Complexrelated

Complexrelated is a term used in discussions of complex-valued data to describe a particular kind of relation between complex numbers or complex-valued signals. It is not part of a fixed, universally adopted formal framework, but several authors describe a similar idea: that a relation should reflect both amplitude and phase information and be stable under basic complex operations.

In a minimal informal sense, a binary relation R on a set of complex-valued objects is called complexrelated if it satisfies two intuitive invariances: 1) homogeneity under nonzero complex scaling: if (x, y) ∈ R then (a x, a y) ∈ R for all a ≠ 0. 2) conjugation invariance: if (x, y) ∈ R then (conj(x), conj(y)) ∈ R. In practice, researchers use complex-relatedness as a qualitative descriptor of dependencies whose form persists when signals are rotated in the complex plane or scaled.

One common context is complex-valued time series or phasor data, where a complexrelated relation captures phase-locked or amplitude-correlated structure beyond what real-valued correlations can express.

As an example, two complex sinusoids with a fixed phase difference are complexrelated, because their relationship remains the same under global phase shifts.

Relation to other concepts: complex correlation, cross-spectral density, and holomorphic dependence are related ideas. The term complexrelated is informal and not yet standardized, appearing mainly in niche literature and online discussions.

See also: complex analysis; complex-valued random variables; correlation; phase.