Abstract

An analytical model is presented firstly in this paper to formulate the link bandwidth of non-line-of-sight (NLOS) ultraviolet (UV) channel. The link bandwidth is characterized by three geometrical parameters including transmitter (Tx) elevation angle, receiver (Rx) field of view (FOV), and transceiver separation distance, and further expressed as a closed-form through software-aided numerical fitting. Comparison with the link bandwidth obtained via a Monte Carlo model is done to verify the feasibility of this model. Based on this model, we investigate the diversity reception on the NLOS UV communication from a new perspective. A spatially squared distributed Rx array is customized for the NLOS UV channel. Lower temporal broadening is enabled, leading to a higher link bandwidth. Numerical results suggest that over 100% improvement of the link bandwidth is predicted by the square array reception and the ratio grows rapidly with the narrowing of Tx beam divergence. Therefore, this paper provides a guide for link analysis and receiver design for NLOS UV communication.

© 2017 Optical Society of America

1. Introduction

Scattered optical wireless link, solar-blind (SB) radiation feature and recent progress in low cost semiconductor devices make short-range non-line-of-sight (NLOS) ultraviolet (UV) communication attractive [1–3]. During the propagation in atmospheric medium, an intensity modulated UV optical pulse signal can be defined by two dimensions: the amplitude domain and the frequency domain [4,5]. Accordingly, two affects are resulted in these two domains of UV optical links due to the photons’ random trajectories: the atmospheric absorption induced path loss and the temporal broadening induced bandwidth deficiency [3,6].

In terms of path loss, an analytical model is firstly derived considering single scattered geometry in [7]. The single scattered model is then extended to noncoplanar geometry with generalized closed-form [8–10] and supported by empirical outdoor test-bed [11]. On the other hand, temporal impulse response of NLOS UV channel is experimentally measured in [5] and theoretically examined via Monte Carlo based statistical simulation [12,13]. In [12], impulse response is approximated by Gamma function for given geometry parameters and the link bandwidth is characterized by 3dB bandwidth obtained through Fourier transform. In [14], impulse response is further simplified to a closed-form under narrow Tx beam assumption and verified with the Gamma functional fitting. However despite the previous studies on temporal response, a generalized analytical model of the link bandwidth applicable for universal geometry similar to the path loss is still absent. Besides, a profound comprehension of the link bandwidth has significance to the tradeoff between data rate and range as well as potential spatial multiplexing for NLOS UV communication [15–17].

In this work, an analytical bandwidth model of NLOS UV links is developed by software aided numerical fitting of the 3dB bandwidth by time domain impulse response width. This model is testified with the theoretical results by fast Fourier transform (FFT) on the impulse responses obtained via Monte Carlo method. Meanwhile, we discover that the non-central chi-square functional fitting is more accurate for the impulse response curve than the Gamma functional fitting.

Moreover, many attentions have been paid to spatial diversity e.g. imaging receiving, multiple output reception in NLOS UV systems and the detector noise suppression is mainly focused on [18,19]. Based on the presented bandwidth model, we propose a square array reception to investigate the spatial diversity from a new perspective. For NLOS UV systems, wide FOV brings high optical power gains, but the link bandwidth is also restricted. Through this receiver design, the FOV in each Rx of the receiver can be reduced significantly without losing signal power. As a result, lower temporal broadening is generated leading to higher link bandwidth. Numerical simulation is done to prove this scheme and results suggest that over 100% improvement of the link bandwidth is possible.

The organization of this paper is as follows. The analytical link bandwidth model is described by tractable expressions in Section II. Section III provides details of the square array reception along with performance analysis considering different system parameters. Some conclusions are drawn in Section IV.

2. Analytical link bandwidth model

In this section, we use the 3dB bandwidth Bc to estimate the bandwidth of NLOS UV links [12]. Due to the short range of the NLOS UV communication link, single scattered channel model is adopted in this paper [9]. Therefore, the system geometry and parameters can be depicted by Fig. 1 where the LED beam divergence βT is initialized as 17° [12] and the receiver elevation angle θR is fixed at 90° to make omnidirectional receiving [2]. Three cases are classified to numerically model Bc, where the impacts of three parameters are considered including the Tx elevation angle θT, Rx FOV βR and transceiver distance r. However, it is difficult for us to define an object by a closed-form expression mathematically with three degrees of freedom. From [5], we know the reciprocal relation between the link bandwidth and channel response temporal width. In [5], four types of the channel temporal width are given including the root mean square (rms) width, 3-dB equivalent width, full width half FWHM, and 5% width. However, these four types of width are all estimated (3-dB width, FWHM, 5% width) or calculated statistically (rms width) via the measured channel impulse responses. Therefore, in order to analytically develop a closed-form expression of the link bandwidth, the time domain impulse response Td is used in our model as it can be obtained by tmax-tmin. The reciprocal relation between the link bandwidth and time domain impulse response width also convinces this approach [5]. Hence we decompose this problem into two major steps: 1) construct the connection between time domain impulse response width Td with θT, βR, r. 2) do numerical fitting of Bc by Td.

 figure: Fig. 1

Fig. 1 Schematic drawing of the short-range NLOS UV communication system geometry. Rx (PMT) is placed at distance r from the Tx (LED). Each photon escapes from the LED Tx to the PMT Rx across a random scattered path. The scattered propagation path is marked by the green dashed line for an example. Three geometry parameters: Tx/Rx distance r, Tx elevation angle θT, Rx FOV βR are focused on in the model as shown by the three cases, where θT, βR and r’ denote the corresponding changes in these three parameters. The red solid lines denote the scattered path of the first photon and last photon arriving at the Rx by time tmin and tmsx. The time domain impulse response width is defined by Td = tmax-tmin.

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2.1 Step 1): closed-form of Td with θT, βR, r

By common communication theory, Bc is negatively correlated to the temporal broadening written as Td = tmax- tmin for NLOS UV channels as shown in Fig. 1

Bc1/(tmaxtmin),
where tmin and tmax denote the arriving time of the first photon and the last photon counting at Rx. Define terms dmin and dmax in Fig. 1, tmin and tmax are expressed as follows
tmin=[dminsin(θTβT/2)+dmincos(βR/2)]/c,
tmax=[dmaxsin(θT+βT/2)+dmaxcos(βR/2)]/c.
where c is the light speed, dmin = r/[cot(θT-βT/2) + tan(βR/2)], dmax = r/[cot(θT + βT/2)-tan(βR/2)]. Then Eq. (2) and (3) can be furthered reformulated as

tmax=rc[1cos(θT+βT/2)tan(βR/2)sin(θT+βT/2)+1cot(θT+βT/2)cos(βR/2)sin(βR/2)],
tmin=rc[1cos(θTβT/2)+tan(βR/2)sin(θTβT/2)+1cot(θTβT/2)cos(βR/2)+sin(βR/2)].

Thereby, Td is given below

Td=rc[F1(θT,βR)+F2(θT,βR)],
where F1(θT,βR) and F2(θT,βR) are arranged by symmetrical form

F1(θT,βR)=1cos(θT+βT/2)tan(βR/2)sin(θT+βT/2)1cos(θTβT/2)+tan(βR/2)sin(θTβT/2),
F2(θT,βR)=1cot(θT+βT/2)cos(βR/2)sin(βR/2)1cot(θTβT/2)cos(βR/2)+sin(βR/2).

Hence, Td is finally integrated as a composite analytical expression by substituting Eq. (7) and (8) into Eq. (6). As shown by Eq. (6), since UV photons propagate across the common volume to the Rx by random scattered multipath, Td is uniquely determined by θT, βR and r. Thus the abovementioned three cases can also be characterized by three limitations: L(θT), L(βR), L(r). Particularly, if narrow Tx beam is assumed [10] where βT<<θT, the above Eq. (7) and (8) can be simplified as follows by trigonometric functional transformation

F1(θT,βR)=2tanβR2sinθTcos2θTtan2βR2sin2θT,
F2(θT,βR)=2sinβR2cos2θTcos2βR2sin2βR2.

2.2 Step 2): software aided numerical fitting of Bc by Td

In this step, Bc is numerically fitted by the above Td. Among the derivation of Bc, we employ the Monte Carlo method to simulate the impulse response h(t) of NLOS UV channel. Details of the Monte Carlo method can be found in our previous works [3] and [13]. Single scattered is approximated in the short-range scenario [7]. FFT is adopted on h(t) to generate simple representation for the frequency response and Bc.

In [12], the impulse response curve obtained via Monte Carlo method is numerically fitted by Gamma distribution function as follows

f(t,β,α)=P0βαΓ(α)tα1eβx,t>0,
where α and β can be estimated by the statistical feature of arriving time t of the photons, P0 is the normalized scale factor. According to the Gamma distribution function, the mean E(t) and variance Var(t) of t can be written as E(t) = α/β, Var(t) = α/β2 [20]. Thus β = E(t)/Var(t), α = βE(t). Given the impulse response h(t) = ∑Hiσ(t-tmin-i*τ), where i = 1…L, L is the number of resolved multipath, τ is the time delay unit, Hi is the amplitude coefficient of each resolved path, E(t) and Var(t) are expressed as

E(t)=1Hiτ[Hiτ(tmin+iτ)],
Var(t)=1Hiτ[Hiτ(tmin+iτE(t))2].

However, from the simulated results which will be given in the next section, we observe that the curve of the impulse response is non-central symmetric, whereas the Gamma function is central symmetric. Thus, we use the non-central chi-square distribution function instead to approximate the impulse response curves [20], as its numerical pattern is closer to the impulse response curve. The non-central chi-square distribution function is given below

p(t)=12e(t+λ)/2(xλ)k/41/2Ik/21(λt),
where k is the degrees of freedom, λ is the non-central parameter. Based on the general non-central chi-square distribution function, we can further compress the range of t from T0 where p(T0)~0 to Td and move the origin of t from 0 to tmin to make the function curve consistent with the actual impulse response. Examples of the impulse responses and comparison with the Gamma function/non-central chi-square function will be demonstrated in section 2.3 to prove this modification.

Next, using the post-processed impulse responses, Bc is obtained through FFT where the maximum frequency position at 1.67 × 108Hz (1/6ns) assuming the response time of the PMT is 6ns [21] and the frequency resolution is 1/Td. As mentioned above, Bc is negatively related to Td. In order to precisely estimate the functional connection between Bc and Td, we take advantage of the cftool toolbox of MATLAB to execute the curve fitting. During the testament, the data set of (Bc,Td) under various βR and θT is chosen as the input. Power function fitting is output to coincide with the simulated results. Therefore, Bc is expressed as follows

Bc=aTdb,
where a and b depends on the specific (βR,θT). The optimal values of a and b for different (βR,θT) are produced through numerous curve fittings with different r and listed in Fig. 2. Although Fig. 2 merely lists the values of a and b when βR = 0~45 o, θT = 30°, 45° and 60°, a and b can be obtained by the abovementioned means for every geometry parameter.

 figure: Fig. 2

Fig. 2 Lists of coefficient a and b in the expression of Bc = aTd-b. θT is selected at 30°, 45°, 60°. Different ranges of βR [0°~5°], [5°~10°], [15°~25°], [25°~35°], [35°~45°] are measured.

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Consequently, substituting Eq. (6) into Eq. (15), an analytical model of Bc is presented as

Bc=a[rc(F1(θT,βR)+F2(θT,βR))]b,
where F1(∙) and F2(∙) are calculated respectively by Eq. (7) and Eq. (8). Mean square error ratio (MSER) defined by Eq. (17) will be used to evaluate the accuracy of this fitting operation in the next section.
MSER=1num(Bc)[m(Bc)BcBc]2,
where m(Bc) means Bc by the analytical model, num(Bc) is the number of Bc.

2.3 Numerical results and validation

Firstly, in this section, we use the Monte Carlo method to simulate the impulse responses of NLOS UV channel and compare them with the fitted approximate functional curves by Gamma function and non-central chi-square distribution. The atmospheric and geometric parameters are listed in Fig. 3. Examples of the impulse responses are plotted in Fig. 4 to testify the abovementioned functional approximation by non-central chi-square distribution. In our simulation, θT = [30°, 45°], βR = [30°, 45°], r = 100m.

 figure: Fig. 3

Fig. 3 The atmospheric and geometric parameters used in Monte Carlo method

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 figure: Fig. 4

Fig. 4 Demonstration of the simulated impulse responses and comparison with the results by numerical fitting. In our simulations, θT is 30° and 45°, βR is 30° and 45°, r = 100m. Gamma, chi-square and MC denote the Gamma functional fitting, non-central chi-square distribution functional fitting and Monte Carlo method.

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In the simulations, α and β of the Gamma distribution function are calculated by the method described in section 2.2. In the non-central distribution functional fitting, to approach optimal fitting, k ranges at [4–6] and λ = 1. In Fig. 4, k = 6 when βR = 30°, θT = 30°, k = 5 when βR = 45°, θT = 30°. k = 4 when βR = 45°, θT = 45°. The results prove that the non-central chi-square distribution functional fitting outperforms the Gamma distribution functional fitting. When θT = 30°, βR = 30° or 45°, the results by non-central chi-square fitting match the impulse response curve better where the tail of impulse response curve induced by the photons scattering from relatively longer migrating path makes it non-central symmetric. In particular, if θT increases to 45°, the deviation of the Gamma fitting becomes server because the longest scattering path gets larger with higher θT. Furthermore, as shown by the results, we find that the Rx FOV does not influence the performance of the non-central chi-square distribution functional fitting significantly because the average arrival time of scattered photons is approximately equivalent with different Rx FOV. On the other hand, when the Tx elevation angle rises from 30° to 45°, the performance of the non-central chi-square distribution matching degrades a bit because the axis of the impulse response shifts with various Tx elevation angle. Meanwhile the tails of the impulse responses get longer due to larger Td. In particularly, the performance of Gamma distribution functional fitting becomes extremely worse in this case. The above analysis proves that the non-central chi-square distribution functional fitting is more universal than the Gamma distribution functional fitting. Besides, lower Tx elevation angle is helpful for the numerical fitting.

Then let r = [20m, 40m, 60m, 80m, 100m], the results of the link bandwidth Bc through FFT are plotted in Fig. 5 where βR = 30° and 45°, θT = 30° and 45°. From the results, we find that Bc decreases with r, θT and βR. Compared with βR, θT has a more significant impact on Bc. In addition, if θT stays constant, Bc increases with the narrowing of βR, which also interprets the reason why we design the square array receiver in the next section. As shown by the results, near average 1MHz bandwidth gains is output when βR decreases from 45° to 30°. Moreover, note that Bc will be lower than 2MHz if r rises up to 100m.

 figure: Fig. 5

Fig. 5 Comparison of the link bandwidth by the analytical model and Monte Carlo method. βR = 30°, 45° and θT = 30°, 45°. r ranges from 20m to 100m. MSER is given to indicate the accuracy of the numerical functional approximation.

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Comparison of the above analytical model with Bc is also given in Fig. 5. The results convince that the analytical model is highly reliable and the predicted Bc is consistent with the results via Monte Carlo method. The MSERs in different cases are calculated by Eq. (17) and plotted to examine the accuracy of this analytical model. They indicate that the analytical model is effective. The MSER is merely 1.57 × 10−5, 7.72 × 10−4, 1.97 × 10−5 and 5.40 × 10−3 respectively when (βR, θT) is (30°, 30°), (45°, 30°), (30°, 45°) and (45°, 45°).

3. Square array reception design for NLOS UV channel

3.1 Theorem of the square array reception

According to the above analysis, the link bandwidth of NLOS UV channel is limited by the scattered propagation induced temporal broadening. From Fig. 5, we find that Bc is highly dependent on the Rx FOV βR. Larger βR produces lower Bc. Thus to overcome the restriction, we investigate the diversity reception on NLOS UV links from a new perfective, where the receiver is designed by a square array structure to reduce βR without losing signal power.

The system infrastructure is depicted in Fig. 6 where N × N Rx are distributed by a matrix structure in the square array receiver. βR(1) and βR(N) denotes the FOV of a single Rx and each Rx in the square array receiver. L denotes the channel path loss. d is the Rx spacing of the multiple output receiver. dr is the projected width of the diffusing Tx beam at the receiver side as shown by Fig. 6. For UV photon detector e.g. PMT, the received signal noise ratio (SNR) can be express as follows regardless of the background radiation [21]

SNR=j=0Pk(j/λs)j(ζAe)2+(2keTo/RL)Tp,
where Pk(j/λs) denotes the Poisson distribution with arriving rate of λs, ζ is the PMT factor, A is the amplified gain, ke is the Boltzman constant, To is the receiver temperature, RL is the load resistance, Tp is the pulse interval of the OOK or PPM signals. λs is given by ηPt/Lhv, where η is the quantum efficiency of the receiver, Pt is the transmission power, L is the channel path loss, h is the Planck constant, v is the spectral frequency. Thus the receiver performance is negatively proportional to L.

 figure: Fig. 6

Fig. 6 Schematic drawing of the square array receiver where N × N Rx are distributed by a matrix structure. βR(1) and βR(N) denotes the FOV of a single Rx and each Rx in the square array receiver. L denotes the path loss. d is the Rx spacing of the multiple output receiver. dr is the projected width of the diffusing Tx beam at the receiver side. d is chose as 1~10cm in this receiver design.

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Considering the diameter of the PMT receiver, the Rx spacing is chose as 1~10cm in the square array receiver. In this case, dr is relatively far greater than d. In [22], the subchannel correlation coefficient is found to be more than 0.8 for two receivers in NLOS UV channel, even if they are separated by several meters. Thus, L can be approximated to be equal for all the Rx members under above criterion. Assuming the system is working with geometrical parameter of (βR, θT) at separation distance of r, the link bandwidth can be calculated by Eq. (16). In order to maintain the equal receiver performance, L needs to be the same with the single receiver condition if using the square array receiver. As a result, the tolerable signal attenuation for each Rx in the array receiver could increase by N2 times. In [7], L is analytically modeled as a closed-form, where the interrelation of L and βR is expressed as

L1βR(12+βR2sinθT).

Therefore given the size of the square array receiver N × N, the corresponding βR(N) can be estimated by following equation

1βR(N)[12+βR2(N)sinθT]=N2βR(1)[12+βR2(1)sinθT].

As βR2(1)sinθT and βR2(N)sinθT are relatively much smaller than 12, thus the above equation can be further simplified and βR(N) is finally expressed as

βR(N)=βR(1)N2.

By Eq. (21), N2 times less βR can enable the same path loss with the single receiver case if using the square array reception. Thus the link bandwidth will increase accordingly as described by Eq. (16).

3.2 Numerical results and discussions

Finally in this section, we numerically simulate and compare the link bandwidth in different situations where θT = 30° and 45°, r = 60m and 100m. The initialized βR for single receiver situation is assumed to be 40° in all cases. To ensure nearly equal path loss at all Rx of the square array receiver, N is selected from [1–4].

Let θT = 30°, r = 60m/100m and using the geometry parameters listed in section 2, Fig. 7(a) and 7(b) give the link bandwidth by different N. N = 1 denotes the single Rx situation. From the results, we can find that the square array reception helps to increase the link bandwidth of NLOS UV channel significantly. Take r = 60m for example, the link bandwidth is 2.51MHz in single Rx situation. It grows to 3.72MHZ, 4.83MHz and 5.12MHz respectively when N = 2, 3 and 4. Thus, when N is 4, over 100% improvement in the link bandwidth of NLOS UV channel is brought by the square array reception. Besides, from the results, we should note that this improvement is not unlimited as βR(N) will be close to 0° when N gets larger. Moreover, similar conclusions are summarized when θT = 45°. Figure 7(c) and 7(d) illustrate the numerical results in these cases. The results show that over 100% improvement of the channel link bandwidth is satisfied as well in case N = 4 when θT = 45°.

 figure: Fig. 7

Fig. 7 Demonstration of the link bandwidth of NLOS UV channel using the square array receiver with size N × N. N = 1 represents the single Rx situation. In out simulations, βR = 30° and 45°, r = 60m and 100m. Comparison of the link bandwidth is done by different N to evaluate the performance of square array reception.

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Then, from the system geometry in section 2, we can see that the temporal broadening of NLOS UV channel is closely related to the Tx beam width βT. If βT gets smaller, the shortest scattering path and longest scattering path of the photons will become closer to each other when βR is reduced by the square array receiver. Thus we further investigate the performance of the square array reception under smaller βT conditions. Let βT = 10° and 5°, Fig. 8(a) and 8(b) demonstrate the results of the link bandwidth when θT = 30° and r = 100m. From the results, we discover that smaller βT actually promote the performance of the square array receiver. Higher improvement of the link bandwidth is achieved by narrower Tx beam. The link bandwidth increases by over 150% and 300% when βT = 10° and 5° if N = 4. Additionally, from the discussions in section 2, the link bandwidth is approaching 9MHz when r is 20m if θT = 30° and βT = 17°. Whereas, supposing βT gets lower to 5°, the link bandwidth will surpass 9MHz only if r is 100m as show by Fig. 8 when a 4 × 4 square array receiver is applied. Therefore, this square array reception is valuable for NLOS UV optical links and modest narrowing of the Tx beam is advantageous for its performance.

 figure: Fig. 8

Fig. 8 Results of the link bandwidth of NLOS UV channel using square array reception when βT = 10° and 5°. Compared with the cases when βT = 17° as listed in the above Fig. 7, better performance of the square array reception is found.

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4. Conclusions

Since the link bandwidth of NLOS UV channel for various geometry similar to the path loss model are still studied insufficiently, an analytical bandwidth model of NLOS UV links is developed in this paper. Geometrically-based channel temporal width and software aided numerical fitting are used to formulate the link bandwidth. This model is verified with the theoretical results by fast FFT on the channel impulse responses obtained via Monte Carlo method. Non-central chi-square functional fitting is found to be more accurate for the impulse response curve than the Gamma functional fitting. Moreover, based on the presented bandwidth model, we propose a square array reception for NLOS UV communication systems. By this receiver structure design, the FOV in each Rx of the receiver can be reduced significantly without losing signal power. Consequently, lower temporal broadening is enabled leading to higher link bandwidth. Numerical simulation is done to prove this scheme and the results show that over 100% improvement of the link bandwidth is possible through the square array receiver. Therefore, this work provides a comprehensive understanding of the link bandwidth of NLOS UV channel and constructive tutorial for the practical NLOS UV communication system design.

Funding

National Natural Science Foundation of China (NSFC) (61571067).

References and links

1. Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-Art,” IEEE Commun. Mag. 46(5), 67–73 (2008). [CrossRef]  

2. X. Zhang, Y. Tang, H. Huang, L. Zhang, and T. Bai, “Design of an omnidirectional optical antenna for ultraviolet communication,” Appl. Opt. 53(15), 3225–3232 (2014). [CrossRef]   [PubMed]  

3. H. Qin, Y. Zuo, D. Zhang, Y. Li, and J. Wu, “Received response based heuristic LDPC code for short-range non-line-of-sight ultraviolet communication,” Opt. Express 25(5), 5018–5030 (2017). [CrossRef]   [PubMed]  

4. Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009). [CrossRef]  

5. G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18(10), 10500–10509 (2010). [CrossRef]   [PubMed]  

6. Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010). [CrossRef]   [PubMed]  

7. Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008). [CrossRef]   [PubMed]  

8. L. Wang, Z. Xu, and B. M. Sadler, “An approximate closed-form link loss model for non-line-of-sight ultraviolet communication in noncoplanar geometry,” Opt. Lett. 36(7), 1224–1226 (2011). [CrossRef]   [PubMed]  

9. Y. Zuo, H. Xiao, J. Wu, Y. Li, and J. Lin, “A single-scatter path loss model for non-line-of-sight ultraviolet channels,” Opt. Express 20(9), 10359–10369 (2012). [CrossRef]   [PubMed]  

10. Y. Zuo, H. Xiao, J. Wu, W. Li, and J. Lin, “Closed-form path loss model of non-line-of-sight ultraviolet single-scatter propagation,” Opt. Lett. 38(12), 2116–2118 (2013). [CrossRef]   [PubMed]  

11. P. Luo, M. Zhang, D. Han, and Q. Li, “Performance analysis of short-range NLOS UV communication system using Monte Carlo simulation based on measured channel parameters,” Opt. Express 20(21), 23489–23501 (2012). [CrossRef]   [PubMed]  

12. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009). [CrossRef]  

13. H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012). [CrossRef]  

14. Y. Sun and Y. Zhan, “Closed-form impulse response model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 33(4), 752–757 (2016). [CrossRef]   [PubMed]  

15. M. A. El-Shimy and S. Hranilovic, “Binary-Input Non-Line-of-Sight Solar-Blind UV Channels: Modeling, Capacity and Coding,” J. Opt. Commun. Netw. 4(12), 1008–1017 (2012). [CrossRef]  

16. A. Vavoulas, H. G. Sandalidis, and D. Varoutas, “Node Isolation Probability for Serial Ultraviolet UV-C Multi-hop Networks,” J. Opt. Commun. Netw. 3(9), 750–757 (2011). [CrossRef]  

17. G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

18. C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wirel. Commun. 14(10), 5338–5349 (2015). [CrossRef]  

19. M. A. El-Shimy and S. Hranilovic, “Spatial-Diversity Imaging Receivers for Non-Line-of-Sight Solar-Blind UV Communications,” J. Lightwave Technol. 33(11), 2246–2255 (2015). [CrossRef]  

20. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions-2 (NJ: John Wiley & Sons, 1995).

21. Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010). [CrossRef]   [PubMed]  

22. C. Gong, B. Huang, and Z. Xu, “Correlation and outage probability of NLOS SIMO optical wireless scattering communication channels under turbulence,” J. Opt. Commun. Netw. 8(12), 928–937 (2016). [CrossRef]  

References

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  1. Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-Art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
    [Crossref]
  2. X. Zhang, Y. Tang, H. Huang, L. Zhang, and T. Bai, “Design of an omnidirectional optical antenna for ultraviolet communication,” Appl. Opt. 53(15), 3225–3232 (2014).
    [Crossref] [PubMed]
  3. H. Qin, Y. Zuo, D. Zhang, Y. Li, and J. Wu, “Received response based heuristic LDPC code for short-range non-line-of-sight ultraviolet communication,” Opt. Express 25(5), 5018–5030 (2017).
    [Crossref] [PubMed]
  4. Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
    [Crossref]
  5. G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18(10), 10500–10509 (2010).
    [Crossref] [PubMed]
  6. Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010).
    [Crossref] [PubMed]
  7. Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
    [Crossref] [PubMed]
  8. L. Wang, Z. Xu, and B. M. Sadler, “An approximate closed-form link loss model for non-line-of-sight ultraviolet communication in noncoplanar geometry,” Opt. Lett. 36(7), 1224–1226 (2011).
    [Crossref] [PubMed]
  9. Y. Zuo, H. Xiao, J. Wu, Y. Li, and J. Lin, “A single-scatter path loss model for non-line-of-sight ultraviolet channels,” Opt. Express 20(9), 10359–10369 (2012).
    [Crossref] [PubMed]
  10. Y. Zuo, H. Xiao, J. Wu, W. Li, and J. Lin, “Closed-form path loss model of non-line-of-sight ultraviolet single-scatter propagation,” Opt. Lett. 38(12), 2116–2118 (2013).
    [Crossref] [PubMed]
  11. P. Luo, M. Zhang, D. Han, and Q. Li, “Performance analysis of short-range NLOS UV communication system using Monte Carlo simulation based on measured channel parameters,” Opt. Express 20(21), 23489–23501 (2012).
    [Crossref] [PubMed]
  12. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
    [Crossref]
  13. H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012).
    [Crossref]
  14. Y. Sun and Y. Zhan, “Closed-form impulse response model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 33(4), 752–757 (2016).
    [Crossref] [PubMed]
  15. M. A. El-Shimy and S. Hranilovic, “Binary-Input Non-Line-of-Sight Solar-Blind UV Channels: Modeling, Capacity and Coding,” J. Opt. Commun. Netw. 4(12), 1008–1017 (2012).
    [Crossref]
  16. A. Vavoulas, H. G. Sandalidis, and D. Varoutas, “Node Isolation Probability for Serial Ultraviolet UV-C Multi-hop Networks,” J. Opt. Commun. Netw. 3(9), 750–757 (2011).
    [Crossref]
  17. G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).
  18. C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wirel. Commun. 14(10), 5338–5349 (2015).
    [Crossref]
  19. M. A. El-Shimy and S. Hranilovic, “Spatial-Diversity Imaging Receivers for Non-Line-of-Sight Solar-Blind UV Communications,” J. Lightwave Technol. 33(11), 2246–2255 (2015).
    [Crossref]
  20. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions-2 (NJ: John Wiley & Sons, 1995).
  21. Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010).
    [Crossref] [PubMed]
  22. C. Gong, B. Huang, and Z. Xu, “Correlation and outage probability of NLOS SIMO optical wireless scattering communication channels under turbulence,” J. Opt. Commun. Netw. 8(12), 928–937 (2016).
    [Crossref]

2017 (1)

2016 (2)

2015 (2)

C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wirel. Commun. 14(10), 5338–5349 (2015).
[Crossref]

M. A. El-Shimy and S. Hranilovic, “Spatial-Diversity Imaging Receivers for Non-Line-of-Sight Solar-Blind UV Communications,” J. Lightwave Technol. 33(11), 2246–2255 (2015).
[Crossref]

2014 (1)

2013 (2)

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Y. Zuo, H. Xiao, J. Wu, W. Li, and J. Lin, “Closed-form path loss model of non-line-of-sight ultraviolet single-scatter propagation,” Opt. Lett. 38(12), 2116–2118 (2013).
[Crossref] [PubMed]

2012 (4)

2011 (2)

2010 (3)

2009 (2)

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[Crossref]

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

2008 (2)

Bai, T.

Chen, G.

H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012).
[Crossref]

G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18(10), 10500–10509 (2010).
[Crossref] [PubMed]

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
[Crossref] [PubMed]

Ding, H.

H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012).
[Crossref]

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
[Crossref] [PubMed]

El-Shimy, M. A.

Gong, C.

C. Gong, B. Huang, and Z. Xu, “Correlation and outage probability of NLOS SIMO optical wireless scattering communication channels under turbulence,” J. Opt. Commun. Netw. 8(12), 928–937 (2016).
[Crossref]

C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wirel. Commun. 14(10), 5338–5349 (2015).
[Crossref]

Han, D.

He, Q.

Hranilovic, S.

Huang, B.

Huang, G.

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Huang, H.

X. Zhang, Y. Tang, H. Huang, L. Zhang, and T. Bai, “Design of an omnidirectional optical antenna for ultraviolet communication,” Appl. Opt. 53(15), 3225–3232 (2014).
[Crossref] [PubMed]

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Li, Q.

Li, W.

Li, Y.

Lin, J.

Luo, P.

Majumdar, A. K.

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

Ni, G.

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Qin, H.

Sadler, B. M.

H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012).
[Crossref]

L. Wang, Z. Xu, and B. M. Sadler, “An approximate closed-form link loss model for non-line-of-sight ultraviolet communication in noncoplanar geometry,” Opt. Lett. 36(7), 1224–1226 (2011).
[Crossref] [PubMed]

Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010).
[Crossref] [PubMed]

G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18(10), 10500–10509 (2010).
[Crossref] [PubMed]

Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010).
[Crossref] [PubMed]

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[Crossref]

Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-Art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
[Crossref]

Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
[Crossref] [PubMed]

Sandalidis, H. G.

Sun, Y.

Tang, Y.

X. Zhang, Y. Tang, H. Huang, L. Zhang, and T. Bai, “Design of an omnidirectional optical antenna for ultraviolet communication,” Appl. Opt. 53(15), 3225–3232 (2014).
[Crossref] [PubMed]

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Varoutas, D.

Vavoulas, A.

Wang, L.

Wu, J.

Xiao, H.

Xu, Z.

C. Gong, B. Huang, and Z. Xu, “Correlation and outage probability of NLOS SIMO optical wireless scattering communication channels under turbulence,” J. Opt. Commun. Netw. 8(12), 928–937 (2016).
[Crossref]

C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wirel. Commun. 14(10), 5338–5349 (2015).
[Crossref]

H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012).
[Crossref]

L. Wang, Z. Xu, and B. M. Sadler, “An approximate closed-form link loss model for non-line-of-sight ultraviolet communication in noncoplanar geometry,” Opt. Lett. 36(7), 1224–1226 (2011).
[Crossref] [PubMed]

Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010).
[Crossref] [PubMed]

G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18(10), 10500–10509 (2010).
[Crossref] [PubMed]

Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18(12), 12226–12238 (2010).
[Crossref] [PubMed]

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[Crossref]

Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-Art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
[Crossref]

Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
[Crossref] [PubMed]

Zhan, Y.

Zhang, D.

Zhang, L.

Zhang, M.

Zhang, X.

X. Zhang, Y. Tang, H. Huang, L. Zhang, and T. Bai, “Design of an omnidirectional optical antenna for ultraviolet communication,” Appl. Opt. 53(15), 3225–3232 (2014).
[Crossref] [PubMed]

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Zuo, Y.

Appl. Opt. (1)

IEEE Commun. Mag. (1)

Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-Art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27(9), 1535–1544 (2009).
[Crossref]

IEEE Trans. Wirel. Commun. (1)

C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wirel. Commun. 14(10), 5338–5349 (2015).
[Crossref]

IET Commun. (1)

H. Ding, G. Chen, Z. Xu, and B. M. Sadler, “Channel modeling and performance of non-line-of-sight ultraviolet scattering communication,” IET Commun. 6(5), 514–524 (2012).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Commun. Netw. (3)

J. Opt. Soc. Am. A (1)

Opt. Express (6)

Opt. Lett. (3)

Proc. SPIE (2)

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[Crossref]

G. Huang, Y. Tang, G. Ni, H. Huang, and X. Zhang, “Application of MIMO technology in ultraviolet communication,” Proc. SPIE 9043, 237–244 (2013).

Other (1)

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions-2 (NJ: John Wiley & Sons, 1995).

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Figures (8)

Fig. 1
Fig. 1 Schematic drawing of the short-range NLOS UV communication system geometry. Rx (PMT) is placed at distance r from the Tx (LED). Each photon escapes from the LED Tx to the PMT Rx across a random scattered path. The scattered propagation path is marked by the green dashed line for an example. Three geometry parameters: Tx/Rx distance r, Tx elevation angle θT, Rx FOV βR are focused on in the model as shown by the three cases, where θT, βR and r’ denote the corresponding changes in these three parameters. The red solid lines denote the scattered path of the first photon and last photon arriving at the Rx by time tmin and tmsx. The time domain impulse response width is defined by Td = tmax-tmin.
Fig. 2
Fig. 2 Lists of coefficient a and b in the expression of Bc = aTd-b. θT is selected at 30°, 45°, 60°. Different ranges of βR [0°~5°], [5°~10°], [15°~25°], [25°~35°], [35°~45°] are measured.
Fig. 3
Fig. 3 The atmospheric and geometric parameters used in Monte Carlo method
Fig. 4
Fig. 4 Demonstration of the simulated impulse responses and comparison with the results by numerical fitting. In our simulations, θT is 30° and 45°, βR is 30° and 45°, r = 100m. Gamma, chi-square and MC denote the Gamma functional fitting, non-central chi-square distribution functional fitting and Monte Carlo method.
Fig. 5
Fig. 5 Comparison of the link bandwidth by the analytical model and Monte Carlo method. βR = 30°, 45° and θT = 30°, 45°. r ranges from 20m to 100m. MSER is given to indicate the accuracy of the numerical functional approximation.
Fig. 6
Fig. 6 Schematic drawing of the square array receiver where N × N Rx are distributed by a matrix structure. βR(1) and βR(N) denotes the FOV of a single Rx and each Rx in the square array receiver. L denotes the path loss. d is the Rx spacing of the multiple output receiver. dr is the projected width of the diffusing Tx beam at the receiver side. d is chose as 1~10cm in this receiver design.
Fig. 7
Fig. 7 Demonstration of the link bandwidth of NLOS UV channel using the square array receiver with size N × N. N = 1 represents the single Rx situation. In out simulations, βR = 30° and 45°, r = 60m and 100m. Comparison of the link bandwidth is done by different N to evaluate the performance of square array reception.
Fig. 8
Fig. 8 Results of the link bandwidth of NLOS UV channel using square array reception when βT = 10° and 5°. Compared with the cases when βT = 17° as listed in the above Fig. 7, better performance of the square array reception is found.

Equations (21)

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B c 1/( t max t min ),
t min =[ d min sin( θ T β T /2) + d min cos( β R /2) ]/c,
t max =[ d max sin( θ T + β T /2) + d max cos( β R /2) ]/c.
t max = r c [ 1 cos( θ T + β T /2)tan( β R /2)sin( θ T + β T /2) + 1 cot( θ T + β T /2)cos( β R /2)sin( β R /2) ],
t min = r c [ 1 cos( θ T β T /2)+tan( β R /2)sin( θ T β T /2) + 1 cot( θ T β T /2)cos( β R /2)+sin( β R /2) ].
T d = r c [ F 1 ( θ T , β R )+ F 2 ( θ T , β R ) ],
F 1 ( θ T , β R )= 1 cos( θ T + β T /2)tan( β R /2)sin( θ T + β T /2) 1 cos( θ T β T /2)+tan( β R /2)sin( θ T β T /2) ,
F 2 ( θ T , β R )= 1 cot( θ T + β T /2)cos( β R /2)sin( β R /2) 1 cot( θ T β T /2)cos( β R /2)+sin( β R /2) .
F 1 ( θ T , β R )= 2tan β R 2 sin θ T cos 2 θ T tan 2 β R 2 sin 2 θ T ,
F 2 ( θ T , β R )= 2sin β R 2 cos 2 θ T cos 2 β R 2 sin 2 β R 2 .
f(t,β,α)= P 0 β α Γ(α) t α1 e βx ,t>0,
E(t)= 1 H i τ [ H i τ( t min +iτ) ] ,
Var(t)= 1 H i τ [ H i τ ( t min +iτE(t)) 2 ] .
p(t)= 1 2 e (t+λ)/2 ( x λ ) k/41/2 I k/21 ( λt ),
B c =a T d b ,
B c =a [ r c ( F 1 ( θ T , β R )+ F 2 ( θ T , β R )) ] b ,
MSER= 1 num( B c ) [ m( B c ) B c B c ] 2 ,
SNR= j=0 P k (j/ λ s ) j ( ζAe ) 2 +(2 k e T o / R L ) T p ,
L 1 β R (12+ β R 2 sin θ T ) .
1 β R (N)[12+ β R 2 (N)sin θ T ] = N 2 β R (1)[12+ β R 2 (1)sin θ T ] .
β R (N)= β R (1) N 2 .

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