Group delay shaping - introduction.
The CMS software is an effective tool for equalization  - or shaping - of the group delay (GD)characteristic in band-pass filters. In the following this feature is demonstrated by some examples.

A more advanced group delay shaping example can also be found here.

Example 1
The starting point is the 5'th order filter shown in the CMS screen-dump below. It has a center frequency of 14.5 GHz and 0.5 GHz ripple bandwidth. All the transmission zeroes are placed at infinity - represented by the frequency 1e12 MHz. No GD shaping is therefore applied. The resulting S21 and Group Delay characteristics are also shown in the output window.
Example 2
A pair of complex transmission poles are now inserted. When complex transmission zeroes are introduced they must always appear as complex conjugate pairs, i.e. same real parts - and imaginary parts which have opposite signs. If this is not fulfilled the calculated characteristics are invalid.

The effect of a pair of complex transmission zeroes is a 'bump' in the GD characteristic as shown in the screen-dump below.
The real part of the complex transmission zeros defines the frequency at which the bump appears.
The imaginary part defines the 'height' of the bump.

If the imaginary part is small relative to the bandwidth the 'height' of the GD bump will be big (as in the example below).
An imaginary part which is comparable to the bandwidth gives a small GD bump.
Example 3
This example is identical to Example 2, except for the imaginary parts, which are doubled from 100 MHz to 200 MHz.
The result is a much smaller 'bump' at 14.5 GHz.

Do also note the effect of complex transmission zeroes on the S21 characteristic - the stop band rejection is decreased quite a lot as compared with Example 1.
Example 4
In order to get a very flat group delay additional complex pairs may be introduced. In the example below two complex pairs are used. The positions and imaginary values were easily found in a few trials.
The addition of the extra complex pair has decreased the stop-band attenuation further.
Example 5
A simpler single pair example is shown below.
The topology matrix has been set up to allow at folded topology, and from the coupling matrix it is noted that two X-couplings are needed in order to get the required shaping. The X-couplings (marked with red circle below) are inserted between resonators 2 & 4 and 2 & 5 and are both positive.
The X-coupling between resonator 2 & 4 is much weaker then the one between resonator 2 & 5 (77 times). It is reasonable to assume that such a weak coupling has limited influence on the transfer characteristics. This is investigated in Example 6.
Example 6
CMS allows manual editing of the coupling matrix values. In this example the weak coupling addressed in example 5 is manually set to zero. This is done by double clicking on the coupling coefficient in cell (2,4), deleting the old value and inserting '0' as the new value. The new characteristics are plotted by selecting the "From Coupling matrix" radio button in the plot window.

It is seen on the screen-dump below that zeroing the coupling between resonator 2 & 4 has almost no influence on the filter characteristics and this coupling can therefore be omitted, which leads to a considerably simpler filter - if manufactured.

The only penalty by removing the weakest coupling, is a slight decrease in S11 - approx. 0.6 dB.
Another Example:

A more advanced group delay shaping example can be found here.