White Noise Research Articles

Robust Direction-of-Arrival Estimation in the Presence of Outliers and Noise Nonuniformity

In direction-of-arrival (DOA) estimation with sensor arrays, the background noise is usually modeled to be uncorrelated uniform white noise, such that the related algorithms can be greatly simplified by making use of the property of the noise covariance matrix being a diagonal matrix with identical diagonal entries. However, this model can be easily violated by the nonuniformity of sensor noise and the presence of outliers that may arise from unexpected impulsive noise. To tackle this problem, we first introduce an exploratory factor analysis (EFA) model for DOA estimation in nonuniform noise. Then, to deal with the outliers, a generalized extreme Studentized deviate (ESD) test is applied for outlier detection and trimming. Based on the trimmed data matrix, a modified EFA model, which belongs to weighted least-squares (WLS) fitting problems, is presented. Furthermore, a monotonic convergent iterative reweighted least-squares (IRLS) algorithm, called the iterative majorization approach, is introduced to solve the WLS problem. Simulation results show that the proposed algorithm offers improved robustness against nonuniform noise and observation outliers over traditional algorithms.

Design and Analysis of a Contact Piezo Microphone for Recording Tracheal Breathing Sounds.

Analysis of tracheal breathing sounds (TBS) is a significant area of study in medical diagnostics and monitoring for respiratory diseases and obstructive sleep apnea (OSA). Recorded at the suprasternal notch, TBS can provide detailed insights into the respiratory system's functioning and health. This method has been particularly useful for non-invasive assessments and is used in various clinical settings, such as OSA, asthma, respiratory infectious diseases, lung function, and detection during either wakefulness or sleep. One of the challenges and limitations of TBS recording is the background noise, including speech sound, movement, and even non-tracheal breathing sounds propagating in the air. The breathing sounds captured from the nose or mouth can interfere with the tracheal breathing sounds, making it difficult to isolate the sounds necessary for accurate diagnostics. In this study, two surface microphones are proposed to accurately record TBS acquired solely from the trachea. The frequency response of each microphone is compared with a reference microphone. Additionally, this study evaluates the tracheal and lung breathing sounds of six participants recorded using the proposed microphones versus a commercial omnidirectional microphone, both in environments with and without background white noise. The proposed microphones demonstrated reduced susceptibility to background noise particularly in the frequency ranges (1800-2199) Hz and (2200-2599) Hz with maximum deviation of 2 dB and 2.1 dB, respectively, compared to 9 dB observed in the commercial microphone. The findings of this study have potential implications for improving the accuracy and reliability of respiratory diagnostics in clinical practice.

Modeling the Long-Term Variability in the Surfaces of Three Lakes in Morocco with Limited Remote Sensing Image Sources

Satellite imagery has become a widespread resource for modeling variability in lake surfaces. However, the extended monitoring of a lake’s perimeter faces significant challenges due to atmospheric obstacles that cannot be rectified. Due to the atmosphere’s everchanging opacity, only half of the acquired satellite images have reliable qualitative accuracy making it possible to identify a lake’s contour. Consequently, approximately 50% of the monthly lake outline values can be determined using remote sensing methods, leaving the remaining 50% unknown. This situation is applicable to three lakes in Morocco (Abakhan, Ouiouan, and Tiglmanine), the subjects of the current research for the period between 1984 and 2022. What can we do if, during a period of time in which we monitored the evolution of the surface of a lake by satellite means, we obtain only about 50% of the possible images? Shall we just settle for this and stop the analysis? Although it may be challenging to believe, the present study introduces two statistical methods for interpolating and validating the monthly values of the lake outline: the iterative ratio method based on the autocorrelation of the monthly water balance and the Kalman filter. We estimated the reconstruction errors of the missing values and validated the methodology using an inverse philosophy, reconstructing the initial data from the table of the simulation results. Given that the difference between the initial values and the reconstructed initial values resembles white noise or an AR (1) process with a low coefficient, we deemed the methodological approach acceptable. Several comparison criteria between the two interpolation methods were employed, yet determining the more appropriate one remains challenging. Based on our surface reconstruction method, Lake Abakhan, with an average area of 22 hectares, experienced significant fluctuations, ranging from a maximum of 34 hectares in 2010 to a minimum of 0.8 hectares in 2022. Lake Ouiouan, with an average area of 14 hectares, displayed much lower variation, with a maximum of 17 hectares in 2020 and a minimum of 6.5 hectares in 1988. Lake Tiglmanine showed a pattern similar to that of Lake Abakhan but with less pronounced fluctuations. With an average area of 6.1 hectares, its maximum was 9.2 hectares in 2011 and its minimum was 4.1 hectares in 1984.

LSTM-BIGRU based Spectrum Sensing for Cognitive Radio

There is a shortage of wireless spectrum due to developments in the area of wireless communications as well as the number of users that are using resources. Spectrum sensing is a method that solves the issue of shortage. Deep learning surpasses classical methods in spectrum sensing by enabling autonomous feature learning, which enables the adaptive identification of complicated patterns in radio frequency data for cognitive radio in wireless sensor networks. This innovation increases the system's capacity to manage dynamic, real-time circumstances, resulting in increased accuracy over traditional approaches. Spectrum sensing (SS) using LSTM-BIGRU with gaussian noise has been suggested in this article. Long-term dependencies in sequential data are well- preserved by LSTM due to its dedicated memory cells. In addressing and man- aging long-term dependencies in sequential data, BIGRU's integration enhances the efficacy of the model as a whole. To conduct the investigation, RadioML2016.04C.multisnr open-source dataset was utilized. Whereas, by using RadioML2016.10b open-source dataset, QAM64, QPSK and QAM16 performance evaluation has been investigated. The experimental findings demonstrate that the suggested Spectrum Sensing has better accuracy on the dataset particularly at lower SNRs. The improved spectrum sensing (SS) performance of our suggested model is shown by the evaluation of performance indicators, such as the F1 Score, CKC and Matthew's correlation coefficient, highlighting its potency in the field of spectrum sensing applications.

Estimation of the Impulse Response of the AWGN Channel with ISI within an Iterative Equalization and Decoding System That Uses LDPC Codes.

In this paper, new schemes have been proposed for the estimation of the additive white Gaussian noise (AWGN) channel with intersymbol interference (ISI) in an iterative equalization and decoding system using low-density parity check (LDPC) codes. This article explores the use of the least squares algorithm in various scenarios. For example, the impulse response of the AWGN channel h was initially estimated using a training sequence. Subsequently, the impulse response was calculated based on the training sequence and then re-estimated once using the sequence estimated from the output of the LDPC decoder. Lastly, the impulse response was calculated based on the training sequence and re-estimated twice using the sequence estimated from the output of the LDPC decoder. Comparisons were made between the performances of the three mentioned situations, with the situation in which a perfect estimate of the impulse response of the channel is assumed. The performance analysis focused on how the bit error rate changes in relation to the signal-to-noise ratio. The BER performance comes close to the scenario of having a perfect estimate of the impulse response when the estimation is performed based on the training sequence and then re-estimated twice from the sequence obtained from the output of the LDPC decoder.

Control and stochastic dynamic behavior of Fractional Gaussian noise-excited time-delayed inverted pendulum system

In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H∈(1/2,1), we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic Itoˆ equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system.

Effects of high-intensity chronic noise on spatial memory in male versus female rats.

The detrimental effects of high-intensity noise on the auditory system and emotional status, including the induction of anxiety, are well documented. Preclinical as well as epidemiological and clinical studies have solidly established differential responses between males and females to various stressful stimuli, including high-intensity white noise (HIWN). However, whether chronic exposure to noise affects cognitive functions and whether this effect is sex dependent has not been adequately addressed. In this study, we used two cognitive test paradigms, such as the Morris water maze (MWM) and the multi-branch maze (MBM), to test the effect of chronic HIWN on indices of spatial learning and memory in both male and female Wistar rats. Our findings indicate that daily (1 h) exposure to 100 dB of noise for 30 consecutive days induces different task-dependent responses in male versus female rats. For example, in the acquisition phase of MWM, female rats exposed to noise outperformed their male counterparts at twice the speed. Similarly, in the MBM test, noise-exposed female rats outperformed the male rats in reaching the nest box. It is clear from these studies that noise impairs cognitive functions twice as negatively in male rats as in female rats. Thus, sex-related differences in spatial learning and memory in response to HIWN must be taken into consideration when investigating the neurobiological components and/or treatment modalities.

Self-similarity and probability density function of the transient response of fractional compound motion

In recent years, there has been an increasing use of fractional differential equations due to their ability to effectively represent various engineering phenomena, including viscoelasticity, heat transport, non-local continuum, and others. These equations take into account certain effects that cannot be accurately predicted using classical differential equations.This paper provides a comprehensive analysis of the fractional compound motion, specifically focusing on the response of a one-term fractional differential equation that is excited by a Poissonian white noise process. The present study introduces a straightforward equation for the probability density function of fractional compound motion. The validity of this equation is subsequently confirmed by the execution of various numerical simulations. Furthermore, a comprehensive analysis is conducted on the self-similarity of fractional compound motion, demonstrating that the phenomenon can be regarded as self-similar in weak sense. This characteristic can be effectively employed to mitigate the loss of Markovianity in fractional differential equations.

DP-FedEwc: Differentially private federated elastic weight consolidation for model personalization

Federated learning (FL) has become a prevalent paradigm for training a model collaboratively on multiple clients with the coordination of a central server. As traditional FL suffers from client-drift due to data heterogeneity across clients, many personalized FL (PFL) techniques have been proposed. However, the issue of privacy leakage within PFL remains inadequately addressed. When incorporating differential privacy (DP) directly into PFL to provide rigorous privacy guarantees, it amplifies heterogeneity among clients and introduces high variance in uploaded information, significantly compromising model’s utility. In this paper, we propose a novel privacy-preserving PFL framework called Differentially Private Federated Elastic weight consolidation (DP-FedEwc), to achieve effective model personalization for each client under sample-level DP. We focus on a practical setting where the server is honest-but-curious. We first implement a FedEwc algorithm in a communication-efficient manner, and provide privacy guarantees by perturbing models and their parameter importance (PI). We show that FedEwc is robust to DP-introduced heterogeneity caused by noisy models, especially when the model is a deep neural network. Since excessive noise may render PI invalid, we present an Adaptive Parameter importance Perturbation (APP) method to adaptively add Gaussian noise to different coordinates of PI, thereby alleviating the negative effect of DP noise. Moreover, to accurately calibrate the privacy cost resulting from querying heterogeneous data across various clients when computing PI through APP, we adapt a Bayesian Accountant (BA) method to DP-FedEwc. We conduct experiments on standard benchmark datasets, and the experimental results confirm the superiority of DP-FedEwc over DP-PFL baselines.

Image dejittering on the perspective of spatially-varying mixed noise removal

Jittery image is visually abnormal in jags of edge and loss of coherence. The problem of image dejittering is challenging to resolve due to the ubiquitous blur and/or noise in jittery data. In this paper, we devote to the pixel-jitter (possibly blurry) image recovery on the perspective of spatially-varying mixed noise removal. By viewing jittery image as the corruption of ideal image with outliers and spatially-varying Gaussian noise, we proposed a two-phase (including filtering and diffusing phases) image dejittering approach. In the filtering phase, outliers posed by jitters around edges are inspected by median filters. In the diffusing phase, structure tensor based anisotropic diffusion is exploited to reduce the perturbations in piecewise smooth image regions. Upon the spectral decomposition of structure tensor, the variational model in diffusing phase can be solved by some state-of-the-art optimization methods. Numerical simulations on synthetic and real jittery data demonstrate the compelling performance of the proposed approach. The Matlab source codes of the proposed approach are available at the repositories of https://github.com/WenxingZhang.

Nonlinear vibro-acoustic modulation for microcrack detection of steel strands based on S-transform bispectrum

Fatigue crack detection is an important issue to ensure the safety of steel strands. The key to solve this problem is to extract the nonlinear response generated by fatigue cracks. In this paper, the nonlinear VAM method is used to detect the structural cracks. The structure with fatigue cracks is excited using low-frequency pumping and high-frequency probing, and the power spectrum analysis of the modulated signal is carried out. In view of the shortcomings of spectral analysis, a new method combining S-transform and bispectrum is proposed, which is called S-transform bispectrum. S transform contains the phase factor, which can retain the absolute phase characteristics of each frequency, and has good time–frequency multi-scale focusing performance. The bispectrum can suppress Gaussian noise, retain phase information, and quantitatively describe the quadratic phase coupling in the signal. Then the simulation and experiment of damaged straight rod, helical rod, and steel strands are carried out. The results show that the proposed method can effectively detect the nonlinear features by using the sideband peaks in the S-transform bispectrum three-dimensional plot, and the nonlinear features are important to identify the structure with damage. At the same time, in order to prove the ability of S-transform bispectrum, an S-transform bispectrum detector is used to verify it, which is superior to the spectrum in terms of its ability to localize modulation sidebands. The proposed S-transform bispectrum has a good application prospect and provides a new tool for structural damage detection.

Stochastic effects on plankton dynamics: Insights from a realistic 0-dimensional marine biogeochemical model

Marine ecosystems exist in a noisy and uncertain environment, not governed by deterministic laws. The development of ecological communities is significantly influenced by variability, and the interaction between nonlinearity and stochastic processes can lead to phenomena that deterministic models cannot explain. Plankton, forming the base of the marine food web, are highly affected by stochastic fluctuations due to their short reproductive timescales. Investigating the effects of noise on plankton growth is essential for accurately describing and predicting marine health. We present a realistic biogeochemical model where multiplicative white noise represents environmental stochasticity affecting plankton. The model suggests ergodic properties in the presence of stochastic fluctuations, with temporal and ensemble distributions being coherent. Analytical and numerical analyses reveal that, given sufficiently low noise intensity, dynamics near equilibrium resemble an Ornstein-Uhlenbeck additive process. With higher noise intensities, resonance occurs, particularly when endogenous dynamics are periodic. The results indicate that low noise intensity can positively influence plankton persistence with an higher number of species coexisting, while higher noise intensity can establish a new equilibrium in the system.

Detection Method of Stator Coating Quality of Flat Wire Motor Based on Improved YOLOv8s.

The stator of a flat wire motor is the core component of new energy vehicles. However, detecting quality defects in the coating process in real-time is a challenge. Moreover, the number of defects is large, and the pixels of a single defect are very few, which make it difficult to distinguish the defect features and make accurate detection more difficult. To solve this problem, this article proposes the YOLOv8s-DFJA network. The network is based on YOLOv8s, which uses DSFI-HEAD to replace the original detection head, realizing task alignment. It enhances joint features between the classification task and localization task and improves the ability of network detection. The LEFG module replaces the C2f module in the backbone of the YOLOv8s network that reduces the redundant parameters brought by the traditional BottleNeck structure. It also enhances the feature extraction and gradient flow ability to achieve the lightweight of the network. For this research, we produced our own dataset of stator coating quality regarding flat wire motors. Data augmentation technology (Gaussian noise, adjusting brightness, etc.) enriches the dataset, to a certain extent, which improves the robustness and generalization ability of YOLOv8s-DFJA. The experimental results show that in the performance of YOLOv8s-DFJA compared with YOLOv8s, the mAP@.5 index increased by 6.4%, the precision index increased by 1.1%, the recall index increased by 8.1%, the FPS index increased by 9.8FPS/s, and the parameters decreased by 3 Mb. Therefore, YOLOv8s-DFJA can be better realize the fast and accurate detection of the stator coating quality of flat wire motors.

Edge–cloud collaborative estimation lithium-ion battery SOH based on MEWOA-VMD and Transformer

The State of Health (SOH) of lithium-ion batteries significantly impacts the performance, safety, and reliability of the battery, making it a crucial component of the battery management system. Addressing the issues of inadequate accuracy and lack of robustness in current SOH estimation methods, this study introduces a novel methodology for estimating SOH in lithium-ion batteries. It leverages the multi-population evolution whale optimization algorithm optimized variational mode decomposition (MEWOA-VMD) in conjunction with Transformer architecture. This framework enhances the efficiency and accuracy of SOH estimation by leveraging the computational capabilities of edge devices for real-time data processing, as well as the robust data processing power and model training advantages offered by cloud computing. Specifically, MEWOA is utilized to optimize VMD parameters, enabling MEWOA-VMD to fully decompose the capacity signal of lithium-ion batteries. This results in a component showing a global attenuation trend and a set of fluctuating components that represent capacity regeneration, thereby minimizing the impact of capacity regeneration on SOH estimation. Subsequently, all components are collectively input into the Transformer, marking the first application of this method for input. To enhance convergence speed and training efficiency, the layer normalization (LN) layer within the neural network architecture is proactively advanced. Finally, various artificial neural networks are compared and validated on three publicly available datasets. Furthermore, Gaussian noise is introduced into the original data to validate robustness. To confirm the practical applicability of the proposed method, real-world vehicle data is used for SOH estimation. The results indicate that the proposed method achieves a maximum MSE of no more than 0.009% across three publicly available datasets, showcasing improved accuracy and stability in SOH estimation. The practical applicability is further validated using real-world vehicle data, proving the proposed method’s potential for application in edge cloud-based battery management systems.

Incorporating memory effect into a fractional stochastic diffusion particle tracking model for suspended sediment using Malliavin-Calculus-based fractional Brownian Motion

This study aims to develop a Fractional Stochastic Diffusion Particle Tracking Model (FSDPTM) to illustrate suspended sediment’s long-term dependence (LTD) trajectories immersed in fully developed turbulent flow. Carried by eddies in the open channel, individual suspended sediment behavior exhibits dependence on its past increments. The Fractional Brownian Motion (FBM) is incorporated into the FSDPTM to account for the randomness of turbulent flow and to include the time-dependent increments as a memory effect. Additionally, to enhance the capacity for differentiating FBM, this study introduces Malliavin Calculus, the stochastic calculus of variation, to construct the square -differentiable FBM. For the numerical realizations of square differentiable FBM, the integration of fractional Gaussian white noise and Wiener–Ito Chaos expansion with the Hermite sequence is employed. The FSDPTM offers probabilistic LTD trajectories as solutions to the Ito-type Langevin equation. It also provides particle velocity, acceleration, and ensemble statistics, collectively demonstrating the application of FBM to sediment transport modeling. In conclusion, the key feature of the FSDPTM is its ability to translate LTD derived from FBM into the trajectory of suspended sediment particles in fully developed turbulent flow. Additionally, it expands the solutions from non-differentiable to square-differentiable stochastic processes.

Generative adversarial networks with deep blind degradation powered terahertz ptychography

Ptychography is an imaging technique that uses the redundancy of information generated by the overlapping of adjacent light regions to calculate the relative phase of adjacent regions and reconstruct the image. In the terahertz domain, in order to make the ptychography technology better serve engineering applications, we propose a set of deep learning terahertz ptychography system that is easier to realize in engineering and plays an outstanding role. To address this issue, we propose to use a powerful deep blind degradation model which uses isotropic and anisotropic Gaussian kernels for random blurring, chooses the downsampling modes from nearest interpolation, bilinear interpolation, bicubic interpolation and down-up-sampling method, and introduces Gaussian noise, JPEG compression noise, and processed detector noise. Additionally, a random shuffle strategy is used to further expand the degradation space of the image. Using paired low/high resolution images generated by the deep blind degradation model, we trained a multi-layer residual network with residual scaling parameters and dense connection structure to achieve the neural network super-resolution of terahertz ptychography for the first time. We use two representative neural networks, SwinIR and RealESRGAN, to compare with our model. Experimental result shows that the proposed method achieved better accuracy and visual improvement than other terahertz ptychographic image super-resolution algorithms. Further quantitative calculation proved that our method has significant advantages in terahertz ptychographic image super-resolution, achieving a resolution of 33.09 dB on the peak signal-to-noise ratio (PSNR) index and 3.05 on the naturalness image quality estimator (NIQE) index. This efficient and engineered approach fills the gap in the improvement of terahertz ptychography by using neural networks.

Prognostic Properties of Instantaneous Amplitudes Maxima of Earth Surface Tremor.

A method is proposed for analyzing the tremor of the earth's surface, measured by GPS, in order to highlight prognostic effects. The method is applied to the analysis of daily time series of vertical displacements in Japan. The network of 1047 stations is divided into 15 clusters. The Huang Empirical Mode Decomposition (EMD) is applied to the time series of the principal components from the clusters, with subsequent calculation of instantaneous amplitudes using the Hilbert transform. To ensure the stability of estimates of the waveforms of the EMD decomposition, 1000 independent additive realizations of white noise of limited amplitude were averaged before the Hilbert transform. Using a parametric model of the intensities of point processes, we analyze the connections between the instants of sequences of times of the largest local maxima of instantaneous amplitudes, averaged over the number of clusters and the times of earthquakes in the vicinity of Japan with minimum magnitude thresholds of 5.5 for the time interval 2012-2023. It is shown that the sequence of the largest local maxima of instantaneous amplitudes significantly more often precedes the moments of time of earthquakes (roughly speaking, has an "influence") than the reverse "influence" of earthquakes on the maxima of amplitudes.

Schizophrenia diagnosis based on diverse epoch size resting-state EEG using machine learning.

Schizophrenia is a severe mental disorder that impairs a person's mental, social, and emotional faculties gradually. Detection in the early stages with an accurate diagnosis is crucial to remedying the patients. This study proposed a new method to classify schizophrenia disease in the rest state based on neurologic signals achieved from the brain by electroencephalography (EEG). The datasets used consisted of 28 subjects, 14 for each group, which are schizophrenia and healthy control. The data was collected from the scalps with 19 EEG channels using a 250 Hz frequency. Due to the brain signal variation, we have decomposed the EEG signals into five sub-bands using a band-pass filter, ensuring the best signal clarity and eliminating artifacts. This work was performed with several scenarios: First, traditional techniques were applied. Secondly, augmented data (additive white Gaussian noise and stretched signals) were utilized. Additionally, we assessed Minimum Redundancy Maximum Relevance (MRMR) as the features reduction method. All these data scenarios are applied with three different window sizes (epochs): 1, 2, and 5 s, utilizing six algorithms to extract features: Fast Fourier Transform (FFT), Approximate Entropy (ApEn), Log Energy entropy (LogEn), Shannon Entropy (ShnEn), and kurtosis. The L2-normalization method was applied to the derived features, positively affecting the results. In terms of classification, we applied four algorithms: K-nearest neighbor (KNN), support vector machine (SVM), quadratic discriminant analysis (QDA), and ensemble classifier (EC). From all the scenarios, our evaluation showed that SVM had remarkable results in all evaluation metrics with LogEn features utilizing a 1-s window size, impacting the diagnosis of Schizophrenia disease. This indicates that an accurate diagnosis of schizophrenia can be achieved through the right features and classification model selection. Finally, we contrasted our results to recently published works using the same and a different dataset, where our method showed a notable improvement.

Dynamical behavior of obligatory mutualistic-cheater interactions under the influence of white noise

Intraspecific mutualism is vital for ecosystems. Some interactions involve one species benefiting without reciprocating. Recent experiments suggest stable co-existence between mutualists and cheaters. This paper focuses on interactions between obligate mutualistic species and their cheaters, using a modified Lotka-Volterra model to analyze the negative impact of cheaters. We establish conditions for stability and verify a diffusive system's stability under spatial effects. Additionally, we study population dynamics fluctuations in the presence of Gaussian noise.

Optimization of tuned mass dampers – minimization of potential energy of elastic deformation

A tuned mass damper (TMD) optimization can be performed under various assumptions and objectives. All the variables of the optimization, such as structural model, performance index and load type affect the optimal parameters of the TMD. This paper presents a new optimization method that implements straightforward performance index and allows taking load spectral characteristics into account. Thanks to the usage of modal coordinates, the method allows fast numerical optimization of TMD attached to large or complicated structures with numerous degrees of freedom. One of the complicated tasks while optimizing TMD is the choice of a performance index. In this paper, the mean value of potential energy stored in the elastic deformation of a structure under periodic load serves as a performance index, which leads to a low numerical complexity task if the optimization is performed in the frequency domain. The new method also allows a simple inclusion of load spectral characteristics and permits TMD optimization for any loading spectral range. When applied to a structure with a single degree of freedom, this method leads to H2 optimization in the case of white noise excitation. However, it is applicable to multiple degrees of freedom structures with single or multiple TMDs and any given load. The paper also presents several examples of numerical optimization of the TMD attached to both single and multiple degrees of freedom structures under various loads, including white noise excitation, pedestrian load, and earthquake strong motion.