# Statistical Non-linear Model, Achievable Rates and Signal Detection for Photon-level Photomultiplier Receiver

We characterize the practical receiver in a wide range of signal intensity for optical wireless communication, from discrete pulse regime to continuous waveform regime. We first propose a statistical non-linear model based on the photomultiplier tube (PMT) multi-stage amplification and Poisson channel, and then derive the optimal and tractable suboptimal duty cycle with peak-power and average-power constraints for on-off key (OOK) modulation in linear regime. Subsequently, a threshold-based classifier is proposed to distinguish the PMT working regimes based on the non-linear model. Moreover, we derive the approximate performance of mean power detection with infinite sampling rate and finite over-sampling rate in the linear regime based on small dead time and central-limit theorem. We also fomulate a signal model in the non-linar regime. Furthermore, the performance of mean power detection and photon counting detection with maximum likelihood (ML) detection for different sampling rates is evaluated from both theoretical and numerical perspectives. We can conclude that the sample interval equivalent to dead time is a good choice, and lower sampling rate would significantly degrade the performance.

## Authors

• 5 publications
• 26 publications
• 6 publications
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• ### On the Achievable Rate of a Sample-based Practical Photon-counting Receiver

We investigate the achievable rate for a practical receiver under on-off...
01/18/2019 ∙ by Zhimeng Jiang, et al. ∙ 0

• ### SPAD-Based Optical Wireless Communication with Signal Pre-Distortion and Noise Normalization

In recent years, there has been a growing interest in exploring the appl...
01/22/2021 ∙ by Shenjie Huang, et al. ∙ 0

• ### Minimax Optimal Sparse Signal Recovery with Poisson Statistics

We are motivated by problems that arise in a number of applications such...
01/21/2015 ∙ by Mohammad H. Rohban, et al. ∙ 0

• ### The ALAMO approach to machine learning

ALAMO is a computational methodology for leaning algebraic functions fro...
05/31/2017 ∙ by Zachary T. Wilson, et al. ∙ 0

• ### The Degraded Discrete-Time Poisson Wiretap Channel

01/11/2021 ∙ by Morteza Soltani, et al. ∙ 0

• ### Interference Queueing Networks on Grids

Consider a countably infinite collection of coupled queues representing ...
10/26/2017 ∙ by Abishek Sankararaman, et al. ∙ 0

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## I Introductions

Optical wireless communication can work in both line-of-sight and non-line-of-sight scenarios. On some specific occasions where conventional RF is prohibited and direct-link transmission cannot be guaranteed, NLOS ultra-violet optical scattering communication provides an alternative solution to achieve certain information transmission rate, which shows a large path loss. Optical wireless communication may work in a wide range of signal intensity, from continuous waveform regime to discrete pulse regime. For the latter, it is difficult to detect the received signals using a conventional continuous waveform receiver, such as photon-diode (PD) and avalanche photondiode (APD). Instead, a photon-level photomultiplier tube (PMT) receiver needs to be employed, which converts the received photons to electronic pulse signals applying an multi-stage amplification.

Using a photon-level receiver, the number of detected photoelectrons satisfies a Poisson distribution, which forms a Poisson channel. For Poisson channel, existing works mainly focus on the channel capacity, such as the continuous Poisson channel capacity

[1], discrete Poisson channel capacity [2], wiretap Poisson channel capacity [3], as well as the Poisson interference channel capacity [4], system characterization and optimization [5], and signal processing [6]

. A variety of channel estimation approaches have been proposed for indoor visible light communication in

[7, 8] and photon-counting PMT receiver in [9].

Most information theory and signal process works focus on perfect photo-counting receiver. A practice receiver typically consists of a PMT and the subsequent processing blocks [10]. A practical solution is to adopt PMT to detect the arrival photon and generate a series of pulses with certain width, which incurs certain dead time. The receiver parameters optimization including pulse holding time and threshold has been inverstigated in [11], which shows negligible thermal noise compared with shot noise in experiments. Practical PMT signal characterization based on three regimes related the non-linear effect has been investigated in [12]. [13]

shows that the non-linear takes place between the last dynode and anode due to space charge effect. A novel PMT shot noise model with asymmetric probability density has been inverstigated in

[14] and outperforms Gaussian model reported by experimental PMT gain data [15].

In this work, we are interested in the signal characterization, achievable transmission rate, and signal detection for a wide range of signal intensity, including discrete pulse regime, the continuous waveform regime, and the transition regime between them. More specifically, we propose a practical PMT model with a finite sampling rate ADC. We assume no electical thermal noise since it is negligible comparion with shot noise in PMT and propose a statistical non-linear model on the PMT receiver. Based on the asymmetric shot noise model, we investigate the achievable rate of single and multiple sampling rates in the PMT linear regime, along with the optimal duty cycle and tractable suboptimal duty cycle. We propose a threshold-based classifier related to non-linear function to distinguish the three work regimes, the discrete pulse regime, the continuous waveform regime, and the transition regime between them. Futhermore, we consider on-off keying (OOK) modulation and inverstigate the error probability of mean power detection (MPD) and photon counting detection (PCD) with different sampling rates. We derive approximate performance of mean power detection with “infinite” sampling rate and different over-sampling rate in linear regime based on short dead time compare with symbol duration and central-limit theorem. The theoretical results on the detection error probability are also evaluated by simulation results.

The remainder of this paper is organized as follows. In Section II, we present the PMT statistical non-linear model under consideration. In Section III, we derive the optimal duty cycle and tractable suboptimal duty cycle for single- and multiple-sampling rates. In Section IV, we propose a threshold-based classifier to distinguish the three work regimes. In Sections V and VI, we inverstigate the error probability of MPD with “infinite” and finite sampling rates. Numerical results are given in Section VII. Finally, Section VIII provides the concluding remarks.

## Ii System Model

Consider a practical receiver with finite sampling rate for optical wireless communication with photon-level signals, which contains a PMT detector, an ADC, and a diginal signal processing unit. When a photon arrives, the PMT detector generates a continuous pulse with certain width resulting from multi-stage ampification in the receiver dynode. The PMT output signal is sampled by the ADC.

Assume OOK modulation at the transmitter. The photons arrival times follows a Poisson process with arrival rate proportional to the instantaneous incident power intensity at the receiver surface. Let be the general expression on the gain of PMT. The normalized single photon response is related to PMT characteristics. Denote as the symbol duration and the number of arrival photons in a symbol is given as follows,

 P(N=n)=e−ΛTb(ΛTb)nn!, (1)

where is the arrival rate of Poisson process. Letting denote the th arrival time of photons, the corresponding single photon response is given by

 s(t;τj)=Gjh(t−τj), (2)

where gains for different s are independently and identically distributed (i.i.d) and satisfying Fokker-Planck equation [14]

. Its moment generate function (MGF) is given by,

 MG(μ)=exp(−Aμ1+Bμ), (3)

where ,

, and denotes the number of dynode stages.

Considering the PMT saturation effect caused by space charge effects in the last dynode stage of the electron multiplier, we model it using a non-linear function . Assume that each photon response is linearly superposed in the PMT linear regime and saturated out of it. Denoting as the photon impulse response in interval , we have

 Xj(t)=Nj∑k=1s(t;τjk), (4)

where denotes the total arrival photons number in with photon arrival time for . Hence, the total photocurrent at time resulting from all photons can be compactly written as follows,

 x(t)=C(+∞∑j=0Xj(t)). (5)

For simplicity, assume that and , where is the step function. Define as the normalized sample value by average photon amplification , where denotes the ADC sample value. According to [14], we have the following moment generate function of ,

 M~Z(μ|Λ)=eλ(exp(−μ1+a−1μ)−1), (6)

where and . Furthermore, the probability density of is given by

 f~Z(z|Λ)=eλ(e−a−1)δ(z)+a√1ze−(λ+az)∞∑n=0√n(λe−a)nn!I1(2a√nz), (7)

where is the modified Bessel function of the first kind.

In the remainder of this paper, we consider finite dead time and sampling rate for the receiver. We call under-sampling if the sampling interval is longer than the dead time and over-sampling otherwise.

## Iii The achievable rate for finite sampling rate

Consider OOK modulation, where and denote the photon arrival rates of the background radiation component and signal component, respectively. Let denote the transmitted symbol, denote the number of detected photoelectrons with the corresponding arrival time , denote the PMT output analog signal, and denote the number of detected photoelectrons from samples , where denotes the number of samples. We have that

forms a Markov chain.

Based on Section II, is determined by photon arrival and the PMT characterization. The output samples are i.i.d. if the sample interval . Note that is equivalent to for and constant gain , which implies . Considering the nonlinearity effect, shot noise of amplification gain and thermal noise, we have the general expression .

Considering function which models the linear regime, we have , where is the indicator function.

We first provide the following peak-power and average-power constraints,

 0≤Λs≤ΛA,E[Λ]≤ηΛA. (8)

where reflects the maximum peak power and reflects the maximum average power. Note that can be maximized by with the optimal distribution of binary power levels with zero and maximum peak power. Thus, we assume OOK modulation for the transmitted symbols with duty cycle .

### Iii-a Single Symbol-rate Sampling

According to above model, we have the conditional probability function of as follows:

 f~z|b=i=f~Z(~z|λi),i=0,1, (9)

where and . For sufficiently small , we have the following approximation up to the first order of ,

 f~Z(~z|λ0) = +∞∑i=0f~Z(~z|N=i)P(N=i|λ0) = e−λ0δ(~z)+λ0e−λ0[e−aδ(~z)+ae−a(1+~z)√1~zI1(2a√~z)]+o(λ0) = (a0+o(λ0))δ(~z)+(λ0+o(λ0))g(~z),

where and . Similarly to the scenario for , based on Equation (7

) we can obtain the probability density function for

. Note that probability functions can be decomposed into discrete part and continuous part with corresponding conditional cumulative distribution and , respectively.

To further characterize the mutual information, we have the uniform continuity of is formalized as follows:

###### Lemma 1

Function is uniformly continuous with respect to . Defining , and sequence of functions , then we have . According to Equation (2.4) in [16], we have for . Thus for . Noting that there exists sufficiently large so that for , we have . Thus partial sum of is uniformly bounded.

Noting that for fixed , attenuates to 0 as approaches infinity. According to Dirichlet’s test, uniformly converges in . As each term of is continuous, function is uniformly continuous with respect to .

We have that the mutual information can be expressed as the sum of the continuous part and the discrete part, given by

 I(b;~Z)=∫logdFb,~Z(b,~z)dFb(b)dF~Z(~z)dFb,~Z(b,~z)=ID(b;~Z)+IC(b;~Z), (11)

where the mutual information for the discrete part is given as follows,

 ID(b;~Z)=−(μa1+(1−μ)a0)log(μa1+(1−μ)a0)+[μa1loga1+(1−μ)a0loga0], (12)

and ; and the mutual information for the continuous part is given as follows,

 IC(b;~Z)=H(μfC~Z|λ1(~z)+(1−μ)fC~Z|λ0(~z))−μH(fC~Z|λ1(~z))−(1−μ)H(fC~Z|λ0(~z)), (13)

where . Define such that . Since calculating is intractable, we resort to Taylor’s theorem to approach with small . Note that

 log(μfC~Z|λ1(~z)+(1−μ)fC~Z|λ0(~z)) = log(μfC~Z|λ1(~z))+1−μμfC~Z|λ0(~z)fC~Z|λ1(~z)+o(λ0), (14) ∫∞0fC~Z|λ(~z)d~z = 1−e−λ(1−e−a). (15)

The following Lemma illuminates the relationship of and .

###### Lemma 2

For any fixed and sufficiently small , there exists independent of such that for any , . Since , it suffices to have that if for any . In the case of , for we have for any . In the case of , for , we have for any . In summary, define the condition for both and can be satisfied, and thus we complete the proof.

Based on Lemma 2, for we have

 ∫+∞0[fC~Z|λ0(~z)]2fC~Z|λ1(~z)d~z<ϵ∫+∞0{fC~Z|λ0(~z)}d~z=ϵ(1−e−λ0(1−e−a)), (16)

which implies that . According to Cauchy-Schwarz inequality, we have the following

 ∫+∞0[fC~Z|λ0(~z)]2fC~Z|λ1(~z)d~z∫+∞0fC~Z|λ1(~z)d~z≥(∫+∞0fC~Z|λ0(~z)d~z)2=(1−e−λ0(1−e−a))2 (17)

which implies for for certain constant . Thus, the output signal entropy is given by

 H(μfC~Z|λ1(~z)+(1−μ)fC~Z|λ0(~z)) = H(μfC~Z|λ1)+H2((1−μ)fC~Z|λ0,μfC~Z|λ1)−(1−μ)∫+∞0fC~Z|λ0(~z)d~z −(1−μ)2μ∫+∞0[fC~Z|λ0(~z)]2fC~Z|λ1(~z)d~z+o(λ0) = −(1−μ)logμ(1−(e−λ0(1−e−a)))−(1−μ)(1−(e−λ0(1−e−a)))+o(λ0).

Plugging Equation (III-A) into Equation (13), we have

 IC(b;~Z) = (1−μ)H2(fC~Z|λ0,fC~Z|λ1)−(1−μ)H(fC~Z|λ0)−μlogμ(1−(e−λ1(1−e−a))) −(1−μ)logμ(1−e−λ0(1−e−a))−(1−μ)(1−e−λ0(1−e−a))+o(λ0).

Noted that

 H2(fC~Z|λ0,fC~Z|λ1)=λ0C1+o(λ0), (20)

where and

 H(fC~Z|λ0) = −(λ0+o(λ0))∫+∞0g(~z)logg(~z)d~z−(λ0logλ0+o(λ0))∫+∞0g(~z)d~z = λ0C2−λ0logλ0(1−e−a)+o(λ0),

where . Plugging Equations (20) and (III-A) into Equation (III-A), we have

 IC(b;~Z) = −μlogμ(1−(e−λ1(1−e−a)))−(1−μ)logμλ0(1−e−a) −(1−μ)λ0(1−e−a)+λ0(1−μ)(C1−C2)+o(λ0).

Noting that , we have

 ID(b;~Z) = −(μa1+1−μ)log(μa1+1−μ)+μa1loga1 +(1−μ)(1−e−a)log(μa1+1−μ)λ0+o(λ0),

and thus the mutual information with both continuous part and discrete part is given as follows,

 I(b;~Z) = −μlogμ(1−a1)−(μa1+1−μ)log(μa1+1−μ)+μa1loga1 +(1−μ)(1−a0)log(a1−1+1μ)+λ0(1−μ)(C1−C2)+o(λ0).

Due to small in practical scenario, we omit the terms that attenuates with , which lead to the following approximation on the optimal duty cycle ,

 μ∗1=argmax0≤μ≤η−μlogμ(1−a1)−(μa1+1−μ)log(μa1+1−μ)+μa1loga1. (25)
###### Theorem 1

For , a suboptimal duty cycle is given by , where . Moreover, we have , . Defining with , we have

 G′(μ) = (1−a1)[log(μa1+1−μ)−log(μ)]+a1loga1; (26) G′′(μ) = −(1−a1)μ(μa1+1−μ)<0. (27)

Since and , we have that equation with the unique solution .

From above arguments it is seen that is achieved for , provided . When (from the concavity of ), this maximum is achieved for . Thus, we have

 μ∗1=min{η,μ0}. (28)

Furthermore, since

 limλA→0a−a11−a11=lima1→1[(1+(a1−1))1a1−1]a1=e,limλA→+∞a−a11−a11=elima1→0−a11−a1lna1=1, (29)

we have and .

Note that the above asymptotic result is different from that reported in [1] where for the optimal duty cycle is given by . Note that in this work, since random PMT amplification is incorporated, the optimal duty cycle varies and its asymptotic value as approaches infinity changes to . Moreover, we have the following on the optimal solution to Problem (25).

###### Theorem 2

For , the optimal duty cycle , where is the unique solution to the following equation

 ∫+∞0(fC~Z|λ0(~z)−fC~Z|λ1(~z))log[μfC~Z|λ1(~z)+(1−μ)fC~Z|λ0(~z)]d~z−H(fC~Z|λ1) (30) +H(fC~Z|λ0)−(a1−a0)log[μa1+(1−μ)a0]+a1loga1−a0loga0=0.

Define with . Based on Equation (12) and Equation (13) we have

 G′2(μ) = ∫+∞0(fC~Z|λ0(~z)−fC~Z|λ1(~z)){1+log[μfC~Z|λ1(~z)+(1−μ)fC~Z|λ0(~z)]}d~z −H(fC~Z|λ1)+H(fC~Z|λ0)−(a1−a0){1+log[μa1+(1−μ)a0]}+a1loga1−a0loga0.

Noting that , we have

 G′2(μ) = ∫+∞0(fC~Z|λ0(~z)−fC~Z|λ1(~z))log[μfC~Z|λ1(~z)+(1−μ)fC~Z|λ0(~z)]d~z−H(fC~Z|λ1) +H(fC~Z|λ0)−(a1−a0)log[μa1+(1−μ)a0]+a1loga1−a0loga0,