  The maximum likelihood detection rule for a four-dimensional direct-detection
optical front-end is derived. The four dimensions are two intensities and two
differential phases. Three different signal processing algorithms, composed of
symbol-by-symbol, sequence and successive detection, are discussed. To remedy
dealing with special functions in the detection rules, an approximation for
high signal-to-noise ratios (SNRs) is provided. Simulation results show that,
despite the simpler structure of the successive algorithm, the resulting
performance loss, in comparison with the other two algorithms, is negligible.
For example, for an 8-ring/8-ary phase constellation, the complexity of
detection reduces by a factor of 8, while the performance, in terms of the
symbol error rate, degrades by 0.5 dB. It is shown that the high-SNR
approximation is very accurate, even at low SNRs. The achievable rates for
different constellations are computed and compared by the Monte Carlo method.
For example, for a 4-ring/8-ary phase constellation, the achievable rate is 10
bits per channel use at an SNR of 25 dB, while by using an 8-ring/8-ary phase
constellation and an error correcting code of rate 5/6, this rate is achieved
at an SNR of 20 dB.
