Exact evaluation of the bit- and symbol-error rates in a 2-D constellation has closed solutions for particular modulations and/or bits-to-symbol mapping. To solve this fundamental problem of digital communications, we propose a general method which yields the exact results for arbitrary modulation symbols' set, arbitrary bits-to-symbol mapping, and deals with the case of nonuniform signaling. These three conditions define any digital transmission using memoryless modulation, so the proposed method is a general tool solving all problems tackled in the literature, under constraints imposed on one or more of the parameters defining the modulation or signaling type. Such an evaluation tool is of practical importance during the design of the modulation. Our analysis and numerical simulations show the advantages offered by the new method when compared with the bounding techniques.

Exact evaluation of bit- and symbol-error rates in a 2-D constellation is a fundamental problem of digital communications, which only for particular modulations and/or bits-to-symbol mapping has closed-form solutions. Here, we propose a general, numerically efficient algorithmic method, which yields the exact results for arbitrary modulation symbols' set and arbitrary bits-to-symbol mapping, and which deals with the case of nonuniform signaling. These three conditions define any digital transmission using memoryless modulation, so the proposed method is a general tool solving all problems tackled in the literature under constraints imposed on one or more of the parameters defining the modulation or signaling type. Such an evaluation tool is of practical importance during the design of the modulation. Our analysis and numerical simulations show the advantages offered by the new method when compared with the bounding techniques

Exact evaluation of the bit- and symbol error rates in two dimensional constellation is a fundamental problem of digital communications which has closed-form solutions for particular modulations and/or bits-to-symbol mapping. In this paper we propose a general method which yields the exact results for arbitrary modulation constellation, arbitrary bits-to-symbol mapping and deals with the case of non-uniform signalling. These three conditions define any digital transmission using memoryless modulation thus, our method is a general tool solving all problems tackled in the literature under constraints imposed on one or more of the parameters defining the modulation or signalling type. Such evaluation tool is of practical importance during the design of the modulation. Through numerical simulations we illustrate the advantages offered by the new method when compared to the bounding technique.

During the so-called ARQ the same date is retransmitted many times to reduce the probability of error. In this paper, we consider the case when each transmission carries exactly the same information content (bits) as the first one and we focus on the constellation design aimed at improvement of the receiver performance. The performance for high SNR depends on the minimum Euclidean distance between symbols which grows as fast as the number of transmissions if the same modulation and bit-to-symbol mapping are used. It was shown in the literature that by modifying the bit-to-symbol mapping the Euclidean distance may be made to increase faster than the number of transmissions. Despite possible performance gains, no method was proposed to optimally design the mapping constellation for each of the transmissions, mainly, because the exhaustive search is out of reach of modern computers for constellations with more than 8 symbols. In this paper, we propose an algorithm which efficiently finds optimal (unlike the known heuristics which yield sub-optimal solutions) mappings for constellation sizes 16 and 32.