Time-reversal waveform design for underwater wireless optical communication systems

Jiale Wang; Jie Lian

doi:10.1364/OE.493813

1. Introduction

Compared with underwater acoustic wireless communication (UAWC) techniques using acoustic waves to transmit data with a limited bandwidth up to a few kHz [1,2], underwater wireless optical communications (UWOC) has attracted attention for many high-rate data transmission applications in military and civilian areas, such as underwater navigation, sensor networks, ocean resources explorations, and oceanographic studies [3,4]. The development of UWOC in recent years focuses on optical devices, signal detection algorithms, and underwater optical channel modeling techniques, which illustrates the current and future research hot topics in UWOC area [5–9]. Many researchers investigated the underwater optical transmissions using the light with different wavelengths, and found that the wavelength around 522 nm has the lowest attenuation rate [10,11]. In recent years, some researches of red light transmission in highly turbid water is also studied, which can provide a wide channel bandwidth, as investigated numerically by Xu et al. [12]. In this paper, we focus on waveform designing to combat the intersymbol interference caused by the dispersive underwater optical channel.

Although UWOC has a promising application in many aspects, the scattering nature of ocean water results in severe attenuation, multiple paths, and turbulence effects [13–16]. In ocean water, the propagation of photons inevitably suffers from absorption and scattering, which leads to severe intersymbol interference (ISI). Thus, eliminating the ISI is a considerable challenge for UWOC system design. Time-reversal (TR) waveforming is a power-efficient and effective technique to mitigate ISI when the channel impulse response (CIR) contains abundant dispersive randomness. This approach has been proposed and applied in UAWC and radio frequency (RF) systems [17–22]. Sending the reversed CIR as the transmitted waveform is the basic idea of TR waveform design. This time reversed CIR propagates through the dispersive channel resulting in a power-concentrated signal at the receiver without severe ISI. This approach utilizes the auto-correlation characteristic of the random signals to reduce the ISI and achieves a significant processing gain. From the preliminary research on underwater optical transmissions, the photons emitted from the light sources are scattered by the suspended particles, which leads to numerous independent and random multiple paths [23]. This phenomenon makes the TR waveform design naturally suitable for UWOC systems. Unlike other researchers who develop precoding algorithms or equalizer techniques to combat the ISI caused by the UWOC scattering channel [24–27], we utilize the random dispersive multi-path UWOC channel to construct a TR transmitted waveform. Then, the received signals are automatically concentrated and easily equalized after propagating through the channel.

In this paper, we develop a TR waveform design technique for UWOC systems over the optical scattering dispersive channel. Considering the differences between RF and UWOC systems, we model the UWOC channel as the combination of an exponential function and a sequence of random dispersive path losses. After that, this work analyzes the system performances using the designed TR waveform with the proposed channel model and compares the bit error rate (BER) at different transmitted data rates. In general, our contributions are summarized as follows:

• A new channel model decomposition approach is proposed to analyze the performance of UWOC systems using TR waveform design technique.
• Based on the performance analysis and the numerical results shown in this paper, we find a squeezing effect when the TR waveform is transmitted through the UWOC dispersive channel. This phenomenon guides us on how to design a post-equalizer to combat the ISI.
• Due to the squeezing effect explored in this work, an equalizer is necessary at the receiver to further reduce the ISI. In addition, TR waveform can dramatically enhance the system performance when the equalizer is applied at the receiver due to the symmetric received signal using TR waveforms.

The rest of this paper is organized as follows. In Section 2 the UWOC system model using TR waveforming technique is introduced. Then, the UWOC system performance of using TR waveform is analyzed in Section 3. In Section 4 we present the corresponding numerical and simulation results based on the given parameters. Finally, this paper is concluded, and the future work is discussed in Section 5.

2. UWOC system model using TR waveform

Since the propagation of the photons inevitably experiences strong scattering in the ocean water, the multi-path channel caused by the random scattering of the numerous photons has a sound auto-correlation characteristic. Different from the multi-path channel in RF systems, UWOC channel is significantly affected by absorption in the ocean water. Therefore, the dispersive UWOC CIR exponentially decays over time. In addition, due to the intensity modulation and direct detection (IM/DD) used in UWOC systems, the UWOC channel model should only contain non-negative and real values.

In this section, we first propose the effective model of the UWOC channel impulse response with random scatterings and absorption. Using the proposed channel model, we introduce the TR waveform design technique in UWOC systems. Then, the transmitted and received signal model is described.

2.1 Effective model of UWOC channel impulse response

The propagation of photons in an underwater environment is a complicated process that contains absorption and scattering. The randomly scattered photons have different propagation delays and transmission paths, which result in a dispersive channel response. Since IM/DD is used, the CIR is real-valued and non-negative. Thus, the discrete-time domain version of the CIR can be modeled as

(1)$$\begin{aligned} h[k] & =\alpha(t) \left(h_{\ell}[k]+c[k]\right)\\ & \approx \alpha \left(h_{\ell}[k]+c[k]\right) \end{aligned} ~~k=0,1,\ldots,L-1,$$

where $h[k]$ represents the $k$th sample of the CIR, $c[k]$ is a added bias to guarantee $h[k]$ greater or equal to $0$. $\alpha (t)$ is the time-varying fading coefficient induced by the turbulence. Comparing with the transmitted symbol rate, $\alpha (t)$ generally is slow-varying for high-speed UWOC systems. Thus, it can be treated as a constant [28]. In (1), $L$ represents the length of the truncated discrete channel. To model the random scattering process of the photons, $h_{\ell }[k]$ in (1) is assumed a Gaussian distributed random variable with zero mean, which is represented as

(2)$$h_{\ell}[k]\sim \mathcal{N}(0,\sigma_{k}^2).$$

$\sigma _{k}^2$ denotes the variance of $h_{\ell }[k]$, which exponentially decreases as the time indice $k$ increases, and it can be calculated from [20,29] as

(3)$$\sigma_{k}^2 = e^{\frac{-kT_s}{D_T}},$$

where $T_s$ is the sampling period of this system, and $D_T$ is the delay spread of $h[k]$, which is calculated by $D_T=L\cdot T_s$ [30].

In (1), since the variance of $h[k]$ decays, $c[k]$ is modeled as an exponential function, which represents the optical power absorption with the CIR time increasing, shown as

(4)$$c[k]= a_0e^{{-}b_0kT_s},\quad k=0,~1,\ldots,L-1,$$

where $a_0$ and $b_0$ respectively depend on the transmission distance between the light source and the receiver, and the sea-water quality. The values of $a_0$ and $b_0$ can be obtained by fitting the real experimental CIR data. In Section 4.1, the Monte-Carlo simulation of the CIR supports our idea and the proposed channel model. Note that the channel model in (1) is only used to analyze the performance of the TR waveform in UWOC systems and the CIRs based on the Monte-Carlo simulation are still utilized in the performance validation of the proposed TR waveforms.

2.2 TR waveform design for UWOC systems

In this section, we derive the TR waveform design according to the channel model proposed in Section 2.1. To simplify the notations, we only discuss the point-to-point transmission case, in which the light source and the receiving photo-detector (PD) straightly face to each other.

A block diagram of a point-to-point UWOC system using the TR waveform is shown in Fig. 1. At the transmitter, $M$-ary pulse magnitude modulation ($M$-PAM) is applied to carry the binary data, where $x[k]\in \{\pm \frac {M-1}{M},~\pm \frac {M-3}{M},~\cdots,~\pm \frac {1}{M}\}$, and $M$ denotes the modulation constellation size. Although a more advanced modulation scheme than the PAM can be adopted, such as optical-OFDM [31], we mainly focus on the performance of the TR waveform instead of the modulation schemes. Since the binary information is assumed to be randomly distributed, $x[k]$ is assumed to have a uniform probability distribution.

Fig. 1. The schematic diagram of TR based UWOC system with an equalizer.

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To eliminate the ISI, we use the time-reversed CIR as the transmitted waveform, and the actual CIR can be pre-estimated. Therefore, the TR waveform can be obtained by

(5)$$g(t)=\frac{h(T-t)}{\int_{0}^{T}h^2(t)d t},$$

where $h(t)$ represents the CIR, and $T$ is the duration of the channel. $g(t)$ is actually the normalized time-reversal $h(t)$. Then, the transmitted signal $s(t)$ is modeled as

(6)$$s(t) = \sum_{k=0}^{\infty}x[k]g(t-kT_s),$$

where $T_s$ represents the transmitted symbol period, which is assumed to be the same as the system sampling period shown in (3). To avoid sending negative signals, a DC bias is added before $s(t)$ is transmitted.

For the sake of convenience and simplifying the notations, we analyze the received signals and the system performances in the discrete-time domain. After the DC bias is removed at the receiver, the $k$th sample of the received signal is represented as

(7)$$\begin{aligned} r[k] & = (s\otimes h) [k]+n[k]\\ & = \sum_{k_1=0}^\infty x[k_1]g[k-k_1] \otimes h[k]+n[k]\\ & = \rho(x\otimes g\otimes h)[k]+n[k],~~ \end{aligned}$$

where $\otimes$ is the convolution operation in time domain. $\rho$ represents the responsivity of the light source. In the following analysis, we assume $\rho = 1$ for simplicity. $n[k]$ is the $k$th sample of the added noise at the receiver, which is a combination of shot and thermal noises. Usually, we model $n[k]$ as an additive Gaussian noise with zero-mean and variance $\sigma _{n}^2$. Plugging (5) into (7), the received signal can be rewritten as

(8)$$\begin{aligned} r[k] & = \frac{(x\otimes \overleftarrow{h} \otimes h)[k]}{\sum_{l=0}^{L-1}h^2[l]} +n[k] \\ & = \left(x\otimes R\right)[k] +n[k], \end{aligned}$$

where $\overleftarrow {h}$ denotes the reversal order of $h$, and

(9)$$\begin{aligned} R[k] & = \frac{(\overleftarrow{h} \otimes h)[k]}{\sum_{l=0}^{L-1}h^2[l]}\\ & = \frac{\sum_{l=0}^{L-1}h[L-k-1+l]h[l]}{\sum_{l=0}^{L-1}h^2[l]}. \end{aligned}$$

3. Performance analysis

In this section, we analyze the system performance for UWOC using TR waveforms. Although the TR waveform is proven to have an advantage on ISI reduction in RF communication systems, the differences between UWOC and RF systems make the performance analysis of using TR waveforms in UWOC systems significantly meaningful.

3.1 Equivalent channel model

Since the CIR is modeled as the combination of an exponential function and a sequence of random pulses as shown in (1), the received signal model is different from that in RF systems. To better analyze the system performance, we define the numerator of (9) as

(10)$$\tilde{R}[k]=(\overleftarrow{h} \otimes h)[k].$$

Only considering the impact of the random channel, we analyze the expectation of $\tilde {R}[k]$ instead of $r[k]$ from the statistical perspective. Therefore, the expectation of $\tilde {R}[k]$, called equivalent channel, is represented as

(11)$$\begin{aligned} \mathbb{E}_h\left[\tilde{R}[k]\right] & = \mathbb{E}_h\left[(\overleftarrow{h}\otimes h)[k]\right]\\ & = \mathbb{E}_h\left[[(\overleftarrow{h_{\ell}+{c}})\otimes (h_{\ell}+{c})][k]\right]\\ & =\mathbb{E}_h\Big[\underbrace{ (\overleftarrow{h_{\ell}} \otimes h_{\ell})[k]}_{\mathrm{random~ part}}+\underbrace{({\overleftarrow{c}}\otimes {c})[k]}_{\mathrm{squeezing~part}}\\ & +\underbrace{({c}\otimes\overleftarrow{h_{\ell}})[k]+(\overleftarrow{c}\otimes{h_{\ell}})[k]}_{\mathrm{null~part}}\Big], \end{aligned}$$

where $\mathbb {E}_h[\cdot ]$ denotes the expectation with respect to $h$. Since the scattering of the numerous photons results in plenty of transmission paths, and $h_{\ell }[k]$ is assumed as a Gaussian distributed random variable, the $\mathrm {random~part}$ shown in (11) can be calculated as the theorem in [29],

(12)$$\mathbb{E}_h\left[ (\overleftarrow{h_{\ell}} \otimes h_{\ell})[k]\right]= \sigma ^2 \delta(k-L+1),$$

where $\sigma ^2=\frac {1}{L}\sum _{k=1}^{L}\sigma _k^2$, which leads to an auto-correlation processing gain at the receiver. The value of the term $\mathrm {null~part}$ is zero, which can be easily canceled. The calculation of $\mathrm {null~part}$ term is shown as

(13)$$\begin{aligned} & \mathbb{E}_h\left[({c}\otimes\overleftarrow{h_{\ell}})[k]+(\overleftarrow{c}\otimes{h_{\ell}})[k]\right] \\ & = \mathbb{E}_h\left[\sum_{n={-}\infty}^{\infty}{c}[n]\overleftarrow{h_{\ell}}[k-n]+{\overleftarrow{c}[n]}{h_{\ell}}[k-n]\right]\\ & =\sum_{n={-}\infty}^{\infty}{c}[n]\mathbb{E}_h\left[\overleftarrow{h_{\ell}}[k-n]\right]+{\overleftarrow{c}[n]}\mathbb{E}_h\left[{h_{\ell}}[k-n]\right]\\ & = 0. \end{aligned}$$

The term $\mathrm {squeezing~part}$ shown in (11) leads to a concentration of the ISI power, which degrades the system performance. To better analyze how this term impacts the system performance, we define a new variable $t_0$ as the continuous version of $k$, and $T_0$ represents the continuous version of $L$. Then, we can represent the squeezing part in (11) as

(14)$$\begin{aligned} \epsilon(t_0) & ={\overleftarrow{c}}\otimes {c}\\ & =\begin{cases} & \int_{0}^{T_0-t_0}c(\tau)c(t_0+\tau)d\tau ,\quad 0<t_0<T_0\\ & \frac{a_0^2}{b_0}(1-e^{{-}2b_0T_0})\approx \frac{a_0^2}{b_0} ,\quad t_0 = 0\\ & \int_{0}^{T_0-t_0}c(\tau)c({-}t_0+\tau)d\tau ,\quad -T_0<t_0<0 \end{cases}. \end{aligned}$$

Since (14) is symmetric with respect to $t_0 = 0$, we can only discuss the case that $0<t_0<T_0$, which is function of $t_0$ and $b_0$, represented as

(15)$$\begin{aligned} \epsilon(t_0,b_0) & = \int_{0}^{T_0-t_0}a_0^2e^{{-}b_0\tau}e^{{-}b_0(t_0+\tau)}d\tau\\ & ={-}\frac{a_0^2 e^{{-}b_0 T_0} \sinh [b_0 (t_0-T_0)]}{b_0},\quad 0<t_0<T_0, \end{aligned}$$

where $\sinh (x)= \frac {e^x-e^{-x}}{2}$. From (15), $a_0$ is a coefficient that is irrelevant to the shape of the equivalent channel waveform, therefore, this work only discusses how $b_0$ and $t_0$ affect the squeezing part. We first take the partial derivative of $\epsilon (t_0,b_0)$ to $t_0$, which is

(16)$$\begin{aligned} \lvert \epsilon^{'}_{(t_0)}(t_0,b_0)\rvert & = \lvert\frac{\partial{\epsilon(t_0)}}{\partial{t_0}}\rvert\\ & = a_0^2 e^{{-}b_0 T_0} \cosh [b_0 (t_0-T_0)],\quad 0<t_0<T_0, \end{aligned}$$

where $\cosh (x)= \frac {e^x+e^{-x}}{2}$. In (16), it evaluates how ISI varies over $t_0$. The larger value of (16) is, the less ISI is obtained with an increasing $t_0$. To further explore the influence of $b_0$ on ISI, we take the derivative of $\epsilon ^{'}_{(t_0)}(t_0,b_0)$ to $b_0$, which is

(17)$$\begin{aligned} \epsilon^{\prime\prime}_{(t_0,b_0)}(t_0) & = \frac{\partial{ \lvert \epsilon^{'}_{(t_0)}(t_0,b_0)\rvert}}{\partial{b_0}}\\ & = a_0^2 e^{{-}b_0 T_0} (t_0-T_0) \sinh [b_0 (t_0-T_0)]\\ & -a_0^2 T_0 e^{{-}b_0 T_0} \cosh [b_0 (t_0-T_0)]\\ & \begin{cases} & >0,\quad t_0>\frac{\ln{\frac{T_0+1}{1-T_0}}}{2b_0}+T_0\\ & =0,\quad t_0=\frac{\ln{\frac{T_0+1}{1-T_0}}}{2b_0}+T_0,\quad \quad 0<t_0<T_0.\\ & <0,\quad t_0<\frac{\ln{\frac{T_0+1}{1-T_0}}}{2b_0}+T_0\\ \end{cases} \end{aligned}$$

Here, we can obtain when $b_0$ increases, the curve in (15) becomes more dispersed for $~t_0<\frac {\ln {\frac {T_0+1}{1-T_0}}}{2b_0}+T_0$, and has a more concentrated waveform for $t_0>\frac {\ln {\frac {T_0+1}{1-T_0}}}{2b_0}+T_0$. In other words, the ISI is gathered within the range of $~t_0<\frac {\ln {\frac {T_0+1}{1-T_0}}}{2b_0}+T_0$. Thus, we define a metric, time-varying signal interference ratio (TSIR) $\eta$, to represent the ratio of the signal power from the direct path to that from the multi-path component at time $t_0$, which describes how ISI varies over sampling time. According to the proposed channel model shown in (11), the lower the TSIR we obtain, the more severe ISI the system experiences. Without using the TR waveform, TSIR can be represented as a function of $t_0$

(18)$$\eta_{wo}(t_0) = \frac{a_0}{a_0e^{{-}b_0t_0}} = e^{b_0t_0}.$$

Using the proposed TR waveforms, TSIR is calculated as

(19)$$\begin{aligned} \eta_{w}(t_0) & =\frac{1}{2} \sinh (b_0 T_0) \text{csch}[b_0 (t_0-T_0)] , \end{aligned}$$

where $\text {csch}(x) = 1/\sinh (x)$, and the coefficient $1/2$ comes from the symmetry of the equivalent channel due to the employment of the TR waveform. To analyze how the squeezing part impacts the ISI with and without using TR waveforms, we obtain that

(20)$$\frac{\eta_{w}(t_0)}{\eta_{wo}(t_0)} ={-}\frac{1}{2}e^{{-}b_0 t_0} \sinh (b_0 T_0) \text{csch}[b_0 (t_0-T_0)].$$

Let $\frac {\eta _{w}(t_0)}{\eta _{wo}(t_0)}<1$, we can find the range of $t_0$ making the ISI with TR waveform is worse than without using TR waveform. Correspondingly, we assume the range of $t_0$ is $t_0\in [0,~t_0^*)$, where ${\eta _{w}(t_0^*)}/{\eta _{wo}(t_0^*)}=1$. Therefore, for $t_0<t_0^*$, the ISI of using TR waveforms is greater than the case without TR. We define this phenomenon as the “squeezing effect", which is resulted from the strong multi-path components centralizing around the direct path after the TR waveform. This phenomenon is consistent with (17). Thus, an equalizer is necessary to remove the ISI.

We illustrate a numerical result to present the squeezing effect as shown in Fig. 2. In this figure, the normalized channel is truncated when its instantaneous power is lower than 40 dB, i.e., $T_0=(\ln 100)/b_0$, then show the (20) with different $b_0$ in Fig. 2. When $t_0<t_0^*$, using TR waveforms makes the ISI greater than without using TR, and (20) is less than 1, as shown in the part below the red line.

Fig. 2. Squeezing effect for different $b_0$, with $T_0=\ln {100}/b_0$.

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3.2 Received SINR

For a communication system, SINR is a representative metric to evaluate the system performance. In this subsection, we analyze the received signal structure and describe how the squeezing effect impacts the received SINR. The received signal described in (8) before applying the equalizer can be rewritten as

(21)$$\begin{aligned} r[k] = \underbrace{x[k-L+1]}_{\mathrm{Signal}} + \underbrace{\sum_{l=0,l\neq L-1}^{2L-2} x[k-l] R[l]}_{\mathrm{ISI}} +\underbrace{n[k]}_{\mathrm{Noise}}, \end{aligned}$$

which can be decomposed into three parts. Then, the average SINR over the transmitted signal can be calculated as

(22)$$\text{SINR} = \frac{\bar{P}_{\mathrm{s}}}{\bar{P}_{\mathrm{ISI}}+\bar{P}_{\mathrm{n}}}$$

where $\bar {P}_{\mathrm {ISI}}$ and $\bar {P}_{\mathrm {n}}$ denote the average power of ISI and the additive noise, respectively. Since $M$-PAM is used, the average power of the target signal is calculated as

(23)$$\bar{P}_{\mathrm{s}} = \frac{M^2-1}{3M^2}.$$

The average power of the ISI over the data is calculated by

(24)$$\begin{aligned} \bar{P}_{\mathrm{ISI}} & = \mathbb{E}_x\left[\left(\sum_{l=0,l\neq L-1}^{2L-2} x[k-l] R[l]\right)^2\right]\\ & = \mathbb{E}_x\left[\sum_{p=0,p\neq L-1}^{2L-2}\sum_{q=0,q\neq L-1}^{2L-2} x[k-p]x[k-q] R[p]R[q]\right]\\ & = \sum_{p=0,p\neq L-1}^{2L-2}\sum_{q=0,q\neq L-1}^{2L-2} R[p]R[q] \mathbb{E}_x\big[x[k-p]x[k-q]\big], \end{aligned}$$

where $\mathbb {E}_x[\cdot ]$ denotes the expectation respect to $x$. Since we have the following equation

(25)$$\begin{aligned} & \mathbb{E}_x\big[x[k-p]x[k-q] \big]\\ & = \begin{cases} \mathbb{E}_x\big[x[k-p]x[k-p] \big] = \bar{P}_{\mathrm{s}},~ & p=q\\ 0,~ & p \neq q \end{cases}, \end{aligned}$$

the average power of the ISI can be reduced as

(26)$$\begin{aligned} \bar{P}_{\mathrm{ISI}} = \bar{P}_{\mathrm{s}}\left(\sum_{l=0,l\neq L-1}^{2L-2}R^2[l]\right). \end{aligned}$$

From (11), (12) and (15), we can calculate the expectation of $R[k]$ as

(27)$${\mathbb{E}}_h\left[ R[k] \right] \begin{cases} = \frac{a_0^2e^{{-}b_0T_0}\sinh [b_0 (k-L+1-T_0)]}{-a_0^2-b_0\sigma^2},\quad & k>L-1\\ = 1,\quad & k=L-1\\ = \frac{a_0^2e^{{-}b_0T_0}\sinh [b_0 (L-1-k-T_0)]}{-a_0^2-b_0\sigma^2},\quad & k<L-1 \end{cases}.$$

Then, the average SINR over the channel and the transmitted signals can be calculated as

(28)$$\begin{aligned} & \text{SINR}_{e} \\ & = \mathbb{E}_h\left[\frac{\bar{P}_{\mathrm{s}}}{\bar{P}_{\mathrm{ISI}}+\bar{P}_{\mathrm{n}}}\right]\\ & = \frac{1}{\sum_{k=0,k\neq L-1}^{2L-2}\left[\frac{a_0^2e^{{-}b_0T_0}\sinh [b_0 (L-1-k-T_0)]}{-a_0^2-b_0\sigma^2}\right]^2+\frac{3M^2\sigma^2_{n}}{M^2-1}}. \end{aligned}$$

Similarly, from the SINR metric we also note that the squeezing part of the equivalent channel impacts the SINR significantly. Focusing on analyzing the interference introduced by the UWOC channel, we only discuss the received SINR without applying the equalizer in this paper. With the help of the transmitted time-reversal waveform, the received signal is more concentrated than the case without using the TR waveform. In addition, the received signal has a symmetric waveform with respect to the sampling time. Therefore, some commonly used equalizers can be applied to reduce the ISI easily.

3.3 Equalizer design

Different from applying TR waveforms in RF systems, an equalizer is required to reduce the residual ISI in UWOC systems. In this paper, we respectively derive zero forcing (ZF) and minimum mean squared error (MMSE) equalizers at the receiver, and compare their performances.

When the channel state information is estimated, the ZF equalizer with $2N+1$ taps can be designed as

(29)$$\boldsymbol{e}_{\text{ZF}} = \left({\bf{H}}^T{\bf{H}}\right)^{{\dagger}}{\bf{H}}^T\bf{d},$$

where $(\cdot )^\dagger$ is pseudo-inverse operation. ${\bf {H}}$ is obtained from (9) as

(30)$$\begin{aligned} & {\bf{H}} = \\ & \begin{bmatrix} \begin{matrix} R[L-1] & R[L] & \cdots & R[L+2N-1] \\ R[L] & R[L-1] & \cdots & R[L+2N-2]\\ \vdots & \vdots & \ddots & \vdots \\ R[L+2N-1] & R[L+2N-2] & \cdots & R[L-1] \end{matrix} \end{bmatrix} . \end{aligned}$$

$\bf {d}$ is a $2N+1$ by $1$ column vector, representing as

(31)$${\bf{d}} = [0~ \cdots~0 \quad 1 \quad 0 ~ \cdots~0]^T,$$

where $(\cdot )^T$ represents the transpose operation. Note that, the sampling period is assumed to be $T_s$, the idea of which is the same as applying a matched filter and sampling the data at the peak values of the correlation signals at the receiver.

Considering the impact of the additive noise, MMSE equalizer is alternative. Similar to the ZF equalizer, we also design a $(2N+1)$-tap MMSE equalizer by solving the function

(32)$$\arg~\min_{\boldsymbol{e}_{\text{MMSE}}}\mathbb{E}_{\bf{x,n}}\left[(\boldsymbol{e}_{\text{MMSE}}^T\left(\bf{Hx+n}\right)-x[N+1])^2\right],$$

where $\mathbb {E}_{\bf {x.n}}\left [\cdot \right ]$ denotes the expectation with respect to $\bf {x}$ and $\bf {n}$. ${\bf {x}}=\left [x[1]~ x[2] ~\cdots ~x[2N-1]\right ]^T,{\bf {n}}=\left [n[1]~ n[2] ~\cdots ~n[2N-1]\right ]^T$. After solving (32), the MMSE equalizer can be represented as

(33)$$\boldsymbol{e}_{\text{MMSE}} = {\bf{H}}^T\left({\bf{H}}{\bf{H}}^T+\frac{\sigma^2_n}{\bar{P}_{\mathrm{s}}}{\bf{I}}\right)^{{\dagger}}\bf{d},$$

where ${\bf {I}}$ is the unit matrix.

4. Numerical results

In this section, we discuss the UWOC system performance using the proposed TR waveform numerically and analytically. Unless otherwise noted, the parameters used to obtain the CIR are shown in Table 1, which is typical for UWOC systems. In this work, we assume the harbor water is used, and the corresponding seawater parameters, such as absorption and scattering coefficients can be found from [32].

Table 1. Parameters of Monte Carlo simulation for the underwater optical channel.

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To fully analyze and discuss the system performances using the designed TR-waveform, we consider two typical scenarios: band-unlimited and band-limited cases. For a system using a laser diode and avalanche photodiode (APD) with a high bandwidth [33–37], we can assume the system bandwidth is unlimited. For a given transmitted bit rate and the used modulation scheme, the symbol rate and the occupied spectrum are obtained. For clarity, we first illustrate the channel impulse responses for different propagation distances and discuss the properties of the equivalent channel. Then, the BER performances for bandwidth-unlimited light sources are shown. After that, a more practical case that takes the low-pass light sources into account is applied.

4.1 UWOC channel and TR waveforms

Monte-Carlo photon tracing algorithm is an effective and commonly used technique for UWOC CIR modelling [23,38,39]. Figure 3 illustrates the normalized channel impulse response, the equivalent channel response and the proposed channel model under the three scenarios, labeled as CIR1-3 shown in Tabel 1. From the first column of Fig. 3, we notice that with the increase of the beam width, the delay spread of the channel becomes larger for the same propagation distance, as the comparison of CIR2 and CIR3. In addition, the longer the propagation distance the photons transmit, the larger number of scattering is introduced. Therefore, the multi-path effect is severer. From the second column of Fig. 3, the simulation results of the channel impulse response are shown to be consistence with the channel model proposed in (1). It suggests that the TR waveform is feasible for the UWOC system due to the randomness component. Observing the equivalent channel, we find its waveform is more concentrated and smooth than the original channel impulse response, which implies that the received signals using TR waveforms are easier for equalization than that without using the TR waveforms.

Fig. 3. Underwater optical CIR and its equivalent channel. The first column subfigures are original CIR as labeled in Table 1; The second column is the decomposition of the first column as shown in (1); The third column presents the equivalent channel, the green and red lines are random and squeezing parts in the equivalent channel model (11), respectively.

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In this section, we also display the spatial distribution of the normalized light intensity at the receiving plane in Fig. 4. This figure shows the three cases using the parameters from Tabel 1. From the result, the light spots arrived at the receiving plane have a relatively large size compared with the regular diameter of the photo-detector. Therefore, an accurate alignment is neglected in this work. In some high-dynamic moving scenarios, the alignment is a critical issue needed to be addressed [40,41]. Some useful alignment techniques are discussed in [42], which is beyond the scope of this work.

Fig. 4. The spatial distribution of normalization light intensity at the receiving plane, (a) CIR1 case, (b) CIR2 case, (c) CIR3 case.

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4.2 Unlimited bandwidth

To better analyze the system performance, we first assume the bandwidth of the light sources is wide enough and the overall channel impulse response only depends on the propagation of the numerous photons. Figure 5 shows the BER comparison with different CIR scenarios. The cases with and without an equalizer at different transmitted bit rates are also tested. In this work, the ZF and MMSE equalizers with $x$ numbers of taps are employed and denoted as "ZF-$x$" and "MMSE-$x$", respectively. In addition, "TR" and "NTR" denoted in the figure respectively represent the system using and without using the TR waveforms. Particularly, "0-TR/NTR" is the case without using any equalizers.

Fig. 5. BER performance comparison under 4-PAM modulation with unlimited bandwidth using ZF and MMSE equalizers. (a) Bit Rate is 4 Gbps over CIR1, (b) Bit Rate is 4 Gbps over CIR2, (c) Bit Rate is 4 Gbps over CIR3, (d) Bit Rate is 8 Gbps over CIR1,(e) Bit Rate is 8 Gbps over CIR2, (f) Bit Rate is 8 Gbps over CIR3.

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Figure 5 shows the BER performances of 4-PAM with 4-Gbps and 8 Gbps. In Figs. 5(a) and (d) with the scenario CIR1, we can obtain that only using TR waveforms cannot achieve an acceptable performance to guarantee a usable communication quality. With the help of TR waveforms, only a few taps of the equalizer are needed to improve the BER by reducing the ISI. From Figs. 5(a) and (d), a ZF and MMSE equalizer with 21 taps is sufficient to reduce the ISI. The results shown in Figs. 5(b) and (e) similarly illustrate that an equalizer is necessarily after the TR waveforms are transmitted. For the scenario CIR2, the relatively poor channel quality requires an equalizer with a large number of taps. The BER performances for the CIR3 scenario are shown in Figs. 5(c) and (f). From the figure, a consistent phenomenon with the scenarios CIR1 and CIR2 are illustrated. Comparing the performances using ZF and MMSE equalizers, the BER is similar. In Fig. 5(c) and (d), the BER performance of MMSE equalizer with 11 taps is better than ZF equalizer a litter.

Figure 5 also shows the system performances at different bit rates. Since the ISI in the scenario CIR1 is fully reduced by using TR waveform and the equalizer with 11 taps or 21 taps, the cases of 4 Gbps and 8 Gbps achieve a similar BER as illustrated. For a relatively poor channel quality in the scenarios CIR2 and CIR3, a higher transmitted bit rate in this system results in a worse BER performance than transmitting data in the scenario CIR1 at the same data rate. However, if a larger tap equalizer is applied, the ISI can also be significantly mitigated.

The BER performances with 8-PAM modulation are shown in Fig. 6, the phenomenon of which is similar to the case of 4-PAM shown in Fig. 5. From this result, the performance of BER with TR waveforming is better than without TR waveform in all the tested three scenarios. However, due to the improvement in modulation size, the BER also increases. The higher-order equalizers can be used to improve the BER of the system.

Fig. 6. BER performance comparison under 8-PAM modulation with unlimited bandwidth using ZF and MMSE equalizers. (a) Bit Rate is 4 Gbps over CIR1, (b) Bit Rate is 4 Gbps over CIR2, (c) Bit Rate is 4 Gbps over CIR3, (d) Bit Rate is 8 Gbps over CIR1,(e) Bit Rate is 8 Gbps over CIR2, (f) Bit Rate is 8 Gbps over CIR3.

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4.3 Limited bandwidth

After discussing the performances of unlimited bandwidth of the light sources, a more practical case with bandlimited light sources is considered in this paper. The state-of-the-art light sources such as high-speed LEDs and laser diodes (LD) can usually achieve a few tens hundred of MHz 3 dB bandwidth [33–37]. In this paper, we discuss the system performances with the 3 dB bandwidth of 100 MHz and 1 GHz in Figs. 7 and 8, respectively. Considering the optical-electrical properties of the light sources, we model the bandlimited light source as a first-order low-pass filer.

Fig. 7. BER performance comparison under 4-PAM modulation with limited bandwidth using ZF and MMSE equalizers. The transmitted bit rate is 1 Gbps. (a) $f_c$ = 100 MHz with CIR1, (b) $f_c$ = 100 MHz with CIR2, (c) $f_c$ = 100 MHz with CIR3, (d) $f_c$ = 1 GHz with CIR1, (e) $f_c$ = 1 GHz with CIR2, (f) $f_c$ = 1 GHz with CIR3.

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Fig. 8. BER performance comparison under 4-PAM modulation with limited bandwidth using ZF and MMSE equalizers. The transmitted bit rate is 2 Gbps. (a) $f_c$ = 100 MHz with CIR1, (b) $f_c$ = 100 MHz with CIR2, (c) $f_c$ = 100 MHz with CIR3, (d) $f_c$ = 1 GHz with CIR1, (e) $f_c$ = 1 GHz with CIR2, (f) $f_c$ = 1 GHz with CIR3.

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We conduct simulations to evaluate the bit error rate (BER) at 1 Gbps and 2 Gbps, as illustrated in Figs. 7 and 8, respectively. Overall, our proposed TR waveform combined with an equalizer performs effectively in bandlimited underwater optical communication systems. In Figs. 7(a), (b) and (c), the system that does not use TR waveform is only effective with equalizers having a large number of taps, whose LEDs have a narrow 3 dB bandwidth. However, when TR waveform is utilized, BER performance significantly improves in the three tested scenarios. For the 3 dB bandwidth case at 1 GHz, it is observed that the non-TR system outperforms the TR system, as depicted in Fig. 7(e), (f). This can be attributed to the fact that with an increase in bandwidth up to 1 GHz, both TR and non-TR channels tend towards a delta function. At a lower bit rate of 1 Gbps, the absence of ISI renders noise as the primary obstacle to BER improvement. The system using TR waveforms exhibits a greater sensitivity to the noise than the non-TR system. This discrepancy leads to a worse BER for the system using TR waveforms than not using TR waveforms, as illustrated in Fig. 7(e), (f).

For the scenarios CIR2 and CIR3 with 100 MHz bandwidth shown in Figs. 8(b) and (c), the system BER is significantly worse than the case with a wider bandwidth shown in Figs. 8(e) and (f). Due to the great channel quality of the scenario CIR1, its BER using TR waveform and an equalizer is better than the scenarios CIR2 and CIR3 with the same 3 dB bandwidth.

5. Conclusion

In this paper, we propose a TR waveforming technique in UWOC systems for ISI reduction. Due to the optical scattering properties in the ocean, the dispersive channel impulse response of UWOC is caused by the multi-path effects of numerous delayed photons. Based on the analysis and simulation results shown in this paper, TR waveforming is well-suited for UWOC systems. In addition, transmitting TR waveforms through the dispersive channel makes the received pulses with a symmetric shape. Thus, the ISI can be effectively mitigated by equalizing. Since only the intensity modulation and direct detection can be used for UWOC systems, we derive the UWOC channel as an exponential bias with the random scattering effects. From the discussions shown in this work, a phenomenon called the squeezing effect is found, which explains the influence of non-negative valued channel responses in the UWOC systems using TR waveforms. Due to the squeezing effect, only transmitting the TR waveforms cannot fully reduce the ISI. Furthermore, the SINR sometimes is degraded because of the squeezing effect. However, TR waveforming provides a significant processing gain and results in a symmetric equivalent channel shape, which facilitates the equalizer’s ability to reduce residual ISI. Based on the numerical and simulation results presented in this paper, TR waveforms are highly suitable for UWOC systems. However, an equalizer is necessary due to the squeezing effect.

Funding

National Natural Science Foundation of China (62001392); Key Research and Development Projects of Shaanxi Province (501100015401) (2022GY-097); Scientific Research Plan Projects of Shaanxi Education Department (501100013101) (20JK0873).

Acknowledgment

The authors would like to appreciate Professor Yingmin Wang for his effort and suggestions on this paper.

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Name	Propagation distance	BeamWidth (BW)
CIR1	10 m	$\frac{π}{6} rad$
CIR2	15 m	$\frac{π}{6} rad$
CIR3	15 m	0.75 $\times 10^{- 3} rad$

Time-reversal waveform design for underwater wireless optical communication systems

Abstract

1. Introduction

2. UWOC system model using TR waveform

2.1 Effective model of UWOC channel impulse response

2.2 TR waveform design for UWOC systems

3. Performance analysis

3.1 Equivalent channel model

3.2 Received SINR

3.3 Equalizer design

4. Numerical results

4.1 UWOC channel and TR waveforms

4.2 Unlimited bandwidth

4.3 Limited bandwidth

5. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (33)

Optics Express