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Since the trail-blazing paper of C. Shannon in 1948, channel capacity has been regarded as the fundamental information-theoretic performance measure to predict the maximum information rate of a communication system. However, in contrast with the analysis of other important performance measures of wireless communication systems, a unified and general approach for computing the channel capacity over fading channels has yet to be proposed. Motivated by this consideration, we propose a novel and unified communication-theoretic framework for the analysis of channel capacity over fading channels. It is shown that the framework can handle various fading channel models, communication types, and adaptation transmission policies. In particular, the specific contributions of this paper are as follows: (1) We introduce a transform operator, called the E i-transform, which is shown to provide a unified tool to compute the channel capacity with either side information at the receiver or side information at the transmitter and the receiver, directly from the moment-generating function (MGF) or the MGF and the truncated MGF of the Signal-to-Noise-Ratio (SNR) at the receiver, respectively; (2) we show that when either a channel inversion or a truncated channel inversion adaptation policy is considered, the channel capacity can readily be computed from the Mellin or the Hankel transform of the MGF of the received SNR, respectively; (3) a simple yet effective numerical method for the analysis of higher order statistics (HOS) of the channel capacity with side information at the receiver is introduced; and (4) some efficient and ad hoc numerical methods are explicitly introduced to allow the efficient computation of the proposed frameworks. Numerical and simulation results are also shown and compared to substantiate the analytical derivation.