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 for a non-contiguous frequency domain resource allocation. To this end, we introduce a framework which describes a recursion evolved with two linear operators at each step as an encoding operation. The framework algebraically determines how the operators applied at each recursion step are distributed to the coefficients of a polynomial via binary construction sequences. By applying the introduced framework to a recursive Golay complementary pair (GCP) construction relying on Budi$\v{s}$in's methods, we 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 obtain an encoder which generates non-contiguous CSs, i.e., CSs with zero symbols, by deriving the algebraic normal forms (ANFs) for the shifting factors. Thus, three important aspects for communications, i.e., frequency diversity, coding gain, and low PAPR, are achieved simultaneously for OFDM symbols. As another result, we extend the standard sequences by separating the encoders that control the amplitude and the phase of the elements of a CSs. We also demonstrate that the proposed encoder generates CSs with quadrature amplitude modulation (QAM) alphabet.