On sequency-complete and sequency-ordered matrices

Abstract

The concept of sequency holds a fundamental significance in signal analysis using Walsh basis functions. In this study, we closely examine the concept of sequency and explore the properties of sequency-complete and sequency-ordered matrices. We obtain results on cardinalities of sets containing sequency-complete and sequency-ordered matrices of arbitrary sizes. We present methods for obtaining interesting classes of sequency-complete and sequency-ordered matrices of arbitrary sizes. We also provide results on the sequencies of columns in tensor products involving two or more matrices.

Figures (5)

• Figure 1: Walsh basis functions in sequency-order for $N=8$.
• Figure 2: The matrices in the top row are sequency-complete and the matrices in the bottom row are sequency-incomplete. These matrices were generated using Eq. \ref{['eq:example1']} for $3 \leq n \leq 8$. We note that for each $n$, $\mathop{\mathrm{Seq}}\nolimits_n$ denotes the sequency of the columns of the corresponding $n \times n$ matrix.
• Figure 3: Some examples of sequency-complete matrices. These matrices were generated using Eq. \ref{['eq:example2']} for $2 \leq n \leq 7$. We note that for each $n$, $\mathop{\mathrm{Seq}}\nolimits_n$ denotes the sequency of the columns of the corresponding $n \times n$ matrix.
• Figure 4: Illustration of the segments $L$ (blue) and $R$ (red) as describe in the proof of Theorem \ref{['thm:Aijone']}. We note that the boundary points $0$, $2m$, $\ldots$, $2jm \in L$ and $m$, $3m$, $\ldots$, $(2j-1)m \in R$.
• Figure 5: Some examples of sequency-ordered matrices. These matrices were generated using Eq. \ref{['eq:example3']} for $2 \leq n \leq 7$.

Theorems & Definitions (26)

• Definition 1.1: Sequency
• Definition 1.2: Sequency-complete
• Remark 1.3
• Example 1.4
• Definition 1.5: Sequency-ordered
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3