Decomposition of higher-order spectra for blind multiple-input deconvolution, pattern identification and separation

Abstract

a b s t r a c tLike the ordinary power spectrum, higher-order spectra (HOS) describe signal properties that are invariant under translations in time. Unlike the power spectrum, HOS retain phase information from which details of the signal waveform can be recovered. Here we consider the problem of identifying multiple unknown transient waveforms which recur within an ensemble of records at mutually random delays. We develop a new technique for recovering filters from HOS whose performance in waveform detection approaches that of an optimal matched filter, requiring no prior information about the waveforms. Unlike previous techniques of signal identification through HOS, the method applies equally well to signals with deterministic and non-deterministic HOS. In the non-deterministic case, it yields an additive decomposition, introducing a new approach to the separation of component processes within non-Gaussian signals having non-deterministic higher moments. We show a close relationship to minimum-entropy blind deconvolution (MED), which the present technique improves upon by avoiding the need for numerical optimization, while requiring only numerically stable operations of time shift, element-wise multiplication and averaging, making it particularly suited for real-time applications. The application of HOS decomposition to real-world signals is demonstrated with blind denoising, detection and classification of normal and abnormal heartbeats in electrocardiograms.

A Slice of Trispectrum for Patterns of Modulation

“…For simplicity, it will be assumed that all signals are univariate and real valued, though the extension to multivariate and complex signals is straightforward [10]. The problem can be formalized as a nonstationary system driven by a stationary white noise orthogonal increment process, Z:…”

“…The modulogram represents a windowing of the 4 th -order spectrum, restricted to the diagonal slice. A recently described technique for decomposing HOS of any order (HOSD) [10], is applicable to windowed HOS of this type. It is motivated by observing that a filter matched to an unknown waveform, f , may be recovered from the HOS of a signal that contains the waveform in Gaussian noise at an unknown delay, x j (t) = f (t − ∆t j ) + n j (t), given the K th -order deterministic HOS for the waveform…”

“…An estimate ĝ may therefore be obtained by plugging an estimate M K it into (14). Further details of the algorithm and its application to non-deterministic HOS are given in [10] and Appendix C.…”

“…Progress on questions of interpretation can be found in recent work exploring the Corresponding author: C.K. Kovach relationships between the third-order spectrum (bispectrum) and measures of phase-amplitude coupling widely adopted in electrophysiological literature [5]- [9], and in new strategies and techniques for exploiting the information in HOS to identify and recover transient or harmonically related signal components [10]- [12]. The present work seeks a similar development for the blind identification of characteristic patterns of modulation from the fourth-order spectrum, also known as the trispectrum.…”

Adaptive Multiscale Symbolic-Dynamics Entropy for Condition Monitoring of Rotating Machinery

“…In addition, Reference [ 22 ] demonstrated the usefulness of HOS in detecting a fatigue crack of a straight beam and in analyzing vibration signals of rolling bearings, Reference [ 23 ] displayed the potential of bispectrum and trispectrum for fault diagnosis of rotating machinery, Reference [ 24 ] made use of HOS to distinguish between cracks and misalignment in a rotating shaft and Reference [ 25 ] made a comparison between the results of HOS and higher order coherence for fault diagnosis of rotating machinery. However, HOS performs unequally well in deterministic and nondeterministic cases and may produce obscure spectra for a narrowband signal [ 26 ]. Additionally, results acquired by HOS generally lack clear physical meaning [ 17 ].…”