Gain Control

📐 Definition

For $\beta \ge 0$ and scalar $x$ (componentwise for vectors),

$$G_{\beta}(x) = \frac{x}{1 + \beta |x|}$$

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Domain and Codomain

Applies to real or complex inputs (using $|x|$ for magnitude). Output retains the input sign or phase with bounded magnitude.

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⚙️ Key Properties

Monotone in $|x|$; linear near the origin with slope $1$; saturates to $\operatorname{sgn}(x)/\beta$ as $|x| \to \infty$ when $\beta>0$. For real $x$, the map is differentiable everywhere (including at $0$), but is not twice differentiable at $0$ due to the cusp in $|x|$.

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$\beta = 0$ recovers identity. Large $\beta$ enforces strong compression. Replacing $|x|$ with $x^2$ yields a smooth even variant.

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🎯 Special Cases and Limits

• $\beta=0$ is the identity.
• Large $\beta$ yields stronger compression with smaller effective output magnitude.

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🔗 Related Functions

Normalization rescales to unit norm globally; smooth saturation functions offer alternative soft knees; thresholding can be combined to mute small signals before gain control.

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Usage in Oakfield

Oakfield does not currently expose this exact rational “soft limiter” as a standalone operator, but similar gain-control behavior exists in several places:

• Bounded nonlinearities: analytic_warp (TANH profile) and remainder (TANH) provide smooth saturation to prevent runaway amplitudes.
• Energy regulation: the thermostat operator provides feedback-style gain control based on $|u|^2$.
• Spectral damping: linear_dissipative applies per-mode exponential decay in Fourier space (a frequency-dependent gain control).
• Utility hooks like sim_clamp_complex support phase-preserving magnitude limiting when a kernel wants explicit clamping.

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Historical Foundations

📜 Dynamic Range Control

Gain-control style nonlinearities are standard in signal processing and numerical stabilization, enforcing bounded magnitude while preserving sign or phase.

🌍 Modern Perspective

They are commonly used as “soft limiters” in iterative updates to prevent runaway amplitudes.

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📚 References

• Oppenheim & Schafer, Discrete-Time Signal Processing