In this paper, a certified synthesis technique is presented for the design of matching filters that combines convex optimisation with the Fano-Youla matching theory. This technique is applicable to any rational load and provides lower hard bounds for the best matching level, as well as a practical synthesis of a matching filter approaching those bounds. Furthermore, if the load is a rational function of degree 1, the optimal matching filter is synthesized, yielding in this case an extension of the classical filter synthesis for resistive loads. As example, a dual-band matching filter is conceived for a dual-band antenna. Additionally a single-band filter is implemented in SIW technology to match a single-band antenna.