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In this paper, the effect of imperfect channel state information at the receiver, which is caused by noise and other interference, on the multi-access channel capacity is analyzed by means of a statistical-mechanical approach. Replica analysis focus on analytically studying how the minimum mean square error (MMSE) channel estimation error appears in the multiuser channel capacity formula and mathematical expressions are derived. At the same time, numerical simulation results are demonstrated to validate the replica analysis. The simulation results show how the system parameters, such as channel estimation error, system load and signal-to-noise ratio, effect on the channel capacity.