Phase-Preserving Nonlinearities

📐 Definition

For $z \in \mathbb{C}\setminus\{0\}$ and a scalar nonlinearity $g : [0,\infty) \to [0,\infty)$ , a phase-preserving (radial) nonlinearity can be written as

T(z) = g(|z|)\,\frac{z}{|z|}

If $g(0)=0$ , then defining $T(0)=0$ makes $T$ continuous at the origin.

⚙️ Key Properties

Phase Invariance

For $z \ne 0$ ,

\operatorname{Arg}(T(z)) = \operatorname{Arg}(z)\qquad\text{whenever }g(|z|)>0

Equivalently, $T$ commutes with multiplication by unit-magnitude complex phases: $T(e^{i\phi}z)=e^{i\phi}T(z)$ .

Magnitude Response

|T(z)| = g(|z|)

The choice of $g$ controls smoothness, saturation, and growth while leaving phase untouched.

🎯 Special Cases and Limits

Identity: $g(r)=r$ gives $T(z)=z$ .
Hard clipping: $g(r)=\min(r,r_{\max})$ clips magnitudes but is nondifferentiable at $r_{\max}$ .
Smooth saturation: $g(r)=r_{\max}\tanh(r/r_{\max})$ limits magnitude while remaining differentiable.

Complex Magnitude Input to radial functions

Complex Phase Phase is preserved by radial mappings

Phase Coherence Measures Diagnostics for phase alignment after applying radial mappings

Usage in Oakfield

Oakfield implements several phase-preserving (radial) behaviors for complex fields:

Analytic warp (polar mode) updates complex samples by computing a scalar response at $|z|$ and applying it along the current direction $z/|z|$ (ANALYTIC_WARP_COMPLEX_MODE_POLAR).
Remainder (polar mode) computes magnitude-only residues between warped/reference fields and writes the result along the warped phase direction (SIM_REMAINDER_COMPLEX_MODE_POLAR).
Thermostat regulation uses $|u|^2$ and applies either radial “push/pull” along $u/|u|$ (ADD mode) or proportional scaling of $u$ (MULT mode), both of which preserve phase away from the origin.
Phase feature extraction effectively produces a weighted unit phasor (thresholded $z/|z|$ scaled by a magnitude-derived weight).
A general-purpose helper sim_clamp_complex exists for magnitude clamping via phase-preserving rescaling.

Historical Foundations

📜 Radial Mappings

Amplitude-only (radial) mappings are a standard construction in complex-valued signal processing and wave modeling, where phase encodes timing/geometry and magnitude encodes envelope or energy.
🌍 Modern Perspective

These mappings are widely used as “phase-safe” nonlinearities: they modify amplitude while minimizing spurious phase distortion.

📚 References

Cohen, Time-Frequency Analysis
Mallat, A Wavelet Tour of Signal Processing