1. Analog Filters: Network Functions Franco Maloberti

2. Analog Filters: Network Functions2 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

3. Franco MalobertiAnalog Filters: Network Functions3 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(  2 ) and that for obtaining Q(s) are the same

4. Franco MalobertiAnalog Filters: Network Functions4 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(  2 ) are Quadrant symmetry

5. Franco MalobertiAnalog Filters: Network Functions5 General Procedure (iii) In A(  2 ) replace  2 by -s 2 Factor A(-s 2 ) 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

6. Franco MalobertiAnalog Filters: Network Functions6 General Procedure (iv) EXAMPLE one NO

7. Franco MalobertiAnalog Filters: Network Functions7 Butterworth Network Functions Remember that therefore: The zeros of Q are obtained by Therefore

8. Franco MalobertiAnalog Filters: Network Functions8 Butterworth Network Functions

9. Franco MalobertiAnalog Filters: Network Functions9 Chebyshev Network Functions Remember that Therefore The zeros of Q are obtained by Let

10. Franco MalobertiAnalog Filters: Network Functions10 Chebyshev Network Functions

11. Franco MalobertiAnalog Filters: Network Functions11 Chebyshev Network Functions (ii) Equation Becomes Equating real and imaginary parts For a real v this is > 1

12. Franco MalobertiAnalog Filters: Network Functions12 Chebyshev Network Functions (iii) Remember that Therefore The real and the imaginary part of  k are such that Zeros lie on an ellipse.

13. Franco MalobertiAnalog Filters: Network Functions13 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

14. Franco MalobertiAnalog Filters: Network Functions14 Elliptic Filter

15. Franco MalobertiAnalog Filters: Network Functions15 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

16. Franco MalobertiAnalog Filters: Network Functions16 Bessel-Thomson Filter Function (ii) Find an approximation of in the form And set Approximations of Example

17. Franco MalobertiAnalog Filters: Network Functions17 Bessel-Thomson Filter

18. Franco MalobertiAnalog Filters: Network Functions18 Different Filter Comparison

19. Franco MalobertiAnalog Filters: Network Functions19 Different Filter Comparison

20. Franco MalobertiAnalog Filters: Network Functions20 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

21. Franco MalobertiAnalog Filters: Network Functions21 Examples