Spectral Complexity Index

The Spectral Complexity Index is the information entropy, in bits, of how a harmonic system's energy is spread across its harmonics. Each harmonic's share of the total energy is treated as a probability, and the entropy of that distribution measures how much information the allocation of energy encodes. A spectrum whose energy sits in one harmonic has low complexity; one spread over many has high complexity. The book uses it as the first of three spectral complexity measures, with the Recursive Depth Indicator and the Harmonic Clustering Coefficient, to quantify the computational intricacy of recursive harmonic structures and distinguish them from classical ones. Among the Chapter 7.1 predictions, this entropy is to be measured in recursive antenna arrays through signal processing. The formal statement is C⚚ = −sum_n p_n log_2(p_n), where C⚚ is the Spectral Complexity Index, n the harmonic index, p_n the normalized energy distribution E_n/E⊚, E_n the energy of the nth harmonic and E⊚ the total energy across all harmonics. Dimensions: C⚚ [1ᵇ], p_n and n [∅], E_n and E⊚ [J]; the book's check is consistent. It connects to the Recursive Depth Indicator, Harmonic Clustering Coefficient, Convergence Parameter and Information Density Criterion.