2. Frequency-Domain Adaptive Filters Wu, Yihong EE 491D 5-12-2005

3. Overview Advantages of FDAF Block-Adaptive Filters (BAF) Convergence Properties Choice of Block Size Methods Computational Complexity

4. Advantages of FDAF Providing a possible solution to the computational complexity problem Attaining a more uniform convergence rate by exploiting the orthogonality properties of DFT

5. Block-Adaptive Filters (BAF) BAF Diagram BLMS Equations Convergence behavior

6. BAF Diagram Figure 1: Block-adaptive filter

7. BLMS Equations  From the conventional LMS, we could get the following equations for BLMS

8. BLMS Equations(cont.) Tap-weight updated once after collection of every block of data samples Error is the difference between the output and the desired signal L is the block length is the effective step-size parameter

9. BLMS Equations(cont.) If written in matrices format:

10. Convergence Properties Average time constant equation: If is the same for both conventional LMS and BLMS, they would have the same average time constant. For zero-order formula, the will be small while comparing to 1/. For fast adaptation, is small and L is large, so will be so large that the higher order effects would cause instability problems. BLMS could overcome the problem.

11. Choice of Block Size L = M: optimal choice for computational complexity L < M: offers the advantage of reduced processing delay L > M: gives rise to redundant operations in the adaptive process Overall, L = M is the best choice

12. Methods Overlap-Save Sectioning Overlap-Add Sectioning Circular Convolution

13. Overlap-Save Sectioning  Linear convolution between a finite-length sequence [w(n)]and an infinite-length sequence [x(n)].  Zero-padding w(n) from N-point to 2N-point  FFT both x(n) and w(n) to get Y(k)  First N point data would be ignored  Only looking for the last N point data

14. Overlap-Save Sectioning Figure 2: Overlap-Save Sectioning.

15. Overlap-Save Sectioning Figure 3: Overlap-Save FDAF

16. Overlap-Save Sectioning FDAF Algorithm Based on Overlap-Save Sectioning

17. Overlap-Add Sectioning Input signal is different Output y(k) Error E(k)

18. Overlap-Add Sectioning Figure 4: Overlap-Add FDAF.

19. Overlap-Add Sectioning FDAF Algorithm Based on Overlap-Add Sectioning

20. Circular Convolution Reducing complexity Causing additional degradation Constraint matrices have been eliminated Rank of F is M = N  Data is not overlapping any more  Error is always linear between Y(K) and D(K)

21. Circular Convolution Figure 5: Circular-Convolution FDAF.

22. Circular Convolution FDAF Algorithm Based on Circular Convolution

23. Computational Complexity Equation Depending on the filter size Greatest: Linear convolution Smallest: Circular convulution

24. Computational Complexity Table 1: FDAF computational complexity ratios

25. Conclusion Advantages of FDAF Computational Complexity Some methods Potential to be developed in the future

26. Thank you very much! Have a nice Summer!