Higher-Order Spectra and their Unbiased Estimation in the GPU-accelerated SignalSnap Library

The analysis of time-dependent data poses a fundamental challenge in many fields of science and engineering. While concepts for higher-order spectral analysis like Brillinger's polyspectra for stationary processes have long been introduced, their applications have been limited probably due to high computational cost and complexity of implementation. Here we discuss the theoretical background of estimating polyspectra with our open-source GPU-accelerated SignaSnap library and highlight its advantages over previous implementations: (i) The calculation of spectra is unprecedentedly based on unbiased and consistent estimators that suppress the appearance of false structures in fourth-order spectra. (ii) SignalSnap implements cross-correlation spectra for up to four channels. (iii) The spectral estimates of SignalSnap have a clear relation to Brillinger's definition of ideal spectra of continuous stochastic processes in terms of amplitude and spectral resolution. (iv) SignalSnap estimates the variance of each spectral value. We show how polyspectra reveal, e.g., the correlations between different channels or the breaking of time-inversion symmetry and discuss how quasi-polyspectra uncover the non-stationarity of signals.