![Example
• F={ [0.5,2], [4,5], [8,8.5] } fmax=12
• λ(F)=1.5+1+.5=3
• Ω=0.25
4](https://image.slidesharecdn.com/mscpart1-150713071005-lva1-app6892/85/Non-Uniform-sampling-and-reconstruction-of-multi-band-signals-4-320.jpg)
![Periodic Non-Uniform sampling
• t= (nL +ci)T!!
• x[c1T], x[c2T],…,x[cpT], x[(L+c1)T], x[(L+c2)T],…,x[(L+cp)T],...
6](https://image.slidesharecdn.com/mscpart1-150713071005-lva1-app6892/85/Non-Uniform-sampling-and-reconstruction-of-multi-band-signals-6-320.jpg)
![Reconstruction
• y(f)= AC(k) z(f) if p > q =>
• Time domain: x [n]= h[n] * x [n]• Time domain: xhi[n]= h[n] * xi[n]
14](https://image.slidesharecdn.com/mscpart1-150713071005-lva1-app6892/85/Non-Uniform-sampling-and-reconstruction-of-multi-band-signals-14-320.jpg)
Non-Uniform sampling and reconstruction of multi-band signals
- 1. Non-Uniform sampling and reconstruction of multi-band signals M. R. Avendi Supervisor: Prof. M. Viberg & Prof. L. Svensson M. R. Avendi June 2010
- 2. Agenda • Multiband Signal Model • Nonuniform Sampling • Reconstruction • Efficient Reconstruction 2
- 3. Signal Model & Definitions signal model: Spectral support : 3 Spectral Occupancy: Launda lower bound:
- 4. Example • F={ [0.5,2], [4,5], [8,8.5] } fmax=12 • λ(F)=1.5+1+.5=3 • Ω=0.25 4
- 5. Uniform Sampling Why not? •As the sparsity increase it will not efficient any more. •Cost in power, calculation,Storage. 5
- 6. Periodic Non-Uniform sampling • t= (nL +ci)T!! • x[c1T], x[c2T],…,x[cpT], x[(L+c1)T], x[(L+c2)T],…,x[(L+cp)T],... 6
- 7. Sampling parameters • T= 1/fmax base sample time • L > 0 period of pattern • p : number of samples per block, p<L • C={ c , c , …, c } : sample pattern set > 0 • C={ c1, c2 , …, cp} : sample pattern set 0 ≤ c1 ≤ c2 ≤ … ≤ cp ≤ L-1 • The average sampling ratio = p/L 7
- 10. Sampling Formulation • Categorize p sequences 10
- 11. Sampling Formulation • Express in matrix form • y is a known p*1 vector • AC is a sub-DFT matrix , p*L • s is unkonwn a L*1 vector p < L ! ! 11
- 12. Active slots and Spectral set • Active slots : 1 & 4 • Spectral index set: k={1,4} • q= number of active slots =|k|=2 12
- 13. Reconstruction • Reduced order model y(f)=AC s(f) => y(f)=AC(k) z(f) ,, 13
- 14. Reconstruction • y(f)= AC(k) z(f) if p > q => • Time domain: x [n]= h[n] * x [n]• Time domain: xhi[n]= h[n] * xi[n] 14
- 16. Simulation Ω=0.25 , fmax=5 L=32,p=12 , D=1.8!, RMSE=1.9%, 16 Time domain and frequency domain of original and reconstructed signal
- 17. Extreme case In an extreme case each of the L spectral cells may be partly occupied by small sliver of the spectrum. Ω=0.2 L=20 , q=20
- 19. What is the differences between Sampling Patterns RMSE=5% RMSE=105%