Use Of Window Functions Research Articles

Construction of a window function for estimating the parameters of sinusoidal signals with non-harmonic frequencies

Discrete Fourier Transform (DFT) allows you to determine the discrete spectrum of a signal. Due to the presence of its high-speed implementation, called Fast Fourier Transform (FFT), this transform is widely used in digital signal processing (DSP). Most DSP tasks that deal with analogy signal and spectrum adapt the DFT to find the signal spectrum between harmonics. One of the most commonly used ways of such adaptation – the use of window functions. The analysis of standard window functions (Kaiser, etc.) showed that their direct application to solving the problem of estimating the parameters (frequency, amplitude, and phase) of nonharmonic sinusoidal components of signals leads both to the need for additional corrections of the estimation results and to additional errors in determining the phase of the signal. The paper proposes a method that allows building a window function without the indicated drawbacks based on standard window functions. The essence of the method is to transform the standard window function so that its spectrum does not contain imaginary components, and the amplitude of the fundamental harmonic would be equal to 1. The results of modeling the proposed method on the example of the Kaiser window showed that the phase estimate of the nonharmonic components of the spectrum using the obtained window function, in contrast to the estimate using standard window functions, is not displaced.

Research on automatic recognition algorithm of piano music based on convolution neural network

Convolutional neural network is a typical deep neural network, which combines the advantages of deep learning and image processing, improving the accuracy of feature recognition and greatly reducing the calculation of neural network. Window function is a signal with finite width in time domain. In signal processing, the use of window function can reduce the leakage of spectrum. Combining the window function and convolution neural network, the effective input in the model can be increased, and the accuracy and speed can be improved. In this paper, the music score of piano music is processed by window function as the input image of convolutional neural network. Through the analysis of the results, it is found that the algorithm has a 25.8% increase in rate compared with the traditional recognition method in rate, and the average value of F value reaches 89.24% in the accuracy test. The test results show that the piano music recognition rate of the algorithm has been greatly improved, and good results have been achieved in accuracy.

Периодограммная оценка спектральной плотности мощности на основе бинарно-знакового стохастического квантования сигналов с использованием оконных функций

Spectral analysis of signals is used as one of the main methods for studying systems and objects of various physical natures. Under conditions of a priori statistical uncertainty, the signals are subject to random changes and noise. Spectral analysis of such signals involves the estimation of the power spectral density (PSD). One of the classical methods for estimating PSD is the periodogram method. The algorithms that implement this method in digital form are based on the discrete Fourier transform. Digital multiplication operations are mass operations in these algorithms. The use of window functions leads to an increase in the number of these operations. Multiplication operations are among the most time consuming operations. They are the dominant factor in determining the computational capabilities of an algorithm and determine its multiplicative complexity. The paper deals with the problem of reducing the multiplicative complexity of calculating the periodogram estimate of the PSD using window functions. The problem is solved based on the use of binary-sign stochastic quantization for converting a signal into digital form. This two-level signal quantization is carried out without systematic error. Based on the theory of discrete-event modeling, the result of a binary-sign stochastic quantization in time is considered as a chronological sequence of significant events determined by the change in its values. The use of a discrete-event model for the result of binary-sign stochastic quantization provided an analytical calculation of integration operations during the transition from the analog form of the periodogram estimation of the SPM to the mathematical procedures for calculating it in discrete form. These procedures became the basis for the development of a digital algorithm. The main computational operations of the algorithm are addition and subtraction arithmetic operations. Reducing the number of multiplication operations decreases the overall computational complexity of the PSD estimation. Numerical experiments were carried out to study the algorithm operation. They were carried out on the basis of simulation modeling of the discrete-event procedure of binary-sign stochastic quantization. The results of calculating the PSD estimates are presented using a number of the most famous window functions as an example. The results obtained indicate that the use of the developed algorithm allows calculating periodogram estimates of PSD with high accuracy and frequency resolution in the presence of additive white noise at a low signal-to-noise ratio. The practical implementation of the algorithm is carried out in the form of a functionally independent software module. This module can be used as a part of complex metrologically significant software for operational analysis of the frequency composition of complex signals.

SPICE Model of Current Polarity-Dependent Piecewise Linear Window Function for Memristors

Memristor and memristive systems are nonlinear systems. It is important to model them accurately. There are different memristor models and most of the models make use of window functions. In literature, there are various window functions. Recently, a piecewise linear (PWL) window function is used to model a memristor and memristive systems. Such a memristor with a PWL window function lacks a SPICE model. Also, in literature, there is current polarity dependent window functions proposed for memristors to model polarity dependent drift speed within the thin-film memristors. In this study, an alternative current-polarity dependent PWL window function is suggested to model a memristor, a different PWL function one for each current polarity is used, its SPICE model is made in LTSpice and also its simulation results are given. Such a model can be used to model the polarity dependent drift speed within the thin-film memristors.

Window functions for self-consistency evaluation of optical constants.

Optical-constant data of a material typically come from various sources, which may result in inconsistent data. Sum rules are tests to evaluate the self-consistency of optical constant data sets. Standard sum rules provide collective self-consistency evaluation of an optical-constant set in the full electromagnetic spectrum, but they give no information on the specific spectral range originating the inconsistency. Spectrally-resolved self-consistency information can be obtained with the use of window functions (WFs). Window functions can give more weight to the desired spectral range in the calculation of the sum rule. A previously developed WF was successfully used to evaluate self-consistency over the spectrum, but since it involves steep transition at the window edges and center, it has a trend to turn unstable in the calculation of sum-rule integrals for a fast decaying WF outside the window band. Two new WFs have been developed to reduce such instability. They use weight functions that smoothly cancel at the two window edges and center. The two new WFs use a weight function with three straight lines or with two 4-degree polynomials. The new WFs have been tested on exact optical constants with a coarse sampling, and they provide a strong instability reduction in self-consistency evaluation compared with the old WF. The new WFs have been also tested on experimental data sets of Al and Au reported in the literature, which unveils ranges of inconsistency. The large stability of the new WFs compared with the old one helps decide that the inconsistency calculated with the new WFs on experimental data must be attributed to inconsistency of the data sets, and not to poor sampling rate. A WF that has been used in the literature in the calculations of the dielectric function at imaginary energies for the thermal Casimir effect is also analyzed in terms of self-consistency when it is applied to sum rules involving optical constant at real (not imaginary) energies.

Effects of motion related outliers in dynamic functional connectivity using the sliding window method

BackgroundIt has been suggested that the use of window functions, other than the rectangular, in the sliding window method, may be beneficial for reducing the effects of motion-related outliers in the time-series, when assessing dynamic functional connectivity (dFC) in resting-state fMRI (rs-fMRI). MethodologyTen window functions for a wide range of window lengths (20−150 s) combined with Pearson and Kendall correlation metrics, were investigated. One hundred high quality rs-fMRI datasets from healthy controls, were used to systematically assess the effect of varying the window function and length on dFC assessment. To this end, two approaches were implemented: a) simulated outliers were added to the experimental data and b) the experimental data were divided into low and high motion subgroups. ResultsThe presence of experimental motion-noise tended to inflate the number of dynamic connections for longer (≥100 s) wide-shaped windows, while shorter (20−30 s) narrow-shaped windows exhibited increased sensitivity in the presence of simulated outliers. Moreover, window sizes from 60 s to 90 s were mildly affected by motion-related effects. In most cases, the number of dynamic connections increased, and gradually lower frequencies were captured, with an increasing window size. ConclusionsSubject motion considerably affects the obtained dFC patterns; thus, it is preferable to perform motion artefact removal in the pre-processing stage rather than using alternative window functions to mitigate their effects. Provided that motion-noise is not excessive, the choice of a rectangular window is adequate. Finally, low frequency oscillations in functional connectivity seem to play an important role in the context of dFC assessment.

On the physical definition of dynamic impedance: How to design an optimal strategy for data extraction

Dynamic electrochemical impedance spectroscopy (DEIS) has attracted the interest of researchers due to its capability of acquiring impedance spectra of non-stationary systems in a broad range of frequencies. Although developed in the 70's, a physical definition of dynamic impedance, and in general of dynamic transfer function, is not present in literature. In this manuscript the general concept of dynamic frequency response is given in correlation to the Volterra series expansion: this can be used to define the ideal dynamic impedance response of an electrochemical system. It is clarified that DEIS is the intermodulation of the ac signal with the dc signal. Starting from this, different strategies for the extraction of the dynamic impedance spectra are compared, which are based on different heuristic definitions of DEIS. There are two main methodologies, one based on the use of window functions and the other on quadrature filters. Limits and merits of the different methodologies are discussed in the frame of the data analysis and data handling and show that window functions suffers at recovering low-frequencies impedances while quadrature filters work best in experiments which have a periodic or quasi-periodic trend.

Toothwise Fault Identification for a Planetary Gearbox Based on a Health Data Map

Vibration-based fault diagnosis of a planetary gearbox is challenging due to the revolving planet gears inducing modulation of vibration signals. For accurate fault diagnosis, researchers have suggested that vibration signals should be extracted by window function when the planet gears are positioned under the sensor. However, vibration modulation characteristics can be affected by operational and geometrical uncertainties. Thus, fault-related features can be inadvertently discarded when the window function is employed. Alternatively, periodicity analysis of anomalies can be employed with the entire vibration signal without the use of window function. However, it is challenging in a planetary gearbox because the fault-related features are also modulated. This paper proposes a health data map for toothwise fault identification in a planetary gearbox. In the proposed approach, samplewise health data are aligned in the domains of a pair of gear teeth of the planetary gearbox. The proposed approach represents the health state of every pair of gear teeth, while making it possible to isolate the location of any faulty teeth in the gears. For demonstration of the proposed method, two case studies are presented: an analytical model and a 2-kW testbed. The results suggest that the proposed method performs well even under unexpected vibration modulation characteristics.

A Detection Method for Induction Motor Bar Fault Using Sidelobes Leakage Phenomenon of the Sliding Discrete Fourier Transform

The Fourier transform is widely used to diagnose induction motor faults through the monitoring of fault signatures from measured signals such as stator currents. For a good frequency resolution, Fourier transform needs a long signal acquisition time that increases the probability of speed fluctuations which, leads to fault signatures variations. In addition, limited acquisition time and acquired points generate unwanted sidelobes leakage phenomenon, caused by step frequency resolution. In signal processing, the use of window functions allows the avoidance of this phenomenon with the cost of losing a part of signal information. In this paper, the authors propose a new method for the diagnosis of induction motor broken bar fault based on sliding window discrete Fourier transform and the effect of sidelobes of sideband frequencies on the fundamental component amplitude of stator current. The main advantage of the proposed method is that one can detect the amplitude of the fault indicator frequency in vicinity of the fundamental one in shorter time and with good precision even if the motor turns at no-load when compared to used methods, as fast Fourier transform, zoom fast Fourier transform, multiple signal classification, and zoom multiple signal classification. The simulation and experimental results validate the effectiveness of the proposed method.

Super-resolution estimation of cyclic arrival rates

Exploiting the fact that most arrival processes exhibit cyclic behaviour, we propose a simple procedure for estimating the intensity of a nonhomogeneous Poisson process. The estimator is the super-resolution analogue to Shao (2010) and Shao and Lii [J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 99–122], which is a sum of $p$ sinusoids where $p$ and the amplitude and phase of each wave are not known and need to be estimated. This results in an interpretable yet flexible specification that is suitable for use in modelling as well as in high resolution simulations. Our estimation procedure sits in between classic periodogram methods and atomic/total variation norm thresholding. Through a novel use of window functions in the point process domain, our approach attains super-resolution without semidefinite programming. Under suitable conditions, finite sample guarantees can be derived for our procedure. These resolve some open questions and expand existing results in spectral estimation literature.

Radon emission rate and analysis of its influencing parameters

Abstract The geological and structural conditions define the radon situation inside a building. While the geological realities can be specified by the content of radium-226 and the ratio of radon-222 emitted from the ground the structural conditions are defined by the tightness of the building envelope. The radon concentration inside has an unsteady character, which is caused by meteorological conditions outside and the air change rate (ACH or ACR), which in turn is influenced by the residents’ behaviour such as venting and heating. For the assessment of the radon exposition, it is necessary to perform measurements for a long time. An approach to reduce this time by eliminating the inhabitants influence on the radon concentration is the radon emission rate, also known as radon entry rate. This variable is based on the measurement of the radon concentration and the parallel determination of the air change rate via a tracer gas method, the result expresses a released activity per time. Due to their noisy character, it is necessary to apply a smoothing algorithm to the input parameters. In addition to mean values, the use of window functions, known from digital signal processing, was analysed. For the verification of the whole calculation procedure, simulations and measurements under defined conditions were used. Furthermore, measurements in an uninhabited house showed proof of the capability of the assessment of the radon potential. First examinations of influencing parameters of the radon emission rate showed a possible dependence on the temperature difference inside and outside the building.

Tuning sum rules with window functions for optical constant evaluation

Sum rules are a useful tool to evaluate the global consistency of a set of optical constants. We present a procedure to spectrally tune sum rules to evaluate the local consistency of optical constants. It enables enhancing the weight of a desired spectral range within the sum-rule integral. The procedure consists in multiplying the complex refractive index with an adapted function, which is named window function. Window functions are constructed through integration of Lorentz oscillators. The asymptotic decay of these window functions enables the derivation of a multiplicity of sum rules akin to the inertial sum rule, along with one modified version of f-sum rule. This multiplicity of sum rules combined with the free selection of the photon energy range provides a double way to tune the spectral contribution within the sum rule. Window functions were applied to reported data of SrF2 and of Al films in order to check data consistency over the spectrum. The use of window functions shows that the optical constants of SrF2 are consistent in a broad spectrum. Regarding Al, some spectral ranges are seen to present a lower consistency, even though the standard sum rules with no window function did not detect inconsistencies. Hence window functions are expected to be a helpful tool to evaluate the local consistency of optical constants.

Histogram Equalization Utilizing Window-Based Smoothed CDF Estimation for Feature Compensation

In this letter, we propose a new histogram equalization method to compensate for acoustic mismatches mainly caused by corruption of additive noise and channel distortion in speech recognition. The proposed method employs an improved test cumulative distribution function (CDF) by more accurately smoothing the conventional order statistics-based test CDF with the use of window functions for robust feature compensation. Experiments on the AURORA 2 framework confirmed that the proposed method is effective in compensating speech recognition features by reducing the averaged relative error by 13.12% over the order statistics-based conventional histogram equalization method and by 58.02% over the mel-cepstral-based features for the three test sets.

Introduction of multiple nulls in otherwise omnidirectionalpattern of circular dipole array

The results are presented of a synthesis procedure for omnidirectional radiation patterns with multiple nulls for circular arrays of vertical dipoles. The gain ripple in the omnidirectional region can be controlled by the use of window functions in the design procedure. The results are verified using the numerical electromagnetics code.

BOUNDSTATES OF TWO‐ELECTRON SYSTEMS IN QUANTUM CONFINED GEOMETRIES

With the advent of sophisticated growth techniques such as Molecular Beam Epitaxy and Metal Organic Chemical Vapor Deposition, the calculation of the energy boundstates and electron wave‐functions of the one‐electron Schrödinger equation has received a lot of attention over the last decade. With the more recent fabrication of quantum wires and dots, it seems now imperative to extend the boundstates calculation to systems containing only a few electrons. Hereafter, we investigate the effect of electron exchange and Coulomb interactions on the boundstates of a two‐electron system in a square quantum well. The technique is based on a general Alternating Direction Implicit algorithm ( T. Singh and M. Cahay, SPIE Vol. 1675, Quantum Wells and Superlattice Physics IV (1992), p.11) combined with a Fourier spectrum analysis of the two‐particle wavefunction correlation , <ψ(χ1,χ2;0)/ψ(χ1,χ2;τ)> , where χ1, χ2 are the coordinates of the two electrons. The precise location of the energy eigenvalues requires the appropriate use of window functions before calculating the Fourier transform of the correlation function. We also compare our results for the boundstate energies with those obtained using a first order time‐independent perturbation theory.

A class of algorithms for identification in H/sub infinity /: continuous-time case

The problem of system identification in H/sub infinity / for the continuous-time case is investigated. It is shown that the class of systems with a lower bound on the relative stability, an upper bound on the steady-state gain, and an upper bound on the roll-off rate is admissible. This allows one to develop a class of robustly convergent nonlinear algorithms. The algorithms in this class have a two-stage structure and are characterized by the use of window functions. Explicit worst-case error bounds in H/sub infinity / norm between the identified model and the unknown system are given for a particular algorithm. An example is provided to illustrate the application of the results obtained. >

Top-resolution frequency analysis of electrocardiogram with adaptive frequency determination. Identification of late potentials in patients with coronary artery disease.

Frequency analysis of the electrocardiogram with Fourier transform is a sensitive method of detecting late potentials. However, information about localization of late potentials is lost, frequency resolution is poor, and window functions have to be applied. We therefore analyzed multiple segments (25 msec long) of the surface electrocardiogram ("spectrotemporal mapping") with adaptive frequency determination (AFD), an autoregressive algorithm that is characterized by high-frequency resolution in very short segments without the use of window functions. Results were compared with those from Fourier transform and the Simson method. We studied 38 patients after myocardial infarction (MI) with sustained ventricular tachycardia (VT), 21 patients after MI without VT, and 18 healthy subjects. Frequency peaks could be clearly differentiated until a minimal interval of 6 Hz; with fast Fourier transform (Blackman Harris window) in a much longer segment (80 msec), the spectral peaks merged into one another at an interval of about 30 Hz. AFD revealed high-frequency components as narrow peaks in the range of 40-160 Hz in 28 of 38 patients (74%) after MI with VT. Because of the short segment size, exact localization of late potentials was possible; in most of the patients, the peaks occurred in segments inside the QRS complex and ended 20 +/- 10 msec after termination of the QRS complex. In patients after MI without VT, only four of 21 patients (19%) had spectral peaks in segments after the end of the QRS complex; however, 13 of 21 patients demonstrated microvolt potentials in segments within the QRS complex. These potentials did not extend beyond the end of normal ventricular activation. Only two of 18 healthy subjects showed abnormal AFD results. Patients with bundle branch block did not need to be excluded. AFD allowed good differentiation between late potentials and noise by a characteristic pattern of the spectral peaks. For the Simson method, patients with bundle branch block had to be excluded, and overall sensitivity was 42%. In five cases, the cause of failure of the Simson method could be identified as incorrect determination of the QRS limits due to noise. Thus, AFD is a promising method for detailed analysis of late potentials; it combines the advantages of frequency analysis (good differentiation between signal and noise and high-pass filters not necessary) and time domain analysis (localization of late potentials).