Log-amplitude Variance Research Articles

Performance of Non-Line of Sight Underwater Optical Wireless Communication Links with Spatial Diversity

Line-of-sight (LOS) underwater optical wireless communication (UOWC) transmission may suffer blocking and are not always possible due to obstructions from sea creatures, bubbles, large suspended particles and features of the seabed, especially in coastal and turbid water environments. Thus, we present the performance of a spatially diverse non-line-of-sight (NLOS) UOWC system employing continuous phase modulation (CPM), which is shown to offer sensitivity benefits of several dBs over on–off keying (OOK) without coherent reception. We obtain the channel impulse response (CIR) by using Monte Carlo simulation, including absorption and multiple scattering. Turbulence is included by conditioning the CIR on log-normal statistics. To mitigate the resultant fading, we exploit spatial diversity with equal gain combining at the receiver side. Photon counting at the receiver is employed to accommodate shot noise. We compare the saddlepoint and Gaussian approximations for bit error rate (BER) calculations, using the latter for later calculations as it delivers excellent results and is simpler. Our results show that spatial diversity offers performance improvements, for example an 8 dB sensitivity gain at 10^-9 BER using 1 Gbps 3×1 multiple-input single-output (MISO) transmission over a 20 m link with 0.16 log-amplitude variance. We determine using an upper bound that Intersymbol Interference (ISI) has a significant impact at high bit rates, producing error floors for multiple-output arrangements.

Vertical and slanted sound propagation in the near-ground atmosphere: Amplitude and phase fluctuations.

Sound propagation along vertical and slanted paths through the near-ground atmosphere impacts detection and localization of low-altitude sound sources, such as small unmanned aerial vehicles, from ground-based microphone arrays. This article experimentally investigates the amplitude and phase fluctuations of acoustic signals propagating along such paths. The experiment involved nine microphones on three horizontal booms mounted at different heights to a 135-m meteorological tower at the National Wind Technology Center (Boulder, CO). A ground-based loudspeaker was placed at the base of the tower for vertical propagation or 56 m from the base of the tower for slanted propagation. Phasor scatterplots qualitatively characterize the amplitude and phase fluctuations of the received signals during different meteorological regimes. The measurements are also compared to a theory describing the log-amplitude and phase variances based on the spectrum of shear and buoyancy driven turbulence near the ground. Generally, the theory correctly predicts the measured log-amplitude variances, which are affected primarily by small-scale, isotropic turbulent eddies. However, the theory overpredicts the measured phase variances, which are affected primarily by large-scale, anisotropic, buoyantly driven eddies. Ground blocking of these large eddies likely explains the overprediction.

Accurate Computation of Scintillation in Tropospheric Turbulence With Parabolic Wave Equation

Parabolic wave equation (PWE) propagation models using multiple phase screens (MPS) are widely used to predict the log-amplitude variance of the signal propagating in tropospheric turbulence. Previous literature demonstrates the limitations of the MPS technique, usable only when the signal frequency is low (typically below 5 GHz for radio-wave propagation in the troposphere). In this article, we demonstrate that the accuracy of the MPS technique depends not only on the signal frequency but also on the outer scale length ( L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> ) of the turbulent eddy. For example, when L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> is on the order of tens of meters (e.g., in most terrestrial communication links), the MPS method is indeed unusable at high frequencies, but, when L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> is on the order of hundreds of meters (e.g., in air-ground and satellite links), the method can be used even at millimeter-wave frequencies. Next, a correlated phase screen (CPS) implementation within the PWE is introduces, which avoids the condition of statistical noncorrelation of the phase screens in the MPS technique, and as such, it can be used for any signal frequency. Finally, an ad hoc error metric is provided, allowing for the selection of the appropriate phase screen model based on the signal frequency, turbulence outer scale length, and the electromagnetic (EM) link distance. The simulated signal log-amplitude variances as a function of range at microwave and millimeter-wave frequencies using the CPS method are reported here for the first time and are shown to be in excellent agreement with the analytically obtained values.

Effect of the Zenith Angle on Optical Wave Propagation in Anisotropic Non-Kolmogorov Atmospheric Turbulence: A New Experiment-Based Model

The spectral properties of the structural function of atmospheric turbulence can differ in the vertical and horizontal directions. Taking into account the influence of this anisotropy on the signal along the direction of propagation at an angle y, to the horizon, can be useful in modeling and interpreting the results of measurements. Usually when modeling the propagation of an electromagnetic signal in the anisotropic turbulent atmosphere, the effective anisotropic parameter η is introduced. In the latter case, the dependence of the wave vector is given by the expression k = √ηk <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ρ</sub> +k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , where k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> is a vertical component of k, and k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ρ</sub> is horizontal. However, the spectral index of the structural function, α, remains constant and independent of y. Here, we consider the structural function of atmospheric turbulence with dependence on the propagation angle y. The existing experimental data show that the spectral index α is a function of height and y. The dependence of the α parameter on y is introduced, and it is determined from the experimental measurements of atmospheric turbulent spectra. As an example of using the proposed approach, we present the results of calculations of the log-amplitude variance and propagation of a Gaussian beam through the anisotropic turbulent atmosphere.

Wave-optics investigation of turbulence thermal blooming interaction: II. Using time-dependent simulations

Part II of this two-part paper uses wave-optics simulations to look at the Monte Carlo averages associated with turbulence and time-dependent thermal blooming (TDTB). The goal is to investigate turbulence thermal blooming interaction (TTBI). At wavelengths near 1 μm, TTBI increases the amount of constructive and destructive interference (i.e., scintillation) that results from high-power laser beam propagation through distributed-volume atmospheric aberrations. As a result, we use the spherical-wave Rytov number, the number of wind-clearing periods, and the distortion number to gauge the strength of the simulated turbulence and TDTB. These parameters simply greatly given propagation paths with constant atmospheric conditions. In addition, we use the log-amplitude variance and the branch-point density to quantify the effects of TTBI. These metrics result from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and TDTB. Overall, the results show that the log-amplitude variance and branch-point density increase significantly due to TTBI. This outcome poses a major problem for beam-control systems that perform phase compensation.

Wave-optics investigation of turbulence thermal blooming interaction: I. Using steady-state simulations

Part I of this two-part paper uses wave-optics simulations to look at the Monte Carlo averages associated with turbulence and steady-state thermal blooming (SSTB). The goal is to investigate turbulence thermal blooming interaction (TTBI). At wavelengths near 1 μm, TTBI increases the amount of constructive and destructive interference (i.e., scintillation) that results from high-power laser beam propagation through distributed-volume atmospheric aberrations. As a result, we use the spherical-wave Rytov number and the distortion number to gauge the strength of the simulated turbulence and SSTB. These parameters simplify greatly given propagation paths with constant atmospheric conditions. In addition, we use the log-amplitude variance and the branch-point density to quantify the effects of TTBI. These metrics result from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and SSTB. Overall, the results show that the log-amplitude variance and branch-point density increase significantly due to TTBI. This outcome poses a major problem for beam-control systems that perform phase compensation.

Electromagnetic Wave Propagation in the Turbulent Atmosphere With an Anisotropic Exponent of the Spectrum

In this communication, we propose a new model for describing the anisotropy of turbulence relative to the vertical and horizontal directions, through the anisotropy of the structure function exponent. Using the obtained anisotropy spectrum, we calculate various parameters such as log-amplitude variance for various atmospheric communication channels with comparison to the classical Kolmogorov model and newer models developed in previous works.

Performance Studies of Underwater Wireless Optical Communication Systems With Spatial Diversity: MIMO Scheme

In this paper, we analytically study the performance of multiple-input multiple-output underwater wireless optical communication (UWOC) systems with ON–OFF keying modulation. To mitigate turbulence-induced fading, which is amongst the major degrading effects of underwater channels on the propagating optical signal, we use spatial diversity over UWOC links. Furthermore, the effects of absorption and scattering are considered in our analysis. We analytically obtain the exact and an upper bound bit error rate (BER) expressions for both optimal and equal gain combining. In order to more effectively calculate the system BER, we apply Gauss-Hermite quadrature formula as well as approximation to the sum of lognormal random variables. We also apply the photon-counting method to evaluate the system BER in the presence of shot noise. Our numerical results indicate an excellent match between the exact and upper bound BER curves. Also, a good match between the analytical results and numerical simulations confirms the accuracy of our derived expressions. Moreover, our results show that spatial diversity can considerably improve the system performance, especially for channels with higher turbulence, e.g., a $3\times 1$ multiple-input single-output transmission in a 25 m coastal water link with a log-amplitude variance of 0.16 can introduce 8 dB performance improvement at the BER of 10−9.

A test of deep water Rytov theory at 284 Hz and 107 km in the Philippine Sea.

Predictions of log-amplitude variance are compared against sample log-amplitude variances reported by White, Andrew, Mercer, Worcester, Dzieciuch, and Colosi [J. Acoust. Soc. Am. 134, 3347-3358 (2013)] for measurements acquired during the 2009 Philippine Sea experiment and associated Monte Carlo computations. The predictions here utilize the theory of Munk and Zachariasen [J. Acoust. Soc. Am. 59, 818-838 (1976)]. The scattering mechanism is the Garrett-Munk internal wave spectrum scaled by metrics based on measured environmental profiles. The transmitter was at 1000 m depth and the receivers at nominal range 107 km and depths 600-1600 m. The signal was a broadband m-sequence centered at 284 Hz. Four classes of propagation paths are examined: the first class has a single upper turning point at about 60 m depth; the second and third classes each have two upper turning points at roughly 250 m; the fourth class has three upper turning points at about 450 m. Log-amplitude variance for all paths is predicted to be 0.04-0.09, well within the regime of validity of either Born or Rytov scattering. The predictions are roughly consistent with the measured and Monte Carlo log-amplitude variances, although biased slightly low. Paths turning in the extreme upper ocean (near the mixed layer) seem to incorporate additional scattering mechanisms not included in the original theory.

Evaluating a linearized Euler equations model for strong turbulence effects on sound propagation

Sound propagation outdoors is strongly affected by atmospheric turbulence. Under strongly perturbed conditions or long propagation paths, the sound fluctuations reach their asymptotic behavior, e.g., the intensity variance progressively saturates. The present study evaluates the ability of a numerical propagation model based on the finite-difference time-domain solving of the linearized Euler equations in quantitatively reproducing the wave statistics under strong and saturated intensity fluctuations. It is the continuation of a previous study where weak intensity fluctuations were considered. The numerical propagation model is presented and tested with two-dimensional harmonic sound propagation over long paths and strong atmospheric perturbations. The results are compared to quantitative theoretical or numerical predictions available on the wave statistics, including the log-amplitude variance and the probability density functions of the complex acoustic pressure. The match is excellent for the evaluated source frequencies and all sound fluctuations strengths. Hence, this model captures these many aspects of strong atmospheric turbulence effects on sound propagation. Finally, the model results for the intensity probability density function are compared with a standard fit by a generalized gamma function.

Comparison of 2D and 3D Electromagnetic Approaches to Predict Tropospheric Turbulence Effects in Clear Sky Conditions

Two-dimensional electromagnetic simulations are often used to evaluate the atmospheric turbulence effects on radiowave propagation in clear sky conditions. However, turbulence is clearly a three-dimensional atmospheric process. Therefore, errors potentially introduced by 2D propagation schemes to predict 3D scintillation effects have to be quantitatively assessed. On the one hand, as part of an analytical approach and starting from the Kolmogorov-von Karman turbulent spectrum, 2D formulations for log-amplitude and phase variances and for log-amplitude and phase temporal power spectra are derived from the 2D scalar propagation equation. They are compared asymptotically to their classical 3D counterparts. On the other hand, as part of a numerical approach, the scintillation effects are evaluated from 3D and 2D parabolic wave equation (PWE) approaches associated with 2D and 1D multiple phase screen (MPS), respectively. It is then shown that 2D propagation schemes underestimate by a factor 1.86 the log-amplitude variances in Fresnel regime and can lead to significant errors in predicting log-amplitude and phase temporal spectra at low frequencies. It is then suggested that the dimensional reduction should be limited to the prediction of log-amplitude and phase variances in Fraunhofer configurations, or to the evaluation of log-amplitude and phase power spectra at high frequencies.

Optical propagation through non-Kolmogorov turbulence

In this paper, the effects of the generalized exponent, the height and the zenith angle on the log-amplitude variance in the weak fluctuation are investigated. The theoretical results indicate that for the downlink, the log-amplitude variance of the Kolmogorov model is always smaller than that of the three-layer model, while for the uplink, there is a point of intersection in the log-amplitude variance curves of the two models. The different phenomena for the downlink and uplink are analyzed in detail. Further, we find a method to ascertain precise values of the boundary layer altitudes for the three-layer model under various atmospheric conditions through the analysis for the point of intersection. And at the point of intersection, the Kolmogorov model can be used to replace the three-layer model to simplify the analysis of the system performance. Moreover, the log-amplitude variance increases with the increase of the zenith angle.

Computer Backplane With Free Space Optical Links: Air Turbulence Effects

Over recent decades, there have been increasing demands for high data-rate backplanes for enhanced performance in computers. The present technology of electronic backplanes cannot meet the required data-rate demands for next-generation computers, indicating a need for a technology shift from the electronic to the optic domain. In this paper, we develop a mathematical model to describe the effect of air turbulence on a short range free space optical communication link, such as would be found in computer backplane communication, and conduct experiments to validate the model. The air turbulence under discussion resembles that which would result from the high temperature of chips on the board together with the ventilation of the air by the chip fan. In our experiment, the communication performance, expressed as bit error rate (BER), is presented as a function of the location of the turbulence source and the log amplitude variance. The log amplitude variance was evaluated by two independent methods. One of the main results of this study is the indication that the performance of the backplane deteriorates even for very small values of air turbulence strength. This is exacerbated by the fact that extremely low BER performance is required at the backplane (e.g., from 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-14</sup> to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">- 18</sup> ). We also demonstrate that increasing the distance between the turbulence source and the optical link reduces the influence of the air turbulence in an exponential manner. This fact is important for the design of future-generation optical backplanes. It is noted that operation at wavelengths around 1550 nm yields communication performance that is less severely degraded by air turbulence effects than at 670 nm.

The relative effects of particles and turbulence on acoustic scattering from deep-sea hydrothermal vent plumes

Acoustic methods are applied to the investigation and monitoring of a vigorous hydrothermal plume within the Main Endeavor vent field at the Endeavor segment of the Juan de Fuca Ridge. Forward propagation and scattering from suspended particulates using Rayleigh scattering theory is shown to be negligible (log-amplitude variance σ(χ) (2)~10(-7)) compared to turbulence induced by temperature fluctuations (σ(χ) (2)~0.1). The backscattering from turbulence is then quantified using the forward scattering derived turbulence level, which gives a volume backscattering strength of s(V)=6.5 × 10(-8) m(-1). The volume backscattering cross section from particulates can range from s(V)=3.3 × 10(-6) to 7.2 × 10(-10) m(-1) depending on the particle size. These results show that forward scatter acoustic methods in hydrothermal vent applications can be used to quantify turbulence and its effect on backscatter measurements, which can be a dominant factor depending on the particle size and its location within the plume.

Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence

In the past decades, both the increasing experimental evidences and some results of theoretical investigation on non-Kolmogorov turbulence have been reported. This has prompted the study of optical propagation in non-Kolmogorov atmospheric turbulence. In this paper, using a non-Kolmogorov power spectrum which owns a generalized power law instead of standard Kolmogorov power law value 11/3 and a generalized amplitude factor instead of constant value 0.033, the log-amplitude variances for a Gaussian-beam wave are derived in the weak-fluctuation regime for a horizontal path. The analytic expressions are obtained and then used to analyze the effect of spectral power-law variations on the log-amplitude fluctuations of Gaussian-beam wave.

LOG-AMPLITUDE VARIANCE OF LASER BEAM PROPAGATION ON THE SLANT PATH THROUGH THE TURBULENT ATMOSPHERE

Based on the altitude-dependent model of ITU-R slant atmospheric turbulence structure constant, the log-amplitude variance of laser beam propagation on the slant path through turbulent atmosphere is obtained with transmitter and receiver parameters and can be degenerated to the result of the horizontal path with atmospheric structure constant as a flxed value. These expressions are convenient tools for beam-wave analysis. Finally, we apply the ITU-R turbulence structure constant model to calculate collimated, divergent and convergent beam log-amplitude variance. The numerical conclusions indicate the log-amplitude variance of laser beam propagation on slant path is generally smaller than those on horizontal path.

Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model

Turbulence properties of communication links (optical and microwave) in terms of log-amplitude variance are studied on the basis of a three-layer model of refractive index fluctuation spectrum in the free atmosphere. We suggest a model of turbulence spectra (Kolmogorov and non-Kolmogorov) changing with altitude on the basis of obtained experimental and theoretical data for turbulence profile in the troposphere and lower stratosphere.

Optical phase unwrapping in the presence of branch points

Strong turbulence causes phase discontinuities known as branch points in an optical field. These discontinuities complicate the phase unwrapping necessary to apply phase corrections onto a deformable mirror in an adaptive optics (AO) system. This paper proposes a non-optimal but effective and implementable phase unwrapping method for optical fields containing branch points. This method first applies a least-squares (LS) unwrapper to the field which isolates and unwraps the LS component of the field. Four modulo-2pi-equivalent non-LS components are created by subtracting the LS component from the original field and then restricting the result to differing ranges. 2pi phase jumps known as branch cuts are isolated to the non-LS components and the different non-LS realizations have different branch cut placements. The best placement of branch cuts is determined by finding the non-LS realization with the lowest normalized cut length and adding it to the LS component. The result is an unwrapped field which is modulo-2pi -equivalent to the original field while minimizing the effect of phase cuts on system performance. This variable-range 'phi LS +phi non phi LS' unwrapper, is found to outperform other unwrappers designed to work in the presence of branch points at a reasonable computational burden. The effect of improved unwrapping is demonstrated by comparing the performance of a system using a fixed-range phi 'LS + phi non--LS' realization unwrapper against the variable-range 'phi LS +phi non--LS' unwrapper in a closed-loop simulation. For the 0.5 log-amplitude variance turbulence tested, the system Strehl performance is improved by as much as 41.6 percent at points where fixed-range 'phi LS + phi non phi LS' unwrappers result in particularly poor branch cut placement. This significant improvement in previously poorly performing regions is particularly important for systems such as laser communications which require minimum Strehl ratios to operate successfully.

Statistical analysis of measured free-space laser signal intensity over a 2.33 km optical path

Experimental research is conducted to determine the characteristic behavior of high frequency laser signal intensity data collected over a 2.33 km optical path. Results focus mainly on calculated power spectra and frequency distributions. In addition, a model is developed to calculate optical turbulence intensity (C(n)/2) as a function of receiving and transmitting aperture diameter, log-amplitude variance, and path length. Initial comparisons of calculated to measured C(n)/2 data are favorable. It is anticipated that this kind of signal data analysis will benefit laser communication systems development and testing at the U.S. Army Research Laboratory (ARL) and elsewhere.

Correctability limitations imposed by plane-wave scintillation in multiconjugate adaptive optics

Plane-wave scintillation is shown to impose multiconjugate adaptive optics (MCAO) correctability limitations that are independent of wavefront sensing and reconstruction. Residual phase and log-amplitude variances induced by scintillation in weak turbulence are derived using linear (diffraction-based) diffractive MCAO spatial filters or (diffraction-ignorant) geometric MCAO proportional gains as open-loop control parameters. In the case of Kolmogorov turbulence, expressions involving the Rytov variance and/or weighted C(2)(n) integrals apply. Differences in performance between diffractive MCAO and geometric MCAO resemble chromatic errors. Optimal corrections based on least squares imply irreducible performance limits that are validated by wave-optic simulations.