More on DFT

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3F3 – Digital Signal Processing (DSP), January 2009-2010, lecture slides 2a, Dr Elena Punskaya, Cambridge University Engineering Department

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More on DFT

  1. 1. More on DFT Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 39
  2. 2. DFT Interpolation normalised 40
  3. 3. Zero padding 41
  4. 4. Padded sequence 42
  5. 5. Zero-padding π N 43
  6. 6. Zero-padding just visualisation, not additional information! 44
  7. 7. Circular Convolution xxxxxxxxx m circular convolution 45
  8. 8. Example of Circular Convolution Circular convolution of x1={1,2,0} and x2={3,5,4} clock-wise anticlock-wise 1 3 5 4 0 2 folded sequence y(0)=1×3+2×4+0×5 y(1)=1×5+2×3+0×4 y(2)=1×4+2×5+0×3 1 1 1 3 5 4 0 spins 1 spin 2 spins … 5 4 4 3 3 5 0 2 0 2 0 2 x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3 46
  9. 9. Example of Circular Convolution clock-wise anticlock-wise 47
  10. 10. IDFT m + + + + + 48
  11. 11. Standard Convolution using Circular Convolution It can be shown that circular convolution of the padded sequence corresponds to the standard convolution 49
  12. 12. Example of Circular Convolution clock-wise anticlock-wise 1 3 0 2 5 0 4 folded sequence y(0)=1×3+2×0+0×4+0×5 y(0)=1×5+2×3+0×0+0×4 0 y(0)=1×4+2×5+0×3+0×0 1 1 1 3 5 4 0 spins 1 spin 2 spins 0 5 0 2 0 4 3 2 0 0 5 2 4 0 3 … x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3 50 0 0 0
  13. 13. Standard Convolution using Circular Convolution 51
  14. 14. Proof of Validity Circular convolution of the padded sequence corresponds to the standard convolution 52
  15. 15. Linear Filtering using the DFT FIR filter: Frequency domain equivalent: DFT and then IDFT can be used to compute standard convolution product and thus to perform linear filtering. 53
  16. 16. Summary So Far •  Fourier analysis for periodic functions focuses on the study of Fourier series •  The Fourier Transform (FT) is a way of transforming a continuous signal into the frequency domain •  The Discrete Time Fourier Transform (DTFT) is a Fourier Transform of a sampled signal •  The Discrete Fourier Transform (DFT) is a discrete numerical equivalent using sums instead of integrals that can be computed on a digital computer •  As one of the applications DFT and then Inverse DFT (IDFT) can be used to compute standard convolution product and thus to perform linear filtering 54

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