Analog Filters: Network Functions

1. Analog Filters: Network Functions Franco Maloberti

2. Introduction • Magnitude characteristic • Network function • Realizability • Can be implemented with real-world components • No poles in the right half-plane • Instability: • goes in the non-linear region of operation of the active or passive components • Self destruct Analog Filters: Network Functions

3. General Procedure • The approximation phase determines the magnitude characteristics • This step determines the network function H(s) • Assume that • The procedure to obtain P(s) for a given A(w2) and that for obtaining Q(s) are the same Analog Filters: Network Functions

4. General Procedure (ii) • P(s) is a polynomial with real coefficients • Zeros of P(s) are real or conjugate pairs • Zeros of P(-s) are the negative of the zeros of P(s) • Zeros of A(w2) are Quadrant symmetry Analog Filters: Network Functions

5. General Procedure (iii) • In A(w2) replace w2 by -s2 • Factor A(-s2) and determine zeros • Split pair of real zeros and complex mirrored conjugate Example Four possible choices, but …. B(s) must be Hurwitz, for a the choice depends on minimum-phase requirements • The polynomial A(s) [or B(s)] results Analog Filters: Network Functions

6. General Procedure (iv) • EXAMPLE one NO Analog Filters: Network Functions

7. Butterworth Network Functions • Remember that • therefore: The zeros of Q are obtained by Therefore Analog Filters: Network Functions

8. Butterworth Network Functions Analog Filters: Network Functions

9. Chebyshev Network Functions • Remember that • Therefore • The zeros of Q are obtained by • Let Analog Filters: Network Functions

10. Chebyshev Network Functions Analog Filters: Network Functions

11. Chebyshev Network Functions (ii) • Equation • Becomes • Equating real and imaginary parts For a real v this is > 1 Analog Filters: Network Functions

12. Chebyshev Network Functions (iii) • Remember that • Therefore • The real and the imaginary part of wk are such that • Zeros lie on an ellipse. Analog Filters: Network Functions

13. NF for Elliptic Filters • Obtained without obtaining the prior magnitude characteristics • Based on the use of the Conformal transformation • Mapping of points in one complex plane onto another complex plain (angular relationships are preserved) • Mapping of the entire s-plane onto a rectangle in the p-plane • sn is the Jacobian elliptic sine function • Derivation complex and out of the scope of the Course • Design with the help of Matlab Analog Filters: Network Functions

14. Elliptic Filter Analog Filters: Network Functions

15. Bessel-Thomson Filter Function • Useful when the phase response is important • Video applications require a constant group delay in the pass band • Design target: maximally flat delay • Storch procedure Analog Filters: Network Functions

16. Bessel-Thomson Filter Function (ii) • Find an approximation of in the form • And set • Approximations of • Example Analog Filters: Network Functions

17. Bessel-Thomson Filter Analog Filters: Network Functions

18. Different Filter Comparison Analog Filters: Network Functions

19. Different Filter Comparison Analog Filters: Network Functions

20. Delay Equalizer • It is a filter cascaded to a filter able to achieve a given magnitude response for changing the phase response • It does not disturb the magnitude response • Made by all-pass filter • The magnitude response is 1 since Moreover Analog Filters: Network Functions

21. Examples Analog Filters: Network Functions