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An analysis for the Shannon channel capacity of digital receivers operating over η-μ fading channels is presented under four adaptive policies: constant power with optimal rate adaptation, optimal power and rate adaptation, channel inversion with fixed rate and truncated channel inversion with fixed rate. In this context, useful formulae for the average channel capacity for maximal-ratio combining receivers with not necessarily identically distributed branches are derived. The analysis also includes the performance of single-branch receivers. These expressions provide an effective tool to assess the spectral efficiency of the aforementioned adaptive transmission techniques over practical fading channels. Our newly derived expressions include several special cases that arise from the η-μ distribution, namely the Nakagami-m and the Hoyt distribution. Extensive numerical and computer simulation results are also presented that illustrate the proposed mathematical analysis.