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:
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 Type 1 
 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. 
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.
