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In this paper, we firstly derive two approximations of the achievable uplink rate with the perfect/imperfect channel state information (CSI) in cell-free massive multi-input multi-output (MIMO) systems, and all these approximations are not only in the simple, but also converge into the classical bounds achieved in conventional massive MIMO systems where the base-station (BS) antennas are co-located. It is worth noting that the obtained two approximations with perfect CSI could be regarded as the special cases of the obtained two approximations with imperfect CSI when the pilot sequence power becomes infinite, respectively. Moreover, the theory analysis shows that all obtained approximations with perfect/imperfect CSI have an asymptotic lower bound &#x03B1;/2 log<sub>2</sub> L thanks to the extra distance diversity offered by massively distributed antennas, where L is the number of BS antennas and the path-loss factor &#x03B1; &gt; 2, except for the free space environment. Obviously, these results indicate that the cell-free massive MIMO system has huge potential of spectral efficiency than the conventional massive MIMO system with the asymptotically tight bound log<sub>2</sub> L.

Spectral Efficiency Analysis of Cell-Free Massive MIMO Systems With Zero-Forcing Detector