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 {πœ—π‘—},0≀𝑗<∞, such that πœ—π‘—+1β‰₯πœ—π‘—,forall𝑗. These {πœ—π‘—}𝑠, 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 πœ†(πœ—π‘—) 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 𝑐) correspond to a physical field, for example, a coherent or a thermal state, then it is reasonable to admit that intermediate values of 𝑐 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 πœƒπ‘—=πœ—π‘—+1βˆ’πœ—π‘—, called residual time (or lifetime sometimes). Its probability distribution function (PDF) 𝑀(πœƒ) will be called here a triggered PDF. When πœ—π‘—=𝑑0 is arbitrary, that is to say not a point of the (RPP), the 𝑣(πœƒ), PDF of the corresponding πœƒπ‘— 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 𝑃(𝑛,𝑑) of the random variable (RV) 𝑁(𝑑0,𝑑0+𝑑) which is the number of instants occurring within the time interval [𝑑0,𝑑0+𝑑],𝑑0andπ‘‘β‰ πœ—π‘—,forall𝑗. As previously we may as well define 𝑄(𝑛,𝑑) a triggered PND of the RV 𝑁(πœ—π‘—,πœ—π‘—+𝑑),π‘‘β‰ πœ—π‘—, 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 𝑐. 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 𝑐: 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 𝑓(𝑑)ξ€œπœŒ(𝑠)=ℒ𝑓(𝑑)β‰œβˆž0eβˆ’π‘ π‘‘π‘“(𝑑)d𝑑(1) and the functions𝜌1(𝑠)=𝜌(𝑠),πœŒπ‘ (1βˆ’πœŒ(𝑠))2𝜌(𝑠)=(𝑠)(1+𝜌(𝑠)).𝑠(1βˆ’πœŒ(𝑠))(2) The PDF of the number of events registered between 0 and 𝑑, denoted by PND, reads𝑃𝑑(𝑛;𝑑)=Probπ‘‘βˆˆπ‘‡=𝑛,𝑑𝑛+1ℐ𝑑=𝖀𝑛<𝑑<𝑑𝑛+1ξ€Έξ€»,(3) where ℐ is the indicator function ℐ(π‘‘βˆˆπ‘‡)=1(0), in case the event occurs or not within π‘‘βˆˆπ‘‡. Therefore𝑃(𝑛;𝑑)=β„’βˆ’1𝑃(𝑛;𝑠)=β„’βˆ’1ξ‚΅1βˆ’πœŒ(𝑠)π‘ πœŒ(𝑠)𝑛,(4) where the symbol β„’βˆ’1 denotes the inverse Laplace transform. From (4), it is easily seen that (see e.g., [4, 5])𝖀[𝑁](𝑑)=β„’βˆ’1𝜌1𝖀𝑁(𝑠),2ξ€»(𝑑)=β„’βˆ’1𝜌2(𝑠).(5) The symbol 𝖀 denotes the mathematical expectation. In the following 𝑃(𝑛;𝑑) will be called the relaxed PDF. Another interesting PDF that can be derived from (4),𝑄(𝑛;𝑑)=𝑛+1𝖀[𝑁]𝑃(𝑛+1;𝑑),(6) 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 densityξ€œπ’₯(𝑑)=𝑑0𝑑+𝑑0πœ†(πœ—)dπœ—,(7)π’₯(𝑑)=πœ†π‘‘ for the Poisson RPP. Therefore1𝑃(𝑛;𝑑)=𝖀π’₯𝑑𝑛!0ξ€Έ+𝑑𝑛eβˆ’π’₯(𝑑0+𝑑)ξ€»,1𝑄(𝑛;𝑑)=𝖀[𝑁]1𝖀π’₯𝑑𝑛!0ξ€Έ+𝑑𝑛+1eβˆ’π’₯(𝑑0+𝑑)ξ‚„.(8) For all RPPs that we deal with here, all the PDFs depend only on 𝑑 (not on 𝑑0) due to the stationary property. When 𝑑 is a parameter, 𝑃(𝑛;𝑑) and 𝑄(𝑛;𝑑) are simply denoted 𝑃(𝑛) and 𝑄(𝑛).

In terms of TIDs, we just recall the basic formulas of 𝑀(𝑑) and 𝑣(𝑑) (see [6])1𝑀(𝑑)=𝖀[π’₯]𝖀π’₯𝑑0ξ€Έπ’₯𝑑0ξ€Έe+π‘‘βˆ’π’₯(𝑑0+𝑑)ξ€»,ξ€Ίπ’₯𝑑𝑣(𝑑)=𝖀0ξ€Έe+π‘‘βˆ’π’₯(𝑑0+𝑑)ξ€».(9) Notice that 𝑔(0)β‰œπ‘€(𝑑=0)/𝑣(𝑑=0) the ratio of the values at 𝑑=0 is related to 𝜎2π’₯, the variance of π’₯, such that 𝑔(0)βˆ’1=𝜎2π’₯/𝖀[π’₯]2, where 𝜎2π’₯ 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]𝑓(𝑑,𝑒)=𝐾eβˆ’(π‘Žπ‘‘+𝑏𝑒+𝑐𝑑𝑒),(10) where π‘Ž,𝑏,𝑐>0. The time intervals 𝑑,𝑒 are of course positive RVs in [0,∞[. The constant of normalization is𝐾=𝑐eβˆ’π‘Žπ‘/π‘βŸΆβŽ§βŽͺ⎨βŽͺ⎩𝐾Ei(1,π‘Žπ‘/𝑐)ℓ𝑐=π‘Žπ‘+π‘βˆ’2πΎπ‘Žπ‘for𝑐β‰ͺ1,β„Ž=𝑐1βˆ’π›Ύβˆ’log(π‘Žπ‘)for𝑐≫1,(11) where Ei(π‘š,π‘₯) are the exponential integral functions [9] for π‘š=1,2,…, and π‘₯βˆˆβ„+. They verify the approximate expressions given in (A.1a)–(A.2a) in Appendix A.1. The constant 𝛾=0.5772157 is the Euler constant.

The marginal distribution of 𝑑 is deduced from (10) and is given byξ€œπ‘“(𝑑)=∞0e𝑓(𝑑,𝑒)d𝑒=πΎβˆ’π‘Žπ‘‘.𝑏+𝑐𝑑(12) In the following, we report some calculations that can be obtained under closed forms depending on the value of the parameter 𝑐. Let us first consider the case where 𝑐β‰ͺ1.

3.1. 𝑐β‰ͺ1

In what follows, most of the calculations are done up to 𝑂(𝑐3).

3.1.1. TID

It can be shown that (see, e.g., [6])𝑀e(𝑑)=πΎβˆ’π‘Žπ‘‘,ξ€œπ‘+𝑐𝑑𝑣(𝑑)β‰œπœ†βˆžπ‘‘π‘€(πœƒ)dπœƒ.(13) Given 𝖀[𝑑], the first moment of 𝑑, the average value of the density of the process πœ†=1/𝖀[𝑑] is given by π‘πœ†β‰ƒπ‘Ž+π‘π‘βˆ’22π‘Žπ‘2.(14) Equation (13) yields 𝜈(0)=πœ† and d𝜈/d𝑑=βˆ’πœ†π‘€(𝑑) which prove that 𝜈(𝑑) is a monotonic continuously decreasing function of 𝑑. More precisely, we can show that𝑀(𝑑)β‰ƒπ‘Ž3𝑏2βˆ’π‘π‘π‘‘+𝑐2𝑑2π‘Ž2𝑏2βˆ’π‘Žπ‘π‘+2𝑐2eβˆ’π‘Žπ‘‘,π‘Ž(15)𝜈(𝑑)β‰ƒπ‘Ž2𝑏2βˆ’π‘Žπ‘π‘+2𝑐2βˆ’π‘Žπ‘(π‘Žπ‘βˆ’2𝑐)𝑑+π‘Ž2𝑐2𝑑2π‘Ž2𝑏2βˆ’2π‘Žπ‘π‘+6𝑐2eβˆ’π‘Žπ‘‘.(16) From (15), we deduce𝜎2𝑑≃1π‘Ž2ξ‚΅1βˆ’2𝑐+π‘Žπ‘9𝑐2π‘Ž2𝑏2ξ‚Ά.(17) On the other hand, we can see thatβ„Žπ‘‘πΎ(0,𝑐β‰ͺ1)=𝑀(0)βˆ’π‘£(0)=β„“π‘π‘βˆ’πœ†β‰ƒ2π‘Žπ‘2,(18) which positivity is a characteristic of classical processes [10].

3.1.2. PND

For a simple approximation, at a first order of 𝑐, we may use the following:1𝑓(𝑑)βˆπ‘Žπ‘π‘Žeβˆ’π‘Žπ‘‘βˆ’π‘π‘Ž2𝑏2π‘Ž2𝑑eβˆ’π‘Žπ‘‘,𝑃(𝑛,𝑑)∝(π‘Žπ‘‘)𝑛e𝑛!βˆ’π‘Žπ‘‘βˆ’π‘π‘Ž2𝑏2Γ—ξ‚΅(π‘Žπ‘‘)2𝑛+1(2𝑛)!2ξ‚΅(π‘Žπ‘‘)2𝑛+1+(2𝑛+1)!(π‘Žπ‘‘)2π‘›βˆ’1e(2π‘›βˆ’1)!ξ‚Άξ‚Άβˆ’π‘Žπ‘‘,(19) where 𝑃(0,𝑑)=(1+π‘Žπ‘‘/2)eβˆ’π‘Žπ‘‘ (see [1] pages 789–792).

When only a few values of 𝑃(𝑛,𝑑) or only the moments are needed, it is however, better, to use (3) and (5). Based on their expansion as series in 𝑠, we obtain, at the second order of 𝑐,𝖀[𝑁]β‰ƒπ‘Ž2π‘Žπ‘Žβˆ’πœ”π‘‘+22(π‘Žβˆ’πœ”)2πœ”2𝑑2,𝖀𝑁2ξ€»β‰ƒπ‘Ž2π‘Žπ‘Žβˆ’πœ”π‘‘+2ξ€·2π‘Ž2+πœ”2ξ€Έ(π‘Žβˆ’πœ”)2πœ”2𝑑2,𝜎2𝑛1β‰ƒπ‘Ž+πœ”+2+1π‘Žξ‚πœ”βˆ’π‘Ž2,(𝑑=1),(20) where we denote πœ”=𝑐/𝑏.

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 𝑛=0,1,2 and 𝑑=1, as given in (A.3) in Appendix A.2. Therefore, we can prove thatβ„Žπ‘›(0,𝑐β‰ͺ1)=𝑃(0)βˆ’π‘ƒ(1)≃eβˆ’π‘Žξ‚€1βˆ’π‘Žβˆ’2βˆ’π‘Žπ‘ξ‚=𝑐,π‘Ž=1,𝑏=1βˆ’1𝑒2π‘βˆ’π‘3𝑒2,(21) which is negative forall𝑐β‰ͺ1. We will numerically illustrate this property by simulations. When we choose the triggered processing, we obtainπ‘˜π‘›1(0,𝑐β‰ͺ1)=𝑄(0)βˆ’π‘„(1)β‰ƒβˆ’π‘’1π‘βˆ’π‘4𝑒2,(22) which is here again negative forall𝑐.

The case 𝑐≫1 (but finite) is perhaps more interesting although closed forms of the moments are difficult to obtain.

3.2. 𝑐≫1

The calculations are now done up to 𝑂(1/𝑐3).

3.2.1. TID

We can show that the normalized 𝑀(𝑑) is given by𝑀(𝑑)βˆΌπ‘eβˆ’π‘Žπ‘/𝑐eEi(1,π‘Žπ‘/𝑐)βˆ’π‘Žπ‘‘βŸΆπ‘π‘‘+𝑏𝑑≠0eβˆ’π‘Žπ‘‘ξ‚€π‘π‘‘(πœ’βˆ’log𝑐)1βˆ’π‘ξ‚€1π‘Ž+𝑑,(23) and the unnormalized PDF, evaluated up to 𝑂(1/𝑐2) and 𝑂(𝑑),𝑣(𝑑)βˆΌπœ†Ei(1,π‘Ž(𝑑+𝑏/𝑐)),⟢Ei(1,π‘Ž+𝑏/𝑐)0<𝑑β‰ͺ1πœ†Ei(1,π‘Žπ‘‘)𝑏log(𝑐/π‘Žπ‘)βˆ’π›Ύβˆ’πœ†π‘1elog(π‘Žπ‘/𝑐)+π›Ύπ‘Žπ‘‘π‘‘βˆ’πœ†π‘Žπ‘π‘Ei(1,π‘Žπ‘‘)(log(π‘Žπ‘/𝑐)+𝛾)2,βˆΌβˆ’πœ†π›Ύ+log(π‘Žπ‘‘),log(π‘Žπ‘/𝑐)βˆ’π›Ύ(24) leading toβ„Žπ‘‘πΎ(0,𝑐≫1)=𝑀(0)βˆ’π‘£(0)=β„Žπ‘π‘βˆ’πœ†β‰ƒπ‘(1βˆ’πœ’)βˆ’π‘Žlog𝑐,(25) where we denote πœ’=𝛾+log(π‘Žπ‘) and assume that πœ†β‰ƒπ‘Ž(logπ‘βˆ’πœ’). It is seen that β„Žπ‘‘(0,𝑐≫1) given by (25) is now positive and increases more slowly than β„Žπ‘‘(0,𝑐β‰ͺ1)βˆΌπ‘2 given by (18). From (23), we can also deduce that𝖀[𝑑]∼1π‘Ž1,𝜎logπ‘βˆ’πœ’2π‘‘βˆΌ1π‘Ž21=𝖀[𝑑]logπ‘βˆ’πœ’π‘Ž.(26) Both moments tend to 0 when π‘β†’βˆž.

3.2.2. PND

The approximate expressions given in (A.4) in Appendix A.2  lead toβ„Žπ‘›(0,𝑐≫1)=𝑃(0)βˆ’π‘ƒ(1)≃1+π›Ύβˆ’log𝑐𝑐.(27) We can calculate the first two moments of the number for the specific case π‘Ž=1 and 𝑏=1, 𝖀[𝑁]𝜎∝log(𝑐),(28a)2𝑛[𝑁]βˆπ–€2.(28b)As for the previous case, it can be shown that π‘˜π‘›(𝑐)≀0,forall𝑐.

4. Simulation and Results

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

In Figure 1, results of the simulated data of the TIDs are plotted. The 𝑀(𝑑), the triggered PDF follows very well (23). The 𝑣(𝑑), the relaxed one has been well fitted by the expression𝑣(𝑑)=𝜈0πœ†Ei(1,𝑑)⟢Ei(1,1+1/𝑐)0<𝑑β‰ͺ1𝜈1πœ†log𝑑,log(1/𝑐)(29) for π‘Ž=1,𝑏=1,𝑐=30 and where 𝜈0=0.07 and 𝜈1=0.2.


Figure 1 
Plots of the Time Interval Distributions 𝑀 ( 𝑑 ) (curve 1) and 𝑣 ( 𝑑 ) (curve 2) versus 𝑑 . The parameters are chosen equal to π‘Ž = 1 , 𝑏 = 1 , where the value of 𝑐 = 3 0 is moderate. The points are the simulated results and the theoretical curves given by (23) and (29) are plotted in solid lines.

By the way, it is interesting to remark that the PDF of 𝑆ℓ=βˆ‘β„“π‘—=0πœƒπ‘—,ℓ≫1, 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 𝑐, the theoretical approximations of the PNDs are given by1𝑃(𝑛)=𝒩𝑏1eβˆ’π‘Ž1𝑛+𝑏2eβˆ’π‘Ž2π‘Žπ‘›2ξ‚Ά,𝑛!(30) where𝒩=𝑏2+𝑏11βˆ’eβˆ’π‘Ž1,𝖀[𝑁]=1(31a)π’©ξ‚΅π‘Ž2𝑏2+𝑏1eπ‘Ž1(eπ‘Ž1)βˆ’12ξ‚Ά,(31b) leading to𝑄(𝑛)=𝑛+1𝖀[𝑁]𝑃(𝑛+1).(32) The values of the parameters that correctly fit the simulated results of the PDFs, as seen in Figure 2, are 𝑏1=0.2,π‘Ž1=1,𝑏2=0.62,π‘Ž2=5.2,and𝖀[𝑁]=3.64.

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Figure 2 
Plots of the counting PNDs 𝑃 ( 𝑛 ; 𝑑 ) and 𝑄 ( 𝑛 ; 𝑑 ) versus 𝑛 for the parameters chosen equal to π‘Ž = 1 , 𝑏 = 1 , 𝑐 = 1 0 0 and 𝑑 = 1 . The points are the simulated results and the theoretical approximations given by (30)–(32) are plotted in solid lines. Only integer values of 𝑛 are meaning. For 𝑐 = 0 , the PDFs are identical 𝑃 ( 𝑛 ) = 𝑄 ( 𝑛 ) and are of Poisson type which is drawn in solid line (curve at the top).

The variations of the moments with respect to 𝑐 are derived from the expressions recalled in the Section 2. We obtained𝖀[𝑁]1≃1+𝛾+2𝜎log𝑐,(33a)2𝑛1≃1+𝛾+4(1+log𝑐)2.(33b)Regarding the reduced moments β„Žπ‘› and π‘˜π‘›, an excellent fit of the simulated results of Figure 3 is carried out with 𝛼1=4,𝛽1=0.17 and 𝛼2=3,𝛽1=0.24β„Žπ‘›β‰ƒ1βˆ’π›Ύ2+12ξ€·π›Όπ›Ύβˆ’log1𝑐𝛼1π‘βˆ’π›½1,π‘˜(34a)𝑛≃1βˆ’π›Ύ2+12ξ€·π›Όπ›Ύβˆ’log2𝑐𝛼2π‘βˆ’π›½2,(34b)leading to β„Žπ‘›β‰ƒπ‘2/2log𝑐 and π‘˜π‘›β‰ƒβˆ’π‘4/24(log𝑐)3 which are only in qualitative agreement.

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Figure 3 
The points and circles are the numerical results of the variance 𝜎 2 𝑛 and the average number ⟨ 𝑛 ⟩ versus 𝑐 for the values of the parameters chosen equal to π‘Ž = 1 , 𝑏 = 1 , 𝑑 = 1 . The theoretical approximations calculated for 𝑐 ≫ 1 are given by (33a) and (33b) respectively (a) and drawn in solid lines. The points and circles are the numerical results of the differences β„Ž 𝑛 = 𝑃 ( 0 ) βˆ’ 𝑃 ( 1 ) and π‘˜ 𝑛 = 𝑄 ( 0 ) βˆ’ 𝑄 ( 1 ) versus 𝑐 for π‘Ž = 1 , 𝑏 = 1 , 𝑑 = 1 . The theoretical approximations calculated for 𝑐 ≫ 1 are given by (34a) and (34b) respectively (b) and drawn in solid lines.

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 that𝑛≦𝑛𝑠(𝑐)⟢H0βŸΆπ‘ƒπ‘=0(𝑛)=eβˆ’π‘Žπ‘Žπ‘›,𝑛!(35a)𝑛>𝑛𝑠(𝑐)⟢H1βŸΆπ‘ƒπ‘β‰ 0(𝑛),(35b)based on the binary hypotheses: H0 (no correlation) and H1 (correlation with the parameter 𝑐). The threshold is obtained from the likelihood ratio Ξ›(𝑛),𝑃Λ(𝑛)=𝑐≠0(𝑛)𝑃𝑐=0(𝑛)>1βŸΆπ‘›π‘ (𝑐).(36) We can also utilize the TIDs as a useful tool of processing [12]. The threshold would be based on the likelihood ratio Θ(𝑑)=𝑀𝑐(𝑑)/πœ†eβˆ’πœ†π‘‘>1→𝑑𝑠(𝑐) and the decision would operate as follows𝑑≀𝑑𝑠(𝑐)⟢H1βŸΆπ‘€π‘e(𝑑)=πΎβˆ’π‘Žπ‘‘,𝑏+𝑐𝑑𝑑β‰₯𝑑𝑠(𝑐)⟢H0βŸΆπ‘Žeβˆ’π‘Žπ‘‘,(37) where 𝑑𝑠(𝑐)=(1/c)(𝐾/aβˆ’π‘), 𝐾 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 by𝑃err=121βˆ’π‘›π‘ ξ“π‘›=0𝑃𝑐=0(𝑛)+𝑛s𝑛=0𝑃𝑐≠0ξƒͺ.(𝑛)(38) Similarly, we have for the processing with the triggered PND𝑄err=121βˆ’π‘›π‘ ξ“π‘›=0𝑃𝑐=0(𝑛)+𝑛𝑠𝑛=0𝑄𝑐≠0ξƒͺ.(𝑛)(39) It is first seen that 𝑄err≀𝑃err, the inequality which is demonstrated for a special case in the Appendix B. Furthermore, both probabilities of error in detection decrease with 𝖀[𝑁], then with 𝑐 because 𝖀[𝑁]∝log𝑐 as it is calculated in (28a), (33a) and is shown in Figure 3. This is within the range of π‘βˆΆ0≀𝑐≲100. For instance, with the help of (30)–(31b) and (38)-(39) calculated for 𝑛𝑠=1 and 𝑐=100, we obtained 𝑃err≃0.28 and 𝑄err≃0.16, the values which are in excellent agreement with the simulated results of Figure 4.


Figure 4 
Plots of the probabilities of error 𝑃 e r r and 𝑄 e r r versus the average number 𝖀 [ 𝑁 ] , respectively for the relaxed and the triggered processing. The values of the parameters are chosen equal to π‘Ž = 1 , 𝑏 = 1 , 𝑑 = 1 . The average number 𝖀 [ 𝑁 ] ( 𝑐 ) is the average number of the PDF 𝑃 𝑐 β‰  0 ( 𝑛 ) for 0 ≀ 𝑐 ≀ 1 0 0 , with 𝖀 [ 𝑁 ] ( 0 ) = π‘Ž . The curves quoted β€œ1” and β€œ2” have been obtained from (41) and (42) respectively.

On the other hand, the bounds to 𝑄err and 𝑃err can easily be calculated. In fact, denoting𝖀[𝑁]𝜈=[𝑁]1+𝖀,(40) where 𝖀[𝑁]=π‘Ž+πœ‰. Now, taking into account (35a), we have up to 𝑂(πœ‰2),𝑃1(𝑛)=(1βˆ’πœˆ)πœˆπ‘›,𝑃(1)err1≃1βˆ’π‘’βˆ’πœˆ221≃1βˆ’π‘’βˆ’πœ‰8,(41) for 0β‰€πœˆβ‰ 1. Therefore 𝑃(1)err is the approximation of the exact 𝑃err for 𝑛𝑠=1 plotted as the curve quoted β€œ1” in Figure 4 where π‘Ž=1. Similarly, taking into account (35a), we have𝑃1(𝑛)=eβˆ’π–€[𝑁]𝖀[𝑁]𝑛,𝑃𝑛!(2)err≃12βˆ’1𝑒+[𝑁]1+𝖀2eβˆ’π–€[𝑁]≃12βˆ’1π‘’πœ‰.(42) Again, 𝑃(2)err is the approximation of the exact 𝑃err for 𝑛𝑠=1 plotted as the the curve quoted β€œ2” in Figure 4 for π‘Ž=1. Finally, we have𝑃(2)err≀𝑄err≀𝑃err≀𝑃(1)err.(43) 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 β€œ0” and β€œ1” is done via the probabilities 𝑃𝑐=0 and 𝑃𝑐≠0 respectively (the probabilities a priori are taken equal to 1/2) the cut-off rate expresses as2𝑅=logβˆ‘1+βˆžπ‘›=0βˆšπ‘ƒπ‘=0(𝑛)𝑃𝑐≠0,(𝑛)(44) which is generally interpreted as a lower bound to the channel capacity [14]. The 𝑃𝑐=0(𝑛) is the Poisson PND of parameter π‘Ž=1 and 𝑃𝑐≠0(𝑛) will be, as above, either the relaxed or the triggered PND yielding the cut-off rates 𝑅𝑝 and π‘…π‘ž, 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 PNDs𝑃0ξ€·(𝑛)=1βˆ’πœˆ0ξ€Έπœˆπ‘›0𝑅,(45a)12=log1+ξ€·1βˆ’πœˆ0ξ€Έξ‚€βˆš(1βˆ’πœˆ)/1βˆ’πœˆ0πœˆξ‚β‰ƒ0β‰€πœ‰β‰²2πœ‰2ξ‚΅321βˆ’3πœ‰4+πœ‰22ξ‚Ά,(45b)where 𝜈 is given by (40) and 𝜈0=π‘Ž/(1+π‘Ž). This is a lower bound to 𝑅𝑝.

The second bound is obtained using the Poisson PNDs𝑅22=log1+eβˆšβˆ’1/2(βˆšπ–€[𝑁]βˆ’π‘Ž)2≃0β‰€πœ‰β‰²2πœ‰2ξ‚΅πœ‰161βˆ’2+9πœ‰2ξ‚Ά,32(46) which is an upperbound to 𝑅𝑝.

In Figure 5, the results of the simulation 𝑅𝑝 and π‘…π‘ž and the bounds (curves quoted β€œ1” and β€œ2”) given by (45b) and (46) are plotted versus 𝖀[𝑁]. Here again, it is seen that the processing with the triggered PND performs much better that the processing with the relaxed PND𝑅1≀𝑅𝑝≀𝑅2β‰€π‘…π‘ž,(47) in the limit of 0≀𝑐≀100. However, because the cut-off rate is a monotonic increasing function with respect to 𝖀[𝑁], the inequalities (47) may be extrapolated to for all 𝑐.


Figure 5 
Plots of the cut-off rates 𝑅 𝑝 and 𝑅 π‘ž versus the average number 𝖀 [ 𝑁 ] , respectively for the processing with the relaxed and the triggered PNDs. The values of the parameters are chosen equal to π‘Ž = 1 , 𝑏 = 1 , 𝑑 = 1 . The average number 𝖀 [ 𝑁 ] ( 𝑐 ) is the average number of the PDF 𝑃 𝑐 β‰  0 ( 𝑛 ) for 0 ≀ 𝑐 ≀ 1 0 0 , 𝖀 [ 𝑁 ] ( 0 ) = π‘Ž . The curves quoted β€œ1” and β€œ2” have been obtained from (45b) and (46) respectively.

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 𝑂(π‘₯3), π‘₯Ei(1,π‘₯)β‰ƒβˆ’π›Ύβˆ’logπ‘₯+π‘₯βˆ’24,π‘₯(A.1a)Ei(2,π‘₯)≃1+(𝛾+logπ‘₯βˆ’1)π‘₯βˆ’22,1(A.1b)Ei(3,π‘₯)≃2ξ‚€1βˆ’π‘₯βˆ’21𝛾+23logπ‘₯βˆ’4π‘₯2,(A.1c)where 𝛾 is the Euler constant. For π‘₯β†’βˆžβ€‰eEi(1,π‘₯)β‰ƒβˆ’π‘₯π‘₯ξ‚€11βˆ’π‘₯,e(A.2a)Ei(2,π‘₯)β‰ƒβˆ’π‘₯π‘₯ξ‚€21βˆ’π‘₯e(A.2b)Ei(3,π‘₯)β‰ƒβˆ’π‘₯π‘₯ξ‚€31βˆ’π‘₯.(A.2c)

A.2. Probability of Number Distribution

Depending on the values of 𝑐, we have the following approximations of the first values of 𝑃(𝑛):𝑃(0)≃eβˆ’π‘Žξ‚€11βˆ’πœ”βˆ’π‘Žπœ”2,𝑃(1)≃eβˆ’π‘Žξ‚€ξ‚€1π‘Ž+(1βˆ’π‘Ž)πœ”+π‘Ž+π‘Ž6βˆ’32ξ‚πœ”2,𝑃(2)≃eβˆ’π‘Žξ‚΅π‘Ž22ξ‚€π‘Ž+π‘Ž1βˆ’2ξ‚ξ‚΅π‘Žπœ”+28+32ξ‚Άπœ”βˆ’2π‘Ž2ξ‚Ά,(A.3) for 𝑐β‰ͺ1 and where πœ”=𝑐/𝑏. Similarly, it can be shown that𝑃(0)≃1+π›Ύβˆ’log𝑐𝑐,𝑃(1)≃logπ‘βˆ’π›Ύπ‘,𝛾𝑃(2)≃2βˆ’2(1+𝛾)log𝑐𝑐2,(A.4) for 𝑐≫1.

B. Inequality between Probabilities of Error in Binary Detection

To demonstrate that the triggered processing yields better performance than the relaxed processing, Δ𝑒=𝑄errβˆ’π‘ƒerr≀0, we begin with (8)1𝑃(𝑛;𝑑)=𝖀𝑛!π’₯(𝑑)𝑛eβˆ’π’₯(𝑑)ξ€»,1𝑄(𝑛;𝑑)=𝖀[𝑁]1𝖀𝑛!π’₯(𝑑)𝑛+1eβˆ’π’₯(𝑑)ξ€»,(B.1) and show that for𝑃err=121βˆ’π‘›π‘ ξ“π‘›=0𝑃𝑐=0(𝑛)+𝑛𝑠𝑛=0𝑃𝑐≠0ξƒͺ,𝑄(𝑛)err=121βˆ’π‘›π‘ ξ“π‘›=0𝑃𝑐=0(𝑛)+𝑛𝑠𝑛=0𝑄𝑐≠0ξƒͺ,(𝑛)(B.2) we haveΔ𝑒=𝑄errβˆ’π‘ƒerr,=12𝑛𝑠𝑛=0𝖀eβˆ’π’₯ξ‚΅π’₯𝑛+1𝖀[π’₯]βˆ’π’₯𝑛ξƒͺ=1ξ‚Άξ‚Ή2𝖀𝑛𝑠𝑛=0eβˆ’π’₯ξ‚΅π’₯𝑛+1𝖀[π’₯]βˆ’π’₯𝑛≀12𝖀𝑛𝑠𝑛=0ξ‚΅π’₯𝑛+1𝖀[π’₯]βˆ’π’₯𝑛=12𝑛𝑠𝑛=0𝖀π’₯𝑛+1𝖀[π’₯]ξ€Ίπ’₯βˆ’π–€π‘›ξ€»ξƒͺ.(B.3) where we used eβˆ’π’₯≀1,π’₯β‰₯0.

For 𝑛𝑠=1, noticing that 𝖀[π’₯2]β‰₯𝖀[π’₯]2, we have Δ𝑒≀0.

For higher values of 𝑛𝑠, this method seems not useful because it requires to prove that 𝖀[π’₯𝑛+1]/𝖀[π’₯]β‰₯𝖀[π’₯𝑛],forall𝑛, 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.