Improved Synchronization Scheme for the Photon- Counting-Based Underwater Optical Wireless Communication System

To address existing issues in conventional correlation method, we propose an improved synchronization method for photon-counting-based underwater optical wireless communication system employing on-off keying modulation.The proposed method jointly estimates the synchronization position, signal intensity, background intensity and data sequence based on the received vector in the period of a frame length. To reduce the complexity for solving the estimation problem, several simplifications are performed based on the characteristics of the data sequence. Experimental results show that the proposed method achieves an optical power gain of at least 1 dB over the correlation method at the bit error rate of $10^{-3}$ under dynamic channel.

Synchronization is an important part in UOWC. For UOWC systems with intensity detector, there have been plenty of studies on synchronization [11]. However, for UOWC system with photon-counting detector, there are only a few works on synchronization. In [12], a correlation-based synchronization method was proposed for the photon-counting based system, which divides the received signal into small chips and estimates the position of the sync word by maximizing the correlation value between the sync word and the number of received photons within the synchronization window. Despite the implementation simplicity, the correlation-based method neglects the signals surrounding the sync word and thus leads to a transmission performance degradation. Meanwhile, some unfavorable factors (e.g., bubble and turbulence) in real underwater scenarios can cause drastic fluctuation of the optical link, bringing challenges to synchronization in photon-counting-based UOWC systems [13].
To address the above issues, we propose an improved synchronization method for photon-counting-based UOWC system employing on-off keying (OOK) modulation. The proposed method jointly estimates the synchronization position, signal intensity, background intensity and data sequence based on the received vector in the period of a frame length. To reduce the complexity for solving the estimation problem, several simplifications are performed using the characteristics of the data sequence. Experimental results verify the performance improvement by the proposed method over the correlation method. The performance superiority makes the proposed method appealing in real applications.

II. SYSTEM MODEL
Consider a photon-counting-based UOWC system adopting a blue/green LED at the transmitter and a photon-counting detector (e.g., PMT) at the receiver. Non-return-to-zero-OOK modulation is employed for its simplicity. As in Refs. [14], [15], [16], authors assume that the data is transmitted in successive frames with frame length N , as illustrated in Fig. 1. Each frame consists of an L-length fixed sync word and an (N − L)-length random data sequence.
To realize synchronization, we consider the received signals in N successive symbol durations and divide each symbol duration Let r i,j denote the number of received photons in the i-th symbol interval starting from phase j. In a photon-counting-based UOWC system, the Poisson model can well characterize the number of received photons during each symbol period. Specifically, assuming the sync word starts from symbol position n + 1 and phase k in the received signals, the probability of r n+i,k given the transmitted symbol x i in the i-th symbol period is [17], [18], [19] Pr(r n+i, where λ s and λ b are signal photon mean and background photon mean, respectively. To facilitate the descriptions, the subscripts i on r i,j will hereafter be taken modulo N , i.e., r i+N,j = r i,j . From (1), the numbers of received photons in the N successive symbol durations can be written in a vector form as

III. CORRELATION-BASED SYNCHRONIZATION METHOD
A correlation-based synchronization method has been proposed in [12], [20] and [21], which estimates the position of sync word by maximizing the correlation value between the sync word and the number of received photons, i.e., After synchronization, the signal photon mean λ s and background photon mean λ b can be estimated using the sync word asλ , and "| · |" denotes the cardinality of a set.
With the results of synchronization and channel estimation, we can perform symbol detection using the log-likelihood ratio (LLR) criterion as LLR ≷ 1 0 0. According to (1), the LLR is given by where the estimated values are adopted for (n, k, λ b , λ s ). Despite the implementation simplicity, the correlation-based method neglects the signals surrounding the sync word and thus leads to a transmission performance degradation. In the following, we propose an improved synchronization method for photoncounting-based UOWC system.

A. Problem Formulation
We consider joint estimation of sync word position (n, k), signal photon mean λ s , background photon mean λ b and data sequence d based on the received vector r k . The joint maximum likelihood (ML) estimation problem is formulated as Under the Poisson distribution given by (1), Pr(r k |n, k, λ s , λ b , d) can be expressed as Using (7) and taking logarithm, problem (6) can be reformulated as To reduce the number of jointly estimated unknowns in problem (8), we first maximize F 1 (n, k, λ s , λ b , d) with respect to {λ s , λ b } for given (n, k) and d, yieldinĝ By substituting (10) into (9), problem (8) can be equivalently transformed into the following problem: where Note that exhaustive search for the optimal solution of problem (11) requires high computational complexity. To reduce the complexity, we propose several simplifications to solve problem (11) based on characteristics of data sequence. (11) 1) Simplification for Symbol Detection:

B. Simplified Method for Solving Problem
Given signal photon mean λ s , background photon mean λ b and sync word position (n, k) in Poisson channel, the ML detection for data d is equivalent to a threshold-based detection. It implies that d can be determined by detecting the Δ d largest elements in {r L+n+1,k , . . . , r N +n,k } to be symbol "1" and the remaining elements to be symbol "0". Based on this idea, we sort the signals {r L+n+1,k , . . . , r N +n,k } in descending order such thatr (1) n,k ≥r (2) n,k ≥ · · · ≥r (N −L) n,k . In the special case By plugging (13) into (12), problem (11) is equivalent to where Problem (14) n,k ≥r (2) n,k ≥ · · · ≥ r (N −L) n,k . Level ii) For each n, find the optimal Δ d maximizing F 3 (n, k, Δ d ) in (15), where the maximum value is denoted by F 3 (n, k). Level iii) Find the optimal n ∈ {0, 1, . . . , N − 1} and k ∈ {0, 1, . . . , K − 1} maximizing F 3 (n, k). We can perform simplifications for sorting A n,k in Level i) and searching Δ d in Level ii), as presented below.
2) Simplification for Sorting A n,k : Observing A q+1,k ∩ A q,k = {r j,k } N +q j=L+q+2 , we can further simplify the sort operation in Level i) as n changes from q to q + 1. Specifically, A q+1,k can be quickly sorted by deleting r L+q+1,k from B q,k and then inserting r N +q+1,k , in which the former can be readily realized with known position of r L+q+1,k in B q,k after sorting A q,k , and the latter can be realized through the bisection method. Consequently, the N sort operations for n ∈ {0, . . . , N − 1} can be simplified to single sort operation for n = 0, together with the deletion and insertion operations for n ∈ {1, . . . , N − 1}. The simplification for sorting A n,k is illustrated in Fig. 3.
3) Simplification for Searching Δ d : In Level ii), the search for the optimal Δ d ∈ {0, 1, . . . , N} can be further simplified when n varies form q to q + 1. Assume the number of d j = 1 in d is Δ q d for n = q. Since A q+1,k and A q,k only differ in the two elements r N +q+1 and r L+q+1 , Δ q+1 d can only take values   (14) is summarized in Algorithm 1. Note that the proposed method and correlation method both estimate the sync word position (n, k), where n represents the frame synchronization position and k represents the bit synchronization position. Consequently, these two methods realize both the bit synchronization and frame synchronization simultaneously [12], [20], [21].

C. Complexity Analysis
For the correlation method, the correlation value is calculated K × N times and each calculation requires a complexity of O(L), so the total complexity is O(K × N × L). For the proposed method, the major complexity comes from calculating F 3 (n, k, Δ d ) for each (n, k), which involves a complexity of O(N ). Consequently, the total complexity of the proposed method is O(K × N 2 ) when n varies from 0 to N − 1 and k varies from 0 to K − 1. Fig. 4 shows the experimental photon-counting UOWC system. At the transmitter, a green LED ("Signal-LED", Jred, 520 nm ∼ 525 nm) is driven by an arbitrary waveform generator (AWG, Keysight 33612 A) to send the OOK signal with a data rate of 1.25 Mbps, considering the limited bandwidth of the adopted LED and PMT [12], [21], [22]. An optical power meter (Thorlabs PM100D with sensor S130 C) is placed in front of the LED to measure the transmit optical power, which can be adjusted by varying the output voltage of AWG [23], [24], [25]. The total transmit optical power of LED can be calculated from The temperature is about 20 • C. A green LED ("Background-LED", Jred, 520 nm ∼ 525 nm) is placed near the signal-LED acting as the background light source. At the receiver, a PMT (Hamamatsu CR315) is adopted to detect the number of received photons. To attenuate the optical power and simulate long-distance transmission, an optical attenuator (Thorlabs NE30 A with an optical density of 3.0) is placed in front of the PMT [26], [27], [28]. An optical filter (Smerock FF01-523/20-25) is also employed to filter out the light of other wavelengths. The output of the PMT is recorded by an oscilloscope (OSC, Tektronix MSO64B) with a sampling rate of 250 MSa/s. In order to emulate a dynamic UOWC environment, a wave maker is installed on one side of the water tank. The transmitter and receiver in the water sway with the waves generated by the wave maker, which affects the alignment of the optical link.

A. Experimental Setup
In the experiment, a maximum-length sequence of length 31 (i.e., L = 31) is chosen as the sync word, and the length of data sequence is set to be N − L = 400. The value of K is set to be 200, which is the number of samples during a symbol period. This setting is advantageous for accurate synchronization, as stated in [12], [20] and [21]. To calculate the synchronization error, we adopt an auxiliary signal transmitted through a cable connecting the AWG and OSC as reference signal. This signal helps to provide the exact position (n, k) of the sync word. Then the estimated position (n,k) of the sync word is compared with the exact position (n, k) to obtain a calculated synchronization error as e = |(n − n)K +k − k|. Considering the maximum record length of OSC, we sample 1450 frames at the receiver and obtain the synchronization error of the correlation method and the proposed method. The running time adopting the proposed synchronization method is 8.5 times of that adopting the correlation method, which roughly coincides with complexity analysis in Section IV-C.

B. Experimental Results
Fig . 5 shows the Poisson distribution fitting results for the number of detector output pulses which is typically used to represent the number of received photons [12], [21], [22]. It is observed that the numbers of pulses in symbol "1" duration and symbol "0" duration both can be well fitted by the Poisson distribution. The fitting results demonstrate the applicability of the Poisson model adopted in this work. Fig. 6 shows the cumulative distribution function (CDF) of the synchronization error for the correlation method and the proposed method. It is seen that the synchronization performances of both methods improve as the transmit optical power increases. Moreover, compared with the correlation method, the synchronization error of the proposed method is more concentrated around 0, demonstrating the synchronization performance improvement by the proposed method. Table I lists the number of frame synchronization errors using different methods. It is seen that when the transmit optical power is -4.84 dBm (-3.34 dBm), the correlation method yields 13 (5) frame synchronization errors. In contrast, the proposed method yields no frame synchronization error under all tested transmit optical powers. Fig. 7 shows the experimental bit error rate (BER) using the correlation method and the proposed method under dynamic channel with various signal optical power. It is observed that the proposed method can provide an optical power gain exceeding 1 dB over the correlation method. Specifically, at the BER of 10 −3 , the required optical powers of the correlation method and the proposed method are 0.9 dBm and -0.4 dBm, respectively. Note that the actual running time of the proposed synchronization method can be reduced in several aspects: (1) Adopting parallel computations; (2) Approximating the number of d j = 1 in d as Δ d ≈ (N − L)/2; (3) Properly choosing the length of data sequence in a frame. For instance, in our experiment, the running time of the proposed method can be reduced by 55% using the approximation Δ d ≈ (N − L)/2. As shown in Fig. 7, using the approximation Δ d ≈ (N − L)/2 (labeled as "Suboptimal") can still achieve a noticeable optical power gain over the correlation method. Fig. 8 shows the experimental BER using the correlation method and the proposed method under dynamic channel with various background optical power. Again, we can see that the proposed method outperforms the correlation method under different background intensities. For the same BER target, the proposed method can tolerate larger background noise than the correlation method.
Note that typical issues in UOWC may include the dispersion, turbulence and misalignment. The above experimental results verify the performance improvement by the proposed synchronization method under the dynamic UOWC environment generated by the wave maker. Such improvement stems from the utilization of the data sequence in a frame in addition to the sync word. Consequently, the proposed synchronization method can be superior to the conventional correlation method in the photon-counting-based UOWC system regardless of the environmental factors.

VI. CONCLUSION
We have proposed an improved synchronization method for the photon-counting-based UOWC system, which jointly estimates the sync word position, signal intensity, background intensity and data sequence based on the received vector in the period of a frame length. Three simplifications have been performed based on the characteristics of data sequence to reduce the complexity for solving the estimation problem. Experimental results have demonstrated the performance superiority of the proposed method over the traditional correlation method.