Channel likelihood correction for photon-counting array receivers in the presence of dead time and jitters

Jingsong Xiang; Yuanming Jia; Chenhui Deng; Sheng Huang; Changping Li; Xueli Chen; Pei Zhang; Yimiao Wu

doi:10.1364/OE.26.002848

1. Introduction

Lunar exploration and other deep space explorations require increasing data transmission capability. Laser communication has many advantages, such as large capacity, low power consumption and small antenna size, indicating its bright future in long-distance space communication. NASA’s Lunar Laser Communication Demonstration (LLCD) mission was a great success in 2013, when the lunar orbiter was able to realize the 622Mb/s laser communication between the moon and the ground by only 0.5 watts of laser power and 10 cm diameter optical antenna [1]. Compared with microwave communications, laser communications show great benefits in Lunar-Earth communications and other deep space communications.

PPM combined with photon-counting detector arrays of direct-detection laser communication is an effective means to solve the massive data transmission problem. This is a great potential for deep-space optical communication. However, in order to reach the link to the desired communications rate, the receiver must be sufficiently sensitive to collect and probe enough photons. The single-photon detector, GM-APD, operates on its breakdown voltage under the Geiger mode. A photon arrival will thus cause an avalanche of charge carriers, which makes the photon effectively detected. This has motivated the recent researches on a number of applications, including deep-space optical communications, three-dimensional laser imaging, laser ranging, quantum cryptography and visible light communication (VLC) [2–5]. However, after a photon is detected, a quenching circuit will be utilized to recover the detector from the excess charge carriers. The quenching process will introduce a finite recovery time, called dead time, during which the device will never respond to another arrived photons [6]. The BER performance of the photon-counting detector array receiver in the presence of dead time needs to be evaluated.

The performance of ground-based optical receivers has been enhanced by the use of photon-counting detector arrays together with optimum signal processing [7]. Investigation and design of adaptive optics (AO) subsystems are presented in [8], to mitigate the coupled effects of background noise and atmospheric turbulence on telescope array-based receivers used in deep-space optical communications links. A synchronization scheme based on Kalman filters was applied to a telescope array-based optical communication receiver [9]. In [10], the effects of detector jitter and timing estimation error have been incorporated in the probability model for the likelihood ratio. The jitter models were presented in [11] for three detectors, i.e. InGaAsP PMT, InGaAsP GM-APD and NbN SSPD. Recent work by Elham Sarbazi and Harald Haas [12] investigated the impact of the detector dead time on the error performance of an on-off key (OOK) modulation optical communication system and it was shown that the bit error rate (BER) degraded rapidly with increasing dead time. The author in [13] analyzed the effect of dead time on the detection efficiency and ranging performance of photon counting, proposed a universal recursive model of the detection probability of discrete time with various dead-times and verified with controlled parameters. However, in terms of the photon-counting array receiver decoding, the above works did not provide a solution to reduce the receiver BER performance loss caused by dead time.

This paper first gives a brief description of the Poisson channel model and the channel likelihood. Secondly, a modified channel model in the presence of the dead time is developed and an improved computation method is derived for the channel likelihood. Moreover, a generalized channel likelihood expression is proposed to solve the problem of the simultaneous existence of both detector dead time and photon jitter. Finally, BER performance comparisons are made between ideal Poisson channel likelihood and improved channel likelihood.

2. Symbol likelihood

2.1 Poisson channel

The use of PPM combined with the single-photon detection technology is extensively recommended in deep-space optical communications. At the transmitter, each PPM symbol consists of M slots with a slot-width of $T_{s}$ and contains $l o g_{2} M$ information bits. Considering that a photon-counting array receiver consists of N detector elements, each element can realize the measurement of the arrival time of a signal photon. The time resolution is $Δ t$ , then a PPM slot can be divided into about $H = r o u n d (T_{s} / Δ t)$ bins, where round(.) represents rounding to the nearest integer. At first, the ideal Poisson channel case is considered. It is assumed that the transmitted signal is uniformly distributed among the signal slot, the background is uniformly distributed among PPM symbols, detector dead time is not taken into account, and a detector element can detect multiple photons simultaneously. Under this ideal case, in the n-th detector, i-th slot, j-th bin, if the received photon number is $k_{n, j}^{i}$ , the corresponding priori probability in signal slots and in noise slots can be given by:

{\begin{cases} p_{n, j}^{i} (k_{n, j}^{i} | 1) = \frac{{[(n_{s} + n_{b}) / N H]}^{k_{n, j}^{i}}}{k_{n, j}^{i}!} \exp [- (n_{s} + n_{b}) / N H] \\ p_{n, j}^{i} (k_{n, j}^{i} | 0) = \frac{{(n_{b} / N H)}^{k_{n, j}^{i}}}{k_{n, j}^{i}!} \exp (- n_{b} / N H) \end{cases}

where

n_{s}

is the average signal photons over the array receiver in one PPM signal slot,

n_{b}

is the total background noise arriving at the receiver in one PPM symbol slot, including background radiation in the channel and dark-counting noise from photon detectors. Hence, the channel log-likelihood ratio of the i-th slot for the array receiver is:

L L R^{i} = \ln \frac{\prod_{n = 1}^{N} \prod_{j = (i - 1) H + 1}^{i H} p_{n, j}^{i} (k_{n, j}^{i} | 1)}{\prod_{n = 1}^{N} \prod_{j = (i - 1) H + 1}^{i H} p_{n, j}^{i} (k_{n, j}^{i} | 0)} = \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} k_{n, j}^{i} \ln (1 + \frac{n_{s}}{n_{b}}) - n_{s}

In the above equation, the notation $\sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} k_{n, j}^{i}$ represents the total photons which the array has detected in the i-th slot. Hence, the computation of channel likelihood Eq. (2) equals to the standard expression proposed in [14].

2.2 Channel with detector dead time

In Eq. (2), the detector dead time is not considered. During the dead time, the detector element cannot respond to the arrived photons any longer. The current detection characteristics have relation to the previous detection state, and the channel will have memory. Assuming that a detector element has a dead time $T_{d} = H_{d} Δ t$ , and $H_{d}$ is an integer, a rule can be given. Namely, when $H_{d} = 0$ , only the current $Δ t$ bin will be affected. During this bin, at most one photon can be detected. When $H_{d} = 1$ , two bins will be affected. In the current bin, at most one photon can be detected. In the next bin, no photon can be detected if a photon has been detected in the previous bin. The above rule applies to all cases where $H_{d} = 2, 3, 4 \dots$ .

Considering the dead time, the work patterns of a detector element can be divided into three categories. As to the first category, outside the dead time, no photon has arrived during $Δ t$ , therefore the detector element will not detect any photons. When ‘1’ or ‘0’ pulse is emitted, the priori probability is given by:

{\begin{cases} P_{01} = p_{n, j}^{i} (k_{n, j}^{i} = 0 | 1) = \exp [- (n_{s} + n_{b}) / N H] \\ P_{00} = p_{n, j}^{i} (k_{n, j}^{i} = 0 | 0) = \exp (- n_{b} / N H) \end{cases}

As to the second category, outside the dead time, one or more photons have arrived during, however the detector element only can detect one photon. When ‘1’ or ‘0’ pulse is emitted, the priori probability is given by:

{\begin{cases} P_{11} = 1 - P_{01} = 1 - \exp [- (n_{s} + n_{b}) / N H] \\ P_{10} = 1 - P_{00} = 1 - \exp (- n_{b} / N H) \end{cases}

As to the third category, during the dead time, the detector element cannot detect any photons. Whenever ‘1’ or ‘0’ pulse is emitted, the number of detected photons must be zero. The priori probability is given by:

{\begin{cases} P_{01}^{'} = p_{n, j}^{i} (k_{n, j}^{i} = 0 | 1) = 1 \\ P_{00}^{'} = p_{n, j}^{i} (k_{n, j}^{i} = 0 | 0) = 1 \end{cases}

According to Eqs. (3)-(5), the channel likelihood ratio of the array receiver for the i-th slot is given by:

\begin{array}{l} L R^{i} = \frac{\prod_{n = 1}^{N} \prod_{j = (i - 1) H + 1}^{i H} p_{n, j}^{i} (k_{n, j}^{i} | 1)}{\prod_{n = 1}^{N} \prod_{j = (i - 1) H + 1}^{i H} p_{n, j}^{i} (k_{n, j}^{i} | 0)} \\ = \frac{p_{1, (i - 1) H + 1}^{i} (k_{1, (i - 1) H + 1}^{i} | 1) p_{1, (i - 1) H + 2}^{i} (k_{1, (i - 1) H + 2}^{i} | 1) \cdot \cdot \cdot p_{N, i H}^{i} (k_{N, i H}^{i} | 1)}{p_{1, (i - 1) H + 1}^{i} (k_{1, (i - 1) H + 1}^{i} | 0) p_{1, (i - 1) H + 2}^{i} (k_{1, (i - 1) H + 2}^{i} | 0) \cdot \cdot \cdot p_{N, i H}^{i} (k_{N, i H}^{i} | 0)} \\ = {(\frac{P_{01}}{P_{00}})}^{H_{1}^{i}} {(\frac{P_{11}}{P_{10}})}^{H_{2}^{i}} {(\frac{P_{01}^{'}}{P_{00}^{'}})}^{H_{3}^{i}} \end{array}

where

k_{n, j}^{i} \in {0, 1},

P_{01}^{'} / P_{00}^{'} = 1

, and

H_{1}^{i}

,

H_{2}^{i}

,

H_{3}^{i}

satisfy the following relationship,

{\begin{cases} H_{1}^{i} + H_{2}^{i} + H_{3}^{i} = N H \\ H_{2}^{i} = \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} k_{n, j}^{i} \\ H_{3}^{i} = \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} \sum_{t = j - H_{d}}^{j -1} k_{n, t}^{i} \\ H_{1}^{i} = N H - (\sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} k_{n, j}^{i} + \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} \sum_{t = j - H_{d}}^{j -1} k_{n, t}^{i}) \\ = N H - \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} (\sum_{t = j - H_{d}}^{j -1} k_{n, t}^{i} + k_{n, j}^{i}) \\ = N H - \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} \sum_{t = j - H_{d}}^{j} k_{n, t}^{i} \end{cases}

where

H_{1}^{i}

refers to the total process when the detector elements can detect but no photon is detected in the i-th slot,

H_{2}^{i}

denotes the total process when the photon has been detected in the i-th slot.

H_{3}^{i}

represents the total processes that cannot respond to the arrived photons due to those detector elements in the dead time. According to Eqs. (3)-(7), the channel log-likelihood ratio in the presence of detector dead time for the i-th slot is given by:

\begin{matrix} L L R^{i} = H_{1}^{i} \ln \frac{P_{01}}{P_{00}} + H_{2}^{i} \ln \frac{P_{11}}{P_{10}} \\ = \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} k_{n, j}^{i} \ln \frac{1 - \exp [- (n_{s} + n_{b}) / N H]}{1 - \exp (- n_{b} / N H)} + \frac{n_{s}}{N H} \sum_{n = 1}^{N} \sum_{j = (i - 1) H + 1}^{i H} \sum_{t = j - H_{d}}^{j} k_{n, t}^{i} - n_{s} \end{matrix}

2.3 Channel with photon jitter and detector dead time

The following limiting factors, changes in atmospheric conditions, synchronization and tracking errors, and the response time of each detector, all contribute to the unknown location of photons (i.e. slot boundaries). These uncertainties are collectively called the photon jitter. Photon jitter will generate the laser pulse broadening effect, which cannot be ignored in very high throughput channels. Let $s_{j}$ denote the arrival time of the j-th photon. In the presence of photon jitter, it can be observed at time $t_{j} = s_{j} + δ$ , where $δ$ is independent of one another and obeys a probability density function of $f_{δ} (δ)$ . In [11], the jitter of InGaAsP PMT, InGaAsP GM-APD and NbN SSPD single photon detectors was experimented, and the experimental results show that the probability density function of InGaAsP PMT and NbN SSPD single photon detectors is similar to that of Gaussian distribution. Considering that a transmitted laser pulse P(t), and the integral of P(t) is normalized to one, at the receiver, the distribution of signal photons can be given by:

f (t) = (P * f_{δ} (δ)) (t)

where

*

represents a convolution operation. Figure 1 shows the diagrams of P(t),

f_{δ} (δ)

and

f (t)

.

Fig. 1 Left: The transmitted signal pulse P(t). Middle: The Gaussian jitter’s probability density function $f_{δ} (δ)$ . Right: The probability density function $f (t)$ of the received signal.

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In the presence of jitter, the signal photons may be observed in the adjacent slots. If the i-th slot is a signal slot, then in the j-th bin of the n-th detector, the average arrived photons can be expressed as,

k_{n, j}^{i} = w_{n, j}^{i} n_{s} / N + n_{b} / N H

where

w_{n, j}^{i}

is the weight of the signal photons in the j-th bin, which is given by:

w_{n, j}^{i} = \int_{(j - 1) Δ t - i T_{s}}^{j Δ t - i T_{s}} f (t - i T_{s}) d t

In general, the probability of the signal pulse widening beyond one slot adjacent to the signal slot is almost zero. So similar to the method described in [10], this method only considers the signal weight contributions over the current signal slot and the two adjacent slots. Both in the presence of photon jitter and the dead time, the channel log-likelihood in the i-th slot for the receiver can be expressed as,

\begin{matrix} L R^{i} \approx L R_{- 1}^{i} L R_{0}^{i} L R_{1}^{i} = \frac{\prod_{n = 1}^{N} \prod_{j = (i - 2) H + 1}^{(i + 1) H} p_{n, j}^{i} (k_{n, j}^{i} | 1)}{\prod_{n = 1}^{N} \prod_{j = (i - 2) H + 1}^{(i + 1) H} p_{n, j}^{i} (k_{n, j}^{i} | 0)} \\ = {\sum_{j = (i - 2) H + 1}^{(i + 1) H} [\frac{\exp [- (w_{n, j}^{i} n_{s} / N + n_{b} / N H)]}{\exp (- n_{b} / N H)}]}^{N - \sum_{n = 1}^{N} \sum_{t = j - H_{d}}^{j} k_{n, t}^{i}} • \\ {(\frac{1 - \exp [- (w_{n, j}^{i} n_{s} / N + n_{b} / N H)]}{1 - \exp (- n_{b} / N H)})}^{\sum_{n = 1}^{N} k_{n, j}^{i}} \end{matrix}

where

L R_{- 1}^{i}

,

L R_{0}^{i}

and

L R_{1}^{i}

signify the channel likelihood provided by signal photons which are distributed in the previous slot, the current slot, and the next slot, respectively.

Converting the Eq. (12) to the logarithmic domain, we have,

\begin{array}{l} L L R^{i} = \ln L R^{i} \approx \sum_{n = 1}^{N} \sum_{j = (i - 2) H + 1}^{(i + 1) H} k_{n, j}^{i} \ln (\frac{1 - \exp [- (w_{n, j}^{i} n_{s} / N + n_{b} / N H)]}{1 - \exp (- n_{b} / N H)}) + \\ \frac{n_{s}}{N} \sum_{n = 1}^{N} \sum_{j = (i - 2) H + 1}^{(i + 1) H} \sum_{t = j - H_{d}}^{j} w_{n, j}^{i} k_{n, t}^{i} - n_{s} \end{array}

The expression of Eq. (13) describes the channel output of a photon-counting detector array receiver responding to a generalized photon jitter model in the presence of detector dead time.

3. Performance

In this section, the performance of a photon-counting detector array receiver is evaluated in terms of bit-error rate (BER) with a serially concatenated pulse-position modulation (SCPPM) [14]. A SCPPM frame contains 2520 PPM symbols, PPM order M = 64 and slot-width T_s = 16ns.

Figure 2 shows the BER performance of the photon-counting detector array receivers under different detector dead time T_d, detector components N and background noise n_b when the time resolution of each detector $Δ t$ = 1.6ns. The line of circles indicates the BERs when the channel likelihood is given by Eq. (2). The line of small squares represents the BER performance of the improved channel likelihood obtained by Eq. (8).

Fig. 2 BER performance of photon-counting detector array receiver under different detector dead time, detector components and background noise.

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From Figs. 2(a)-2(d), it can be seen that the modified channel likelihood expression proposed in this paper can obtain a more obvious performance gain especially under the context of a smaller number of detectors, the stronger background noise, and the longer detector dead time.

Figure 3 shows the BER performance of the photon-counting detector array receiver when the time resolution of each detector $Δ t$ is 8ns, 4ns, 1.6ns, 0.8ns, respectively. From Fig. 3, it is easily demonstrated that it obtains gains of 0.35 and 0.3dB given by Eq. (8) under the time resolution of each detector $Δ t$ of 8ns-4ns and or 4ns-1.6ns at a BER of $10^{- 4}$ . In addition, little gain is obtained when the time resolution of each detector ranges from 1.6ns to 0.8ns.

Fig. 3 BER performance of photon-counting detector array receiver under N = 8, n_b = 1.0 and T_d = 64ns.

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Figure 4 illustrates in the presence of photon jitter and detector dead time, the bit-error rates for channel likelihood calculated by Eq. (2) and improved Eq. (13). We modeled the photon jitter $f_{δ} (δ)$ in a Gaussian distribution with a mean of zero and a standard deviation $σ_{δ}$ of 0.1T_s and 0.2T_s, respectively. Besides, the shape of the transmitted laser pulse P(t) is regarded as a unit pulse on [0,T_s].

Fig. 4 BER performance of photon-counting detector array receiver for N = 8, n_b = 1.0, T_d = 64ns and the standard deviation $σ_{δ}$ : (a) $σ_{δ} = 0.1 T_{s}$ . (b) $σ_{δ} = 0.2 T_{s}$ .

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Figures 4(a) and 4(b) show the BER performance of the photon-counting detector array receivers for the jitter standard deviation $σ_{δ}$ of 0.1T_s, 0.2T_s, respectively. Based on Figs. 4(a) and 4(b), it can be seen that the improved channel likelihood computed by Eq. (13) recovers 0.7dB and 0.85 dB of losses when the jitter standard deviation $σ_{δ}$ is 0.1T_s and 0.2T_s, respectively, compared to the standard likelihood calculation method calculated by Eq. (2) at a BER of $10^{- 4}$ . Then compared with Eq. (8), the improved channel likelihood computed by Eq. (13) still represents gains of 0.45dB and 0.55 dB when photon jitter and detector dead time exist simultaneously. As a result, we can conclude that a more obvious performance gain can be achieved especially under the bigger jitter.

4. Conclusion

In this paper, a modified channel likelihood model is proposed for photon-counting array receivers in the presence of dead time and jitters. It is demonstrated that the BER performance degradation caused by the dead time should not be neglected. In the presence of dead time, the photon-detected process will have memory. Through a detailed analysis, we developed a mathematical model for the dead time photon-counting statistics and obtained a modified notation for the channel likelihood. According to the simulation results, the improved method can obtain a more obvious performance under the context of a stronger background noise, smaller number of detectors, and longer detector dead time.

Furthermore, considering the pulse broadening effect caused by the uncertain photon arrival delay due to various factors in the channel, we derive the generalized channel likelihood both in the presence of photon jitter and dead time. The gains of the improved channel likelihood are more than those of the conventional channel likelihood without modification, especially under the context of the bigger jitter.

Funding

National Natural Science Foundation of China (No. 61571072, No. 61371096).

References and links

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Channel likelihood correction for photon-counting array receivers in the presence of dead time and jitters

Abstract

1. Introduction

2. Symbol likelihood

2.1 Poisson channel

2.2 Channel with detector dead time

2.3 Channel with photon jitter and detector dead time

3. Performance

4. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (13)

Optics Express