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Digitization of analogue filters signal/response

This article discusses the digitization of analogue filters using the Bilinear Transform technique, including its advantages and disadvantages. It also explores filter implementation methods, such as Direct Form I and II, and the CSOS structure.

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Digitization of analogue filters signal/response

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  1. TS Digitization of analogue filters signal/response jW Img Re  z=-1 s=0 z=1  TS z plane s plane

  2. are mapped to nowhere, Frequencies beyond Digitization of analogue filters signal/response Disadvantage of the Pole Mapping technique leading to aliasing error Bilinear Transform A nonlinear one-to-one mapping from the j axis of the s-plane to the unit circle in the z plane

  3. Bilinear Transform    

  4. Bilinear Transform is mapped to The left half s plane is mapped onto the inside of the unit circle in the z plane The stability of a system is preserved in the transform

  5. Bilinear Transform With z = ejand s = j, we have For small value of 

  6. Pole Mapping in Bilinear Transform Consider a pole at s = spis mapped to a pole at

  7. Bilinear Transform TABLE I f   4fs 0.95 2fs 0.90 f fs 0.80 fs/2 0.64 fs = 1/TS fs/4 0.42 fs/8 0.24 fs/16 0.12 0 0

  8. Bilinear Transform TABLE I f   4fs 0.95 1.1 2fs 0.90 1.2 fs 0.80 1.3 fs/2 0.64 1.5 fs/4 0.42 1.8 fs/8 0.24 2.0 fs/16 0.12 0 0

  9. Bilinear Transform The conversion listed Table I is not realistic Consider the sampling lattice in the frequency domain for N = 16           fs 2fs 3fs 4fs 5fs 6fs 7fs fs 7fs 6fs 5fs 4fs 3fs 2fs fs 0 0 16 16 16 16 16 16 16 2 16 16 16 16 16 16 16 Even for low analogue frequencies, the accuracy of mapping from  to  is not uniform

  10. Bilinear Transform The conversion listed Table I is not realistic Consider the sampling lattice in the frequency domain for N = 16           fs 2fs 3fs 4fs 5fs 6fs 7fs fs 7fs 6fs 5fs 4fs 3fs 2fs fs 0 0 16 16 16 16 16 16 16 2 16 16 16 16 16 16 16 is wasted as the input signal cannot reach beyond fs/2 after sampling

  11. Frequency Prewarping The behaviour of bilinear transform is nonlinear and can be compensated with frequency prewarping

  12. Frequency Prewarping The behaviour of bilinear transform is nonlinear and can be compensated with frequency prewarping

  13. Frequency Prewarping TABLE II f *  fs 0  2.0 fs/2  2.0 fs/4 0.32fs  2.0 fs/8 0.13fs  2.0 fs/16 0.06fs  0 0 0

  14. Frequency Prewarping The conversion listed Table II Consider the sampling lattice in the frequency domain for N = 16           fs 2fs 3fs 4fs 5fs 6fs 7fs fs 7fs 6fs 5fs 4fs 3fs 2fs fs 0 0 16 16 16 16 16 16 16 2 16 16 16 16 16 16 16 Even for low analogue frequencies, the accuracy of mapping from  to  is not uniform

  15. Filter Implementation - Direct Form I NF M bk z-kx(n) y(n)= ak z-ky(n) + k= -NF k=1 b-NF z x(n) y(n) b0 + + z-1 z-1 a1 bNF z-1 aM

  16. Filter Implementation - Direct Form II NF M bk z-kx(n) y(n)= ak z-ky(n) + k= -NF k=1 LET , then

  17. Filter Implementation - Direct Form II

  18. Filter Implementation - Direct Form II Filter Implementation - Direct Form II

  19. Filter Implementation - Direct Form II + x(n) u(n) z-1 a1 z-1 aM

  20. Filter Implementation - Direct Form II b-NF z u(n) y(n) b0 + z-1 bNF

  21. b-NF z y(n) u(n) b0 + z-1 bNF Filter Implementation - Direct Form II x(n) + z-1 a1 z-1 aM Canonical structure - minimum number of delays

  22. Filter Implementation - CSOS CSOS - Cascade Combination of Second-Order Sections

  23. Filter Implementation - CSOS Template

  24. Filter Implementation - CSOS

  25. Filter Implementation - CSOS

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