Superlets: time-frequency super-resolution using wavelet sets

Abstract

Time-frequency analysis is ubiquitous in many fields of science. Due to the Heisenberg-Gabor uncertainty principle, a single measurement cannot estimate precisely the location of a finite oscillation in both time and frequency. Classical spectral estimators, like the short-time Fourier transform (STFT) or the continuous-wavelet transform (CWT) optimize either temporal or frequency resolution, or find a tradeoff that is suboptimal in both dimensions. Following concepts from optical super-resolution, we introduce a new spectral estimator enabling time-frequency super-resolution. Sets of wavelets with increasing bandwidth are combined geometrically in a superlet to maintain the good temporal resolution of wavelets and gain frequency resolution in the upper bands. Superlets outperform the STFT, CWT, and other super-resolution methods on synthetic data and brain signals recorded in humans and rodents, resolving time-frequency details with unprecedented precision. Importantly, superlets can reveal transient oscillation events that are hidden in the averaged time-frequency spectrum by other methods.