Abstract

We consider the design of the optimum detector for two-dimensional amplitude/phase modulated signals received in additive white Gaussian noise (AWGN) and a Gaussian distributed phase reference error (PRE) due to imperfect carrier phase estimation. We propose a novel approach of using the amplitude and phase information of the received signal, based on viewing the AWGN as an equivalent additive observation phase noise (AOPN) whose statistics is derived in our earlier work. This allows the AOPN to be combined with PRE, and the maximum $a\ posterior$ probability (MAP)/maximum-likelihood (ML) detection scheme to be readily derived in amplitude-phase form. This amplitude-phase approach is simpler and more convenient than the conventional method of using the in-phase and quadrature components of signals in phase noise. For $M$ -ary phase-shift keying, which only has one ring of signal points, the ML detector here turns out to be the same as the conventional minimum Euclidean distance (MED) detector which is derived without taking the PRE into account, and leads to angular bisector decision boundaries (DB). However, for constellations which have multiple rings, e.g., $M$ -ary quadrature amplitude modulation ( $M$ -QAM) and amplitude phase-shift keying ( $M$ -APSK), the ML detector here is very computationally inefficient. Thus, simpler and closed-form approximations to the ML detector for equiprobable signals are given, which can be easily implemented online. All these simplified ML detectors are shown via simulations to perform almost the same as the exact one and perform much better than the MED detector. The approximate ML DB for both 8-star QAM and rotated 8-star QAM are illustrated as examples and shown to be not necessarily straight lines. As the variance of PRE or the signal-to-noise ratio (SNR) or both increases, the DB between two signal rings asymptotically becomes circular. This leads to an annular sector as the decision region for each signal point. For annular-sector decision regions, simple, accurate, and closed-form approximations to the symbol error probability (SEP) are obtained for both 8-star QAM and the rotated case, 16QAM and even general $M$ -APSK. These expressions provide explicit insight into how the PRE variance affects the performance. Within a wide range of PRE variances, our SEP approximations agree very well with the Monte Carlo simulations for all SNR values of interest. For the special case of an $M$ -APSK constellation which has the same number of points and the same phase values on each ring, such as 8-star QAM, one of the suboptimal ML detectors further simplifies to a structure that performs ring detection and phase detection separately, and the decision regions are always annular sectors.

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