1. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU Multirate Processing of Digital Signals (II): Short-Length FIR Filter VLSI Signal Processing 台灣大學電機系 吳安宇

2. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 2 Outline Concept of Fast Filtering Algorithm Short-Length FIR Filter (Fast Convolution) Matrix Representation Polynomial Multiplication Systematic Derivation Cascade Structure

3. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 3 Concept of Fast Convolution Reduce the complexity or improve the efficiency in performing the filtering algorithm z -1

4. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 4 Concept of Fast Convolution From Software Implementation Point of View Required computation complexity (CC) < N Multiplication per unit sample (MPU) From Hardware Implementation Point of View M-parallel structure Required hardware complexity (HC) < MN Number of multipliers

5. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 5 Types of Fast Convolution Cook-Toom Algorithm Winograd Algorithm Iterated Convolution Cyclic Convolution Transform-Based Convolution

6. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 6 Transform-based approach Overlap-save and Overlap-add (FFT) Characteristics Block processing Redundancy information exists Overlap-Based Approach FFT Network Transform Domain Multiplication IFFT Network Serial/ParallelParallel/Serial

7. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 7 Overlap-Based Approach Disadvantage MAC structure is dead Hard to pipeline (large amount of memory) Not suitable for DSPs Irregular Not good for VLSI implementation

8. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 8 Concept of Short-Length FIR Filter Block processing

9. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 9 Concept of Short-Length FIR Filter Divide h(n) into two groups (even and odd) where

10. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 10 Polynomial Multiplication Polynomial multiplication  Filtering 

11. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 11 Product of two polynomials of order N requires as least 2N-1 general multiplication Decomposition is not unique Different versions result in Various addition counts Various multiplication counts Various sensitivities to round-off noise Various sensitivities to quantization noise Polynomial Multiplication

12. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 12 Polynomial Multiplication 

13. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 13 Short-Length FIR Filter z -1 H0+H1H0+H1 H0H0 -H 1

14. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 14 Computational Complexity Index: Multiplication Per Unit sample (MPU) z -1 H0+H1H0+H1 H0H0 -H 1 MPU = 3N/4

15. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 15 Fast Convolution based on this approach require at least N(2M-1)/M2 multiplication per output, i.e. Computational Complexity where N is the original filter tap-length and M is decimation factor.

16. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 16 Systematic Derivation z-domain manipulation

17. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 17 Systematic Derivation Direct implementation H0(z2)H0(z2) H1(z2)H1(z2) H1(z2)H1(z2) H0(z2)H0(z2) z -2

18. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 18 Systematic Derivation H1(z2)H1(z2) H0(z2)H0(z2) H 0 (z 2 ) + H 1 (z 2 ) z -2

19. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 19 Transposed Form More input adders, and less output adders More robust against quantization noise H0(z2)H0(z2) H0(z2)H0(z2) H 0 (z 2 ) + H 1 (z 2 ) z -2

20. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 20 Higher Decimation Factor Consider M=3

21. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 21 Higher Decimation Factor (M=3)

22. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 22 Cascade Structure Larger decimation factor, M, greater than 2, 3 are used For large decimation factor, the optimum fast FIR algorithm involves too many adders and interconnections to be of practical interest Cascade of small decimation factor are used instead of direct implementation