By employing the Fourier-Bessel theorem to plane wave functions and through a series of deduction and inverse Fourier transform

one kind of series expansions of the product of Bessel functions and sine/cosine function in Cartesian coordinates is obtained.This expression can be used to compute the rectangular to circular waveguide junction and to analytically deduce its generatized S-parameters

which are in good agreement with results from other papers and HFSS.The coupling coefficient of rectangular iris with thickness are computed in Ku and Ka band

and numerical evaluations are presented

which show close agreement with experimental data and other paper.As a consequence

a 6th-order quasi-elliptic function filter in Ku band is realized through rigorous calculation of the coupling iris dimensions

and the measured data agree well with the predicted values.