GoKawiilResonate is a low latency, low memory footprint, and low computational cost algorithm to evaluate perceptually relevant spectral information from audio (and other) signals.
Overview
The Resonate model, developed over a number of years, was first officially introduced in (François, 2025) and extended in (François, 2026).
Resonate builds on a resonator model that accumulates the signal contribution around its resonant frequency in the time domain using the Exponentially Weighted Moving Average (EWMA), also known as a low-pass filter in signal processing. Consistently with on-line perceptual signal analysis, the EWMA gives more weight to recent input values, whereas the contributions of older values decay exponentially. A compact, iterative formulation of the model affords computing an update at each signal input sample, requiring no buffering and involving only a handful of arithmetic operations.
Each resonator (indexed \(k\)), characterized by its natural resonant frequency and its instantaneous resonant frequency \(f_k(t) = \frac{\omega_k(t)}{2\pi}\), is described by a complex number \(R_k(t)\) whose amplitude captures the contribution of the input signal component around frequency \(f_k(t)\). The formulas below capture the recursive update formula for \(R_k(t)\) by way of a phasor \(P_k(t)\), applied for each sample of a real-valued input signal \(x(t) \in [-1,1]\), regularly sampled at sampling rate \(sr\). \(\Delta t=1/sr\) is the sample duration.
\[P_k(t) = P_k(t-\Delta t) e^{-i \omega \Delta t}\] \[R_k(t) = (1-\alpha_k) R_k(t-\Delta t) + \alpha_k x(t) P_k(t)\]
The parameter \(\alpha_k \in [0,1]\) dictates how much each new measurement affects the accumulated value; it can be expressed as a function of the system’s time constant, set heuristically as a function of the resonant frequency \(f\) (intuitively inversely proportional to the frequency). For the frequency range of interest in audio applications (20-20000 Hz), \(\alpha_k = 1-e^{-\Delta t\frac{f_k}{log(1+f_k)} }\) is a reasonable heuristic.
The smoothed state \(\tilde{R}_k\) is produced by applying the EMWA to \(R_k\) with parameter \(\beta_k\) to dampen power and phase oscillations.
\[\tilde{R}_k(t) = (1-\beta_k) \tilde{R}_k(t-\Delta t) + \beta_k R_k(t)\]
This formulation is consistent with the first steps in the filterbank interpretation of the phase vocoder analysis as described by Dolson in his 1986 paper “The Phase Vocoder: A Tutorial”, namely heterodyning followed by lowpass filtering.
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