Envelope estimation using geometric properties of a discrete real signal

Abstract

Despite being an elusive concept, the temporal amplitude envelope of a signal is essential for its complete characterization, being the primary information-carrying medium in spoken voice and telecommunications, for example. Envelope detection techniques have applications in medicine, sound classification, sound synthesis, seismology, and speech recognition. Nevertheless, a general approach to envelope detection does not exist. Most methods involve manual intervention, in the form of filter design, smoothing, and other specific design choices based on prior knowledge of the signals under investigation. To address this problem, we propose an algorithm that uses characteristics of a signal to estimate its envelope, eliminating the necessity of parameter tuning. The method draws inspiration from geometric concepts: the discrete signal is mapped to a set of points in the Cartesian plane, where a new measure of discrete curvature is used to identify the samples that are part of the envelope. Due to its geometric nature, the method can optionally generate the superior and inferior envelopes of a given wave. The algorithm compares favorably with classic envelope detection techniques based on smoothing, filtering, and the Hilbert transform, besides being physically plausible. We provide visualizations of the envelope extracted for various real-world signals with diverse characteristics, such as voice, spoken and sang, and pitched and unpitched musical instruments, and discuss some approaches to objectively assess the quality of the obtained envelopes. An optimized implementation is available via the Python Package Index (PyPI), while interactive visualizations and source code are available from the accompanying GitHub repository.