Phase Probability Density Function Research Articles

SU‐FF‐J‐98: Motion Encoded Beamlets for Optimization and Evaluation in Four‐Dimensional (4D) Radiotherapy

Purpose: To develop a simple model that fully utilizes the information of 4DCT and accurately incorporates a priori knowledge of patient motion into plan optimization and evaluation in 4D radiotherapy. Methods and Materials: We model the IMRT optimization as two independent schemes, the beamlet dose calculation scheme and the beamlet weights updating scheme. All motion information is encoded in the beamlet dose calculation scheme. A 4DCT is used to calculate beamlet dose associated with each phase. Deformable registration determines the voxel‐to‐voxel map from the arbitrary phase to the reference phase, which is used to map beamlet dose of the arbitrary phase to the reference phase. A priori knowledge of respiration is modeled as (time‐dependant) probability density function of respiration phases. The motion encoded beamlet (MEB) is the probability‐weighted summation of the deformed beamlet of the arbitrary phase. Dose calculation and plan evaluation is always in the reference phase. Beamlet weights updating scheme is same as static optimization. We studied different optimization and delivery schemes in 4D radiotherapy. These studies include static optimization with various margins, free breathing delivery (FBD), gating deliveries (GD) with uncertainty and breathing synchronized delivery (BSD) with uncertainty. Results: As for the FBD considered, optimization using MEB show superior DVHs with reduced OAR dose than the margin‐based optimization. GD is better than the FBD if proper gating phase is chosen and gating uncertainty is low. BSD is superior to FBD if synchronization is enforced during delivery. On the other hand, FBD shows greater tolerance to delivery uncertainty than GD and BSD method. Conclusions: Motion encoding through probability‐weighted summation of beamlets is a flexible and powerful technique to incorporate arbitrary patient motion into optimization and evaluation. Through MEB, evaluations of different delivery schemes in 4D radiotherapy can be done in the same framework.

Importance sampling of Gordon-Mollenauer soliton phase noise in optical fibers

We develop an importance sampling method to perform the direct numerical computation of the probability density function of the random optical soliton phase under the influence of both amplifier spontaneous emission noise and nonlinear conversion of amplitude to phase fluctuations owing to the Kerr effect, or nonlinear phase noise.

Minimum entropy approach for carrier frequency recovery

This paper introduces a new carrier frequency recovery approach. It can be applied before modulation classification and/or demodulation of the M-ary PSK signals. It relies on the entropy of the instantaneous phase probability density function, and uses the fact that it reaches minimum when the receiver is fine-tuned to the unknown carrier frequency. This estimator is applicable to algorithms requiring high accuracy without any a priori knowledge concerning modulation scheme, signal contents (bit-stream), or its timing parameters. Simulation results have proved the robustness of the algorithm: for low carrier to noise ratios (CNR), corresponding variances are proportional and close to Cramer-Rao lower bounds (CRLB). For CNR greater than 20 dB, they are constant and limited by the resolution of the algorithm.

An Improved Scheme for Channel Parameter Estimation in Mobile Communication Systems

In [I], an algorithm based on phase variations of received pilot symbols was proposed to estimate one of the most important channel parameters, maximum Doppler shift, f d . However, AWGN (Additive white gauss noise) will cause large estimation error in some cases. In order to analyze the influence of noise, we extended the phase probability density function (pdf) in [1] to the scenario with both fading and AWGN, then the estimation error is characterized in closed-form expression. By this error expression, we found that power control will affect the estimator of [1] and we proposed a modification method based on SNR estimation to obtain accurate Doppler shift estimation in moderate low SNRs (signal-to-noise ratio). Simulation results show high accuracy in wide range of velocities and SNRs.

Algorithms for iterative decoding in the presence of strong phase noise

We present two new iterative decoding algorithms for channels affected by strong phase noise and compare them with the best existing algorithms proposed in the literature. The proposed algorithms are obtained as an application of the sum-product algorithm to the factor graph representing the joint a posteriori probability mass function of the information bits given the channel output. In order to overcome the problems due to the presence in the factor graph of continuous random variables, we apply the method of canonical distributions . Several choices of canonical distributions have been considered in the literature. Well-known approaches consist of discretizing continuous variables or treating them as jointly Gaussian, thus obtaining a Kalman estimator. Our first new approach, based on the Fourier series expansion of the phase probability density function, yields better complexity/performance tradeoff with respect to the usual discretized-phase method. Our second new approach, based on the Tikhonov canonical distribution, yields near-optimal performance at very low complexity and is shown to be much more robust than the Kalman method to the placement of pilot symbols in the coded frame. We present numerical results for binary LDPC codes and LDPC-coded modulation, with particular reference to some phase-noise models and coded-modulation formats standardized in the next-generation satellite Digital Video Broadcasting (DVB-S2). These results show that our algorithms achieve near-coherent performance at very low complexity without requiring any change to the existing DVB-S2 standard.

The phase probability distribution of general Clarke model and its application in Doppler shift estimator

The phase probability density function (PDF) of the channel coefficients is analyzed for general Clarke channel model, and the close-form expressions are obtained under the von Mises/Tikhonov distribution. Then the simplified expressions for Rayleigh and Rice channels are derived. In addition, these expressions are applied to analyze the bias of a Doppler shift estimator based on phase difference proposed firstly by Sheng. The theoretic and simulation results show the good performance and consistency of this estimator.

Automatic classification of interference patterns in driven event series: application to single sympathetic neuron discharge forced by mechanical ventilation.

This study proposes a method for the automatic classification of nonlinear interactions between a strictly periodical event series modelling the activity of an exogenous oscillator working at a fixed and well-known rate and an event series modelling the activity of a self-sustained oscillator forced by the exogenous one. The method is based on a combination of several well-known tools (probability density function of the cyclic relative phase, probability density function of the count of forced events per forcing cycle, conditional entropy of the cyclic relative phase sequence and a surrogate data approach). Classification is reached via a sequence of easily applicable decision rules, thus rendering classification virtually user-independent and fully reproducible. The method classifies four types of dynamics: full uncoupling, quasiperiodicity, phase locking and aperiodicity. In the case of phase locking, the coupling ratio (i.e. n: m) and the strength of the coupling are calculated. The method, validated on simulations of simple and complex phase-locking dynamics corrupted by different levels of noise, is applied to data derived from one anesthetized and artificially ventilated rat to classify the nonlinear interactions between mechanical ventilation and: (1) the discharges of two (contemporaneously recorded) single postganglionic sympathetic neurons innervating the caudal ventral artery in the tail and (2) arterial blood pressure. Under central apnea, the activity of the underlying sympathetic oscillators is perturbed by means of five different lung inflation rates (0.58, 0.64, 0.76, 0.95, 1.99 Hz). While ventilation and arterial pressure are fully uncoupled, ventilation is capable of phase locking sympathetic discharges, thus producing 40% of phase-locked patterns (one case of 2:5, 1:1, 3:2 and 2:2) and 40% of aperiodic dynamics. In the case of phase-locked patterns, the coupling strength is low, thus demonstrating that this pattern is sliding. Non-stationary interactions are observed in 20% of cases. The two discharges behave differently, suggesting the presence of a population of sympathetic oscillators working at different frequencies.

Statistical analysis of multilook SAR interferograms for CFAR detection of ground moving targets

This paper examines the statistics of the phase and magnitude of multilook synthetic aperture radar (SAR) interferograms toward deployment of along-track interferometry (ATI) for slow ground moving-target indication (GMTI). While the known probability density function (pdf) of the interferogram's phase (derived under the assumption of Gaussian backscatter) is shown to agree almost perfectly for a wide variety of backscatter conditions, the corresponding magnitude's pdf tends to deviate strongly in most cases. Motivated by this discrepancy, a novel distribution is derived for the interferogram's magnitude. This pdf, called the polynomial or p-distribution, matches the real data much more accurately, particularly for heterogeneous composite terrain. For extremely heterogeneous terrain, such as urban areas, both pdfs for interferometric phase and magnitude fail and are extended. Based on these statistics, a completely automatic detection scheme with constant false-alarm rates (CFARs) for slow moving targets is proposed. All involved parameters required to determine the detection thresholds are estimated from the sample data. It is demonstrated, on the basis of experimental airborne SAR data, that this detector is capable of detecting slow moving vehicles within severe ground clutter.

Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics

Fixed broadband beamformers using small-size microphone arrays are known to be highly sensitive to errors in the microphone array characteristics. The paper describes two design procedures for designing broadband beamformers with an arbitrary spatial directivity pattern, which are robust against gain and phase errors in the microphone array characteristics. The first design procedure optimizes the mean performance of the broadband beamformer and requires knowledge of the gain and the phase probability density functions, whereas the second design procedure optimizes the worst-case performance by using a minimax criterion. Simulations with a small-size microphone array show the performance improvement that can be obtained by using a robust broadband beamformer design procedure.

Stochastic resonance in a sinusoidally forced LIF model with noisy threshold

In this report, the LIF neural model driven by underthreshold sinusoidal signals but with a gaussian-distributed noise on the threshold, is approximated by suitably defining an instantaneous firing (or escape) rate, which depends only on the momentary value of the voltage variable. This allows us to obtain, by analytically solving the relevant equations, the main statistical functions describing the “firing activity”; namely, the probability density function of firing phases and that of interspike intervals. From these functions two quantities can be derived, whose dependence on the noise intensity allows the Stochastic Resonance (SR) to be demonstrated. Besides the “regular” SR, the analysed system was found to produce, either for low frequencies and large amplitudes of modulation or for high modulation frequencies, resonance curves displaying two peaks. This bimodal feature of the resonance curves is accounted for on the basis of phase locked firing patterns.

First-order statistical properties of ultrasonic speckles back-scattered from a kind of weak scattering interface

First-order statistical properties of ultrasonic speckles back-scattered from an interface with tiny pints on, which is considered as one kind of weak scatterin g interface, are studied in this paper. Based on four assumptions, theoretical a nalyses indicate that the amplitude of ultrasonic speckles in space is in the fo rm of Rice distribution, and the phase probability density function of the ultra sonic speckles is a compound function of Gauss distribution function and Gauss p robability integral function. When the density of the pits on the interface is v ery high, the amplitude of ultrasonic speckles becomes Rayleigh distributed. To verify the theoretical analysis conclusion, the amplitude probability distributi on of ultrasonic speckles is measured by using a set of experimental systems wit h step scanning controlled by a computer, which is built by the authors. The exp erimental results are in good agreement with the theoretical predicted results. So, the assumptions proposed in this paper are suitable for the theoretical anal yses.

An airborne synthetic aperture radar (SAR) experiment to support RADARSAT-2 ground moving target indication (GMTI)

As part of the preparatory work for the RADARSAT-2 moving object detection experiment (MODEX special operating mode), an airborne experiment was conducted at Canadian Forces Base Petawawa in July 1999 to study synthetic aperture radar (SAR) along-track interferometry (ATI) and SAR displaced phase centre antenna (DPCA) as candidate measurement techniques. The CV 580 SAR system operated by Environment Canada was configured as a two-aperture temporal interferometer, and data were collected over an experiment site that contained coordinated, global positioning system (GPS) monitored, moving targets. A robust ATI phase probability density function was identified and used in conjunction with a moving-target matched filter to extract and measure target motion. Corner reflector and vehicle velocities were measured for target speeds between 9 and 100 km/h to within 5% accuracy. SAR ATI and SAR DPCA performed equally well over this range of velocities, but ATI appears to be the more robust processing algorithm.

Estimating the effective number of looks in interferometric SAR data

The probability density function (pdf) of the multi-look interferometric phase between two complex synthetic aperture radar (SAR) images is parameterized by the number of looks and the complex correlation coefficient. In practice, adjacent pixels in a real SAR interferogram, are statistically dependent due to filtering, and hence, the number of looks is usually smaller than the number of samples averaged. It has been shown that compensation with an effective number of looks, rather than an intractable rederivation of the pdf, can account for the statistical dependence. This paper addresses the challenge of how to determine a suitable value for the effective number of looks. It is shown that an optimum value can be found via a maximum-likelihood estimator (MLE) based on the interferometric phase pdf. However, since such an MLE is computationally intensive and numerically unstable, an estimator based on the method of moments (MoM) possessing similar fidelity is proposed. MoM is fast and robust and can be used in operational applications, such as determining constant false alarm rate (CFAR) detection thresholds for moving-target detection in SAR along-track interferometry.

Hilbert transformer and time delay: statistical comparison in the presence of Gaussian noise

We have shown that in the presence of independent and identically distributed (i.i.d.) additive Gaussian noise, the performance of a time delay as a phase shifter, which produces a signal-dependent phase shift, is comparable with the performance of a Hilbert transformer which produces a signal-independent 90/spl deg/ phase shift, especially for high signal-to-noise ratios (SNRs) and the proper choice of the delay. Sinusoidal signals are considered. It is shown that the time-delay phase process in Gaussian noise can be approximated by the noise-free phase plus a non-Gaussian phase noise. As the SNR increases, the expected value of the time-delay phase estimator approaches the true value of the phase, and its variance substantially decreases and converges to the Cramer-Rao bound for all ranges of the effective parameters: the phase shift, the deterministic phase, and SNR. A symmetric transformation of the phase probability density function (PDF) that improves the performance of the time delay in Gaussian noise is also proposed.

非定常確率過程を条件として含む確率場のシミュレーション手法について

The objective of this study is to develop a theoretical framework for conditional random fields (CRFs) which consists of non-stationary stochastic processes. For this purpose, we use the probability density function of Fourier amplitudes and phases in frequency domain. The problem area of CRFs in this study is limited in the estimation of stochastic processes conditioned by realized values of time series. To represent the properties of non-stationary processes, we will introduce group delay time spectra, which are gradient of phase spectra with respect to frequency. Using the style of likelihood method, the conditional probability density functions of Fourier phases are updated by information of group delay time. On this basis, a method to generate numerically the conditional random fields containing non-stationary processes is developed and it is verified through the numerical examples that the method can give reasonable results.

Unsteady Drag Consideration in Stochastic Eulerian-Lagrangian Formulation of Two-Phase Flow

The unsteady effect of drag coefficient in the stochastic Eulerian-Lagrangian computation of a two-phase, turbulent, planar mixing layer flow of droplets-in-gas type is examined. The use of an unsteady drag coefficient in modeling two-phase turbulent flows with the stochastic Eulerian-Lagrangian formulation is necessary to obtain complete information of the dispersed-phase turbulence characteristics. The use of the quasisteady drag coefficient, which is mostly adopted in two-phase turbulent flow computations, does not cause too much deviation in the determination of mean quantities of the droplet velocity through use of the ensemble-averaging method weighting with the probability density function of the carrier phase

An extended Suzuki model for land mobile satellite channels and its statistical properties

This paper deals with the statistical characterization of a stochastic process which is a product of a Rice and lognormal process. Thereby, we consider the more general case where the two Gaussian noise processes describing the Rice process are correlated. The resulting process are named as extended Suzuki process, which can be used as a suitable statistical model for describing the fading behavior of large classes of frequency nonselective land mobile satellite channels. In particular, the statistical properties (e.g., probability density function (pdf) of amplitude and phase, level-crossing rate, and average duration of fades) of the Rice process with cross-correlated components as well as of the proposed extended Suzuki process are investigated. Moreover, all statistical model parameters are optimized numerically to fit the cumulative distribution function and the level-crossing rate of the underlying analytical model to measured data collected in different environments. Finally, an efficient simulation model is presented which is in excellent conformity with the proposed analytical model.

An asymptotic optimal algorithm for modulation classification

A suboptimal algorithm for modulation classification was proposed for classifying the modulation type of general M-ary phase-shifted keying (MPSK) signals. Yang and Soliman (see IEEE Trans. Aerosp. Electron. Syst., vol.33, no.1, p.38-45, 1997) approximated the phase probability density function of a received signal to be the Tikhonov function. Instead, we employ the exact phase density function and derive an asymptotic optimal classification algorithm. We show a structure of this proposed classifier for continuous wave (CW), binary phase-shift keying (BPSK), quaternary phase-shift keying (QPSK), and 8PSK. Besides, we give an example to demonstrate the capability of this algorithm and compare its performance to that in Yang and Soliman. It is shown that the performance is more effective.

A stochastic model for wave localization in one-dimensional disordered structures

This paper presents a probabilistic study of the effects of structural irregularity on wave propagation along an infinite 1-D chain. A general integral equation method based on Markov chain theory is used to determine the phase probability density function (pdf) at the scatterers distributed irregularly along the chain. The scatterers could be atoms in a one-dimensional crystal, or ribs on a flat plate or membrane. The integral equation derived for the phase pdf is simplified considerably when the scatterers are distributed completely randomly or quasi-periodically. In these cases, the integral equations may be asymptotically solved for the phase density functions in the limit of weak or strong scattering; the localization factors are then obtained. The present approach is quite general and is directly applicable to any disordered one-dimensional system consisting of identical scatterers that are arranged according to a probability distribution function. The validity of the present asymptotic solutions is examined and verified by comparing against the existing analytical solutions for simple atomic or mechanical disordered systems.

On the statistical properties of deterministic simulation models for mobile fading channels

Rice's (9144, 1945) sum of sinusoids can be used for an efficient approximation of colored Gaussian noise processes and is therefore of great importance to the software and hardware realization of mobile fading channel models. Although several methods can be found in the literature for the computation of the parameters characterizing a proper set of sinusoids, less is reported about the statistical properties of the resulting (deterministic) simulation model. In this paper, not only is the simulation model's amplitude and phase probability density function (PDF) investigated, but also higher order statistics [e.g., level-crossing rate (LCR) and average duration of fades (ADFs)]. It is shown that due to the deterministic nature of the simulation model, analytical expressions for the PDF of the amplitude and phase, autocorrelation function (ACF), crosscorrelation function (CCF), LCR, and ADFs can be derived. We also propose a new procedure for the determination of an optimal set of sinusoids, i.e., the method results for a given number of sinusoids in an optimal approximation of Gaussian, Rayleigh, and Rice processes with given Doppler power spectral density (PSD) properties. It is shown that the new method can easily be applied to the approximation of various other kinds of distribution functions, such as the Nakagami (1960) and Weibull distributions. Moreover, a quasi-optimal parameter computation method is presented.