A Generic Complementary Sequence Encoder

In this study, we propose a generic complementary sequence (CS) encoder that limits the peak-to-average-power ratio (PAPR) of an orthogonal frequency division multiplexing (OFDM) symbol. We first establish a framework which determines the coefficients of a polynomial generated through a recursion with linear operators via Boolean functions. By applying the introduced framework to a recursive Golay complementary pair (GCP) construction based on Budisin's methods, we then show the impact of the initial sequences, phase rotations, signs, real scalars, and the shifting factors applied at each step on the elements of the sequences in a GCP explicitly. Hence, we provide further insights into GCPs. As a result, we extend the standard sequences by separating the encoders that control the amplitude and the phase of the elements of a CS. We obtain the algebraic structure of an encoder which generates non-contiguous CSs. Thus, three important aspects of communications, i.e., frequency diversity, coding gain, and low PAPR, are achieved simultaneously for OFDM symbols. By using an initial GCP, we algebraically synthesize CSs which can be compatible with the resource allocation in major wireless standards. We also demonstrate the compatibility of the proposed encoder with quadrature amplitude modulation (QAM) constellation and its performance.Comment: This work has been submitted to IEEE Transactions on Communication

A complementary construction using mutually unbiased bases

We present a construction for complementary pairs of arrays that exploits a set of mutually-unbiased bases, and enumerate these arrays as well as 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 $\frac{1}{2}$. Moreover, a subset of the set can be viewed as a codebook that asymptotically achieves $\sqrt{\frac{3}{2}}$ times the Welch bound.Comment: 25 pages, 1 figur

New Construction of Complementary Sequence (or Array) Sets and Complete Complementary Codes (I)

A new method to construct $q$-ary complementary sequence (or array) sets (CSSs) and complete complementary codes (CCCs) of size $N$ is introduced in this paper. An algorithm on how to compute the explicit form of the functions in constructed CSS and CCC is also given. A general form of these functions only depends on a basis of functions from $\Z_N$ to $\Z_q$ and representatives in the equivalent class of Butson-type Hadamard matrices. Surprisingly, all the functions fill up a larger number of cosets of a linear code, compared with the existing constructions. From our general construction, its realization of $q$-ary Golay pairs exactly coincides with the standard Golay sequences. The realization of ternary complementary sequences of size $3$ is first reported here. For binary and quaternary complementary sequences of size 4, a general Boolean function form of these sequences is obtained. Most of these sequences are also new. Moreover, most of quaternary sequences cannot be generalized from binary sequences, which is different from known constructions. More importantly, both binary and quaternary sequences of size 4 constitute a large number of cosets of the linear code respectively.Comment: This paper and another is merged together. And the merged paper is onlin

Golay Layer: Limiting Peak-to-Average Power Ratio for OFDM-based Autoencoders

In this study, we propose a differentiable layer for OFDM-based autoencoders (OFDM-AEs) to avoid high instantaneous power without regularizing the cost function used during the training. The proposed approach relies on the manipulation of the parameters of a set of functions that yield complementary sequences (CSs) through a deep neural network (DNN). We guarantee the peak-to-average-power ratio (PAPR) of each OFDM-AE symbol to be less than or equal to 3 dB. We also show how to normalize the mean power by using the functions in addition to PAPR. The introduced layer admits auxiliary parameters that allow one to control the amplitude and phase deviations in the frequency domain. Numerical results show that DNNs at the transmitter and receiver can achieve reliable communications under this protection layer at the expense of complexity.Comment: This paper is accepted for presentation at IEEE International Conference on Communications (ICC) 202

New Constructions of Complementary Sequence Pairs over 4q4^q-QAM

The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as $4^q$-QAM GCSs of length $2^{m}$ by Li \textsl{et al.} (the generalized cases I-III for $q\ge 2$) in 2010 and Liu \textsl{et al.} (the generalized cases IV-V for $q\ge 3$) in 2013 respectively. Those sequences are presented as the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer $q$, we proposed two new constructions of $4^q$-QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs (including the generalized cases IV-V) are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. For $q=q_{1}\times q_{2}\times \dots\times q_{t}$ ($q_k>1$), the number of new offsets in our first construction is lower bounded by a polynomial of $m$ with degree $t$, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial of $m$ and a quadratic polynomial of $m$, respectively. In particular, the numbers of new offsets in our first construction is seven times more than that in the generalized cases IV-V for $q=4$. We also show that the numbers of new offsets in our two constructions is lower bounded by a cubic polynomial of $m$ for $q=6$. Moreover, our proof implies that all the mentioned GCSs over QAM in this paper can be regarded as projections of Golay complementary arrays of size $2\times2\times\cdots\times2$