Studies on fractional order differentiators and integrators: A survey

The 1/f family of fractal random processes model a truly extraordinary range of natural and man-made phenomena, many of which arise in a variety of signal processing scenarios. Yet despite their apparent importance, the lack of convenient representations for 1/f processes has, at least until recently, strongly limited their popularity. In this paper, we demonstrate that 1/f processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases. Specifically, via a useful frequency domain characterization for 1/f processes, we develop the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes. As an illustration of potential, we show that wavelet based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes

Wavelet-based representations for the 1/f family of fractal processes : Fractals in electrical engineering

We present a new method for automatic detection of peaks in noisy periodic and quasi-periodic signals. The new method, called automatic multiscale-based peak detection (AMPD), is based on the calculation and analysis of the local maxima scalogram, a matrix comprising the scale-dependent occurrences of local maxima. The usefulness of the proposed method is shown by applying the AMPD algorithm to simulated and real-world signals.

https://www.zora.uzh.ch/id/eprint/70606/1/Scholkmann_Algorithm_for_automatic_peak_detection_NeoUSZ.pdf

An Efficient Algorithm for Automatic Peak Detection in Noisy Periodic and Quasi-Periodic Signals

Cardiovascular diseases are the number one cause of death worldwide. Currently, portable batteryoperated systems such as mobile phones with wireless ECG sensors have the potential to be used in continuous cardiac function assessment that can be easily integrated into daily life. These portable point-of-care diagnostic systems can therefore help unveil and treat cardiovascular diseases. The basis for ECG analysis is a robust detection of the prominent QRS complex, as well as other ECG signal characteristics. However, it is not clear from the literature which ECG analysis algorithms are suited for an implementation on a mobile device. We investigate current QRS detection algorithms based on three assessment criteria: 1) robustness to noise, 2) parameter choice, and 3) numerical eciency, in order to target a universal fast-robust detector. Furthermore, existing QRS detection algorithms may provide an acceptable solution only on small segments of ECG signals, within a certain amplitude range, or amid particular types of arrhythmia and/or noise. These issues are discussed in the context of a comparison with the most conventional algorithms, followed by future recommendations for developing reliable QRS detection schemes suitable for implementation on battery-operated mobile devices.

/pdf/revisiting-qrs-detection-methodologies-for-portable-wearable-37qan10upr.pdf

Revisiting QRS detection methodologies for portable, wearable, battery-operated, and wireless ECG systems

Cardiovascular diseases are the number one cause of death worldwide. Currently, portable battery-operated systems such as mobile phones with wireless ECG sensors have the potential to be used in continuous cardiac function assessment that can be easily integrated into daily life. These portable point-of-care diagnostic systems can therefore help unveil and treat cardiovascular diseases. The basis for ECG analysis is a robust detection of the prominent QRS complex, as well as other ECG signal characteristics. However, it is not clear from the literature which ECG analysis algorithms are suited for an implementation on a mobile device. We investigate current QRS detection algorithms based on three assessment criteria: 1) robustness to noise, 2) parameter choice, and 3) numerical efficiency, in order to target a universal fast-robust detector. Furthermore, existing QRS detection algorithms may provide an acceptable solution only on small segments of ECG signals, within a certain amplitude range, or amid particular types of arrhythmia and/or noise. These issues are discussed in the context of a comparison with the most conventional algorithms, followed by future recommendations for developing reliable QRS detection schemes suitable for implementation on battery-operated mobile devices.

/pdf/revisiting-qrs-detection-methodologies-for-portable-wearable-1zu9zzg9yx.pdf

Revisiting QRS detection methodologies for portable, wearable, battery-operated, and wireless ECG systems.

The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Pade, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Pade method which is equivalent to continued fraction expansion technique.

Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling

In this paper, a fractional digital differentiation-based algorithm for detecting R wave in QRS complex of electrocardiogram (ECG) is developed. A FIR bandpass filter, whose coefficients only depend on fractional orders, reduces various noises present in ECG signals and generates peaks corresponding to the ECG parts with high slopes. This filter is followed by nonlinear transforms and smoothing to enhance peaks corresponding to R waves. Algorithm tests on the Massachusetts Institute of Technology/Beth Israel Hospital (MIT/BIH) ECG database illustrate the capability of this novel approach to recognizing QRS complexes in very noisy ECG signals. The algorithm’s performances are comparable to those of the most efficient QRS detectors tested on this database.

R wave detection using fractional digital differentiation

The paper presents a new algorithm based on digital fractional order differentiation for P and T-waves detection and delineation. It is simple to implement and performs in one step high frequency noise filtering and differentiation. The characteristics of the resulting filter depend only on two parameters: the fractional negative order (fractional integration) and the filter order. Tests of the algorithm on ECG signals taken from the Massachusetts Institute Of Technology/Beth Israel Hospital (MIT/BIH) database prove its capability to detect and delineate P-waves and T-waves in noisy ECG as well as low amplitude P-wave and inverted T-wave.

Digital fractional order differentiation-based algorithm for P and T-waves detection and delineation

The goal of this paper is to describe some applications of fractional order calculus to biomedical signal processing with emphasis on the ability of this mathematical tool to remove noise, enhance useful information, and generate fractal signals. Three types of digital filters are considered, namely, lowpass differentiation filter, smoothing filter, and 1/fβ-noise generation filter. The filter impulse responses are functions of the fractional order and the sampling period only, and thus can be computed easily. Application examples are presented for illustrations.

Youcef Ferdi

Papers

Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling

R wave detection using fractional digital differentiation

Digital fractional order differentiation-based algorithm for P and T-waves detection and delineation

Some applications of fractional order calculus to design digital filters for biomedical signal processing

Piecewise linear correction of ECG baseline wander: a curve simplification approach