All-Pole Approximations

  • Hercules G. DimopoulosEmail author
Part of the Analog Circuits and Signal Processing book series (ACSP)


In this chapter the all-pole transfer functions are addressed. These are rational functions which in the lowpass case have no zeros at finite positions. The design of normalized lowpass filters from given specifications follows. It is shown that the corresponding gain function (magnitude response) involves a polynomial, the approximating polynomial, which defines the approximation and determines its characteristics. For the classical Butterworth, Chebyshev and Pascal approximations, the corresponding polynomials are employed. Each of these approximations is presented in detail and all design equations are given explicitly. Moreover, their optimal use is explained and demonstrated through several design examples.


Transfer Function Chebyshev Polynomial Group Delay Gain Function Chebyshev Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Butterworth, S.: On the theory of filter amplifiers. Wirel. Eng. 7, 536–541 (1930) Google Scholar
  2. 2.
    Chebyshev, P.L.: Theorie des mechanisms connus sous le nom de parallelogrammes. Oeuvres, vol. I, St. Petersburg (1899) Google Scholar
  3. 3.
    Chen, W.-K.: Passive, Active and Digital Filters. The Circuits and Filters Handbook, vol. 5. CRC Press, Boca Raton (2009) Google Scholar
  4. 4.
    Chirlian, P.M.: Signals and Filters. Springer, Berlin (1994) Google Scholar
  5. 5.
    Dimopoulos, H.G.: Optimal use of some classical approximations in filter design. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 54(9), 780–784 (2007) CrossRefGoogle Scholar
  6. 6.
    Dimopoulos, H.G., Sarri, E.: The modified pascal polynomial approximation and filter design method. Int. J. Circuit Theory Appl. (2010). doi: 10.1002/cta.711 Google Scholar
  7. 7.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Dover, New York (2002) Google Scholar
  8. 8.
    Schaumann, R., Van Valkenburg, M.E.: Design of Analog Filters. Oxford University Press, London (2001) Google Scholar
  9. 9.
    Temes, G.C., LaPatra, J.W.: Introduction to Circuit Synthesis and Design. McGraw-Hill, Boca Raton (1977) Google Scholar
  10. 10.
    Van Valkenburg, M.E.: Analog Filter Design. Oxford University Press, London (1982) Google Scholar
  11. 11.
    Wanhammar, L.: Analog Filters Using MATLAB. Springer, Berlin (2009) zbMATHCrossRefGoogle Scholar
  12. 12.
    Williams, A., Taylor, F.: Electronic Filter Design Handbook. McGraw-Hill, New York (2006) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department ElectronicsTechnol. Educ. Inst. of Piraeus (T.E.I.)EgaleoGreece

Personalised recommendations