Resonansform

Resonansform is a proposed transform used in signal processing and applied physics to extract and highlight the resonant structure of a time-domain signal. It projects the signal onto a family of damped harmonic-oscillator kernels, so that components corresponding to natural frequencies and their associated damping are emphasized. The idea is to provide a spectrum-like representation that reflects how a system would respond to driven oscillations rather than just how the signal behaves in time or frequency alone.

Mathematically, the resonansform of a signal x(t) is defined by an integral of the form R(ω) = ∫_0^∞ x(t) K(t, ω) dt, where K(t, ω) is a resonant kernel that models a damped oscillator at frequency ω. A common choice is K(t, ω) = e^{-β(ω) t} e^{i ω t}, with β(ω) representing the damping term related to the quality factor of the mode. The resulting transform is generally complex, and its magnitude and phase convey how strongly the signal excites a mode at ω and with what phase.

Properties and relationships: The resonansform is linear and, under suitable conditions on the kernel, invertible, yielding a reconstruction of x(t) from R(ω). It complements Fourier and wavelet analyses by prioritizing resonant contributions and damping, which can yield sharper identification of modal content in noisy or highly damped systems. The resolution depends on the accuracy of the damping estimates β(ω) and the chosen kernel family.

Applications and use cases: In vibration and structural health monitoring, resonansforms help identify natural frequencies and damping in beams, plates, and machinery. In acoustics and electrical circuits, they assist in characterizing resonant modes and energy distribution. The method is often used as a diagnostic tool alongside traditional spectral methods.

Limitations: The reliability of a resonansform hinges on an appropriate kernel model and known or estimated damping parameters; mismatches can blur or misrepresent resonant content. See also Fourier transform, Laplace transform, and Prony methods.