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An asynchronous binary DS-CDMA system using random spreading sequences is considered when communicating over various fading channels. New closed-form expressions are derived for the conditional characteristic function (CF) of the multiple access interference. A unified analysis is provided for calculating the exact average bit error rate (BER) expressed in the form of a single numerical integration based on the CF approach. The numerical results obtained from our exact BER analysis are verified by our simulation results and are also compared to those obtained by the standard Gaussian approximation (SGA), confirming the accuracy of the SGA for most practical conditions, except for high signal-to-noise ratios (SNR) and for a low number of interferers.