The radiation of a dielectric-filled thick-walled parallel-plate impedance waveguide junctioned with a thick dielectric half-plane is treated by using the mode-matching method in conjunction with the Wiener-Hopf technique. The hybrid method that we adopt here reduces the related boundary value problem to two scalar modified Wiener-Hopf equations of the second kind. This bypasses the most challenging stage of the classical formulations, which lead to an intractable matrix Wiener-Hopf factorization. The solution involves infinitely many unknown constants satisfying an infinite system of linear algebraic equations susceptible to a numerical treatment. Some computational results illustrating the various values of the dielectric constant, plate impedances, plate thickness, and plate to plate separation distance on the radiation phenomenon are also presented.