1. Introduction

Ultraviolet communications (UVC) exploiting the strong scattering effect of solar-blind ultraviolet light with wavelength from 200 nm to 280 nm caused by atmospheric molecules and aerosols can realize non-line-of-sight (NLOS) transmission [1–3]. NLOS UVC links can easily bypass obstacles and even achieve omnidirectional transmission and reception [4–6]. That is, the alignment between transceivers is not essentially required in NLOS UVC systems, unlike other optical wireless communications (OWC), such as visible light communications and free-space optical communications [7–9]. This merit can reduce the complexity of practical OWC systems.

However, the ultraviolet light is scattered one or more times when it transmits from a light source to a detector via NLOS links, which makes building an analytical channel model of NLOS UVC knotty. To assess the characteristics of NLOS UVC links, numerical channel models based on Monte-Carlo (MC) methods with multiple scattering events considered have been proposed [10–12]. MC-based NLOS UVC channel models can obtain both path loss (PL) and channel impulse response (CIR), both of which are critical indicators of wireless channels. In addition, MC-based NLOS UVC channel models were experimentally validated and have been widely accepted by current studies [10,12–17]. Nevertheless, those models are time-consuming because they must generate and simulate a considerable number of photons to attain stable results. Fortunately, NLOS UVC is mainly designed for short-range communications where the distance between transceivers is up to tens of meters [18]. The total received optical energy of ultraviolet light in NLOS UVC is dominated by that from the first scattering event under short-range conditions. For this reason, single-scatter channel models of NLOS UVC have attracted widespread attention in recent years because they can achieve nearly identical results as MC-based ones with significantly reduced computational complexity [8,14–17,19–21].

The single-scatter channel models of NLOS UVC were initially developed in the prolate-spheroidal coordinate system [15,19]. In those models, a transmitter (Tx) and a receiver (Rx) are placed at two focal points of a prolate-spheroidal coordinate system, respectively. Using the fact that the sum of the distance from each focal point to a prolate-spheroid surface is constant, the CIR of NLOS UVC links can be obtained. By accumulating the CIR results over time, the PL of NLOS UVC can be further obtained. However, the single-scatter channel model of NLOS UVC proposed in [19] can only be applied to the coplanar geometry, which is a special case of the noncoplanar geometry. Additionally, that proposed in [15] cannot explicitly provide the theoretical expression because determining an upper bound of integration in the channel model requires solving a nonconvex optimization problem. Furthermore, not only in MC-based channel models but also in practical experiments, the geometry of the NLOS UVC systems is described by the inclination (or elevation) and azimuth angles in the spherical coordinate system [10,14]. Therefore, the single-scatter channel models of NLOS UVC are increasingly investigated based on the spherical coordinate system, which is more natural and intelligible. Very recently, some single-scatter PL models of NLOS UVC in the spherical coordinate system have been reported [8,14,16,17,20,22]. Nevertheless, only [3] and [23] have approximately studied the temporal characteristics of single-scatter NLOS UVC links based on the Riemann sum method in the spherical coordinate system. Thus, the theoretical single-scatter CIR model of NLOS UVC has not been well addressed so far.

To measure the CIR of the NLOS UVC, Chen et al. developed an outdoor experiment using a narrow-pulsed ultraviolet laser as the light source [24]. This experimental setup and its corresponding results have been further utilized in subsequent works [25,26]. However, ultraviolet lasers have a narrow beam divergence angle and must be pointed in the direction where the receiver locates, reducing the coverage of the NLOS UVC. In recent decades, with the fast development of solid-state lighting technology, ultraviolet light-emitting diodes (LEDs) have been prevalently employed in UVC [17]. Nevertheless, due to the limited modulation bandwidth of LEDs, the NLOS UVC with an LED as the light source cannot emit a pulse as narrow as ultraviolet lasers. Therefore, the experimental setup proposed in [24] is not feasible for measuring the CIR of the NLOS UVC with an LED as the light source. In addition, since the LEDs-based NLOS UVC link is commonly weak and always employs photon-counting receivers, traditional methods adopted for measuring the CIR in the radio-frequency (RF) or indoor infrared channel, such as frequency-domain channel sounding [27] and pseudorandom binary sequence (PRBS)-based techniques [28], are also not suitable for the LEDs-based NLOS UVC channels. Measuring the LEDs-based NLOS UVC channels’ temporal characteristics with photon-counting receivers has not been reported yet.

Motivated by the above investigation, a theoretical single-scatter CIR model of NLOS UVC in noncoplanar geometry is proposed based on the spherical coordinate system in this work, which is a substantial contribution to filling the research gap in modeling the NLOS UVC channel. Specifically, we establish the relationship between the time that a photon needs to travel from the Tx to the Rx via a single-scatter event and the radial distance of the spherical coordinate system at first. Subsequently, the expression of the total received optical energy at a given time is obtained, which is essentially a cumulative distribution function (CDF) of received optical energy in the time dimension. By differentiating that CDF with respect to time, the expression of the single-scatter CIR of NLOS UVC can be derived. The proposed theoretical single-scatter CIR model is further verified by the widely accepted MC-based channel model of NLOS UVC. Finally, an outdoor experiment is also designed and implemented to measure the temporal characteristics of the LEDs-based NLOS UVC channel with a photon-counting receiver, thereby validating the effectiveness of the proposed theoretical single-scatter CIR model.

2. Propagation model and single-scatter channel impulse response analysis

2.1 Propagation model of single-scatter NLOS UVC link

A single-scatter NLOS UVC link is illustrated in Fig. 1, where the Rx and the Tx are placed at the origin and the point of $(0,r,0)$, respectively. The vector from the Rx to the Tx can be written as ${\mathbf {r}} = {\left [ {0,r,0} \right ]^{\textrm {T}}}$, where the superscript “T” denotes the transpose operator. ${P_s}$ denotes an arbitrary single-scatter event. The vectors from the Tx and the Rx to the ${P_s}$ are represented by ${{\mathbf {r}}_1}$ and ${{\mathbf {r}}_2}$, respectively. Here, coordinates $\theta$, $\phi$, and ${r_2} = \left \| {{{\mathbf {r}}_2}} \right \|$ are the inclination angle, azimuth angle, and the radial distance, respectively, of the spherical coordinate system utilized for the single-scatter CIR model, where $\left \| \cdot \right \|$ is the Euclidean norm. Therefore, we have ${{\mathbf {r}}_2} = {r_2}{\left [ {\sin \theta \cos \phi,\sin \theta \sin \phi,\cos \theta } \right ]^{\textrm {T}}}$ and ${{\mathbf {r}}_1} = {{\mathbf {r}}_2} - {\mathbf {r}}$. The Euclidean norm of ${{\mathbf {r}}_1}$ can be further obtained as

 

Fig. 1. Diagram of a single-scatter NLOS UVC link.

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The field-of-view (FOV) of the Rx is conical with the apex angle equaling ${\beta _R}$, which also refers to the full FOV angle. The direction vector of the Rx’s FOV is represented by ${{\mathbf {{\boldsymbol {\mathrm{\mu} }} }}_R} = {\left [ {\sin {\theta _R}\cos {\phi _R},\sin {\theta _R}\sin {\phi _R},\cos {\theta _R}} \right ]^{\textrm {T}}}$, where $\theta _R$ and ${\phi _R}$ are the inclination and the azimuth angles of the FOV’s axis, respectively. Let ${\cal T}\left ( {{\gamma _T}} \right )$ be the emission pattern of the light source at the Tx, where ${\gamma _T} = \arccos \left [ {\frac {{\sin {\theta _T}\left ( {{r_2}\sin \theta \cos \left ( {\phi - {\phi _T}} \right ) - r\sin {\phi _T}} \right ) + {r_2}\cos \theta \cos {\theta _T}}}{{{r_1}}}} \right ]$ is the angle from the ${{\mathbf {{\boldsymbol {\mathrm{\mu} }} }}_T}$ to the ${{\mathbf {r}}_1}$, ${{\mathbf {{\boldsymbol {\mathrm{\mu} }} }}_T} = {\left [ {\sin {\theta _T}\cos {\phi _T},\sin {\theta _T}\sin {\phi _T},\cos {\theta _T}} \right ]^{\textrm {T}}}$ is the unit vector of the Tx’s pointing direction, ${\theta _T}$ and ${\phi _T}$ are the inclination and the azimuth angles of the Tx beam’s axis, respectively. ${\cal T}\left ( {{\gamma _T}} \right )$ specifies the distribution of the emitted optical power per unit solid angle and refers to the emission pattern of the light source. ${\cal T}\left ( {{\gamma _T}} \right )$ is supposed to be selected based on the practically utilized light source. Two distributions are used in this work to characterize the emission pattern:

1) Uniform distribution (UD) [14,19]. It is demonstrated by orange solid lines at the Tx in Fig. 1. UD is the most representative distribution to describe the emission pattern of the light source in the field of NLOS UVC at present. It can be expressed as

2) Lambertian distribution (LD) [17,29]. It is illustrated by purple dashed lines at the Tx in Fig. 1. LD is recognized as the emission pattern of LEDs, which are widely adopted in short-range NLOS UVC links in recent years. It can be written as

When an optical impulse is transmitted at time 0 with the energy of $Q_T$, the total received optical energy after photons propagating through the single-scatter link of NLOS UVC can be expressed as [17]

2.2 Proposed single-scatter CIR model

We noticed that the CIR of the single-scatter NLOS UVC channel is intrinsically a probability density function (PDF), which indicates how much optical energy the Rx can receive during an infinitesimal time interval when an optical impulse is transmitted at time 0. Additionally, the total received optical energy from time 0 to $t$, which is essentially a CDF of received optical energy, depends on the length of the single-scatter trajectory, i.e., ${r_1} + {r_2}$, given $\theta$ and $\phi$. Therefore, we will derive the total received optical energy from time 0 to $t$ at first. By differentiating it with respect to time, the CIR of the single-scatter NLOS UVC channel in the spherical coordinate system can be further achieved. The above process will be given below in detail.

Let ${r_a}$ be the length of the single-scatter trajectory, i.e., ${r_a} = {r_1} + {r_2}$, and

Lemma 1 $\tilde t\left ( {{r_2}} \right )$ is an increasing function of ${r_2}$ given $\theta$ and $\phi$.

Proof In terms of Eq. (1), $r_a$ can be expressed as

By differentiating Eq. (9) with respect to $r_2$, we have

We noticed that

Based on the Lemma 1, the shortest and longest time that the single-scatter trajectory costs is determined by the ${r_{{2_{\min }}}}$ and ${r_{{2_{\max }}}}$, respectively, given $\theta$ and $\phi$. Therefore, the total received optical energy from time 0 to $t$, i.e., the CDF, can be obtained by adjusting the interval of integration for ${r_2}$ as follows:

It should be noted that ${\tilde r_2}(t)$ of interest locates within the interval of integration for ${r_2}$. This is because if $t \le \tilde t\left ( {{r_{{2_{\min }}}}} \right )$, then the interval of integration for ${r_2}$ is $\left [ {{{\tilde r}_2}(t),{{\tilde r}_2}(t)} \right ]$ and the integral equals zero; if $t > \tilde t\left ( {{r_{{2_{\max }}}}} \right )$, then the interval of integration for ${r_2}$ is $\left [ {{r_{{2_{\min }}}},{r_{{2_{\max }}}}} \right ]$, where ${\tilde r_2}(t)$ is not involved.

The derivative of ${F_{{Q_R}}}\left ( t \right )$ with respect to time and subsequent normalization by the detector area yield the CIR of the single-scatter NLOS UVC channel with the unit of W/m$^2$, as given in the following Theorem.

Theorem 1 The CIR of the single-scatter NLOS UVC channel in the spherical coordinate system can be expressed as

Proof Using the relationship between PDF and CDF, we have

According to Eq. (13), $\frac {{{\mathop {\rm d}\nolimits } {{\hat r}_{{2_{\max }}}}\left ( t \right )}}{{{\mathop {\rm d}\nolimits } t}} = {\tilde r'_2}(t)$ if $t \le \tilde t\left ( {{r_{{2_{\max }}}}} \right )$, otherwise $\frac {{{\mathop {\rm d}\nolimits } {{\hat r}_{{2_{\max }}}}\left ( t \right )}}{{{\mathop {\rm d}\nolimits } t}} = 0$. Similarly, based on Eq. (14), $\frac {{{\mathop {\rm d}\nolimits } {{\hat r}_{{2_{\min }}}}\left ( t \right )}}{{{\mathop {\rm d}\nolimits } t}} = {\tilde r'_2}(t)$ if $t \le \tilde t\left ( {{r_{{2_{\min }}}}} \right )$, otherwise $\frac {{{\mathop {\rm d}\nolimits } {{\hat r}_{{2_{\min }}}}\left ( t \right )}}{{{\mathop {\rm d}\nolimits } t}} = 0$. In addition, applying the technique of implicit differentiation to Eq. (15), the first-order derivative of ${\tilde r_2}(t)$ can be obtained as

Substituting $\frac {{{\mathop {\rm d}\nolimits } {{\hat r}_{{2_{\max }}}}\left ( t \right )}}{{{\mathop {\rm d}\nolimits } t}}$, $\frac {{{\mathop {\rm d}\nolimits } {{\hat r}_{{2_{\min }}}}\left ( t \right )}}{{{\mathop {\rm d}\nolimits } t}}$, and Eq. (19) into Eq. (18), we can achieve Theorem 1.■

3. Experimental setup

In order to validate the proposed theoretical single-scatter CIR model, we developed an outdoor experiment to measure the temporal characteristics of the NLOS UVC channel. The experimental scheme for measuring the CIR is not unique. For example, the CIR can be obtained by directly sending a very narrow pulse and then collecting the broadened waveform at the receiver side [24], by sending a PRBS and then using its autocorrelation property to process the collected signal [28], or by using a vector network analyzer to measure the frequency response function of the channel and then applying the inverse Fourier transform to it [27]. However, due to the long carrier lifetime, LEDs face challenges in emitting very narrow pulses, resulting in limited modulation bandwidth. At the same time, NLOS UVC links are so weak that photon-counting receivers are usually used at the receivers. Hence the PRBS and vector network analyzer-based methods are no longer suitable. During the experiment, we observed a rapid rise in the emitted optical power of the LED upon powering on. Therefore, we decided to measure the channel step response of the NLOS UVC link at first. Then, the CIR can be obtained by taking the derivative of the channel step response. The detailed design of the experiment is provided below.

Figure 2 exhibits the experimental system for measuring the temporal characteristics of the NLOS UVC channel. The schematic representation of the experimental system is given in Fig. 2(a). Figure 2(b) illustrates the outdoor scene of the experiment. The composition of the transmitter and the receiver are provided in Fig. 2(c) and (d), respectively. The transmitter consists of a waveform generator, a driver, and an ultraviolet LED module. The waveform generator is used to produce the input signal for measuring the temporal characteristics of the NLOS UVC channel, as detailed in Fig. 3. The signal from the waveform generator is passed to the driver, which can produce sufficient current to drive the customized ultraviolet LED. The ultraviolet LED has an output optical power of 50 mW around the central wavelength of 260 nm and its full angle at half of the maximum intensity is 60 degrees. The ultraviolet LED is mounted on an aluminum heat sink to efficiently dissipate the thermal power generated during light emission, ensuring a stable current-voltage characteristic. The driver has a monitoring interface connected to the receiver via a coaxial cable, enabling it to trigger the oscilloscope to capture the signal. That coaxial cable is labeled "Sync Cable" in Fig. 2(c) and ensures that when the transmitter sends out a signal, the receiver can record the received signal synchronously. The receiver consists of a photomultiplier tube (PMT) (Hamamatsu R7154), an optical filter, a transimpedance amplifier, and an oscilloscope. The optical filter is mounted in front of the PMT to suppress the background light. The quantum efficiency of the PMT at 260 nm is about 30%, and the transmittance of the filter at 260 nm is about 35%. The output signal of the PMT is passed through a transimpedance amplifier and then captured by an oscilloscope. Power supplies are used to provide power for the transimpedance amplifier and the PMT. The oscilloscope is triggered by a synchronization signal and then records the waveform of the received signal, which will be copied to a computer for offline processing to obtain the CIR results. It should be noted that the spectral response of the adopted PMT, which ranges from 160 nm to 320 nm, is broader than the solar-blind ultraviolet band. Despite the use of an optical filter, background noise outside the solar-blind ultraviolet band during the daytime can degrade the precision of measurements. Consequently, the experiment was conducted at night to mitigate the impact of background noise on the measurements.

 

Fig. 2. Experimental system for measuring the temporal characteristics of the NLOS UVC channel. (a) demonstrates the block diagram of the experimental system. (b) shows the photograph of the experimental site. (c) and (d) give the components of the transmitter and the receiver, respectively.

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Fig. 3. Recording the input signal of the NLOC UVC channel. (a) exhibits the photograph of the experimental site. (b) is the input signal captured on an oscilloscope. (c) gives the input signal after averaging.

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In order to minimize the influence of the response of the circuits and devices at the transmitter and the receiver on experimental results, we capture the signal transmitted via a line-of-sight (LOS) link at a negligible range, as shown in Fig. 3. Figure 3(a) presents the photograph of the experimental setup, where the distance between the transmitter and the receiver is about 1.5 meters. The transmitter emits a pulse with a certain width, and the corresponding signal of interest captured on the oscilloscope is depicted in Fig. 3(b). The blue curve denotes the output signal of the monitoring interface from the driver, which triggers the oscilloscope, and the red curve represents the received signal. Due to the presence of noise, the receiver captures the waveform multiple times and averages them to obtain the smoothed waveform, as shown in Fig. 3(c). It can be observed that the rising edge of the received signal is very steep. Since this waveform is obtained after the signal passes through the whole circuits and devices from the transmitter to the receiver via an extremely short LOS link, it can be considered as the input signal of the system, which is exactly the NLOS channel. In this case, only a segment of the signal after the rising edge of the pulse and before its descent is measured. That signal represents the channel step response.

Since the detected signal at the receiver of the NLOS UVC system is particularly weak, and the number of photons within a pulse may even be zero, we collected $10^5$ samples for each measurement. The whole observation time $T$ is divided into successive periods, each lasting $T_c$. The total number of photons in each period $T_c$ is stored in the corresponding entry of a counting vector $\mathbf {v_c}$, whose number of elements equals $\lfloor \frac {T}{T_c} \rfloor$, where $\lfloor \cdot \rfloor$ is the floor function. Processing each sample offline involves counting the detected photons in each period and aggregating the counts to the corresponding element in $\mathbf {v_c}$. Then, the channel step response of the NLOS UVC link can be obtained until $10^5$ samples have been handled. Since that channel step response is discrete, the CIR can be obtained by taking the difference between adjacent elements, i.e., $h[n]=\mathbf{v}_{\mathbf{c}}[n+1]-\mathbf{v}_{\mathbf{c}}[n]$, where $\mathbf{v}_{\mathbf{c}}[n]$ denotes the $n$-th element of $\mathbf{v}_{\mathbf{c}}$.

4. Results and discussion

The CIR results of NLOS UVC computed through the proposed theoretical single-scatter CIR model are demonstrated and discussed in this section. The emission patterns of the light sources obeying UD and LD are both considered. Additionally, the delay spread (DS) is also provided to quantitatively evaluate the temporal characteristics of NLOS UVC channels and defined as [33,34]

Besides, the results of CIR and PL obtained through the MC-based NLOS UVC channel model [10] with the first-order scattering events (labeled “1st order” in legends) and multiple scattering events (labeled “overall” in legends) considered are exhibited to verify the corresponding theoretical results. The scattering order in the MC-based channel model is set to 3 under the multiple scattering condition. The number of photons in MC simulations is set to $10^8$ to mitigate the jitter of the results caused by the randomness of discrete photon arrivals at the receiver. Geometric parameters of NLOS UVC systems and atmospheric parameters used in the computation are extracted from [8,17] and listed in Table 1 unless otherwise specified.

Table 1. Geometric and atmospheric parameter settings

View Table

The influence of changing the range between the Tx and the Rx on the channel characteristics of NLOS UVC is shown in Fig. 4. Here, we set ${\theta _T} = {\theta _R} = {60^ \circ }$. The CIR of NLOS UVC links with the emission pattern of the light source obeying UD and LD are given in Fig. 4(a) and (b), respectively, where both the conditions of $r=60$ m and $r=100$ m are considered. Theoretical CIR results obtained through the proposed theoretical single-scatter CIR model demonstrate good agreement with those got by the MC-based channel model with both first-order and multiple scattering events considered because the total received optical energy is dominated by that from the first-order scattering event, which verifies the proposed theoretical single-scatter CIR model. Additionally, for both emission patterns, the CIR curve of $r = 60$ m rises at about 0.2 $\mu {\textrm {s}}$ whereas that of $r = 100$ m at roughly 0.33 $\mu {\textrm {s}}$, i.e., the former rises earlier than the latter. This is because the longer the baseline range, the more time it takes for the light to travel from the Tx to the Rx. The DS and PL of NLOS UVC links versus the baseline range are further given in Fig. 4(c) and (d), respectively. Figure 4(c) reveals that as the baseline range between the Tx and the Rx extends, the DS of NLOS UVC links increases, resulting in a reduction of the coherence bandwidth. This is because, with the increase of the baseline range, the common volume between the beam and the FOV where single-scatter events take place becomes larger, which makes the potential time interval for a photon to reach the Rx wider. It can be seen from Fig. 4(d) that with the extension of the baseline range, the PL increases, indicating a decrease in received optical energy. Therefore, when the baseline range increases, the CIR curve shortens and broadens, and the area under it becomes less, as evidenced by the CIR results presented in Fig. 4(a) and (b). In other words, the channel quality of NLOS UVC deteriorates with the increase of the baseline range between the Tx and Rx.

 

Fig. 4. The CIR of NLOS UVC links when the emission pattern of the light source obeys (a) LD and (b) UD with both $r = 60$ m and $r = 100$ m considered. (c) DS and (d) PL of the NLOS UVC links versus the baseline range between the Tx and the Rx.

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Figure 5 illustrates the relationship between channel characteristics of NLOS UVC and ${\phi _R}$. Specifically, the CIR results of NLOS UVC links with ${\phi _R}$ equaling ${50^ \circ }$ and ${90^ \circ }$ are given in Fig. 5(a) and (b), respectively. Figure 5(c) and (d) plot the DS and PL results of NLOS UVC links, respectively, versus ${\phi _R}$. Here, we set $r = 100$ m and ${\theta _T} = {\theta _R} = {30^ \circ }$. It can be observed that the theoretical single-scatter CIR results in Fig. 5(a) and (b) match well with the corresponding CIR results obtained by the MC-based channel model when the first-order scattering events are only considered. Nevertheless, the overall CIR versus time curves with multiple scattering events considered are jagged and slightly higher than single-scatter ones. This is because considering multiple scattering events enhances the randomness of discrete photon arrivals in the MC-based channel model, increases the received optical energy during a given time slot, and extends the time interval within which a photon can arrive at the Rx. Hence, the DS of the overall CIR results obtained by the MC-based channel model is also slightly larger than that of single-scatter CIR results, as shown in Fig. 5(c). At the same time, more optical energy can be received by the Rx via higher-order scattering events, leading to lower PL, as shown in Fig. 5(d). However, the difference between the CIR curves in Fig. 5(a) and (b) of the single-scatter and multiple-scatter cases is unobvious and acceptable in practice. It is worth noting that the computational complexity of the proposed theoretical single-scatter CIR model is much less than that of the MC-based one. Specifically, to obtain Fig. 5(a), the MC-based CIR model costs 1225.8 s whereas the proposed theoretical single-scatter CIR model costs about 6.8 s, less than 0.6% of the former, on a laptop with a CPU of 2.8 GHz and memory of 16 GB, where we set the temporal resolution to 2 ns and the number of photons in the MC-based channel model to ${10^8}$. In other words, the computational time cost by the proposed theoretical single-scatter CIR model is reduced by about three orders of magnitude compared with the MC-based one. In addition, it can be seen from Fig. 5(c) and (d) that both DS and PL have a minimum value when ${\phi _R}$ equals ${90^ \circ }$. This is because when ${\phi _R}$ equals ${90^ \circ }$, the NLOS UVC link is in coplanar geometry, where the axes of the beam and FOV are in the same plane. Besides, comparing Figs. 4 and 5 shows that increasing the inclination angle could improve the channel condition of NLOS UVC links. For example, when ${\theta _T} = {\theta _R} = {30^ \circ }$ and ${\phi _R} = {90^ \circ }$, the single-scatter DS and PL for the emission pattern obeying LD are roughly 0.41 $\mu {\textrm {s}}$ and 111.5 dB, respectively, as shown in Fig. 5(c) and (d). However, when ${\theta _T} = {\theta _R} = {60^ \circ }$ and $r = 100$ m, the single-scatter DS and PL for the emission pattern obeying LD are about 0.044 $\mu {\textrm {s}}$ and 106 dB, respectively, as shown in Fig. 4(c) and (d), which manifests an obvious improvement in both DS and PL. This is because as the inclination angles of the Tx and the Rx increase, the pointing direction of the Tx and the Rx changes from upward to toward each other. Therefore, the received optical energy can be boosted due to the enhancement of directional transmission, resulting in low PL. In addition, the common volume between the beam and the FOV becomes small, leading to narrow CIR curves and low DS.

 

Fig. 5. The CIR of NLOS UVC links with (a) ${\phi _R} = {50^ \circ }$ and (b) ${\phi _R} = {90^ \circ }$. (c) DS and (d) PL of the NLOS UVC links versus ${\phi _R}$.

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Figure 6 presents the influences of changing ${\beta _R}$ on the channel characteristics of NLOS UVC. Figure 6(a) gives the CIR curves of NLOS UVC links when ${\beta _R}$ equals ${60^ \circ }$. DS and PL of NLOS UVC links against ${\beta _R}$ are depicted in Fig. 6(b) and (c), respectively. Here, we set $r = 100$ m and ${\theta _T} = {\theta _R} = {30^ \circ }$. It can be found from Fig. 6(b) and (c) that DS slightly increases, but PL obviously decreases, as ${\beta _R}$ increases. For example, from ${\beta _R} = 10^ \circ$ to ${\beta _R} = 80^ \circ$, the DS of the single-scatter NLOS UVC link with the emission pattern of light source obeying LD and UD increases by roughly 17% and 100%, respectively, whereas the PL of them both reduces by almost 17 dB, nearly 50 times. That is, although a large full FOV angle could enlarge the common volume between the beam and the FOV and further increase DS, it is also beneficial to increase the received optical energy and further reduce the PL. More insightfully, it can be found that the area under the CIR curves in Fig. 6(a) ($\beta _R = {60^ \circ }$) is obviously larger than that in Fig. 5(b) ($\beta _R = {30^ \circ }$), but the broadening of CIR curves is less pronounced. In addition, from Fig. 6(b), we can further observe that the impact of increasing ${\beta _R}$ on the DS performance for the light source with the uniformly distributed emission pattern is more severe than that for the light source with the emission pattern obeying LD. This happens because the light radiation is uniformly distributed within the beam divergence angle and suddenly disappears when ${\gamma _T}$ exceeds ${\beta _T}$ for the emission pattern following UD. Under this condition, the common volume between the beam and the FOV could obviously vary as ${\beta _R}$ changes, resulting in a distinct change in DS. On the contrary, in the case of a light source with the emission pattern obeying LD, the light radiation is distributed within a hemisphere. The intensity of light gradually diminishes from the center axis to the base of the hemisphere, which is independent of the ${\beta _T}$ for LD, leading to a gentle change in DS. Furthermore, the difference in the total received optical power scattered from the common volume between those two emission patterns is not such obvious, leading to nearly identical trends in the PL curves shown in Fig. 6(c).

 

Fig. 6. (a) The CIR of NLOS UVC links with ${\beta _R} = {60^ \circ }$. (b) DS and (c) PL of the NLOS UVC links versus ${\beta _R}$.

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The channel characteristics of NLOS UVC versus the azimuth angle of the Tx ($\phi _T$) are exhibited in Fig. 7. The CIR results of NLOS UVC links with ${\phi _T=-50^\circ }$ are given in Fig. 7(a). Figure 7(b) and (c) show the DS and PL results of NLOS UVC links, respectively, against ${\phi _T}$. Here, we set $r = 100$ m and ${\theta _T} = {\theta _R} = {30^ \circ }$. It can be seen from Fig. 7(a) that the results of the proposed theoretical single-scatter CIR model match well with those of the MC-based model, which manifests that the proposed theoretical single-scatter CIR model can also work well when changing the parameters of the Tx. Additionally, comparing Fig. 7(a) and Fig. 5(b), it can be observed that the area under the CIR curve with ${\phi _T=-50^\circ }$ is less than that with ${\phi _T=-90^\circ }$. More specifically, it can be further found from Fig. 7(b) and (c) that when the NLOS UVC link is in the coplanar geometry, i.e., ${\phi _T=-90^\circ }$, the results of DS and PL both reach the minimum. In other words, when the axes of the beam and FOV are located in the identical plane, the channel condition is the best, which is also consistent with the conclusion of Fig. 5.

 

Fig. 7. (a) The CIR of NLOS UVC links with ${\phi _T} = {-50^ \circ }$. (b) DS and (c) PL of the NLOS UVC links versus ${\phi _T}$.

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Figure 8 demonstrates the channel characteristics of NLOS UVC against $\beta _T$. Here, we set $r = 100$ m and ${\theta _T} = {\theta _R} = {30^ \circ }$. Figure 8(a) shows the CIR results of NLOS UVC links with ${\beta _T=30^\circ }$. The DS and PL results of NLOS UVC links versus ${\beta _T}$ are provided in Fig. 8(b) and (c), respectively. We can observe that the CIR curve of ${\beta _T=30^\circ }$ (Fig. 8(a)) is narrower than that of ${\beta _T=60^\circ }$ (Fig. 5(b)) for both emission patterns of the light source. For example, when the emission pattern of the light source follows LD, the CIR curve of ${\beta _T=30^\circ }$ starts at $0.4 \mu {\textrm {s}}$ and ends at $1.2 \mu {\textrm {s}}$ whereas that of ${\beta _T=60^\circ }$ begins at $0.33 \mu {\textrm {s}}$ and ends at $1.7 \mu {\textrm {s}}$. In addition, the area under the CIR curve of ${\beta _T=30^\circ }$ (Fig. 8(a)) is larger than that under the CIR curve of ${\beta _T=60^\circ }$ (Fig. 5(b)) regardless of the irradiation pattern of the light source. That phenomenon is further clearly revealed by Fig. 8(b) and (c). It can be seen from Fig. 8(b) and (c) that with the increase of $\beta _T$, both the DS and PL ascend. That is, as the divergence angle of the light beam increases, the channel condition will deteriorate. This is primarily because, as $\beta _T$ becomes larger, the common volume between the beam and the FOV becomes greater, resulting in a wider CIR curve and hence long DS; whereas the optical power density reduces, leading to less received optical power within the same FOV of the Rx and hence high PL.

 

Fig. 8. (a) The CIR of NLOS UVC links with ${\beta _T} = {30^ \circ }$. (b) DS and (c) PL of the NLOS UVC links versus ${\beta _T}$.

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Although we have briefly demonstrated the influence of changing the inclination angle of the Tx and the Rx on the channel condition of NLOS UVC by comparing Figs. 4 (${\theta _T} = {\theta _R} = {60^ \circ }$) and 5 (${\theta _T} = {\theta _R} = {30^ \circ }$), here the detailed results of the channel characteristics of NLOS UVC versus the inclination angle of the Tx and the Rx are further provided in Fig. 9. Here, we set $r = 100$ m. The CIR results of NLOS UVC links with ${\theta _T} = {\theta _R} = {0^ \circ }$ are exhibited in Fig. 9(a). The DS and PL results of NLOS UVC links against ${\theta _T}$/${\theta _R}$ are depicted in Fig. 9(b) and (c), respectively, where we set ${\theta _T}$ and ${\theta _R}$ equal. Comparing Fig. 5(b) and Fig. 9(a), it can be found that the CIR curve of ${\theta _T} = {\theta _R} = {0^ \circ }$ is much broader than that of ${\theta _T} = {\theta _R} = {30^ \circ }$. This is mainly because as the inclination angles decrease, the orientations of the Tx and the Rx gradually point upward; hence the common volume between the beam and the FOV becomes increasingly large. Due to that fact, the DS of NLOS UVC grows as the inclination angles of the Tx and the Rx reduce, which is also demonstrated in Fig. 9(b). Besides, it can be further observed from Fig. 9(c) that the PL of NLOS UVC also increases when the inclination angles of the Tx and the Rx decrease. On the contrary, the channel condition of NLOS UVC can be ameliorated through the utilization of large inclination angles of the Tx and the Rx.

 

Fig. 9. (a) The CIR of NLOS UVC links with ${\theta _T} = \theta _R = {0^ \circ }$. (b) DS and (c) PL of the NLOS UVC links versus $\theta _T$ and $\theta _R$.

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The experimental results to validate the proposed theoretical single-scatter CIR model are provided in Figs. 10 and 11, where we set ${\phi _T} = - {90^ \circ }$, ${\phi _R} = {90^ \circ }$, $\beta _R={60^ \circ }$. In the offline processing of the measured data, we set the counting period $T_c=20$ ns for the channel step response. In addition, the measured results of the channel step response demonstrate noticeable random fluctuations due to the shot noise in weak NLOS UVC links. To obtain relatively smoothed CIR results, the counting period $T_c$ is extended to 200 ns for Fig. 10(d), Fig. 10(e), Fig. 11(b), and 300 ns for Fig. 10(f). The predicted results of the channel step response are obtained by convolving the input signal waveform in Fig. 3(c) with the theoretical CIR results obtained from the proposed theoretical single-scatter CIR model and then normalizing it. The experimental results of the channel step response are normalized using the average value of the received step signal at the top stable region. Both experimental and theoretical results of CIR are normalized by their own maximum value.

 

Fig. 10. (a), (b), and (c) demonstrate the channel step response of the NLOS UVC link with the distance between the Tx and the Rx equaling 10m, 20m, and 30m, respectively. (d), (e), and (f) exhibit the CIR of the NLOS UVC link with the distance between the Tx and the Rx equaling 10m, 20m, and 30m, respectively.

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Fig. 11. (a) and (b) present the channel step response and the CIR of NLOS UVC link, respectively, with $\theta _T$ and $\theta _R$ both equaling ${30^ \circ }$.

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Figure 10 demonstrates the experimental and theoretical results of channel step response and CIR of NLOS UVC links with different distances between transceivers. Here, we set $\theta _T=\theta _R={0^ \circ }$. The results of the channel step response with $r=10$ m, $20$ m, and $30$ m are provided in Fig. 10(a), (b), and (c), respectively. The results of the CIR with $r=10$ m, $20$ m, and $30$ m are further illustrated in Fig. 10(d), (e), and (f), respectively. It can be found that the predicted results of the channel step response exhibit good agreement with the corresponding experimental results, which confirms the validity and efficacy of the proposed theoretical single-scatter CIR model. In addition, the theoretical results of CIR comparatively match the measurements. The measurement errors are mainly attributed to the fact that the experimental results of the channel step response are highly influenced by shot noise, even though we have collected $10^5$ samples for each measurement. That quantity has already reached the limitation of our experimental equipment. Therefore, it is also indicated that the performance of the NLOS UVC is dominantly constrained by received optical power but not the coherence bandwidth of the channel. Furthermore, it can be seen from Fig. 10(d) to (f) that, with the increase of $r$, the theoretical CIR undergoes apparent broadening. That phenomenon is also reflected in Fig. 10(a) to (c), where both the experimental and theoretical curves of the channel step response increasingly rise slowly as $r$ increases.

The impact of changing the inclination angles of the Tx and the Rx on the channel step response and CIR of NLOS UVC links has been experimentally revealed in Fig. 11. Here, we set $r=30$ m and $\theta _T=\theta _R={30^ \circ }$. The proposed theoretical CIR model also accurately predicts the temporal behavior of the NLOS UVC link when the Tx and the Rx point toward each other. Comparing Fig. 11(a) with Fig. 10(c), it is evident that, with the increase of the inclination angle, both the experimental and theoretical curves of the channel step response become steeper. This indicates that the delay of the CIR can be mitigated by increasing the inclination angle, as shown in Fig. 11(b) and Fig. 10(f). In addition, the measured PL of the NLOS UVC link decreases by approximately 4.7 dB when $\theta _T$ and $\theta _R$ increase from ${0^ \circ }$ to ${30^ \circ }$. Hence, increasing the inclination angles of the Tx and the Rx enhances the channel quality of NLOS UVC.

5. Conclusion

In this work, a novel theoretical single-scatter CIR model of NLOS UVC in noncoplanar geometry was proposed based on the spherical coordinate system. The results of the proposed theoretical single-scatter CIR model of NLOS UVC were verified by the widely accepted MC-based NLOS UVC channel model. Besides, an outdoor experiment for measuring the temporal characteristics of the LED-based NLOS UVC is designed and performed to validate the proposed theoretical single-scatter CIR mode. The results demonstrate that the proposed theoretical single-scatter CIR model of NLOS UVC could substantially reduce the computational time in comparison with the MC-based one, accounting for approximately a mere 0.6% of the latter. In addition, reducing the baseline range between the Tx and Rx, adjusting the geometry of the NLOS UVC system to the coplanar case, narrowing the beam divergence angle, and increasing the inclination angles of Tx and Rx can lower both the DS and the PL, improving the channel condition of the NLOS UVC. Nevertheless, widening the full FOV angle can reduce the PL but increase the DS. This work will be helpful for analyzing the temporal characteristics of the NLOS UVC channels.