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 
 Maximally flat near the center of the band.
 Smooth transition from passband to stopband.
 Moderate outofband rejection.
 Low group delay variation near center of band.
 Moderate group delay variation near band edges.
 Table of poles for N=1 to 10.


Chebyshev

 Equiripple in passband.
 Abrupt transition from passband to stopband.
 High outofband rejection.
 Rippled group delay near center of band.
 Large group delay variation near band edges.
 Table of poles for N=1 to 10.


Bessel

 Rounded amplitude in passband.
 Gradual transition from passband to stopband.
 Low outofband rejection.
 Very flat group delay near center of band.
 Flat group delay variation near band edges^{[1]}.
 Table of poles for N=1 to 10.

Note: B_{N}, P_{N}, and bo_{N} must be placed
in a loop from 0 through N.

Ideal

 Flat in the passband.
 Step function transition from passband to stopband.
 Infinite outofband rejection.
 Zero group delay everywhere.

(Heaviside step function)

[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.
