Spectral Density Model Research Articles

Probing the rest-frame wavelength dependence of quasar variability. Insights from the Zwicky Transient Facility Survey

Quasar variability can potentially unlock crucial insights into the accretion process. Understanding how this variability is influenced by wavelength is crucial for validating and refining quasar variability models. This paper aims to enhance the understanding of the dependence of variability on the rest-frame wavelength (λ_RF) by isolating the variance in different timescales in well-defined wavelength bins and examining the corona-heated accretion disk (CHAR) model. We investigated the relation between variance and rest-frame wavelength (λ_RF) using optical g- and r-band light curves from the Zwicky Transient Facility (ZTF) Data Release 15 (DR15) for ∼ 5000 quasars within narrow ranges of the black hole mass (M_BH) and Eddington ratio (R_Edd). A spectral model taking into account disk continuum emission, Balmer transitions, Fe II pseudo-continuum emission, and other emission lines is necessary to best interpret the variance spectrum. Our analysis indicates a strong anticorrelation between median variance and λ_RF for quasars with M_ BH M_ ⊙ and R_ Edd at different timescales. This anticorrelation is more pronounced at shorter timescales. The results align well with a bending power-law power spectral density (PSD) model with both the damping timescale and the high-frequency slope of the PSD depending on the wavelength. The predictions provided by the CHAR model on the variance spectrum across most timescales studied showcase its potential in constraining temperature variations within the accretion disk.

A progress in the inverse matrix method in QCD sum rules

In traditional QCD sum rules, the simple hadron spectral density model of the “delta-function-type ground state + theta-function-type continuous spectrum” determines that there is no perfect parameter selection. In recent years, inverse problem methods, particularly the inverse matrix method, have shown better handling of QCD sum rules. This work continues to develop the inverse matrix method. Considering that the narrow-width approximation may still be a good approximation, we separate the ground state from the spectral density. We then follow the general steps of the inverse matrix method to extract physical quantities such as decay constants that we may be more interested in.

A hybrid non‐parametric ground motion model of power spectral density based on machine learning

AbstractIn the fields of engineering seismology and earthquake engineering, researchers have predominantly focused on ground motion models (GMMs) for intensity measures. However, there has been limited research on power spectral density GMMs (PSD‐GMMs) that characterize spectral characteristics. PSD, being structure‐independent, offers unique advantages. This study aims to construct PSD‐GMMs using non‐parametric machine learning (ML) techniques. By considering 241 different frequencies from 0.1 to 25.12 Hz and evaluating eight performance indicators, seven highly accurate and stable ML techniques are selected from 12 different ML techniques as foundational models for the PSD‐GMM. Through mixed effects regression analysis, inter‐event, intra‐event, and inter‐site standard deviations are derived. To address inherent modeling uncertainty, this study uses the ratio of the reciprocal of the standard deviation of the total residuals of the foundational models to the sum of the reciprocals of the total residuals of the seven ML GMMs as weight coefficients for constructing a hybrid non‐parametric PSD‐GMM. Utilizing this model, ground motion records can be simulated, and seismic hazard curves and uniform hazard PSD can be obtained. In summary, the hybrid non‐parametric PSD‐GMM demonstrates remarkable efficacy in simulating and predicting ground motion records and holds significant potential for guiding seismic hazard and risk analysis.

Optical Variability Properties of Southern TESS Blazars

We present a study of high-cadence, high-precision optical light curves from the TESS satellite of 67 blazars in the southern sky. We provide descriptive flux statistics, power spectral density (PSD) model parameters, and characteristic variability timescales. We find that only 15 BL Lacertae objects (BLLs) and 18 flat spectrum radio quasars (FSRQs) from the initial 26 and 41, respectively, exhibit statistically significant variability. We employ an adapted power spectral response method to test the goodness of fit for the PSD function to three power-law variant models. From our best-fitting description of the PSD, we extract the high-frequency power-spectral slopes, and if present, determine the significant bend or break in the model to identify characteristic timescales. We find no significant difference in the excess variance or rms scatter between blazar subpopulations. We identify a linear rms–flux relation in ∼69% of our sample, in which ∼20% show a strong correlation. We find that both subpopulations of blazars show power spectral slopes of α ∼ 2 in which a broken power-law best fits five BLLs and six FSRQs and a bending power-law best fits one BLL and five FSRQs. The shortest timescales of variability in each light-curve range widely from minutes to weeks. Additionally, these objects’ characteristic timescales range from ∼0.8 to 8 days, consistent with optical variability originating in the jet.

Temperature history reconstruction in steel box girders using limited data and proper orthogonal decomposition-based dimension reduction representation

The monitoring temperature history of steel box girders inevitably contains gaps or anomalies due to the malfunctions of the structural health monitoring system. Traditional methods for reconstructing temperature histories face the challenge in accounting for the randomness of temperature variations caused by the uncertainty of complex environmental factors. This research proposes a framework for reconstructing the long-term temperature of steel box girders by combining the limited measurements with a proper orthogonal decomposition (POD) based dimension reduction representation approach, to account for the stochastic nature of daily temperature variations. Unlike the conventional Monte Carlo-based POD method, the developed POD-based dimension reduction representation approach effectively reduces the number of elementary random variables from thousands to two by introducing random functions serving as constraints, overcoming the challenge of high-dimensional random variables inherent in the Monte Carlo methods. The proposed approach divides the measured temperature histories into random high-frequency (HF) and deterministic low-frequency (LF) components and establishes the theoretical models of power spectral density and coherence functions of HF temperature components to accommodate the generation of HF temperature component samples, and finally reconstructs temperature samples by superimposing the LF components and generated HF component samples. The results from a practical example demonstrate that the statistical characteristics of representative HF temperature component samples generated by the POD-based dimension reduction representation align well with the corresponding targeted values, and the proposed method outperforms the traditional POD method, yielding a 60% efficiency enhancement without compromising computational accuracy. The developed framework owns apparent superiority in accuracy compared to the traditional POD and the long short-term memory methods, particularly in continuous and extensive missing data. Moreover, the reconstructed temperature samples with assigned probabilities present complete probability information from the level of total probability. These results advance the probabilistic methods in tackling long-term temperature history reconstruction.

Non-stationary modeling and simulation of strong winds

Wind velocity is usually assumed to obey a stationary stochastic process in wind engineering, and this may cause significant bias in describing extremely severe strong wind such as typhoons and thunderstorms. To take into account the non-stationary characteristics of extreme wind, a novel evolutionary power spectral density (EPSD) model is proposed, and the spectral representation method (SRM) is introduced to simulate the whole process of strong winds. Firstly, the wavelet transform (WT) method is adopted to capture the three-dimensional time-varying properties of the low-frequency mean winds, and the associated turbulence features, including turbulent intensity, gust factor, probability density function, and power spectrum, are analyzed in depth. Secondly, the measured horizontal EPSD of strong winds are estimated. Thirdly, the performance of the proposed EPSD model is validated. Finally, the whole process of non-stationary strong winds are simulated and discussed. The results show that the proposed EPSD models are in good agreement with the measured EPSD, and the time-frequency features of the power spectrum of the simulated winds are well reproduced, which provides a powerful tool for large eddy simulation and wind engineering studies under non-stationary extreme wind climate.

Simulation Method for the Impact of Atmospheric Wind Speed on Optical Signals in Satellite–Ground Laser Communication Links

To analyze the intensity of atmospheric turbulence in a satellite–ground laser communication link, it is important to consider the effect of increased atmospheric turbulence caused by wind speed. Atmospheric turbulence causes a change in the refractive index, which negatively impacts the quality and focusing ability of the laser beam by altering its phase front. To simulate the changes in amplitude and phase characteristics of laser beam propagation in atmospheric turbulence caused by wind speed, a transverse translation phase screen is used. To better understand and address the influence of atmospheric wind speed on the phase of optical signals in satellite–ground laser communication links, this paper proposes a Monte Carlo simulation method. This method utilizes the spatial and temporal variations in the refractive index in the atmosphere and integrates the principles of optical signal propagation in the atmosphere to simulate changes in the phase of optical signals under different wind speed conditions. By analyzing the variations in the received optical signal’s power, the Monte Carlo method is employed to simulate phase screens and logarithmic amplitude screens. Additionally, it models the probability density of the statistical behavior of received optical signal’s fluctuations, as well as the time autocorrelation coefficient of optical signals. This paper, under the coupling condition in satellite–ground laser communication links, conducted a Monte Carlo simulation experiment to analyze the characteristics of the optical signal’s fluctuations in the link and discovered that atmospheric wind speed affects the shape of the power spectral density model of the received optical signal. Increasing wind speed leads to a decrease in the time autocorrelation coefficient of the received optical signal and affects the coupling efficiency. The paper then used a cubic spline interpolation fitting method to verify the models of the power spectral density and the autocorrelation time coefficient of the optical signal. This provides a theoretical foundation and practical guidance for the optimization of satellite–ground laser communication systems.

Low‐frequency noise model development of MoS2 field effect transistor and its analysis with respect to different traps

AbstractIn this paper, an analytical drain current model of double gate MoS2 FET Transistor for low‐frequency noise (LFN) power spectral density is developed, and compared its response with the device simulation of the proposed device structure. The developed low‐frequency noise power spectral density (PSD) model is analyzed with respect to different Trap densities, Oxide thicknesses, and high K‐dielectric materials using the TCAD Device simulation tool. Low‐frequency noise is one of the most important noises in any device since it degrades the device's performance. So, in this paper, we used a unique charge‐control model for analyzing the low‐frequency noise by considering the mobility degradation, trapping, and de‐trapping effects caused by trap densities and device lattice structure. Usually, all the previous studies on drain currents with low frequency are based on fluctuations caused by mobility and carrier variations. In our paper, we even considered the charge density fluctuation to analyze the drain current equation. Using a Charge Control Model, we can compute the parameters like gate capacitance (Cgs, Cgd), channel length modulation, and body effect. While other LFN models like the DC model, small signal model, and hybrid model will focus mainly on the static characteristics of the device such as its DC current–voltage (I‐V) relationship, terminal resistances, and input/output conductance of the devices. All the analytical model results were calibrated with the device simulation and it is observed that both are matches each other. Further, it is observed that with an increase in Trap density, the low‐frequency noise increases and low‐frequency noise is dominant at lower frequencies only (1–10 Hz), and after this frequency range the thermal noise will be the dominant noise factor.

High-speed digital-to-analogue converters phase noise modelling

The paper presents a mathematical model of the phase noise power spectral density of a high-speed digital-to-analogue converter. The shapes of the envelopes of the spectrum of the output signal in RFZ3, RFZ4 operating modes are given. The phase noise transfer coefficients for RFZ3, RFZ4 operating modes are obtained. The modelling of the power spectral density of phase noises in the studied operating modes was carried out. The analysis of the received dependences is carried out. It is concluded that RFZ3, RFZ4 modes of operation allow to obtain a lower level of PSD phase noise for images n = 1, 3, 4, 5, which indicates the possibility of generating high-frequency signals with a frequency that is 5-11 times higher than the generated frequency in NRZ mode without noise degradation.

Analytical modelling of adaptive optics systems: Role of the influence function

Context. Adaptive optics (AO) is now a tool commonly deployed in astronomy. The real time correction of the atmospheric turbulence that AO enables allows telescopes to perform close to the diffraction limit at the core of their point spread function (PSF). Among other factors, AO-corrected PSFs depend on the ability of the wavefront corrector (WFC), generally a deformable mirror, to fit the incident wavefront corrugations. Aims. In this work, we focus on this error introduced by the WFC, the so-called fitting error. To date, analytical models only depend on the WFC cut-off frequency, and Monte Carlo simulations are the only solution for studying the impact of the WFC influence function shape on the AO-corrected PSF. We aim to develop an analytical model accounting for the influence function shape. Methods. We first obtain a general analytical model of the fitting error structure function. With additional hypotheses, we then derive an analytical model of the AO-corrected power spectral density. These two analytical solutions are compared with Monte Carlo simulations on different ideal profiles (piston, pyramid, Gaussian) as well as with real hardware (DM192 from ALPAO). Results. Our analytical predictions show a very good agreement with the Monte Carlo simulations. We show that in the image plane, the depth of the correction as well as the transition profile between the AO-corrected area and the remaining turbulent halo depend on the influence functions of the WFC. We also show that the generally assumed hypothesis of stationarity of the AO correction is actually not met. Conclusions. As the fitting error is the intrinsic optimal limit of an AO system, our analytical model allows for the assessment of the theoretical limits of extreme AO systems limited by the WFC in high-contrast imaging through a context where other errors become comparable.

Common-clock GPS single differences: An improved correlation model for GPS phase observations based on turbulence theory

Microwave signals, for example, those from Global Navigation Satellite System (GNSS) and Very Long Baseline Interferometry (VLBI), are affected by tropospheric turbulence in such a way that the random fluctuations of the atmospheric index of refractivity correlate the phase measurements. A proper modeling of correlations is mandatory to avoid biased analysis, particularly when statistical tests are used. In this contribution, we analyze single differences (SD) computed from Global Positioning System (GPS) phase observations for which the between receiver clock error could be strongly mitigated by a specific common clock setting. We estimate specific parameters from the power spectral density (psd), which is directly related to the correlation function, with the debiased Whittle maximum likelihood and investigate their dependencies with the satellite geometry (elevation, azimuth angles) and the time of the day. We show that (i) the estimated slopes of the psd follow the one predicted by the Kolmogorov turbulence theory and (ii) the cut-off at high frequencies shows daily variations that may be linked with the strength of the turbulence. Based on these findings, we derive an improved spectral density model for GPS phase SD. The results of this study contribute to improving the stochastic description of random effects impacting VLBI and GNSS phase observations by studying variations of parameters from the von Karman spectrum.

Prediction of soil bulk density in agricultural soils using mid-infrared spectroscopy

Soil bulk density (BD) is a key physical parameter in soil quality control and in the calculation from soil organic carbon (SOC) mass (g/kg) content to area stock (kg/ha). However, BD laboratory analysis is time-consuming, labour intensive and expensive, especially for a national-scale soil assessment. Hence, how to fill the omissions of BD values for all or some records in soil databases is widely discussed. This study employed different chemometric and machine learning algorithms to estimate BD in Irish soil from 671 horizon-based samples from MIR spectral libraries by partial least square regression (PLSR), random forest, Cubist and support vector machine (SVM). The best performance was observed for the SVM model with a higher ratio of performance to interquartile distance (RPIQ = 3.61) and R2 (0.81) values and lower root mean square error of prediction (RMSEP = 0.132). Moreover, BD highly correlated wavenumber bands were determined by principal components analysis (PCA) and variable importance analysis. Soil organic matter (SOM) was identified as the primary factor in the spectral soil BD model. The generalisation error of predicting unknown samples using a spectral soil bulk density (BD) model was calculated by employing leave-one-out cross-validation (LOO-CV) on SVM. Estimation of BD by the spectral BD model was compared with published traditional pedo-transfer functions (PTFs), results were then compared for the overall models, different horizon types and specific depth categories. The spectral soil BD model is significantly better than traditional PTFs overall, with RMSEP equalling 0.132 g/cm3 and 0.196 g/cm3 respectively. The spectral soil BD model showed a similar accuracy on the A horizon, but considerable performance improvements were found on the other types of horizon. As for different depth categories, there is no significant accuracy difference between shallow (A-Samples: 5–20 cm) and deep (S-Samples: 35–50 cm) topsoil for the spectral soil BD model, which differs from traditional PTFs. Hence, high accuracy and the homogeneity of performance on different depth layers above 50 cm could be noteworthy strengths of spectral modelling techniques when carrying out national soil surveys and large-scale carbon stock assessments.

The Constant Power Spectral Density Model for Capacity Optimization of Submarine Links

We review the fundamentals of the recently proposed Constant Power Spectral Density (CPSD) model for very long-haul space-division multiplexed submarine links, and highlight its use for Erbium-doped fiber amplifier (EDFA) optimization to maximize the achievable information rate (AIR) of the link. The CPSD line is an abstraction of modern submarine lines, where all EDFAs have identical physical parameters, among which the doped-fiber length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> , and the same pump and Erbium inversion, with identical gain-shaping filters that reproduce the line input power spectral density at the output of each span. The key idea in the CPSD line analysis is to use the hidden state-variable of the EDFA, namely the Erbium population inversion <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> , as a free variable. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> is known, so is the EDFA gain and its noise figure frequency profiles. Thus in the CPSD line we derive a simple expression of the received signal to noise ratio and thus of the AIR in the assumption of Gaussian noises. Among the set of input wavelength-division multiplexed signals that achieve inversion x at EDFA length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> we can find analytically the one maximizing AIR(x, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> ). We finally look numerically for the best (x, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> ) values that maximize AIR, i.e., we optimize the line EDFAs for maximum AIR. The major novelty of this invited paper is the extension of the analysis to include nonlinear effects into the AIR optimization.

Mapping the X-Ray Corona Evolution of IRAS 13224-3809 with the Power Spectral Density

We develop the power spectral density (PSD) model to explain the nature of the X-ray variability in IRAS 13224–3809, including the full effects of the X-ray reverberation due to the lamppost source. We utilize 16 XMM-Newton observations individually as well as group them into three different luminosity bins: low, medium, and high. The soft (0.3–1 keV) and hard (1.2–5 keV) PSD spectra are extracted and simultaneously fitted with the model. We find that the corona height changes from h ∼ 3 r g during the lowest luminosity state to ∼25 r g during the highest luminosity state. This provides further evidence that the source height from the reverberation data is significantly larger than what is constrained by the spectral analysis. Furthermore, as the corona height increases, the energy spectrum tends to be softer while the observed fractional excess variance, F var, reduces. We find that the PSD normalization is strongly correlated with F var and moderately correlated with the PSD bending index. Therefore, the normalization is dependent on the accretion rate that controls the intrinsic shape of the PSD. While the intrinsic variability of the disk is manifested by the reverberation signals, the disk and corona may evolve independently. Our results suggest that, as the source height increases, the disk itself generates less overall variability power but more high-frequency variability resulting in the PSD spectrum that flattens out (i.e., the inner disk becomes more active). Using the luminosity-bin data, the hint of the Lorentzian component is seen, with the peak appearing at lower frequencies with increasing luminosity.

Spatial Selectivity of Rice Channel Based on 3-D Joint Multipath Shape Factors Theory

Based on the 3-D joint multipath shape factors theory, this paper studies the spatial selectivity of Rice channels of wireless communications. A 3-D joint multipath angular power spectral density (APD) model is proposed, and then the closed expressions of the 3-D double spherical harmonic coefficients and 3-D joint multipath shape factors are derived. The errors in the existing formulas of 3-D joint multipath shape factors theory are further pointed out and corrected. The results can provide theoretical support for quantifying and simplifying the analysis and design of 3-D multiple input multiple output (MIMO) beamforming technology and smart antenna arrays for the 5G communication.

Advances in the Prediction of the Statistical Properties of Wall-Pressure Fluctuations under Turbulent Boundary Layers

Analytical or empirical models of the wall-pressure power spectral density under a turbulent boundary layer are often validated on test cases in an incompressible flow regime. In this work, an analytical model based on the compressible Poisson equation for the unsteady pressure in a turbulent boundary layer is developed. The Large Eddy Simulation of the flow over a controlled-diffusion airfoil at Mach 0.5 is used to validate the assumptions made on the statistical properties of the boundary layer turbulence and to validate the prediction of the statistics of the wall-pressure fluctuations. The predicted wall-pressure spectrum also compares favorably with experimental data.

Maximum dynamic response of linear elastic SDOF systems based on an evolutionary spectral model for thunderstorm outflows

The study aims to estimate the maximum dynamic response of linear elastic SDOF systems subjected to thunderstorm outflows. Starting from a recently developed Evolutionary Power Spectral Density (EPSD) model for the wind velocity, the dynamic response is decomposed into a time-varying mean and a non-stationary random fluctuation. The EPSD and the Non-Geometrical Spectral Moments (NGSMs) of the random fluctuation are derived both accounting and neglecting the transient dynamics due to the modulating function of the load. The mean value of the maximum nonstationary fluctuating component of the response is estimated based on the definition of an equivalent stationary process following an approach proposed in the literature. In order to mitigate the overestimations of the maximum dynamic response due to the Poisson approximation, analogously to the formulation developed by Der Kiureghian for withe noise excitation, an equivalent expected frequency is introduced for thunderstorm excitation. Finally, the maximum dynamic response to thunderstorms is estimated as the sum of the maximum mean and fluctuating parts and a numerical validation of the results against real recorded thunderstorms is provided, highlighting the reliability of adding up the mean and fluctuating contributions and the advantages and limits of neglecting the transient dynamics.

Low and High Broadband Spectral Models of Atmospheric Pressure Fluctuation

AbstractThis paper presents new low and high power spectral density models for pressure fluctuations at the Earth’s surface over the frequency range of (10−5 – 8) Hz. Previously proposed models often included limitations, such as a much narrower frequency range, the inclusion of erroneous and non-calibrated data or recorded data not deconvolved from the measurement system responses. The progress recently made with response modeling and field calibration of pressure fluctuation measurement systems now allows to propose more realistic power spectral density models over an extremely large frequency band. This paper describes how the data were selected, processed, and analyzed to obtain the final global models. In addition, the intermediate results allow the characterization of several atmospheric mechanisms, such as gravity wave saturation, limits of the buoyancy and acoustic cut-off frequencies or wind turbulence modes. The two proposed low and high power spectral density models are planned to be used for a wide range of applications, including assessing the quality of measured pressure fluctuations, verifying the validity of modeled pressure fluctuations and supporting the design, testing and calibration of a new generation of measurement systems. The models presented in this paper are made available to the scientific community.

Stochastic Dynamic Response Analysis and Probability Evaluation of Subway Station Considering Subjected to Stochastic Earthquake Excitation

As a relatively new means of transportation, the subway has become an important tool for the sustainable development of many cities. Being buried deep in soil under the weight of vital infrastructure, subway stations can be vulnerable to seismic excitations. Considering the high randomness of ground motions, it is important to research the failure probability and seismic performance of the subway station based on stochastic dynamic analysis. In this paper, a probability density evolution method (PDEM) coupled with a spectral representation random function is used to analyze the stochastic dynamic response and seismic probability of a subway station. First, according to the improved power spectral density model and the seismic design code of urban rail transit structures in China (GB 50909-2014), a set of nonstationary ground motions consistent with the code spectrum are obtained. Then, a great deal of deterministic dynamic calculations for Daikai subway station considering soil–structure interaction based on elastic–plastic methods are performed. In addition, the nonlinear stochastic response analysis and the dynamic probability analysis are obtained for the subway station by solving the PDEM equation. Finally, the probability density function (PDF) and cumulative distribution function (CDF) of the subway station under stochastic earthquake excitations are obtained based on three performance indices, including drift angle in the middle column, relative vertical displacement between floor and roof, and damage area ratio (DAR). The results show that the stochastic dynamic analysis and the probability density evolution method can analyze seismic response and evaluate seismic performance of subway stations effectively. The proposed method will serve as an effective tool for the seismic design of underground structures.

An improved ellipsoid optimization algorithm in subspace predictive control

PurposeThe purpose of this paper is to derive the output predictor for a stationary normal process with rational spectral density and linear stochastic discrete-time state-space model, respectively, as the output predictor is very important in model predictive control. The derivations are only dependent on matrix operations. Based on the output predictor, one quadratic programming problem is constructed to achieve the goal of subspace predictive control. Then an improved ellipsoid optimization algorithm is proposed to solve the optimal control input and the complexity analysis of this improved ellipsoid optimization algorithm is also given to complete the previous work. Finally, by the example of the helicopter, the efficiency of the proposed control strategy can be easily realized.Design/methodology/approachFirst, a stationary normal process with rational spectral density and one stochastic discrete-time state-space model is described. Second, the output predictors for these two forms are derived, respectively, and the derivation processes are dependent on the Diophantine equation and some basic matrix operations. Third, after inserting these two output predictors into the cost function of predictive control, the control input can be solved by using the improved ellipsoid optimization algorithm and the complexity analysis corresponding to this improved ellipsoid optimization algorithm is also provided.FindingsSubspace predictive control can not only enable automatically tune the parameters in predictive control but also avoids many steps in classical linear Gaussian control. It means that subspace predictive control is independent of any prior knowledge of the controller. An improved ellipsoid optimization algorithm is used to solve the optimal control input and the complexity analysis of this algorithm is also given.Originality/valueTo the best knowledge of the authors, this is the first attempt at deriving the output predictors for stationary normal processes with rational spectral density and one stochastic discrete-time state-space model. Then, the derivation processes are dependent on the Diophantine equation and some basic matrix operations. The complexity analysis corresponding to this improved ellipsoid optimization algorithm is analyzed.