Why when the SINR of transmitting signal is very large ,the device cannot decode the received signal at the same time?
A transmit x and receive the y at the same time,so of course it is a full duplex,i think your thinking is right.

If two antennas,A &amp; B, are very close,and A is for transmitting,B is for receiving.B can receive a very strong signal from A,that is,if B is not going to to receive the signal from A,but C.The SINR for receiving C will be very small. I mean

$SNR_{AinB}=\frac{A \ power}{C \ power +noise}$,if A and B are very close,the $A \ power$ will become very large,that is,$A \ power &gt;&gt; C \ power$

And if B want to receive the signal from C in fact,the SINR will become

$SNR_{CinB}=\frac{C \ power}{A \ power +noise}$,and as i just mentioned, $A \ power &gt;&gt; C \ power$,so we can know $SNR_{CinB}$ is very small.

SNR is defined as the ratio of signal power to the noise power,so i think it can also apply on transmitting,not only on receiving.Nevertheless,most people will apply on receiving

If SNR is very small,the information cannot transmit to the receiver correctly,so of course we cannot know,or decode, the real information