The coupling matrix in low-pass prototype filters

Narrow band-pass filters can be designed from a normalized coupled-resonator low-pass prototype network that is completely characterized by its coupling matrix. Matthei, Young, and Jones have explained the detailed transformation of the low-pass prototype elements into band-pass structures. The Chebysheb low-pass prototypes in coupled-resonators form duplicates the response that can be achieved by a simple LC-ladder low-pass filter, but additional couplings between nonadjacent resonators can also be included. A general coupled-resonator low-pass prototype is not restricted to having responses realizable by LC-ladder low-pass networks, which must be symmetric and have at least one loss pole at infinite frequency, because the coupling matrix represents elements that are frequency-independent reactances or susceptances.