# Statistical Properties and Applications of Correlated Length Intervals of Some Stationary Random Point Processes

Authors: Cherif Bendjaballah
Journal: ISRN Optics
 (2012)

---

## 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.


---

## Body

## 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{ϑj},0≤j<∞, such that ϑj+1≥ϑj,forallj. These {ϑj}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 λ(ϑj) 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 herec) 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 θj=ϑj+1-ϑj, called residual time (or lifetime sometimes). Its probability distribution function (PDF) w(θ) will be called here a triggered PDF. When ϑj=t0 is arbitrary, that is to say not a point of the (RPP), the v(θ), PDF of the corresponding θj will be called a relaxed PDF.If we now choose to characterize the RPP by the PND, we can as above, define therelaxed PND P(n,t) of the random variable (RV) N(t0,t0+t) which is the number of instants occurring within the time interval [t0,t0+t],t0andt≠ϑj,forallj. As previously we may as well define Q(n,t) a triggered PND of the RV N(ϑj,ϑj+t),t≠ϑj, 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 correlationc. The one-dimensional processes, such as the Poisson and the geometric PNDs, are finally utilized for establishing bounds to the performances.In Section2, 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 off(t)(1)ρ(s)=Lf(t)≜∫0∞e-stf(t)dt
and the functions(2)ρ1(s)=ρ(s)s(1-ρ(s)),ρ2(s)=ρ(s)(1+ρ(s))s(1-ρ(s)).
The PDF of the number of events registered between 0 and t, denoted by PND, reads(3)P(n;t)=Prob(t∈T=[tn,tn+1])=E[I(tn<t<tn+1)],
where ℐ is the indicator function ℐ(t∈T)=1(0), in case the event occurs or not within t∈T. Therefore(4)P(n;t)=L-1P(n;s)=L-1(1-ρ(s)sρ(s)n),
where the symbol ℒ-1 denotes the inverse Laplace transform. From (4), it is easily seen that (see e.g., [4, 5])(5)E[N](t)=L-1ρ1(s),E[N2](t)=L-1ρ2(s).
The symbol E denotes the mathematical expectation. In the following P(n;t) will be called the relaxed PDF. Another interesting PDF that can be derived from (4),(6)Q(n;t)=n+1E[N]P(n+1;t),
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(7)J(t)=∫t0t0+tλ(ϑ)dϑ,𝒥(t)=λt for the Poisson RPP. Therefore(8)P(n;t)=1n!E[J(t0+t)ne-J(t0+t)],Q(n;t)=1E[N]1n!E[J(t0+t)n+1e-J(t0+t)].
For all RPPs that we deal with here, all the PDFs depend only on t (not on t0) due to the stationary property. When t is a parameter, P(n;t) and Q(n;t) are simply denoted P(n) and Q(n).In terms of TIDs, we just recall the basic formulas ofw(t) and v(t) (see [6])(9)w(t)=1E[J]E[J(t0)J(t0+t)e-J(t0+t)],v(t)=E[J(t0+t)e-J(t0+t)].
Notice that g(0)≜w(t=0)/v(t=0) the ratio of the values at t=0 is related to σ𝒥2, the variance of 𝒥, such that g(0)-1=σ𝒥2/E[𝒥]2, where σ𝒥2 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](10)f(t,u)=Ke-(at+bu+ctu),
where a,b,c>0. The time intervals t,u are of course positive RVs in [0,∞[. The constant of normalization is(11)K=ce-ab/cEi(1,ab/c)⟶{Kl=ab+c-c2abforc≪1,Kh=c1-γ-log⁡(ab)forc≫1,
where Ei(m,x) are the exponential integral functions [9] for m=1,2,…, and x∈ℝ+. 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 oft is deduced from (10) and is given by(12)f(t)=∫0∞f(t,u)du=Ke-atb+ct.
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 c≪1.


### 3.1.c≪1

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


#### 3.1.1. TID

It can be shown that (see, e.g., [6])(13)w(t)=Ke-atb+ct,v(t)≜λ∫t∞w(θ)dθ.
Given E[t], the first moment of t, the average value of the density of the process λ=1/E[t] is given by
(14)λ≃a+cb-2c2ab2.
Equation (13) yields ν(0)=λ and dν/dt=-λw(t) which prove that ν(t) is a monotonic continuously decreasing function of t. More precisely, we can show that(15)w(t)≃a3b2-bct+c2t2a2b2-abc+2c2e-at,(16)ν(t)≃aa2b2-abc+2c2-ac(ab-2c)t+a2c2t2a2b2-2abc+6c2e-at.
From (15), we deduce(17)σt2≃1a2(1-2cab+9c2a2b2).
On the other hand, we can see that(18)ht(0,c≪1)=w(0)-v(0)=Klb-λ≃c2ab2,
which positivity is a characteristic of classical processes [10].


#### 3.1.2. PND

For a simple approximation, at a first order ofc, we may use the following:(19)f(t)∝1abae-at-ca2b2a2te-at,P(n,t)∝(at)nn!e-at-ca2b2×((at)2n(2n)!+12((at)2n+1(2n+1)!+(at)2n-1(2n-1)!))e-at,
where P(0,t)=(1+at/2)e-at (see [1] pages 789–792).When only a few values ofP(n,t) 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,(20)E[N]≃a2a-ωt+a22(a-ω)2ω2t2,E[N2]≃a2a-ωt+a2(2a2+ω2)(a-ω)2ω2t2,σn2≃a+ω+(12+1a-a)ω2,(t=1),
where we denote ω=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 t=1, as given in (A.3) in Appendix A.2. Therefore, we can prove that(21)hn(0,c≪1)=P(0)-P(1)≃e-a(1-a-2-ab)c,a=1,b=1=-1ec-23ec2,
which is negative forallc≪1. We will numerically illustrate this property by simulations. When we choose the triggered processing, we obtain(22)kn(0,c≪1)=Q(0)-Q(1)≃-1ec-14ec2,
which is here again negative forallc.The casec≫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 toO(1/c3).


#### 3.2.1. TID

We can show that the normalizedw(t) is given by(23)w(t)~ce-ab/cEi(1,ab/c)e-atct+bt≠0⟶e-att(χ-log⁡c)(1-bc(a+1t)),
and the unnormalized PDF, evaluated up to O(1/c2) and O(t),(24)v(t)~λEi(1,a(t+b/c))Ei(1,a+b/c),0<t≪1⟶λEi(1,at)log⁡(c/ab)-γ-λbc1log⁡(ab/c)+γeatt-λabcEi(1,at)(log⁡(ab/c)+γ)2,~-λγ+log⁡(at)log⁡(ab/c)-γ,
leading to(25)ht(0,c≫1)=w(0)-v(0)=Khb-λ≃cb(1-χ)-alog⁡c,
where we denote χ=γ+log(ab) and assume that λ≃a(logc-χ). It is seen that ht(0,c≫1) given by (25) is now positive and increases more slowly than ht(0,c≪1)~c2 given by (18). From (23), we can also deduce that(26)E[t]~1a1log⁡c-χ,σt2~1a21log⁡c-χ=E[t]a.
Both moments tend to 0 when c→∞.


#### 3.2.2. PND

The approximate expressions given in (A.4) in Appendix A.2  lead to(27)hn(0,c≫1)=P(0)-P(1)≃1+γ-log⁡cc.
We can calculate the first two moments of the number for the specific case a=1 and b=1, (28a)E[N]∝log⁡(c),(28b)σn2∝E[N]2.As for the previous case, it can be shown that kn(c)≤0,forallc.


## 3.1.c≪1

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


### 3.1.1. TID

It can be shown that (see, e.g., [6])(13)w(t)=Ke-atb+ct,v(t)≜λ∫t∞w(θ)dθ.
Given E[t], the first moment of t, the average value of the density of the process λ=1/E[t] is given by
(14)λ≃a+cb-2c2ab2.
Equation (13) yields ν(0)=λ and dν/dt=-λw(t) which prove that ν(t) is a monotonic continuously decreasing function of t. More precisely, we can show that(15)w(t)≃a3b2-bct+c2t2a2b2-abc+2c2e-at,(16)ν(t)≃aa2b2-abc+2c2-ac(ab-2c)t+a2c2t2a2b2-2abc+6c2e-at.
From (15), we deduce(17)σt2≃1a2(1-2cab+9c2a2b2).
On the other hand, we can see that(18)ht(0,c≪1)=w(0)-v(0)=Klb-λ≃c2ab2,
which positivity is a characteristic of classical processes [10].


### 3.1.2. PND

For a simple approximation, at a first order ofc, we may use the following:(19)f(t)∝1abae-at-ca2b2a2te-at,P(n,t)∝(at)nn!e-at-ca2b2×((at)2n(2n)!+12((at)2n+1(2n+1)!+(at)2n-1(2n-1)!))e-at,
where P(0,t)=(1+at/2)e-at (see [1] pages 789–792).When only a few values ofP(n,t) 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,(20)E[N]≃a2a-ωt+a22(a-ω)2ω2t2,E[N2]≃a2a-ωt+a2(2a2+ω2)(a-ω)2ω2t2,σn2≃a+ω+(12+1a-a)ω2,(t=1),
where we denote ω=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 t=1, as given in (A.3) in Appendix A.2. Therefore, we can prove that(21)hn(0,c≪1)=P(0)-P(1)≃e-a(1-a-2-ab)c,a=1,b=1=-1ec-23ec2,
which is negative forallc≪1. We will numerically illustrate this property by simulations. When we choose the triggered processing, we obtain(22)kn(0,c≪1)=Q(0)-Q(1)≃-1ec-14ec2,
which is here again negative forallc.The casec≫1 (but finite) is perhaps more interesting although closed forms of the moments are difficult to obtain.


## 3.1.1. TID

It can be shown that (see, e.g., [6])(13)w(t)=Ke-atb+ct,v(t)≜λ∫t∞w(θ)dθ.
Given E[t], the first moment of t, the average value of the density of the process λ=1/E[t] is given by
(14)λ≃a+cb-2c2ab2.
Equation (13) yields ν(0)=λ and dν/dt=-λw(t) which prove that ν(t) is a monotonic continuously decreasing function of t. More precisely, we can show that(15)w(t)≃a3b2-bct+c2t2a2b2-abc+2c2e-at,(16)ν(t)≃aa2b2-abc+2c2-ac(ab-2c)t+a2c2t2a2b2-2abc+6c2e-at.
From (15), we deduce(17)σt2≃1a2(1-2cab+9c2a2b2).
On the other hand, we can see that(18)ht(0,c≪1)=w(0)-v(0)=Klb-λ≃c2ab2,
which positivity is a characteristic of classical processes [10].


## 3.1.2. PND

For a simple approximation, at a first order ofc, we may use the following:(19)f(t)∝1abae-at-ca2b2a2te-at,P(n,t)∝(at)nn!e-at-ca2b2×((at)2n(2n)!+12((at)2n+1(2n+1)!+(at)2n-1(2n-1)!))e-at,
where P(0,t)=(1+at/2)e-at (see [1] pages 789–792).When only a few values ofP(n,t) 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,(20)E[N]≃a2a-ωt+a22(a-ω)2ω2t2,E[N2]≃a2a-ωt+a2(2a2+ω2)(a-ω)2ω2t2,σn2≃a+ω+(12+1a-a)ω2,(t=1),
where we denote ω=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 t=1, as given in (A.3) in Appendix A.2. Therefore, we can prove that(21)hn(0,c≪1)=P(0)-P(1)≃e-a(1-a-2-ab)c,a=1,b=1=-1ec-23ec2,
which is negative forallc≪1. We will numerically illustrate this property by simulations. When we choose the triggered processing, we obtain(22)kn(0,c≪1)=Q(0)-Q(1)≃-1ec-14ec2,
which is here again negative forallc.The casec≫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 toO(1/c3).


### 3.2.1. TID

We can show that the normalizedw(t) is given by(23)w(t)~ce-ab/cEi(1,ab/c)e-atct+bt≠0⟶e-att(χ-log⁡c)(1-bc(a+1t)),
and the unnormalized PDF, evaluated up to O(1/c2) and O(t),(24)v(t)~λEi(1,a(t+b/c))Ei(1,a+b/c),0<t≪1⟶λEi(1,at)log⁡(c/ab)-γ-λbc1log⁡(ab/c)+γeatt-λabcEi(1,at)(log⁡(ab/c)+γ)2,~-λγ+log⁡(at)log⁡(ab/c)-γ,
leading to(25)ht(0,c≫1)=w(0)-v(0)=Khb-λ≃cb(1-χ)-alog⁡c,
where we denote χ=γ+log(ab) and assume that λ≃a(logc-χ). It is seen that ht(0,c≫1) given by (25) is now positive and increases more slowly than ht(0,c≪1)~c2 given by (18). From (23), we can also deduce that(26)E[t]~1a1log⁡c-χ,σt2~1a21log⁡c-χ=E[t]a.
Both moments tend to 0 when c→∞.


### 3.2.2. PND

The approximate expressions given in (A.4) in Appendix A.2  lead to(27)hn(0,c≫1)=P(0)-P(1)≃1+γ-log⁡cc.
We can calculate the first two moments of the number for the specific case a=1 and b=1, (28a)E[N]∝log⁡(c),(28b)σn2∝E[N]2.As for the previous case, it can be shown that kn(c)≤0,forallc.


## 3.2.1. TID

We can show that the normalizedw(t) is given by(23)w(t)~ce-ab/cEi(1,ab/c)e-atct+bt≠0⟶e-att(χ-log⁡c)(1-bc(a+1t)),
and the unnormalized PDF, evaluated up to O(1/c2) and O(t),(24)v(t)~λEi(1,a(t+b/c))Ei(1,a+b/c),0<t≪1⟶λEi(1,at)log⁡(c/ab)-γ-λbc1log⁡(ab/c)+γeatt-λabcEi(1,at)(log⁡(ab/c)+γ)2,~-λγ+log⁡(at)log⁡(ab/c)-γ,
leading to(25)ht(0,c≫1)=w(0)-v(0)=Khb-λ≃cb(1-χ)-alog⁡c,
where we denote χ=γ+log(ab) and assume that λ≃a(logc-χ). It is seen that ht(0,c≫1) given by (25) is now positive and increases more slowly than ht(0,c≪1)~c2 given by (18). From (23), we can also deduce that(26)E[t]~1a1log⁡c-χ,σt2~1a21log⁡c-χ=E[t]a.
Both moments tend to 0 when c→∞.


## 3.2.2. PND

The approximate expressions given in (A.4) in Appendix A.2  lead to(27)hn(0,c≫1)=P(0)-P(1)≃1+γ-log⁡cc.
We can calculate the first two moments of the number for the specific case a=1 and b=1, (28a)E[N]∝log⁡(c),(28b)σn2∝E[N]2.As for the previous case, it can be shown that kn(c)≤0,forallc.


## 4. Simulation and Results

We have used the algorithm recently described [11] for several values of c.In Figure1, results of the simulated data of the TIDs are plotted. The w(t), the triggered PDF follows very well (23). The v(t), the relaxed one has been well fitted by the expression(29)v(t)=ν0λEi(1,t)Ei(1,1+1/c)0<t≪1⟶ν1λlog⁡tlog⁡(1/c),
for a=1,b=1,c=30 and where ν0=0.07 and ν1=0.2.Figure 1
Plots of the Time Interval Distributionsw(t) (curve 1) and v(t) (curve 2) versus t. The parameters are chosen equal to a=1,b=1, where the value of c=30 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 ofSℓ=∑j=0ℓθj,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 ofc, the theoretical approximations of the PNDs are given by(30)P(n)=1N(b1e-a1n+b2e-a2a2nn!),
where(31a)N=b2+b11-e-a1,(31b)E[N]=1N(a2b2+b1ea1(ea1-1)2), leading to(32)Q(n)=n+1E[N]P(n+1).
The values of the parameters that correctly fit the simulated results of the PDFs, as seen in Figure 2, are b1=0.2,a1=1,b2=0.62,a2=5.2,andE[N]=3.64.Plots of the counting PNDsP(n;t) and Q(n;t) versus n for the parameters chosen equal to a=1,b=1,c=100 and t=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).
(a)(b)The variations of the moments with respect toc are derived from the expressions recalled in the Section 2. We obtained(33a)E[N]≃1+γ+12log⁡c,(33b)σn2≃1+γ+14(1+log⁡c)2.Regarding the reduced moments hn and kn, an excellent fit of the simulated results of Figure 3 is carried out with α1=4,β1=0.17 and α2=3,β1=0.24(34a)hn≃1-γ2+12γ-log⁡(α1c)α1c-β1,(34b)kn≃1-γ2+12γ-log⁡(α2c)α2c-β2,leading to hn≃c2/2logc and kn≃-c4/24(logc)3 which are only in qualitative agreement.The points and circles are the numerical results of the varianceσn2 and the average number 〈n〉 versus c for the values of the parameters chosen equal to a=1,b=1,t=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 hn=P(0)-P(1) and kn=Q(0)-Q(1) versus c for a=1,b=1,t=1. The theoretical approximations calculated for c≫1 are given by (34a) and (34b) respectively (b) and drawn in solid lines.
(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 that(35a)n≦ns(c)⟶H0⟶Pc=0(n)=e-aann!,(35b)n>ns(c)⟶H1⟶Pc≠0(n),based on the binary hypotheses: H0 (no correlation) and H1 (correlation with the parameter c). The threshold is obtained from the likelihood ratio Λ(n),(36)Λ(n)=Pc≠0(n)Pc=0(n)>1⟶ns(c).
We can also utilize the TIDs as a useful tool of processing [12]. The threshold would be based on the likelihood ratio Θ(t)=wc(t)/λe-λt>1→ts(c) and the decision would operate as follows(37)t≤ts(c)⟶H1⟶wc(t)=Ke-atb+ct,t≥ts(c)⟶H0⟶ae-at,
where ts(c)=(1/c)(K/a-b), 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 therelaxed PND is given by(38)Perr⁡=12(1-∑n=0nsPc=0(n)+∑n=0nsPc≠0(n)).
Similarly, we have for the processing with the triggered PND(39)Qerr⁡=12(1-∑n=0nsPc=0(n)+∑n=0nsQc≠0(n)).
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 E[N], then with c because E[N]∝logc as it is calculated in (28a), (33a) and is shown in Figure 3. This is within the range of c:0≤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 errorPerr and Qerr versus the average number E[N], respectively for the relaxed and the triggered processing. The values of the parameters are chosen equal to a=1,b=1,t=1. The average number E[N](c) is the average number of the PDF Pc≠0(n) for 0≤c≤100, with E[N](0)=a. The curves quoted “1” and “2” have been obtained from (41) and (42) respectively.On the other hand, the bounds toQerr and Perr can easily be calculated. In fact, denoting(40)ν=E[N]1+E[N],
where E[N]=a+ξ. Now, taking into account (35a), we have up to O(ξ2),(41)P1(n)=(1-ν)νn,Perr⁡(1)≃1-1e-ν22≃1-1e-ξ8,
for 0≤ν≠1. Therefore Perr(1) is the approximation of the exact Perr for ns=1 plotted as the curve quoted “1” in Figure 4 where a=1. Similarly, taking into account (35a), we have(42)P1(n)=e-E[N]E[N]nn!,Perr⁡(2)≃12-1e+1+E[N]2e-E[N]≃12-1eξ.
Again, Perr(2) is the approximation of the exact Perr for ns=1 plotted as the the curve quoted “2” in Figure 4 for a=1. Finally, we have(43)Perr⁡(2)≤Qerr⁡≤Perr⁡≤Perr⁡(1).
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 Pc=0 and Pc≠0 respectively (the probabilities a priori are taken equal to 1/2) the cut-off rate expresses as(44)R=log⁡21+∑n=0∞Pc=0(n)Pc≠0(n),
which is generally interpreted as a lower bound to the channel capacity [14]. The Pc=0(n) is the Poisson PND of parameter a=1 and Pc≠0(n) will be, as above, either the relaxed or the triggered PND yielding the cut-off rates Rp and Rq, 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 PNDs(45a)P0(n)=(1-ν0)ν0n,(45b)R1=log⁡21+(1-ν0)(1-ν)/(1-ν0ν)0≤ξ≲2≃ξ232(1-3ξ4+ξ22),where ν is given by (40) and ν0=a/(1+a). This is a lower bound to Rp.The second bound is obtained using the Poisson PNDs(46)R2=log⁡21+e-1/2(E[N]-a)20≤ξ≲2≃ξ216(1-ξ2+9ξ232),
which is an upperbound to Rp.In Figure5, the results of the simulation Rp and Rq and the bounds (curves quoted “1” and “2”) given by (45b) and (46) are plotted versus E[N]. Here again, it is seen that the processing with the triggered PND performs much better that the processing with the relaxed PND(47)R1≤Rp≤R2≤Rq,
in the limit of 0≤c≤100. However, because the cut-off rate is a monotonic increasing function with respect to E[N], the inequalities (47) may be extrapolated to for all c.Figure 5
Plots of the cut-off ratesRp and Rq versus the average number E[N], respectively for the processing with the relaxed and the triggered PNDs. The values of the parameters are chosen equal to a=1,b=1,t=1. The average number E[N](c) is the average number of the PDF Pc≠0(n) for 0≤c≤100, E[N](0)=a. The curves quoted “1” and “2” 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.



---

*Source: 604074-2012-01-23.xml*