Narrow Results

We apply two nonparametric methods to further test the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The term "parametric" refers here to the use of the log-periodic power law formula to fit the data; in contrast, "nonparametric" refers to the use of general tools such as Fourier transform, and in the present case the Hilbert transform and the so-called (H, q)-analysis. The analysis using the (H, q)-derivative is applied to seven time series ending with the October 1987 crash, the October 1997 correction and the April 2000 crash of the Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq indices. The Hilbert transform is applied to two detrended price time series in terms of the ln(t_c-t) variable, where t_c is the time of the crash. Taking all results together, we find strong evidence for a universal fundamental log-frequency f=1.02±0.05 corresponding to the scaling ratio λ=2.67±0.12. These values are in very good agreement with those obtained in earlier works with different parametric techniques. This note is extracted from a long unpublished report with 58 figures available at , which extensively describes the evidence we have accumulated on these seven time series, in particular by presenting all relevant details so that the reader can judge for himself or herself the validity and robustness of the results.

This study presents part of an experimental and analytical survey of candidate methods for damage detection of composite structural. Embedded piezoceramic (PZT) sensors were excited with the high power ultrasonic wave generator generating a propagation of stress wave along the composite plate. The same embedded piezoceramic (PZT) sensors are used as receivers for acquiring stress signals. The effects of center frequency of embedded sensor were evaluated for the damage identification capability with known localized defects. The study was carried out to assess damage in composite plate by fusing information from multiple sensing paths of the embedded network. It was based on the Hilbert transform, signal correlation and probabilistic searching. The obtained results show that satisfactory detection of defects could be achieved by proposed method.

The signal phase differences of Coriolis sensor and the mass flow are proportional. To improve the measurement accuracy of flow signal processing for Coriolis mass flowmeter, a novel method based on Hilbert Transform algorithm was proposed. The main method is as follows, two signals enhanced by LZ-filter with noise-canceling were considered as Hilbert Transform, then, the phase difference of the two filtered signals was calculated by a triangle characteristic of sin function. Synchronously, the instantaneous frequency was estimated by the structure of the analytical signal. Finally, the mass flow value was obtained by calculating the phase difference. The simulation experiments and Digital Signal Processing system (DSP) verification testing demonstrate that the new signal processing method, which is the LZ algorithm, has better characteristics of real-time and high-precision than the others. The experimental results show that the accuracy of phase difference measurement is 0.02% and the tracking accuracy of frequency is better than 0.07%.

Smart Grid is expected to provide a reliable power supply with fewer and briefer outages, cleaner power, and self-healing power systems through advanced Power Quality (PQ) monitoring, analysis and diagnosis of the PQ measurements and identification of the root cause, and timely automated controls. It is important to understand that signal processing has been an integral part of advancing and expanding the horizons of this PQ research significantly and the capabilities and applications of signal processing for PQ are continually evolving. This paper thus presents a survey on the proven and emerging signal applications for enhancing PQ, focusing on algorithms for estimating system modal parameters because resonant frequencies and their damping information are critical signatures in evaluating the PQ. In particular, we discuss the need for investigating time-varying and nonlinear characteristics of the modal parameters due to dynamic changes in system operating conditions and introduce promising signal processing techniques for this purpose.

Initial value problems as well as stationary solitary and periodic waves are investigated for a perturbed KdV equation including the Hilbert transform; u_t + uu_x + βu_xxx + η(ℋu_x - u_xx) = 0 (η > 0). Multi-hump stationary solitary and periodic wave solutions are numerically identified. Furthermore, the close relation between the structure of the stationary waves and the behavior of the temporal evolutions is discussed in comparison with other perturbed KdV equations with different instability and dissipation terms. The results support some general features common to this type of nonlinear evolution equations.

The aim of this study is to find evidence for repetitive global phase transitions occurring simultaneously over multiple areas of cortex during normal behavior. EEGs were recorded from multiple high-density arrays of 14–16 electrodes surgically fixed on the visual, auditory, somatomotor, and entorhinal cortices of trained cats and rabbits, and from a linear array of 64 electrodes on the scalp of human volunteers. Analytic phase relations between gamma EEG signals from multiple cortices were examined with high temporal resolution provided by the Hilbert transform. An index of synchronization was applied to intercortical pairs of signals to detect and display epochs of engagement between pairs. The measure was adapted to derive an index of global synchronization among all 4 cortices that was calculated as a t-value. Global epochs of phase stabilization ("locking") were found to involve all cortices under observation. The phase values were not clustered at zero but were in distributions about nonzero means. Episodic destabilization (decoherence) occurred aperiodically at intervals corresponding to rates in the delta range, with equal likelihood before the onset of the conditioned stimuli (CSs) and in post-stimulus test periods including performance of conditioned responses (CRs). Preferential pairwise phase stabilization was sought but not found between the sensory cortex receiving the auditory or visual CSs and the entorhinal or somatomotor cortex at times of CSs or CRs. The cospectrum from cross-correlating the global synchronization index with the global spatial ensemble average of the unfiltered EEG peaked in the delta range (1–3 Hz) near 2.5 Hz in cat and below 2 Hz in rabbit. The cospectrum of the EEG with the derivative of the analytic phase in humans peaked in the alpha range (7–12 Hz) The results indicated that macroscopic states of synchronized neural activity related to Gestalts formed during perception, that included the primary sensory and limbic areas and perhaps the entire neocortex of each cerebral hemisphere.

Timer options are barrier style options in the volatility space. A typical timer option is similar to its European vanilla counterpart, except with uncertain expiration date. The finite-maturity timer option expires either when the accumulated realized variance of the underlying asset has reached a pre-specified level or on the mandated expiration date, whichever comes earlier. The challenge in the pricing procedure is the incorporation of the barrier feature in terms of the accumulated realized variance instead of the usual knock-out feature of hitting a barrier by the underlying asset price. We construct the fast Hilbert transform algorithms for pricing finite-maturity discrete timer options under different types of stochastic volatility processes. The stochastic volatility processes nest some popular stochastic volatility models, like the Heston model and 3/2 stochastic volatility model. The barrier feature associated with the accumulated realized variance can be incorporated effectively into the fast Hilbert transform procedure with the computational convenience of avoiding the nuisance of recovering the option values in the real domain at each monitoring time instant in order to check for the expiry condition. Our numerical tests demonstrate high level of accuracy of the fast Hilbert transform algorithms. We also explore the pricing properties of the timer options with respect to various parameters, like the volatility of variance, correlation coefficient between the asset price process and instantaneous variance process, sampling frequency, and variance budget.

In this paper, we offer a network model that derives the expected counterparty risk of an arbitrary market after netting in a closed-form expression. Graph theory is used to represent market participants and their relationship among each other. We apply the powerful theory of characteristic functions (c.f.) and Hilbert transforms to determine the expected counterparty risk. The latter concept is used to express the c.f. of the random variable (r.v.) $max(Y,0)$ in terms of the c.f. of the r.v. $Y$. This paper applies this concept for the first time in mathematical finance in order to generalize results of Duffie & Zhu (2011), in several ways. The introduced network model is applied to study the features of an over-the-counter and a centrally cleared market. We also give a more general answer to the question of whether it is more advantageous for the overall counterparty risk to clear via a central counterparty or classically bilateral between the two involved counterparties.

Touchdown bearing (TDB) is one of the key elements in active magnetic bearings (AMBs). When the magnetic bearing fails or is exposed to an overload, touchdown events will occur. According to ISO 14839, three typical orbit responses (pendulum vibration, combined rub and bouncing, and full rub) can be detected in touchdown events. The magnitude of the dynamic forces between the rotor and TDBs has been considered as an indicator to identify these orbit responses. However, the contact forces may not be easy to measure with precision. The instantaneous frequency of the rotor vibration is another significant difference between them. In this paper, a performance analysis model has been established that considers the dynamic and thermal effects in touchdown events. Simulations under different working conditions have been conducted and a recommendation for safety working conditions of the AMB system has been developed. The instantaneous frequency of the rotor vibration has been computed using Hilbert transform and the frequency characteristics of the three orbit responses have been analyzed and discussed. The results reveal that the instantaneous frequency of the pendulum vibration is almost fixed. The instantaneous frequency of the combined rub and bouncing fluctuates vehemently in most cases. The instantaneous frequency of the backward whirl shows periodic regularity.

Early fault detection and diagnosis of rolling element bearing is of paramount importance in wind turbines as it contributes to around 70% of gearbox and 21%–70% of generator failure. When a rolling element bearing strikes a local fault in the inner or outer race, a shock (high frequency) is introduced, and repetitive impact occurs due to continuous rotation. Extracting the fault-sensitive repetitive impact frequency component from the measured signal containing multiple frequencies (discrete gear and shaft frequency, bearing fault frequency and high-frequency noise) is challenging. This paper presents two vibration techniques based on enhanced envelope analysis and blind deconvolution technique for bearing fault identification. The improved envelope analysis diagnosis bearing faults using the three-step process of removing gear and shaft frequency components by auto-regression model, followed by spectral kurtosis to extract fault-sensitive features and envelope analysis to identify bearing faults.

On the contrary, the enhanced blind deconvolution extracts the fault-sensitive component by finding the best inverse finite impulse response filter from the measured vibration signal by adaptively demodulating resonance bands due to repetitive impact and reducing the periodic noise component in a single step. The application of the two bearing fault diagnosis techniques and their comparative study has been demonstrated through numerical simulations and two industrial testrig bearing benchmark datasets. Investigations concluded that both methods extract the transient impulse features due to bearing fault; however, the enhanced blind deconvolution technique outperforms the envelope analysis in the case of measured vibration signal with outliers.

In this paper, a simple formula is derived for the modal damping ratio of the bridge using the correlation between the instantaneous amplitudes of the related front and rear contact responses of a two-axle test vehicle by the Hilbert transform (HT). To start, closed-form solutions were derived for the dynamic response of the damped bridge and vehicle-bridge contact responses. Next, the HT was employed to generate the instantaneous amplitudes of the two contact points. Based on their correlation, a simple formula is derived for the bridge damping ratio. Finally, the reliability of the derived formula was verified in the numerical study. It was demonstrated that the proposed formula can be successfully used to determine the first bridge damping ratio, even in the presence of rough pavement, but with the aid of random traffic.

In this paper, we propose a new complexity computation based on the complex function. This measure exploits the dispersion Lempel–Ziv complexity (DLZC) and the dispersion statistical complexity measure based on Jensen–Shannon divergence (DCJS) of the analytic signal and constructs the binary complex function, called complex number complexity (CNC). The statistical measure depicts the complex system from different angle and mines more potential information. Through simulation experiments, we prove that the proposed method is able to detect data fluctuations more sensitively and accurately. For an application, the CNC is applied to fault detection which has different fault diameters and rotational speeds. The results deliver that the CNC measure can well represent the complexity of faulty bearings and has significant differences in any two fault types. The technique is effective to characterize different fault.

The heart rate increases during inspiration and decreases during expiration; the study of this variation and the change of the second heart sound split (a change related to inspiration and expiration) can determine at what time in a cardiac cycle is the inspiration and the expiration. It would also be interesting to study the variation in systolic pulmonary artery pressure (SPAP) estimated over several cardiac cycles and to understand its evolution as its variation is related to the pulmonary valve, on the one hand, and inspiration and expiration, on the other hand. The algorithm developed based on the Hilbert transform and the energy of Shannon gives the second heart sound split. The SPAP will be estimated from spectral parameters of the second heart S2. The results show an excellent performance of the algorithm proposed to extract different information on the variation of heart rate. The results of the change in pressure and split are encouraging and promising for the use of the proposed method in a clinical context of hypertension in non-invasive pulmonary pathways, for example.

Due to low physical workout, high-calorie intake, and bad behavioral character, people were affected by cardiological disorders. Every instant, one out of four deaths are due to heart-related ailments. Hence, the early diagnosis of a heart is essential. Most of the approaches for automated classification of the heart sound need segmentation of Phonocardiograms (PCG) signal. The main aim of this study was to decline the segmentation process and to estimate the utility for accurate and detailed classification of short unsegmented PCG recording. Based on wavelet decomposition, Hilbert transform, homomorphic filtering, and power spectral density (PSD), the features had been obtained using the beginning 5 second PCG recording. The extracted features were classified using nearest neighbors with Euclidean distances for different values of $K$ by bootstrapping 50% PCG recording for training and 50% for testing over 100 iterations. The overall accuracy of 100%, 85%, 80.95%, 81.4%, and 98.13% had been achieved for five different datasets using KNN classifiers. The classification performance for analyzing the whole datasets is 90% accuracy with 93% sensitivity and 90% specificity. The classification of unsegmented PCG recording based on an efficient feature extraction is necessary. This paper presents a promising classification performance as compared with the state-of-the-art approaches in short time less complexity.

In order to calculate the continuous relative phase (CRP) between joints, the portrait method based on the joint angle and angular velocity and the Hilbert transform method based on the analytical signal have been widely used. However, there are few comparisons of these methods. Therefore, the aim of this study is to quantitatively compare these methods by calculating the CRP in the lower-limb joints of the elderly during level free walking. Eighteen elderly female adults ($76.4\pm 5.6$ year-old, $150.1\pm 2.8$cm, $54.4\pm 8.4$kg) wearing a Helen Hayes full-body marker set walked 10m on level ground at a self-selected velocity. The angles of the hip, knee, and ankle were measured. To calculate the CRP using the portrait method, the angular velocities were measured. Then, the phases between the angle and the angular velocity were calculated. To calculate the CRP using the Hilbert transform method, analytical signals were acquired. Then, the phases between the real and imaginary parts were calculated. A CRP was calculated as the difference between the phase in the proximal joint and the phase in the distal joint. To evaluate the similarity in the shape between the portrait and Hilbert transform methods, the cross-correlation was calculated. Bland–Altman plot analyses were performed to assess the agreement between these methods. For the root mean squares (RMSs) and standard deviations (SDs), a paired $t$-test and the Pearson correlation between methods were evaluated. There were similarities in the in-phase or out-of-phase features and in the RMS and SD between the methods. Additionally, a higher cross-correlation and agreement between them were found. These results indicated the similarity between the portrait and Hilbert transform methods for the calculation of the CRP. Therefore, either method can be used to evaluate joint coordination.

QRS complex detection plays an important role in electrocardiogram (ECG) automatic analysis. The accuracy and robustness of the detection algorithm greatly affect its practicability. However, the existing detection algorithms are greatly affected by ECG signal quality, and some detection algorithms cannot even work properly due to the poor signal quality. In this paper, a robust QRS complex detection algorithm is proposed based on Shannon energy envelope and Hilbert transform. The detection algorithm extracts the Shannon energy envelope of the preprocessed ECG signal, performs Hilbert transform on the envelope signal, then detects the suspected $R$-peaks on the envelope by detecting the position of zero pass and screens the real $R$-peaks by using a combination of ECG refractory period and backtracking mechanism. The proposed detection algorithm is validated using MIT-BIH Arrhythmia Database, and achieves the average detection accuracy of 99.69%, sensitivity of 99.81% and positive predictivity of 99.88%. Experimental results show that the proposed detection algorithm can still detect QRS complex correctly under complex interference, and the performance of the algorithm is hardly affected.

Formulas are derived for expressing Cauchy and Hilbert transforms of a function f in terms of Cauchy and Hilbert transforms of f(x^r). When r is an integer, this corresponds to evaluating the Cauchy transform of f(x^r) at all choices of z^1/r. Related formulas for rational r result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard Cauchy transform. These formulas are used to regularize the behavior of functions that are slowly decaying or oscillatory, in order to facilitate numerical computation and extend asymptotic results.

In this paper, we establish local existence of solutions of a variant of a system derived by Choi and Camassa [Weakly nonlinear internal waves in a two-fluids system, J. Fluid Mech.313 (1996) 83–103] to describe the propagation of an internal wave at the interface of two immiscible fluids with constant densities. We also present a numerical solver to approximate the solutions of the Cauchy problem.

This paper mainly focuses on decomposition of signals in terms of mono-component signals which are analytic with strictly increasing nonlinear phase. The properties of Blaschke basis and the approximation behavior of Blaschke basis expansions are studied. Each Blaschke product is analytic and mono-component. An explicit expression of the phase function of Blaschke product is given. The convergence results for Blaschke basis expansions show that it is suitable to approximate a signal by a linear combination of Blaschke products. Experiments are presented to illustrate the general theory.

The concepts of intrinsic mode functions and mono-components are investigated in relation to the empirical mode decomposition. Mono-components are defined to be the functions for which non-negative analytic instantaneous frequency is well defined. We show that a great variety of functions are mono-components based on which adaptive decomposition of signals are theoretically possible. We justify the role of empirical mode decomposition in signal decomposition in relation to mono-components.