The point process, a sequence of random univariate random variables derived from correlated bivariate random variables as modeled by Arnold and Strauss, has been examined. Statistical properties of the time intervals between the points as well as the probability distributions of the number of points registered in a finite interval have been analyzed specifically in function of the coefficient of correlation. The results have been applied to binary detection and to the transmission of information. Both the probability of error and the cut-off rate have been bounded. Several simulations have been generated to illustrate the theoretical results.

It is known that the detection of an optical field at a low level of power is a sequence of events which is a set of distinct time instants

Reciprocally, the existence of a physical optical field, given the knowledge of all of the properties of such an RPP, is not an easy problem to solve. For example, given an RPP of nonclassical properties [

Nevertheless, using a parameterized RPP, whose special values of such a parameter (denoted here

As already pointed out, there are two types of processing that can be utilized to characterize such RPP: the time interval distributions (TIDs) and the probability of number distributions (PNDs) [

If we now choose to characterize the RPP by the PND, we can as above, define the

The purpose of this paper is firstly to calculate the statistical properties of the RPPs, both in terms of the TIDs and the PNDs. Secondly we apply the results to the calculation of the performances, namely, the probability of error in binary detection and the cut-off rate in the binary transmission of the information, and their variations with respect to the coefficient of correlation

In Section

We need to define the Laplace transform of

In terms of TIDs, we just recall the basic formulas of

Among the several models proposed to deal with correlated variables, we consider the Arnold-Strauss model [

The marginal distribution of

In what follows, most of the calculations are done up to

It can be shown that (see, e.g., [

For a simple approximation, at a first order of

When only a few values of

Seeking the exact expressions of the PDFs of the number seems difficult to obtain. However, as just seen, approximations of closed expressions are simple. Thus, using (

The case

The calculations are now done up to

We can show that the normalized

The approximate expressions given in (

We have used the algorithm recently described [

In Figure

Plots of the Time Interval Distributions

By the way, it is interesting to remark that the PDF of

On the other hand, for high values of

Plots of the counting PNDs

The variations of the moments with respect to

The points and circles are the numerical results of the variance

As an application of these results to communications, we consider a system of communication processed with a direct threshold detector. The decision device operates such that_{0} (no correlation) and H_{1} (correlation with the parameter

Here, we focus on the method based on PNDs because it is generally more efficient. To simplify the calculations, the decision is not randomized [

Now, the probability of error in detection when processing with the

Plots of the probabilities of error

On the other hand, the bounds to

The second bound is obtained using the Poisson PNDs

In Figure

Plots of the cut-off rates

In conclusion, the binary performances, as summarized by the inequalities (

The exponential integral functions can be expressed, up to

Depending on the values of

To demonstrate that the

For

For higher values of

Laboratoire des Signaux et Systèmes is a joint laboratory (UMR 8506) of CNRS. and École Supérieure d'Électricité is and associated with the Université Paris-Orsay, France.