Quantized Constant-Q Gabor Atoms for Sparse Binary Representations of Cyber-Physical Signatures

Abstract

Increased data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present new challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different continuous wavelet transform (CWT) reconstruction formulas are presented and tested under different signal to noise ratio (SNR) conditions. A sparse superposition of Nth order Gabor atoms worked well against a synthetic blast transient using the wavelet entropy and an entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for sparse feature extraction and dictionary-based machine learning across multiple sensor modalities.

Quantized Information in Spectral Cyberspace

“…The transformation of digital time-series into spectral information can enhance signal detection, exploration, and feature extraction for machine learning (ML) applications. This paper expands and generalizes a standardized [ 1 ], quantized computational framework [ 2 ] within the context and nomenclature of Gabor [ 3 ] and Cohen [ 4 , 5 , 6 , 7 , 8 , 9 ], and integrates the ensuing concepts and methods with wavelet [ 10 , 11 , 12 , 13 , 14 , 15 , 16 ] and Stockwell [ 17 , 18 , 19 , 20 , 21 ] transforms. The resulting spectral power metrics are then aligned with Shannon information and entropy metrics [ 9 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 ] to facilitate the fusion of multi-modal data streams from heterogeneous sensor systems [ 37 ].…”

“…Relative entropy metrics are developed and evaluated for time–frequency representations obtained from the continuous wavelet transform (CWT) [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 21 ], the Stockwell transform (STX) [ 17 , 18 , 19 , 20 , 21 ], and the short-time Fourier transform (STFT) [ 38 , 39 , 40 ], using readily available open-source FFT [ 41 , 42 , 43 ] algorithms. The concept of an information and entropy signal-to-noise ratio e.g., [ 2 ] is further standardized relative to a uniform distribution, with possible extension to any other reference distribution.…”

“…The canonical Gabor–Morlet wavelet, also referred to as the Gabor logon, atom, or grain, satisfies the minimum uncertainty requirement with the general canonical form where with standard deviation (e.g., [ 2 ] and Appendix A ).…”

“…For a given reference time scale and a logarithmic scale base G > 1 , a quantized center time scale and frequency can be defined as where n is the band number and N is the band order [ 1 , 2 ]. Appendix A demonstrates that the critical number of cycles in a Gaussian window with standard deviation scales with the angular frequency as with time support per wavelet …”

“…The temporal support window represents the minimum number of digital points required to process a signal using a quantized constant-Q Gabor atom [ 2 ] with a center time scale of order . The window should be a power of two when using FFT.…”