1. Introduction

Recent advances in constructing low-cost and reliable ultraviolet (UV) transceivers, as well as its inherent advantages (e.g., solar-blind [1–4] and non-line-of-sight (NLOS) [1, 2, 5]), have made the ultraviolet communications (UVCs) a promising supplement to the traditional wireless communications. From academic perspectives, increasing efforts have been spent on model formulations [2, 6–8], and scheme designs [9–12], which aim to enhance the accuracy of informative signal detections, or monitor the parameters of interest, e.g., transmission distance, apex or directional angles. In practice, detecting signals and estimating channels may be severely affected by the inter-symbol-interference (ISI), which needs to be addressed.

In the context of UVCs, signal detections and channel estimations can be viewed as a mixed detection and estimation (MDE) problem. The detection of informative signals and the estimations of channel characteristics may mutually interact with each other. To be specific, one erroneous signal detection may mislead the inference of channel characteristics, which in turns may affect the next-round detection. Most existing works focusing on either channel estimation or signal detection unfortunately, did not consider this coupling interactions between estimation and detection, thereby becoming less attractive in applications. For instance, [11,13,14] providing effective UV channel codes under known channel characteristics, may not perform as expected if the channel impulse response (CIR) is unattainable, let alone the aforementioned interactions. Similarly, with assuming a perfectly known CIR, [15,16] proposing receivers and detectors to combat ISI, although theoretically significant, may not function well if the channel parameters are unknown. Channel estimation schemes in [17] can derive precise channel estimations via a designed pilot sequence, however may not be able to pursue estimations from the random informative signals, e.g., random on/off keying (OOK) signals. To the best knowledge, [9] was the first to consider both channel estimations and signal detections, designing a UV channel-related pilot sequence to acquire the CIR, which will then be used in maximum-likelihood sequence detection (MLSD) for acquisition of signal stream. However, sending and processing pilot sequences may cost extra time and energy. Besides, the time-delay may be underestimated in acquisitions of real-time channel characteristics. As such, by exploiting the imperfectly known UV channel parameters, the detection performance of the MLSD (referred as imperfect MLSD) may be subsequently compromised, and therefore simultaneously pursuing channel estimation as well as signal detection is of necessity.

The MDE problem has been widely studied in the traditional wireless communications [18–22]. The term deep sensing was first proposed in [21, 22], to detect a primary-user’s spectrum utilization, and estimate its time-varying channel in cognitive radio scenarios. However, there is little research on how to cope with the MDE problem in the UVC scenarios, especially when encountering the specific UV ISI driven by the photon scattering phenomenon.

In this paper, to combat the ISI effect on the MDE problem, we focus on devising an UV channel-related Bayesian scheme, which can detect informative signals and estimate the channel characteristics simultaneously. To sum up, the main contributions are listed as follows:

2. UV single-scattering model and problem formulation

In general UVC scenarios, a transmitter and a receiver is deployed as shown in Fig. 1. As for the receiver, we assume we know most of the channel parameters (i.e., the emitted energy in terms of the photons number N_T, the atmospheric coefficients k_e and k_a, the receiver half-field of view θ_R, and the receiver apex angle β_R), except for parameters that are relevant with the transmitter (i.e., the transmission distance r, the transmitter beam divergence θ_T, the transmitter apex angle β_T). Here, we assume the atmospheric coefficients k_e and k_a are constants within the detection and estimation period (or at least remain unchanged with a coherence time in the order of 100ms compared with signal interval T_b in the order of μs) [23]. The formula of CIR, denoted as h(t, r, θ_T, β_T) is known from [6], referred as single-scatter photon model (see in Fig. 1).

 

Fig. 1 UV system structure. a) gives the UVC system with channel parameters, i.e., the transmission distance r, the transmitter beam divergence θ_T, the transmitter apex angle β_T, the receiver half-field of view θ_R, and the receiver apex angle β_R. b) illustrates the CIR h(t) under conditions of the emitted photons number N_T = 1.7 × 10¹³, the atmospheric coefficients k_a = 5 × 10⁻⁴m⁻¹ and k_s = 4.9 × 10⁻⁴m⁻¹, and channel parameters illustrated in (a), i.e., r = 250m, β_T = β_R = π/4, and θ_T = θ_R = π/12. When T_b = 0.1μs, we have M = 7 and the value of q_m.

Download Full Size | PDF

For each discrete time-slot k, we characterize the average number of received photons, denoted as λ_k, as follows $\sum$:

The number of received photons (i.e., n_k ∈ ℕ⁺), modeled as a Poisson random variable with average λ_k [2,24], will be measured in terms of the voltage measurement, denoted as z_k, i.e. [3]:

3. UV channel-related Bayesian scheme

In this section, we will elaborate the UV channel-related Bayesian scheme, aiming to simultaneously detect the informative signal x_k, as well as estimate unknown channel parameters i.e., the transmission distance r, the transmitter beam divergence θ_T, and the transmitter apex angle β_T. To this end, we firstly construct the UV CSI, denoted as u_k which involves informative signal and the parameters of our interest. Then, a transitional probability matrix (TPM) of the UV CSI in order to describe the dynamic evolution from u_k−1 to u_k is provided. Relying on the TPM, the sequential Bayesian process is suggested to sequentially estimate the requested CSI. Finally, the signal and channel parameters will be inferred from the derived CSI.

3.1. Construction of UV CSI

In order to characterize the random informative signals and unknown channel parameters, we build an union UV CSI, defined as u_k ≜ [x_k−M:k, r, θ_T, β_T]^T which contains the elements waiting to be inferred.

It is noteworthy that u_k and u_k−1 have an overlapping part, suggesting an evolution of u_k from u_k−1. In other words, by defining the space of the UV CSI as u_k ∈ 𝕊 ≜ {s₁, s₂..., s_L}, L = 2^M+1, we construct the transitional TPM, denoted as P to characterize the CSI evolution from (k − 1)th to kth time-slot, i.e.:

For the likelihood density of the UV CSI, denoted as φ_k(z_k|u_k), we measure u_k in terms of the λ_k given by Eqs. (1) and (2), i.e.:

3.2. Bayesian framework

Given the transitional PDF of the CSI as ξ_k(u_k|u_k−1), its likelihood density, φ_k(z_k|u_k), and the measurement vector i.e., z_1:k ≜ [z₁, z₂, ..., z_k]^T, we consider the Bayesian framework in order to derive the UV CSI, i.e.:

After the acquisition of the posterior density, the estimated CSI, denoted as û_k can be derived by maximizing the posteriori, i.e.:

With the help of the obtainment of û_k, we can subsequently derive the x̂_k, r̂, θ̂_T, and β̂_T.

3.3. Particle filter realization of Bayesian framework

Note from Eqs. (7) and (8), the derivation of predicted density ϱ_k|k−1(u_k|z_1:k−1) relies on the complicated integral computations, which will lead to computational infeasibility for the UVCs. In order to cope with this difficulty, we design the PF-based algorithm to implement the Bayesian framework. In essence, PF uses a group of random discrete particles, denoted as u⁽ⁱ⁾, i = 1, 2, ..., I with evolving probability weights, denoted as w⁽ⁱ⁾ to approximate the complex distribution, (e.g., $\varrho (\mathbf{u})\backsimeq {\sum }_{i=1}^I\delta \left(\mathbf{u}-{\mathbf{u}}^{(i)}\right)\times w^{(i)}$).

For this UVC scenarios, we assign the particles as the UV CSI, i.e., ${\mathbf{u}}_k^{(i)}={\left[{\mathbf{x}}_{k-M:k}^{(i)},r^{(i)},{\theta }_T^{(i)},{\beta }_T^{(i)}\right]}^T$, which will be then propagated to approximate the predict and update stages of the Bayesian framework.

3.3.1. Implementation for predict-stage

At the beginning of the kth time-slot, the initial particles are drawn from the posterior PDF at (k − 1)th time-slot, i.e.:

With the initial particles and their weights, the predict-stage can be pursued by simulating the transitional evolutions of particles at (k − 1)th time-slot, i.e.:

3.3.2. Implementation for update-stage

As we acquire the predicted particles and their weights, i.e., ${\left\{{\mathbf{u}}_{k|k-1}^{(i)},w_{k|k-1}^{(i)}\right\}}_{i=1}^I$, the update-stage will be performed by updating the weights, i.e.:

Note that in Eq. (14), the likelihood PDF, ${\phi }_k\left(z_k|{\mathbf{u}}_{k|k}^{(i)}\right)$ is relevant with particle ${\mathbf{u}}_{k|k}^{(i)}$, so we need to compute it by considering the specific elements of the particle. In this view, the likelihood for particle ${\mathbf{u}}_{k|k}^{(i)}$ is derived by taking ${\lambda }_{k|k}^{(i)}$ into Eq. (6). Here, ${\lambda }_{k|k}^{(i)}$ is the intermediate dependent with the current CSI ${\mathbf{u}}_{k|k}^{(i)}$, i.e.:

With the help of Eqs. (16) and (17) to obtain the updated particles and their weights in Eqs. (14) and (15), the posterior PDF can be therefore approximated as:

After the acquisition of the posterior PDF and the estimative result, re-sample process should be pursued in order to eliminate particles with negligible weights and reproduce those whose weights are of importance [22,25,26]. Finally, as the accomplishment of the re-sample process, all particles remained are of identical importance, and should be assigned as identical weights, which will be used for the next-round (i.e. (k + 1)th time-slot) PF algorithm.

3.4. Algorithm flow of the UV channel-related Bayesian scheme

With the help of the PF-based algorithm, a schematic flow of the UV channel-related Bayesian scheme is illustrated by Algorithm 1.

Algorithm 1. PF-based UV channel-related Bayesian scheme

View Table

At the kth time-slot, the particles ${\mathbf{u}}_{k-1|k-1}^{(i)}$ and their weights $w_{k-1|k-1}^{(i)}$ at the (k − 1)th time-slot, as well as the current measurement z_k serve as the input. Step 1–2 derive the predicted particles and their weights, which will be used to obtain the updated particles and the corresponding weights in step 3–4. Step 5 computes the approximated posterior PDF. Step 6 acquires the estimative UV CSI by maximizing the posteriori. Step 7 gives the re-sample process, which will reproduce particles with relatively large weights, and eliminate those whose weights are trivial. Step 8 finally normalizes the weights as $w_{k|k}^{(i)}=1/I$. The outputs are estimative CSI, and the posterior particles that will be used for the next-round Bayesian scheme.

4. Numerical analysis

In this section, we evaluate the performance of the proposed UV channel-related Bayesian scheme on detecting informative signals and estimating channel parameters, as well as the computational complexity.

Configurations in this analysis are assigned as follows. The number of emitted photons for each time-slot is assigned as N_T = 1.7 × 10¹³. The atmospheric coefficients are k_a = 5 × 10⁻⁴m⁻¹ and k_s = 4.9 × 10⁻⁴m⁻¹ respectively [6]. The transmitter beam divergence and the receiver half-field of view are assigned as θ_T = θ_R = π/12, while the transmitter and receiver apex angles are β_T = β_R = π/4. The transmission distance is r = 250m. As aforementioned, we do not know r, θ_T and β_T at the receiver. We assign signal interval T_b for four different cases, i.e., T_b = 0.4μs, T_b = 0.3μs, T_b = 0.2μs, and T_b = 0.1μs. As the T_b decreases, the ISI memory are respectively M = 1, M = 3, M = 4 and M = 7, suggesting the gradually deteriorating ISI contamination.

As for the measurement function, we use signal-to-noise ratio (SNR) to describe the thermal noise level, i.e.:

4.1. Performance of proposed scheme on UV channel estimations

We firstly examine the performance of the proposed scheme on estimating channel parameters (i.e., r, θ_T, β_T), in terms of the relative error (RE), i.e.:

It is observed in Fig. 2 that RE(r), RE(θ_T) and RE(β_T) decrease with either the increment of signal-noise ratio (SNR) or the augment of T_b (indicating the dwindle of ISI memory M), suggesting that the measurement noise and the ISI are the two factors that affect the estimation performance. For instance, given a fixed T_b = 0.1μs, RE(r) decreases from 11% to 2%, as SNR rises from 0dB to 20dB. Then, if we fix the SNR as 10dB, RE(r) decreases from 6% to 4% with the changes of T_b from 0.1μs to 0.4μs. Similarly, as we assign T_b = 0.4μs, RE(θ_T) and RE(β_T) will both decrease from 10% to nearly 0.1% with the rise of SNR from 0dB to 20dB. Then, given the SNR as 10dB, RE(θ_T) and RE(β_T) will decline from 6% to 4%, and from 4% to 2% respectively, as we rise T_b from 0.1μs to 0.4μs.

 

Fig. 2 Performance of proposed scheme on UV channel estimation, where T_b represents the signal interval, r is the transmission distance, θ_T is the transmitter’s beam divergence, and β_T is the transmitter’s apex angle.

Download Full Size | PDF

Secondly, we can observe that the proposed scheme gains a promising estimation performance. Although the errors increase as the ISI and measurement noise grow stronger, the REs are still tolerant. For instance given T_b = 0.1μs, RE(r) is no more than 12%, which will be even lower if the SNR gets larger. At a fixed SNR as 10dB, our scheme can determine θ_T, and β_T respectively, within RE(θ_T) < 6% and RE(β_T) < 4.5%.

4.2. Performance of proposed scheme on signal detections

In the second experiment, we illustrate the performance of the proposed scheme on signal detections, in contrast with the imperfect MLSD (i.e., the MLSD algorithm with imperfectly known channel parameters r, θ_T, and β_T). Here, we use the bit error rate (BER) to represent the detection performance.

We can firstly observe from Fig. 3 that the BER becomes higher as the SNR becomes lower, suggesting that the measurement noise can deteriorate the communication performance. For instance, under a fixed T_b = 0.4μs, when the SNR decreases from 15dB to 10dB, the BER of the proposed scheme increases from 10⁻⁸ to 10⁻³. Besides, as the T_b decreases (which indicates the stronger effect of the ISI), the BER will also get larger, showing that the ISI can also severely disturb the detection performance. For instance, given the SNR=15dB, the BER derived from the proposed scheme increases from 10⁻⁸ to nearly 9 × 10⁻³, when T_b declines from 0.4μs to 0.1μs. This can be explained as that the decrease of bit interval T_b will make the ISI memory M increase, which will subsequently impact more on the signal detection process.

 

Fig. 3 Performance of proposed scheme on signal detections.

Download Full Size | PDF

Secondly, it is noticeable from Fig. 3 that the proposed scheme outperforms the MLSD when the channel estimations are not perfect. For instance, given T_b = 0.1μs and SNR= 20dB, the BER of the proposed scheme is nearly 10⁻⁷, while the value derived from the MLSD is above 10⁻⁶. Then, as the T_b grows to 0.2μs, the proposed scheme gains nearly 4dB in contrast with the imperfect MLSD. This gain will be further increasing as the T_b gets larger. For instance, when T_b = 0.4μs, the BER of the proposed scheme approaches to nearly 10⁻⁸ at the point SNR=15dB, whereas the imperfect MLSD needs extra 5dB SNR in order to get a BER as 10⁻⁸. Hence, the numerical analysis shows that the detection performance of the proposed UV channel-related Bayesian scheme is better than the one of the imperfect MLSD. This may be attributed to that the proposed scheme can estimate the UV channel parameters (i.e., transmission distance r, transmitter’s beam divergence θ_T, and transmitter’s apex angle β_T as shown in Fig. 2), which if remaining unknown, will deteriorate the signal detection process.

4.3. Complexity analysis

We further evaluate the computational complexity of the proposed scheme, in contrast with the imperfect MLSD.

In general, the complexity of Bayesian scheme is roughly measured by the total times of likelihood density computations, each of which has been given as ϑ [27], which is related with the representative precision and various adopted algorithms. For our proposed UV channel-related Bayesian scheme, the times of computations are proportional to the size of simulated particles (i.e., I) of the PF algorithm. Therefore, the total complexity of the proposed scheme can be described as O(I · ϑ).

By contrast, the complexity of the MLSD scheme (which is also measured by the total times of the likelihood computations) depends on the discretion of λ_k. For each possible λ_k, one likelihood will be computed. Therefore, by enumerating all possible λ_k from the L = 2^M+1-dimension space 𝕊, the complexity can be subsequently computed as L × ϑ.

Given a requested BER as 10⁻⁶, and SNR=20dB, we compare the times of likelihood computations between the proposed scheme and the imperfect MLSD, related to the ISI memory M. It is seen from Fig. 4 that as the ISI memory M increases from M = 4 to M = 9, both the computation times from the proposed scheme and the MLSD are rising in order to approach the requested BER. For instance, the times of the proposed scheme increases from 50 to 500, whereas the times of the MLSD grows from 32 to 1024. Moreover, we can observe that the times from the MLSD grows exponentially, much quicker than that of the proposed scheme. For instance, when M = 8, we need only I = 300 computations for the proposed scheme, as opposed to nearly L = 500 computations in the MLSD scheme.

 

Fig. 4 Complexity of proposed scheme as opposed to imperfect MLSD.

Download Full Size | PDF

Therefore, numerical analysis demonstrates that our proposed UV channel-related Bayesian scheme, by simultaneously estimating the channel and detecting signals, can outperform the MLSD in the cases when channel characteristics are unknown (as shown in Fig. 3). Meanwhile, the proposed scheme consumes less energy as opposed to the MLSD scheme. These two suggest that the proposed scheme can gain the detection and estimation performance in an energy-efficient pattern.

5. Conclusion

For most communication applications, it is significant to obtain channel characteristics and acquire informative signals. As far as the UVC scenarios are concerned, acquisition of such two features becomes a core challenge when encountering the ISI disturbance. To address this, we have proposed the UV channel-related Bayesian scheme which can simultaneously derive both the channel characteristics and the informative signals. Numerical analysis indicates our proposed scheme, which gains an extra 4dB detection performance in contrast with that of the imperfect MLSD, and derives an estimation performance under 4% relative errors. Our new scheme, therefore, may provide a great promise to the future UVC scenarios.