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Lecture 6 : DFT

Lecture 6 : DFT. Xiliang Luo 2014/10. Periodic Sequence. Discrete Fourier Series. For a sequence with period N, we only need N DFS coefs. Discrete Fourier Series. DFS. Synthesis. Analysis. Example. DFS of periodic impulse. DFS Properties. Linearity:. Shift:. DFS Properties.

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Lecture 6 : DFT

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  1. Lecture 6: DFT Xiliang Luo 2014/10

  2. Periodic Sequence • Discrete Fourier Series For a sequence with period N, we only need N DFS coefs

  3. Discrete Fourier Series

  4. DFS Synthesis Analysis

  5. Example • DFS of periodic impulse

  6. DFS Properties Linearity: Shift:

  7. DFS Properties Duality: Periodic Convolution:

  8. DTFT of Periodic Signals

  9. Sampling Fourier Transform Sample the DTFT of an aperiodic sequence: Let the samples be the DFS coefficients:

  10. Sampling Fourier Transform DTFT definition: Synthesized sequence:

  11. Sampling Fourier Transform Synthesized sequence:

  12. Sampling Fourier Transform Sampling the DTFT of the above sequence with N=12, 7

  13. Discrete Fourier Transform For a finite-length sequence, we can do the periodic extension: or DFT definition:

  14. Discrete Fourier Transform DFT is just sampling the unit-circle of the DTFT of x[n]

  15. DFT Properties • Linearity • Circular shift of a sequence • Duality

  16. DFT Properties • Circular convolution

  17. Compute Linear Convolution In DSP, we often need to compute the linear convolution of two sequences. Considering the efficient algorithms available for DFT, i.e. FFT, we typically follow the following steps:

  18. Compute Linear Convolution Linear convolution of two finite-length sequences of length L & P: How about circular convolution using length N=L+P-1?

  19. Compute Linear Convolution Sampling DTFT of x[n] as DFS: one period

  20. Compute Linear Convolution

  21. Compute Linear Convolution DFT/IDFT linear conv w/ aliasing

  22. Compute Linear Convolution Circular convolution becomes linear convolution!

  23. LTI System Implementation

  24. LTI System Implementation Block convolution

  25. LTI System Implementation

  26. LTI System Implementation Overlap-Add Method

  27. Overlap-Save Method P-point impulse response: h[n] L-point sequence: x[n] L > P We can perform an L-point circular convolution as: Observation: starting from sample: P-1, y[n] corresponds to linear convolution!

  28. Overlap-Save Method

  29. Overlap-Save Method

  30. Overlap-Save Method

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