1. ALGORITHMIC S-Z TRANSFORMATIONS FOR CONTINUOUS-TIME TO DISCRETE- TIME FILTER CONVERSION D. Biolek, V. Biolkova Brno University of Technology Czech Republic http://www.vabo.cz/stranky/biolek

2. Generalized Pascal matrix: Using this matrix, the coefficients of transfer functions of the continuous-time and discrete-time linear circuits can be converted on the assumption that both circuits are related by a general first-order s-z transformation. Effective numerical procedure of computing all matrix elements for arbitrary first-order s-z transformation. Introduction

3. S-Z Transformations

5. M- order of s-z transformation (normally M=1) First-order linear s-z transformation: One-to-one linear correspondence {a k }  {c k } and {b k }  {d k }

6. S-Z Transformations linear first-order BILINEAR (BL) BACKWARD DIFFERENCE (BD) FORWARD DIFFERENCE (FD) PARAMETRIC BD-BL …..

7. S-Z Transformations linear first-order GENERAL First-Order (GFO) BL BD BD-BL FD u, v, w ….. Preview GFO Preview BD-BL

8. S-Z Transformations linear first-order {a k }  {c k } (and {b k }  {d k }): Generalized Pascal matrix

9. S-Z Transformations Generalized Pascal matrix (GPM)

10. BL Example: N=5

11. S-Z Transformations Generalized Pascal matrix (GPM) Example: N=5 BD

12. S-Z Transformations Generalized Pascal matrix (GPM) Example: N=5 FD

13. S-Z Transformations Generalized Pascal matrix (GPM) Example: N=5 BD-BL

14. S-Z Transformations Algorithmic compilation of GPM … vector of DFT coefficients of i-th column of the GPM.

15. S-Z Transformations Algorithmic compilation of GPM 1. i = 0. 2. Compiling vectors a i and b i of a size 1 x (N+1) according to the rule 3. Completing both vectors by as much zeros as needed to reach their length to integer power of two. New vectorsare obtained.

16. S-Z Transformations Algorithmic compilation of GPM 4. Performing FFT of vectors 5. Multiplying vectors above element per element and performing inverse FFT: 6. First N+1 elements give the i-th column of the GPM. 7. i = i+1; if i<=N then go to the step 2 else end.

17. S-Z Transformations Algorithmic compilation of GPM - example Example: Parametric BD-BL, r=0.5, N=5. Calculation of GPM in MATLAB: 1. i = 0; 2. a i = [1 2.5 2.5 1.25 0.3125 0.03125]; b i = [1 0 0 0 0 0]; 3. = [1 2.5 2.5 1.25 0.3125 0.03125 0 0]; = [1 0 0 0 0 0 0 0];

18. S-Z Transformations Algorithmic compilation of GPM - example 4. 5-6. 7.59375 1.54929-5.12955i -1.1875-1.28125i -0.17429-0.12955i 0.03125 0.17429+0.12955i -1.1875+1.28125i 1.54929+5.12955i 1 1 1 1 1 2.5 2.5 1.25 0.3125 0.03125 0 0 % i =0

19. Conclusions The proposed algorithm generates all elements of GPM of arbitrary first-order s-z transformation. The described approach is based on the FFT algorithm, which ensures the computation efficiency. The structure of this procedure is not dependent on the type of s-z transformation.