MWC-System Expander Implementation using Filter Banks Final Presentation

1. MWC-SystemExpander Implementation using Filter BanksFinal Presentation • Supervisors: Professor YoninaEldar • Deborah Cohen RazLifshitz, 052856721 AssafBismut, 300316684

2. Goals • Learn the principles of the Sub Nyquist theory and MWC system • Comprehend the purpose of the expander • Comprehend the theory behind the implementation • Implement the expander using filter banks • Comparethe performance between the old implementation and the new implementation Goals MWC Expander Filter Banks Theory Comparison Conclusions

3. Goals • Learn the principles of the Sub Nyquist theory and MWC system MWC Expander Filter Banks Theory Comparison Conclusions

4. MWC – Part I: Analog to Digital Component Input: The sparse signal (original signal) Output: M digital channels Goals MWC Expander Filter Banks Theory Comparison Conclusions

5. MWC – Part II: DSP Component Input: M digital channels Output: The final recovered signal Expander Goals MWC Filter Banks Theory Comparison Conclusions

6. Comprehend the Purpose of the Expander Goals MWC Expander Filter Banks Theory Comparison Conclusions

7. Preventing burden on hardware The main issue: In order to reconstruct the original signal, M equations are necessary, where every equation is represented by a physical channel. M must fulfill: M>2N, where N is the number of bands of the original signal. Thus, the burden on hardware becomes significant. Trading channels for sampling rate The solution: In order to reduce hardware overload, we will combine every “Q” channels into one. This will be done by increasing the bandwidth of each channel by Q. As a result the sampling rate is expedited. For separating back the channels, we will use the Expander. Goals MWC Expander Filter Banks Theory Comparison Conclusions

8. Expander – General Architecture Yi[n] M channels Expander Yi,Q[n] Q*M channels For example, Q=3, in the Frequency domain: Xi,1[n] Expander Yi[n] Xi,2[n] Xi,3[n] Goals MWC Expander Filter Banks Theory Comparison Conclusions

9. Comprehend the theory behind the implementation Goals MWC Expander Filter Banks Theory Comparison Conclusions

10. Expander Implementations Hybrid implementation Primary implementation Q LPF Q LPF Q Q Q Q Goals MWC Expander Filter Banks Theory Comparison Conclusions

11. Expander Implementation Filter Banks Goals MWC Expander Filter Banks Theory Comparison Conclusions

12. Polyphase - Theory Given FIR filter order N The filter can be written as Goals MWC Expander Filter Banks Theory Comparison Conclusions

13. Or The result is M decimated filters Goals MWC Expander Filter Banks Theory Comparison Conclusions

14. Q Q Q Q Goals MWC Expander Filter Banks Theory Comparison Conclusions

15. Filter Banks - Theory We will observe the case: Goals MWC Expander Filter Banks Theory Comparison Conclusions

16. Filter Banks - Theory We will observe the case: H0 can be written as a sum of its polyphase parts: Goals MWC Expander Filter Banks Theory Comparison Conclusions

17. Set it to the Hk (z) phrase (the wanted filter): DFT of (P0,P1….Pq-1 ) Goals MWC Expander Filter Banks Theory Comparison Conclusions

18. Implement the Expander using filter banks Goals MWC Expander Filter Banks Theory Comparison Conclusions

19. Expander Implementation Final Filter Banks Architecture Goals MWC Expander Filter Banks Theory Comparison Conclusions

20. Circular Signal Shift Output Input = IDFT Output Fast Frequencies DC Low Frequencies Low Frequencies DC Fast Frequencies Low Frequencies Circular Signal Shift Low Frequencies Fast Frequencies Goals MWC Expander Filter Banks Theory Comparison Conclusions

21. Initial Phase Fixer Unlike we assumed in theory, as a result of the analog filter, the first sample of the signal doesn't arrive in t=0. In the simulator we implement the analog filter by using an digital filter. The order of the digital filter is 5,000. Initial Phase Fixer Goals MWC Expander Filter Banks Theory Comparison Conclusions

22. Expected Expander output Expander output (Before phase fix) Initial Phase Fixer * Looking at one equation Goals MWC Expander Filter Banks Theory Comparison Conclusions

23. Initial Phase Fixer Output Input Initial Phase Fixer . . . Goals MWC Expander Filter Banks Theory Comparison Conclusions

24. Initial Phase problem – solution #2 • Another solution to fix the initial phase problem is by setting the analog filter length in a way that : • t0 MOD qT =0 • When t0 is the filter delay Goals MWC Expander Filter Banks Theory Comparison Conclusions

25. DFT VS FFT Problem: A unit that uses the FFT algorithm to calculate the IDFT can’t provide Wqkn twiddles because q is an odd number (Therefore q isn’t a power of 2). Though it is faster to calculate the IDFT of a signal using the FFT algorithm than the direct way. It is negligible because we calculate the IDFT of a short signal (q=3,5…15). Hardware that calculates the IDFT directly is less common then the ones that uses the FFT algorithm and therefore more expensive, less accurate, and less efficient. Goals MWC Expander Filter Banks Theory Comparison Conclusions

26. DFT VS FFT A way to use the FFT unit: Though Q is an odd number. The expander can be designed for every number including those in the power of two. Example: Q=4 Expander (Q=4) (Using FFT) * Looking at a single channel. Goals MWC Expander Filter Banks Theory Comparison Conclusions

27. We can use Interpolation/ Decimation before the expander that uses the FFT unit to get the same output EXP for q=3 L=4 LPF M=3 Expander (Q=4) (Using FFT) Goals MWC Expander Filter Banks Theory Comparison Conclusions

28. NFAA (No Filter at All) Goals MWC Expander Filter Banks Theory Comparison Conclusions

29. NFAA (No Filter at All) Goals MWC Expander Filter Banks Theory Comparison Conclusions

30. Goals MWC Expander Filter Banks Theory Comparison Conclusions

31. Compare the performance between the old implementation and the new implementation Goals MWC Expander Filter Banks Theory Comparison Conclusions

32. Quantity of CM (Complexity Multiplies) -Theory Primary: Calculate CM using the fast rate frequency for both LPF’s and the exponents. Hybrid: Calculate CM using the fast rate frequency only for the exponents. While it use a lower rate to calculate the CM of the LPF’s Ours: Calculate CM using the low rate frequency for the LPF, IDFT, and the exponents. Table: Number of CM per second (*) CM with one was not included, therefore there are only q-1 CM with exponents M: Hardware Channels ; L: Digital Filter length q: Parameter q of the MWC system ; CM: Complex Multiplies Goals MWC Expander Filter Banks Theory Comparison Conclusions

33. Quantity of CM (Complexity Multiplies) -Theory Polyphase Polyphase Filter Banks Filter Banks No Filter No Filter Primary Goals MWC Expander Filter Banks Theory Comparison Conclusions

34. Quantity of CM (Complexity Multiplies) - Simulation N = 4 Goals MWC Expander Filter Banks Theory Comparison Conclusions

35. Quantity of CM (Complexity Multiplies) - Simulation Goals MWC Expander Filter Banks Theory Comparison Conclusions

36. Correlations - Simulation Goals MWC Expander Filter Banks Theory Comparison Conclusions

37. SNR Simulation In the case that Mq >2N, there is a 100 % success. For the special case that Mq =2N: Average of: (q, N, M) = (3,6,4), (5,10,4), (7,14,4) Goals MWC Expander Filter Banks Theory Comparison Conclusions

38. SNR Simulation (q, N, M) = (3,6,4) (q, N, M) = (5,10,4) Goals MWC Expander Filter Banks Theory Comparison Conclusions

39. Primary VS Filter Banks Goals MWC Expander Filter Banks Theory Comparison Conclusions

40. Appendix Plot of the polyphase filter: q = 3 q = 5 q = 7 Goals MWC Expander Filter Banks Theory Comparison Conclusions