Chebyshev Type Research Articles

Design of Infinite Impulse Response Filters Based on Multi-Objective Particle Swarm Optimization

The goal of this study is to explore the effectiveness of applying multi-objective particle swarm optimization (MOPSO) algorithms in the design of infinite impulse response (IIR) filters. Given the widespread application of IIR filters in digital signal processing, the precision of their design plays a significant role in the system’s performance. Traditional design methods often encounter the problem of local optima, which limits further enhancement of the filter’s performance. This research proposes a method based on multi-objective particle swarm optimization algorithms, aiming not just to find the local optima but to identify the optimal global design parameters for the filters. The design methodology section will provide a detailed introduction to the application of multi-objective particle swarm optimization algorithms in the IIR filter design process, including particle initialization, velocity and position updates, and the definition of objective functions. Through multiple experiments using Butterworth and Chebyshev Type I filters as prototypes, as well as examining the differences in the performance among these filters in low-pass, high-pass, and band-pass configurations, this study compares their efficiencies. The minimum mean square error (MMSE) of this study reached 1.83, the mean error (ME) reached 2.34, and the standard deviation (SD) reached 0.03, which is better than the references. In summary, this research demonstrates that multi-objective particle swarm optimization algorithms are an effective and practical approach in the design of IIR filters.

Refinements of Pólya-SzegŐ and Chebyshev type inequalities via different fractional integral operators

Various differential and integral operators have been introduced and applied for the generalization of several integral inequalities. The purpose of this article is to create a more generalized fractional integral operator of Saigo type. This operator will be used alongwith the existing Saigo type and Q-Saigo type fractional integral operators to establish extended and generalized versions of several inequalities, including Pólya-Szegő and Chebyshev type inequalities.

New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators

Chebyshev’s inequality is a well-known inequality in the field of inequality theory and yields remarkable results for functions that exhibit synchronous behavior. The study, designed to obtain Chebyshev type inequalities with the help of the Atangana–Baleanu fractional integral operator, which has become the most popular concept of fractional analysis in recent years, aims to obtain general forms for synchronous functions. In the process of obtaining results for synchronized functions within the scope of the study, functional tools such as Atangana–Baleanu fractional integral operator, various mathematical analysis operations and induction were used. The most striking aspect of the findings is that for some special values of the parameter of the fractional integral operator, some of the results can be reduced to the Chebyshev inequality. Compared to the studies in the literature, this study has a unique aspect in that the findings obtained with the help of the Atangana–Baleanu fractional integral operator contain general forms, as well as bringing together a popular concept of synchronous functions and fractional analysis.

Estimation of phase distortions of the photoplethysmographic signal in digital IIR filtering

Pre-processing of the photoplethysmography (PPG) signal plays an important role in the analysis of the pulse wave signal. The task of pre-processing is to remove noise from the PPG signal, as well as to transmit the signal without any distortions for further analysis. The integrity of the pulse waveform is essential since many cardiovascular parameters are calculated from it using morphological analysis. Digital filters with infinite impulse response (IIR) are widely used in the processing of PPG signals. However, such filters tend to change the pulse waveform. The aim of this work is to quantify the PPG signal distortions that occur during IIR filtering in order to select a most suitable filter and its parameters. To do this, we collected raw finger PPG signals from 20 healthy volunteers and processed them by 5 main digital IIR filters (Butterworth, Bessel, Elliptic, Chebyshev type I and type II) with varying parameters. The upper cutoff frequency varied from 2 to 10 Hz and the filter order—from 2nd to 6th. To assess distortions of the pulse waveform, we used the following indices: skewness signal quality index (SSQI), reflection index (RI) and ejection time compensated (ETc). It was found that a decrease in the upper cutoff frequency leads to damping of the dicrotic notch and a phase shift of the pulse wave signal. The minimal distortions of a PPG signal are observed when using Butterworth, Bessel and Elliptic filters of the 2nd order. Therefore, we can recommend these filters for use in applications aimed at morphological analysis of finger PPG waveforms of healthy subjects.

A Wearable Inertial Sensor Approach for Locomotion and Localization Recognition on Physical Activity.

Advancements in sensing technology have expanded the capabilities of both wearable devices and smartphones, which are now commonly equipped with inertial sensors such as accelerometers and gyroscopes. Initially, these sensors were used for device feature advancement, but now, they can be used for a variety of applications. Human activity recognition (HAR) is an interesting research area that can be used for many applications like health monitoring, sports, fitness, medical purposes, etc. In this research, we designed an advanced system that recognizes different human locomotion and localization activities. The data were collected from raw sensors that contain noise. In the first step, we detail our noise removal process, which employs a Chebyshev type 1 filter to clean the raw sensor data, and then the signal is segmented by utilizing Hamming windows. After that, features were extracted for different sensors. To select the best feature for the system, the recursive feature elimination method was used. We then used SMOTE data augmentation techniques to solve the imbalanced nature of the Extrasensory dataset. Finally, the augmented and balanced data were sent to a long short-term memory (LSTM) deep learning classifier for classification. The datasets used in this research were Real-World Har, Real-Life Har, and Extrasensory. The presented system achieved 89% for Real-Life Har, 85% for Real-World Har, and 95% for the Extrasensory dataset. The proposed system outperforms the available state-of-the-art methods.

Design of Wide‐Beam Leaky‐Wave Antenna Arrays Based on the Bilinear Transformation of IIR Digital Filters and the Z Transform

Abstract In the pioneering work, the radiation diagrams of leaky wave antenna arrays can achieve attenuation nulls and gains at specific angles by manually placing the zeros and the poles in the Z domain of the corresponding discrete linear time‐invariant (LTI) system. This handcrafted design procedure does not allow radiation diagrams with wide beams since the interaction between poles involved in the wide beam and their corresponding leaky modes cannot be easily handled. To overcome this limitation, this paper describes a novel method for designing radiation diagrams of leaky‐wave antenna arrays based on the theory of IIR discrete filters. The proposed method relies on the design of discrete filters with the prototypes of analog low‐pass filters defined by Butterworth and Chebyshev type I polynomials, whose roots along with the bilinear transformation provide the location of the poles and the zeros of the discrete LTI system and, therefore, the parameters of the leaky‐wave antenna array. Results with different designs and a comparison with other approaches show the utility and effectiveness of this novel method to design wide‐beam leaky‐wave antenna arrays.

Identification of Nonlinear Systems Using the Hammerstein-Wiener Model with Improved Orthogonal Functions

Hammerstein-Wiener systems present a structure consisting of three serial cascade blocks. Two are static nonlinearities, which can be described with nonlinear functions. The third block represents a linear dynamic component placed between the first two blocks. Some of the common linear model structures include a rational-type transfer function, orthogonal rational functions (ORF), finite impulse response (FIR), autoregressive with extra input (ARX), autoregressive moving average with exogenous inputs model (ARMAX), and output-error (O-E) model structure. This paper presents a new structure, and a new improvement is proposed, which is consisted of the basic structure of Hammerstein-Wiener models with an improved orthogonal function of Müntz-Legendre type. We present an extension of generalised Malmquist polynomials that represent Müntz polynomials. Also, a detailed mathematical background for performing improved almost orthogonal polynomials, in combination with Hammerstein-Wiener models, is proposed. The proposed approach is used to identify the strongly nonlinear hydraulic system via the transfer function. To compare the results obtained, well-known orthogonal functions of the Legendre, Chebyshev, and Laguerre types are exploited.

Detection of Supra Ventricular arrhythmia using LSTM, BI-LSTM & GRU

Deep learning techniques have made early strides in the analysis about complex ECG signals, particularly in the classification about heartbeats & the detection about arrhythmias. Nonetheless, there is still more work toward be done in terms about the analysis about health-related data. This study offers dual structured & bidirectional approaches for classifying arrhythmias that deal with the drawbacks about multilayered dilated models. The data is first preprocessed using the quicker Chebyshev Type II filtering method, which does not make use about statistical properties. Using the Daubechies wavelet, which may resolve fractal issues & signal discontinuities, noise from the preprocessed filter is additionally eliminated. In this paper, the proposed models LSTM, BILSTM, & GRU were employed toward provide fusion features. The signals are categorized by fully connected layer before. The suggested model is trained & validated using the dataset for supra-ventricular arrhythmias. The learnt model considerably enhances the classification performance & interpretability by fusing dilated BILSTM with fusion features. The results about the experiment indicate that arrhythmia can be recognized using a highperformance automated recognition system. Our future development will concentrate on the automatic & cloud-based ECG classification about various arrhythmia signal-based data.

Optimization Method of Acoustic Filter Structures Composed of Helmholtz Resonators Based on Genetic Algorithm

This paper constructs an iteration method based on Genetic Algorithm to optimize the acoustic filter structure comprising Helmholtz resonators, providing a new approach to optimize the low-frequency filtering performance of the acoustic structure under the constraint of keeping the resonator volume unchanged. First, acoustic-electrical analogy is used to design a fifth-order Chebyshev type II acoustic filter structure, which comprises two rectangular Helmholtz resonators; second, an iteration method is constructed based on Genetic Algorithm to optimize the shape of Helmholtz resonator under the constraint of keeping the resonator volume unchanged; finally, an optimal acoustic filter structure is obtained by implementing the iteration procedure to design shapes of the two Helmholtz resonators. The simulation results show that, compared with the original rectangular Helmholtz resonator, the optimal Helmholtz resonator has a reduction of about 10% in the resonant frequency and an obvious shift of the transmission loss curve towards low frequencies; the optimal acoustic filter structure has an effective attenuation frequency range of 84-232Hz, showing a significantly improved low-frequency filtering performance, verifying feasibility and effectiveness of the iteration optimization based on Genetic Algorithm.

Performance Evaluation of Selected Adaptive Filters for Acoustic Echo Cancelation over Voice over Internet Protocol

Advancement of technology has increased the value of the telecommunications industry. One of the services with the rapid growth is Voice over Internet Protocol (VoIP). However, echo is a common disturbing effect in VoIP calls. Instead of suppressing the presence of echo in the VoIP communication channel, echo cancellation works better in full-duplex mode. This paper compares performances of some selected adaptive filters for cancelation of acoustic echo in VoIP channel. The framework for adaptive echo cancellation is built around finite impulse response and fourth-order infinite impulse response Chebyshev type II filters for the modeling of echo and acoustic environment, respectively. Selected adaptive algorithms for analysis are least mean square, recursive least square and frequency domain adaptive filter. The analysis is carried out in MATLAB 2018a environment. Echo return loss enhancement, error estimate and convergence rate are the performance indices employed for performance comparison of selected adaptive algorithms. It was found that the FDAF algorithm performed best based on computed error return loss enhancement and error estimate figures. In addition, FDAF converges faster than LMS and RLS with average time of 0.026431 s over 2048 filter lengths. Thus, FDAF shows a better promise over the two variants of time domain adaptive algorithm (LMS and RLS) analyzed in this paper.

Anti-Gaussian quadrature formulae of Chebyshev type

We prove that there is no positive measure d σ d\sigma on the interval [ a , b ] [a,b] such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure d σ ( t ) = d σ ( − t ) d\sigma (t)=d\sigma (-t) on the symmetric interval [ − a , a ] [-a,a] , for which the anti-Gaussian formula has the form ∫ − a a f ( t ) d σ ( t ) = μ 0 2 [ f ( a ) + f ( − a ) ] + R 2 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{2}[f(a)+f(-a)]+R_{2}^{AG}(f) for n = 1 n=1 and ∫ − a a f ( t ) d σ ( t ) = w 1 f ( a ) + w ∑ μ = 2 n f ( t μ ) + w 1 f ( − a ) + R n + 1 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=w_{1}f(a)+w\sum _{\mu =2}^{n}f(t_{\mu })+w_{1}f(-a)+R_{n+1}^{AG}(f) for all n ≥ 2 n\geq 2 , is the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt . It turns out that the formula for n ≥ 2 n\geq 2 is the ( n − 1 ) (n-1) -point Gauss-Lobatto quadrature formula for the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt , which is a generalization of what happens in the case of the Chebyshev measure of the first kind. Moreover, we compute the anti-Gaussian formulae for any one of the four Chebyshev measures.

On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type

In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss–Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170–1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss–Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.

Some new Chebyshev type inequalities for fractional integral operator containing a further extension of Mittag-Leffler function in the kernel

Some new Chebyshev type inequalities for fractional integral operator containing a further extension of Mittag-Leffler function in the kernel

Generalized k-Fractional Integral Operators Associated with Pólya-Szegö and Chebyshev Types Inequalities

Inequalities related to derivatives and integrals are generalized and extended via fractional order integral and derivative operators. The present paper aims to define an operator containing Mittag-Leffler function in its kernel that leads to deduce many already existing well-known operators. By using this generalized operator, some well-known inequalities are studied. The results of this paper reproduce Chebyshev and Pólya-Szegö type inequalities for Riemann-Liouville and many other fractional integral operators.

On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities

<abstract><p>In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.</p></abstract>

New Hybrid Deep Learning Approach Using BiGRU-BiLSTM and Multilayered Dilated CNN to Detect Arrhythmia

Deep learning methods have shown early progress in analyzing complicated ECG signals, especially in heartbeat classification and arrhythmia detection. There is still a great deal of space for further research on this area before reaching a definite decision. This study introduced a novel hybrid framework based on a bidirectional recurrent neural network (BiRNN) with a multilayered dilated convolution neural network (CNN) for arrhythmia classification. Initially, the raw ECG signals are filtered using Chebyshev Type II method and the Daubechies wavelet method is used to solve fractal problems and signal discontinuities. Then, a synthetic signal is generated using a generative adversarial network (GAN) to handle imbalanced signal classes. The proposed Bidirectional RNN with Dilated CNN (BRDC) architecture takes advantage of multilayered dilated CNN and bidirectional RNN units (bidirectional gated recurrent Units, BiGRU, bidirectional long short-term memory, BiLSTM) to generate fusion features and then, fusion features are classified in the fully connected layer. The PhysioNet 2017 challenge (MIT-BIH) dataset is used to train and validate the proposed approach. By combining fusion features with dilated CNN, the proposed approach outperforms the existing model for arrhythmia detection with 99.90 % accuracy, 98.41 % F1-score, 97.96 % precision, and 99.90 % recall. Overall, our hybrid BRDC model provides a cost-effective ECG signal reduction and high-performance automated recognition technique to identify arrhythmia. In the future, an automatic and cloud-based system with more arrhythmia data variance to test the model’s robustness will be given the highest priority.

Generalizations of some concentration inequalities

For a real-valued measurable function f and a nonnegative, nondecreasing function ϕ, we first obtain a Chebyshev type inequality which provides an upper bound for ϕ(λ1)μ({x∈Ω:f(x)≥λ1})+∑k=2nϕ(λk)−ϕ(λk−1)μ({x∈Ω:f(x)≥λk}), where 0<λ1<λ2<⋯<λn<∞. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé–Chebyshev, Cantelli and Hoeffding inequalities are obtained.

Novel aspects of discrete dynamical type inequalities within fractional operators having generalizedℏ-discrete Mittag-Leffler kernels and application

Discrete fractional calculus (DFC) has had significant advances in the last few decades, being successfully employed in the time scale domain ℏZ. Understanding of DFC has demonstrated a valuable improvement in neural networks and modeling in other terrains. In the context of Riemann form (ABR), we discuss the discrete fractional operator influencing discrete Atangana-Baleanu (AB)-fractional operator having ℏ-discrete generalized Mittag-Leffler kernels. In the approach being presented, some new Pólya-Szegö and Chebyshev type inequalities introduced within discrete AB-fractional operators having ℏ-discrete generalized Mittag-Leffler kernels. By analyzing discrete AB-fractional operators in the time scale domain Z, we can perform a comparison basis for notable outcomes derived from the aforesaid operators. This type of discretization generates novel outcomes for synchronous functions. The specification of this proposed strategy simply demonstrates its efficiency, precision, and accessibility in terms of the methodology of qualitative approach of discrete fractional difference equation solutions, including its stability, consistency, and continual reliance on the initial value for the solutions of many fractional difference equation initial value problems. The repercussions of the discrete AB-fractional operators can depict new presentations for various particular cases. Finally, applications concerning bounding mappings are also illustrated.

Multi‐spurious harmonics suppression in folded hairpin line bandpass filter by meander spur‐line

The present article exhibits the improvement in wide stopband harmonic suppression techniques of fourth-order compact hairpin-line Chebyshev type narrowband (4%) bandpass filter centered at 2.5 GHz by meander spur-line. Initially, the compactness of 42% over the conventional hairpin-line cell has been achieved by folding the open-end arms of the hairpin line to folded hairpin-line cell. Accordingly, the improvement of the skirt characteristics and the shifting of the harmonics to higher frequencies have been observed with the cost of harmonics attenuation levels' degradation. Subsequently, an improvement in the stopband rejection level of 40 dB up to 4f0 has been achieved by incorporating meander spur-line to all the coupled edges of the adjacent folded cells of the compact filters along with compactness of 38%.

Chebyshev type inequalities with fractional delta and nabla h-sum operators

The aim of this study is to establish new discrete inequalities for synchronous functions using fractional order delta and nabla h-sum operators. We give examples to illustrate our results.