
Your RF Cafe

An infinite number of filter transfer functions exist. A handful are commonly used as a starting point due to certain characteristics. The table following the plots lists properties of the filter types shown below. Not given  due to complex numerical methods required  are the Cauer (Elliptical) filters that exhibit equiripple characteristic in both the passband and the stopband.
Phase information may be gleaned from the transfer functions by separating them in to real and imaginary parts and then using the relationship:
Phase: θ = tan^{1} (Im / Re)
Group delay is defined as the negative of the first derivative of the phase with respect to frequency, or
Group Delay:
Type  Properties  Transfer Function (Lowpass) 
Butterworth 


Chebyshev 


Bessel 


Ideal 


[1] Filters with a large BW will exhibit sloped group delay across the band. This usually is not a problem since group delay deviation tends to be specified for variation in some subsection of the band. 
Band Translations 
These equations are used to convert the lowpass prototype filter equation into
equations for highpass, bandpass, and bandstop filters. They work for all three functions  Butterworth, Chebyshev,
and Bessel. Simply substitute the highpass, bandpass, or bandstop transformation of interest for the f_{r}
term in the lowpass equation.

Microwave Filters, Couplers and Matching Network
by Robert J. Wenzel
This CDROM course contains approximately 12hours of instruction on the fundamentals of microwave filters, couplers and matching networks. Included is a thorough review of the common types of filter responses and calculations, filter realization, and various methods of filter design, including bandpass, network theory and Kuroda. Subsequent sessions cover the fundamentals of directional couplers. A final session describes distributed element matching networks and a matching network design example.