Prototype filter
They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed.
However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.
Filters are required to operate at many different frequencies, impedances and bandwidths.
Bandform here is meant to indicate the category of passband that the filter possesses.
The usual bandforms are lowpass, highpass, bandpass and bandstop, but others are possible.
Likewise, the nominal or characteristic impedance of the filter is set to R′ = 1 Ω.
In principle, any non-zero frequency point on the filter response could be used as a reference for the prototype design.
For example, for filters with ripple in the passband, the corner frequency is usually defined as the highest frequency at maximum ripple rather than 3 dB.
Another case is in image parameter filters (an older design method than the more modern network synthesis filters) which use the cut-off frequency rather than the 3 dB point since cut-off is a well-defined point in this type of filter.
A passive lumped low-pass prototype filter of fifth order and the T-topology might have the reactance: To convert them to 50 Ohm multiply the given values by 50.
Example: The resistance shall be 75 Ohm and the corner frequency shall be 2 MHz.
Filter types with adjustable ripple can not be easily tabulated as such as they depend on more than just the impedance and frequency.
The prototype filter is scaled to the frequency required with the following transformation:
It can readily be seen that to achieve this, the non-resistive components of the filter must be transformed by:
It can readily be seen that to achieve this, the non-resistive components of the filter must be scaled as:
However, it is usual to combine the frequency and impedance scaling into a single step:[1]
where ωc is the point on the highpass filter corresponding to ωc′ on the prototype.
the primed quantities being the component value in the prototype.
If ω1 and ω2 are the lower and upper frequency points (respectively) of the bandpass response corresponding to ωc′ of the prototype, then,
Note that frequency scaling the prototype prior to lowpass to bandpass transformation does not affect the resonant frequency, but instead affects the final bandwidth of the filter.
Filters with multiple passbands may be obtained by applying the general transformation:
Lowpass and highpass filters can be viewed as special cases of the resonator expression with one or the other of the terms becoming zero as appropriate.
In his treatment of image filters, Zobel provided an alternative basis for constructing a prototype which is not based in the frequency domain.
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.
For this reason, filters of all classes are given in terms of U(ω) for a constant k, which is notated as,
Uk(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:
