Background and Objectives: Substance use disorders (SUDs) are associated with maladaptive neuroplasticity and chronic dysregulation of cortical arousal. EEG provides a non-invasive tool for quantifying these neurophysiological alterations through spectral power and reactivity indices. Prior research consistently reports elevated beta and diminished alpha activity in SUD, reflecting cortical hyperarousal and reduced inhibitory control. This study sought to identify EEG-based markers of dependence-related neurophysiological alterations by integrating rule-based and score-based models incorporating the theta/beta ratio (TBR), alpha and beta powers, the hyperarousal index, and alpha-blocking measures. Materials and Methods: EEG recordings from 47 individuals with SUD were systematically analyzed, focusing on frontal and central cortical regions. Spectral parameters were derived using power spectral density estimation, and composite indices were computed via Python-based signal analysis. A rule-based Dependence Likelihood variable and a continuous Dependence Score (0-1 scale) classified cases as dependence-related (≥0.7), borderline (0.5-0.7), or normal (<0.5). Results: Low alpha power and an elevated hyperarousal index (mean = 3.45) characterized most participants. Dependence-related EEG profiles were identified in 87.2% of cases (mean score = 0.86). Alpha blocking remained intact in 46.8% of cases, whereas post-hyperventilation recovery was attenuated in 61.7% of cases. Segmental analysis indicated sustained cortical activation with low TBR (0.37) and elevated beta across all conditions. Conclusions: Quantitative EEG analysis revealed consistent hyperarousal and inhibitory deficits in SUD. The combined Dependence Likelihood and Score framework provides an interpretable, reproducible approach for identifying dependence-related EEG signatures and holds promise as a biomarker in addiction neurophysiology.

The a posteriori speech presence probability (SPP) plays a critical role in noise power spectral density (PSD) estimation, which is essential for both speech enhancement and recognition systems. While existing SPP estimators perform well under stationary noise conditions, challenges remain in accurately estimating SPP in non-stationary environments and in reducing the computational complexity of deep learning-based methods. In this paper, we build upon a previously proposed hybrid global–local information-based SPP estimation framework and extend its analysis through a comprehensive experimental study. The framework incorporates joint global and local spectral representations and includes targeted refinements aimed at improving robustness in low signal-to-noise ratio (SNR) scenarios. Beyond standalone SPP estimation, the proposed approach is evaluated in downstream applications, including noise PSD estimation and speech enhancement, across multiple datasets and noise conditions. Experimental results demonstrate consistent performance improvements over conventional approaches and provide clear evidence of the robustness and practical effectiveness of the SPP-based framework in non-stationary environments.

Abstract Depression is a severe mental disorder that influences the emotions, reasoning ability, and behaviour of a person pessimistically. It is associated with several cognitive deficits, including difficulties with attention and concentration. The purpose of this study is to examine the impact of depression on three types of attention, divided, selective, and alternative, during the performance of respective attention tasks using electroencephalogram (EEG) analytics.Twenty-four adults aged 18–21 volunteered for the study. The Welch power spectral density estimation method is adopted to compute the spectral powers of alpha, beta, and theta bands, followed by the analysis of event-related synchronization/desynchronization. The changes in functional connectivity of the brain during the attention tasks are visualized through the magnitude of event-related change from three band powers.The results revealed the reduced alpha desynchronization (p &lt; 0.05) and theta synchronization (p &lt; 0.05) in the experimental group for the provided tasks. It is inferred that the association of alpha desynchronization with attention to task stimulus and the association of theta synchronization with the encoding and retrieval of information illustrates the attention deficit in the experimental group and poor functional connectivity between lobes.

The paper proposes a practical, scalable, and non-intrusive system for the automatic detection of tool wear under real industrial conditions that does not require process metadata (e.g., spindle speed, feed rate) or specialised equipment. The proposed method is based solely on the analysis of triaxial vibration signals and combines multiple signal-processing methods (time, frequency, and time-frequency) to enable deeper analysis. Various time-domain features, such as RMS, standard deviation, kurtosis, and crest factor, are combined with spectral analysis via Welch power spectral density (PSD) estimation and the continuous wavelet transform (CWT) for time-frequency analysis. To make features comparable across machines and operating conditions, the features are combined using median statistics and normalised relative to the median (Δ%). Experimental validation was performed with multiple machines and measurement axes on real industrial datasets that were heavily imbalanced. It was shown that there is a direct and consistent correlation between the condition of worn tools and the overall increase in tool vibration energy, as evidenced by significantly higher RMS and standard deviation values. On the other hand, higher-order statistical measures, such as kurtosis and crest factor, were less consistent when used alone due to their sensitivity to operational variability. Frequency domain analysis showed that the wear of the tool could not only be described by a general rise in energy, but also by the significant increase of certain frequency components in the spectrum. Most notably, the peaks centred at 200 Hz were found to be significantly raised, along with their harmonics at around 600 Hz and 800 Hz, when the tool was worn, and this was true for all the axes of the measurements. These frequencies can thus be considered reliable and repeatable indicators of tool wear. The continuous wavelet transforms (CWT), a complementary method to time-localised time-frequency representations, confirmed the results and, notably, revealed that the signal's high-energy bursts are time-recurrent and running; thus, they can be seen as a series of bright spots closely packed in time in the scalogram. The strength of the proposal is its rational and transparent integration of several well-known signal-processing methods into a single, coherent concept that encompasses the full spectrum of changes, from the global increase in vibration energy to the localised, frequency-specific excitations that are the hallmark of wear progression. By combining global energy indicators with spectral and time frequency features, the approach enhances diagnostic reliability while improving physical understanding of tool-wear mechanisms. It is robust, practical, and readily transferable, offering strong potential for scalable predictive maintenance applications in machining environments.

Physiological time series reflect the underlying behavior of physiological systems. In this paper, we introduce a novel patching with sequential updating for Bayesian nonparametric spectral estimation (PBNSE) to enhance spectral estimation and interpretation of imperfect physiological time series with fragmented, noncontiguous segments. PBNSE incorporates four key strategies: (1) modeling patches as patch-specific Gaussian processes (GPs); (2) patch-dependence, where each patch involves a joint GP with a shared kernel, capturing both observation and spectral dependencies across all patches; (3) sequential parameter shift that transfers knowledge between patches while maintaining computational traceability; and (4) aggregating patch-level posterior spectra into a unified power spectral density (PSD) estimate and computing the expectation of the PSD in a closed form. Extensive experiments demonstrate significant improvements in spectral accuracy and robustness compared to state-of-the-art methods such as BNSE, multitaper, periodogram, Lomb-Scargle, functional kernel learning (FKL), and variational sparse spectrum (SVSS). PBNSE addresses key challenges in physiological signal analysis, including irregular sampling, incomplete signal, and varying noise. The widespread adoption of PBNSE in physiological signal research has the potential to enhance the accuracy of spectral estimation and improve the robustness of interpreting complex, real-world physiological time series.

The research investigated Electroencephalography (EEG) based biomarkers to distinguish Alzheimer’s disease (AD) from healthy controls (HC). The study applied graph theoretical analysis of directed functional connectivity. A dataset of 65 participants, including 36 with AD and 29 HC, was used. Resting-state, eyes-closed EEG data were obtained from the participants. The preprocessing steps included notch and band-pass filtering, resampling, artifact removal, and re-referencing. The preprocessed data were segmented into overlapping 5-second windows. The segments were filtered into three frequency bands: alpha (8–12 Hz), beta (13–30 Hz), and gamma (30–45 Hz). Dynamic functional connectivity was quantified using the directed phase lag index (dPLI) across 19 channels. Graph theoretical metrics were computed for each subject and frequency band. The metrics included global efficiency, modularity, characteristic path length, clustering coefficient, and rich-club coefficient. Group differences were evaluated with Mann–Whitney U tests and false discovery rate (FDR) correction. The correlation between metrics and cognitive status were evaluated using Spearman correlation with Mini-Mental State Examination (MMSE) scores. Then logistic regression with receiver operating characteristic (ROC) analysis quantified the discriminative performance of significant features. Additional analyses included channel-wise connectivity comparisons, power spectral density (PSD) estimation, and grand average waveform analysis at midline electrodes. The findings indicated significant changes were limited to the alpha band. Participants with AD exhibited higher mean global efficiency, lower temporal variability (CV) and reduced modularity in the alpha band. The results reflect altered network integration with reduced temporal variability (lower CV) and weakened functional segregation in the alpha band. ROC analysis showed moderate discrimination from the alpha band biomarkers (area under curve AUC up to 0.74); using the same biomarkers with alternative classification methods, AUCs reached up to 0.88. Channel-wise analysis revealed selective frontoparietal and occipitofrontal disruptions. The PSD results showed flattened alpha peaks, consistent with EEG slowing in AD. Compared to HC group, AD group exhibited altered amplitudes, latencies, and polarity shifts after grand average waveform analysis. The results of the study also indicates that alpha band network disruptions are reliable EEG markers of AD, specifically in global efficiency and modularity. The results show the frequency specific network dysfunction in AD. The findings support the use of graph theoretical EEG analysis as a noninvasive biomarker method for clinical applications.

A dynamic model of a coupled shaft-bearing system was developed to examine the vibrational behavior caused by Hertzian contact mechanics and localized defects on the inner race. The model integrates mechanical energy balance, nonlinear Hertzian contact stiffness, and the Lagrangian formalism, considering the radial loads, system damping, and geometric characteristics. Numerical simulations executed via the fourth-order Runge–Kutta method reveal distinct vibrational signatures between the healthy and cracked bearing conditions. Under healthy operating conditions, broadband spectral energy is observed without significant peaks, indicating a uniform vibration behavior. The introduction of an inner race crack generates amplitude modulation, increased vibration levels, and pronounced spectral peaks between 600 Hz and 800 Hz. These peaks are consistent with defect-passing frequencies and nonlinear impact phenomena. This study reveals that the local defects significantly impact the dynamic response by producing complex, nonstationary vibration patterns that can be detected through time-domain analysis and power spectral density estimation using the power spectrum of the Fast Fourier Transform (FFT) technique.

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Abstract We establish the consistency and the asymptotic distribution of the least squares estimators of the coefficients of a subset vector autoregressive process with exogenous variables (VARX). Using a martingale central limit theorem, we derive the asymptotic normal distribution of the estimators. Diagnostic checking is discussed using kernel‐based spectral density estimators. Multi‐step conditional prediction intervals are developed. In a small empirical study, subset VARX models are fitted and compared when lags are chosen according to some popular model selection techniques. Coverage properties of the prediction intervals are illustrated. An application to the monthly production of cheese in Canada for the time period 2003–2021 illustrates the methodology.

The neurodevelopmental disorder known as autism spectrum disorder (ASD) is typified by aberrant brain development that is impacted by ‎environmental, biological, and genetic variables. To maximize outcomes and to intervene promptly, it is necessary to identify ASD early. ‎The current methods of screening rely heavily on behavioral assessments, which introduce bias and other limitations. Therefore, objective ‎approaches that mitigate subjectivity should be employed. This study presents using EEG signals and Deep Learning (DL) models to ‎identify ASD via Artificial Intelligence (AI)-based Computer-Aided Diagnostic Systems (CADS.‎ ‎ This study examined how well deep ‎learning classifiers (DL classifiers) could classify people with ASD using only their raw EEG data and no manual feature extraction. ‎Several deep learning models, such as FFNN and WST-ASDNet with AlexNet, were employed to categorize people with ASD and ‎controls. As additional signal processing methods for ASD classification, time-frequency representations, Power Spectral Density (PSD) ‎estimates, and Wavelet Scattering Transform (WST) were used. When AleXnet was combined with the other DL model, the suggested ‎Wavelet Scattering Transform-based ASD diagnostic networks (WST-ASDNets) produced accuracies of 98% and 95%, respectively, in ‎separating ASD from standard controls. The AI-based CADS created for ASD diagnosis showed encouraging outcomes and provided a ‎quicker and less labor-intensive method in terms of computation. These techniques have promise as screening instruments for the early ‎identification and classification of ASD. Given the differences in EEG signals among people with ASD, future research may concentrate on ‎expanding these algorithms to categorize ASD subjects into varying severity levels‎.

ABSTRACT Ambient seismic noise from human activities, site conditions, meteorology, and instrument self-noise limits earthquake monitoring. We characterize noise in the Pohang Community Seismograph Network (PCSN), deployed after the 2017 Mw 5.5 Pohang earthquake, using power spectral density (PSD) estimates. The PCSN stations were categorized into four groups based on their location and depth: rural, urban, coastal, and borehole array. We compared PSDs across different period ranges. At very short periods (&amp;lt;0.03 s), we could occasionally identify elevated noise levels caused by everyday weather conditions. The seismic signature of rainfall was clearly recorded at surface stations, but barely recorded at 500 m depth. At short periods (0.02–1.00 s), anthropogenic noise was predominant. The noise levels were highest in urban areas, followed by coastal areas, and lowest in rural areas. At long periods (1–10 s), we investigated the effect of typhoons and ocean waves. As typhoons Hinnamnor and Nanmadol approached the Korean Peninsula, the background noise level increased and reached a peak as it passed the Pohang region. The geological setting affected the ambient seismic noise: the PSDs of the horizontal component are higher than those of the vertical component in areas with thick sediments. The detection thresholds map in the Pohang region shows the smallest magnitude in the middle of the network in regions surrounded by stations with low-ambient noise levels. The recently compiled PCSN earthquake catalog demonstrates the effect of background noise on earthquake detection capability. The number of detected earthquakes under the magnitude of completeness (0.34) was 53 at night and 5 during the day. These findings underscore the importance of reducing ambient noise levels by relocating or installing seismic stations at depth on bedrock for effective microearthquake observations in a densely populated metropolitan area.

BACKGROUND: One of the areas for further improvement of spectral filter analysis of random signals is the transition from conventional narrow-band filters to non-stationary narrow-band dynamic filters operating in a transient mode, allowing to increase the measurement accuracy of the estimated spectral power density with conventional and dynamic filters of the same order. AIM: This work aimed to improve procedural algorithms, generalized expressions for the spectral window function, the relative dispersion of the measured spectral power density estimate, and optimal synthesis of the tuning laws for the narrow-band dynamic filters, including the attenuation ratio, carrier frequency, and transmission ratio in the analysis bandwidth, for spectral analysis of random signals. METHODS: To determine the relative dispersion of the estimated spectral power density, a correlation filter method using narrow-band dynamic second-order filters is used. RESULTS: The paper develops and specifies theoretical results in relation to narrow-band second-order dynamic filters to optimize their tuning (measurement) laws with the focus on a more advanced correlation filter method of spectral analysis of random signals. CONCLUSION: The findings show that narrow-band dynamic filters reordered in the analysis bandwidth allow for higher accuracy of spectral analysis compared to conventional steady-state filters of the same order.

This paper presents a comprehensive study of the use of software-defined radio (SDR) technology as an effective instrument for detecting radiofrequency activity associated with unmanned aerial vehicles (UAVs). The research is focused not only on the general principles of SDR but also on the practical implementation of algorithms and methods aimed at enhancing the efficiency of UAV signal detection in complex and dynamic radio environments. The architecture of SDR systems was analyzed in detail, highlighting their flexibility, scalability, and adaptability compared to conventional radio monitoring tools. Functional aspects, including wideband signal acquisition, reconfigurable software modules, and multi-channel processing capabilities, were examined as key enablers for UAV detection tasks. Special emphasis was placed on the development and application of mathematical tools and algorithms. Fourier transform, wavelet analysis, and power spectral density estimation were applied to perform spectral analysis and identify characteristic frequency components of UAV transmissions. Furthermore, methods for modulation classification and anomaly detection were implemented to distinguish UAV-related signals from background noise and interference. Synchronization techniques and preamble detection were tested to increase the reliability of identification under low signal-to-noise conditions. Based on the conducted analysis, a detection algorithm was designed and optimized. The algorithm takes into account noise resistance, multi-channel processing, and adaptive real-time data analysis. Its performance demonstrates the capability to reliably identify UAV communication signals even in the presence of intentional jamming and rapidly changing electromagnetic conditions. The results of this work create a methodological foundation for the design of advanced radiofrequency monitoring systems intended for electronic warfare and counter-UAV applications. The proposed SDR-based approach not only improves detection accuracy but also offers adaptability to future challenges in the field of UAV threat mitigation.

ABSTRACT Harmonizable processes are a class of nonstationary time series, that are characterized by their dependence between different frequencies of a time series. The covariance between two frequencies is the dual frequency spectral density, an object analogous to the spectral density function. Local stationarity is another popular form of nonstationarity, though thus far, little attention has been paid to the dual frequency spectral density of a locally stationary process. The focus of this paper is on the dual frequency spectral density of local stationary time series and locally periodic stationary time series, its natural extension. We show that there are some subtle but important differences between the dual frequency spectral density of an almost periodic stationary process and a locally periodic stationary time series. Estimation of the dual frequency spectral density is typically done by smoothing the dual frequency periodogram. We study the sampling properties of this estimator under the assumption of locally periodic stationarity. In particular, we obtain a Gaussian approximation for the smoothed dual frequency periodogram over a group of frequencies, allowing for the number of frequency lags to grow with sample size. These results are used to test for correlation between different frequency bands in the time series. The variance of the smooth dual frequency periodogram is quite complex. However, by identifying which covariances are the most pertinent we propose a nonparametric method for consistently estimating the variance. This is necessary for constructing confidence intervals or testing aspects of the dual frequency spectral density. Simulations are given to illustrate our results.

IntroductionCaffeine is the most widely consumed psychoactive substance, and its stimulant properties are well documented, but few investigations have examined its acute effects on brain and cardiovascular responses during cognitively demanding tasks under ecologically valid conditions.MethodThis study used wearable biosensors and machine learning analysis to evaluate the effects of moderate caffeine (162 mg) on heart rate variability (HRV), entropy, pulse transit time (PTT), blood pressure, and EEG activity. Twelve healthy male participants (20–30 years) completed a within-subjects protocol with pre-caffeine and post-caffeine sessions. EEG was recorded from seven central electrodes (C3, Cz, C4, CP1, CP2, CP5, CP6) using the EMOTIV EPOC Flex system, and heart rate (HR) and blood pressure (BP) were continuously monitored via the Huawei Watch D. Data analysis included power spectral density (PSD) estimation, FOOOF decomposition, and unsupervised k-means clustering.ResultsPaired-sample t-tests assessed physiological and EEG changes. Although systolic and diastolic BP showed a non-significant upward trend, HR decreased significantly after caffeine intake (77 ± 5.3 bpm to 72 ± 2.5 bpm, p = 0.027). There was a significant increase in absolute alpha power suppression (from −5.1 ± 0.8 dB to −6.9 ± 0.9 dB, p = 0.04) and beta power enhancement (−4.7 ± 1.2 dB to −2.3 ± 1/1, p = 0.04). The surface data from FOOOF shows these are real oscillatory changes. Based on the changes in clustering prior and post-caffeine, a machine-learning change in the brain activity differentiated pre/post-caffeine states with unsupervised clustering. The study results show that moderate caffeine resulted in synchronized EEG and cardiovascular changes, indicating increased arousal and cortical activation that are detectable with wearable biosensors and classifiable with machine learning.ConclusionA fully integrated, non-invasive methodology based on a wearable device for real-time monitoring of cognitive states holds promise in the context of digital health, fatigue detection, and public health awareness efforts.

The aim of this study is to analyze the signal generated in the brain for a specific motor task and to identify the region where it occurs. For this purpose, electroencephalography (EEG) signals were divided into delta, theta, alpha, and beta frequency sub-bands, and feature extraction was performed by looking at the time-frequency characteristics of the signals belonging to the obtained sub-bands. The epoch corresponding to motor imagery or action and the signal source in the brain were determined by power spectral density features. This study focused on a hand open–close motor task as an example. A machine learning structure was used for signal recognition and classification. The highest accuracy of 92.9% was obtained with the neural network in relation to signal recognition and action realization. In addition to the classification framework, this study also incorporated advanced preprocessing and energy analysis techniques. Eye blink artifacts were automatically detected and removed using independent component analysis (ICA), enabling more reliable spectral estimation. Furthermore, a detailed channel-based and sub-band energy analysis was performed using fast Fourier transform (FFT) and power spectral density (PSD) estimation. The results revealed that frontal electrodes, particularly Fp1 and AF7, exhibited dominant energy patterns during both real and imagined motor tasks. Delta band activity was found to be most pronounced during rest with T1 and T2, while higher-frequency bands, especially beta, showed increased activity during motor imagery, indicating cognitive and motor planning processes. Although 30 s epochs were initially used, event-based selection was applied within each epoch to mark short task-related intervals, ensuring methodological consistency with the 2–4 s windows commonly emphasized in the literature. After artifact removal, motor activity typically associated with the C3 region was also observed with greater intensity over the frontal electrode sites Fp1, Fp2, AF7, and AF8, demonstrating hemispheric symmetry. The delta band power was found to be higher than that of other frequency bands across T0, T1, and T2 conditions. However, a marked decrease in delta power was observed from T0 to T1 and T2. In contrast, beta band power increased by approximately 20% from T0 to T2, with a similar pattern also evident in gamma band activity. These changes indicate cognitive and motor planning processes. The novelty of this study lies in identifying the electrode that exhibits the strongest signal characteristics for a specific motor activity among 64-channel EEG recordings and subsequently achieving high-performance classification of the corresponding motor activity.

In this article, we present a novel method for testing the similarity of two time series by comparing their spectral density functions, without assuming independence between the series. Our hypothesis testing framework builds on a previous result that assessed the similarity of two time series using the multitaper cross-spectrum estimator at a specific frequency, where the test statistic was shown to follow a Beta distribution. By generalizing this approach, we enable comparisons without requiring prior knowledge of the covariance structure. Under the assumption of spectral similarity, the proposed test statistic is distributed as a product of beta random variables. The effectiveness of our method is demonstrated through a comprehensive simulation study. The new test is employed to evaluate whether each pair of time series for commodity price data shares the same underlying processes.

The spectral density of the time series analysis under long-range dependence (LRD) is an important area in investigating the properties of the parameter estimators of interest and observing the periodic characteristics of the time series data. As model assumptions of time-dependent data strongly influence the time series analysis, the paper proposed modification techniques of nonparametric spectral density estimation under long-range dependence to consider the boundary effects using periodicity characteristics of the periodograms and spectral density in the frequency domain. We demonstrated the uniform and pointwise consistency of the modified version of nonparametric spectral density estimator (NPSDE) under LRD satisfying mild conditions. We also provided the uniform and pointwise optimal bandwidth orders to minimize the uniform and pointwise absolute relative estimation errors, respectively. Numerical studies were carried out to compare the general NPSDE under LRD and the modified version of NPSDE under LRD.

Abstract Due to differences between air and debris motions, debris centrifuging creates bias in wind estimates based on Doppler velocities and radar wind retrievals in tornadoes. Anomalous radial divergence, azimuthal wind underestimation, and vertical velocity bias associated with debris centrifuging can lead to erroneous interpretations of tornado intensity and structure from radar data. A novel spectral velocity correction technique is developed to reduce bias by identifying rain and debris motion in radar signals using dual-polarization spectral density estimation and fuzzy logic classification. This technique successfully improves Doppler velocity estimates in simulated S-band polarimetric time series data, although debris concentration modulates both the magnitude and correctability of velocity bias. Large bias magnitudes associated with high debris concentrations are the most difficult to fully correct using this technique, especially at low elevation angles and near the center of the tornado. However, the magnitudes of corrections applied are proportional to the original bias magnitudes, suggesting that the technique performs consistently across low and high debris concentrations. Spectral correction results in an overall 84% reduction in bias in simulations. The spectral correction technique is also applied to dual-polarization S-band radar observations of the 20 May 2013 Moore, Oklahoma, tornado. Overall increases in Doppler velocity magnitudes, especially at lower elevation angles, imply that spectral correction can successfully reduce centrifuging bias in observed Doppler velocities. Significance Statement Much of our knowledge about low-level tornado structure comes from radar observations. Because radars observe object motion, not air motion, centrifuging of lofted debris can cause bias in radar-estimated wind fields. We develop a technique to correct this bias by identifying and filtering debris motion from velocity estimates using polarimetric data. We find that the technique consistently reduces bias by approximately 84% in radar simulations, although bias can remain large when debris concentrations are high. In observations, the changes in velocity magnitudes after correction are larger overall at lower elevations, where debris concentrations tend to be higher. These results suggest that this technique can successfully reduce debris-related bias in Doppler velocities to more accurately estimate tornado wind speeds.

Resolvent-based modelling and estimation is critically dependent on the nonlinear forcing input and hence understanding its role in the flow response is of great significance. This study quantifies the nonlinear forcing input in the resolvent formulation and investigates its characteristics for compressible turbulent boundary layers at Mach number 5.86 and friction Reynolds number 420 subject to adiabatic- and cold-wall conditions. Results show that, with the addition of the eddy viscosity to the resolvent operator, the cross-spectral density (CSD) of the forcing tends to exhibit a spatially uncorrelated distribution, which suggests that the spatial cross-coherence may be neglected and makes the modelling of the forcing input potentially easier. Aiming to quantify the different importance of each forcing component in generating turbulent fluctuations, contributions of the eddy-viscosity-corrected forcing to the flow responses are investigated through reduced-order analysis and matrix decomposition. The streamwise motions are almost insensitive to the temperature-related forcing, and can be oppositely influenced by the wall-normal and spanwise forcing components. By retaining only the diagonal components in the CSD of the forcing input, the assumption of forcing decorrelation in space and among components is also examined in the input–output framework. It is found that this simplified input is able to capture the dominant turbulence features and the local forcing is observed to cause inner-layer responses. That is, present results suggest adequate modelling of the CSD of the forcing can be achieved retaining only its diagonal components. On the basis of the current findings, the forcing input in the resolvent-based framework is thus modelled, with the wall-normal dependence and amplitude ratio between forcing components designed for compressible turbulent boundary layers. Through an algebraic Lyapunov equation, improved estimations of the statistical spectral densities of velocity and temperature fluctuations are finally obtained, in contrast to the results by simply assuming the forcing CSD to be an identity matrix.