Passive Analogue Filter Development - Approximation

Approximation

The approximation problem in network synthesis is to find functions which will produce realisable networks approximating to a prescribed function of frequency within limits arbitrarily set. The approximation problem is an important issue since the ideal function of frequency required will commonly be unachievable with rational networks. For instance, the ideal prescribed function is often taken to be the unachievable lossless transmission in the passband, infinite attenuation in the stopband and a vertical transition between the two. However, the ideal function can be approximated with a rational function, becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was Stephen Butterworth (1930) using his Butterworth polynomials. Independently, Cauer (1931) used Chebyshev polynomials, initially applied to image filters, and not to the now well-known ladder realisation of this filter.