Theoretical Power Spectral Density Research Articles

Improving the surface metrology accuracy of optical profilers by using multiple measurements

The performance of high-resolution optical systems is affected by small angle scattering at the mid-spatial-frequency irregularities of the optical surface. Characterizing these irregularities is, therefore, important. However, surface measurements obtained with optical profilers are influenced by additive white noise, as indicated by the heavy-tail effect observable on their power spectral density (PSD). A multiple-measurement method is used to reduce the effects of white noise by averaging individual measurements. The intensity of white noise is determined using a model based on the theoretical PSD of fractal surface measurements with additive white noise. The intensity of white noise decreases as the number of times of multiple measurements increases. Using multiple measurements also increases the highest observed spatial frequency; this increase is derived and calculated. Additionally, the accuracy obtained using multiple measurements is carefully studied, with the analysis of both the residual reference error after calibration, and the random errors appearing in the range of measured spatial frequencies. The resulting insights on the effects of white noise in optical profiler measurements and the methods to mitigate them may prove invaluable to improve the quality of surface metrology with optical profilers.

Generating Stationary Gaussian Processes Using The Spectral Representation Theorem

Abstract The paper presents the Spectral Representation Theorem of a stationary process and two simulation methods derived from it. In order to test the accuracy of the two simulation methods two numerical applications are employed. First, using a theoretical power spectral density (PSD), two sets of sample functions, corresponding to each method, are generated and a comparison of the obtained numerical results with the analytical PSD is carried out. The second example is more complex and consists in using the stationary zone of the strong motion of the recorded NS acceleration registered at INCERC during the 1977 Vrancea earthquake. The corresponding Fourier Spectrum is calculated. In order to obtain a smoother PSD representation for the real Fourier spectrum, a specific barrier model spectrum (SBM) is fitted to it and the corresponding PSD calculated. This PSD is used to generate two sets of samples. The mean PSD obtained using both methods of simulation is compared with that characterizing the registered acceleration. The paper shows that the generated time series possess all the theoretical probabilistic characteristics discussed below, when the number of terms used in the simulation formulas is large. Three types of estimators are employed in the numerical evaluation of both simulation methods

Doppler Characterization of a Mobile GNSS Receiver in Multipath Fading Channels

Global Navigation Satellite Systems (GNSS) Doppler measurements are commonly used for velocity-based relative positioning and aiding Inertial Navigation Systems (INS) in signal degraded environments. The aim of this paper is to characterise the Doppler measurements in GNSS harsh multipath environments. In multipath fading situations such as indoor and urban canyon environments, multipath components arrive to the receiver antenna from different paths and directions. These give rise to various Doppler shifts that cause errors in the velocity solution. In this work the Doppler measurements discrepancy characterised by Doppler spread in multipath environments is investigated. By assuming a ‘sphere of scatterers’ model and considering the antenna gain pattern, the theoretical Power Spectral Density (PSD) observed by a receiver is formulated. The theoretical findings are examined using two sets of measurements in dense multipath environments. Global Positioning System (GPS) live signals using non-isotropic antennas with different orientations are used for this purpose. Different motion directions are also examined using different data sets. An Assisted GPS (A-GPS) approach is utilised where the code phase and the navigation data bits are provided by a nearby outdoor antenna. By applying a ‘Block Processing’ technique, an epoch-by-epoch Doppler and velocity estimation is implemented. Herein, the Doppler and velocity measurements accuracy in addition to the Doppler spread characterization are studied. As shown both theoretically and experimentally, in harsh multipath environments the PSD of the observed signals is a function of the scatterers' geometry and the antenna gain pattern. The Doppler estimation accuracies in multipath and multipath-free cases are compared for different ranges of Carrier-to-Noise ratio (C/N0). Theoretical and experimental results revealed inaccurate Doppler estimation and poor Doppler-derived velocity solutions in dense multipath environments.

Low-Cost Digital Beacon Receiver Based on Software-Defined Radio

In this paper, we present a low-cost digital beacon receiver based on software-defined radio. A free software toolkit, GNU radio, was used to develop the beacon receiver's functionality. The hardware used was a parabolic antenna, a low-noise block converter, an Ettus universal software-radio peripheral, a Linux PC, and a Global Positioning System receiver. The total hardware cost of the developed beacon receiver was low compared to commercially available analog receivers. To validate the receiver, we compared the measured preliminary results of amplitude scintillation at 11.2 GHz with reported theoretical models. Comparisons between the measured values and the ITU-R model for the amplitude-scintillation distribution showed that the ITU-R model somewhat underestimated the statistic. The measured probability density function was also compared with the Moulsley-Vilar model. For small log-amplitude values, the Moulsley-Vilar model fit the measured statistic well, but slightly overestimated the statistic for large log-amplitude values. Good agreement was found between the measured and theoretical power spectral density of amplitude scintillation. The diurnal variation of the amplitude scintillation's standard deviation was calculated. It was largest at noontime, which was consistent with previously reported measurement results. To some extent, this may validate the performance of the developed receiver. The developed software is available from a software repository (http://www.sourceForge.net). enabling collaboration, development, and use of the software.

Spectral analysis of line edge and line-width roughness with long-range correlation

Large-scale integrations (LSIs) are facing an ever-growing problem of device variability. One of the origins that cause the variability is line-width roughness (LWR) caused by line edge roughness (LER). Accurate characterization of the LWR plays an essential role in controlling the LWR. To do this, we report a methodology, named the “assembly method,” that enables to analyze LWR statistics beyond the conventional correlation length limit, basing on the previous “patchwork” method and recent discrete power spectral density (PSD) method. The methodology virtually assembles a long line by gathering line segments that are randomly scattered on a single line or equally processed different lines. The virtual lines are repeatedly assembled by randomly changing the combination of the segments and the order of the gathered segments while permitting overlaps of the segments between the assembled lines. Squared Fourier transforms of their widths are averaged over the assembled lines to obtain the PSD. By these steps, the statistical noise, which is inherent to experimental PSDs, is markedly reduced. Furthermore, to extract LWR statistics by comparing experimental and theoretical PSDs, we derived an analytic formula of the assembled-line PSD. In the derivation, the randomness of the segment collections played a key role. The PSDs calculated using the formula almost completely fitted experimental PSDs that were obtained by the assembly method. The parameters used in the best-fitted calculation revealed that the photoresist LWR of this study contained a component that had a correlation length of 2780 nm in addition to the previously reported LWR of 35 nm. The LWR variance of the component accounted for approximately 10% of the total variance. The formula also enabled us to evaluate the accuracy of experimentally obtained averages of widths. We find two distinct features in the PSDs by the assembly method. One is the oscillatory structure that shows up in the case when the correlation length is larger than half the length of the segments. A trace of this structure was actually observed in the experimental PSDs of this study. The other is the spikes that are periodically observed as a function of wave number. The spikes originate from a nonstochastic width variation that exists in all the segments in common. Their intensity is proportional to the number of gathered segments in the assembled lines. Because the spikes are excluded from the analysis, the LWR parameters determined by the assembly method are not affected by the nonstochastic variation, unlike the conventional methods. By all these results, we confirm that the assembly method of this study extends the upper limit of analyzable correlation lengths by a factor of approximately 20 and enhances the accuracy as well. This feature also has a practical significance that the widely observed LWR with a correlation length of approximately 35 nm can be analyzed by the assembly method using a conventional critical-dimension scanning-electron-microscope, without resorting to a specially designed one. Accordingly, the method will be a key tool for investigating LER and LWR in developing and manufacturing LSIs. It will also help analyze other stochastic processes in many research and development settings.

Rayleigh fading channel simulator based on inner–outer factorization

The paper deals with the design of Rayleigh fading channel simulators based on the inner–outer factorization. The core of the approach is to approximate the outer spectral factor of the channel power spectral density (PSD) by either finite-order polynomials or rational functions. This, respectively, leads to MA or AR/ARMA models. The parameter estimation operates in two steps: the outer factor, which leads to a minimum-phase filter, is first evaluated inside the unit disk of the z-plane. Then, we propose to compute the Taylor expansion coefficients of the outer factor because they coincide with the model parameters. Unlike other simulation techniques, this has the advantage that the first p parameters remain unchanged when one increases the model order from p to p+1. A comparative study with existing channel simulation approaches points out the relevance of our ARMA model-based method. Moreover, the ARMA model weakens the oscillatory deviations from the theoretical PSD in the case of AR models, or low peaks at the Doppler frequencies for MA models.

Statistical analysis of mobile radio reception: an extension of clarke's model

Statistical analysis and modeling of wireless channels is essential to wireless communication systems. Clark's model [1] and the corresponding statistical analysis of mobile radio reception has been widely accepted in numerous wireless applications. Since the component phases in Clarke's model are assumed to be constant in time, the well-known results of statistical analysis based on this model, such as the autocorrelation and Doppler power spectrum, are not appropriate to describe real wireless channels for which the random environments (radio propagation paths) are time-varying and accordingly for which the channel is non-constant in the absence of Doppler frequency shift. In this paper, we extend the traditional Clarke's model incorporating the effect of fluctuations in the component phases, and perform the statistical analysis which results in a closed-form expression of the autocorrelation of the fading. The theoretical power spectral density function, which is the Fourier transform of the resultant autocorrelation of the fading, is shown to fit the practical measured spectra, which is in contrast to the traditional theoretical flat fading channel spectra (Jake's spectrum in [2]). The proposed model and statistical results should have important implications for detailed spectral analysis and channel simulations for real wireless communications systems in random fluctuating electromagnetic propagation environments.

Evolution of the multiscale statistical properties of corotating streams from 1 to 95 AU

This paper discusses the multiscale structure of the large‐scale speed fluctuations between 1 and 95 AU during the declining phase of the solar cycle, when corotating streams are dominant structures close to the Sun. A deterministic, multifluid, one‐dimensional MHD model with the WIND data at 1 AU during 1995 as input was used to compute the time series of daily averages of speed, V(ti), with a length on the order of 1 year at R = 5, 10, … 95 AU, from which we calculated various statistical functions describing the solar wind speed. The probability distribution functions (PDFs) of the running speed differences dVn(ti) were calculated at scales τ from 1 to 64 days. The theoretical PDFs at 50 AU have the same qualitative forms as those observed by Voyager 2 (V2) between 46 and 49 AU. The theoretical PDF at a scale of 1 day at 15 AU agrees with the corresponding PDF observed by V2 between 13 and 16 AU. With increasing distance from the Sun, the width of the predicted PDF with τ = 1 decreases and the tail becomes more prominent. The standard deviation of dVn(ti) at various R computed from the model, SD(τ, R), decreases nearly exponentially with increasing R, and it is consistent with the observations of SD(τ, R) made by V2 near 15 and 50 AU. The skewness at a scale of 1 day increases almost linearly with R beyond ≈35 AU, but it shows structure between 5 and 35 AU; it is consistent with the observations of V2 near 15 and 50 AU. The theoretical power spectral density of the speed fluctuations, PSD(f, R), agrees with the observations of V2 near 15 and 50 AU. For frequencies f > 8.5 × 10−7 Hz, PSD(f, R) ∝ f−s(R). The observed and predicted s are ≤−2 beyond 40 AU, consistent with the jump‐ramp structure of V(t) in the distant heliosphere. We predict that the PDF for τ = 1 which will be observed by V2 at 70 AU in 2003 will have a core whose width is only ≈5 km/s and a tail extending up to 60 km/s that represents the major jumps in the speed profile. The predicted PDF at 95 AU is very similar to that at 70 AU. The model predicts that most speed jumps at 15 and 70 AU will have the signature of forward shocks and reverse shocks. The amplitudes of the large and moderate speed jumps diminish as they move from 45 to 95 AU. Some speed jumps move through this region without interacting with other speed jumps. In a few cases, neighboring speed jumps merge between 45 and 95 AU. The structure of the solar wind changes slowly as it moves from 45 to 95 AU.

Computing short-term amplitude fluctuations from statistical and spectral characteristics of atmospheric scintillation

A model is presented for generating time series to emulate the rapid fluctuations induced by scintillation and affecting waves propagating in the atmosphere. The model time series are generated by taking into account the theoretical power spectral density characteristics of scintillation derived from the theory of propagation in turbulent media. The model also combines the short-term statistical characteristics of scintillation estimated from measurements. A study of the statistics of the scintillation intensity, carried over periods of hours and days, highlights a strong diurnal cycle and is taken into consideration in the model. The first-order statistics of the model time series are compared to measured data and show a good agreement.

Power spectral analysis for evaluating optical near-field images of 20 nm gold particles

We have demonstrated that the power spectral analysis could be employed as a potential tool for evaluation of near-field optical images of colloidal gold particles of size 20 nm fixed on a glass substrate. It has been found that the power spectral density of the near-field image resembles the Airy pattern due to a circular dot with the first minimum occurring at the spatial frequency 5.6 × 10 7m −1. This corresponds to an estimated particle diameter of 22 nm. From a comparison with the theoretical power spectral density of randomly distributed circular dots, the transfer function of the near-field optical microscope has been estimated. It has been found that the resolution estimated using −3 dB cutoff value obtained from the transfer function was 0.8 nm which is much higher than that determined by aperture size. The high resolving ability of the present microscopic system is believed to be due to the confinement of very high spatial frequency evanescent fields around the edges of the metal-glass fiber boundary at the apex of the apertured probe.