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The description and evaluation of track geometry irregularities are commonly performed in the distance domain or the frequency domain, and therefore one cannot simultaneously obtain precise measure...

We present a multiscale analysis of the price dynamics of U.S. sector exchange-traded funds (ETFs). Our methodology features a multiscale noise-assisted approach, called the complementary ensemble empirical mode decomposition (CEEMD), that decomposes any financial time series into a number of intrinsic mode functions from high to low frequencies. By combining high-frequency modes or low-frequency modes, we show how to filter the financial time series and estimate conditional volatilities. The results show the different dynamics of the sector ETFs on multiple timescales. We then apply Hilbert spectral analysis to derive the instantaneous energy-frequency spectrum of each sector ETF. Using historical ETF prices, we illustrate and compare the properties of various timescales embedded in the original time series. Through the new metrics of the Hilbert power spectrum and frequency deviation, we are able to identify differences among sector ETF and with respect to SPY that were not obvious before.

Patterns in external sensory stimuli can rapidly entrain neuronally generated oscillations observed in electrophysiological data. Here, we manipulated the temporal dynamics of visual stimuli with cross-frequency coupling (CFC) characteristics to generate steady-state visual evoked potentials (SSVEPs). Although CFC plays a pivotal role in neural communication, some cases reporting CFC may be false positives due to non-sinusoidal oscillations that can generate artificially inflated coupling values. Additionally, temporal characteristics of dynamic and non-linear neural oscillations cannot be fully derived with conventional Fourier-based analyses mainly due to trade off of temporal resolution for frequency precision. In an attempt to resolve these limitations of linear analytical methods, Holo-Hilbert Spectral Analysis (HHSA) was investigated as a potential approach for examination of non-linear and non-stationary CFC dynamics in this study. Results from both simulation and SSVEPs demonstrated that temporal dynamic and non-linear CFC features can be revealed with HHSA. Specifically, the results of simulation showed that the HHSA is less affected by the non-sinusoidal oscillation and showed possible cross frequency interactions embedded in the simulation without any a priori assumptions. In the SSVEPs, we found that the time-varying cross-frequency interaction and the bidirectional coupling between delta and alpha/beta bands can be observed using HHSA, confirming dynamic physiological signatures of neural entrainment related to cross-frequency coupling. These findings not only validate the efficacy of the HHSA in revealing the natural characteristics of signals, but also shed new light on further applications in analysis of brain electrophysiological data with the aim of understanding the functional roles of neuronal oscillation in various cognitive functions.

Purpose The comfort of lift passengers has a significant effect on their general health condition as well as stress levels during travel. This study reports the results of vibration measurements taken during travel in a passenger lift. Methods Vibration signals were analyzed by the empirical mode decomposition method and the Hilbert transform. Results Selected modes from the Hilbert spectral analysis were compared with the resonance frequencies of human body organs (range 20–90 Hz) as well as with the resonance frequencies of lift components. Conclusion The use of Hilbert spectral analysis enables the isolation of individual signal components and the determination of the dominant frequency in the signal. This, in turn, allows for the isolation of raw vibration frequencies from the signal that are particularly significant for passenger comfort assessment (resonance frequencies of human body organs) and analysis of their occurrence.

A modified masking empirical mode decomposition (EMD) for effectively improving the mode separation capability of standard EMD while analysing power system signals with closely spaced modes of frequency is proposed here. A multi-frequency test signal resembling the dynamic power system signal and the actual data recorded by wide area measurement systems (WAMS) from Northern Grid, India, on 1 June 2010 due to generation loss are analysed by both the existing method and the proposed method. Various components of modal frequency extracted by masking signal base EMD are compared with extracted components by the proposed modified masking EMD. Further, the time–frequency representation of extracted modes by Hilbert spectral analysis is implemented to know mode behaviour with time reference. Simulation results prove that the EMD with proposed modification is capable of separating various modes of frequencies present in the WAMS signal.

In this study, multiple analytical mode decompositions (M-AMD) are proposed to identify the parameters of nonlinear oscillators from free vibration. The time-varying damping (stiffness) coefficient of an oscillator is divided into slow- and fast-varying components by a bisecting frequency in reference to velocity (displacement). The slow-varying damping and stiffness components are estimated from the oscillator’s responses and their Hilbert transforms, and filtered with the adaptive low-pass filter AMD. Each fast-varying component is estimated from the oscillator’s responses and the determined slow-varying components, and corrected with AMD using an approximate bisecting frequency of the two fast-varying components. The computational efficiency and accuracy of the proposed M-AMD are illustrated with three characteristic oscillators described by Duffing, Bouc-Wen, and spherical bearing models. The errors in estimation of all model parameters of the representative nonlinear oscillators are less than 3% from uncontaminated displacement responses and 9% when the root-mean-square value of displacement noises correspond to 0.05% of the peak displacement. In the case of Duffing oscillator, the error of the proposed method is less than 1.2% unless the ratio between the nonlinear and linear terms of the restoring force (degree of nonlinearity) exceeds 25. In the case of spherical bearing, the error of the proposed method is less than 1.2% when the coefficient of friction is less than 0.017. The instantaneous stiffness determined from Hilbert spectral analysis differs from the system stiffness by amount that rapidly increases with the degree of nonlinearity.

We present the method of complementary ensemble empirical mode decomposition (CEEMD) and Hilbert-Huang transform (HHT) for analyzing nonstationary financial time series. This noise-assisted approach decomposes any time series into a number of intrinsic mode functions, along with the corresponding instantaneous amplitudes and instantaneous frequencies. Different combinations of modes allow us to reconstruct the time series using components of different timescales. We then apply Hilbert spectral analysis to define and compute the associated instantaneous energy-frequency spectrum to illustrate the properties of various timescales embedded in the original time series. Using HHT, we generate a collection of new features and integrate them into machine learning models, such as regression tree ensemble, support vector machine (SVM), and long short-term memory (LSTM) neural network. Using empirical financial data, we compare several HHT-enhanced machine learning models in terms of forecasting performance., 28 pages, 10 figures. arXiv admin note: text overlap with arXiv:2105.08133

We study the price dynamics of cryptocurrencies using adaptive complementary ensemble empirical mode decomposition (ACE-EMD) and Hilbert spectral analysis. This is a multiscale noise-assisted approach that decomposes any time series into a number of intrinsic mode functions, along with the corresponding instantaneous amplitudes and instantaneous frequencies. The decomposition is adaptive to the time-varying volatility of each cryptocurrency price evolution. Different combinations of modes allow us to reconstruct the time series using components of different timescales. We then apply Hilbert spectral analysis to define and compute the instantaneous energy-frequency spectrum of each cryptocurrency to illustrate the properties of various timescales embedded in the original time series., 20 pages, 8 figures

The electric power systems together with the entire energy sector are rapidly evolving towards a low-carbon, secure, and competitive economy facing revolutionary transformations from technical structure to economic value chain. Pathways to achieve sustainability led to the development of new technologies, accommodation of larger shares of unpredictable and stochastic electricity transfer from sources to end-users without loss of reliability, new business models and services, data management, and so on. The new technologies and incentives for local energy communities along with large development of microgrids are main forces driving the evolution of the low voltage energy sector changing the context and paradigm of rigid contractual binding between utilities and end-user customers (now progressing to flexible prosumers with generation and storage capabilities). The flexibility and operation of a prosumer can be enhanced by a non-intrusive time-frequency analysis of distorted power quality waveforms for both generation and demand at the point of common connection. Therefore, it becomes of importance to discriminate among successive quasi-steady-state operation of a given local system using only the aggregated waveforms information available in the PCC. This paper focuses on the Hilbert–Huang method with modifications such as empirical mode decomposition improved with masking signals based on the Fast Fourier Transform, Hilbert spectral analysis, and a post-processing method for separating components and their amplitudes and frequencies within distorted power signals for a low-voltage prosumer operation. The method is used for a time-frequency-magnitude representation with promising localization capabilities enabling efficient operation for prosumers.

Frequency attributes embedded in nonlinear geophysical signals such as well-log data or seismic data can provide vital clues in the effective characterization of the subsurface reservoir. Therefore, it is always important to recognize suitable mathematical techniques that need to be employed to extract such vital information from the nonlinear geophysical data. Several nonlinear mathematical techniques include wavelet analysis, fractal and multifractal analyses, and the fully data-adaptive Hilbert-Huang transform (HHT) technique. In the present study, we employ the HHT and the wavelet analysis techniques to seismic data of the Dutch sector of southern North Sea. We use a small portion of the seismic data, consisting of inline section 133 and time slice section 520 ms of the F3 block, to extract the instantaneous frequency associated with the subsurface gas zone. HHT constitutes two independent techniques, namely, the empirical mode decomposition (EMD) technique and the Hilbert spectral analysis (HSA). While the EMD technique facilitates to decompose the data into different signals of varied frequencies, called intrinsic mode functions (IMFs), the HSA helps to determine the instantaneous amplitudes and frequencies of the IMFs. EMD technique, together with HSA, has delineated a low-frequency spectral signature, believed to be a subsurface gas zone in the chosen data, corroborating the results of earlier studies of the same data sets. The instantaneous frequency range of the gas zone estimated by HSA was validated with the two-dimensional continuous wavelet transform (CWT) of the data sets using a suitable frequency range of Morlet wavelet. Results of the combined HHT and CWT analyses confirmed that the subsurface gas zone could be well identified in the frequency range 14–18 Hz. Results have direct implications in situ, in the sense that, by performing CWT analysis of seismic data using a Morlet wavelet in the frequency range 14–18 Hz, one can easily identify the location of subsurface gas zones.

The arbitrary order Hilbert spectral analyses are applied to study the intermittency and multifractality of Global Horizonal Irradiation (GHI) based on one available high sampling rate of 1-year GHI records located at Saint-Denis (Moufia) over Reunion Island. The scaling exponents $$\xi \left( q \right)$$ ξ q is estimated through the arbitrary order Hilbert spectral analyses, and three parameters: Hurst exponent (H), the fractal co-dimension (C1), and Lévy parameter ($$\alpha$$ α ) are taken to study the multifractal process of the GHI in the sub-daily, daily fluctuations and also in seasonal variations. A power law behaviour with a spectral exponent β = 1.68 close to the Kolmogorov spectrum is detected through Fourier spectrum analysis, which indicates that the sub-daily fluctuations of GHI are nonstationary. The scaling exponent ζ(q) is then estimated by the arbitrary order Hilbert spectral analysis and the multifractal properties is detected. The log-stable model parameters $$H,{ }C_{1}$$ H , C 1 and $$\alpha$$ α characterize the concavity of the scaling exponent ζ(q) for analysing the intermittency of GHI. The classification method is applied to the daily GHI for analysing the distribution of the daily intermittency process and five classes with GHI and Kb are obtained.

This paper presents the application of the Hilbert-Huang transform as a tool for signal analysis. The Hilbert-Huang transform (HHT) enables the analysis of nonstationary and nonlinear data being a combination of the Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis (HSA). A noise assisted data analysis method, called Ensemble Empirical Mode Decomposition (EEMD), based on the statistical properties of white noise, was used to overcome EMD problems like mode mixing, and successfully deal with the scale separation. The EEMD was applied to each velocity signal to obtain a collection of Intrinsic Mode Functions (IMF). Vortex shedding in turbulent flows past circular cylinders represents a canonical problem suitable for validating new approaches in experimental analysis, as it has well known features. Experimental data in form of time series from hot wire signals of the turbulent flow past circular cylinders in an aerodynamic channel are employed for the analysis. Cylinder had a diameter of 32 mm. The Reynolds numbers varied from 1.05 × 104 to 3.28 × 104. EEMD provided the localization of particular events in time-frequency space, allowing a complete physical interpretation of the dynamic processes at the corresponding flow scales. The Hilbert Spectral analysis provided an accurate representation of the amplitude-frequency-time distribution of the flow. The most energetic IMFs and the corresponding flow scales were identified from the Mean Square Energy of the IMF components. The application of HHT to the analysis of the hot wire signals obtained in the turbulent wake behind cylinders showed that it is a useful tool for the analysis of turbulent data, allowing the identification of turbulent structures and flow scales.

Motion capture data are widely used in different research fields such as medical, entertainment, and industry. However, most motion researches using motion capture data are carried out in the time-domain. To understand human motion complexities, it is necessary to analyze motion data in the frequency-domain. In this paper, to analyze human motions, we present a framework to transform motions into the instantaneous frequency-domain using the Hilbert-Huang transform (HHT). The empirical mode decomposition (EMD) that is a part of HHT decomposes nonstationary and nonlinear signals captured from the real-world experiments into pseudo monochromatic signals, so-called intrinsic mode function (IMF). Our research reveals that the multivariate EMD can decompose complicated human motions into a finite number of nonlinear modes (IMFs) corresponding to distinct motion primitives. Analyzing these decomposed motions in Hilbert spectrum, motion characteristics can be extracted and visualized in instantaneous frequency-domain. For example, we apply our framework to (1) a jump motion, (2) a foot-injured gait, and (3) a golf swing motion.

In this paper, a novel pre-treatment technique Hilbert Huang Transformation with filtering (HHTF) that is coupling of the Hilbert Huang Transformation and the digital filtering is proposed for the measurement of glucose from near infrared spectroscopy. HHTF comprises of the Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis. In Hilbert spectral analysis, Butterworth filtering was used to eliminate the noise present in the Intrinsic Mode Functions (IMFs). The traditional Partial Least squares Regression (PLSR) has been used as the regression method. The proposed HHTF with the PLSR method has been assessed to determine the concentration of glucose from near infrared spectra of two distinct compositions that are prepared by mixing triacetin, urea and glucose in a phosphate buffer solution (PBS) and another composition of glucose and human serum albumin in a PBS. The efficiency of the proposed method has been compared with the standard normal variate and the 1st derivative preprocessing methods and is shown to outperform both.

Despite growing evidence of aberrant neuronal complexity in Alzheimer’s disease (AD), it remains unclear how this variation arises. Neural oscillations reportedly comprise different functions depending on their own properties. Therefore, in this study, we investigated details of the complexity of neural oscillations by decomposing the oscillations into frequency, amplitude, and phase for AD patients. We applied resting-state magnetoencephalography (MEG) to 17 AD patients and 21 healthy control subjects. We first decomposed the source time series of the MEG signal into five intrinsic mode functions using ensemble empirical mode decomposition. We then analyzed the temporal complexities of these time series using multiscale entropy. Results demonstrated that AD patients had lower complexity on short time scales and higher complexity on long time scales in the alpha band in temporal regions of the brain. We evaluated the alpha band complexity further by decomposing it into amplitude and phase using Hilbert spectral analysis. Consequently, we found lower amplitude complexity and higher phase complexity in AD patients. Correlation analyses between spectral complexity and decomposed complexities revealed scale-dependency. Specifically, amplitude complexity was positively correlated with spectral complexity on short time scales, whereas phase complexity was positively correlated with spectral complexity on long time scales. Regarding the relevance of cognitive function to the complexity measures, the phase complexity on the long time scale was found to be correlated significantly with the Mini-Mental State Examination score. Additionally, we examined the diagnostic utility of the complexity characteristics using machine learning (ML) methods. We prepared a feature pool using multiple sparse autoencoders (SAEs), chose some discriminating features, and applied them to a support vector machine (SVM). Compared to the simple SVM and the SVM after feature selection (FS + SVM), the SVM with multiple SAEs (SAE + FS + SVM) had improved diagnostic accuracy. Through this study, we 1) advanced the understanding of neuronal complexity in AD patients using decomposed temporal complexity analysis and 2) demonstrated the effectiveness of combining ML methods with information about signal complexity for the diagnosis of AD.

The fractional Hilbert transform (FHT) is a linear combination of a signal and its Hilbert transform, and invites extensions of the classical analytic signal. Mathematical properties of the FHT and three possible versions of fractional analyticity are explored in detail.

Nowadays, decomposition of multi-component signals has gained popularity in time-frequency analysis (TFA) of non-stationary signals. Eigenvalue decomposition (EVD) is one such technique which decomposes signals into mono-components. In this paper, a new approach named sliding EVD for non-stationary signal decomposition has been proposed. The sliding EVD comprises short duration EVD of signals and an unsupervised grouping of obtained components. This proposed algorithm surpasses other EVD based techniques by successfully decomposing the signals which are overlapped in frequency domain and separated in time-frequency domain. Later, Hilbert spectral analysis has been used on decomposed mono-components for obtaining time-frequency distribution (TFD). At the end, proposed method has been compared with Hilbert Huang transform and is found to be providing better TFD.

The statistical properties of fast Alfv&eacute, nic solar wind turbulence have been analyzed by means of empirical mode decomposition and the associated Hilbert spectral analysis. The stringent criteria employed for the data selection in the Wind spacecraft database, has made possible to sample multiple k‖ field-aligned intervals of the three magnetic field components. The results suggest that the spectral anisotropy predicted by the critical balance theory is not observed in the selected database, whereas a Kolmogorov-like scaling (E(k‖)&sim, k&minus, 5/3) and a weak or absent level of intermittency are robust characteristics of the Alfv&eacute, nic slab component of solar wind turbulence.

We discuss the method of complementary ensemble empirical mode decomposition (CEEMD) for analyzing nonstationary financial time series. This noise-assisted approach decomposes any time series into a number of intrinsic mode functions, along with the corresponding instantaneous amplitudes and instantaneous frequencies. Different combinations of modes allows us to reconstruct the time series based on different timescales. Using Hilbert spectral analysis, we compute the associated instantaneous energy-frequency spectrum to illustrate and interpret the properties of various timescales embedded in the original time series.

In this study, we applied the Holo-Hilbert spectral analysis (HHSA) method to analyze HRV in electrocardiography (ECG) signals. The HHSA intends to interpret the measured interbeat-interval (IBI) signal as the multiplication of an oscillatory signal (carrier wave) with a modulatory signal (envelope wave). Instantaneous frequency analysis (IFA) was adopted to calculate the frequency modulation (FM) and amplitude modulation (AM) frequencies at each time point. The probability of FM-AM distribution across the whole measured IBI signal was accumulated and displayed on a Holo-Hilbert spectrum (HHS). Twenty healthy subjects were recruited in our ECG study. Each subject was requested to view two video clips, including one relaxing video clip and one horror video clip. We observed the very high-frequency (VHF) band was modulated by a ∼0.05 Hz AM frequency on the calculated HHS. The value of VHF power was significantly decreased (p

Three-dimensional computations have been carried out to investigate flow past a sphere performing rotation and rotary oscillation about streamwise axis. The effect of streamwise rotation has been analysed for fixed Reynolds number of 300, with non-dimensional rotational rate taking values 0, 0.5, 1, 1.5 and 2. The transitions in the near wake have been depicted with the help of time-averaged path lines, iso- λ 2 surfaces and vortex core transformation. This study is further extended by considering forced rotational oscillations of the sphere for two different amplitudes of oscillation and reduced oscillation frequency ranging from 0.5 to 2. Effect of these parameters on flow structure has been described using λ 2 -surfaces and streamlines at different locations in the flow field. The response of mean hydrodynamic forces to variation in amplitude and frequency of oscillation has also been reported. Temporal behaviour of wake has been disclosed through Hilbert spectral analysis of time dependent force coefficient signals. The growth and evolution of wake nonlinearities have been elaborated with the help of marginal spectra, Hilbert spectra and distribution of non-stationarity.

In this study, multiple analytical mode decompositions (M-AMD) are proposed to identify the parameters of nonlinear structures from forced vibration. For the time-varying damping (or stiffness) coefficient of a weakly-to-moderately nonlinear system, the slow-varying part is first estimated from the system responses and their Hilbert transforms, which is corrected with an adaptive low-pass filter referred to as analytical mode decomposition (AMD). The fast-varying part can then be identified from the responses together with the estimated slow-varying part, which is again corrected with the AMD. The computational efficiency and accuracy of the proposed M-AMD are demonstrated with a Duffing oscillator subjected to harmonic loading. The errors in estimation of all model parameters are less than 3% from uncontaminated displacement responses, which is more accurate compared with the results from Hilbert spectral analysis. Changes of the fast-varying stiffness part have been taken into account with high accuracy. The M-AMD algorithm is then validated with a ¼-scale, 3-story building with one piezoelectric friction damper under earthquake excitations. The parameters of such a semi-active damper are identified with less than 1% error on average.

Neural oscillatory activities existing in multiple fre-quency bands usually represent different levels of neurophysiolog-ical meanings, from micro-scale to macro-scale organizations. In this study, we adopted Holo-Hilbert spectral analysis (HHSA) to study the amplitude-modulated (AM) and frequency-modulated (FM) components in sensorimotor Mu rhythm, induced by slow- and fast-rate repetitive movements. The HHSA-based approach is a two-layer empirical mode decomposition (EMD) architecture, which firstly decomposes the EEG signal into a series of frequency-modulated intrinsic mode functions (IMF) and then decomposes each frequency-modulated IMF into a set of amplitude-modulated IMFs. With the HHSA, the FM and AM components were incor-porated with their instantaneous power to achieve full-informa-tional spectral analysis. We observed that the instantaneous power induced by slow-rate movements was significantly higher than that induced by fast-rate movements (p0.01, Wilcoxon signed rank test). The alpha-band AM frequencies induced by slow-rate movements were higher than those induced by fast-rate move-ments, while no statistical difference was found in beta-band AM frequencies. In addition, to study the functional coupling between the primary sensorimotor area and other brain regions, spectral coherence was applied and statistical difference was found in frontal area in slow-rate versus fast-rate movements. The discrep-ancy between slow- and fast-rate movements might be owing to the change of motor functional modes from default mode network (DMN) to automatic timing with the increase of movement rates. The use of HHSA for oscillatory activity analysis can be an effi-cient tool to provide informative interaction among different fre-quency bands.

This study is concerned with the intrinsic temporal scales of the variability in the surface solar irradiance (SSI). The data consist of decennial time series of daily means of the SSI obtained from high-quality measurements of the broadband solar radiation impinging on a horizontal plane at ground level, issued from different Baseline Surface Radiation Network (BSRN) ground stations around the world. First, embedded oscillations sorted in terms of increasing timescales of the data are extracted by empirical mode decomposition (EMD). Next, Hilbert spectral analysis is applied to obtain an amplitude-modulation–frequency-modulation (AM–FM) representation of the data. The time-varying nature of the characteristic timescales of variability, along with the variations in the signal intensity, are thus revealed. A novel, adaptive null hypothesis based on the general statistical characteristics of noise is employed in order to discriminate between the different features of the data, those that have a deterministic origin and those being realizations of various stochastic processes. The data have a significant spectral peak corresponding to the yearly variability cycle and feature quasi-stochastic high-frequency variability components, irrespective of the geographical location or of the local climate. Moreover, the amplitude of this latter feature is shown to be modulated by variations in the yearly cycle, which is indicative of nonlinear multiplicative cross-scale couplings. The study has possible implications on the modeling and the forecast of the surface solar radiation, by clearly discriminating the deterministic from the quasi-stochastic character of the data, at different local timescales.

In this paper, a hybrid method is proposed for noise reduction in MEMS gyro signal. To ensure rapid response rate and small steady-state error, and by simultaneously considering the motion state complexity of noisy signal especially under dynamic state, denoising scheme is well-designed, which can be divided into three steps: distinguishing different IMFs modes, determining current motion state, and selecting proper denoising method. Two carefully selected indexes divide the IMFs into three parts, noisy IMFs, mixed IMFs and information IMFs, with the mixed IMFs needed further processing. Sample variances based on AMA are used to determine current motion state. Accordingly, soft interval thresholding, soft thresholding, or forward-backward linear prediction is selected to reduce noise components contained in the mixed IMFs. Denoised mixed IMFs and information IMFs constitute final denoised signal. Practical MEMS gyro signal under different motion conditions are employed to validate the effectiveness of the proposed method. Hilbert spectral analysis and Allan variance further verify the proposed method from qualitative and quantitative point of view. Besides, computational time complexity is also analyzed.

Proper analysis of bearing faults is a necessity as it increases their working life and helps us in predicting their modes of failure beforehand. In the current study, paper deal with collecting and analyzing the vibration data of different faults of a journal bearing and comparison of the results thus obtained with a healthy bearing. To realize this objective, the vibration data for each fault condition as well as the healthy bearing is collected from the experimental setup. Journal bearing used for this study are made of gun metal. The signal obtained is split into Intrinsic Mode Functions (IMFs) by Empirical Mode Decomposition (EMD) method. Instantaneous frequency is calculated to meet the mono-component requirement. This is followed by the Hilbert Spectral Analysis from which the peaks are obtained for instantaneous frequency. Various faults can be determined by comparing these peaks with the known data. Although, Hilbert Huang Transform is an empirical method, it is more adaptive and can be used for the feature extraction of discrete and continuous data. As this method is based on the localized time scale of the data, it is applicable to both non-linear and non-stationary processes.

This paper first discusses the limitation that the intrinsic mode functions (IMFs) decomposed by the empirical mode decomposition (EMD) in Hilbert-Huang transform (HHT) are not orthogonal. As an improvement to the HHT method, three orthogonal techniques (the forward, backward and arbitrary sequence orthogonalization algorithms) based on the Gram-Schmidt method are then proposed to obtain the completely orthogonal IMFs. According to the orthogonal index and the energy index, the effectiveness of the proposed technique and algorithms is validated through a synthetic signal generated by the combination of three sinusoidal waves with different frequencies and the El Centro (1940, N-S) earthquake accelerogram. By taking the El Centro (1940, N-S) earthquake accelerogram as an example, the problem that whether the orthogonal IMFs satisfy the requirements of IMF is discussed, then the backward and the arbitrary sequence orthogonalization algorithms are recommended. Three historic earthquake accelerograms are analyzed by using the recommended orthogonalization algorithms combined with the Hilbert spectral analysis. The results show that the orthogonal Hilbert spectrum and the orthogonal Hilbert marginal spectrum can produce more faithful representation of earthquake accelerograms than the Hilbert spectrum and the Hilbert marginal spectrum, and they can be used to quantitatively characterize the energy distribution of earthquake accelerograms at different frequency regions.