Quantifying diversity for wireless systems with finite-bit representation

Diversity order is widely adopted as a critical metric on the reliability of wireless transmissions over fading channels. Numerous designs have been proposed to collect the diversity from different types of channels. However, since the simulators and practical systems can only afford finite bits to represent the real/complex numbers, the definition of diversity needs to be revisited. In this paper, we thoroughly investigate the effects of finite-bit representation on the statistical properties of the channel and the diversity with different transmitters/receivers. We show that finite-bit representation may cause diversity loss relative to the infinite precision representation. For different transmission systems, the diversity loss follows different patterns as the number of bits decreases. Furthermore, we find that the number of non-vanishing eigenvalues plays an important role in quantizing the diversity for infinite lattice/constellation. Numerical examples verify our theoretical findings.