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In this paper, we analyze the performance of code division multiple access (CDMA) systems with orthogonal frequency division multiplexing (OFDM) that employ random spreading sequences in an additive white Gaussian noise (AWGN) channel. We obtain the probability density function (pdf) of the multiple access interference and extend the results to OFDM-CDMA systems to determine the pdf of multiple access and inter-carrier interference in terms of the number of users, the spreading length, the number of sub-carriers, and the frequency offset. We consider the synchronous downlink of cellular multi-carrier CDMA and derive a Gaussian approximation of the multiple access and inter-carrier interference. Overall the effect of frequency offset is shown to vary with the system loading. The analysis in this paper is critical for further development into fading channels and frequency selective multipath channels.