Complementary Set Matrices Satisfying a Column Correlation Constraint

TL;DR

This paper presents a recursive design method for complementary set matrices with column sequences satisfying correlation constraints, aiming to reduce PAPR in transmitted signals, and introduces a new approach for selecting companion pairs to optimize correlation properties.

Contribution

It introduces a recursive construction algorithm for complementary set matrices with correlation constraints and proposes a novel method for selecting companion pairs to optimize column correlations.

Findings

Recursive construction of complementary set matrices from companion pairs.

A new cost function for selecting optimal companion pairs.

Conditions for the existence of suitable companion pairs based on Welch bound.

Abstract

Motivated by the problem of reducing the peak to average power ratio (PAPR) of transmitted signals, we consider a design of complementary set matrices whose column sequences satisfy a correlation constraint. The design algorithm recursively builds a collection of $2^{t+1}$ mutually orthogonal (MO) complementary set matrices starting from a companion pair of sequences. We relate correlation properties of column sequences to that of the companion pair and illustrate how to select an appropriate companion pair to ensure that a given column correlation constraint is satisfied. For $t=0$, companion pair properties directly determine matrix column correlation properties. For $t\geq 1$, reducing correlation merits of the companion pair may lead to improved column correlation properties. However, further decrease of the maximum out-off-phase aperiodic autocorrelation of column sequences is not possible once the companion pair correlation merit is less than a threshold determined by $t$. We also reveal a design of the companion pair which leads to complementary set matrices with Golay column sequences. Exhaustive search for companion pairs satisfying a column correlation constraint is infeasible for medium and long sequences. We instead search for two shorter length sequences by minimizing a cost function in terms of their autocorrelation and crosscorrelation merits. Furthermore, an improved cost function which helps in reducing the maximum out-off-phase column correlation is derived based on the properties of the companion pair. By exploiting the well-known Welch bound, sufficient conditions for the existence of companion pairs which satisfy a set of column correlation constraints are also given.