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Utilizing an empirical path loss model proposed in the first paper of a two-part series, the bit error rate performance of short-range non-line-of-sight ultraviolet communication receivers is analyzed. Typical photodetector models and modulation formats are considered. Our results provide semi-analytical prediction of the achievable communication performance as a function of system and channel parameters, and serve as a basis for system design.
©2010 Optical Society of America
1. Introduction
Ultraviolet (UV) technology is a candidate for wireless optical communications (WOC) due to the unique features of non-line-of-sight (NLOS) connectivity and range-dependent attenuation. Aided by recent device technologies in deep UV light emitting diodes [1,2] and avalanche photodiodes (APDs) [3,4], short range NLOS UV communications has significant potential [5].
We consider a NLOS communication system using LED sources in the deep UV, or UV-C, that features a very low noise condition at ground level outdoors due to the absorption of solar radiation by the upper ozone layer in the atmosphere. Receiver performance strongly depends on the detector characteristics, as well as the received signal power as a function of the channel path loss. For NLOS UV communication the apex angles of the transmitter beam and receiver field of view may vary, complicating the development of a general path loss model. Models based on a single scattering assumption, and approximations based on large apex angles, have been developed [6–9]. Recently a Monte Carlo technique was suggested to simulate the multiple scattering effects on path loss [10]. Also an empirical path loss model for communication range up to a few hundred meters, validated by field experiments, was proposed in the first paper of this two-part series [11]. Based on extensive measurements, the path loss exponent and attenuation factor for different link geometries have been tabulated. The model was used to study bit error rate (BER) as a function of system parameters, assuming on-off keying (OOK) modulation and quantum limited direct detection. The empirical path loss model [11] was also compared against the Monte Carlo model in [10].
In this paper, we extend the performance analysis presented in [11]. Both photomultiplier tube (PMT) and avalanche photodiode (APD) based receiver structures are considered. The device and circuit random characteristics are incorporated along with the channel path loss model from [11]. To start with, the system architecture and UV NLOS channel path loss model are described in Section 2. With the ideal photon counting receiver, performance results for intensity modulation and direct detection (IM/DD) using OOK and pulse position modulation (PPM) are summarized, and tradeoffs between communication range and data rate are studied to assist the design process, both in Section 3. Following up, we analyze the performance of the practical receivers built upon a PMT or APD to handle weak signal detection. The thermal noise in the electric circuit and the shot noise due to the background radiation add elements to the theoretical analysis. From our analysis, we predict the interactions among different system and geometric parameters in determining the BER performance. Our analytical results and techniques are generally applicable to other WOC systems given a corresponding path loss model.
2. System architecture and NLOS UV channel path loss model
The NLOS UV communication system is depicted in Fig. 1 . The transmitter consists of a modulated UV LED array, with pointing optics. The receiver employs either a PMT or APD aided by a solar-blind optical filter. The post-processing circuit involves integration and detection processing. The NLOS UV channel configuration is uniquely specified by a set of parameters noted in the figure. θ1 and θ2 are the transmitter (Tx) and receiver (Rx) apex angles, and ϕ1 and ϕ2 are the Tx full beam divergence angle and Rx field of view (FOV), respectively. For the purposes of this paper, ϕ1 and ϕ2 are fixed to be 10° and 30°, respectively [11]. The Tx-Rx baseline separation is r. V is referred to as the optical common volume, and r1 and r2 are the distances of the common volume to the Tx and Rx, respectively.In order to characterize the system performance, we utilize the channel model to predict received optical power, and incorporate stochastic models for the detector and noise sources. We ignore atmospheric turbulence effects, which are generally negligible for ranges on the order of a few hundred meters or less. We model the photodetector output as a Poisson point process with the counting rate λ determined by the instantaneous received optical power such that λS = ηPr/(hν) = ηPt/(Lhν). η, Pr, h, ν, Pt, and L denote the quantum efficiency of the detector including optical filter and photodetector, received power, Planck’s constant, the frequency of the optical field, transmitted power and path loss, respectively. In particular, path loss L can be used to predict the received power given the transmission power.
Experimental results on short-range (up to a few hundred meters) NLOS UV channel path loss for different pointing angles are reported in [12]. We employ an empirical path loss model given by Eq. (1) [11]
where path loss exponent α and path loss factor ξ are unknown non-linear functions of the apex angles (cf. Figure 1). These parameters are summarized in Tables 1 and 2 below.
Table 1. Path loss factor ξ of the NLOS UV channel model
Table 2. Path loss exponent α of the NLOS UV channel model
The random effects of background radiation, shot noise due to the photon detection process (that differs depending on the type of detector), and thermal noise of the post-processing electric circuit, must all be accounted for in the analysis of system performance. Including both shot noise and thermal noise, the photodetector output statistics are governed by a mixture of a Poisson point process and a Gaussian process. In the following we develop these models and present a unified performance analysis for short range UV-C NLOS communications.
3. Ideal photon counting receiver performance
We first examine the performance of NLOS UV communications within the framework of an ideal Poisson counting receiver that does not require photon multiplication. The integrator output is proportional to the photon count k1 per pulse time out of each modulation symbol period, which also complies with a Poisson distribution with photon arrival rate λ. The probability is given by Eq. (2)
where λ = λS + λb when the signal is present and λ = λb when signal is absent, and λb denotes the background radiation photon count rate. Although the ozone layer in the upper atmosphere absorbs most of the radiation in the deep UV, there remains some background radiation, some percentage of which will leak through the optical filter.From the previous section, it is clear that λS depends on the path loss L which in turn is a function of the system configuration parameters. Assuming threshold-based direct detection with OOK modulation, the symbol error probability is given by Eq. (3) [13]
The optimal detection threshold can be obtained as mT = floor(λS/ln(1 + λS/λb)) by minimizing Pe_OOK. For the quantum-limited case when λb = 0, Eq. (3) boils down to Pe-OOK = exp(-λS)/2. After simple manipulations by substitution of λS, the achievable range r and the uncoded bit rate Rb can be related as follows, yielding the range-rate tradeoff as in Eq. (4) [14]We next extend this to the PPM case. M-PPM is an orthogonal modulation which transmits a pulse in one of M slot positions. With narrower pulse duration time Tp and the number of noise photons per second Nn depending on the operating conditions, it is desirable to minimize the noise photon count λb = NnTp for improved signal detection performance as long as the peak power constraint of the light source is met. More specifically, the bit error probability is given by Eq. (5) [15]
Nn typically varies over zero to 14.5 kHz from night time to noon [12], so we may reduce Tp by increasing the order of the PPM modulation to achieve a very small value of λb. As λb approaches zero, Eq. (5) then evolves into Pe-PPM = exp(-λS)/2, which appears the same as the limiting case for OOK. However, the expression of λs for M-PPM becomes λs = (ηPt log2M)/(LRb hυ). Therefore, for a given bit error probability requirement or equivalently the same λs, M-PPM can achieve a data rate of log2M times that of OOK. In Section 5 we use Eqs. (3) and (5) to illustrate the range-rate performance tradeoffs.4. Performance of PMT/APD based receivers with thermal noise
The NLOS UV propagation channel incurs severe attenuation. Therefore, it is interesting to investigate a receiver distorted by the random photon amplification process and thermal noise in the post-processing circuit.
4.1 Statistical modeling of the photodetector output
The photomultiplication process is subject to random variation in the output number of secondary photoelectrons k2 in response to the primary photoelectrons k1. The number of secondary photoelectrons is described by the probability mass function in Eq. (6) [13]
where Pk1(k1) is the primary photoelectron probability described by Eq. (2), and Pk2(k2|k1) is the conditional probability of the secondary photoelectrons given k1. There has been a rich amount of literature since the 1960s on modeling Pk2(k2) for PMT and APD photodetectors. For brevity, we provide only the main results here.The PMT’s output secondary photoelectrons have the probability Pk2(k2) given by Eq. (7) [13]
In the equation, C is the normalizing factor, and ζ is the PMT spreading factor that describes the variance of PMT gain A.
For an APD, McIntyre and Conradi have shown that Pk2(k2) follows Eq. (8) [16,17]
whereand F is the excess noise factor jointly determined by the ionization factor γ and gain A. For PMT and APD cases, F is given by 1 + ζ2 and γA + [2-(1/A)](1-γ), respectively. With the typical values of ζ = 0.1 and γ = 0.028, F takes the values 1.01 and 4.7343 [13] for a PMT and APD (with gain 100), respectively.Next we consider additive thermal noise. The decision variable for detection of ν from z takes the form in Eq. (9)
where n is the thermal noise normal random variable and v is the output current of the detector. The noise realization can usually be described by a zero mean Gaussian random variable with variance given by Eq. (10) [18] where ke, To and RL denote the Boltzmann constant, the receiver temperature (Kelvin), and the load resistance. Tp is the pulse interval for pulse-based modulation such as OOK and PPM.Coupled with the output of the photodetector, the number of secondary photoelectrons k2 shifts the pdf of v by changing the mean of n to the form of Eq. (11)
where k2 is a random variable, leading to a random mean. Conditioned on the average count of primary photoelectrons per pulse, denoted as λ, the conditional density of z, pz(z|λ), is the average of the continuous Gaussian distribution over the secondary photoelectron probability given by Eq. (12) [13]In the above, G(z,a,b) denotes the pdf of the Gaussian random variable z with mean a and variance b. Pk2(j) is given in Eq. (7) for a PMT, and in Eq. (8) for an APD.However, note that the application of Eq. (11) to obtain a numerical solution may call for a large quantity of computations in any analysis utilizing the distribution (even with truncating the summation at the higher end). For example, the photoelectron output of a high gain PMT can be on the order of 105~108, such that a probability accumulation will have a corresponding high computational complexity. So, as an alternative, we directly model the random gain effect with the thermal noise, as proposed in [15].
Consider the realizations of gain {Ai} to be independent and identically distributed (i.i.d.) random variables representing the multiplication gain for each photoelectron. Then z can be written as Eq. (13)
Approximating the random gain as a Gaussian random variable, z is the summation of k1 Gaussian random variables each with mean Ae and variance (F-1)(Ae)2, plus noise n with zero mean and variance as given in Eq. (10). Conditioned on the primary photon arrival rate which is typically moderate, z is essentially a summation of Gaussian random variables and pz(z|λ) is thus found to follow Eq. (14) In this equation, Pk1(j) assumes the form of Eq. (2), and σ 2 involves the contributions from thermal noise and the random gain effect of the photodetector, which isThe mean is replaced by its expected value, and the conditional probability is conditioned on the photon arrival instead of the secondary photoelectrons. By substituting the expressions of excess noise factor F for PMT or APD, σ 2 takes the following forms in Eq. (15), respectively
4.2 Receiver performance
Aided by the statistics and the path loss model from the previous sections, we are now ready to evaluate the error performance, rates, and ranges that can be achieved by a short range NLOS UV communication system.
For OOK modulation, the receiver adopts threshold-based direct detection, and a general expression of error probability follows Eq. (16)
where P1 and P0 are probabilities of transmitting information “1” and “0”, respectively. Assume P1 = P0 = 0.5. The optimal detection threshold zth to minimize Pe_OOK can be found by setting the derivative to zero as dPe_OOK/dzth = 0, leading to Eq. (17) Given the pdf of z in Eq. (14), an equation for the optimal zth will be a mixture of summation and integral. Since a closed form for the optimal zth is not attainable, a numerical solution will be used. Using Eq. (14) for pz(z) and applying the following identitythe error probability in Eq. (16) can be written in a compact form as Eq. (18)M-ary pulse position modulation (M-PPM) is widely employed as a pulse-based modulation option, where the receiver is based on the maximum likelihood method by comparing the outputs of different slot positions without a need for the threshold. The probability of correct detection for M-PPM is given by Eq. (19)
which is the probability that the slot containing the signal arrival has the most significant value exceeding all other slots within each PPM word. This results in the error probability Pe_PPM = 0.5M/(M-1)(1-PD). Combining our results, the correct detection probability can be expressed as Eq. (20) [15]where erf (x) is the error function defined asThese analytical results will be evaluated numerically to demonstrate NLOS UV communication receiver performance.5. Numerical analysis
By varying the pointing angles from small to large, the NLOS UV channel path loss traverses a huge dynamic range of up to 100dB variation. This necessitates a receiver capable of handling the weak optical signal while minimizing the random noise effects. We propose to use PMT/APD in the deep UV range along with a high impedance amplifier, desiring to maintain low noise operation. The NLOS UV channel is primarily path loss limited, although pulse-broadening occurs that induces a channel bandwidth limitation. Here we focus on rates that are sufficiently low to generally avoid the question of bandwidth limits and induced inter-symbol interference (ISI). Also, note that an equalizer can be introduced to improve the bandwidth of a high impedance amplifier [19] even in the absence of channel induced ISI. Our numerical examples use the typical system values listed in Table 3 .
Table 3. Typical UV communication system parameters
5.1 Rate-range tradeoffs with a photon counting receiver
We begin with an analysis of the rate and range tradeoffs with an ideal photon counting receiver as discussed in Section 3. The path loss is dependent on the geometric setup and baseline range. We adopt three background noise levels considered to be low, medium, and high noise cases, corresponding to 0, 5kHz, 14.5kHz. These levels were selected based on measurements [12].
The performance in the NLOS UV channel is exemplified in Fig. 2 , where results for OOK and 4-PPM are plotted with the high noise level. The inherent tradeoffs between rate and range embedded in Eqs. (3) and (5) have been numerically calculated and the achievable data rate found by numerical search with respect to the range for a given uncoded BER target of 10−3. We see that small pointing angles provide substantially higher data rates, with generally more than 1 Mbps improvement at 100 meters. At large pointing angles, rates of tens of kbps are indicated. This illustrates that while strict pointing is not needed at high noise levels, some knowledge of directionality enhances the communications via lowering the pointing angles and thus the path loss. When the noise condition significantly improves for solar blind operation such that λb = 0, both OOK and 4-PPM performance improves as shown in Fig. 3 . 4-PPM yields twice the data rate of OOK for a given range when λb = 0, as predicted earlier. However, we observe that the data rate performance is more sensitive to noise at large pointing angles by noticing that the data rate increases significantly as λb reduces. For example, at 100m and (40°,40°) pointing, the rate under OOK almost doubles, and 4-PPM has about a 1.5 times data rate increase. This is because the path loss is high and communication performance is then more sensitive to noise, whereas for small angles the path loss is reduced and the data rate shows less sensitivity to noise.
Fig. 2 Achievable rate vs. baseline range for OOK (left) and 4-PPM (right) modulations, high noise case.
Fig. 3 Achievable rate vs. baseline range for OOK (left) and PPM (right) modulations, and no background noise.
Figure 4 depicts a BER versus data rate comparison of OOK versus M-PPM for M = (4,8,16). We show small and large angle cases for a fixed range. With moderate background noise, M-PPM outperforms OOK. Again, performance is very sensitive to pointing due to significant path loss variation. With a small angle geometry, for example (20°, 20°), a data transfer rate up to 2Mbps is possible with an error rate of 10−3 using 16-PPM, while a (40°, 40°) link barely allows for a rate of 50kbps.
5.2 PMT/APD receiver performance with fixed gain
Next we consider BER versus range, with background and thermal noise in Figs. 5 and 6 , according to Eqs. (18) and (20). We select several typical data rates with fixed PMT/APD gain and transmit power. Given the significant path loss at larger pointing angles, a high gain photodetector becomes necessary. For these examples we set the PMT and APD gains to 104 and 250, respectively, and we assume the APD and PMT have the same detection area and quantum efficiency.
Fig. 5 Error probability vs. baseline range for OOK modulation, Pt = 100mW, PMT (left) and APD (right).
Fig. 6 Error probability vs. baseline range for 4-PPM modulation, Pt = 100mW, PMT (left) and APD (right).
We first compare the PMT and APD receivers assuming threshold based detection with OOK modulation in Fig. 5. The high gain PMT receiver excels at achieving higher data rate and/or better error performance with all four pointing geometries. The APD receiver performs much worse than the PMT, with only the (10°, 10°) case barely reaching the 10−3 error probability at 70 meters. We comment that the relatively poorer performance is due to the excess noise with the APD gain that raises the error floor, as well as the limited APD multiplication capability. We address this further in the next sub-section in terms of gain optimization.
Next, in Fig. 6, we consider the more power efficient modulation 4-PPM. For the data rates considered at a range of 60 meters the error performance is typically well below 10−3 for most geometries. Comparing with Fig. 5, the data rate increase with 4-PPM versus OOK is evident for both detector types. A cross-comparison between PMT and APD in both Figs. 5 and 6 also validates the PMT as an effective solution to provide a viable communication link with large pointing angles.
5.3 Improving the performance with detector gain control
Our analysis so far has assumed a fixed detector gain. However, because both the PMT and APD amplification gains can be controlled through the bias voltage, we next take a close look at the gain effect on performance. Intuitively, because of the non-ideal nature of the amplification process, excess noise is dependent on signal level and gain variance. A simple increase of gain may not be a panacea in all circumstances.
Figure 7 depicts BER against PMT and APD detector gains, at a range of 50 meters. For the PMT, the error probability improves monotonically with increasing gain, but becomes saturated when the gain exceeds about 104. The extent of BER improvement differs with pointing geometry, with the most gain occurring for the smaller pointing angles. For very large gain, the error performance is significantly better at small angles. For the APD, the BER first decreases with increasing gain, but then increases beyond a certain gain point. The variation is more pronounced for smaller pointing angles, and an optimal gain exists in each case. The optimal gain point tends to shift to a larger value as the pointing angles increase. The optimal gain is around 150 for (10°,10°), and approaches 300 for the (40°,40°) pointing case. Overall the PMT shows much better detection performance than the APD. A high gain PMT based receiver is practically feasible for the cases considered. For the APD based receiver, excess noise from random gain fluctuation significantly degrades performance, and an automatic gain control mechanism can be incorporated to adjust the APD bias voltage.
6. Conclusions and future work
We evaluated PMT and APD based receiver performance for short range NLOS UV communications both theoretically and numerically. The tradeoffs of data rate, range and error performance were characterized by utilizing an empirical path loss model for the short-range NLOS UV channel based on extensive outdoor measurements. The separate and combined effects of thermal and background noise were shown. A study of PMT and APD gain reveals good operating points, and indicates the benefit from gain control, especially in an APD based system. The results and conclusions form a basis for system design and link analysis. The general techniques may also be applicable to other wireless optical communications systems when a channel path loss model is available.
Further open work in this area includes introduction of more system parameters, including beam angle and field of view into the channel model, offering additional degrees of freedom in the analysis. Pulse broadening due to the NLOS propagation channel may warrant channel equalization to maximize the available rate. Random scattering and possible turbulence need to be experimentally characterized and incorporated into future analytical study, especially when the communication range is extended using a high power source and/or when large pointing angles are adopted. Hence channel model extension from short to medium range to better capture the above mentioned effects is also anticipated. The link level study is closely related to Poisson communication theory using an IM/DD abstraction and consequently new performance limits may be unveiled.
Acknowledgements
The authors extend thanks to the anonymous reviewers for their valuable suggestions and comments. This work was supported in part by the United States Army Research Office (USARO) under grants W911NF-09-1-0293 and W911NF-08-1-0163, and the United States Army Research Laboratory (USARL) under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011.
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