#### Abstract

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.

#### 1. Introduction

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 , such that . These , which are the time instants of interaction between the photons and the detector device, for example, a photomultiplier, constitute a random point process (RPP) for which a positive instantaneous density can be defined. We here consider only simple and homogeneous stationary processes.

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 [1] does not necessarily correspond to a nonclassical physical optical field, although its feasibility can be demonstrated [2]. This problem, mainly due to the quantum nature of the interaction between radiation and matter is not examined here.

Nevertheless, using a parameterized RPP, whose special values of such a parameter (denoted here ) correspond to a physical field, for example, a coherent or a thermal state, then it is reasonable to admit that intermediate values of corresponds to realistic optical fields. Despite these limitations, the properties of the RPPs studied here are important in statistical optics.

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) [3]. If we choose to characterize the RPP by the TID, we can define the time interval between points , called *residual time* (or *lifetime* sometimes). Its probability distribution function (PDF) will be called here a *triggered* PDF. When is arbitrary, that is to say not a point of the (RPP), the , PDF of the corresponding will be called a *relaxed* PDF.

If we now choose to characterize the RPP by the PND, we can as above, define the *relaxed* PND of the random variable (RV) which is the number of instants occurring within the time interval . As previously we may as well define a *triggered* PND of the RV , the counting process being started by a point of the RPP.

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 . The one-dimensional processes, such as the Poisson and the geometric PNDs, are finally utilized for establishing bounds to the performances.

In Section 2, the notations are defined and the basic equations are briefly recalled. Section 3 establishes the main results of the PDFs both those dealing with TIDs and PNDs depending on extreme values of : low values and high values. Finally, Section 4 presents theoretical results and curves that explain and illustrate the results of the numerical simulation.

#### 2. Basic Equations

We need to define the Laplace transform of
and the functions
The PDF of the number of events registered between 0 and , denoted by PND, reads
where is the indicator function , in case the event occurs or not within . Therefore
where the symbol denotes the inverse Laplace transform. From (4), it is easily seen that (see e.g., [4, 5])
The symbol denotes the mathematical expectation. In the following will be called the *relaxed* PDF. Another interesting PDF that can be derived from (4),
will be called the *triggered* PDF. In fact, it is known [3, 6] that these PDFs can be expressed using the moments of the time-integrated density for the Poisson RPP. Therefore
For all RPPs that we deal with here, all the PDFs depend only on (not on ) due to the stationary property. When is a parameter, and are simply denoted and .

In terms of TIDs, we just recall the basic formulas of and (see [6]) Notice that the ratio of the values at is related to , the variance of , such that , where is the variance of the time-integrated density.

#### 3. Model for Correlated Variables

Among the several models proposed to deal with correlated variables, we consider the Arnold-Strauss model [7, 8] where . The time intervals are of course positive RVs in . The constant of normalization is where are the exponential integral functions [9] for , and . They verify the approximate expressions given in (A.1a)–(A.2a) in Appendix A.1. The constant is the Euler constant.

The marginal distribution of is deduced from (10) and is given by In the following, we report some calculations that can be obtained under closed forms depending on the value of the parameter . Let us first consider the case where .

##### 3.1.

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

###### 3.1.1. TID

It can be shown that (see, e.g., [6]) Given , the first moment of , the average value of the density of the process is given by Equation (13) yields and which prove that is a monotonic continuously decreasing function of . More precisely, we can show that From (15), we deduce On the other hand, we can see that which positivity is a characteristic of classical processes [10].

###### 3.1.2. PND

For a simple approximation, at a first order of , we may use the following: where (see [1] pages 789–792).

When only a few values of or only the moments are needed, it is however, better, to use (3) and (5). Based on their expansion as series in , we obtain, at the second order of , where we denote .

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 (4), it is easy to calculate approximate expressions of the PDFs of the number for and , as given in (A.3) in Appendix A.2. Therefore, we can prove that
which is negative . We will numerically illustrate this property by simulations. When we choose the *triggered* processing, we obtain
which is here again negative .

The case (but finite) is perhaps more interesting although closed forms of the moments are difficult to obtain.

##### 3.2.

The calculations are now done up to .

###### 3.2.1. TID

We can show that the normalized is given by and the unnormalized PDF, evaluated up to () and (), leading to where we denote and assume that . It is seen that given by (25) is now positive and increases more slowly than given by (18). From (23), we can also deduce that Both moments tend to 0 when .

###### 3.2.2. PND

The approximate expressions given in (A.4) in Appendix A.2 lead to We can calculate the first two moments of the number for the specific case and , As for the previous case, it can be shown that .

#### 4. Simulation and Results

We have used the algorithm recently described [11] for several values of .

In Figure 1, results of the simulated data of the TIDs are plotted. The , the *triggered* PDF follows very well (23). The , the *relaxed* one has been well fitted by the expression
for and where and .

By the way, it is interesting to remark that the PDF of , which is the addition of several correlated identically distributed positive random time interval, deviates from the Gaussian profile. We may conclude that the application of the central limit theorem requires the addition of a very high number of correlated random variables.

On the other hand, for high values of , the theoretical approximations of the PNDs are given by where leading to The values of the parameters that correctly fit the simulated results of the PDFs, as seen in Figure 2, are .

**(a)**

**(b)**

The variations of the moments with respect to are derived from the expressions recalled in the Section 2. We obtainedRegarding the reduced moments and , an excellent fit of the simulated results of Figure 3 is carried out with and leading to and which are only in qualitative agreement.

**(a)**

**(b)**

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 thatbased on the binary hypotheses: H_{0} (no correlation) and H_{1} (correlation with the parameter ). The threshold is obtained from the likelihood ratio ,
We can also utilize the TIDs as a useful tool of processing [12]. The threshold would be based on the likelihood ratio and the decision would operate as follows
where , being given by (11).

Here, we focus on the method based on PNDs because it is generally more efficient. To simplify the calculations, the decision is not randomized [13].

Now, the probability of error in detection when processing with the *relaxed* PND is given by
Similarly, we have for the processing with the *triggered* PND
It is first seen that , the inequality which is demonstrated for a special case in the Appendix B. Furthermore, both probabilities of error in detection decrease with , then with because as it is calculated in (28a), (33a) and is shown in Figure 3. This is within the range of . For instance, with the help of (30)–(31b) and (38)-(39) calculated for and , we obtained and , the values which are in excellent agreement with the simulated results of Figure 4.

On the other hand, the bounds to and can easily be calculated. In fact, denoting
where . Now, taking into account (35a), we have up to ,
for . Therefore is the approximation of the exact for plotted as the curve quoted “1” in Figure 4 where . Similarly, taking into account (35a), we have
Again, is the approximation of the exact for plotted as the the curve quoted “2” in Figure 4 for . Finally, we have
Let us conclude this analysis with a brief comment on information. We will concentrate on the cut-off rate which is known as a useful criterion for evaluating the performances of a channel. Thus, for a binary noiseless channel, when the transmission of messages “0” and “1” is done via the probabilities and respectively (the probabilities *a priori* are taken equal to ) the cut-off rate expresses as
which is generally interpreted as a lower bound to the channel capacity [14]. The is the Poisson PND of parameter and will be, as above, either the *relaxed* or the *triggered* PND yielding the cut-off rates and , respectively. Because exact closed expressions seem difficult to attain, the bounds are very useful and can easily be established. The first bound is obtained using geometric PNDswhere is given by (40) and . This is a lower bound to .

The second bound is obtained using the Poisson PNDs which is an upperbound to .

In Figure 5, the results of the simulation and and the bounds (curves quoted “1” and “2”) given by (45b) and (46) are plotted versus . Here again, it is seen that the processing with the *triggered* PND performs much better that the processing with the *relaxed* PND
in the limit of . However, because the cut-off rate is a monotonic increasing function with respect to , the inequalities (47) may be extrapolated to for all .

In conclusion, the binary performances, as summarized by the inequalities (43) and (47), show that the processing with the *triggered* PND is the preferable mode of operation. Both performances, in detection and information transmission, are improved with the coefficient of correlation.

#### Appendices

#### A. Approximate Expressions

##### A.1. The Exponential Integral Functions

The exponential integral functions can be expressed, up to , where is the Euler constant. For

##### A.2. Probability of Number Distribution

Depending on the values of , we have the following approximations of the first values of : for and where . Similarly, it can be shown that for .

#### B. Inequality between Probabilities of Error in Binary Detection

To demonstrate that the *triggered* processing yields better performance than the *relaxed* processing, , we begin with (8)
and show that for
we have
where we used .

For , noticing that , we have .

For higher values of , this method seems not useful because it requires to prove that , which is not so easy although the inequality is true for several types of density distributions of interest in statistical optics.

#### Disclosure

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.