Range entropy: A bridge between signal complexity and self-similarity

Approximate entropy (ApEn) and sample entropy (SampEn) are widely used for temporal complexity analysis of real-world phenomena. However, their relationship with the Hurst exponent as a measure of self-similarity is not widely studied. Additionally, ApEn and SampEn are susceptible to signal amplitude changes. A common practice for addressing this issue is to correct their input signal amplitude by its standard deviation. In this study, we first show, using simulations, that ApEn and SampEn are related to the Hurst exponent in their tolerance r and embedding dimension m parameters. We then propose a modification to ApEn and SampEn called range entropy or RangeEn. We show that RangeEn is more robust to nonstationary signal changes, and it has a more linear relationship with the Hurst exponent, compared to ApEn and SampEn. RangeEn is bounded in the tolerance r-plane between 0 (maximum entropy) and 1 (minimum entropy) and it has no need for signal amplitude correction. Finally, we demonstrate the clinical usefulness of signal entropy measures for characterisation of epileptic EEG data as a real-world example.Comment: This is the revised and published version in Entrop

Accurate and efficient implementation of the time-frequency matched filter

The discrete time--frequency matched filter should replicate the continuous time--frequency matched filter. But the methods differ. To avoid aliasing the discrete method transforms the real-valued signal to the complex-valued analytic signal. The theory for the time--frequency matched filter does not consider the discrete case using the analytic signal. We find that the performance of the matched filter degrades when using the analytic, rather than real-valued, signal. This performance degradation is dependent on the signal to noise ratio and the signal type. In addition, we present a simple algorithm to efficiently compute the time--frequency matched filter. The algorithm with the real-valued signal, comparative to using the analytic signal, requires one-quarter of the computational load. Hence the real-valued signal---and not the analytic signal---enables an accurate and efficient implementation of the time--frequency matched filter

Proofs for Discrete Time-Frequency Distribution Properties

This technical report contains proofs for a set of mathematical properties of a recently proposed discrete time-frequency distribution class

A Nonstationary Model of Newborn EEG

The detection of seizure in the newborn is a critical aspect of neurological research. Current automatic detection techniques are difficult to assess due to the problems associated with acquiring and labelling newborn electroencephalogram (EEG) data. A realistic model for newborn EEG would allow confident development, assessment and comparison of these detection techniques. This paper presents a model for newborn EEG that accounts for its self-similar and non-stationary nature. The model consists of background and seizure sub-models. The newborn EEG background model is based on the short-time power spectrum with a time-varying power law. The relationship between the fractal dimension and the power law of a power spectrum is utilized for accurate estimation of the short-time power law exponent. The newborn EEG seizure model is based on a well-known time-frequency signal model. This model addresses all significant time-frequency characteristics of newborn EEG seizure which include; multiple components or harmonics, piecewise linear instantaneous frequency laws and harmonic amplitude modulation. Estimates of the parameters of both models are shown to be random and are modelled using the data from a total of 500 background epochs and 204 seizure epochs. The newborn EEG background and seizure models are validated against real newborn EEG data using the correlation coefficient. The results show that the output of the proposed models has a higher correlation with real newborn EEG than currently accepted models (a 10% and 38% improvement for background and seizure models, respectively)

A New Discrete Analytic Signal for Reducing Aliasing in the Discrete Wigner-Ville Distribution

It is not possible to generate an alias-free discrete Wigner--Ville distribution (DWVD) from a discrete analytic signal. This is because the discrete analytic signal must satisfy two mutually exclusive constraints. We present, in this article, a new discrete analytic signal that improves on the commonly used discrete analytic signal's approximation of these two constraints. Our analysis shows that---relative to the commonly used signal---the proposed signal reduces aliasing in the DWVD by approximately 50%. Furthermore, the proposed signal has a simple implementation and satisfies two important properties, namely, that its real component is equal to the original real signal and that its real and imaginary components are orthogonal

A computationally efficient implementation of quadratic time-frequency distributions

Time-frequency distributions (TFDs) are computationally intensive methods. A very common class of TFDs, namely quadratic TFDs, is obtained by time-frequency (TF) smoothing the Wigner Ville distribution (WVD). In this paper a computationally efficient implementation of this class of TFDs is presented. In order to avoid artifacts caused by circular convolution, linear convolution is applied in both the time and frequency directions. Four different kernel types are identified and separate optimised implementations are presented for each kernel type. The computational complexity is presented for the different kernel types

Elaboration of geopolymer cement based on dredged sediment

This work aims to study the feasibility of making a geopolymer cement based on dredged sediments, from the Fergoug dam (Algeria). Sedimentary clays were characterized before and after calcination by X-ray diffraction, ATG / ATD, spectroscopy (FTIR) and XRF analysis. The reactivity of the calcined products was measured using isothermal calorimetric analysis (DSC) on pastes prepared by mixing an alkaline solution of sodium hydroxide (NaOH) 8 M in an amount allowing to have a Na / Al ratio close to 1. Also, cubic mortar samples were prepared with a ratio L / S: 0.8, sealed and cured for 24 hours at 60 ° C and then at room temperature. The results obtained allowed to optimize the calcination time of 5 hours for a better reactivity of these sediments, and a concentration of 8M of sodium hydroxide and more suitable to have the best mechanical performances

Neonatal EEG Seizure Detection using a Time-frequency Matched Filter with a Reduced Template Set

Electroencephalographic (EEG) recordings are an important diagnostic resource in determining the presence or absence of clinical seizures in neonates. These nonstationary signals require some form of nonstationary analysis to detect seizures in the EEG data. A time-frequency (TF) matched lter has been previously proposed to detect seizures in both adult and newborn EEG. A method which constructs a reference or template set from a feature of EEG seizures, rather than the whole EEG seizure, displayed the most promising results. However this method suffered from an inability to adequately represent patient variability in the template set while simultaneously maintaining a low false detection rate. A new method of the TF matched lter is proposed that halves the template set required by approximating the templates with a more general ambiguity domain function representation. This proposed method is also less sensitive to false detections when a larger reference set is used, as evidenced by the ndings on both simulated and real neonatal EEG

A Discrete Time and Frequency Wigner Distribution: Properities and Implementation

Time-frequency distributions are used in the analysis and processing of nonstationary signals. The Wigner-Ville distribution (WVD) is a fundamental time-frequency distribution uniquely satisfying many desirable mathematical properties. The realisation of this distribution for hardware or software platforms requires a discrete version. Historically the majority of the work on deriving discrete versions of the WVD has focused on creating alias-free distributions, often resulting in a loss of some desirable properties. Here a new discrete time and frequency WVD will be presented for nonperiodic signals and will be examined both in terms of its properties and aliasing. In particular unitarity, an assumed property for optimum time-frequency detection and signal estimation, and invertibility, a useful property especially for time-frequency filtering, will be examined. An efficient implementation of the distribution using standard real-valued fast Fourier transforms will also be presented