To obtain the transfer function of the high-pass filters, an equivalent low-pass analog prototype has to be designed first. Then the low-pass prototype transfer function can be mapped to the desired high-pass filter using s-domain frequency transformations. The mapping function that converts low-pass prototype into corresponding high-pass transfer function is given as
where is a low-pass prototype variable and is a high-pass variable. The same function (7.1) must be used to convert the high-pass filter specification into low-pass prototype specification.
The general procedure of obtaining the high-pass filter transfer function includes several steps:
Generate specification for the desired high-pass filter, which includes 4 parameters:
Convert the high-pass specification into the low-pass prototype specification using the mapping function (7.1)
Note, that this conversion should be done with angular frequencies.
Generate the low-pass prototype transfer function
Map the low-pass prototype transfer function into the desired high-pass transfer function