Prototype Filter - Alternative Prototype

Alternative Prototype

In his treatment of image filters, Zobel provided an alternative basis for constructing a prototype which is not based in the frequency domain. The Zobel prototypes do not, therefore, correspond to any particular bandform, but they can be transformed into any of them. Not giving special significance to any one bandform makes the method more mathematically pleasing; however, it is not in common use.

The Zobel prototype considers filter sections, rather than components. That is, the transformation is carried out on a two-port network rather than a two-terminal inductor or capacitor. The transfer function is expressed in terms of the product of the series impedance, Z, and the shunt admittance Y of a filter half-section. See the article Image impedance for a description of half-sections. This quantity is nondimensional, adding to the prototype's generality. Generally, ZY is a complex quantity,

and as U and V are both, in general, functions of ω we should properly write,

With image filters, it is possible to obtain filters of different classes from the constant k filter prototype by means of a different kind of transformation (see composite image filter), constant k being those filters for which Z/Y is a constant. For this reason, filters of all classes are given in terms of U(ω) for a constant k, which is notated as,

In the case of dissipationless networks, i.e. no resistors, the quantity V(ω) is zero and only U(ω) need be considered. U_k(ω) ranges from 0 at the centre of the passband to -1 at the cut-off frequency and then continues to increase negatively into the stopband regardless of the bandform of the filter being designed. To obtain the required bandform, the following transforms are used:

For a lowpass constant k prototype that is scaled:

the independent variable of the response plot is,

The bandform transformations from this prototype are,

for lowpass,

for highpass,

and for bandpass,