Show HN: Resonate – real-time high temporal resolution spectral analysis

Summary

Resonate is a low latency, low memory footprint, and low computational cost algorithm to evaluate perceptually relevant spectral information from audio (and other) signals. Overview 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, characterized by its resonant frequency \(f = \frac{\omega}{2\pi}\), is described by a complex number \(R\) whose amplitude captures the contribution of the input signal component around frequency \(f\). The formulas below capture the recursive update for \(R\) by way of a phasor \(P\), applied for each sample \(x\) 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, and \(\alpha \in [0,1]\) is a constant parameter that dictates how much each new measurement affects the accumulated value. \[P \leftarrow P e^{-i \omega \Delta t}\] \[R \leftarrow (1-\alpha) R + \alpha x P\] The two complex numbers \(P\) and \(R\) capture the full state of the resonator. Updating the state at each input signal sample only requires a handful of arithmetic operations. Calculating the power and/or magnitude is not necessary for the update, and can be carried out only when required by the application, relatively efficiently as well. The single parameter \(\alpha\), which can be related to a time constant, governs the dynamics of the system. For the frequency range of interest in audio applications (20-20000 Hz), the function \(\alpha_f = ...