Steady-state Probability Density Function Research Articles

Performance of Phase-Locked Loops in the Presence of Fading Communication Channels

An approach is presented for the analysis of phase-locked loops whose input signal has passed through time-varying channels. The specific channels considered in detail are the Rice-Nakagami, Rayleigh, and lognormal fading channels. Loop performance is characterized in terms of the steady-state probability density function of the reduced phase error process. The basic parameters which characterize performance include the loop signal-to-noise ratio (SNR) and the variance and bandwidth of the fading components introduced by the channel. Particular channel models are used to illustrate the theory for the firstorder loop. The results are also applied to the analysis of the PSK noisy reference problem in the presence of these time-varying channels.

Statistical Behavior of Second-Order Phase-Locked Loops in the Presence of a Mixture of Impulsive and Gaussian Noise

An equation is found for the steady-state probability density function of the modulo-2π-reduced phase error of a secondorder phase-locked loop (PLL) with a proportional-proportional-plus-integral control type filter. The equation involves a certain conditional expectation which has been evaluated in an approximate way following the same approach used by previous authors in the case of sole Gaussian noise. With this approximation the equation becomes of the same type as that pertinent to a first-order system and therefore it can be solved with a method described by Ohlson in [3]. Theoretical results obtained in this way are complemented by experimental observations gathered by simulating the PPL on a digital computer.

Transient analysis of phase-locked tracking systems in the presence of noise

This paper is concerned with the problem Of obtaining time-dependent solutions to a class of Fokker-Planck equations that arise in the analysis and synthesis of a variety of first-order synchronization systems employing the phase-lock principle. These include the classical sinusoidal phase-locked loop, squaring and Costas loops, data-aided loops, hybrid loops, various symbol synchronizer mechanizations, and tunnel-diode oscillators. By analyzing the spectral properties of the associated time-dependent Fokker-Planck boundary value problem, eigenfunction expansions of the reduced modulo- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 \pi</tex> phase-error-transition probability-density function are developed for a class of first-order synchronization systems. Whereas previous work treated the steady-state probability density function, this approach yields a complete statistical description of the phase-error process reduced modulo- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 \pi</tex> .

A Double-Loop Tracking System

A nonlinear analysis which can be used to assess certain statistical characteristics of double-loop tracking systems is presented. It takes into account the mutual coupling effects of the loops in the system. Two approaches are taken to obtain steady-state probability density functions (pdf's) of the system phase errors, φ 1 and φ 2 From these pdf's, important system performance statistics, e.g., the phaseerror variances, can be calculated, thus illustrating the application and usefulness of the analysis. The analysis is applied to a satellite transponder as an example.

An alternative expression for the probability density function of the phase error in a phase-locked loop

An alternative formula is given for the steady-state probability density function of the phase error in a phase-locked loop. This formula is of interest because it involves simple functions available in all books of mathematical tables.

Phase-density distribution of phase-locked loops in cascade

Based upon methods outlined by the author in 1968, he presents in this correspondence the steady-state probability density function of a phase measurement made by two phase-locked loops (PLLs) in cascade. He treats the special case of two first-order loops. However, he expects to report upon the general theory presently under development in a later paper.