Harmonic Analysis of Binary Data: A Multi-Scale Structural Metric

Traditional methods for comparing binary sequences often rely on bit-by-bit analysis, which fails to account for the underlying structural patterns and is highly sensitive to simple alignment shifts. This paper introduces a novel structural metric that bridges the gap between discrete data and continuous signal analysis. By mapping binary sequences into a functional space through a multi-scale block decomposition, we represent data as a continuous "harmonic signature" using Fourier series expansion. Our analysis shows that this functional approach is inherently robust against common sequence transformations, such as cyclic shifts and localized errors, which typically defeat traditional comparison techniques. By evaluating the area between these generated functions, we provide a qualitative measure of similarity that prioritizes the "rhythm" and motifs of the data over its exact bitwise positions. Furthermore, we demonstrate the practical utility of this framework by proposing a structural hashing algorithm derived from functional critical points and a cryptographic scheme based on frequency-domain phase shifts. This work suggests a new direction for data verification and encryption by treating binary information as a continuous mathematical landscape.