Abstract

Crosstalk is the main challenge for active collision avoidance LIDAR. Conventional strategy on static discrimination threshold and unified number of accumulated pulses is not valid against crosstalk with varying number of sources and varying intensity. This paper presents an adaptive strategy to distinguish the target echo from dynamic crosstalk with unknown intensity in the scenario of intelligent vehicles and autonomous ground vehicles. The adaptive strategy is theoretically and experimentally verified to have much better performance on dynamic crosstalk than conventional strategy, and the detection rate is enhanced by 82.4% as well.

© 2017 Optical Society of America

1. Introduction

Automotive radar is widely applied due to its unique performance and reasonable cost. Such a system could ease the pressure on drivers and reduce the probability of accidental collisions. However, the main challenge and key requirement in radar performance is to distinguish the target echo from the complicated received signal that contains the solar background noise, vehicle-to-vehicle (V2V) crosstalk, internal electrical noise of receiver, etc. Unlike solar background noise that is easy to be filtered out because of its continuity in time and space domain, V2V crosstalk originating from vehicles has similar physical characteristics with signal echo. Hence, V2V crosstalk reduces the sensitivity of the sensor, or even increase the difficulty to correctly identify the echo, given that the correct echo has already been attenuated and distorted by the weather conditions and other factors [1]. Therefore, accurately extracting the signal echo against the complex crosstalk becomes a critical issue for active collision avoidance [2].

Capability of detecting the target against crosstalk enables a diverse range of modulation-based solutions, such as Frequency Modulated Continuous Wave (FMCW) modulation [3,4] and Chaotic Pulse Position Modulation (CPPM) [5–8]. Although LIDAR has been made more and more suitable for use in intelligent vehicles from a performance point of view, the dynamic crosstalk is still a major drawback for these perceptions. Recently, the statistics of dynamic radar crosstalk has been performed in stochastic geometry [9], and a study supports the observable interference power at an automotive radar antenna for different traffic situations [10]. Alternatively, Gunzung Kim reported in 2015 the probability of LIDAR scanner detecting target distance with mutual interferences considering spatial and temporal overlaps [11]. However, to the best of our knowledge, no strategy that effectively distinguishes the target echo from dynamic crosstalk has been reported before.

In our approach, a novel adaptive strategy has been put forward to detect the target against the dynamic crosstalk with unknown intensity in the scenario of intelligent vehicles and autonomous ground vehicles. The technology used is based upon the single photon detector (SPD) with Geiger-mode avalanche photodiode (GM-APD) [12] as well as time correlated single photon counting (TCSPC) approach [13,14]. The adaptive strategy of extracting the target echo from complicated received signal using CPPM scheme is performed theoretically and experimentally to verify the excellent anti-crosstalk capability. Section 2 describes the theory of conventional strategy and the adaptive strategy. Section 3 demonstrates the experiment system and the collected data, and then discusses how the detecting efficiency can be improved.

2. Theory

2.1 Types of crosstalk

A complicated and practical crosstalk model with multiple sources is demonstrated, with the scenario depicted in Fig. 1. The solid line indicates the signal echo generated from vehicle A. The dash line indicates the line-of-sight crosstalk (LOS-C) from vehicle C heading in the opposite direction to vehicle A, and the dotted line indicates the reflective crosstalk (R-C) from vehicle D heading in the same direction as vehicle A.

 

Fig. 1 Signal echo and two types of crosstalk among vehicles on the road

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In this process, the signal level can be computed using the radar equation [15]. Assuming Necho, Nc_b and Nc_r as the received photon number per pulse from vehicle A, C and D respectively, while PA, PC, and PD represents the intensity of the transmitted laser beam from vehicle A, C and D respectively, the LIDAR equations can be described as

Necho=PAfAλAhc(FOVθT)2ρπARLA2ηRT2
Nc_b=PCfCλChc1πARθT2LC2ηRT2
Nc_r=PDfDλDhc(FOVθT)2ρπARθT2LA2cosθRηRT2
where LA is the distance between vehicle A and B, which is the target range, LC is the distance between vehicle A and C, λ, θT, and f is respectively the wavelength, divergence angle and repetition rate of laser output. The receiver optics system parameters include optical efficiency ηR and the aperture area AR. As for reflected object, the diffuse surface element of AS acts as a Lambertian scatterer, ρ is the diffuse target reflectivity, and T is the single path transmission efficiency. The term FOV2 denotes the field of view of the receiver, which is equal to AS/πL2. When the laser spot area is smaller than the target receiving plane, the term (FOV/θT)2 is regarded as unity. Note that the signal power Necho and Nc_r decrease with L4, and the Nc_b decreases with L2.

To illustrate the properties of different types of crosstalk, an example is given as follow. For the LIDAR system, the laser output is assumed having the average power of 800 nW, pulse repetition rate of 100 kHz and wavelength of 1550 nm. The receive aperture is 10 cm in diameter, while a cross section of 7.58 × 10−3 m−2 and the reflectivity of 0.3 is assumed for the target. Figure 2 shows the predicted photon counts per second recorded by the detector. As for the LOS-C, Nc_b is higher than 105 according to Eq. (2) that saturates the GAPD. Considering that GAPD is saturated without afterpulsing effects, GAPD can record only one photoelectron (PE)/pulse, thus the intensity of crosstalk (IOC), represented by the crosstalk counts per second, remains at 100k c/s for all the time. For Necho and Nc_r, the detector records less than 1 PE/pulse from the target. In this case, the accumulated result Necho and Nc_r is dependent on the photon number received, demonstrating the performance similar to an ideal linear detector, with little portion of signal uncaptured. Therefore, the received photons of R-C are reduced with increasing distance.

 

Fig. 2 Received counts with increasing distance for target echo and two types of crosstalk.

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2.2 Conventional strategy

In accordance with Poisson statistics, the response probability of time bin during TCSPC process relates to the number of photons (Nphoton) fallen in the time bin as well as the detection efficiency of the detector (η), as given by [16,17]

P=1exp(Nphotonη).

For comparison with the new approach, we firstly describe a conventional strategy that enables acquisition of a target in range [15,18,19]. With regard to an input to the strategy, we specify the desired probability of detection PD, the probability of false alarm for each bin Pfa, and the effective signal level n, based on Eqs. (1)-(3). Furthermore, the required threshold and the number of transmitted laser pulse can be obtained in order to achieve the specified performance. For the situation with multiple vehicles, the probability of response for any non-target time bin and the target bin is attained respectively as below.

pc=1exp[(Nc_b+Nc_r+n)η].
ps=1exp[(Necho+Nc_b+Nc_r+n)η].
where pc is the single-pulse probability of a firing if no target is present, and ps is the single-pulse probability of a firing if a target with signal level Necho is present.

Since the response of each bin is independent with each other, the probability of detection and false alarm in a single bin can be obtained according to the Bernoulli process with a given number of pulses U and threshold Th. The probability Pdect and Pfa are given by [15]

Pdect=i=ThUCUipsi(1ps)Ui
Pfa=i=ThUCUipci(1pc)Ui

The total probability of false alarm is then written as

Pfau=1(1Pfa)Nbin
where Nbin is the number of storage bins. A detection probability above 95% is required hereinafter to ensure the system reliability, while the total probability of false alarm is set as below 5%, corresponding a probability of false alarm in a single bin as less than 8.21 × 10−5.

Assuming IOC of 10k c/s that corresponds to the reflective crosstalk from vehicle D at a distance of 150 m, a family of receiver operating characteristics (ROC) curves can be computed by varying U and Th, in order to determine the optimal value of U and Th to identify the target [20]. As illustrated in Fig. 3, along each ROC curve there are a series of discrete data points, whose integer detection thresholds are incremented successively by one, beginning from the right end with the threshold being zero and Pfa-u being unity towards the left. Figure 3 indicates that 400 accumulated pulses with a threshold of 5 is necessary to guarantee target acquisition for IOC of 10k c/s. Figure 4 describes the variation of the ROC curves under different crosstalk intensity, based on the strategy of U = 400 and Th = 5, showing the reduction of detection probability as well as the rise of the probability of false alarm, along with the increase of crosstalk up to 300k c/s. Obviously the optimal strategy for IOC of 10k c/s is not capable of satisfying the detection requirement for the cases of stronger crosstalk. Thus the conventional strategy is not available for target detection in the dynamic crosstalk.

 

Fig. 3 ROC curves with the accumulation number from 300 to 800, the signal repetition rate of 100 kHz, and the crosstalk repetition rate of 10k c/s.

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Fig. 4 ROC curves under different crosstalk with 400 pulses accumulated.

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2.3 Adaptive strategy

For single-photon detection in the context of multiple vehicles, IOC alters randomly and the crosstalk level is not known in advance due to variation of involved vehicle numbers, driving directions, relative locations among vehicles and so on. In order to realize collision avoidance, an adaptive strategy is put forward which constantly adjusts the accumulated number of pulses through self-evaluation to accurately identify the target distance against dynamic and complicated crosstalk condition. The procedure starts with an initial number of accumulated pulse U, then estimates the target bin from the resulting data set to determine the new U, the process of which is repeated until a target being detected. The adaptive strategy can be summarized as follows, with the flow chart demonstrated in Fig. 5:

 

Fig. 5 Flow chart of adaptive strategy.

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  • (a) Launch three successive detection cycles Ci, Ci + 1, Ci + 2 (originally i = 1): in each cycle, a train of laser pulses with the pulse number of u are transmitted, and the data obtained by SPD for each cycle is collected respectively;
  • (b) Generate three frames denoted as Mi, Mi + 1 and Mi + 2: the detected distribution of counts of Ci is regarded as Mi. The detected results of Ci and Ci + 1 are added together in each bin to form the frame Mi + 1. Similarly, Mi + 2 is created by combing results of all three cycles (Ci, Ci + 1 and Ci + 2).
  • (c) For each of the three frames (Mi, Mi + 1 and Mi + 2), record respectively the location of the peak bin, defined as the time bin that collects the highest photon number.
  • (d) Self-evaluation: if the peak bin has same location for successive three frames (Mi, Mi + 1 and Mi + 2), the evaluation ends and the signal echo is figured out, considered to arrive at the timing of peak bin. Otherwise, one more cycle (Ci + 3) is launched and a new frame Mi + 3 is generated by combing results of three cycles (Ci + 1, Ci + 2 and Ci + 3). Let i = i + 1, and repeat steps (c) and (d) until the signal echo is identified.

During the process of successive accumulation, it is assumed that the accumulative laser shot number in one cycle is u while the detecting probability is ps. The process is considered as Bernoulli distribution X~B(u,ps), with the probability of detecting n photons during an accumulating cycle given as

X~[0Cu0ps0(1ps)u01Cu1ps1(1ps)u1nCunpsn(1ps)un]

When the product ups is not negligible compared to unity, where u is intense and ps is tiny enough, the Bernoulli distribution can be approximated as the Poisson distribution, and the expectation value of λ will be equal to ups . Thus the detection probability can be expressed as

Cnkpk(1p)nkλkk!eλ

Taking N cycles into account, the number of accumulated pulse U = Nu, and the probability of detecting k photons in N cycles can be calculated as

P(x=k)=PPoisson(Nups=k)

Similarly, for the probability of detecting photons in the non-target bin, the distribution is considered as Y~Ppoisson(Nupc). Using the adaptive strategy, the photon counts in each bin are compared when selecting the target location per frame. Assuming there are Ns photons in the signal bin when the target is successfully detected during N cycles of pulses accumulating, the photon number of all the non-target bins exceeds Ns. Thus we have

pdect(Ns)=(i=0NsPpoisson(Nupc=i))Nbin

Taking the distribution of photon number Ns into account, the detection probability of frame MN after the accumulation of N cycle pulses can be obtained as

Pdectadp1=i=1Nu[P(Nups=i)pdect(i)]

Given that the target has been detected in the frame MN, the probability of both MN + 1 and MN + 2 frame detecting the target location would be given as

Pdectadp2=i=0uj=1Nu[k=0jq=0min(i+jk,u)P(upc=q)P(Nupc=k)]NbinP(Nups=j)P(ups=i)

In a similar way, the probability of all the three frames (MN, MN + 1 and MN + 2) simultaneously detecting the target location can be given as

Pdectadp3=x=0ui=0uj=1Nu[k=0jq=0min(i+jk,u)y=0min(x+i+jkq,u)P(upc=y)P(upc=q)P(Nupc=k)]NbinP(Nups=j)P(ups=i)P(ups=x)

As for the probability of false alarm, the corresponding expressions can be written as

Pfaadp1=i=0Nu[P(Nups=i)(1(1k=max(1,i)NuP(Nupc=k))Nbin)]
Pfaadp2=i=0uj=0Nu{1[1k=max(1,j)Nuq=max(0,i+jk)uP(upc=q)P(Nupc=k)]Nbin}P(Nups=j)P(ups=i)
Pfaadp3=x=0ui=0uj=0Nu{1[1k=max(1,j)Nuq=max(0,i+jk)uy=max(0,x+i+jkq)uP(upc=y)P(upc=q)P(Nupc=k)]Nbin}P(Nups=j)P(ups=i)P(ups=x)

It should be noted that for the adaptive strategy, in a single frame the detection probability rises and the false alarm probability drops with the pulses accumulated which is similar to the conventional strategy. While adding up the results of three consecutive frames, the probability of detection and false alarms both drop comparing to the single frame. Therefore, the adaptive strategy does sacrifice the detection probability to achieve lower false alarm.

With the crosstalk of 100k c/s, the probability of detection and false alarm under different accumulative numbers are shown in Fig. 6.

 

Fig. 6 The detection probability and probability of false alarm by the adaptive strategy, for different accumulative numbers and different frames.

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For satisfying the requirement of Pd and Pfa-u, the optimal number of accumulated pulse Uopt for two different strategies was respectively shown in Fig. 7, with IOC from 10k to 300k c/s, indicating that less laser shots are needed before detecting the target for adaptive strategy than conventional strategy under the same crosstalk in the static scenario. For example, the reduction ratio of Uopt of the adaptive strategy is 16% for IOC of 10k c/s and 21% for IOC of 300k c/s.

 

Fig. 7 Comparison of two strategy in terms of Uopt under static crosstalk.

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3. Experiment

3.1 Experiment setup

The experimental setup is illustrated in Fig. 8, including the transceiver, crosstalk generator and the optical channel simulator. The transceiver module consists of an FPGA-based signal generator, a laser (Laser-A), and a SPD connected directly to a personal computer (PC) equipped with TCSPC board. In the transceiver module, CPPM approach is adopted having pulse interval of transmitted laser pulse train randomly modulated instead of being fixed. The realization method of CPPM is to additionally introduce a randomized tiny increment δ within a certain scope above the fixed temporal interval. The chaotic time interval pulse sequence can be generated by algorithm. The laser pulses are reflected by the object and detected by the SPD. The timer of TCSPC board is started by the firing of laser pulse and stopped by the arrival of a periodic triggers that are originated from the laser pulse. The time between start and stop signal is recorded while a histogram of photon arrival times builds up. In the measurement, the average count rate from the detector is found to be in the order of 0.16 detected events per laser pulse taking into account the SPD efficiency, and the photon flux is controlled to be 2400 c/s by tuning the attenuation ratio. An incandescent lamp is used to act as background light.

 

Fig. 8 Experimental setup.

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For the crosstalk generator, Laser-B and Laser-C with variable attenuators act as the source of LOS-C and R-C respectively. All the driving signals of FPGAs were independent from each other, avoiding the potential phase correlation caused by generating multiple sets of pulse sequence from the same FPGA. The experiment was generally implemented under different scenarios by switching the corresponding FPGAs on or off. In order to simulate the line-in-sight crosstalk, the photon counts was decreased to 100k c/s. The collected photon counts originating from FPGA-R-C I, II was attenuated to 10k and 50k c/s which is approximately proportional to the intensity of reflective crosstalk at different distances. Different scenarios realized by switched working status of five FPGA boards are summarized in Fig. 9.

 

Fig. 9 Different scenarios realized by switched working status of five FPGA boards.

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3.2 Results and discussion

Conventional and adaptive strategies were experimentally carried out and compared for the task of detecting the target against dynamic crosstalk. Figure 10 illustrates the experimental results using the conventional strategy. Figure 10(a) studies the strategy determined based on the crosstalk level of 10k c/s, having U = 400 and Th = 5. It can be seen that total photon counts is always above the threshold at the target bin for different IOC level, while in the non-target bins, however, it is lower than the threshold for IOC of 10k c/s, and rapidly goes beyond the threshold with increased IOC. Figure 10(a) clearly demonstrates that with the threshold set according to IOC of 10k c/s, the conventional strategy may fail for target identification with stronger IOC such as 300k c/s, in which condition serious false alarms may appear.

 

Fig. 10 Detection results with IOC of 10k, 100k, 200k and 300k c/s by the conventional strategy.

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Figure 11 illustrates the case of dynamic crosstalk, while the target is detected using strategy determined based on the crosstalk level of 10k c/s. Real-time probability of false alarm under different IOC is described in Fig. 11, while the dash line indicates the upper limit of requirement as 5%. The false alarm fluctuates markedly with the change of IOC and increases rapidly as higher crosstalk occurs. Thus the strategy for 10k c/s is suboptimal for detecting target under dynamic crosstalk.

 

Fig. 11 Probability of false alarm under dynamic crosstalk counts.

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Figure 10(b) describes the case with the strategy determined based on the crosstalk level of 300k c/s, having U = 800 and Th = 15, indicating successful target extractions for IOC level from 10k to 300k c/s. The main problem of this case is that the detection rate is seriously limited due to data redundancy. The conventional strategy is based on the total accumulated pulses measurement, demanding both a count above the threshold in the target bin and a count below the threshold in each non-target bins. However, the adaptive strategy focuses on successive frames, requiring a higher count in the target bin than all the non-target bins and ignoring the threshold limitation. That is, as long as the photon number in the target bin exceeds all the non-target bins, even below the nominal threshold set by the conventional strategy, the target can be correctly identified, thus reducing data redundancy and the detection time.

Figure 12 shows the detecting results by adaptive strategy for different frame under the IOC of 10k, 100k and 200k c/s respectively. The number of transmitted laser pulse in a cycle is set as u = 100. The frame data was acquired after each emitting cycle. As can be seen from Fig. 12, higher IOC requires more detecting cycles. The temporal distribution of photon counts per cycle is independent, unable to tell the presence of target. The target photons are fixed in a particular storage bin due to the constant distance, but the time interval between the crosstalk photons and the trigger of cycle start are random due to the application of CPPM. Taking this difference into account, it is probable that the target is correctly detected without false alarms declared during several cycles of obtaining frame statistical results. In practice, the adpative strategy is able to detect the target unaffected by the crosstalk.

 

Fig. 12 Detection results by adaptive strategy with different accumulated pulse number and varying crosstalk, in which the horizontal axis represents the sequence number of storage bins while the circle and square marks represent the location of target and false alarm respectively.

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Total number of accumulated pulses recorded during one minute is shown in Fig. 13, in which the number of oncoming vehicles at the same time are limited up to 3, corresponding to IOC range from 10k to 300k c/s. In the static IOC scenario, with IOC level assumed as 300k c/s that represents the case of traffic jam, statistics by conventional strategy illustrates that the detection rate, that is, the total time of successful target detection during one second, is 125 times/s (shown as dotted line). As for the adaptive strategy, the detection rate is 188 times/s (shown as dash-dotted line). The enhancement ratio of adaptive strategy over conventional strategy in terms of detection rate is represented by Δ1 = 150.4%.

 

Fig. 13 Comparison of two strategy in terms of target detection rate under dynamic crosstalk, (horizontal axis: the sequence number of events generated by exhaustive method).

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In the dynamic IOC scenario, events are generated by exhaustive method with IOC ranging from 10k c/s to 300k c/s, while each IOC scenario has same total duration of one second but independent time of appearance. In Fig. 13 the horizontal axis represents the sequence number of events, while the simulated detection rate of each event as shown by the solid line increases dramatically from 188 times/s to 322 times/s. The average value of 228 times/s corresponds to an average traffic condition as shown by the dotted line. Compared with the scenario of the static IOC through the adaptive strategy, the improvement factor of the case of dynamic IOC (averaged) is defined as Δ2 = 121.3%. Thus the total improving factor for the detection rate of adaptive strategy versus conventional strategy is Δ1Δ2 = 182.4%, calculated based on the detecting probability requirement of 95%.

Furthermore, the variation of average detection rate for different level of probability requirements on detection and false alarm is demonstrated in Fig. 14. It can be seen that the average detection rate rapidly increases while the upper limit of Pfa-u is loosened. Specifically, the average detection rate drops dramatically with more strict requirement on Pd with Pfa-u below 20%, while it holds constant against varying requirement on Pd with Pfa-u below 5%.

 

Fig. 14 Variation of average detection rate for different level of probability requirements on detection and false alarm.

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4. Conclusion

The adaptive strategy of CPPM scheme for extracting the target echo from complex received signal is performed theoretically and experimentally to verify the excellent anti-crosstalk capability. The adaptive strategy applies dynamic accumulation of photon counts and concurrent multiple thresholds to extract the target echo. In the adaptive algorithm, the number of accumulated shots is increased step by step, until the self-evaluation finally identifies the target echo. According to ROC-based simulation and experimental results, the adaptive strategy is proved to have much better performance against crosstalk from varying number of sources, with varying intensity of each source, compared with conventional strategy on static discrimination threshold and unified number of accumulated pulses. Meanwhile the adaptive strategy is experimentally verified to be more practical than the conventional strategy, as it never relies on ROC curve to configure TCSPC and costs less laser shots for the target acquisition, with the detection rate enhanced by 82.4%. A future work is concentrated on a system-on-board solution based on the proposed adaptive strategy.

Funding

Beijing Natural Science Foundation (4172030); Tsinghua University Initiative Scientific Research Program, China (Grant No. 2014z21035).

References and links

1. J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016). [CrossRef]  

2. R. H. Rasshofer, M. Spies, and H. Spies, “Influences of weather phenomena on automotive laser radar systems,” Adv. Radio Sci. 9(2), 49–60 (2011). [CrossRef]  

3. G. M. Brooker, “Mutual interference of millimeter-wave radar systems,” IEEE Trans. Electromagn. Compat. 49(1), 170–181 (2007). [CrossRef]  

4. M. Goppelt, H. L. Blöcher, and W. Menzel, “Analytical investigation of mutual interference between automotive FMCW radar sensors,” Proc. Eur. Microw. Conf., 1–4 (2011).

5. L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003). [CrossRef]  

6. F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009). [CrossRef]  

7. P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015). [CrossRef]  

8. J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016). [CrossRef]  

9. A. Hourani, R. J. Evans, S. Kandeepan, B. Moran, and H. Eltom, “Stochastic geometry methods for modelling automotive radar interference,” ArXiv 1607.02434 (2016).

10. T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015). [CrossRef]  

11. G. Kim, J. Eom, and Y. Park, “Investigation on the occurrence of mutual interference between pulsed terrestrial LIDAR scanners,” in Intelligent Vehicles Symposium (IV) (IEEE, 2000), pp. 437–442.

12. M. D. S. Cavalcanti, F. A. Mendonça, and R. V. Ramos, “Spectral method for characterization of avalanche photodiode working as single-photon detector,” Opt. Lett. 36(17), 3446–3448 (2011). [CrossRef]   [PubMed]  

13. J. S. Massa, A. M. Wallace, G. S. Buller, S. J. Fancey, and A. C. Walker, “Laser depth measurement based on time-correlated single-photon counting,” Opt. Lett. 22(8), 543–545 (1997). [CrossRef]   [PubMed]  

14. M. Ren, X. Gu, Y. Liang, W. Kong, E. Wu, G. Wu, and H. Zeng, “Laser ranging at 1550 nm with 1-GHz sine-wave gated InGaAs/InP APD single-photon detector,” Opt. Express 19(14), 13497–13502 (2011). [CrossRef]   [PubMed]  

15. A. B. Milstein, L. A. Jiang, J. X. Luu, E. L. Hines, and K. I. Schultz, “Acquisition algorithm for direct-detection ladars with Geiger-mode avalanche photodiodes,” Appl. Opt. 47(2), 296–311 (2008). [CrossRef]   [PubMed]  

16. H. Luo, X. Yuan, and Y. Zeng, “Range accuracy of photon heterodyne detection with laser pulse based on Geiger-mode APD,” Opt. Express 21(16), 18983–18993 (2013). [CrossRef]   [PubMed]  

17. Z. Zhang, J. Zhang, L. Wu, Y. Zhang, Y. Zhao, and J. Su, “Photon-counting chirped amplitude modulation lidar using a smart premixing method,” Opt. Lett. 38(21), 4389–4392 (2013). [CrossRef]   [PubMed]  

18. D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt. 42(27), 5388–5398 (2003). [CrossRef]   [PubMed]  

19. P. Gatt, S. Johnson, and T. Nichols, “Geiger-mode avalanche photodiode ladar receiver performance characteristics and detection statistics,” Appl. Opt. 48(17), 3261–3276 (2009). [CrossRef]   [PubMed]  

20. S. E. Johnson, “Target detection with randomized thresholds for lidar applications,” Appl. Opt. 51(18), 4139–4150 (2012). [CrossRef]   [PubMed]  

References

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  1. J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
    [Crossref]
  2. R. H. Rasshofer, M. Spies, and H. Spies, “Influences of weather phenomena on automotive laser radar systems,” Adv. Radio Sci. 9(2), 49–60 (2011).
    [Crossref]
  3. G. M. Brooker, “Mutual interference of millimeter-wave radar systems,” IEEE Trans. Electromagn. Compat. 49(1), 170–181 (2007).
    [Crossref]
  4. M. Goppelt, H. L. Blöcher, and W. Menzel, “Analytical investigation of mutual interference between automotive FMCW radar sensors,” Proc. Eur. Microw. Conf., 1–4 (2011).
  5. L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003).
    [Crossref]
  6. F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009).
    [Crossref]
  7. P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
    [Crossref]
  8. J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
    [Crossref]
  9. A. Hourani, R. J. Evans, S. Kandeepan, B. Moran, and H. Eltom, “Stochastic geometry methods for modelling automotive radar interference,” ArXiv 1607.02434 (2016).
  10. T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
    [Crossref]
  11. G. Kim, J. Eom, and Y. Park, “Investigation on the occurrence of mutual interference between pulsed terrestrial LIDAR scanners,” in Intelligent Vehicles Symposium (IV) (IEEE, 2000), pp. 437–442.
  12. M. D. S. Cavalcanti, F. A. Mendonça, and R. V. Ramos, “Spectral method for characterization of avalanche photodiode working as single-photon detector,” Opt. Lett. 36(17), 3446–3448 (2011).
    [Crossref] [PubMed]
  13. J. S. Massa, A. M. Wallace, G. S. Buller, S. J. Fancey, and A. C. Walker, “Laser depth measurement based on time-correlated single-photon counting,” Opt. Lett. 22(8), 543–545 (1997).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. A. B. Milstein, L. A. Jiang, J. X. Luu, E. L. Hines, and K. I. Schultz, “Acquisition algorithm for direct-detection ladars with Geiger-mode avalanche photodiodes,” Appl. Opt. 47(2), 296–311 (2008).
    [Crossref] [PubMed]
  16. H. Luo, X. Yuan, and Y. Zeng, “Range accuracy of photon heterodyne detection with laser pulse based on Geiger-mode APD,” Opt. Express 21(16), 18983–18993 (2013).
    [Crossref] [PubMed]
  17. Z. Zhang, J. Zhang, L. Wu, Y. Zhang, Y. Zhao, and J. Su, “Photon-counting chirped amplitude modulation lidar using a smart premixing method,” Opt. Lett. 38(21), 4389–4392 (2013).
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  18. D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt. 42(27), 5388–5398 (2003).
    [Crossref] [PubMed]
  19. P. Gatt, S. Johnson, and T. Nichols, “Geiger-mode avalanche photodiode ladar receiver performance characteristics and detection statistics,” Appl. Opt. 48(17), 3261–3276 (2009).
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2016 (1)

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

2015 (2)

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
[Crossref]

2013 (2)

2012 (1)

2011 (3)

2009 (2)

F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009).
[Crossref]

P. Gatt, S. Johnson, and T. Nichols, “Geiger-mode avalanche photodiode ladar receiver performance characteristics and detection statistics,” Appl. Opt. 48(17), 3261–3276 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

G. M. Brooker, “Mutual interference of millimeter-wave radar systems,” IEEE Trans. Electromagn. Compat. 49(1), 170–181 (2007).
[Crossref]

2003 (2)

L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003).
[Crossref]

D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt. 42(27), 5388–5398 (2003).
[Crossref] [PubMed]

1997 (1)

Alonge, F.

F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009).
[Crossref]

Appenrodt, N.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Blöcher, H. L.

M. Goppelt, H. L. Blöcher, and W. Menzel, “Analytical investigation of mutual interference between automotive FMCW radar sensors,” Proc. Eur. Microw. Conf., 1–4 (2011).

Bloecher, H.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Branciforte, M.

F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009).
[Crossref]

Brooker, G. M.

G. M. Brooker, “Mutual interference of millimeter-wave radar systems,” IEEE Trans. Electromagn. Compat. 49(1), 170–181 (2007).
[Crossref]

Buller, G. S.

Cavalcanti, M. D. S.

Dickmann, J.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Du, P.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
[Crossref]

Fancey, S. J.

Fortuna, L.

L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003).
[Crossref]

Fouche, D. G.

Frasca, M.

L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003).
[Crossref]

Fu, X.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

Gatt, P.

Geng, D.

P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
[Crossref]

Gong, M.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
[Crossref]

Goppelt, M.

M. Goppelt, H. L. Blöcher, and W. Menzel, “Analytical investigation of mutual interference between automotive FMCW radar sensors,” Proc. Eur. Microw. Conf., 1–4 (2011).

Gu, X.

Hahn, M.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Hao, J.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

Harter, M.

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

Hines, E. L.

Jiang, L. A.

Johnson, S.

Johnson, S. E.

Klappstein, J.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Kong, W.

Liang, Y.

Lu, B.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

Luo, H.

Luu, J. X.

Massa, J. S.

Mendonça, F. A.

Menzel, W.

M. Goppelt, H. L. Blöcher, and W. Menzel, “Analytical investigation of mutual interference between automotive FMCW radar sensors,” Proc. Eur. Microw. Conf., 1–4 (2011).

Milstein, A. B.

Motta, F.

F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009).
[Crossref]

Nichols, T.

Prophet, S.

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

Ramos, R. V.

Rasshofer, R. H.

R. H. Rasshofer, M. Spies, and H. Spies, “Influences of weather phenomena on automotive laser radar systems,” Adv. Radio Sci. 9(2), 49–60 (2011).
[Crossref]

Ren, M.

Rizzo, A.

L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003).
[Crossref]

Sailer, A.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Schipper, T.

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

Schultz, K. I.

Spies, H.

R. H. Rasshofer, M. Spies, and H. Spies, “Influences of weather phenomena on automotive laser radar systems,” Adv. Radio Sci. 9(2), 49–60 (2011).
[Crossref]

Spies, M.

R. H. Rasshofer, M. Spies, and H. Spies, “Influences of weather phenomena on automotive laser radar systems,” Adv. Radio Sci. 9(2), 49–60 (2011).
[Crossref]

Su, J.

Walker, A. C.

Wallace, A. M.

Wang, W.

P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
[Crossref]

Werber, K.

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

Wu, E.

Wu, G.

Wu, L.

Yuan, X.

Zeng, H.

Zeng, Y.

Zhang, F.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

Zhang, H.

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

Zhang, J.

Zhang, Y.

Zhang, Z.

Zhao, Y.

Zwick, T.

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

Zwirello, L.

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

Adv. Radio Sci. (1)

R. H. Rasshofer, M. Spies, and H. Spies, “Influences of weather phenomena on automotive laser radar systems,” Adv. Radio Sci. 9(2), 49–60 (2011).
[Crossref]

Appl. Opt. (4)

Chin. Phys. B (1)

J. Hao, M. Gong, P. Du, B. Lu, F. Zhang, H. Zhang, and X. Fu, “Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach,” Chin. Phys. B 25(7), 250–257 (2016).
[Crossref]

IEEE Trans. Electromagn. Compat. (2)

T. Schipper, S. Prophet, M. Harter, L. Zwirello, and T. Zwick, “Simulative prediction of the interference potential between radars in common road scenarios,” IEEE Trans. Electromagn. Compat. 57(3), 322–328 (2015).
[Crossref]

G. M. Brooker, “Mutual interference of millimeter-wave radar systems,” IEEE Trans. Electromagn. Compat. 49(1), 170–181 (2007).
[Crossref]

IEEE Trans. Instrum. Meas. (2)

L. Fortuna, M. Frasca, and A. Rizzo, “Chaotic pulse position modulation to improve the efficiency of sonar sensors,” IEEE Trans. Instrum. Meas. 52(6), 1809–1814 (2003).
[Crossref]

F. Alonge, M. Branciforte, and F. Motta, “A novel method of distance measurement based on pulse position modulation and synchronization of chaotic signals using ultrasonic radar systems,” IEEE Trans. Instrum. Meas. 58(2), 318–329 (2009).
[Crossref]

Opt. Eng. (1)

P. Du, D. Geng, W. Wang, and M. Gong, “Laser detection of remote targets applying chaotic pulse position modulation,” Opt. Eng. 54(11), 114102 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Other (4)

J. Dickmann, J. Klappstein, M. Hahn, N. Appenrodt, H. Bloecher, K. Werber, and A. Sailer, “Automotive radar the key technology for autonomous driving: From detection and ranging to environmental understanding,” Radar Conf., 1–6 (2016).
[Crossref]

M. Goppelt, H. L. Blöcher, and W. Menzel, “Analytical investigation of mutual interference between automotive FMCW radar sensors,” Proc. Eur. Microw. Conf., 1–4 (2011).

G. Kim, J. Eom, and Y. Park, “Investigation on the occurrence of mutual interference between pulsed terrestrial LIDAR scanners,” in Intelligent Vehicles Symposium (IV) (IEEE, 2000), pp. 437–442.

A. Hourani, R. J. Evans, S. Kandeepan, B. Moran, and H. Eltom, “Stochastic geometry methods for modelling automotive radar interference,” ArXiv 1607.02434 (2016).

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Figures (14)

Fig. 1
Fig. 1 Signal echo and two types of crosstalk among vehicles on the road
Fig. 2
Fig. 2 Received counts with increasing distance for target echo and two types of crosstalk.
Fig. 3
Fig. 3 ROC curves with the accumulation number from 300 to 800, the signal repetition rate of 100 kHz, and the crosstalk repetition rate of 10k c/s.
Fig. 4
Fig. 4 ROC curves under different crosstalk with 400 pulses accumulated.
Fig. 5
Fig. 5 Flow chart of adaptive strategy.
Fig. 6
Fig. 6 The detection probability and probability of false alarm by the adaptive strategy, for different accumulative numbers and different frames.
Fig. 7
Fig. 7 Comparison of two strategy in terms of Uopt under static crosstalk.
Fig. 8
Fig. 8 Experimental setup.
Fig. 9
Fig. 9 Different scenarios realized by switched working status of five FPGA boards.
Fig. 10
Fig. 10 Detection results with IOC of 10k, 100k, 200k and 300k c/s by the conventional strategy.
Fig. 11
Fig. 11 Probability of false alarm under dynamic crosstalk counts.
Fig. 12
Fig. 12 Detection results by adaptive strategy with different accumulated pulse number and varying crosstalk, in which the horizontal axis represents the sequence number of storage bins while the circle and square marks represent the location of target and false alarm respectively.
Fig. 13
Fig. 13 Comparison of two strategy in terms of target detection rate under dynamic crosstalk, (horizontal axis: the sequence number of events generated by exhaustive method).
Fig. 14
Fig. 14 Variation of average detection rate for different level of probability requirements on detection and false alarm.

Equations (19)

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N echo = P A f A λ A hc ( FOV θ T ) 2 ρ π A R L A 2 η R T 2
N c_b = P C f C λ C hc 1 π A R θ T 2 L C 2 η R T 2
N c_r = P D f D λ D hc ( FOV θ T ) 2 ρ π A R θ T 2 L A 2 cos θ R η R T 2
P=1exp( N photon η ).
p c =1exp[( N c_b + N c_r +n)η].
p s =1exp[( N echo + N c_b + N c_r +n)η].
P dect = i=Th U C U i p s i ( 1 p s ) Ui
P fa = i=Th U C U i p c i ( 1 p c ) Ui
P fau =1 (1 P fa ) N bin
X~[ 0 C u 0 p s 0 ( 1 p s ) u0 1 C u 1 p s 1 ( 1 p s ) u1 n C u n p s n ( 1 p s ) un ]
C n k p k ( 1p ) nk λ k k! e λ
P( x=k )= P Poisson ( Nu p s =k )
p dect ( N s )= ( i=0 N s P poisson ( Nu p c =i ) ) N bin
P dectadp1 = i=1 Nu [ P ( Nu p s =i ) p dect ( i ) ]
P dectadp2 = i=0 u j=1 Nu [ k=0 j q=0 min( i+jk,u ) P( u p c =q ) P( Nu p c =k ) ] N bin P( Nu p s =j )P( u p s =i )
P dectadp3 = x=0 u i=0 u j=1 Nu [ k=0 j q=0 min( i+jk,u ) y=0 min( x+i+jkq,u ) P( u p c =y ) P( u p c =q )P( Nu p c =k ) ] N bin P( Nu p s =j )P( u p s =i )P( u p s =x )
P faadp1 = i=0 Nu [ P ( Nu p s =i )( 1 ( 1 k=max( 1,i ) Nu P( Nu p c =k ) ) Nbin ) ]
P faadp2 = i=0 u j=0 Nu { 1 [ 1 k=max( 1,j ) Nu q=max( 0,i+jk ) u P( u p c =q ) P( Nu p c =k ) ] N bin } P( Nu p s =j )P( u p s =i )
P faadp3 = x=0 u i=0 u j=0 Nu { 1 [ 1 k=max( 1,j ) Nu q=max( 0,i+jk ) u y=max( 0,x+i+jkq ) u P( u p c =y ) P( u p c =q )P( Nu p c =k ) ] N bin } P( Nu p s =j )P( u p s =i )P( u p s =x )

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