## 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 *B _{c}* 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

*β*is initialized as 17° [12] and the receiver elevation angle

_{T}*θ*is fixed at 90° to make omnidirectional receiving [2]. Three cases are classified to numerically model

_{R}*B*, where the impacts of three parameters are considered including the Tx elevation angle

_{c}*θ*, Rx FOV

_{T}*β*and transceiver distance

_{R}*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

*T*is used in our model as it can be obtained by

_{d}*t*-

_{max}*t*. 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

_{min}*T*with

_{d}*θ*,

_{T}*β*,

_{R}*r*. 2) do numerical fitting of

*B*by

_{c}*T*.

_{d}#### 2.1 Step 1): closed-form of T_{d} with θ_{T}, β_{R}, r

By common communication theory, *B _{c}* is negatively correlated to the temporal broadening written as

*T*=

_{d}*t*-

_{max}*t*for NLOS UV channels as shown in Fig. 1

_{min}*t*and

_{min}*t*denote the arriving time of the first photon and the last photon counting at Rx. Define terms

_{max}*d*and

_{min}*d*in Fig. 1,

_{max}*t*and

_{min}*t*are expressed as follows

_{max}*c*is the light speed,

*d*=

_{min}*r*/[cot(

*θ*-

_{T}*β*/2) + tan(

_{T}*β*/2)],

_{R}*d*=

_{max}*r*/[cot(

*θ*+

_{T}*β*/2)-tan(

_{T}*β*/2)]. Then Eq. (2) and (3) can be furthered reformulated as

_{R}Thereby, *T _{d}* is given below

*F*

_{1}(

*θ*,

_{T}*β*) and

_{R}*F*

_{2}(

*θ*,

_{T}*β*) are arranged by symmetrical form

_{R}Hence, *T _{d}* 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,

*T*is uniquely determined by

_{d}*θ*,

_{T}*β*and

_{R}*r*. Thus the abovementioned three cases can also be characterized by three limitations:

*L*(

*θ*),

_{T}*L*(

*β*),

_{R}*L*(

*r*). Particularly, if narrow Tx beam is assumed [10] where

*β*<<

_{T}*θ*, the above Eq. (7) and (8) can be simplified as follows by trigonometric functional transformation

_{T}#### 2.2 Step 2): software aided numerical fitting of B_{c} by T_{d}

In this step, *B _{c}* is numerically fitted by the above

*T*. Among the derivation of

_{d}*B*, we employ the Monte Carlo method to simulate the impulse response

_{c}*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

*B*.

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

*α*and

*β*can be estimated by the statistical feature of arriving time

*t*of the photons,

*P*

_{0}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*/

*β*

^{2}[20]. Thus

*β = E*(

*t*)/

*Var*(

*t*),

*α*=

*βE*(

*t*). Given the impulse response

*h*(

*t*) = ∑

*H*σ(

_{i}*t*-

*t*

_{min}-

*i**

*τ*), where

*i*= 1…

*L*,

*L*is the number of resolved multipath,

*τ*is the time delay unit,

*H*is the amplitude coefficient of each resolved path,

_{i}*E*(

*t*) and

*Var*(

*t*) are expressed as

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

*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

*T*

_{0}where

*p*(

*T*

_{0})~0 to

*T*and move the origin of

_{d}*t*from 0 to

*t*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.

_{min}Next, using the post-processed impulse responses, *B _{c}* is obtained through FFT where the maximum frequency position at 1.67 × 10

^{8}Hz (1/6ns) assuming the response time of the PMT is 6ns [21] and the frequency resolution is 1/

*T*. As mentioned above,

_{d}*B*is negatively related to

_{c}*T*. In order to precisely estimate the functional connection between

_{d}*B*and

_{c}*T*, we take advantage of the

_{d}*cftool*toolbox of

*MATLAB*to execute the curve fitting. During the testament, the data set of (

*B*,

_{c}*T*) under various

_{d}*β*and

_{R}*θ*is chosen as the input. Power function fitting is output to coincide with the simulated results. Therefore,

_{T}*B*is expressed as follows

_{c}*a*and

*b*depends on the specific (

*β*,

_{R}*θ*). The optimal values of

_{T}*a*and

*b*for different (

*β*,

_{R}*θ*) are produced through numerous curve fittings with different

_{T}*r*and listed in Fig. 2. Although Fig. 2 merely lists the values of

*a*and

*b*when

*β*= 0~45

_{R}^{o},

*θ*= 30°, 45° and 60°,

_{T}*a*and

*b*can be obtained by the abovementioned means for every geometry parameter.

Consequently, substituting Eq. (6) into Eq. (15), an analytical model of *B _{c}* is presented as

*F*

_{1}(∙) and

*F*

_{2}(∙) 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.

*m*(

*B*) means

_{c}*B*by the analytical model,

_{c}*num*(

*B*) is the number of

_{c}*B*.

_{c}#### 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, *θ _{T}* = [30°, 45°],

*β*= [30°, 45°],

_{R}*r*= 100m.

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 *β _{R}* = 30°,

*θ*= 30°,

_{T}*k*= 5 when

*β*= 45°,

_{R}*θ*= 30°.

_{T}*k*= 4 when

*β*= 45°,

_{R}*θ*= 45°. The results prove that the non-central chi-square distribution functional fitting outperforms the Gamma distribution functional fitting. When

_{T}*θ*= 30°,

_{T}*β*= 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

_{R}*θ*increases to 45°, the deviation of the Gamma fitting becomes server because the longest scattering path gets larger with higher

_{T}*θ*. 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

_{T}*T*. 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.

_{d}Then let *r* = [20m, 40m, 60m, 80m, 100m], the results of the link bandwidth *B _{c}* through FFT are plotted in Fig. 5 where

*β*= 30° and 45°,

_{R}*θ*= 30° and 45°. From the results, we find that

_{T}*B*decreases with

_{c}*r*,

*θ*and

_{T}*β*. Compared with

_{R}*β*,

_{R}*θ*has a more significant impact on

_{T}*B*. In addition, if

_{c}*θ*stays constant,

_{T}*B*increases with the narrowing of

_{c}*β*, 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

_{R}*β*decreases from 45° to 30°. Moreover, note that

_{R}*B*will be lower than 2MHz if

_{c}*r*rises up to 100m.

Comparison of the above analytical model with *B _{c}* is also given in Fig. 5. The results convince that the analytical model is highly reliable and the predicted

*B*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

_{c}^{−5}, 7.72 × 10

^{−4}, 1.97 × 10

^{−5}and 5.40 × 10

^{−3}respectively when (

*β*,

_{R}*θ*) is (30°, 30°), (45°, 30°), (30°, 45°) and (45°, 45°).

_{T}## 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 *B _{c}* is highly dependent on the Rx FOV

*β*. Larger

_{R}*β*produces lower

_{R}*B*. 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

_{c}*β*without losing signal power.

_{R}The system infrastructure is depicted in Fig. 6 where *N* × *N* Rx are distributed by a matrix structure in the square array receiver. *β _{R}*(1) and

*β*(N) denotes the FOV of a single Rx and each Rx in the square array receiver.

_{R}*L*denotes the channel path loss.

*d*is the Rx spacing of the multiple output receiver.

*d*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]

_{r}*P*(

_{k}*j*/

*λ*) denotes the Poisson distribution with arriving rate of

_{s}*λ*,

_{s}*ζ*is the PMT factor,

*A*is the amplified gain,

*k*is the Boltzman constant,

_{e}*T*

^{o}is the receiver temperature,

*R*is the load resistance,

_{L}*T*is the pulse interval of the OOK or PPM signals.

_{p}*λ*is given by

_{s}*ηP*/

_{t}*Lhv*, where

*η*is the quantum efficiency of the receiver,

*P*is the transmission power,

_{t}*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*.

Considering the diameter of the PMT receiver, the Rx spacing is chose as 1~10cm in the square array receiver. In this case, *d _{r}* 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 (

*β*,

_{R}*θ*) at separation distance of

_{T}*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

*N*

^{2}times. In [7],

*L*is analytically modeled as a closed-form, where the interrelation of

*L*and

*β*is expressed as

_{R}Therefore given the size of the square array receiver *N* × *N*, the corresponding *β _{R}*(

*N*) can be estimated by following equation

As *β _{R}*

^{2}(1)sin

*θ*and

_{T}*β*

_{R}^{2}(N)sin

*θ*are relatively much smaller than 12, thus the above equation can be further simplified and

_{T}*β*(

_{R}*N*) is finally expressed as

By Eq. (21), *N*^{2} times less *β _{R}* 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 *θ _{T}* = 30° and 45°,

*r*= 60m and 100m. The initialized

*β*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,

_{R}*N*is selected from [1–4].

Let *θ _{T}* = 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

*β*(

_{R}*N*) will be close to 0° when

*N*gets larger. Moreover, similar conclusions are summarized when

*θ*= 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

_{T}*N*= 4 when

*θ*= 45°.

_{T}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 *β _{T}*. If

*β*gets smaller, the shortest scattering path and longest scattering path of the photons will become closer to each other when

_{T}*β*is reduced by the square array receiver. Thus we further investigate the performance of the square array reception under smaller

_{R}*β*conditions. Let

_{T}*β*= 10° and 5°, Fig. 8(a) and 8(b) demonstrate the results of the link bandwidth when

_{T}*θ*= 30° and

_{T}*r*= 100m. From the results, we discover that smaller

*β*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

_{T}*β*= 10° and 5° if

_{T}*N*= 4. Additionally, from the discussions in section 2, the link bandwidth is approaching 9MHz when

*r*is 20m if

*θ*= 30° and

_{T}*β*= 17°. Whereas, supposing

_{T}*β*gets lower to 5°, the link bandwidth will surpass 9MHz only if

_{T}*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.

## 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).

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