Spectral efficient frequency division multiplexing (SEFDM) can improve the spectral efficiency for next-generation optical and wireless communications. In this work, we apply SEFDM in beyond 100-Gb/s optical intensity modulation and direct detection transmissions and propose a low-complexity logarithmic-maximum-a-posteriori (log-MAP) Viterbi decoding algorithm to achieve the maximum likelihood (ML) detection. We evaluate the likelihood of detections using a posteriori probability instead of Euclidean distance by taking both noise and inter-carrier interference into consideration. In order to balance the performance and complexity, we then employ Viterbi decoding principle to retain only certain paths with ML detections (a.k.a., the surviving paths) while discarding the others during the decoding procedure. Results show that the proposed log-MAP Viterbi decoding scheme achieves optimal performance due to the precise likelihood evaluation, which guarantees the retention of the global ML detection. By using the proposed decoding scheme, the data rate of SEFDM signals can reach 150-Gb/s in a 2-km standard single mode fiber transmission, using only 28-GHz bandwidth and 16-QAM modulation. Experimental results show that the 16-QAM modulated SEFDM signal with a bandwidth compression factor of 0.8 outperforms 32-QAM modulated OFDM, while both signals have the same bandwidth (28-GHz) and data rate (140-Gb/s), which demonstrate the superiority of SEFDM in optical short reach applications.