The diversity of communication standards implies the use of multi-band and multi-mode radios. Recent years have seen a wide investigation on Software Defined Radio for Cognitive Radio application. But, this is always constrained to multi-standards prospectives while a complete agility of RF transceivers is required. That is why Full Software Radio proposes to challenge a new way of integrating RF circuits and systems by tackling the main issue: transceiving concurrently any RF signal within a very wide band of interest for telecommunication industry, from 0 to 5GHz for instance. It is clearly observed that disruptive solutions are required. The focus of this tutorial will be on the design by mathematics of such RF transceiver design, exploring novel approaches along with a thorough discussion of advanced techniques for these receivers and transmitters towards a revolution in RF integrated circuits and systems. First, a frequency system is presented using a Sampled Analog Signal Processor as a receiver and a Walsh frequency combiner for the transmitter. Then, a temporal system is presented using a wide-band delta analog to digital converter as a receiver and a RF arbitrary waveform generator, named Riemann Pump for the transmitter. The methodology of every approach will be detailed following the same flow: mathematics, trade-off with RF electronics integration, architecture proposal, high level simulation results, circuit design issues, measurement results. Finally, an application to 5G standard will be addressed by demonstrating the feasibility of such systems to carrier aggregation, wide-band capabilities, low power consumption and high order of modulation schemes.