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. DFT/IDFT linear conv w/ aliasing

22. Compute Linear Convolution Circular convolution becomes linear convolution!

23. LTI System Implementation

24. Block convolution

25. LTI System Implementation

26. 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