Realizations of complex pole all-pass networks

It is shown that an all-pass function with complex poles can always be constructed by adding an appropriate all pole function to a first order all-pass function. This method has several advantages: 1) the complex pole sensitivities of the resulting all-pass network are the same as the pole sensitivities of the all pole network; 2) the output of the all pole network is available as an aid in tuning the network as well as for other purposes; 3) the method yields an abundance of networks. Three new all-pass realizations are derived to illustrate this method. One of the realizations is analyzed in detail, and analysis results are given for the other two realizations. For all three examples considered, the zero sensitivity magnitudes are approximately equal to the pole sensitivity magnitudes of the all pole function.