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Introduction to F ast F ourier T ransform

Introduction to F ast F ourier T ransform. 長庚大學電機所 指導教授:黃文傑 博士 研究生:吳濟廷. OUTLINE. What’s FFT Why do we use FFT How’s FFT work Comparison DFT and FFT Conclusion Reference. What ’ s FFT. FFT is a class of algorithms to efficiently compute Discrete Fourier Transform

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Introduction to F ast F ourier T ransform

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  1. Introduction to Fast Fourier Transform 長庚大學電機所 指導教授:黃文傑 博士 研究生:吳濟廷

  2. OUTLINE • What’s FFT • Why do we use FFT • How’s FFT work • Comparison DFT and FFT • Conclusion • Reference

  3. What’s FFT • FFT is a class of algorithms to efficiently compute Discrete Fourier Transform • Classifications • Decimation-in-time • Decimation-in-frequency

  4. Why do we use FFT(1/2) • DFT • Computation Complexity • 1 complex(x)= 4 real(x)+ 2 real(+) Totally N*N*4 real(x)

  5. Why do we use FFT(2/2) • Complex Conjugate Symmetry • Periodicity

  6. How’s FFT work(1/13) • Decimation-in-time FFT • To divide x[n] into several shorter sequence

  7. How’s FFT work (2/13)

  8. How’s FFT work (3/13) • Originally DFT computation : • Two N/2-DFT computation : • When N>2 we can proof that

  9. How’s FFT work (4/13) • Equivalently

  10. How’s FFT work (5/13)

  11. How’s FFT work (6/13)

  12. How’s FFT work (7/13)

  13. How’s FFT work (8/13)

  14. How’s FFT work (9/13)

  15. How’s FFT work (10/13)

  16. How’s FFT work (11/13) • Bit-Reversal => binary

  17. How’s FFT work (12/13)

  18. How’s FFT work (13/13)

  19. Comparison DFT and FFT • For complex(x) • DFT/FFT • N DFT/FFT 32 12.8 1024 204.8 4096 682.67

  20. Conclusion • Advantage • So much lower computation complexity • Disadvantage • Have to calculate all the DFT values

  21. Reference [1]Sophocles J. Orfanidis,”Introduction to Signal Processing” [2]Ren-yuan Lyu,”Digital Signal Processing” [3]Alan V. Oppenheim, Ronald W. Schafer and John R. Buck,”Discrete-time Signal Processing”

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