Wiener Filter Research Articles

Neural network reconstruction of density and velocity fields from the 2MASS Redshift Survey

Aims. Our aim is to reconstruct the 3D matter density and peculiar velocity fields in the local Universe up to a distance of 200 h−1 Mpc from the Two-Micron All-Sky Redshift Survey (2MRS) using a neural network (NN). Methods. We employed an NN with a U-net autoencoder architecture and a weighted mean squared error loss function trained separately to output either the density or velocity field for a given input grid of galaxy number counts. The NN was trained on mocks derived from the Quijote N-body simulations, incorporating redshift-space distortions (RSDs), galaxy bias, and selection effects closely mimicking the characteristics of 2MRS. The trained NN was benchmarked against a standard Wiener filter (WF) on a validation set of mocks before applying it to 2MRS. Results. The NN reconstructions effectively approximate the mean posterior estimate of the true density and velocity fields conditioned on the observations. They consistently outperform the WF in terms of reconstruction accuracy and effectively capture the nonlinear relation between velocity and density. The NN-reconstructed bulk flow of the total survey volume exhibits a significant correlation with the true mock bulk flow, demonstrating that the NN is sensitive to information on “super-survey” scales encoded in the RSDs. When applied to 2MRS, the NN successfully recovers the main known clusters, some of which are partially in the Zone of Avoidance. The reconstructed bulk flows in spheres of different radii less than 100 h−1 Mpc are in good agreement with a previous 2MRS analysis that required an additional external bulk flow component inferred from directly observed peculiar velocities. The NN-reconstructed peculiar velocity of the Local Group closely matches the observed Cosmic Microwave Background dipole in amplitude and Galactic latitude, and only deviates by 18° in longitude. The NN-reconstructed fields are publicly available.

Orthonormal aberration decomposition for circular aperture and rectangular field with extension to multiple-aperture systems

A new set of orthogonal polynomials and the traditional Zernike polynomials are combined to construct the wavefront of the sparse aperture (SA) optical system. The new set of orthogonal polynomials in the rectangular domain is derived based on the Zernike polynomials. The modulation transfer functions (MTFs) of the SA system under different fields of view are calculated. The effects of some wavefront terms on the field-related MTF are analyzed. Imaging simulation and restoration are conducted for the SA system. The results indicate that the wavefront of the SA system can be represented by the combination of the new set of orthogonal polynomials and the Zernike polynomials. The former represents the field term, and the latter represents the pupil term of the wavefront. The field-related Wiener filter is constructed to restore the imaging results of the SA system, and the results show that the image quality can be improved greatly through restoration.

An optimal Bayesian strategy for comparing Wiener–Hunt deconvolution models in the absence of ground truth

Abstract This paper considers the quantitative comparison of several alternative models to perform deconvolution in situations where there is no ground truth data available. With applications to very large data sets in mind, we focus on linear deconvolution models based on a Wiener filter. Although comparatively simple, such models are widely prevalent in large scale setting such as high-resolution image restoration because they provide an excellent trade-off between accuracy and computational effort. However, in order to deliver accurate solutions, the models need to be properly calibrated in order to capture the covariance structure of the unknown quantity of interest and of the measurement error. This calibration often requires onerous controlled experiments and extensive expert supervision, as well as regular recalibration procedures. This paper adopts an unsupervised Bayesian statistical approach to model assessment that allows comparing alternative models by using only the observed data, without the need for ground truth data or controlled experiments. Accordingly, the models are quantitatively compared based on their posterior probabilities given the data, which are derived from the marginal likelihoods or evidences of the models. The computation of these evidences is highly non-trivial and this paper consider three different strategies to address this difficulty—a Chib approach, Laplace approximations, and a truncated harmonic expectation—all of which efficiently implemented by using a Gibbs sampling algorithm specialised for this class of models. In addition to enabling unsupervised model selection, the output of the Gibbs sampler can also be used to automatically estimate unknown model parameters such as the variance of the measurement error and the power of the unknown quantity of interest. The proposed strategies are demonstrated on a range of image deconvolution problems, where they are used to compare different modelling choices for the instrument’s point spread function and covariance matrices for the unknown image and for the measurement error.

Saliency-based video summarization for face anti-spoofing

With the growing availability of databases for face presentation attack detection, researchers are increasingly focusing on video-based face anti-spoofing methods that involve hundreds to thousands of images for training the models. However, there is currently no clear consensus on the optimal number of frames in a video to improve face spoofing detection. Inspired by the visual saliency theory, we present a video summarization method for face anti-spoofing detection that aims to enhance the performance and efficiency of deep learning models by leveraging visual saliency. In particular, saliency information is extracted from the differences between the Laplacian and Wiener filter outputs of the source images, enabling the identification of the most visually salient regions within each frame. Subsequently, the source images are decomposed into base and detail images, enhancing the representation of the most important information. Weighting maps are then computed based on the saliency information, indicating the importance of each pixel in the image. By linearly combining the base and detail images using the weighting maps, the method fuses the source images to create a single representative image that summarizes the entire video. The key contribution of the proposed method lies in demonstrating how visual saliency can be used as a data-centric approach to improve the performance and efficiency for face presentation attack detection. By focusing on the most salient images or regions within the images, a more representative and diverse training set can be created, potentially leading to more effective models. To validate the method’s effectiveness, a simple CNN–RNN deep learning architecture was used, and the experimental results showcased state-of-the-art performance on four challenging face anti-spoofing datasets.

Improved Non-Negative Matrix Factorization-Based Noise Reduction of Leakage Acoustic Signals.

The detection of gas leaks using acoustic signals is often compromised by environmental noise, which significantly impacts the accuracy of subsequent leak identification. Current noise reduction algorithms based on non-negative matrix factorization (NMF) typically utilize the Euclidean distance as their objective function, which can exacerbate noise anomalies. Moreover, these algorithms predominantly rely on simple techniques like Wiener filtering to estimate the amplitude spectrum of pure signals. This approach, however, falls short in accurately estimating the amplitude spectrum of non-stationary signals. Consequently, this paper proposes an improved non-negative matrix factorization (INMF) noise reduction algorithm that enhances the traditional NMF by refining both the objective function and the amplitude spectrum estimation process for reconstructed signals. The improved algorithm replaces the conventional Euclidean distance with the Kullback-Leibler (KL) divergence and incorporates noise and sparse constraint terms into the objective function to mitigate the adverse effects of signal amplification. Unlike traditional methods such as Wiener filtering, the proposed algorithm employs an adaptive Minimum Mean-Square Error-Log Spectral Amplitude (MMSE-LSA) method to estimate the amplitude spectrum of non-stationary signals adaptively across varying signal-to-noise ratios. Comparative experiments demonstrate that the INMF algorithm significantly outperforms existing methods in denoising leakage acoustic signals.

Identification of exon regions in eukaryotes using fine-tuned variational mode decomposition based on kurtosis and short-time discrete Fourier transform

In genomic research, identifying the exon regions in eukaryotes is the most cumbersome task. This article introduces a new promising model-independent method based on short-time discrete Fourier transform (ST-DFT) and fine-tuned variational mode decomposition (FTVMD) for identifying exon regions. The proposed method uses the N/3 periodicity property of the eukaryotic genes to detect the exon regions using the ST-DFT. However, background noise is present in the spectrum of ST-DFT since the sliding rectangular window produces spectral leakage. To overcome this, FTVMD is proposed in this work. VMD is more resilient to noise and sampling errors than other decomposition techniques because it utilizes the generalization of the Wiener filter into several adaptive bands. The performance of VMD is affected due to the improper selection of the penalty factor (α), and the number of modes (K). Therefore, in fine-tuned VMD, the parameters of VMD (K and α) are optimized by maximum kurtosis value. The main objective of this article is to enhance the accuracy in the identification of exon regions in a DNA sequence. At last, a comparative study demonstrates that the proposed technique is superior to its counterparts.

Speckle noise suppression of a reconstructed image in digital holography based on the BM3D improved convolutional neural network

In digital holographic measurement, when light waves pass through inhomogeneous media or surfaces, speckle noise is generated, resulting in random, granular light and dark spots in the hologram, which greatly reduces the image quality. Therefore, in order to improve the image quality of holographic reconstruction, a noise reduction method based on the BM3D improved convolutional neural network (CNN) is proposed in this paper. Firstly, the similarity and important statistical information between blocks can be obtained by using BM3D. Then, the denoising convolutional neural network (DnCNN) is used to learn the relationship between the noise of a large number of samples and the noise image, and further purify the image to retain the details for a better denoising effect. Finally, a reflective off-axis digital holographic optical path system is constructed to collect the holograms of the test samples, and the reconstructed images are obtained by the Fresnel diffraction method to constitute a dataset with the simulated holographic reconstructed images to validate the proposed method in this paper, compared to the other methods, such as DnCNN, convolutional blind denoising network (CBDNet), BM3D, and Wiener filtering. The experimental results of qualitative and quantitative analyses show that the proposed method combines the advantages of traditional algorithms and deep learning, significantly enhances the robustness of the system, optimizes the denoising performance, and preserves the details of the reconstructed image to the greatest extent.

Characterization of Cochlear Implant Artifact and Removal Based on Multi-Channel Wiener Filter in Unilateral Child Patients.

Cochlear implants (CI) allow deaf patients to improve language perception and improving their emotional valence assessment. Electroencephalographic (EEG) measures were employed so far to improve CI programming reliability and to evaluate listening effort in auditory tasks, which are particularly useful in conditions when subjective evaluations are scarcely appliable or reliable. Unfortunately, the presence of CI on the scalp introduces an electrical artifact coupled to EEG signals that masks physiological features recorded by electrodes close to the site of implant. Currently, methods for CI artifact removal have been developed for very specific EEG montages or protocols, while others require many scalp electrodes. In this study, we propose a method based on the Multi-channel Wiener filter (MWF) to overcome those shortcomings. Nine children with unilateral CI and nine age-matched normal hearing children (control) participated in the study. EEG data were acquired on a relatively low number of electrodes (n = 16) during resting condition and during an auditory task. The obtained results obtained allowed to characterize CI artifact on the affected electrode and to significantly reduce, if not remove it through MWF filtering. Moreover, the results indicate, by comparing the two sample populations, that the EEG data loss is minimal in CI users after filtering, and that data maintain EEG physiological characteristics.

Hybrid design scheme for enabling large-aperture diffractive achromat imaging.

Diffractive achromats (DAs) combined with image processing algorithms offer a promising lens solution for high-performance ultra-thin imagers. However, the design of large-aperture DAs that align seamlessly with image processing algorithms remains challenging. Existing sequential methods, which prioritize focusing efficiency in DAs before selecting an algorithm, may not achieve a satisfactory match due to an ambiguous relationship between efficiency and final imaging quality. Conversely, image-quality-oriented end-to-end design often entails high computational complexity for both front-end optics and back-end algorithms, impeding the development of large-aperture designs. To address these issues, we present a hybrid design scheme that begins with end-to-end optimization of the DA with the simplest image processing algorithm, i.e., Wiener filter, significantly reducing the back-end complexity. Subsequently, we apply complex algorithm fine-tuning to further enhance image quality. We validate this hybrid design scheme through extensive investigations on several DA imagers. Our results demonstrate a reduction in memory requirement by approximately 50% while maintaining a high imaging quality with a reasonably large aperture. As a case in point, we simulated a DA imager with a 25 mm diameter aperture. Furthermore, our hybrid design scheme provides two crucial insights. Firstly, we find no strong linear correlation between focusing efficiency and imaging quality, which challenges the conventional understanding. Secondly, we establish a prediction formula for imaging quality, benefiting from the hybrid design scheme.

MVDR Algorithm Based on Loading Factor and Sample Selection Strategy

The MVDR (Minimum Variance Distortionless Response) algorithm is a classic Wiener filtering method used for beamforming in array signal processing. In a one-dimensional linear array high-frequency ground wave radar system, it can be employed to suppress various types of ionospheric clutter. A key step in suppression is the use of maximum likelihood estimation (MLE) to estimate the ionospheric clutter covariance matrix. However, MLE typically assumes that samples are independently and identically distributed (i.i.d.). Traditional MVDR algorithms estimate the clutter covariance matrix using all samples, which may not satisfy the i.i.d. condition. Therefore, this paper proposes a new sample selection strategy that utilizes the Mahalanobis distance method to select samples. Meanwhile, due to the very small numerical values of the ionospheric clutter covariance matrix, even if the matrix is full-rank, numerical instability may occur during inversion. To address this issue, the paper introduces 4 different forms of regularization factors. Empirical data demonstrate that the proposed new method offers better suppression of ionospheric clutter compared to the traditional MVDR algorithm.

Evaluation of Denoising Performance of ResNet Deep Learning Model for Ultrasound Images Corresponding to Two Frequency Parameters.

Ultrasound imaging is widely used for accurate diagnosis due to its noninvasive nature and the absence of radiation exposure, which is achieved by controlling the scan frequency. In addition, Gaussian and speckle noises degrade image quality. To address this issue, filtering techniques are typically used in the spatial domain. Recently, deep learning models have been increasingly applied in the field of medical imaging. In this study, we evaluated the effectiveness of a convolutional neural network-based residual network (ResNet) deep learning model for noise reduction when Gaussian and speckle noises were present. We compared the results with those obtained from conventional filtering techniques. A dataset of 500 images was prepared, and Gaussian and speckle noises were added to create noisy input images. The dataset was divided into training, validation, and test sets in an 8:1:1 ratio. The ResNet deep learning model, comprising 16 residual blocks, was trained using optimized hyperparameters, including the learning rate, optimization function, and loss function. For quantitative analysis, we calculated the normalized noise power spectrum, peak signal-to-noise ratio, and root mean square error. Our findings showed that the ResNet deep learning model exhibited superior noise reduction performance to median, Wiener, and median-modified Wiener filter algorithms.

Effective Noise Reduction Filters for Precise Temperature Measurement Using Brillouin Distributed Optical Fiber Sensors

This study explores the effectiveness of various noise reduction filters in accurately measuring temperature distributions over a 38.2 km single-mode fiber (SMF). Brillouin optical time-domain analysis (BOTDA) sensor is utilized to gather Brillouin gain spectra (BGSs), which are denoised using bilateral filter (BF), guided filter (GF), adaptive Wiener filter (AWF), non-local means filter (NLMF), average filter (AF) and disc filter (DF). The temperature distributions over the SMF are then determined by applying least-squares curve fitting (LSCF). The study assesses the efficacy of noise reduction filters considering signal-to-noise ratio (SNR), uncertainty in temperature measurement (UTM), experimental spatial resolution (ESR) and signal processing speed (SPS). Among six different filters, NLMF outperforms other filters which can provide SNR improvement of 10.22 dB for which the UTM can be improved by 58.93% without deteriorating the ESR of the sensor. The noise reduction using such filter can also provide 6.2% faster SPS. Therefore, NLMF can be considered as an effective noise reduction filter for the precise temperature measurement using BOTDA sensors. DUJASE Vol. 8 (1) 42-50, 2023 (January)

Improved method for drone sound event detection system aiming at the impact of background noise and angle deviation

Drones pose a hidden threat to public safety, and effective and accurate detection technology for the drone that exist in the environment is imminent, in which how to weaken the influence of target sound source localization deviation and strong background interference noise on the detection task is the key to improve the accuracy of drone sound event detection. In this paper, a method has been proposed to improve the accuracy of drone acoustic event detection, named as the linear shrinkage-subspace projection-power spectral density filter method (LSP). This method mainly including covariance matrix reconstruction, steering vector recalibration, and filter coefficient redesign method. Firstly, based on Minimum Variance Distortionless Response, the linear shrinkage method is used to suppress the interference and noise components in the signal plus interference covariance matrix, and the sample covariance matrix is reconstructed to eliminate background interference noise. Then, the correlation between the steering vector and the eigenvector is used to eliminate the angle correlation term, and the subspace projection method is combined to recalibrate the steering vector, so as to improve the ability of the beamforming method to resist the angle deviation and realize the correction of the target source positioning deviation. Next, a redesign method for wiener filter coefficients based on the estimated power spectral density is used to further weaken background interference noise. In order to verify the accuracy of the proposed method, a complete drone sound event detection system is constructed by combining the deep learning drone sound event detection classifier, and the evaluation is carried out according to different angular deviations and interference sound distances. In addition, a new evaluation criterion is proposed, named as the Machine-Human Extreme Hearing Distance Rate (MHDR), which analogizes the system's detection ability with the ear's auditory detection ability. The research results of this article indicate that the detection accuracy of the detection system shows satisfactory accuracy when the proposed method is applied to circular microphone array, that improved by 15.12 % compared to existing methods. The proposed method improves the detection accuracy of the drone acoustic event detection task under the influence of sound source position deviation and strong background speech interference, and provides a reference for the development of anti-drone technology.

An SNR Enhancement Method for Φ-OTDR Vibration Signals Based on the PCA-VSS-NLMS Algorithm.

To improve the signal-to-noise ratio (SNR) of vibration signals in a phase-sensitive optical time-domain reflectometer (Φ-OTDR) system, a principal component analysis variable step-size normalized least mean square (PCA-VSS-NLMS) denoising method was proposed in this study. First, the mathematical principle of the PCA-VSS-NLMS algorithm was constructed. This algorithm can adjust the input signal to achieve the best filter effect. Second, the effectiveness of the algorithm was verified via simulation, and the simulation results show that compared with the wavelet denoising (WD), Wiener filtering, variational mode decomposition (VMD), and variable step-size normalized least mean square (VSS-NLMS) algorithms, the PCA-VSS-NLMS algorithm can improve the SNR to 30.68 dB when the initial SNR is -1.23 dB. Finally, the PCA-VSS-NLMS algorithm was embedded into the built Φ-OTDR system, an 11.22 km fiber was measured, and PZT was added at 10.19-10.24 km to impose multiple sets of fixed-frequency disturbances. The experimental results show that the SNR of the vibration signal is 8.77 dB at 100 Hz and 0.07 s, and the SNR is improved to 26.17 dB after PCA-VSS-NLMS filtering; thus, the SNR is improved by 17.40 dB. This method can improve the SNR of the system's position information without the need to change the existing hardware conditions, and it provides a new scheme for the detection and recognition of long-distance vibration signals.

Prediction method of human defecation based on informer audio data augmentation and improved residual network

Defecation care for disabled patients is a major challenge in health management. Traditional post-defecation treatment will bring physical injury and negative emotions to patients, while existing pre-defecation forecasting care methods are physically intrusive. On the basis of exploring the mechanism of defecation intention generation, and based on the characteristic analysis and clinical application of bowel sounds, it is found that the generation of desire to defecate and bowel sounds are correlated to a certain extent. Therefore, a deep learning-based bowel sound recognition method is proposed for human defecation prediction. The wavelet domain based Wiener filter is used to filter the bowel sound data to reduce other noise. Statistical analysis, fast Fourier transform and wavelet packet transform are used to extract the integrated features of bowel sound in time, frequency and time-frequency domain. In particular, an audio signal expansion data algorithm based on the Informer model is proposed to solve the problem of poor generalization of the training model caused by the difficulty of collecting bowel sound in reality. An improved one-dimensional residual network model (1D-IResNet) for defecation classification prediction is designed based on multi-domain features. The experimental results show that the proposed bowel sound augmentation strategy can effectively improve the data sample size and increase the sample diversity. Under the augmented dataset, the training speed of the 1D-IResNet model is accelerated, and the classification accuracy reaches 90.54 %, the F1 score reaches 83.88 %, which achieves a relatively good classification stability while maintaining a high classification index.

Emotion detection using EEG: hybrid classification approach

The field of emotion research facilitates the development of several applications, all of which aim to precisely and swiftly identify emotions. Speech and facial expressions are the main focus of typical emotion analysis, although they are not accurate indicators of true feelings. Signal analysis, namely the electroencephalograph (EEG) of the brain signals, is the other area in which emotions are analyzed. When compared to other modalities, EEG offers precise and comprehensive data that facilitates the estimation of emotional states. In order to categories the emotions using an EEG signal, this work suggests a hybrid classifier (HC). The input EEG data is preprocessed using the wiener filtering approach to extract the original information from the noisy signal. The preprocessed signal is used to extract features, such as entropy and a new hybrid model that includes models such as Bi-directional long short-term memory (Bi-LSTM) and improved recurrent neural networks (IRNN), which trains using the retrieved features, is included as part of the classification process. Happy, sad, calm, and angry are the categorization findings; the suggested work demonstrates more accurate classification results than the traditional approaches. All these are done on DEAP dataset with 60%, 70%, 80%, and 90% training sets and also a new DOSE dataset is been created similar to DEAP dataset.

An optimal sensor layout method based on noise reduction estimation for active road noise control

Active noise control (ANC) is an effective method to suppress vehicle road noise. The rapid selection of the reference sensor set and rational selection of its layout position are crucial for determining the vibration noise transmission paths with high contributions, which determines the noise reduction performance of the active road noise control (ARNC) system. The reference sensor set selected by the conventional multiple coherence function (MCOH) method does not always ensure the optimal noise reduction performance and the matrix singularity problem based on the Fisher information matrix (FIM) expansion method. To alleviate this problem, this paper proposes a method based on the overall level of noise reduction (OANR) and the FIM to select a set of reference sensors and determine their positions. This method uses an optimal causally constrained Wiener filter to estimate the theoretical OANR for each control area to ensure the reference sensor selected has the optimal noise reduction performance, and it expands efficiently based on the FIM method to select a set of reference sensors. Besides, the matrix singularity problems in the FIM-based expansion method are solved using noise frequency summation and matrix operation, which makes the method more universally applicable. Numerical simulations are performed to analyze and evaluate the noise reduction performance and selection efficiency of the proposed method under different conditions. In addition, a series of real vehicle ANC experiments are conducted. The results show that the reference sensor set selected based on the proposed method has an obvious advantage in noise reduction performance, and only four reference sensors are needed to achieve 4.63 dB(A) noise reduction in ARNC experiments. It fully confirms the effectiveness of the proposed method in real vehicle applications.

Comparing Mass Mapping Reconstruction Methods with Minkowski Functionals

Using higher-order statistics to capture cosmological information from weak lensing surveys often requires a transformation of observed shear to a measurement of the convergence signal. This inverse problem is complicated by noise and boundary effects, and various reconstruction methods have been developed to implement the process. Here we evaluate the retention of signal information of four such methods: Kaiser-Squires, Wiener filter, 𝙳𝚊𝚛𝚔𝙼𝚊𝚙𝚙𝚢, and 𝙳𝚎𝚎𝚙𝙼𝚊𝚜𝚜. We use the higher order statistics 𝑀𝑖𝑛𝑘𝑜𝑤𝑠𝑘𝑖 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑎𝑙𝑠 to determine which method best reconstructs the original convergence with efficiency and precision. We find 𝙳𝚎𝚎𝚙𝙼𝚊𝚜𝚜 produces the tightest constraints on cosmological parameters, while Kaiser-Squires, Wiener filter, and 𝙳𝚊𝚛𝚔𝙼𝚊𝚙𝚙𝚢 are similar at a smoothing scale of 3.5 arcmin. We also study the MF inaccuracy caused by inappropriate training sets in the 𝙳𝚎𝚎𝚙𝙼𝚊𝚜𝚜 method and find it to be large compared to the errors, underlining the importance of selecting appropriate training cosmologies.

Broiler health monitoring technology based on sound features and random forest

The existing broiler health monitoring technology has problems such as low automation, unstable monitoring results, and low practical value, making it difficult to provide timely and reliable broiler health monitoring results. The broiler sound signal can provide feedback on their health. A widely validated and correct experience is to analyze the frequency of coughs in a segment of broiler sound signal to determine the health of the broiler group. Based on this, in this paper, the authors proposed a new broiler health monitoring technology based on sound detection. The broiler health monitoring problem is cleverly transformed into a multi-classification problem, which can be solved by identifying the sound types in broiler sound signals. Specifically, the audio signal collection system was designed to complete signal collection and preliminary signal filtering. Wiener filtering was used for deep signal filtering. The 60-dimensional sound features with good performance from three aspects, time-frequency domain, Mel-Frequency Cepstral Coefficients, and sparse representation were extracted, and a preliminary data set was created. Min-max normalization was used to align the numerical distribution of the data set, and a high-quality data set was created. Multi-classification models based on different classification algorithms and neural networks were trained, and the best-performing Random Forest was obtained, thus parameter optimization was carried out, and the optimal multi-classification model was obtained, achieving a classification accuracy of 91.14%. The visualization platform was built to process the classification results of the multi-classification model, completing majority voting processing and cough rate calculation, thereby achieving broiler health monitoring. In addition, the definitions of cough rate and prediction accuracy were newly proposed. A large number of experiments have verified the feasibility of the broiler health monitoring technology proposed in this paper, with an average prediction accuracy of 98.97% achieved.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.