Generalized bounds on the crest-factor distribution of OFDM signals with applications to code design

In this paper generalized bounds on the crest-factor (CF) distribution in orthogonal frequency-division multiplexing (OFDM) transmission for both independent and dependent subcarriers are derived. Here, the latter situation represents the coded case. For independent subcarriers, a general path for bounding practical constellations is provided. Moreover, a complete characterization of their asymptotic behavior is devised and discussed. The results are shown to carry over to the spherical constellations improving on recent results. For dependent subcarriers, the focus is mainly on binary codes where bounds on the CF distribution are obtained in terms of the distance distributions and their duals. The asymptotic behavior of codes is analyzed and it is shown that the upper bound on the effective crest-factor of a large class of Bose-Chaudhuri-Hocquenghem (BCH) codes behaves asymptotically as root log N. Finally, two applications of the results to code design are presented: first, fixed phase shifts on the subcarriers for all codewords are used and an algorithm to calculate the phase shifts is designed. That way, it is proved that the effective CF of any binary code can be scaled to be of order root log N for large N without sacrificing on rate. Furthermore, the same approach is applied to calculation of the phases of redundant subcarriers for each codeword. It is shown by simulations that the values can be effectively chosen so that the CF is significantly reduced with nonexponential complexity.