Phase Probability Density Function Research Articles

Characterization of Stochastic Bifurcations in a Simple Biological Oscillator

This study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first-passage-time problem of the Ornstein–Uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density. Stochastic phase-locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase. A firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel.

Phase Probability Density Function in Two-Mode Fock Space

Based on the newly found representation in which the phase operator corresponding to Shapiro-Wagner phase measurement scheme manifestly exhibits its phase behavior, we construct the phase probability density function in the two-mode Fock space, which provides an alternative approach for analysing the phase properties of optical fields.

Performance Analysis of an Expanding-Search Algorithm forCoarse Acquisition of DSSS Signals

For coarse acquisition of Direct Sequence Spread Spectrum (DSSS) signals, a search of the possible code phases is required. In cases such as signal reacquisition, where a priori information is available, the serial search may be abandoned in favor of an expanding search. In this paper, an expanding-search algorithm is presented and compared to the expanding-sweep algorithms which have been treated in the literature. Performance results for the serial-search and the expanding-search algorithms are given. A discussion of the optimality of the serial-search and the expanding-search algorithms is presented. Performance comparisons of the serial search and the expanding search are made for different a priori code phase probability density functions (pdfs) and the results are discussed. A technique for determining when the expanding search is an optimal search algorithm is developed and applied to the performance curves shown previously. The results indicate that the expanding search provides a quantifiable improvement over the serial search when the pdf is somewhat peaked. The results also indicate that in many cases the expanding search may be a more effective search algorithm than the more common expanding sweep.

Fluctuations of Differential Phase and Radar Measurements of Precipitation

Abstract The specific differential polarization phase shift KDP has been suggested as a possible estimator of rainfall as well as a rain-hail discriminator in several recent studies. Here, the complex multivariate Gaussian model, used widely in communication theory and statistics optics, is applied for modeling KDP caused by precipitation. Specifically, the width of the marginal phase probability density function is used to estimate ϕDP (cumulative differential phase) fluctuations. This results in suboptimal but simple expressions, which highlight the dependence of ϕDP fluctuations on the magnitude of the HV (horizontal–vertical) cross-correlation coefficient |ρHV|. Statistical arguments are separated from the effects of radar wavelength and Doppler spread. Implications of ϕDP fluctuations on accuracy of rainfall and snow/hail radar measurements are also discussed and the author comments on the feasibility of using cross-polarized differential phase measurements.

Low SNR approximation to phase PDF for PSK signals

For the original article see ibid., vol. 30, no. 16, p. 1279-80 (1994). In the afore-mentioned paper the authors, P.C. Sapiano et al., derived a Fourier series expansion of the probability density function of the signal phase in the presence of AWGN. The commenter shows how Fourier series coefficients may be obtained directly from this expansion. In reply, the authors agree with this but maintain that their alternative route for deriving the Fourier series coefficients, based on series expansions of special functions, is still valid. However, they consider that this does not detract from the main thrust of their argument which was based on the low SNR approximation of a multilevel PSK phase PDF and an efficient method for generating the Fourier series coefficients. >

First-order phase statistics of gaussian speckles produced by a curved surface in the diffraction region

Statistical properties of the speckle phase produced in the diffraction region by a curved rough surface are theoretically studied. The general expression for the probability density function of the speckle phase is derived by introducing an “optical system function” and taking the curvature radius of an object surface into the macroscopic phase component of the optical system function. The effects of the curvature radius of an object on the probability density function of the speckle phase are investigated by using the equiprobability density ellipse from the viewpoint of circularity and the asymmetry of the complex amplitude of the speckle field.

Theoretical study of the VHF radar echo statistics

In this article the statistical characteristics of the VHF radar returns, which are assumed to comprise components generated from the atmospheric isotropic turbulences plus anisotropic irregularities are theoretically studied. By employing a theory of the random variable, the analytic form of the amplitude and phase probability density functions of the VHF radar echoes are derived. After a somewhat tedious and complicated calculation it shows that the conventional Rayleigh distribution, the Rice distribution, and the Hoyt distribution are closely related to the derived generalized probability density function. The Nakagami m parameter corresponding to this generalized probability density function of the VHF radar signal amplitude is derived as well. It indicates that the magnitude of Nakagami m parameter is governed by the radar echo parameters, that is, 2S2, σ2, and μ, where 2S2 is the power of the radar echo component scattered from the isotropic turbulences, μ and σ2 are the mean and variance, respectively, of the radar signal component generated from the anisotropic irregularities. After examining the general behavior of the derived Nakagami m parameter in more detail, it is found that no matter what the S value is, the magnitude of Nakagami m parameter, m, is always in one of the following three categories, depending on the relative changes between μ and σ. Namely, m = 1 if σ2 = (2½ ‐ 1)μ2; 0.5 < m < 1 if σ2 > (2½ ‐ 1)μ2; and m > 1 if σ2 < (2½ ‐ 1)μ2. These results are quite different from those expected with the conventional theories. Therefore great care should be taken when the probability density function of signal amplitude and phase and the Nakagami m parameter are employed to distinguish the echo mechanism of VHF radar.

On the error rates of differentially detected narrowband pi /4-DQPSK in Rayleigh fading and Gaussian noise

Closed-form formulas for the probability density function (PDF) and cumulative distribution function (CDF) of the differential phase of a phase-modulated carrier in Rayleigh-fading and correlated noise is derived and applied to evaluate the bit-error-rate (BER) of pi /4-DQPSK and conventional DQPSK (differential quaternary phase-shift keyed) systems. Under very narrow IF band-limitation and at high SNR, the pi /4-DQPSK outperforms the conventional DQPSK because the former has less spectral splatter during phase transitions between symbol. The performances of these two systems approach one another as the bandwidth parameter BT increases. >

Unwrapped-phase distribution model for speckle and turbulence

A model for the unwrapped phase of a speckle field that propagates through clear air turbulence is proposed, and a formulation for the probability density function of the unwrapped phase is developed. A method is given to obtain the parameters of the unwrapped density function from parameters of the diffuse target, the laser source, and the atmosphere. Unwrapped-phase measurements at an atmospheric test site with a CO(2) laser show agreement with the model.

Probability density functions of sums of sinusoidal waves having nonuniform random phases and random numbers of multipaths

The purpose of this article is to study the first-order statistics (i.e., probability density functions and moments) of finite sums of cosines and of sines, each having nonuniformly distributed random phases governed by a von Mises probability density function. The number of sinusoids N is, first, taken to be deterministic, but arbitrary, and, second, taken to be a discrete random variable governed by a negative binomial distribution (which includes the Poisson distribution as a special case). Since the phase probability density functions are nonuniform, the probability density functions of the cosine sum and the sine sum are different. The probability density functions of these cosine and sine sums are evaluated using Fourier series expansions whose Fourier coefficients are sampled values of the corresponding characteristic functions. Representative numerical calculations have been carried out to illustrate the general features. The corresponding moments have also been evaluated. Finally, the behavior of the probability density functions and moments when N approaches infinity in the deterministic situation and 〈N〉 approaches infinity in the random situation is investigated, the former involving the theory of stable probability density functions, the latter involving the theory of infinitely divisible probability density functions.

Probability density function for the phase and frequency noise in a semiconductor laser

A physical definition for the phase and frequency noise of a semiconductor laser is given, which can be applied in determining the performance of coherent optical communication systems. The definition allows a filtered Poisson model to be developed for the probability density function (pdf) of the laser output phase or frequency noise, based on evaluating the cumulants of the noise processes. A condition is derived to quantify under what laser and system conditions a Gaussian pdf is a good approximation. It is shown that the phase and frequency noise can increase significantly for bit rates approaching the relaxation oscillation frequency of the laser diode.

Error Rates of DCPSK Signlas in Hard-Limited Multilink Systems with Cochannel Interference and Noise

This paper presents an analytical method for evaluation of the error performance of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary differentially coherent phase-shift keyed signals (DCPSK) transmitted through a finite chain of bandpass hard-limiting (BPHL) repeaters. Each link between repeaters is additively corrupted by Gaussian noise and cochannel interference (having the same form as the signal). The system passband is sufficiently wide so that the signal and each interference pass without distortion. The analysis is based on the newly derived phase probability density function of the sum of the signal, noise, and interference. In addition, the performance of the BPHL system is compared with the corresponding performance of a linear system. Also, the interference immunity of CPSK and DCPSK signals is examined. The general observations are similar to those previously discussed for CPSK (referring to the BPHL multilink system) and for DCPSK in the linear channel.

Low-Frequency Correlation Phase Detectors Using Hard Limiting

Low-frequency radio navigation receivers reduce dynamic range by hard limiting and phase is estimated from the outputs of correlation detectors. The detector outputs are averages of a detector characteristic function determined by convolution of the hardlimited signal and reference waveform. The expected value and variance of the output are determined by expanding both the characteristic function and the phase probability density function (pdf) into Fourier series. Results are given for four detectors.

Interference Immunity Effects in CPSK Systems with Hard-Limiting Transponders

Transmission characteristics are described for M-ary coherent phase-shift keyed (CPSK) systems with N-stage hard-limiting transponders where noise and multiple continuous wave (CW) interference sources are additively combined in each link. A general expression for the probability density function of the composite phase at the final link is derived; the overall error probability is then obtained from this general expression. Comparisons with N-cascaded linear amplifier systems clearly show the noise and interference immunity that result from the use of bandpass hard limiters (BPHLs). The BPHL system's error probability improvement versus interference sources is shown to be much larger than the improvement versus noise.

First-order probability density function of the laser speckle phase

An expression for the first-order probability density function of the laser speckle phase is analytically derived under the assumption that the speckle field obeys a non-circular, complex Gaussian, random process with a certain correlation between the real and imaginary parts of its complex amplitude. The probability density function of the speckle phase is actually evaluated for various cases and shown three-dimensionally as a function of the standard deviation of random object phase variations. The effect of random object phase variations on the probability density function is also investigated in detail.

Fourier-series representation for phase density of sum of signal, noise and interference

A new Fourier-series expansion of the phase probability density function for the sum of signal, Gaussian noise and co-channel interference is derived. One of several possible applications of the derived expression is given.

Erratum: Fourier-series representation for phase density of sum of signal, noise and interference

A new Fourier-series expansion of the phase probability density function for the sum of signal, Gaussian noise and co-channel interference is derived. One of several possible applications of the derived expression is given.

Transmission Characteristics of an M-ary Coherent PSK Signal Via a Cascade of N Bandpass Hard Limiters

This paper provides a general analysis of the transmission characteristics of a coherent PSK signal, where an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -ary coherent signal is transmitted via a repeater system composed of a cascade of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> bandpass hard limiters. The probability density function of a composite phase of the signal and noise over the system is newly derived to obtain an error probability. In addition, the comparison with the repeater system composed of a cascade of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> linear amplifiers is made to clarify the features of the transmission characteristics. It is found that the probability density function of the composite phase over the system is symmetric about each input carrier-to-noise power ratio (CNR), and it reduces to some well-known results. It is also shown that for a reasonably large input CNR in each stage, the presence of the bandpass hard limiter (BPHL) provides some improvement, but that for small input CNR it causes degradation in performance.

On the Fourier coefficients for the phase-shift keyed phase density function (Corresp.)

A closed-form expression is derived for the Fourier coefficients in the series expansion of the phase-shift keyed (PSK) receiver output phase probability density function.