It is basically impossible to transmit and recieve at the same time. This is why in some modes, you have to go jabber jabber jabber OVER and then the other party starts talking.
Sending the output from the transmitter into the input of the reciever would destroy it. There is a device called a 'circulator' that allows this though - very expensive and limited to the higher frequencies. The circulator forces the radio waves to move in one direction only so the transmitted waves will move only from the output to the antenna and the recieved waves will move only from the antenna to the reciever.
Some people just built one on a chip. From the Defense Advanced Research Projects Agency:
Novel miniaturized circulator opens way to doubling wireless capacity
Since the advent of the integrated circuit in 1958, the same year the Advanced Research Projects Agency was established, engineers have been jamming ever more microelectronic integration into ever less chip real estate. Now it has become routine to pack billions of transistors onto chips the size of fingernails.
DARPA (the D for Defense was first added in 1972) has played key roles in this ongoing miracle of miniaturization, giving rise to new and sometimes revolutionary military and civilian capabilities in domains as diverse as communication, intelligence gathering, and optical information processing. Now a DARPA-funded team has drastically miniaturized highly specialized electronic components called circulators and for the first time integrated them into standard silicon-based circuitry. The feat could lead to a doubling of radiofrequency (RF) capacity for wireless communications—meaning even faster web-searching and downloads, for example—as well as the development of smaller, less expensive and more readily upgraded antenna arrays for radar, signals intelligence, and other applications.
The work, funded under DARPA’s Arrays at Commercial Timescales (ACT) program, was led by Columbia University electrical engineers Harish Krishnaswamy and Negar Reiskarimian and described in the April 15, 2016 issue of the journal Nature Communications.
Going to stop excerpting here - there is a lot more at the link for those interested. This is very cool! Normally circulators used magnetic ferrite materials, large magnets and resonant cavities (tuned chambers - think about blowing across the top of a coke bottle). Doing this in solid state brings things into the next generation.
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