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This paper is an extension of earlier analysis work relating to a new method of achieving frequency division multiplexing, capable of being effectively integrated with time division switching. The band shifting inherent in frequency division multiplexing and demultiplexing is achieved by placing a sampling gate between a baseband and a sideband filter. When this technique is combined with time division switching, the transmission path also includes the cases of transmission via a switch between sideband filters, that may or may not cover the same sideband region, and between baseband filters. Resonant transfer is incorporated for all cases. Only the case of resonant transfer between baseband filters has been considered in the past. In general, the basic transmission path under analysis is the one between sideband filters, where the term sideband is used in the sense to include baseband. This is logical, since the baseband may be considered to be an upper sideband around zero frequency. Such a basic transmission path is rigorously analyzed on the basis of certain ideal assumptions. The purpose is to provide further insight into the fundamental operation of this new modulation-demodulatian scheme.