT (Eq. 9) for the case 29 http://numericalmethods.eng.usf.edu

  30. The End http://numericalmethods.eng.usf.edu

  31. Acknowledgement 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

  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 MethodsFast Fourier Transform Part: Companion Node Observationhttp://numericalmethods.eng.usf.edu

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

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

  37. 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.

  38. Careful observation of Figure 2 has revealed that each computed vector (where and ) we can always find two (companion) nodes which came from the same pair of nodes in the previous vector. For example, and are computed in terms of and . Similarly, the companion nodes and are computed from the same pair of nodes as and . Lecture # 13 Chapter 11.05 : Companion Node Observation (Contd.) 38 http://numericalmethods.eng.usf.edu

  39. Computation Arrays Data Array L=1 L=2 L=3 L=4 Figure 2. Graphical Form of FFT (Eq. 9) for the case http://numericalmethods.eng.usf.edu

  40. Furthermore, the computation of companion nodes are independent of other nodes (within the –vector). Therefore, the computed and will override the original space of and . Similarly, the computed and will override the space occupied by and which in turn, will occupy the original space of and . Hence, only one complex vector (or 2 real vectors) of length are needed for the entire FFT process. Companion Node Observation cont. 40 http://numericalmethods.eng.usf.edu

  41. a) in the first vector ( ), the companion nodes and are separated by b) in the second vector ( ), the companion nodes and are separated by . Companion Node Spacing Observing Figure 2, the following statements can be made: 41 http://numericalmethods.eng.usf.edu

  42. Companion Node Computation The operation counts in any companion nodes (of the vector), such as and can be explained as (see Figure 2). (15) (16) 42 http://numericalmethods.eng.usf.edu

  43. Thus, the companion nodes and computation will require 1 complex multiplication and 2 complex additions (see Eq. (15) and (16)). The weighting factors for the companion nodes ( and ) are (or ) and (or ), respectively. Companion Node Computation cont. 48 (17) (18) 43 http://numericalmethods.eng.usf.edu

  44. Because the pair of companion nodes and are separated by the “distance” , at the level, after every node computation, then the next nodes will be skipped. (see Figure 2) Skipping Computation of Certain Nodes 44 http://numericalmethods.eng.usf.edu

  45. The End http://numericalmethods.eng.usf.edu

  46. Acknowledgement 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

  47. 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.

  48. The End - Really

  49. Numerical MethodsFast Fourier Transform Part: Determination of http://numericalmethods.eng.usf.edu

  50. For more details on this topic • Go to http://numericalmethods.eng.usf.edu • Click on Keyword • Click on Fast Fourier Transform

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