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The commonly used principle for measuring the depth of anesthesia involves changes in the frequency components of the electroencephalogram (EEG) under general anesthesia. Therefore, it is essential to construct an effective spectrum and spectrogram to analyze the relationship between the depth of anesthesia and the EEG frequency during general anesthesia. This paper reviews the computer programming techniques for analyzing the spectrum and spectrogram derived from a single-channel EEG recorded during general anesthesia. A periodogram is obtained by repeating a Fourier transform on EEG segments separated by short time intervals, but spectral leakage (i.e., dissociation from the true spectrum) occurs as a consequence of unnatural segmentation and noise. While offsetting the securing of the dynamic range, practical analyses of the adaptation of the window function are explained. Finally, the multitaper method, which can suppress artifacts caused by the edges of the analysis segments, suppress noise, and probabilistically infer values that are close to the real power spectral density, is explained using practical examples of the analysis. All analyses were performed and all graphs plotted using Python under Jupyter Notebook. The analyses demonstrated the effectiveness of Python-based programming under the integrated development environment Jupyter Notebook for constructing an effective spectrum and spectrogram for analyzing the relationship between the depth of anesthesia and EEG frequency analysis in general anesthesia.

In classical information processing, the windowed Fourier transform (WFT), or short-time Fourier transform, which is a variant of the Fourier transform by dividing a longer time signal into shorter segments of equal length and then computing the Fourier transform separately on each shorter segment, is proposed to provide a method of signal processing. Up to now the discrete Fourier transform has been successfully applied to the field of quantum information, but the related short-time discrete Fourier transform of this field has not been developed accordingly. To address this problem, we first introduce the concept of quantum window state, and further prove that the quantum Fourier transform of a quantum window state is also a quantum window state. Based on the definition of the quantum window state the local information of a quantum signal is extracted and the corresponding quantum circuits are also given. And then, by applying the quantum Fourier transform to the windowed quantum superposition states, we propose a novel concept called quantum windowed Fourier transform (QWFT). Finally, an application of quantum signal processing is given where QWFT is employed.

Single-window discrete Gabor transform with an adaptive window width selection algorithm (AWWS-DGT) and multi-window discrete Gabor transform (M-DGT) are the current mainstream analysis and processing methods for multi-component signals. However, the time–frequency accuracy in the time–frequency plane obtained by the two algorithms is not very high, and not all components of the signals that contain complex and very different time–frequency components can be displayed accurately. The linear combination window algorithm based on the $$\ell_{1}$$ and $$\ell_{2}$$ norm constraints of the transform coefficients can accurately display all the components of the signal, but the algorithm has poor anti-noise performance. In this paper, an algorithm for finding the best linear combination window under the sparse solution constraint of discrete Gabor transform coefficients is proposed. First, the optimal window number adaptive selection method proposed in this paper is used to select a set of window functions with different window widths, and the goal is to obtain the sparse solution of the transform coefficient. Through the gradient descent method, the best linear combination coefficient is obtained. Second, according to the best linear combination coefficient obtained, the selected window is combined into an analysis window function. Finally, the influence of the parameters on the combined window is further studied, and the results show that the number of windows has a great influence on the result and that the influence of the regularization parameter is negligible. Experiments on two simulated non-stationary signals and real electrocardiogram signals show that the proposed algorithm can display all components of the signals, the time–frequency accuracy is improved by 3% to 10%, and the anti-noise performance is more than 20% higher than that of the existing linear combination window algorithm. These advantages are very important for the analysis and processing of practical signals.

A method of personality recognition by echographic parameters of the human ear is developed based on the naive Bayes classifier in the two following modes: the biometric identification (EER = 0.0053) and the biometric authentication (FRR = 0.0002 at FAR ≤ 0.0001), respectively. A device is developed for recording the biometric characteristics of the external ear, and a set of echographic data is collected from the external ears of 75 subjects. The spectral and cepstral characteristics of the signals reflected from the ear canal are used as biometric parameters. Several window functions for constructing spectra and cepstrograms are considered. It is established that more than 90% of “cepstral” features have a weak correlation, which allows us to use the naive Bayesian classifier and to obtain highly accurate results of user recognition at the same time. An advantage of the Bayesian classification is the possibility of the robust fast learning of the identification system.

The temporal resolution of conventional S transform (ST) is not sufficient for the separation of local coherent noise. We present a revised S transform (RST) which uses an analyzing window function with two control parameters of the scalar σ and the exponential factor γ. Selecting proper parameter values (say σ = 1.1, γ = 1.08), the time–frequency representation (TFR) acquired by our method exhibits a higher temporal resolution. Applying an appropriate filter in the time–frequency domain, we are able to remove specific local noise. Distributed acoustic sensing (DAS) VSP section may suffer from fiber cable coupling noise, hindering the subsequent data processing and geologic interpretation. The real data example shows the coupling noise occurred in the DAS VSP can be removed by the presented RST.

An explicit expression for the Kohn-Nirenberg symbol of a Weyl-Heisenberg frame operator on $L^{2}(\mathbb {R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary Fourier analysis, is independent of the theory of distributions and works strictly inside $L^{2}(\mathbb {R})$ . Kohn-Nirenberg operators are introduced and are shown to be Weyl-Heisenberg frame operators in suitable cases.

The low-pass filtering effect of the Earth results in the absorption and attenuation of the high-frequency components of seismic signals by the stratum during propagation. Hence, seismic data have low resolution. Considering the limitations of traditional high-frequency compensation methods, this paper presents a new method based on adaptive generalized S transform. This method is based on the study of frequency spectrum attenuation law of seismic signals, and the Gauss window function of adaptive generalized S transform is used to fit the attenuation trend of seismic signals to seek the optimal Gauss window function. The amplitude spectrum compensation function constructed using the optimal Gauss window function is used to modify the time-frequency spectrum of the adaptive generalized S transform of seismic signals and reconstruct seismic signals to compensate for high-frequency attenuation. Practical data processing results show that the method can compensate for the high-frequency components that are absorbed and attenuated by the stratum, thereby effectively improving the resolution and quality of seismic data.

Sensing performance of fiber Bragg grating in presence of various apodization functions are studied and compared. Using couple modes theory and matching the fields at various boundaries, the equation of reflectivity of fiber Bragg grating sensors are obtained. The sensors are optimized to get maximum reflectivity with narrow full width half maximum of the resonance peak. It is observed that all considered apodization functions suppress the sidelobes. The suppression of dominating sidelobe is maximum −141.29 dB for Blackman apodization function in all considered cases. The larger sensitivity 6.07 AU/RIU and 6.06 AU/RIU is obtained in Bessel apodization function and Barthann apodization function, respectively. Since, Bessel apodization function shows lower (0.1500 nm) fullwidth half maximum value in comparison to the Barthann apodization function, therefore it gives maximum detection accuracy10329 and quality parameter 40.47 AU/nm-RIU. Followed by Bessel apodization function the Gaussian apodization function also shows high detection accuracy10030 and quality parameter 38.18 AU/nm-RIU due to its lower (0.1545) fullwidth half maximum value. Here the considered Bessel apodization function is recommended for sensing applications.

Reservoir prediction is often a primary objective in seismic exploration, especially for deep hydrocarbon detection with its potential for assessing oil and gas resources. Time–frequency analysis has become a successful method for detecting subsurface features and can also be used to identify potential hydrocarbon reservoirs. However, current applications for hydrocarbon detection often occur at low resolution, due to the influence of the windowing functions and lack of prior constraints, thereby affecting gas prediction for thin reservoirs. To rectify this issue, we investigate the use of sparse inverse spectral decomposition (SISD) based on the wavelet transform, which adopts an lp norm to constrain the time–frequency spectra, and thereby provides a more highly concentrated time–frequency representation than conventional spectral decomposition techniques. The main objective of this paper is to investigate the performance of spectral attributes derived from lp-norm constrained SISD and its application to the characterization of deep-marine dolomite reservoirs in southwest China. The gas-bearing zones in target reservoir are predicted well by extracting and analyzing the spectral amplitude responses of different frequency components. The predicted favorable gas-bearing areas are closely related to local structures in the target reservoir, which are also in good agreement with gas-testing results at three well locations and may be used to guide subsequent exploration well development in this region.

The finite difference scheme is now widely used in the reverse time migration and full waveform inversion. Their results are dependent on the accuracy of finite difference operators. In this paper, we combine the cosine function with the original window function to construct a new window function, in order to obtain higher precision finite difference operators. The absolute error curves of the optimized finite difference operators are close to zero for low wavenumbers. In other words, we do not observe an oscillating curve of absolute errors produced by other optimized methods. In order to overcome the limitations of a single graphics processing unit (GPU), we developed the multiple-GPU method for the elastic wave equation. Numerical experimental results show that our new window function can control the numerical dispersion better than the binomial window and scaled binomial window, and the multiple-GPU computation is very stable.

Metameric problems are variable-length optimization problems whose representations take on an at least partially segmented structure. This is referred to as a metameric representation. Frequently, each of these segments defines one of a number of analogous components in the solution. Examples include the nodes in a coverage network or turbines in a wind farm. Locating optimal solutions requires, in part, determining the optimal number of components. Evolutionary algorithms can be applied but require modifications to the traditional fixed-length operators. This study proposes a new selection operator for metameric problems: length niching selection. First, the population is partitioned into several niches based on solution length. A window function determines at which lengths a niche is formed. Local selection is then applied within each niche independently, resulting in a new parent population formed by a diverse set of solution lengths. A coverage and a wind farm problem are used to demonstrate the effectiveness of the new operator.

Fracture-cave reservoirs in carbonate rocks are characterized by a large difference in fracture and cavity size, and a sharp variation in lithology and velocity, thereby resulting in complex diffraction responses. Some small-scale fractures and caves cause weak diffraction energy and would be obscured by the continuous reflection layer in the imaging section, thereby making them difficult to identify. This paper develops a diffraction wave imaging method in the dip domain, which can improve the resolution of small-scale diffractors in the imaging section. Common imaging gathers (CIGs) in the dip domain are extracted by Gaussian beam migration. In accordance with the geometric differences of the diffraction being quasilinear and the reflection being quasiparabolic in the dip-domain CIGs, we use slope analysis technique to filter waves and use Hanning window function to improve the diffraction wave separation level. The diffraction dip-domain CIGs are stacked horizontally to obtain diffraction imaging results. Wavefield separation analysis and numerical modeling results show that the slope analysis method, together with Hanning window filtering, can better suppress noise to obtain the diffraction dip-domain CIGs, thereby improving the clarity of the diffractors in the diffraction imaging section.

The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy’s uncertainty principle, Beurling’s uncertainty principle and Donoho–Stark’s uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb’s uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff–Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.

Ground motion time-domain stochastic simulations are performed by generating a white or a Gaussian noise, shaping it with a window, loading its normalized spectrum with the spectrum of the motion at a specific site and transforming it back in time-domain (Boore in Pure Appl Geophys 160: 635–676, 2003). The outcome of the resulted simulations is compared to the target parameters, namely amplitude, frequency content and duration parameters. Using SMSIM (Boore in SMSIM: Fortran programs for simulating ground motions form earthquakes, 2005) and EXSIM (Motazedian and Atkinson in Bull Seismol Soc Am 95(3): 995–1010, 2005. https://doi.org/10.1785/0120030207), Coțovanu (J Mil Technol 1(2): 27–32, 2018. https://doi.org/10.32754/jmt.2018.2.05) and Coțovanu and Vacareanu (J Seismol 24(1): 229–241, 2020. https://doi.org/10.1007/s10950-019-09892-5) applied the stochastic simulation method for generating ground motions produced by intermediate-depth Vrancea (Romania) earthquakes and found inconsistent results for duration parameters (significant duration and energy distribution in time-domain). In simulations, the duration of the motion results from the source and path duration, but the significant duration (that contains 90% of the energy) depends on the way the energy is distributed in time, and this distribution is controlled through a noise shaping window. The objective of this paper is to provide an appropriate form of the shaping window that can describe the specific energy release of ground motions generated by Vrancea intermediate depth seismic source (the first 50–70% of energy being rapidly released, the rest of it having a very slow release). For this matter, 371 recorded motions were analyzed and 5 types of energy release were defined. The set with the above described pattern that contained almost half of the analyzed accelerograms was used to define a mean normalized cumulative energy that describes the Vrancea specificity. On this basis, the parameters for the SMSIM implemented exponential window were determined and, because the results were inconsistent, a more adequate window function was defined and implemented in SMSIM. For the newly defined two-interval window function an algorithm for calculating the parameters was described, and the parameters of the mean release of energy were determined. By using the two-interval function for describing the Vrancea specific release of energy, more appropriate ground motion simulations resulted.

In recent decades, with the rapid development of artificial intelligence technologies and bionic engineering, the spiking neural network (SNN), inspired by biological neural systems, has become one of the most promising research topics, enjoying numerous applications in various fields. Due to its complex structure, the simplification of SNN circuits requires serious consideration, along with their power consumption and space occupation. In this regard, the use of SSN circuits based on single-electron transistors (SETs) and modified memristor synapses is proposed herein. A prominent feature of SETs is Coulomb oscillation, which has characteristics similar to the pulses produced by spiking neurons. Here, a novel window function is used in the memristor model to improve the linearity of the memristor and solve the boundary and terminal lock problems. In addition, we modify the memristor synapse to achieve better weight control. Finally, to test the SNN constructed with SETs and memristor synapses, an associative memory learning process, including memory construction, loss, reconstruction, and change, is implemented in the circuit using the PSPICE simulator.

The staggered-grid finite-difference method (SGFDM) has been extensively employed in numerical modeling, seismic imaging, and waveform inversion because of its high efficiency, accuracy, and practicality. Numerical modeling accuracy is crucial for imaging and inversion. Truncating the spatial convolution sequence of the pseudospectral method by window functions can derive the finite-difference method. However, truncation error is inevitable; it can create illusions in numerical modeling, leading to misjudgments. To reduce the truncation error and improve numerical modeling accuracy, enhancing the performance of the truncated window function is imperative. In this study, a new optimal window function was investigated based on the least-squares combination of square windows. Then, the new optimized window function was used to truncate the spatial convolution sequence of the pseudospectral method and obtain an optimized SGFDM. Dispersion analysis showed that the optimized SGFDM has wider spectrum coverage than the conventional SGFDM at the same order. In addition, at low orders, the error range of the new optimized SGFDM could be strictly controlled within 1‱, whereas, at high orders, while ensuring accuracy, the spectrum coverage significantly improved. Finally, numerical modeling for elastic waves demonstrated that the optimized SGFDM could superiorly suppress numerical dispersion and improve calculation efficiency.

In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here, we consider the continuous Kaiser–Bessel, continuous exp-type, sinh-type, and continuous cosh-type window functions with the same support and same shape parameter. We present novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice of the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter. For the considered continuous window functions, the error constants have an exponential decay with respect to the truncation parameter.

We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : Rd → C. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in such a way that it becomes symmetric with respect to a given symmetry group on Zd. The results are illustrated with constructions of dual pairs of Gabor frames for L2(Rd). In addition, we obtain general approaches to construct dual Gabor frames whose window functions are symmetric with respect to an arbitrary symmetry group. Through sampling and periodization, these dual Gabor frames for L2(Rd) lead.

The quality factor (or Q value) is an important parameter for characterizing the inelastic properties of rock. Achieving a Q value estimation with high accuracy and stability is still challenging. In this study, a new method for estimating ultrasonic attenuation using a spectral ratio based on an S transform (SR-ST) is presented to improve the stability and accuracy of Q estimation. The variable window of ST is used to solve the time window problem. We add two window factors to the Gaussian window function in the ST. The window factors can adjust the scale of the Gaussian window function to the ultrasonic signal, which reduces the calculation error attributed to the conventional Gaussian window function. Meanwhile, the frequency bandwidth selection rules for the linear regression of the amplitude ratio are given to further improve stability and accuracy. First, the feasibility and influencing factors of the SR-ST method are studied through numerical testing and standard sample experiments. Second, artificial samples with different Q values are used to study the adaptability and stability of the SR-ST method. Finally, a further comparison between the new method and the conventional spectral ratio method (SR) is conducted using rock field samples, again addressing stability and accuracy. The experimental results show that this method will yield an error of approximately 36% using the conventional Gaussian window function. This problem can be solved by adding the time window factors to the Gaussian window function. The frequency bandwidth selection rules and mean slope value of the amplitude ratio used in the SR-ST method can ensure that the maximum error of different Q values estimation (Q > 15) is less than 10%.

Surface profile is one of the key evaluation indexes in the optical industry. With the development of optical technology, wavelength-tuned phase-shifting interferometry has been widely used for measuring the surface profiles of optical components based on the high accuracy of this technique. However, coupling errors caused by higher harmonics and phase-shift errors may lead to systematic errors in the calculated phase. In this research, to suppress coupling errors effectively, we propose an 8N - 7 phase-shifting algorithm consisting of a novel polynomial window function and discrete Fourier transform. By locating eight multiple roots on the characteristic diagram, the polynomial window function of the proposed algorithm is derived based on the theory of characteristic polynomials. The characteristic polynomial of the 8N - 7 algorithm is estimated based on a Fourier representation. Phase errors are discussed and compared to the errors in other algorithms. The results indicate that the 8N - 7 algorithm has the smallest phase error among the compared algorithms and can compensate for up to sixth-order nonlinearity in phase measurements. Finally, the proposed algorithm was applied to profile the surface of a silicon wafer. The results revealed that the standard deviation across 20 experiments was 3.231 nm, which is much smaller than the standard deviations of conventional phase-shifting algorithms.

The discrete Gabor transform (DGT) is an important and widely used time–frequency analysis tool in signal processing. In this paper, a sparse time–frequency representation (TFR) method based on the DGT is studied using mixed $$\ell _{1/2}$$ – $$\ell _2$$ norm regularization which can overcome the over-sparse problem and permit a better sparsity and stability. Because the width of the window function in the DGT directly affects the time–frequency resolution and concentration of the Gabor spectrum, an adaptive optimal window width (AOWW) selection algorithm based on information entropy theory can be applied to search for the optimal width in the DGT. In this paper, the traditional DGT is converted into a constraint optimization problem by minimizing a mixed $$\ell _{1/2}$$ – $$\ell _2$$ norm sparse constraint of the Gabor coefficients. The numerical experimental results show that the proposed sparsity-based TFR based on the conventional DGT is an effective and powerful tool for analyzing and processing nonstationary signals, obtaining a higher time–frequency concentration and sparsity of the Gabor TFR of a given signal.

Memristors have become promising candidates for the advancement of recent technology as the miniaturization of complementary metal–oxide–semiconductor (CMOS) technology approaches its final stage. Nanoscale size, easy fabrication, compatibility with MOS, and diverse applications have accelerated these devices to new levels. In this paper, we discuss the merits and demerits of existing window functions and propose a novel window function that addresses their limitations. The suggested window function exhibits high nonlinearity at the boundaries and resolves other boundary issues. The results obtained using the proposed window function are compared with data reported in the literature to validate our design approach.

Aiming at the inherently problems that interpolated transformation technique can not work effectively over a large transformation area, and the sidelobe level is high during beamforming. In this paper, we propose a new algorithm by designing the weight of interpolated array to jointly optimize sidelobe level and transforming error. The key feature of this algorithm is that in the case of minimizing the error of output signal power which is caused by transforming error, the sum of the sidelobe power is constrained to minimum by the angle weighting function. Numerical simulations prove the validity of this new algorithm. In comparing with the existing algorithms, this algorithm can achieve low sidelobe beamforming in a large transformation area, and it overcomes high current taper, mainlobe expansion, and inability to use the window function to suppress sidelobe in interpolated array. Besides, it has good reception and suppression effects for the spatial wave signals from the mainlobe and the sidelobe, respectively.

We introduce the short-time fractional Fourier transform with a suitable quaternion valued function as the window function on the space of square integrable quaternion valued functions on $${\mathbb {R}}$$ . We study the properties of the quaternionic short-time fractional Fourier transform including Parseval’s formula, inversion formulae and uncertainty principle.

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983)., Journal of Fourier Analysis and Applications, 27 (2), ISSN:1069-5869, ISSN:1531-5851

Independent low-rank matrix analysis (ILRMA) is the state-of-the-art algorithm for blind source separation (BSS) in the determined situation (the number of microphones is greater than or equal to that of source signals). ILRMA achieves a great separation performance by modeling the power spectrograms of the source signals via the nonnegative matrix factorization (NMF). Such a highly developed source model can solve the permutation problem of the frequency-domain BSS to a large extent, which is the reason for the excellence of ILRMA. In this paper, we further improve the separation performance of ILRMA by additionally considering the general structure of spectrograms, which is called consistency, and hence we call the proposed method Consistent ILRMA. Since a spectrogram is calculated by an overlapping window (and a window function induces spectral smearing called main- and side-lobes), the time-frequency bins depend on each other. In other words, the time-frequency components are related to each other via the uncertainty principle. Such co-occurrence among the spectral components can function as an assistant for solving the permutation problem, which has been demonstrated by a recent study. On the basis of these facts, we propose an algorithm for realizing Consistent ILRMA by slightly modifying the original algorithm. Its performance was extensively evaluated through experiments performed with various window lengths and shift lengths. The results indicated several tendencies of the original and proposed ILRMA that include some topics not fully discussed in the literature. For example, the proposed Consistent ILRMA tends to outperform the original ILRMA when the window length is sufficiently long compared to the reverberation time of the mixing system., Submitted to EURASIP J. Adv. Signal. Process. Accepted on Oct. 30, 2020

This paper concerns the approximation of bivariate functions by using the well-known filtered back projection (FBP) formula from computerized tomography. We prove error estimates and convergence rates for the FBP approximation of target functions from Sobolev spaces $$\mathrm H^\alpha ({\mathbb {R}}^2)$$Hα(R2) of fractional order $$\alpha >0$$α>0, where we bound the FBP approximation error, which is incurred by the application of a low-pass filter, with respect to the weaker norms of the rougher Sobolev spaces $$\mathrm H^\sigma ({\mathbb {R}}^2)$$Hσ(R2), for $${0 \le \sigma \le \alpha }$$0≤σ≤α. In particular, we generalize our previous results to non band-limited filter functions and show that the decay rate of the error saturates at fractional order depending on smoothness properties of the filter’s window function at the origin. The theoretical results are supported by numerical simulations.

Electrocardiogram (ECG) signal processing and analysis is becoming more and more popular as it is useful in diagnosis and prognosis of human heart and clinically automatic machine estimation is based upon it. R-peak is the most important component in ECG beat and is widely used to investigate normal and abnormal subjects (patients). From the last few decades, R-peak detection in ECG has been the most challenging topic in the biomedical research. As QRS complex has high frequency in ECG as compared to other waves (P, T, U-wave), so majority of algorithms estimate QRS complex by either filtering or suppressing the lower frequency waves, including various artifacts like baseline wander, power line interference, and electromyograph noises. This paper demonstrates a new kind of ECG denoising algorithm based on self-convolution window (SCW) concept. The SCW based on Hamming window, herein referred to as Hamming self-convolution window, is used to design a new kind of filter which possesses negligible ripples in the stop band, as compared to the conventional window-based filters. This algorithm is validated on MIT-BIH arrhythmia database and the results outperform in terms of sensitivity, positive predictivity, and error rate obtained as 99.93%, 99.95%, and 0.117%, respectively, as compared to the other well-established works. The approach has also outperformed the results of well-established window-based filters (Hamming and Kaiser) in terms of reduced false negative, false positive, and error rate.

A simple but versatile hyperbolic cosine window function is presented in this paper, which has two terms in the time domain. A parameter p in the window can be varied to make the mainlobe width of the window function to approach that of a rectangular window ( $$\pm 1/T,$$ ) while maintaining higher sidelobe decay (12 dB/octave.) Even though such behavior has been demonstrated by the two-term polynomial window, it suffers from the limitation that only some discrete values of mainlobe widths can be achieved in the range, $$\pm 1/T$$ to $$\pm 1.5/T$$ . The proposed hyperbolic cosine window has no such limitation; one can achieve any desired value of mainlobe width in the above range. The proposed window can be employed for applications involving spectral resolution.

Memristor is a type of memory device representing the relation between charge q and flux $$\varphi $$ . Recently, memristor, as well as other memristive elements and their corresponding fractional elements, have become very attractive in many applications, due to their unique behavior which cannot be obtained by using conventional elements. However, there are few studies about composite behaviors or characteristics of multiple memristive elements connected in series or parallel, especially for the fractional-order memristors. This paper focuses on analyzing the composite characteristics of two fractional-order memristors connected in series as well as concerning the window function. Under the applied sinusoidal current source, two charge-controlled memristors connected in series are adopted to theoretically demonstrate the variation of memristance and v–i curves. The two fractional order $$\alpha $$ and $$\beta $$ are interpreted as two phase offset which makes the analysis much convenient. The obtained formulas of instantaneous memristance of composite memristor are derived and some influential factors are analyzed in detail. The further analyzing demonstrates that the composite memristor circuits behave as a new memristor with higher complexity and some typical phenomena are found especially for the pinched hysteresis loops in v–i plane.

A memristor is a nonlinear polarity-dependent fundamental circuit element. Due to these intrinsic properties of the device, analyzing a circuit that contains multiple memristors becomes complex. In this paper, we study the characteristics of multiple-memristor series (anti-series) and parallel (anti-parallel) connections, including their transient and stable state composite properties. Also, the existing phenomenological and physics-based memristor mathematical modeling techniques have been discussed for use in SPICE simulation environment. For making a standardized comparison between memristor stochastic and deterministic models, all models presented in this paper have been implemented in a single SPICE program. In addition to the well-known previously reported Joglekar and Biolek window functions, a modified Biolek window function and a novel generic scalable window function have been used to model the intrinsic nonlinearity of memristors effectively. Furthermore, electronic synapse circuits based on memristive devices in series and parallel connections and synaptic circuits based on CMOS transistor–memristor architecture have been presented and analyzed. Based on the obtained results, artificial synaptic circuit design limitation using a single memristor has been demonstrated.

Fractional Fourier transform (FRFT) is generalization of Fourier transform. It has an adjustable parameter in the form of $$\alpha$$ rotational angle that makes it more useful in the various fields of science and engineering. In this paper, an analysis of Kaiser and Gaussian window functions is obtained in the FRFT domain. The behavior of these window functions is observed in terms of spectral parameters along with their special cases. The effect of their behavior is applied in FIR filter implementation and tunes its transition band. By changing the order of the FRFT, the variation in the transition width of windowed FIR filter is obtained which makes it possible to vary the stop-band attenuation. While designing a new filter, this tuning method saves significant time to compute filter coefficients. To validate the results, tuning of the FIR filter with Gaussian and Kaiser window functions is achieved in FRFT domain and the performance of the filter is measured in terms of stop-band attenuation.

The GroundBIRD telescope aims to detect B-mode polarization of the cosmic microwave background radiation using the kinetic inductance detector array as a polarimeter. For the readout of the signal from detector array, we have developed a frequency division multiplexing readout system based on a digital down converter method. These techniques in general have the leakage problems caused by the crosstalks. The window function was applied in the field programmable gate arrays to mitigate the effect of these problems and tested it in algorithm level.

The staggered-grid finite-difference (SGFD) method has been widely used in seismic forward modeling. The precision of the forward modeling results directly affects the results of the subsequent seismic inversion and migration. Numerical dispersion is one of the problems in this method. The window function method can reduce dispersion by replacing the finite-difference operators with window operators, obtained by truncating the spatial convolution series of the pseudospectral method. Although the window operators have high precision in the low-wavenumber domain, their precision decreases rapidly in the high-wavenumber domain. We develop a least squares optimization method to enhance the precision of operators obtained by the window function method. We transform the SGFD problem into a least squares problem and find the best solution iteratively. The window operator is chosen as the initial value and the optimized domain is set by the error threshold. The conjugate gradient method is also adopted to increase the stability of the solution. Approximation error analysis and numerical simulation results suggest that the proposed method increases the precision of the window function operators and decreases the numerical dispersion.

A method is proposed for determining the parameters of the smoothing window function in pulsed spectroscopy. It is based on coupling the window to the spectral and energy centers of a pulsed signal recorded in a spectrogram.

The goal of this paper is to find an excellent adaptive window function for extracting the weak vibration signal and high frequency vibration signal under strong noise. The relationship between windowing transform and filtering is analyzed first in the paper. The advantage of adjustable time-frequency window of wavelet transform is introduced. Secondly the relationship between harmonic wavelet and multiple analytic band-pass filter is analyzed. The coherence of the multiple analytic band-pass filter and harmonic wavelet base function is discussed, and the characteristic that multiple analytic band-pass filter included in the harmonic wavelet transform is founded. Thirdly, by extending the harmonic wavelet transform, the concept of the adaptive harmonic window and its theoretical equation without decomposition are put forward in this paper. Then comparing with the Hanning window, the good performance of restraining side-lobe leakage possessed by adaptive harmonic window is shown, and the adaptive characteristics of window width changing and analytical center moving of the adaptive harmonic window are presented. Finally, the proposed adaptive harmonic window is applied to weak signal extraction and high frequency orbit extraction of high speed rotor under strong noise, and the satisfactory results are achieved. The application results show that the adaptive harmonic window function can be successfully applied to the actual engineering signal processing.

The performance of multitaper perceptual linear prediction (PLP) features of speech samples to discriminate healthy and early stage Parkinson diseased subjects is investigated in this paper. The PLP features are conventionally obtained by computing the power spectrum using a single tapered Hamming window. This estimated spectrum exhibits large variance which can be reduced by computing the weighted average of power spectra obtained using a set of tapered windows, leading to multitaper spectral estimation. In this investigation, two multitaper techniques namely Sine wave taper and Thomson multitaper along with the conventional single taper windowing are investigated. Artificial Neural network is then used to classify the PLP features extracted by applying the three types of window tapers on the speech signals of healthy and early stage Parkinson affected people and their respective performances are compared. The results show more accuracy using the multitaper techniques when compared with the conventional single taper technique. It is seen that the accuracy obtained using Sine wave tapers as well as Thomson multitaper is maximum for five tapers. An improvement in the recognition accuracy by 7.5% using the Sine tapers and by 6.9% using the Thomson tapers is obtained when compared with the conventional method. An improvement in other performance measures like Equal error rate, False positive rate, False negative rate, Sensitivity and Specificity is also observed in the multitaper techniques.

Indium tin oxide (ITO) thin films are the important component in display and semiconductor industries because of their excellent electrical and optical properties. To improve the reliability and performance of these films, the surface shape and thickness should be measured accurately. In this research, the surface topography of an ITO thin film on the surface of a supporting plate was measured by wavelength tuned phase shifting interferometer and polynomial window function. The surface topography is measured by compensating for the phase difference error caused by interference of the transparent thin film. Optical interference is formulated by the phase and amplitudes determined by wavelength tuning and polynomial window function. The measurement method allows for noncontact (nondestructive), large aperture, and nanoscale measurement of highly reflective thin films.

A design technique for multiple stop band graded index filters is proposed in this paper using quintic apodization function and by superimposing refractive index profiles optimized for different wavelengths in visible portion of electromagnetic spectrum (400, 500 and 600 nm). The optical performance of the filters is simulated using the Filmstar™ design software for double and triple stop band filters. The results are compared for constant and varying peak-to-peak refractive index. The aim of this work is to attain equal reflectance values at all desired wavelengths (~85%). The results show that reflectance performance for equal peak amplitude for double band and triple band filters is obtained as ~86 and 83%, respectively for each stop band wavelengths when peak-to-peak refractive index is varied.

Even though the rotational ground motion may contribute significantly to the response of certain structures, their effects are generally ignored in seismic design, because of non-availability of appropriate instruments for direct recording of the rotational components. Like many others, a simplified framework was proposed by the authors elsewhere (Rodda and Basu in Int J Earthq Impact Eng 1(3):253–288, 2016) to extract the rotational motion as a temporal derivative of an apparent translational component (ATC) followed by scaling with an apparent velocity. ATC was defined such that its time derivative is closely correlated with the respective rotational motion. But the a priori knowledge of rotational motion is required in estimating the ATC for rocking component. An empirical procedure has been proposed here to bypass the requirement of rotational motion a priori. This paper also assesses the definition of ATC through examining the similitude between the time derivative of ATC and the respective rotational motion (benchmark) quantitatively. Similitude is assessed on smoothened response spectra (by Hamming window) of the time derivative of ATC and that of rotational motion. A new definition of spectral contrast angle (SCA) based on distance correlation has been proposed to assess the spectral similitude. To differentiate the similar from non-similar spectra, SCA corresponding to an acceptable degree of similarity is proposed by studying a large ensemble of ground motions from the PEER database. This similitude study is further extended using relative energy build up and energy spectra.

The $$\alpha $$ -modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl–Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in $$\alpha $$ -modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results.

Image registration is defined as an important process in image processing in order to align two or more images. A new image registration algorithm for translated and rotated pairs of 2D images is presented in order to achieve subpixel accuracy and spend a small fraction of computation time. To achieve the accurate rotation estimation, we propose a two-step method. The first step uses the Fourier Mellin Transform and phase correlation technique to get the large rotation, then the second one uses the Fourier Mellin Transform combined with an enhance Lucas–Kanade technique to estimate the accurate rotation. For the subpixel translation estimation, the proposed algorithm suggests an improved Hanning window as a preprocessing task to reduce the noise in images then achieves a subpixel registration in two steps. The first step uses the spatial domain approach which consists of locating the peak of the cross-correlation surface, while the second uses the frequency domain approach, based on low-frequency (aliasing-free part) of aliased images. Experimental results presented in this work show that the proposed algorithm reduces the computational complexities with a better accuracy compared to other subpixel registration algorithms.

The numerical dispersion and computational cost are high for conventional Taylor series expansion staggered-grid finite-difference forward modeling owing to the high frequency of the wavelets and the large grid intervals. In this study, the cosine-modulated binomial window function (CMBWF)-based staggered-grid finite-difference method is proposed. Two new parameters, the modulated time and modulated range are used in the new window function and by adjusting these two parameters we obtain different characteristics of the main and side lobes of the amplitude response. Numerical dispersion analysis and elastic wavefield forward modeling suggests that the CMBWF method is more precise and less computationally costly than the conventional Taylor series expansion staggered-grid finitedifference method.

Wireless multimedia sensors have been frequently used for detecting events in acoustic rich environments such as protected area networks. Such areas have diverse habitat, frequently varying terrain and are a source of very large number of acoustic events. This work is aimed at detecting the tree cutting event in a forest area, by identifying the acoustic pattern generated due to an axe hitting a tree bole, with the help of wireless multimedia sensors. A series of operations using the hamming window, wiener filter, Otsu thresholding and mathematical morphology are used for removing the unwanted clutter from the spectrogram obtained from such events. Using the sparse nature of the acoustic signals, a compressed sensing based energy efficient data gathering scheme is devised for accurate event reporting. A network of Mica2 motes is deployed in a real forest area to test the validity of the proposed scheme. Analytical and experimental results proves the efficacy of the proposed event detection scheme.

Estimating observation error covariance matrix properly is a key step towards successful seismic history matching. Typically, observation errors of seismic data are spatially correlated; therefore, the observation error covariance matrix is non-diagonal. Estimating such a non-diagonal covariance matrix is the focus of the current study. We decompose the estimation into two steps: (1) estimate observation errors and (2) construct covariance matrix based on the estimated observation errors. Our focus is on step (1), whereas at step (2) we use a procedure similar to that in Aanonsen et al. 2003. In Aanonsen et al. 2003, step (1) is carried out using a local moving average algorithm. By treating seismic data as an image, this algorithm can be interpreted as a discrete convolution between an image and a rectangular window function. Following the perspective of image processing, we consider three types of image denoising methods, namely, local moving average with different window functions (as an extension of the method in Aanonsen et al. 2003), non-local means denoising and wavelet denoising. The performance of these three algorithms is compared using both synthetic and field seismic data. It is found that, in our investigated cases, the wavelet denoising method leads to the best performance in most of the time.

Windowing is a common method to reduce spectral leakage in Fourier analysis. Based on the principle that time domain multiplication corresponds to frequency domain convolution, this paper suggests frequency convolution to implement windowing functions by FIR filter structure. The proposed digital filter has full compatibility for variable length of window and is capable of implementing the three most popular windows, viz. Hanning, Hamming and Blackman. Numerical simulation indicates that the method consumes $$78\,\%$$78% less area, $$11\,\%$$11% higher throughput rate than the CORDIC-based (co-ordinate rotation digital computer) method, and achieves $$33\,\%$$33% lower quantization error for fixed point than the time domain multiplication method. Furthermore, an example of stretch processing in HF radar is shown to reduce the $$99\,\%$$99% computational complexity because of the segmental frequency points windowing.

In this work, two-channel perfect reconstruction quadrature mirror filter (QMF) bank has been proposed based on the prototype filter using windowing method. A novel window function based on logarithmic function along with the spline function is utilized for the design of prototype filter. The proposed window has a variable parameter ‘ $$\alpha $$ ’, which varies the peak side lobe level and rate of fall-off side lobe level which in turn affects the peak reconstruction error (PRE) and amplitude distortion ( $$e_{am}$$ ) of the QMF bank . The transition width of the prototype is controlled by the spline function using the parameter ‘ $$\mu $$ ’. The perfect reconstruction condition is satisfied by setting the cutoff frequency ( $$\omega _{c}$$ ) of the prototype low-pass filter at ‘ $$\pi /2$$ ’. The performance of the proposed design method has been evaluated in terms of mean square error in the pass band, mean square error in the stop band, first side lobe attenuation ( $$A_{1}$$ ), peak reconstruction error (PRE) and amplitude error ( $$e_{am}$$ ) for different values of ‘ $$\alpha $$ ’ and ‘ $$\mu $$ ’. The results are provided and compared with the existing methods.

An enhanced fusion algorithm for generating a super-resolution image from a sequence of low-resolution images captured from identical scene apparently a video based on adaptive normalized convolution has been designed and analyzed. The algorithm for fusing the images is based on the supporting structure of normalized convolution. Here the idea is projection of local signals onto a subspace. The adaptive nature of the window function in adaptive normalized convolution helps to gather more samples for processing and increases signal-to-noise ratio, decreases diffusion through discontinuities. The validation of proposed method is done using simulated experiments and real-time experiments. These experimental results are compared with various latest techniques using performance measures like peak signal-to-noise ratio, sharpness index and blind image quality index. In both the cases of experiments, the proposed adaptive normalized convolution-based super-resolution image reconstruction has proved to be highly efficient which is needed for satellite imaging, medical imaging diagnosis, military surveillance, remote sensing, etc.

A novel method called Impact-synchronous modal analysis (ISMA) was proposed previously which allows modal testing to be performed during operation. This technique focuses on signal processing of the upstream data to provide cleaner Frequency response function (FRF) estimation prior to modal extraction. Two important parameters, i.e., windowing function and impact force level were identified and their effect on the effectiveness of this technique were experimentally investigated. When performing modal testing during running condition, the cyclic loads signals are dominant in the measured response for the entire time history. Exponential window is effectively in minimizing leakage and attenuating signals of non-synchronous running speed, its harmonics and noises to zero at the end of each time record window block. Besides, with the information of the calculated cyclic force, suitable amount of impact force to be applied on the system could be decided prior to performing ISMA. Maximum allowable impact force could be determined from nonlinearity test using coherence function. By applying higher impact forces than the cyclic loads along with an ideal decay rate in ISMA, harmonic reduction is significantly achieved in FRF estimation. Subsequently, the dynamic characteristics of the system are successfully extracted from a cleaner FRF and the results obtained are comparable with Experimental modal analysis (EMA).

Local coupled extreme learning machine (LCELM) (Qu in Neural Comput Appl 27(1):27---33, 2016) is a recently developed variant of ELM and uses the paradigm of local coupled feed-forward neural network (LCFNN) (Sun in Neural Netw 23(1):108---113, 2010) to train an ELM model. This comment is to point out that the window function used in LCELM does not satisfy the necessary conditions of window function required by LCFNN. As a result, LCELM cannot effectively control the activation of hidden-layer node for different instances. It is possible that the improper window function can cause LCELM has the same discriminant with the basic ELM.