The asymmetric random telegraph signal has unequal transition probabilities in the up and down directions. This signal, with and without filtering by a single-time-constant filter, is used in modeling ionic channels in nerve membrane. Formulas are derived for the distribution of number of jumps in an interval, probability density functions of pulse duration, autocorrelation, spectral density, Kolmogorov and Fokker-Planck-Kolmogorov equations, transition-probability functions, and the steady-state probability density function for the filtered signal. This collection of results includes both previously published formulas, here presented in most general form and in uniform notation, and some new results not previously published. These formulas are of use in analyzing records of single channels in nerve membrane.

Statistical properties of the asymmetric random telegraph signal, with applications to single-channel analysis

Since Blake and Lindsey’s [19] comprehensive survey of results and techniques for level-crossing problems for random processes appeared in 1973, a number of interesting new results addressing both classical and new problem areas have been developed. As there are many diverse areas of application of level crossing results, the theoretical literature is fairly widely dispersed. The present survey is intended to be an update of Blake and Lindsey’s and is intended to be similar to theirs with an emphasis on explicit analytical results for continuous parameter processes. As they discussed many of the techniques and methods available in an accessible tutorial fashion, the present survey will be confined to an overview of the results that are available.

A Survey of Recent Progress on Level-Crossing Problems for Random Processes

The problem of calculating the probability density function of the output of an RC filter driven by a binary random process with intervals generated by an equilibrium renewal process is studied. New integral equations, closely related to McFadden's original integral equations, are derived and solved by a matrix approximation method and by iteration. Transformations of the integral equations into differential equations are investigated and a new closed-form solution is obtained in one special case. Some numerical results that compare the matrix and iteration solutions with both exact solutions and approximate solutions based upon the Fokker-Planck equation are presented.

On filtered binary processes

It is shown that for a broad class of random binary processes, the mean of a binary process, conditioned on its zero crossings, is proportional to the derivative of the correlation function of that process. This specific mean, referred to as the implicit mean, can be used both to characterize a random binary waveform and to determine its delay. The proposed empirical estimator of the implicit mean is unbiased, and its variance depends on the number of selected trajectories of the waveform being processed and the correlation between those trajectories.

Implicit averaging and delay determination of random binary waveforms

Equations are derived which describe the time evolution of the probability density and corresponding characteristic function of a telegraph signal which has passed through a detuned Lorentzian filter. A closed form expression for the characteristic function is obtained for the tuned case and the predicted joint statistics and correlation properties are reviewed in the context of earlier results. Low-order correlation properties for the more general detuned case are calculated. It is shown that the stationary single interval statistics can be generated by a random phasor moving in a space of fractional dimensions and that a simple transform of variable leads to distributions which are stable in this space.

http://iopscience.iop.org/article/10.1088/0305-4470/32/50/303/pdf

Statistics of a filtered telegraph signal

A study is made of various statistical properties of a particular non-Gaussian process, the low-pass filtered random telegraph signal. Results are obtained for the distribution of slope, the average numbers of zero and level crossings, the average number of crossings with a straight line, distributions of maxima and minima and the average number of specular points. As a prelude to these specific investigations, general expressions for the average numbers of crossings of random processes with arbitrary curves are derived. In particular, for the average number of zero-crossings of a stationary random process, the result does not explicitly involve the joint distribution of the process and its derivative.

R. F. Pawfjla

Papers

Statistical geometry of the smoothed random telegraph signal