A New Paradigm in Design of IIR Digital Filters

1. Vladislav A. Lesnikov lesnlex@mail.ru Tatiana V. Naumovich ntv_new@mail.ru A New Paradigm in Design of IIR Digital Filters Alexander V. Chastikov alchast@mail.ru Vyatka State University, Russia, Kirov Sergey V. Armishev sarmishev@gmail.com Rondosoft, Inc., USA Speaker of this paper EWDTS-2009 Vladislav A. Lesnikov

2. Motivation • Techniques of synthesis of nonrecursive (FIR) digital filters is well enough try-out and realized in development tools (Altera Quartus, Xilinx System Generator). • Techniques of synthesis of recursive (IIR) digital filters is still a long way off perfection. • This paper is one of steps on a way to perfection of a technique of synthesis of recursive filters EWDTS-2009 Vladislav A. Lesnikov

3. Old Paradigm EWDTS-2009 Vladislav A. Lesnikov

4. The traditional approach to synthesis of digital filters consists of following stages: • Computation of transfer function of the nth order digital filter (functional synthesis) • Choice of the block diagram for implementation of the digital filter(structural synthesis) • Quantization of coefficients of the digital filter • The analysis of characteristics of the filter at finite coefficient word length • Go to stage #... • Exit EWDTS-2009 Vladislav A. Lesnikov

5. The traditional approach to synthesis of digital filters consists of following stages: • Computation of transfer function of the nth order digital filter (Finiteness of word length is not considered.) • Choice of the block diagram for implementation of the digital filter • Quantization of coefficients of the digital filter • The analysis of characteristics of the filter at finite coefficient word length • Go to stage #... • Exit EWDTS-2009 Vladislav A. Lesnikov

6. Computation of transfer function of the nth order digital filter The frequency response of such filter should satisfy to the set tolerance scheme. Thus finiteness of word length is not taken into consideration. Example of tolerance scheme – Low Pass Filter. EWDTS-2009 Vladislav A. Lesnikov

7. Computation of transfer function of the nth order digital filter EWDTS-2009 Vladislav A. Lesnikov

8. Computation of transfer function of the nth order digital filter EWDTS-2009 Vladislav A. Lesnikov

9. EWDTS-2009 Vladislav A. Lesnikov

10. Yet we considered a functional stage of synthesis of filters • At once we pass to structural synthesis EWDTS-2009 Vladislav A. Lesnikov

11. The traditional approach to synthesis of digital filters consists of following stages: • Computation of transfer function of the nth order digital filter • Choice of the block diagram for implementation of the digital filter(structural synthesis) • Quantization of coefficients of the digital filter • The analysis of characteristics of the filter at finite coefficient word length • Go to stage #... • Exit EWDTS-2009 Vladislav A. Lesnikov

12. Choice of the block diagram Strk for implementation of the digital filter Cascaded form Canonical form Direct form Parallel form Ladder structure Ladder structure EWDTS-2009 Vladislav A. Lesnikov

13. Choice of the block diagram Strk for implementation of the digital filter Ladder structure State space structure Scaled normal structure Wave structure Scaled wave structure EWDTS-2009 Vladislav A. Lesnikov

14. Choice of the block diagram Strk for implementation of the digital filter ………………………………………… Infinity of choice!!! EWDTS-2009 Vladislav A. Lesnikov

15. Choice of the block diagram Strk for implementation of the digital filter EWDTS-2009 Vladislav A. Lesnikov

16. Choice of the block diagram Strk for implementation of the digital filter EWDTS-2009 Vladislav A. Lesnikov

17. Choice of the block diagram Strk for implementation of the digital filter EWDTS-2009 Vladislav A. Lesnikov

18. Yet we considered a structural stage of synthesis of filters • At once we pass to quantization of coefficients EWDTS-2009 Vladislav A. Lesnikov

19. The traditional approach to synthesis of digital filters consists of following stages: • Computation of transfer function of the nth order digital filter • Choice of the block diagram for implementation of the digital filter(structural synthesis) • Quantization of coefficients of the digital filter • The analysis of characteristics of the filter at finite coefficient word length • Go to stage #... • Exit EWDTS-2009 Vladislav A. Lesnikov

20. Quantization of coefficients of the digital filter Owing to limitation of word length coefficients of the transfer function calculated on a stage 1 are represented with an error. EWDTS-2009 Vladislav A. Lesnikov

21. Quantization of coefficients of the digital filter As a result zeros and poles also with an error are represented. Displacement of zeros and poles EWDTS-2009 Vladislav A. Lesnikov

22. Quantization of coefficients of the digital filter The frequency response of the filter can fall outside the limits of the tolerance scheme. Distortion of frequency responce EWDTS-2009 Vladislav A. Lesnikov

23. The traditional approach to synthesis of digital filters consists of following stages: • Computation of transfer function of the nth order digital filter • Choice of the block diagram for implementation of the digital filter(structural synthesis) • Quantization of coefficients of the digital filter • The analysis of characteristics of the filter at finite coefficient word length • Go to stage #... • Exit EWDTS-2009 Vladislav A. Lesnikov

24. The analysis of characteristics of the filter at finite coefficient word length At this stage the frequency response, round-off noise are estimated, limit cycles are detected. As a result of the analysis the decision on increase in word length of filter coefficients or on returning to the previous stages can be accepted. EWDTS-2009 Vladislav A. Lesnikov

25. Old paradigm: Conclusions The stage 1 corresponds to functional level of the description of digital filters. The stages 2 – 4 correspond to structural level. Other stages in this paper are not considered. EWDTS-2009 Vladislav A. Lesnikov

26. Old paradigm: Conclusions The considered approach to design possesses the imperfections leading to inefficient practical realization of digital filters. • First, fundamental properties of IIR digital filters with the finite word length are not considered. • Secondly, the small set of possible structures of filters is considered only. • Thirdly, stages of structural level corrupt results of functional level. EWDTS-2009 Vladislav A. Lesnikov

27. New paradigm The purpose of new paradigm to elimination of the specified imperfections The purpose of this paper to bring the contribution to decision this problem EWDTS-2009 Vladislav A. Lesnikov

28. New paradigm At functional level the new approach assumes ultimate determination of transfer function taking into account finite word length of coefficients. Unlike the traditional approach structural synthesis will not corrupt results of functional level. EWDTS-2009 Vladislav A. Lesnikov

29. Number-theoretical aspects of IIR filter design on functional level EWDTS-2009 Vladislav A. Lesnikov

30. Number-theoretical aspects of IIR filter design on functional level EWDTS-2009 Vladislav A. Lesnikov

31. Number-theoretical aspects of IIR filter design on functional level EWDTS-2009 Vladislav A. Lesnikov

32. Number-theoretical aspects of IIR filter design on functional level In our papers number-theoretical aspects of synthesis of filters are described. This results are presented on International conferences: GSPx2004, GSPx2005, GSPx2006 (USA, Santa Clara), IEEE Eurocon 2009 (Russia, Sait-Peterburg) EWDTS-2009 Vladislav A. Lesnikov

33. Number-theoretical aspects of IIR filter design on functional level Quantization of coefficients of the filter leads to z-plane discretization Possible positions of zeros or poles for second-order (a), (c) and fourth-order (b), (d) units in first quadrant of z-plane EWDTS-2009 Vladislav A. Lesnikov

34. Design Flow Diagrams Functional level of synthesis EWDTS-2009 Vladislav A. Lesnikov

35. Algebraic aspects of IIR filter design on structural level As it is described above, at the traditional approach structural synthesis is based on a choice of a variant from the limited number of the block diagrams of digital filters. However the much larger number of structures is known. It is obvious that the set of all possible structures is infinite. EWDTS-2009 Vladislav A. Lesnikov

36. Algebraic aspects of IIR filter design on structural level Realization of algorithms of generation of structures of digital filters is necessary for improvement of procedure of structural synthesis. The algorithms based on representation of digital filters in space of states are known. However, between set of structures and set of state space descriptions of structures there is no biunique correspondence. EWDTS-2009 Vladislav A. Lesnikov

37. Algebraic aspects of IIR filter design on structural level Realization of algorithms of generation of structures of digital filters is necessary for improvement of procedure of structural synthesis. The algorithms based on representation of digital filters in space of states are known. However, between set of structures and set of state space descriptions of structures there is no biunique correspondence. EWDTS-2009 Vladislav A. Lesnikov

38. Algebraic aspects of IIR filter design on structural level In our papers we represent the approach to the structural synthesis, allowing to generate all possible structures of digital filters. This approach allows executing renumbering of generated structures, to define properties of structures under their description. The presented approach is based on a technique introduced in paper of Crochier and Oppenheim (1975). EWDTS-2009 Vladislav A. Lesnikov

39. Algebraic aspects of IIR filter design on structural level At this approach the structure of the digital filter is described by a topological matrix. EWDTS-2009 Vladislav A. Lesnikov

40. Algebraic aspects of IIR filter design on structural level The technique of assignment to each structure of the identifier is developed. Identifier is rich in content. The identifier can be used for extraction of the information on properties of structure of the filter and for generation of a topological matrix. N5z2p2d2p3d2 EWDTS-2009 Vladislav A. Lesnikov

41. Algebraic aspects of IIR filter design on structural level Generation of structures of filters is reduced to generation of topological matrixes. Degree of the algebraic numbers (which are zeros or poles), indicators of complexity and accuracy of the filter are such properties. Studying of fine structure of a topological matrix has allowed executing renumbering of all possible structures of the filter. EWDTS-2009 Vladislav A. Lesnikov

42. Algebraic aspects of IIR filter design on structural level EWDTS-2009 Vladislav A. Lesnikov

43. Algebraic aspects of IIR filter design on structural level EWDTS-2009 Vladislav A. Lesnikov

44. Algebraic aspects of IIR filter design on structural level EWDTS-2009 Vladislav A. Lesnikov

45. Algebraic aspects of IIR filter design on structural level EWDTS-2009 Vladislav A. Lesnikov

46. Algebraic aspects of IIR filter design on structural level EWDTS-2009 Vladislav A. Lesnikov

47. Design Flow Diagrams Structural level of synthesis • At structural level under requirements to accuracy and complexity to filter structure the structure of coefficients is set. The structure of coefficients is used at a choice of structure of the filter. Exact values of zeros and the poles, calculated at functional level, are used for calculation of values of coefficients of the filter. By results of the analysis of results or the structure of coefficients changes, or the filter structure changes. EWDTS-2009 Vladislav A. Lesnikov

48. Conclusions • Thus, in this paper the new paradigm of synthesis of IIR digital filters is presented. This approach allows considering finiteness of word length of coefficients more correctly. The split-hair accuracy at the given approach is reached at comprehensible expenses of computing resources. It is especially important at realization of digital filters with strict requirements to characteristics. In this paper the questions connected with an estimation of level of round off noise and limit cycles are not considered. It is supposed that these estimations are carried out by the known methods. EWDTS-2009 Vladislav A. Lesnikov

49. References • Filter Design ToolboxTM 4.5. Reference Guide. 2009, The MathWorks, Inc., 1432 p. • Signal Processing BlocksetTM 6.9 Reference Guide, 2009, The MathWorks, Inc., 1579 p. • W. Y. Yang, et al, Signals and Systems with MATLAB. Springer-Verlag, Berlin – Heidelberg, 2009, 476 p. • D. Hilbert. The Theory of Algebraic Number Fields. Springer – Verlag, Berlin – Heidelberg – New York, 1998, 359 p. EWDTS-2009 Vladislav A. Lesnikov

50. Thanks! • Q&A EWDTS-2009 Vladislav A. Lesnikov