Relationship Between Smith Normal Form of Periodicity Matrices and Sampling of Two-Dimensional Discrete Frequency Distributions With Tiling Capability

Abstract

It has been found that 2-D discrete diamond-shaped frequency distributions used in the harmonic balance method can be calculated by 1-D sampling, where the number of sampling points can be equal to that of nonzero (NZ) frequency components. The reciprocal vector of the periodicity matrix of tiling is chosen as the unit step of the 1-D sampling. However, in general, reciprocal vectors cannot always be the unit vectors of 1-D samplings for 2-D discrete frequency distributions. The condition for a reciprocal vector to be a unit step of 1-D sampling for 2-D discrete frequency distributions with tiling capability is examined in this brief. The results show that the condition is that two things are satisfied simultaneously for the periodicity matrix of tiling, [M]. When [M] is expressed as [M]=[A][D][B] in the Smith normal form, [A] and [B] are unimodular matrices, and [D] is an integer-valued diagonal matrix. One is that d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> of [D] is 1. The other is that ([A] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> of [D] are congruent. When the condition is satisfied, the reciprocal vector T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> made from [M] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> can be a unit step of 1-D sampling, where 1-D discrete Fourier transform can be used in the calculation, and the number of sampling points is equal to that of NZ frequency components.