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If the constellation at the tarsmitter is symmetric about 0 on the real line then you can simply threshold the output between -1 and +1 with 0 as the threshold. So anything above zero is a logical 1 and everything below zero is a logical 0. This detection is rule is optimal when the noise is additive.


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CRC is a "definite" indicator of the channel impacting the bits and error in received symbol. So deploying a CRC check would definitely help. Length of CRC would depend on the overall system and data packets.


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$R_H$ could be anything other than identity and you would get correlated fading with the definitions you have in place in the question. Defining a kronecker matrix product is one way. $R_H$ should be $N_R \times N_R$ matrix, you can make the kronecker product to result in a non identity matrix of this size.


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