Study On Codes With Low Peak-to-Mean Envelope Power Ratio

Orthogonal frequency-division multiplexing (OFDM) is a communication technique which has been used in several wireless communication standards such as IEEE 802.16 Mobile WiMAX. A major barrier to the widespread acceptance of OFDM is the large peak-to-mean envelope power ratio (PMEPR) of uncoded OFDM signals. Coding approaches can not only solve the problem of high PMEPR, but also enjoy good error-correcting capability. This thesis considers the utility of coding approaches in solving the problem of high PMEPR, mainly focuses on analysis and constructions of large sets of sequences with low PMEPR and high pairwise distance, including the following four parts:The first main topic of this thesis is the construction of near complementary se-quences. To solve the drawback that the code rate of known Golay sequences is low, near-complementary sequences of which PMEPR is bounded by a finite value c> 2 are studied in this thesis. Firstly, by employing some new shortened and extended Golay complementary pairs as the seeds, we enlarge the family size of near-complementary se-quences given by Yu and Gong. We also show that the new set of sequences we obtained is just a reversal of the original set. Numerical results show that the enlarged family size is almost twice of the original one. Besides, the Hamming distances of the binary near-complementary sequences are also analyzed. Secondly, the PMEPR bound of these near-complementary sequences are improved to asymptotically equivalent to 2. Based on the proof, new near-complementary sequences constructed by other seed pairs are proposed. The PMEPR of these near-complementary sequences are still asymptotically equivalent to 2. Thirdly, by a computer program, a set of 256 non-Golay quaternary sequences of length 8 with PMEPR upper bound 1.98 is reported. The set can be used with the existing 768 quaternary Golay sequences of length 8, then the number of total sequences with PMEPR not greater than 2 is 1024, an integer power of 2, which expe-dite implementation. The resultant set can be applicable to the practical OFDM system implemented by the fast Fourier transform.The second main topic of this thesis is the construction of Golay complementary sets. Firstly, we construct new complementary sequence sets of size 4, using a graphical description. We explain how the construction can be seen as a special case of a less explicit array construction by Parker and Riera. Some generalizations of the construction ire also given, which are not in the construction of Parker and Riera. Lower bound and upper bound on the number of sequences in the constructions are analyzed. Secondly, a construction for complementary sets of arrays that exploits a set of mutually-unbiased bases (a MUB) is given. We show that the problem of construction of large sets of complementary sequences with good pairwise distinguishability is naturally solved by seeding the recursive construction with optimal mutually-unbiased bases (MUBs). In particular we present, in detail, the construction for complementary pairs that is seeded py a MUB of dimension 2, where we enumerate the arrays and the corresponding set of complementary sequences obtained from the arrays by projection. We also sketch an algorithm to uniquely generate these sequences. The pairwise squared inner-product of members of the sequence set is shown to be less than or equal to 1/2. Moreover, a subset of the set can be viewed as a codebook that asymptotically achieves 1/23/2 times the Welch bound.The third main topic of this thesis is the construction of Boolean functions which have at least two flat spectra with respect to {H, N}n. Two constructions of Boolean functions which have at least two flat spectra with respect to {H, N}n are proposed. Some known results about Bent-Negabent functions are special cases of our results. Some properties of transforms in {H, N}n are presented. Sufficient and necessary conditions for a Boolean function to have a flat spectrum with respect to transform U ∈{H, N}n are given. Furthermore, some lower bounds on the numbers of flat spectra of Boolean functions with respect to {H, N}n or {I, N}n are given. In particular, we show that any Vlaiorana-McFarland Bent function of n (n even) variables has at least n/2+2n/2 flat spectra with respect to {H, N}nThe fourth main topic of this thesis is to consider the numbers of flat spectra with espect to {I, H, N}n or subsets thereof for some quadratic Boolean functions. We apply the results given by Riera and Parker to develop some formulae for the numbers of flat spectra of some quadratic functions and some sequences given in this thesis constructed by using mutually unbiased bases. The results show that some sequences constructed by using mutually unbiased bases will have the largest known number of flat spectra with respect to {H,N}n.