Quantum limit of optimum four-level ASK signals with direct detection optically preamplified receivers

We analytically obtain the exact fundamental limit of 4-level amplitude shifted keying formats (4ASK) with direct detection optically preamplified receivers. The optimum multilevel spacing and the corresponding decision thresholds, which depend both on the signal to noise ratio and optical bandwidth, are obtained numerically considering the Chi-squared distribution of each level. The quantum limit under the optimum level spacing is 127.5 photons/bit, which is about 0.2 dB smaller than the results by the Gaussian approximation. Over a broad range of the signal to noise ratio and the optical bandwidth, we have found that not only the bit error rate but also the optimum level spacing are well predicted by Gaussian method, although the three decision thresholds are all underestimated.