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A new moment generating function (MGF) is proposed to assess rigorously the performance of optically preamplified receivers with direct-detection employing baseband orthogonal frequency division multiplexing (OFDM). By comparison with Monte Carlo simulation, the proposed MGF-based method shows very good accuracy when predicting the bit error probability for arbitrary optical and electrical filters, different numbers of OFDM subcarriers, and double sideband and single sideband optical OFDM signals. Its good accuracy when estimating the bit error probability in each individual subcarrier is also demonstrated. The new MGF-based method exhibits a significant improvement of accuracy when compared to the MGF-based method previously reported in the literature, especially for filter bandwidths not exceeding the OFDM signal bandwidth, which is the range corresponding to the optimum filter bandwidth.