1. Introduction

Recent advances made in constructing low-cost semiconductor devices [1, 2], together with the scattered phenomenon [3], and the solar-blind (SB) feature [4, 5] have increased growing research activities on ultraviolet communications (UVCs), ranging from theoretical investigations (e.g., single-scatter model [6, 7], multi-scatter model [8,9]) to applications (e.g., networks [10], and military communications [11]).

In the context of applicative scenarios of UVC systems, two main factors may heavily deteriorate the communication performance, i.e., the inter-symbol-interference (ISI), and the path loss propagation. In applications of relatively long-distance and high-rate communications, ISI should be inevitably considered as a significant factor [12]. Increasing efforts have been spent on studying the ISI issue. For instance, [6] provide a single scatter model to describe the channel response function of ISI. In [12], a multi-antenna receiver has been designed to combat the ISI distortion.

In situations where the transmission span is shorter (e.g. < 100m), and the bit rate is lower, the ISI is not significant to have a noticeable impact on the communication performance. In such scenarios, the path loss propagation becomes the major factor [13–15]. To address this problem, a number of schemes utilizing space, frequency, and time resources - based on Shannon famous channel coding theorem - have been proposed [16,17]. However, the space and frequency schemes may be restricted because of the space-resource constrained scenarios and the narrow band of the solar-blind UV channel (i.e., 1.4 MHz).

Therefore, the time-resource schemes, whereby reduce the bit rate at relatively lower comparative to the channel capacity, would be a better option for UVC. In this scheme, channel coding schemes have been adopted to reduce the bit rate via the introduction of additional correlations between information bits, and consequently improve the communication performance. Several channel coding schemes, adopted from the traditional RF wireless communications, have been investigated in UVC. For example, Reed-solomon (RS) and the state-of-the-art low density parity check (LDPC) codes were adopted in [15] and achieved communication distance increase about 32% and 78% respectively. However, in these coding schemes the generator matrix did not consider the UV channel, which is different from the RF wireless channel. In [18] the performance of two improved belief propagation based decoding algorithms for LDPC codes based on the density evolution for the additive white Gaussian noise and Rayleigh channel was reported, which is widely adopted in RF wireless systems. In [19–21] the polar codes, which are based on channel polarization, with low encoding and decoding complexity were investigated with applications in multiple access channels, relay channels, quantum channels and quantum key distribution.

However, to the best of our knowledge, no related research works on the application of polar codes in UVC have been reported. In this paper, we investigate polar codes, with a relatively low computational complexity, in UVC to combat the path loss propagation induced attenuation. In general, the main contributions can be summarized as follows:

The rest of this paper is arranged as follows. Section II gives a path loss photon model by introducing the particle-wave duality. Section III provides the UV channel-related polar code. Numerical simulations and off-line experiments are described in section IV. We finally conclude this paper in Section V.

2. Problem and system formulation

In this section, we outline a SISO-UVC employing the polar codes as the channel coding method. The composition of the intensity modulated - direct detection (IM-DD) SISO-UV system can be divided into three parts; the Tx, the UV channel, and the Rx, as shown in Fig. 1.

 

Fig. 1 Schematic diagram of the SISO-UVC system.

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2.1. System structure

At the Tx, a binary stream is applied to a polar codes encoder, the output of which is passed through an arbitrary express software and an arbitrary waveform generator. Then the signal (in on and off keying format) is generated via the intensity modulation of an encapsulated LED array (e.g., 4 × 9) which is supplied by DC level.

Following propagation through the free space, at the Rx the received photons are detected by a high sensitivity photon multiplier tube (PMT), with a narrow-band UV filter in front of it. The output of PMT is captured using a real time digital oscilloscope for off-line processing. Finally, the received signals are demodulated and decoded to recover the transmitted bit stream.

2.2. Path loss UV photon model

In a general UV channel, the path loss L, which is defined as a ratio between the emitted energy E_t and the residue intensity received by the receiver E_r, is usually employed to describe the average energy loss, and can be specified according to an empirical model, i.e. [14],

On the other hand, this energy loss can be also characterized by the photons statistics [22], where photon’s received status is governed by an independent and identically distributed (i.i.d) Binomial distribution, i.e., $ℬ(N_t,p_r)$ where N_t is the number of emitted photons, and the received probability, which is given by:

Next, we consider the UV channel model in terms of the transitional probability matrix (TPM). In the case of sending 1-bit, the transitional probability, p(0|1) can be determined according to the Binomial distribution $ℬ(N_t,p_r)$, i.e.:

3. UV channel-related polar code

In this section we elaborate the polar code for UVC in terms of the BER performance and the computational complexity. As for the BER performance, we re-compute the error probabilities of the split channels premised on the UV channel, in order to strengthen the code correlations of the 2 × 2 core matrix-based polar code. In this way, the proposed channel-related polar code scheme differs from the existing polar schemes used in traditional wireless communications. Then, to further improve the BER performance, initialization for the SC decoder is operated based on the computed UV channel error rate. Furthermore, we consider the trade-off between the code performance and the computational complexity, which are directly related to the code length N = 2ⁿ, where n > 0 is the channel combination step. A relatively long N leads to improved performance at the cost of increased computational complexity. In order to determine the optimum N, we compute a Bhattacharyya upper-bound by inferring on the constructed binary tree, and fit it to the specific UV channel.

3.1. Polar code encoder

Assuming K input bits, denoted as ${\mathbf{u}}_1^K={\left[u_1,u_2,\dots ,u_K\right]}^T$, the polar encoded symbol ${\mathbf{y}}_1^N={\left[y_1,y_2,\dots ,y_N\right]}^T$ can be derived as:

The generator matrix G_N is from the n^th Kronecker power ${\mathbf{G}}_2=\mathbf{F}=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$, generating an N-input N-output channel (i.e., $W_N:{\mathcal{X}}^N\stackrel{{\mathbf{G}}_N}{\to }{\mathcal{Y}}^N$, where $\mathcal{X}=\{0,1\}$, and $\mathcal{Y}=\{0,1\}$) via combining two N/2-input N/2-output channels (i.e., W_N/2). In this way, the recursive flow of G_N can be described as follows: i) N-input sequentially pass through N/2 F-matrix; ii) the results are permutated by Π_N, whereby the 1^st and 2^nd N/2-element are those with original indexes assigned as odd and even, respectively; iii) the rearranged elements are passed through two W_N/2 with generator matrixes of G_N/2. Therefore, we have [23]:

The combination channel W_N can be equivalently seen as the composition of N splitting channels i.e., $W_N=\left\{W_N^{(i)}:\mathcal{X}\to \mathcal{Y}\times {\mathcal{X}}^{i-1},1\le i\le N\right\}$ with every splitting channel composed of a scaler input, which corresponds to the input of G_N, and a hybrid vector output. That is, for K < N information bits K number of noiseless splitting channels needs to be selected, which are determined by Bhattacharyya parameters $Z(W_1^{(1)})$ of the actual channel W. For a discrete memory less channel, $Z(W_1^{(1)})$ is given by, i.e., [23]:

The recursive properties of splitting channels can be determined by the iteration of Bhattacharyya values, which is given as [23]:

Note that, the splitting channel selection process highly depends on the initialization of $Z(W_1^{(1)})$. If $Z(W_1^{(1)})$ cannot characterize the property of a specific UV channel, then the propagating Bhattacharyya values will be considered unreliable, which will subsequently affect the code correlations and the BER performance. For correct initialization of $Z(W_1^{(1)})$ we introduce the UV TPM by substituting (3)–(4) into Eq. (8), i.e.:

3.2. Successive cancelation polar decoder

Considering the selection process of K splitting channels, there exist code correlations in the received data, therefore a SC decoding can be adopted as a baseline algorithm with a very low complexity. The SC decoder can estimate and decide the information bits based on the recursive structure of polar encoder. Note that, SC decoding is a suboptimal algorithm due to its susceptibility to error propagation. However, its error probability can be made arbitrarily small provided N is sufficiently long and R is less than the capacity. Here, we define the likelihood density per each splitting channel $W_N^{(i)}$ as:

Note that, initialization of $L_1^{(1)}(r_1)$ in (13)–(14) is important, since an improper $L_1^{(1)}(r_1)$ that cannot characterize the stochastic of the UV channel, thus leading to the further propagative likelihood error, and thereby deterioration of the decoding performance. According to (12), the computation of $L_1^{(1)}(r_1)$ can be given as:

Therefore, the SC decoding scheme can be operated by recursively computing the likelihood density of each splitting channels in (13) and (14).

3.3. Optimum polar code length for an UV channel

To determine the optimum N we need to analyze the relations between N and the BER performance. First, we deduce a newly recursive Bhattacharyya upper-bound with respect to n. Such an upper-bound can reflect the BER performance of a general formed polar code. Next, considering the upper-bound in an UV channel we determine the suitable value for n, which trade-offs the computational burden with the code performance.

3.3.1. Bhattacharyya upper-bound of polar code

Here, we focus on analyzing the general form of polar code, and compute a new Bhattaharyya upper-bound. That is, the Bhattacharyya values of the selected splitting channels (i.e., ${\mathcal{A}}_n$) corresponds to the BER of the polar code defined in terms of sum of Bhattacharyya values (SBV) as given by [23]:

Note that, (16) intuitively outlines that the BER of polar code is composed of the erroneous probabilities of the total selected splitting channels. However, form (16) it is difficult to compute the BER upper-bound for higher values of N, e.g., N = 2²⁰. Therefore, a computational approach that can be used to determine N is needed. More specifically, we show a recursive upper-bound for SBV investigating the generating process of each splitting channels.

(a) Binary tree model

The generation process of splitting channels best illustrated with reference to the binary tree model as shown in Fig. 2. As depicted the root node is the channel $W_1^{(1)}$ with Bhattacharyya value of $Z(W_1^{(1)})$, which branches out into the upper and lower channels $W_2^{(1)}$ and $W_2^{(2)}$, respectively with Bhattacharyya values of $Z(W_2^{(1)})=2Z(W_1^{(1)})-Z^2(W_1^{(1)})$ accordingly. This proves of channel branching continuous. Note that, $Z(W_2^{(2)})=Z^2(W_1^{(1)})$ is located at the level n^th kevel of the tree with node numbering in ascending order from the top. The natural indexing of nodes in the tree is based on the bit sequences: i) the root node is indexed with the null sequence; ii) the upper and lower nodes at the level 1 are indexed as 0 and 1, respectively; iii) for a given node at the level n with index of b₁b₂ ⋯ b_n, the proceeding upper and lower nodes will have the sequences of b₁, b₂,·⋯ b_n0 and b₁b₂ ⋯ b_n1, respectively. Based on this process, $W_{2n}^{(i)}$ is located at the node b₁b₂ ⋯ b_n with $i=1+\begin{array}{c}n\\ j=1\end{array}{b}_j{2}^{n-j}$, and labelled as $W_{b_1{b}_2\cdots b_n}$.

 

Fig. 2 Binary tree based process of the UV channel splitting. Each branch denotes a unique formulation of one splitting channel, and links between any two nodes represents the dependence of two channels.

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Using the binary tree, Bhattacharyya value upper-bound can be determined by selecting K-sequence with K smallest Bhattacharyya values. E.g., the channel W_11⋯1 with a sequence of N-element of 1 assigned with the lowest Bhattacharyya value must be selected; whilst W_00⋯0 with the highest Bhattacharyya value should not be selected. However, by doing so there will error i.e., for a given relatively large R which satisfies R < I(W) selecting channels becomes an issue. This is because when assigning Bhattacharyya value the order of N-elements of 0 and 1 must also be considered. E.g., in channels with n/2 number of 1, $W_{b_1=0\cdots b_{n/2}=0b_{n/2+1}=1\cdots b_n=1}$ will have a larger Bhattacharyya value than $W_{b_1=1\cdots b_{n/2}=1b_{n/2+1}=0\cdots b_n=0}$.

(b) Recursive upper-bound

Assume a known channel selection at n step as ${\mathcal{A}}_n$, accompanying with its SBV as $\epsilon ({\mathcal{A}}_n){\triangleq }_{W_N^{(i)}\in {\mathcal{A}}_n}Z(W_N^{(i)})$. We will build a channel selection, denoted as ${\tilde{\mathcal{A}}}_{n+2}$ for the n + 2 step from the known ${\mathcal{A}}_n$. Note that the SBV of ${\tilde{\mathcal{A}}}_{n+2}$ must be greater than the real selected ${\mathcal{A}}_{n+2}$, i.e.,

Remark 1 For some $W_{b_1{b}_2\cdots b_n}\in {\mathcal{A}}_n$ with relatively small $Z(W_{b_1{b}_2\cdots b_n})$, $W_{b_1{b}_2\cdots b_{n}11}$, $W_{b_1{b}_2\cdots b_{n}10}$, $W_{b_1{b}_2\cdots b_{n}01}$ and $W_{b_1{b}_2\cdots b_{n}00}$ should belong to ${\tilde{\mathcal{A}}}_{n+2}$. In this case, all $Z(W_{b_1{b}_2\cdots b_{n}11})$, $Z(W_{b_1{b}_2\cdots b_{n}10})$,$Z(W_{b_1{b}_2\cdots b_{n}01})$ and $Z(W_{b_1{b}_2\cdots b_{n}00})$ are small, we use ${\left(2Z(W_{b_1{b}_2\cdots b_n})-Z^2(W_{b_1{b}_2\cdots b_n})\right)}^2$ to approximate the sum, i.e.,

Remark 2 For other $W_{b_1^{*}{b}_2^{*}\cdots b_n^{*}}\in {\mathcal{A}}_n$ whose $Z(W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}})$ are larger, we firstly select $W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}11}$, and $W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}10}$, given the smaller Bhattacharyya values than their siblings i.e., $W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}01}$, and $W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}00}$. Then, in order to keep the SBV of ${\tilde{\mathcal{A}}}_{n+2}$ low, we replace the selection of $W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}01}$ or $W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}11}$, depending on whether there exists a $W_{b_1^{\prime }{b}_2^{\prime }\cdots b_n^{\prime }}\notin {\mathcal{A}}_n$ whose $Z(W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}11})<Z(W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}01})$ or $Z(W_{b_1^{*}}{ }_{b_2^{*}\cdots b_n^{*}11})$. In this way, we also use ${\left(2Z(W_{b_1{b}_2\cdots b_n})-Z^2(W_{b_1{b}_2\cdots b_{n}01})\right)}^2$ to approximate the sum, i.e.,

Given the Remark 1, and Remark 2, the SBV of ${\tilde{\mathcal{A}}}_{n+2}$, i.e., $\epsilon ({\tilde{\mathcal{A}}}_{n+2})$ can be computed with the approximations in Eqs. (18)–(19), i.e.,

It is noteworthy that a function extracted from the right-hand of (20), i.e., f(z) = (2z − z²)² is a convex formula, in the range z ∈ (0, 1). According to the convex characteristic, the right-hand of (20) can be thereby re-written as:

In Eq. (22), the initial n is relevant with the speed of polarization given different F [24]. Typically with $\mathbf{F}=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$, the initial n should be no lesser than 4 to ensure stability of channel polarization, i.e., we assigned the initial values of n as 4 and 5 for even and odd indexes respectively given by:

3.3.2. Deducing code length

Based on the Bhattacharyya recursive upper-bound, the suitable code length N in UVC can be determined via it’s exponent (n = log₂ N), as follow: (i) we firstly initialize ζ by Eq. (23) i.e., $\epsilon ({\mathcal{A}}_4){=}_{i\in {\mathcal{A}}_4}Z(W_{16}^{(i)})$ and $\epsilon ({\mathcal{A}}_5){=}_{i\in {\mathcal{A}}_5}Z(W_{32}^{(i)})$, with respective $\left|{\mathcal{A}}_4\right|=16R$ and $\left|{\mathcal{A}}_5\right|=32R$. For instance, in the case R = 1/2, we have $\epsilon ({\mathcal{A}}_4)=Z^{16}(W_1^{(1)})+278Z^8(W_1^{(1)})+28Z^4(W_1^{(1)})$ and $\epsilon ({\mathcal{A}}_5)=Z^{32}(W_1^{(1)})+65814Z^{16}(W_1^{(1)})+71548Z^8(W_1^{(1)})$; (ii) for a given allowable BER value ϱ (i.e., BER_n ϱ), we can obtain a proper n by inferring an inequality, i.e.,

According to the upper-bound in (22), relations in (24) can be subsequently described as:

By solving (25), the appropriate step n can be derived as:

4. Numerical and experimental results

In this section, numerical simulations and off-line experimental investigations are performed to evaluate the proposed polar coding scheme from both the communication performance and the complexity perspectives. First of all, we examine the statistical characteristics of the UV channel, which is served as the basis for subsequent designs of the polar code. Next, verification of the appropriate code length is provided. Finally, we evaluate the communication performance as well as the computational complexity of the proposed UV channel-related polar code.

The experimental test-bed is shown in Fig. 3. At the Tx, the emitted power of the UV LED array is assigned as P_t = 0.1mW at the wavelength of 265nm. The data bit duration T_b is 8us. For the path loss model, we assign α = 2.117, β = 0, and ξ = 285.6 in accordance with the measured channel parameters. At the Rx, the transmissivity η_f is 0.2, the transimpedance amplifier gain G_amp = 22 × 10³Ω, the PMT gain G = 1.0 × 10⁷, and the sensitivity of the cathode radiant S_c = 62 × 10⁻³A/W. The signal was captured using a real time oscilloscope with a sampling time is T_s = 0.4µs. The expectation and variance of the thermal noise are µ = 0.0220356 and σ² = 8.87294 × 10⁻⁵, respectively.

 

Fig. 3 Experimental test-bed of UVC system: (a) Tx, and (b) Rx.

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4.1. Statistical characteristic of the path loss based photon channel

In order to examine statistics of the UV path loss channel, we have to validate the use of Binomial distribution on characterizing the number of received photons. To do this, the theoretical expectation of the number N_r is firstly compared with the experimental value (i.e., ${\overline{N}}_r$) in Fig. 4(a). The experimental ${\overline{N}}_r$ is calculated by averaging J = 10⁴ samples of N_r,j, j = 1, …, J, which can be transformed via the voltage V_r,j measured by the oscilloscope, i.e.,

 

Fig. 4 Verification of the statistical characteristics of the UV path loss model.

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Then, we build a histogram of N_r,j in Fig. 4(b) to test the statistics density of the number of received photons. It can be found that the histogram is almost a Binomial distribution, with expectation as N_r. Therefore, these two results provide a verification of the statistics characteristic of the UV channel model, which can be used for designing channel-related polar coding scheme.

4.2. Optimum polar code length for UV channel

In the second experiments, we verify the proposed theoretical code length of the UV channel-related polar code by following steps. Firstly, the proposed recursive Bhattacharyya upper-bound for general polar codes is tested. Then, given the correctness of general cases, we testify the appropriate code length in the UV scenarios.

(a) Test of recursive Bhattacharyya upper-bound

The proposed recursive Bhattacharyya upper-bound given by Eqs. (22)–(23) is shown in Fig. 5, compared with the real SBV in Eq. (16) and a popular used bound in [24]. We can firstly notice that as the code length N = 2ⁿ rises, the Bhattacharyya values decrease, illustrating the fact that the larger code length gives a lower BER. Then, a lower initial $Z(W_1^{(1)})$, meaning a better channel environment, leads to a lower Bhattacharyya values (reflecting a lower BER). Thirdly, with the increment of the code rate (e.g., from R = 0.5 to R = 0.75), the Bhattacharyya value is increasing, which suggests a worse BER performance as the code rate rises.

 

Fig. 5 Demonstration of the proposed recursive Bhattachaeyya upper-bound, which can be used as an easily computed version as real SBVs, reflecting the BER performance.

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More intriguingly, it is seen that the proposed upper-bound matches well with the real SBV; whilst the popular used bound deviates the real SBV, suggesting the proposed upper-bound can be adopted as an easily computed version of the real SBV that reflects the BER of general polar codes. Hence, we can use this proposed recursive upper-bound to compute a proper code length in the context of UVCs.

(b) Test of proper code length in UVCs

We further evaluate the appropriate code length of the polar code under UV scenarios. In Fig. 6(a), the BER performances of the channel-related polar code with respect to the the changes of code length N = 2ⁿ is illustrated. We can easily observe that the BER decreases with the increase of the code length. Then, it is seen that as the distance increases (e.g., from r = 45m, to r = 55m), the BER rises up drastically. Therefore, it is worthy to determine proper code length under different communication distances. And there comes the Fig. 6(b).

 

Fig. 6 Verification of appropriate code length for UV channel.

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Given a requested BER performance as ϱ = 10⁻⁶, theoretical proper code length given by Eq. (27) as well as simulated ones is provided in Fig. 6(b), with respect to the changes of distance r. We can observe that theoretical one almost matches the simulated result, suggesting that we can use the theoretical computation to determine a minimum code length without raising the BER larger than the requested performance. Therefore, a lower complexity of coding and decoding process in realizing the UV channel-related polar code can be achieved.

4.3. Performance of UV channel-related polar code

In this experiment, we compare the BER performances of the UV channel-related polar code with the existing LDPC scheme [15]. As configured, code rate is assigned as R = 0.5 for both the polar code and LDPC code, with code length as 512 and 960 respectively. It can be observed from Fig. 7 that, both polar code and LDPC can lower the BER in the UV scenarios. Moreover, the proposed polar code outperforms the LDPC scheme. From both simulative and experimental results, we can see that the polar code can enhance an extra 15% communication distance at the requested BER as ϱ = 10⁻⁶, in contrast with the LDPC scheme. This result gives a great promising value of the proposed polar code on addressing the UV path loss propagation.

 

Fig. 7 Measured and simulated BER comparison as a function of the transmission span for the polar code, LDPC, and un-code OOK.

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Next, we evaluate the computational complexity by means of the number of multiplications. For the encoder, the number of multiplications for the channel-related polar code is 512 × 512 = 262144; whilst the LDPC scheme takes 480 × 960 = 460800 [18]. For the soft-decision decoder, the number of multiplications for the SC-decoder of polar code is 512 × log₂ 512 = 4608, which is much lower than 921600 for the LDPC.

Therefore, these two comparisons show the promising prospective of this channel-related polar code in UVC systems in order to improve transmission performance.

5. Conclusion

In communication systems, it is advantageous to achieve the desired system performance with a low level of computational complexity. In low data rate and short distance UVC systems the path loss propagation is a major limiting factor. In this paper, we have outlined the design of an UV channel-related polar coding scheme, which is ideal for UVC in addressing the path loss. We considered the complexity issue in terms of the code length. By theoretically deducing Bhattacharyya upper-bound of the general polar code, and fitting it into UV channels, we derived a minimal code length without impairing on the required code performance. Numerical simulations and experimental investigations were performed to validate the proposed polar code scheme. We showed that, the proposed scheme offered an improve performance with reduced computational complexity compared with existing schemes, therefore illustrating the suitability of the proposed scheme for future short distance UVC systems.