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Temporal dispersion compensation for turbid underwater optical wireless communication links
Accepted Manuscript Temporal dispersion compensation for turbid underwater optical wireless communication links Yiming Li, Haodong Liang, Chao Gao, Maoke Miao, Xiaofeng Li
PII: DOI: Reference:
S0030-4018(18)31028-9 https://doi.org/10.1016/j.optcom.2018.11.062 OPTICS 23647
To appear in:
Optics Communications
Received date : 22 October 2018 Revised date : 22 November 2018 Accepted date : 23 November 2018 Please cite this article as: Y. Li, H. Liang, C. Gao et al., Temporal dispersion compensation for turbid underwater optical wireless communication links, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.11.062 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Temporal Dispersion Compensation for Turbid Underwater Optical Wireless Communication Links Yiming Lia , Haodong Lianga , Chao Gaoa , Maoke Miaoa , Xiaofeng Lia, a School
of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731 China
Abstract Temporal dispersion compensation has been considered in the turbid underwater optical wireless communication systems. A comprehensive comparison of different equalization criteria and adaptive equalization strategies has been made in this paper. New results have been shown since there is an optimal field of view for certain channels. Moreover, one can also make a tradeoff among computational complexity, performance and transmit distance by referring to the conclusions in this paper. Keywords: Temporal dispersion, underwater optical wireless communication, equalization
1. Introduction Underwater optical wireless communication (UOWC) is a promising technology due to its much higher data rate, security and lower latency when compared to traditional acoustic communications. Although the UOWC system has a relatively short transmit length as the laser beam suffers from absorption, scattering and turbulence-induced fading, it is still a necessary technology to satisfy the increasing data rate demands in many applications such as underwater wireless sensor networks (UWSNs) [1]. In order to analyze the UOWC channels, inherent optical properties (IOPs) were proposed to describe the absorption and scattering effect of the transmitted Email address: [email protected] (Xiaofeng Li )
Preprint submitted to Optics Communications
November 21, 2018
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laser beams in sea water [2]. In a turbid water environment, the photons will be scattered for more than once. This phenomenon is called as the multiple scattering effect, and the received optical power will be dominated by such an effect in a relatively long transmit length [3]. Although this phenomenon exerts a positive impact on the overall received power, it also introduces a negative effect of increasing the temporal dispersion of the received signals [4]. Recently, numerous researches have been carried out to numerically analyze the temporal dispersion of the UOWC channels [5–8]. Moreover, the MonteCarlo results have also been compared with Mullen’s experimental data to validate the efficiency of the simulation results [6, 9]. On the other hand, impulse response has also been mathematically modelled by a combination of Gamma functions [10–12]. By naturally decompose the absorption and scattering effect, a more accurate model has also been proposed to explain the connections between the IOPs and the impulse response [13]. As is shown in these previous studies, the received signal of the impulse may be stretched to a few nanoseconds in the UOWC channel. And this will introduce severe inter-symbol interference (ISI) in high speed transmission system. Although the compensation of ISI is mentioned in [10], only the most preliminary ZF equalizer is briefly discussed in coastal water where the ISI is not that significant. To the best of our knowledge, there is no comprehensive research on the compensation of ISI in very turbid water channels (i.e. harbor water). In this paper, we carry out a detailed study on the compensation of temporary dispersion in such channels. Different equalization criteria as well as adaptive equalization strategies are discussed. As is seen from the given results, an appropriate algorithm should be choosed to effectively compensate the ISI in different channels. Moreover, it can be concluded from the results that we can make a tradeoff between the ISI and the received power to achieve the optimal system performance by carefully setting the receiver FOVs. This paper is organized as follows: Section 2 describes the modeling for UOWC systems as the preliminaries to our study. In Section 3 we describe different equalization criteria applied in this paper. In Section 4 we describe differ2
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ent algorithms for adaptive equalizers to compensate the time-varying channel parameters. Numerical results and data analysis are presented in Section 5 followed by conclusions in Section 6. 2. Temporal Dispersion Modeling for UOWC systems 2.1. Channel Models of UOWC According to Mobley’s previous work [2], absorption and scattering are two major effects in the UOWC channel, which can be expressed by a (λ) and b (λ) respectively. Moreover, the attenuation coefficient c (λ) = a (λ)+b (λ) is defined to describe the overall energy loss in the channel. It is also worth mentioning that the values of a (λ), b (λ) and c (λ) varies with different types of water as well as the wavelength of the light λ.
Figure 1: The illustration of the multiple scattering effect.
Considering that a large amount of particles are dispersed in the turbid water with a transmit distance L, the scattering effect will be much more significant than that in clear water. As a consequence, the multiple scattering effect, which is shown in Fig. 1, will be dominant in the received power. In order to accurately describe the energy distribution of the light beam, the volumn scattering function (VSF) is introduced as: β (θ, λ) = lim
lim
∆D→0 ∆Ω→0
PS (θ, λ) , ∆D∆Ω
(1)
where PS (θ, λ) is the fraction of incident power scattered out of the beam into a solid angle ∆Ω centered on θ (Fig. 1) when the light propagation distance is 3
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∆D. VSF is the scattered intensity per unit incident irradiance per unit volume of water. Integrating β (θ, λ) over all solid angles gives the scattering coefficient [14]: b (λ) =
Z
β (θ, λ)dΩ =
Z
2π
Z
π
β (θ, φ, λ) sin (θ) dθ dφ,
0
0
(2)
where θ is the polar angle and φ is the azimuthal angle of the scattering respectively, which are also shown in Fig. 1. Normalizing Eq. (2) with the scattering coefficient b, we obtain the scattering phase function (SPF), which is defined as: β (θ, λ) β˜ (θ, λ) = . b (λ)
(3)
Moreover, it is often assumed that the scattering is azimuthally symmetric. According to Eqs. (2) and (3), the SPF can be written as: Z π 2π β˜ (λ, θ) sin (θ) dθ = 1,
(4)
0
There are two most widely used SPF models among all other models. The first model is the long-standing Henyey-Greenstein phase function (HGPF), which is first proposed by Henyey to describe the multiple scattering effect in astrophysics [15]. The HGPF model is defined as: β˜HG (θ) =
1 − g2
3
4π(1 + g 2 − 2g cos (θ)) 2
,
(5)
where g is the average cosine of θ. Although this model is very convenient for numerical calculation, it is somewhat different from Petzold’s measurements in the UOWC channels [16, 17]. Therefore, it is supplanted by a more complicated but more accurate form called the Fournier-Forand phase function (FFPF) [18]. This function is derived under two assumptions: the first is the hyperbolic size distribution of the particles, and the second is the anomalous diffraction approximation to the exact Mie theory. The FFPF is given by [19]: β˜F F (θ) =
1 2
4π(1 − δ) δ ν
{ν (1 − δ) − (1 − δ ν )
θ + [δ (1 − δ ) − ν (1 − δ)] sin 2 ν 1 − δ180 + 3cos2 (θ) − 1 , ν 16π (δ180 − 1) δ180 −2
ν
4
(6)
where ν =
3−µ 2
and δ =
4 sin2 3(n−1)2
θ 2
. Here µ is the slope parameter of
the hyperbolic distribution, n is the refractive index of the water, and δ180 is δ evaluated at θ = 180◦ . 4
10
Petzold’s measurement FFPF model (n=1.10, ν=3.5835) HGPF model (g=0.924) 2
10 Phase function
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0
10
−2
10
−1
10
0
1
10
10
2
10
Scattered angle[degree]
Figure 2: Comparison of different phase functions.
A comparison of the optimal HGPF (g=0.924), the optimal FFPF (n=1.10, ν=3.5835) and the Petzold’s measurements is shown in Fig. 2 [16]. As is seen from Fig. 2, the HGPF is different from Petzold’s experimental data in both small forward scattering angles (θ < 20◦ ) and large backward scattering angles (θ > 130◦ ). As a comparison, the FFPF fits well with the experimental data. Therefore, the FFPF is applied to model the scattering effect in the rest of this paper. 2.2. Temporal Dispersion of UOWC While the analytical solutions of the radiative transfer equations (RTE) are too difficult to be solved in general cases, both Monte-Carlo simulations and modeling functions can be used to depict the temporal dispersion of impulse response in the UOWC channels. By exploiting the inherent optical properties (IOPs) of such channels, the combination of exponential and arbitrary power function (CEAPF) fits well with the Monte-Carlo simulation results and outper5
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forms the weighted double Gamma functions (WDGF) in most realistic cases [11, 13]. Therefore, the CEAPF is applied to model the temporal dispersion of the impulse response in the rest of this paper. The form of the CEAPF can be written as: h (t) = C1
∆tα β
(∆t + C2 )
· e−a·v(∆t+t0 ) ,
(7)
where C1 > 0, C2 > 0, α > −1, and β > 0 are the four parameters to be solved and v is speed of light in water. 3. Digital Equalization Criteria In this section, we assume the channel state information are known at the receiver (CSIR). And some most widely used criteria for the temporal dispersion compensation, which are also called equalization algorithms, are discussed. Consider the UOWC channel with an impulse response {fn }, the k th received sample can be written as: vk =
L0 X
fj Ik−j + nk ,
(8)
j=0
where Ik is the k th transmitted M -ary symbol, L0 is the number of interfering components and nk is the k th additive white Gaussian noise (AWGN). 3.1. Maximum-Likelihood Sequence Estimator (MLSE) When the probability of each transmitted symbol is independent and identically distributed (i.i.d.), the MLSE is the optimum estimator as it minimizes the probability of error. In a discrete time channel, the MLSE criterion is equivalent to a Viterbi algorithm for a corresponding finite-state machine. The metrics of the Viterbi algorithm can be written as [20]: LX 0 +1 P M (I , · · · , I ) = max ln p (vj |Ij , · · · , I1 ) 1 L +1 2 0 I1 j=1 ln p (vL0 +k |IL0 +k , · · · , Ik ) (k ≥ 2) P Mk (IL0 +k , · · · , Ik+1 ) = max Ik +P M (I ,··· ,I ) k−1
L0 +k−1
k
(9)
6
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where Ij is the j th M -ary symbol with M different values. As is seen from Eq. (9), M L0 +1 conditional probabilities corresponding to (IL0 +k , · · · , Ik ) will be calculated in the k th (k ≥ 2) step. After the Ik with the largest probability
is selected and the others are discarded, P Mk has M L0 states represented by (IL0 +k , · · · , Ik+1 ). As an inherent property of the Viterbi algorithm, the delay in detecting each symbol is variable. Fortunately, this phenomenon can be avoided by truncating the remaining sequence to the q most recent symbols (q L0 ) and choosing
the symbol in the most probable sequence. In practice, the performance loss is negligible if q ≥ 5L0 . 3.2. Zero-Forcing (ZF) Equalizer Although the MLSE is the optimum criterion, its computational complexity grows exponentially, which is prohibitively expensive in most realistic cases. Therefore, linear equalizer is introduced to reduce the computational complexity at the cost of a suboptimal performance. The ZF equalizer has the simplest criterion by completely eliminating the intersymbol interference (ISI). Consider the ZF equalizer with an impulse response {c−K , c−K+1 , . . . , cK }, the response of the ZF criterion can be written as: qn =
K X
j=−K
cj fn−j
1 = 0
(n = 0) (n 6= 0)
.
(10)
By applying Eq. (10), the ZF criterion completely eliminates the ISI when K = ∞. However, K is a finite number in realistic systems. In such cases, the equalizer has 2K + 1 adjustable parameters while there are 2K + L0 + 1 functions of response {qn }. Therefore, there is always some residual ISI in realistic systems. 3.3. Mean-Square Error (MSE) Equalizer The ZF equalizer introduces redundant noise when trying to eliminate ISI. Therefore, a tradeoff can be made between the ISI and the additive noise to
7
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achieve a better performance. MSE equalizer optimally trades off this problem by minimizing the mean square value of the error: εk = Ik − Iˆk ,
(11)
where Iˆk is the estimate of Ik . In order to minimize the MSE, εk should be orthogonal to the received ∗ sample vk−l (l ∈ R). Consider the MSE equalizer with an impulse response
{c−K , c−K+1 , . . . , cK }, we obtain: K X ∗ E Ik − cj vk−j vk−l = 0,
(12)
j=−K
or equivalently,
K X
j=−K
∗ ∗ cj E vk−j vk−l = E Ik vk−l .
(13)
By applying Eq. (8) and exploiting the independent property of Ik with different values of k (i.e. E (Ij Ik )=0) when j 6= k), The terms in Eq. (13) can be written as:
L0 X ∗ E vk−j vk−l fn∗ fn+l−j + N0 δlj =
n=0
E
∗ Ik vk−l
=
,
(14)
∗ f−l
where −K ≤ l ≤ K, N0 is the variance of the complex noise nk , and δlj is the Dirichlet function. By substituting Eq. (14) into Eq. (13), cj can be represented by fj . 3.4. Decision-Feedback Equalizer (DFE) When the previous detection is correct, the corresponding ISI can be completely eliminated by a feedback filter without introducing any redundant noise. Therefore, the nonlinear DFE will have a better performance than the linear filters such as ZF equalizer and MSE equalizer. The DFE equalizer is composed by a feedforward transversal filter and a feedback filter with its input from a symbol detector. Assume that the feedforward filter is a MSE equalizer with
8
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K1 + 1 taps and the feedback filter has K2 taps, the criterion of the feedforward filter can be written as: 0 X
cj
j=−K
0 X
fn∗ fn+l−j + N0 δlj
n=−l
!
∗ = f−l
(−K1 ≤ l ≤ 0) .
(15)
And the coefficients of the feedback filter can be calculated by the equation: ck = −
0 X
cj fk−j
j=−K1
(1 ≤ k ≤ K2 ) .
(16)
4. Adaptive Equalization Algorithms In the UOWC channel, the channel characteristics is unknown to the receiver. Considering the random changes of the temperature, salinity and randomly distributed bubbles, the channel characteristics can even be time-variant. Therefore, adaptive algorithm should be considered to determine the coefficients of the equalizers. 4.1. Least-Mean-Square Algorithm (LMSA) By applying the steepest descent algorithm, the LMSA can recursively calculate the coefficient array as below: ˆ Tk Vk Iˆk = C εk = Ik − Iˆk
ˆ k = −εk Vk∗ G ˆ ˆ k − ∆G ˆk Ck+1 = C
.
(17)
ˆ k is the estimation of the coefficient array , Vk = At the k th iteration, C T [vk+K , · · · , vk−K ] is the received sample array, Iˆk is the k th estimated symbol,
ˆ k is the gradient array, Ik is the k th transmitted symbol, εk is the error signal, G and ∆ is a positive number which is small enough to ensure the convergence of the iterative procedure. The tap weight magnitudes of LMSA are shown in Fig. 3. The receiver FOV is 60◦ , the data rate is 1 Gbps, the scale factor ∆ is 0.003, and the SNR is set 9
1.5
Magnitude of Tap Weight
1.5
Magnitude of Tap Weight
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1 LMSA Coeficients Calculated Coefficients 0.5
0 0
0.5
1 Number of Iterations
1.5
2 4
x 10
(a) SNR=15dB
1 LMSA Coeficients Calculated Coefficients 0.5
0 0
0.5
1 Number of Iterations
1.5
2 4
x 10
(b) SNR=40dB
Figure 3: Magnitudes of tap weights when applying LMSA. ∆ = 0.003. Harbor water. FOV= 60◦ . Data rate=1 Gbps.
at 15 dB and 40 dB in Figs. 3(a) and 3(b) respectively. In order to observe the convergence characteristics of the LMSA equalizer, MMSE coefficients calculated by Eqs. (13) and (14) are plotted by dashed lines as the convergence limit. As is seen in Fig. 3, the convergence time of the LMSA tap weights is approximately 10 µs. Moreover, the LMSA is always an unbiased estimate of the coefficients. Furthermore, a relatively low SNR will induce a negative fluctuation on the coefficients. However, this fluctuation will also slightly decrease the convergence time as a positive effect. 4.2. Constant-Modulus Algorithm (CMA) A known training sequence is mandatory in the LMSA. However, this is not preferred or even prohibited in certain cases. In order to solve this problem, blind equalization is proposed to adaptively adjust the coefficients of the equalizers without the benefits of a training sequence. Constant-modulus algorithm, which is also called Godard algorithm, is the most widely used algorithm among all other blind equalization algorithms. This algorithm is a steepestdescent algorithm which was first proposed and studied by Godard. In the
10
UOWC channel, the CMA can be written as: ˆ Tk Vk Iˆk = C 2 , ˆ k+1 = C ˆ k + ∆ · Vk∗ Iˆk R2 − Iˆk C
(18)
. 4 2 where ∆ is the step coefficient and R2 = E |Ik | E |Ik | is a constant
based on the constellation and distribution of the transmitted symbol Ik . 1.5
Magnitude of Tap Weight
1.5
Magnitude of Tap Weight
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1 CMA Coeficients Calculated Coefficients 0.5
0 0
2
4 6 Number of Iterations
8
10 4
x 10
(a) SNR=15dB
1 CMA Coeficients Calculated Coefficients 0.5
0 0
2
4 6 Number of Iterations
8
10 4
x 10
(b) SNR=40dB
Figure 4: Magnitudes of tap weights when applying CMA. ∆ = 0.001. Harbor water. FOV= 60◦ . Data rate=1 Gbps
The tap weight magnitudes of CMA are shown in Fig. 4. As same as Fig. 3, the receiver FOV is 60◦ , the data rate is 1 Gbps, and the SNR is set at 15 dB and 40 dB in Figs. 4(a) and 4(b) respectively. However, the scale factor ∆ is set at a smaller value of 0.001 to compensate for the negative impact of estimating without a training sequence and achieve a similar estimation performance as the LMSA. Moreover, MMSE coefficients are also plotted as the convergence limit. While the system setting is comparable with the LMSA, the convergence time of the CMA tap weights is approximately 30 µs, which is much longer than the LMSA. This phenomenon implies that the absence of the training sequence will induce a negative influence on the adaptive equalization algorithm. Moreover, The coefficients will be less stable and the convergence time will be slightly decreased when the SNR is relatively low. Furthermore, due to the lack of the training sequence, the CMA is not an unbiased estimation algorithm of the 11
coefficients.
5. Numerical Results In this section, we consider a UOWC system with a 514 nm wavelength which is working in harbor water to correspond with Petzold’s measurements [16]. In such channels, the speed of light in water is v = 2.237 × 108 m · s−1
and the IOPs are a = 0.366 m−1 and b = 1.829 m−1 . Moreover, the data rate is set to 1 Gbps and the channel distance is set at 10.93 m and 16.40 m such that the ISI is notable to the system performance. Firstly, the fitting curves of the impulse response under typical channel conditions and system settings are introduced as the preconditions of channels. Secondly, different equalization criteria is discussed and it can be observed from the numerical results that the equalized output is sensitive to channel settings, equalizer types and receiver FOVs. Therefore, it is crucial to carefully choose the system settings in different channels to achieve the optimal performance. Thirdly, we discuss the performance of different adaptive equalizers. By comparing the advantages and disadvantages of LMSA and CMA equalizers, an appropriate algorithm can be chosen to adapt to the given channels and system settings. −7
−9
x 10
2.5 MC
1.8
x 10
°
FOV=20
1.6
°
2
FOV=40
1.2
FOV=40°
MC
CEAPF FOV=40°
°
CEAPF FOV=40 °
MC
FOV=20
CEAPF FOV=20
°
MC
1.4
°
MC
CEAPF FOV=20°
FOV=180
°
MC
CEAPF FOV=180°
Intensity
2
Intensity
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1 0.8
1.5
FOV=60
°
CEAPF FOV=60
FOV=180°
MC
CEAPF FOV=180° 1
0.6 0.4
0.5
0.2 0 0
0.5
1
1.5
2
2.5
∆t[ns]
(a) L=10.93 m
0 0
1
2
3
4 ∆t[ns]
5
6
7
(b) L=16.40 m
Figure 5: Impulse response in harbor water. On-axis. FOV= 20◦ , 40◦ , 180◦ .
Fig. 5 shows the normalized impulse response in different UOWC channels. Relevant data and detailed analysis can be found in [13]. As is seen in Fig. 5, 12
the CEAPF model fits well with the simulation results (RMSE < 1%) with 109 independent transmissions in all typical channel conditions and system settings. Therefore, this model will be applied to our further analysis. 0
0
10
10
−1
−1
10
10
−2
−2
10 BER
BER
10
−3
10
−4
10
−5
10
−6
10
0
MLSE ZF MSE DFE w/o Equ w/o ISI 2
−3
10
−4
10
−5
10
−6
4
6 8 SNR [dB]
10
12
10
14
5
10
15
SNR [dB]
(b) L=16.40 m, FOV= 60◦
0
0
10
10
−1
−1
10
10
−2
−2
10 BER
10
−3
10
−4
10
−5
10
−6
10
MLSE ZF MSE DFE w/o Equ w/o ISI
0
(a) L=16.40 m, FOV= 20◦
BER
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
0
MLSE ZF MSE DFE w/o Equ w/o ISI 5
−3
10
−4
10
−5
10
−6
10 SNR [dB]
15
10
20
(c) L=16.40 m, FOV= 180◦
0
MLSE ZF MSE DFE w/o Equ w/o ISI 2
4
6 8 SNR [dB]
10
12
14
(d) L=10.93 m, FOV= 180◦
Figure 6: BER performance of different equalization criteria in different transmission lengths and receiver FOVs. Harbor water. w/o=without, Equ=Equalizer.
The performance of different equalization criteria is shown in Fig. 6. Different transmission lengths and receiver FOVs have also been compared in Fig. 6. As is seen in Figs. 6 (a) ∼ (c), when the transmission length is relatively long (L = 16.40 m), there will be significant performance discrepancies among different equalization criteria. Although the Linear (MSE and ZF) equalizers suffer from severe SNR penalties (up to 6 dB at the FEC limit when FOV = 180◦ ), the nonlinear DFE equalizer will achieve an acceptable SNR penalty (< 1.5 dB) without increasing intolerant computational complexity. Moreover, the chan13
nel ISI is so severe that the eye diagram is closed when the FOV is only 20◦ . Therefore, the equalization technology is mandatory in such systems. On the other hand, the multiple scattering effect is stronger when the FOV is larger. Therefore, a higher transmit power is necessary to achieve the same BER performance. It is worth mentioning that the larger FOV will also result in higher receiving power (see Fig. 5), and there will be an inherent tradeoff between the channel ISI and the receiving power. This tradeoff will be discussed in details later. On the other hand, as is seen in Fig. 6 (d), when the transmission length is relatively short (L = 10.93 m), there is almost no performance loss at the feedforward error correction (FEC) limit of 10−3 when applying the MLSE equalizer. And the performance discrepancies among different criteria are less than 1 dB at the FEC limit. Moreover, as the eye diagram is not closed in this system, a desired BER can even be achieved without equalization. But this is at the expense of significant SNR penalty. 76 ZF MSE DFE MLSE
74 72 SNRT [dB]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
70 68 66 64 0
50
100 FOV [degree]
150
Figure 7: Transmit power v.s. FOV. BER=10−3 . L=16.40 m. On axis.
As is mentioned earlier, there will be an inherent tradeoff between the channel ISI and the received power. This phenomenon is shown in Fig. 7. SN RT is defined as the ratio of transmitted optical power to background noise. On one 14
hand, the system is mainly restricted by the received power when the receiver FOV is relatively small. when the FOV increases, the total received power will increase by receiving more multiple scattered light, and thus the required transmit power will decrease. On the other hand, when the receiver FOV is relatively large, the system performance is mainly restricted by the ISI which is induced by the multiple scattering effect. When the receiver FOV is decreased, the multiple scattering effect can be suppressed. As a result, the optimal FOV is achieved at approximately 60◦ by balancing the influence of the received power and the multiple scattering effect. It is also worth mentioning that due to the negligible power of the large angle MSE, the SNR performance will be flat and consistent when the FOV is sufficiently large (> 130◦ ). 0
0
10
10
LMSA,∆=0.015 LMSA,∆=0.003 MMSE
RMSE
RMSE
LMSA,∆=0.015 LMSA,∆=0.003 MMSE
SNR=15dB
−1
10
SNR=20dB
−1
10
SNR=40dB
−2
10
SNR=40dB 4 6 Iteration Number
−2
0
0.5
1 Iteration Number
1.5
10
2
0
2
4
x 10
(a) LMSA. FOV= 60◦ . 100 data average.
8 4
x 10
(b) LMSA. FOV= 180◦ . 100 data average.
0
0
10
10 CMA,∆=0.004 CMA,∆=0.001 MMSE
CMA,∆=0.004 CMA,∆=0.001 MMSE
SNR=15dB
−1
RMSE
RMSE
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
10
SNR=20dB
−1
10
SNR=40dB −2
10
0
SNR=40dB
−2
2
4 6 Iteration Number
8
10 4
x 10
(c) CMA. FOV= 60◦ . 400 data average.
10
0
0.5
1
1.5 2 Iteration Number
2.5
3 5
x 10
(d) CMA. FOV= 180◦ . 400 data average.
Figure 8: Comparison of convergence rate for the LMSA and CMA. L=16.40 m. harbor water.
The convergence rate of the LMSA and the CMA is shown in Fig. 8. By 15
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selecting a relatively large scale factor (∆ = 0.015 for LMSA and ∆ = 0.004 for CMA), a relatively fast initial convergence can be achieved. However, this is at the cost of some excess RMSE which is an undesired effect. On the other hand, due to the influence of noise, the convergence time will be shorter and the RMSE will be larger in the small SNR area. Furthermore, by comparing Fig. 8 with Figs. 3 and 4, it can be observed that the convergence rate of the RMSE is consistent with that of the coefficients. As a result, we can conclude from the above discussions that the LMSA is preferred when the channel length is relatively large as it has a faster convergence rate, while the CMA is preferred in a short channel because it is free from a known training sequence.
6. Conclusions In order to compensate the ISI in turbid UOWC channels, the performance of different equalization criteria has been discussed in turbid water. Compared to the previous work with the most preliminary ZF equalizers, we made a comprehensive discussion of ZF, MSE, DFE and MLSE equalization criteria as well as LMSA and CMA adaptive equalizers on different channel and system settings. Furthermore, new results can be obtained and concluded as follows. Firstly, the ISI can be compensated in typical UOWC channels by applying the equalizers. Moreover, there will be significant performance discrepancies when applying different equalization criteria. On the other hand, it is worth mentioning that the DFE equalizer exhibits a preferable tradeoff between the computational complexity and the BER performance by applying the nonlinear algorithm. Secondly, a tradeoff can be made between the ISI and the received power to achieve the optimal system performance. And this is realized by choosing the receiver FOVs. To be specific, the optimal FOV of the given channel is approximately 60◦ and this value may change under different system settings. Thirdly, when refers to the adaptive equalizers, the LMSA equalizer outperforms the CMA equalizer when the ISI is severe due to its relatively fast
16
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convergence rate. However, the CMA equalizer is preferred in a moderate ISI channel as it is free from training sequence. The above mentioned conclusions can provide guidelines to the UOWC systems. In the next step, the research will be focused on the experimental demonstration of the dispersion compensation technology. And the USWNs in harbor water can be thus considered in the near future.
References [1] Z. Zeng, S. Fu, H. Zhang, Y. Dong, J. Cheng, A survey of underwater optical wireless communications, IEEE Communications Surveys & Tutorials 19 (1) (2017) 204–238. [2] C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, 1st Edition, Academic Press, San Diego, CA, USA, 1994. [3] B. M. Cochenour, L. J. Mullen, A. E. Laux, Characterization of the beamspread function for underwater wireless optical communications links, IEEE Journal of Oceanic Engineering 33 (4) (2008) 513–521. [4] B. Cochenour, L. Mullen, J. Muth, Temporal response of the underwater optical channel for high-bandwidth wireless laser communications, IEEE Journal of Oceanic Engineering 38 (4) (2013) 730–742. [5] W. C. Cox Jr, Simulation, modeling, and design of underwater optical communication systems, North Carolina State University, 2012. [6] J. Li, Y. Ma, Q. Zhou, B. Zhou, H. Wang, Channel capacity study of underwater wireless optical communications links based on monte carlo simulation, Journal of Optics 14 (1) (2011) 015403. [7] A.-P. Huang, L.-w. Tao, Monte carlo based channel characteristics for underwater optical wireless communications, IEICE Transactions on Communications 100 (4) (2017) 612–618.
17
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[8] S. K. Sahu, P. Shanmugam, A theoretical study on the impact of particle scattering on the channel characteristics of underwater optical communication system, Optics Communications 408 (2018) 3–14. [9] L. Mullen, A. Laux, B. Cochenour, Propagation of modulated light in water: implications for imaging and communications systems, Applied optics 48 (14) (2009) 2607–2612. [10] S. Tang, Y. Dong, X. Zhang, Impulse response modeling for underwater wireless optical communication links, IEEE transactions on communications 62 (1) (2014) 226–234. [11] Y. Dong, H. Zhang, X. Zhang, On impulse response modeling for underwater wireless optical mimo links, in: Communications in China (ICCC), 2014 IEEE/CIC International Conference on, IEEE, 2014, pp. 151–155. [12] H. Zhang, Y. Dong, Impulse response modeling for general underwater wireless optical mimo links, IEEE Communications Magazine 54 (2) (2016) 56–61. [13] Y. Li, M. S. Leeson, X. Li, Impulse response modeling for underwater optical wireless channels, Applied Optics 57 (17) (2018) 4815. [14] C. Mobley, E. Boss, C. Roesler, Ocean optics web book (2010). [15] L. G. Henyey, J. L. Greenstein, Diffuse radiation in the galaxy, The Astrophysical Journal 93 (1941) 70–83. [16] T. J. Petzold, Volume scattering functions for selected ocean waters, Tech. rep., Scripps Institution of Oceanography La Jolla Ca Visibility Lab (1972). [17] C. Gabriel, M.-A. Khalighi, S. Bourennane, P. L´eon, V. Rigaud, Montecarlo-based channel characterization for underwater optical communication systems, Journal of Optical Communications and Networking 5 (1) (2013) 1–12.
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[18] G. R. Fournier, J. L. Forand, Analytic phase function for ocean water, in: Ocean Optics XII, International Society for Optics and Photonics, 1994, pp. 194–201. [19] G. R. Fournier, M. Jonasz, Computer-based underwater imaging analysis, in: SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation, International Society for Optics and Photonics, 1999, pp. 62–70. [20] J. G. Proakis, M. Salehi, Digital Communications, 5th Edition, McGrawHill, New York, 2008.
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PII: DOI: Reference:
S0030-4018(18)31028-9 https://doi.org/10.1016/j.optcom.2018.11.062 OPTICS 23647
To appear in:
Optics Communications
Received date : 22 October 2018 Revised date : 22 November 2018 Accepted date : 23 November 2018 Please cite this article as: Y. Li, H. Liang, C. Gao et al., Temporal dispersion compensation for turbid underwater optical wireless communication links, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.11.062 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Temporal Dispersion Compensation for Turbid Underwater Optical Wireless Communication Links Yiming Lia , Haodong Lianga , Chao Gaoa , Maoke Miaoa , Xiaofeng Lia, a School
of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731 China
Abstract Temporal dispersion compensation has been considered in the turbid underwater optical wireless communication systems. A comprehensive comparison of different equalization criteria and adaptive equalization strategies has been made in this paper. New results have been shown since there is an optimal field of view for certain channels. Moreover, one can also make a tradeoff among computational complexity, performance and transmit distance by referring to the conclusions in this paper. Keywords: Temporal dispersion, underwater optical wireless communication, equalization
1. Introduction Underwater optical wireless communication (UOWC) is a promising technology due to its much higher data rate, security and lower latency when compared to traditional acoustic communications. Although the UOWC system has a relatively short transmit length as the laser beam suffers from absorption, scattering and turbulence-induced fading, it is still a necessary technology to satisfy the increasing data rate demands in many applications such as underwater wireless sensor networks (UWSNs) [1]. In order to analyze the UOWC channels, inherent optical properties (IOPs) were proposed to describe the absorption and scattering effect of the transmitted Email address: [email protected] (Xiaofeng Li )
Preprint submitted to Optics Communications
November 21, 2018
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laser beams in sea water [2]. In a turbid water environment, the photons will be scattered for more than once. This phenomenon is called as the multiple scattering effect, and the received optical power will be dominated by such an effect in a relatively long transmit length [3]. Although this phenomenon exerts a positive impact on the overall received power, it also introduces a negative effect of increasing the temporal dispersion of the received signals [4]. Recently, numerous researches have been carried out to numerically analyze the temporal dispersion of the UOWC channels [5–8]. Moreover, the MonteCarlo results have also been compared with Mullen’s experimental data to validate the efficiency of the simulation results [6, 9]. On the other hand, impulse response has also been mathematically modelled by a combination of Gamma functions [10–12]. By naturally decompose the absorption and scattering effect, a more accurate model has also been proposed to explain the connections between the IOPs and the impulse response [13]. As is shown in these previous studies, the received signal of the impulse may be stretched to a few nanoseconds in the UOWC channel. And this will introduce severe inter-symbol interference (ISI) in high speed transmission system. Although the compensation of ISI is mentioned in [10], only the most preliminary ZF equalizer is briefly discussed in coastal water where the ISI is not that significant. To the best of our knowledge, there is no comprehensive research on the compensation of ISI in very turbid water channels (i.e. harbor water). In this paper, we carry out a detailed study on the compensation of temporary dispersion in such channels. Different equalization criteria as well as adaptive equalization strategies are discussed. As is seen from the given results, an appropriate algorithm should be choosed to effectively compensate the ISI in different channels. Moreover, it can be concluded from the results that we can make a tradeoff between the ISI and the received power to achieve the optimal system performance by carefully setting the receiver FOVs. This paper is organized as follows: Section 2 describes the modeling for UOWC systems as the preliminaries to our study. In Section 3 we describe different equalization criteria applied in this paper. In Section 4 we describe differ2
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ent algorithms for adaptive equalizers to compensate the time-varying channel parameters. Numerical results and data analysis are presented in Section 5 followed by conclusions in Section 6. 2. Temporal Dispersion Modeling for UOWC systems 2.1. Channel Models of UOWC According to Mobley’s previous work [2], absorption and scattering are two major effects in the UOWC channel, which can be expressed by a (λ) and b (λ) respectively. Moreover, the attenuation coefficient c (λ) = a (λ)+b (λ) is defined to describe the overall energy loss in the channel. It is also worth mentioning that the values of a (λ), b (λ) and c (λ) varies with different types of water as well as the wavelength of the light λ.
Figure 1: The illustration of the multiple scattering effect.
Considering that a large amount of particles are dispersed in the turbid water with a transmit distance L, the scattering effect will be much more significant than that in clear water. As a consequence, the multiple scattering effect, which is shown in Fig. 1, will be dominant in the received power. In order to accurately describe the energy distribution of the light beam, the volumn scattering function (VSF) is introduced as: β (θ, λ) = lim
lim
∆D→0 ∆Ω→0
PS (θ, λ) , ∆D∆Ω
(1)
where PS (θ, λ) is the fraction of incident power scattered out of the beam into a solid angle ∆Ω centered on θ (Fig. 1) when the light propagation distance is 3
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∆D. VSF is the scattered intensity per unit incident irradiance per unit volume of water. Integrating β (θ, λ) over all solid angles gives the scattering coefficient [14]: b (λ) =
Z
β (θ, λ)dΩ =
Z
2π
Z
π
β (θ, φ, λ) sin (θ) dθ dφ,
0
0
(2)
where θ is the polar angle and φ is the azimuthal angle of the scattering respectively, which are also shown in Fig. 1. Normalizing Eq. (2) with the scattering coefficient b, we obtain the scattering phase function (SPF), which is defined as: β (θ, λ) β˜ (θ, λ) = . b (λ)
(3)
Moreover, it is often assumed that the scattering is azimuthally symmetric. According to Eqs. (2) and (3), the SPF can be written as: Z π 2π β˜ (λ, θ) sin (θ) dθ = 1,
(4)
0
There are two most widely used SPF models among all other models. The first model is the long-standing Henyey-Greenstein phase function (HGPF), which is first proposed by Henyey to describe the multiple scattering effect in astrophysics [15]. The HGPF model is defined as: β˜HG (θ) =
1 − g2
3
4π(1 + g 2 − 2g cos (θ)) 2
,
(5)
where g is the average cosine of θ. Although this model is very convenient for numerical calculation, it is somewhat different from Petzold’s measurements in the UOWC channels [16, 17]. Therefore, it is supplanted by a more complicated but more accurate form called the Fournier-Forand phase function (FFPF) [18]. This function is derived under two assumptions: the first is the hyperbolic size distribution of the particles, and the second is the anomalous diffraction approximation to the exact Mie theory. The FFPF is given by [19]: β˜F F (θ) =
1 2
4π(1 − δ) δ ν
{ν (1 − δ) − (1 − δ ν )
θ + [δ (1 − δ ) − ν (1 − δ)] sin 2 ν 1 − δ180 + 3cos2 (θ) − 1 , ν 16π (δ180 − 1) δ180 −2
ν
4
(6)
where ν =
3−µ 2
and δ =
4 sin2 3(n−1)2
θ 2
. Here µ is the slope parameter of
the hyperbolic distribution, n is the refractive index of the water, and δ180 is δ evaluated at θ = 180◦ . 4
10
Petzold’s measurement FFPF model (n=1.10, ν=3.5835) HGPF model (g=0.924) 2
10 Phase function
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0
10
−2
10
−1
10
0
1
10
10
2
10
Scattered angle[degree]
Figure 2: Comparison of different phase functions.
A comparison of the optimal HGPF (g=0.924), the optimal FFPF (n=1.10, ν=3.5835) and the Petzold’s measurements is shown in Fig. 2 [16]. As is seen from Fig. 2, the HGPF is different from Petzold’s experimental data in both small forward scattering angles (θ < 20◦ ) and large backward scattering angles (θ > 130◦ ). As a comparison, the FFPF fits well with the experimental data. Therefore, the FFPF is applied to model the scattering effect in the rest of this paper. 2.2. Temporal Dispersion of UOWC While the analytical solutions of the radiative transfer equations (RTE) are too difficult to be solved in general cases, both Monte-Carlo simulations and modeling functions can be used to depict the temporal dispersion of impulse response in the UOWC channels. By exploiting the inherent optical properties (IOPs) of such channels, the combination of exponential and arbitrary power function (CEAPF) fits well with the Monte-Carlo simulation results and outper5
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forms the weighted double Gamma functions (WDGF) in most realistic cases [11, 13]. Therefore, the CEAPF is applied to model the temporal dispersion of the impulse response in the rest of this paper. The form of the CEAPF can be written as: h (t) = C1
∆tα β
(∆t + C2 )
· e−a·v(∆t+t0 ) ,
(7)
where C1 > 0, C2 > 0, α > −1, and β > 0 are the four parameters to be solved and v is speed of light in water. 3. Digital Equalization Criteria In this section, we assume the channel state information are known at the receiver (CSIR). And some most widely used criteria for the temporal dispersion compensation, which are also called equalization algorithms, are discussed. Consider the UOWC channel with an impulse response {fn }, the k th received sample can be written as: vk =
L0 X
fj Ik−j + nk ,
(8)
j=0
where Ik is the k th transmitted M -ary symbol, L0 is the number of interfering components and nk is the k th additive white Gaussian noise (AWGN). 3.1. Maximum-Likelihood Sequence Estimator (MLSE) When the probability of each transmitted symbol is independent and identically distributed (i.i.d.), the MLSE is the optimum estimator as it minimizes the probability of error. In a discrete time channel, the MLSE criterion is equivalent to a Viterbi algorithm for a corresponding finite-state machine. The metrics of the Viterbi algorithm can be written as [20]: LX 0 +1 P M (I , · · · , I ) = max ln p (vj |Ij , · · · , I1 ) 1 L +1 2 0 I1 j=1 ln p (vL0 +k |IL0 +k , · · · , Ik ) (k ≥ 2) P Mk (IL0 +k , · · · , Ik+1 ) = max Ik +P M (I ,··· ,I ) k−1
L0 +k−1
k
(9)
6
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where Ij is the j th M -ary symbol with M different values. As is seen from Eq. (9), M L0 +1 conditional probabilities corresponding to (IL0 +k , · · · , Ik ) will be calculated in the k th (k ≥ 2) step. After the Ik with the largest probability
is selected and the others are discarded, P Mk has M L0 states represented by (IL0 +k , · · · , Ik+1 ). As an inherent property of the Viterbi algorithm, the delay in detecting each symbol is variable. Fortunately, this phenomenon can be avoided by truncating the remaining sequence to the q most recent symbols (q L0 ) and choosing
the symbol in the most probable sequence. In practice, the performance loss is negligible if q ≥ 5L0 . 3.2. Zero-Forcing (ZF) Equalizer Although the MLSE is the optimum criterion, its computational complexity grows exponentially, which is prohibitively expensive in most realistic cases. Therefore, linear equalizer is introduced to reduce the computational complexity at the cost of a suboptimal performance. The ZF equalizer has the simplest criterion by completely eliminating the intersymbol interference (ISI). Consider the ZF equalizer with an impulse response {c−K , c−K+1 , . . . , cK }, the response of the ZF criterion can be written as: qn =
K X
j=−K
cj fn−j
1 = 0
(n = 0) (n 6= 0)
.
(10)
By applying Eq. (10), the ZF criterion completely eliminates the ISI when K = ∞. However, K is a finite number in realistic systems. In such cases, the equalizer has 2K + 1 adjustable parameters while there are 2K + L0 + 1 functions of response {qn }. Therefore, there is always some residual ISI in realistic systems. 3.3. Mean-Square Error (MSE) Equalizer The ZF equalizer introduces redundant noise when trying to eliminate ISI. Therefore, a tradeoff can be made between the ISI and the additive noise to
7
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achieve a better performance. MSE equalizer optimally trades off this problem by minimizing the mean square value of the error: εk = Ik − Iˆk ,
(11)
where Iˆk is the estimate of Ik . In order to minimize the MSE, εk should be orthogonal to the received ∗ sample vk−l (l ∈ R). Consider the MSE equalizer with an impulse response
{c−K , c−K+1 , . . . , cK }, we obtain: K X ∗ E Ik − cj vk−j vk−l = 0,
(12)
j=−K
or equivalently,
K X
j=−K
∗ ∗ cj E vk−j vk−l = E Ik vk−l .
(13)
By applying Eq. (8) and exploiting the independent property of Ik with different values of k (i.e. E (Ij Ik )=0) when j 6= k), The terms in Eq. (13) can be written as:
L0 X ∗ E vk−j vk−l fn∗ fn+l−j + N0 δlj =
n=0
E
∗ Ik vk−l
=
,
(14)
∗ f−l
where −K ≤ l ≤ K, N0 is the variance of the complex noise nk , and δlj is the Dirichlet function. By substituting Eq. (14) into Eq. (13), cj can be represented by fj . 3.4. Decision-Feedback Equalizer (DFE) When the previous detection is correct, the corresponding ISI can be completely eliminated by a feedback filter without introducing any redundant noise. Therefore, the nonlinear DFE will have a better performance than the linear filters such as ZF equalizer and MSE equalizer. The DFE equalizer is composed by a feedforward transversal filter and a feedback filter with its input from a symbol detector. Assume that the feedforward filter is a MSE equalizer with
8
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K1 + 1 taps and the feedback filter has K2 taps, the criterion of the feedforward filter can be written as: 0 X
cj
j=−K
0 X
fn∗ fn+l−j + N0 δlj
n=−l
!
∗ = f−l
(−K1 ≤ l ≤ 0) .
(15)
And the coefficients of the feedback filter can be calculated by the equation: ck = −
0 X
cj fk−j
j=−K1
(1 ≤ k ≤ K2 ) .
(16)
4. Adaptive Equalization Algorithms In the UOWC channel, the channel characteristics is unknown to the receiver. Considering the random changes of the temperature, salinity and randomly distributed bubbles, the channel characteristics can even be time-variant. Therefore, adaptive algorithm should be considered to determine the coefficients of the equalizers. 4.1. Least-Mean-Square Algorithm (LMSA) By applying the steepest descent algorithm, the LMSA can recursively calculate the coefficient array as below: ˆ Tk Vk Iˆk = C εk = Ik − Iˆk
ˆ k = −εk Vk∗ G ˆ ˆ k − ∆G ˆk Ck+1 = C
.
(17)
ˆ k is the estimation of the coefficient array , Vk = At the k th iteration, C T [vk+K , · · · , vk−K ] is the received sample array, Iˆk is the k th estimated symbol,
ˆ k is the gradient array, Ik is the k th transmitted symbol, εk is the error signal, G and ∆ is a positive number which is small enough to ensure the convergence of the iterative procedure. The tap weight magnitudes of LMSA are shown in Fig. 3. The receiver FOV is 60◦ , the data rate is 1 Gbps, the scale factor ∆ is 0.003, and the SNR is set 9
1.5
Magnitude of Tap Weight
1.5
Magnitude of Tap Weight
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1 LMSA Coeficients Calculated Coefficients 0.5
0 0
0.5
1 Number of Iterations
1.5
2 4
x 10
(a) SNR=15dB
1 LMSA Coeficients Calculated Coefficients 0.5
0 0
0.5
1 Number of Iterations
1.5
2 4
x 10
(b) SNR=40dB
Figure 3: Magnitudes of tap weights when applying LMSA. ∆ = 0.003. Harbor water. FOV= 60◦ . Data rate=1 Gbps.
at 15 dB and 40 dB in Figs. 3(a) and 3(b) respectively. In order to observe the convergence characteristics of the LMSA equalizer, MMSE coefficients calculated by Eqs. (13) and (14) are plotted by dashed lines as the convergence limit. As is seen in Fig. 3, the convergence time of the LMSA tap weights is approximately 10 µs. Moreover, the LMSA is always an unbiased estimate of the coefficients. Furthermore, a relatively low SNR will induce a negative fluctuation on the coefficients. However, this fluctuation will also slightly decrease the convergence time as a positive effect. 4.2. Constant-Modulus Algorithm (CMA) A known training sequence is mandatory in the LMSA. However, this is not preferred or even prohibited in certain cases. In order to solve this problem, blind equalization is proposed to adaptively adjust the coefficients of the equalizers without the benefits of a training sequence. Constant-modulus algorithm, which is also called Godard algorithm, is the most widely used algorithm among all other blind equalization algorithms. This algorithm is a steepestdescent algorithm which was first proposed and studied by Godard. In the
10
UOWC channel, the CMA can be written as: ˆ Tk Vk Iˆk = C 2 , ˆ k+1 = C ˆ k + ∆ · Vk∗ Iˆk R2 − Iˆk C
(18)
. 4 2 where ∆ is the step coefficient and R2 = E |Ik | E |Ik | is a constant
based on the constellation and distribution of the transmitted symbol Ik . 1.5
Magnitude of Tap Weight
1.5
Magnitude of Tap Weight
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1 CMA Coeficients Calculated Coefficients 0.5
0 0
2
4 6 Number of Iterations
8
10 4
x 10
(a) SNR=15dB
1 CMA Coeficients Calculated Coefficients 0.5
0 0
2
4 6 Number of Iterations
8
10 4
x 10
(b) SNR=40dB
Figure 4: Magnitudes of tap weights when applying CMA. ∆ = 0.001. Harbor water. FOV= 60◦ . Data rate=1 Gbps
The tap weight magnitudes of CMA are shown in Fig. 4. As same as Fig. 3, the receiver FOV is 60◦ , the data rate is 1 Gbps, and the SNR is set at 15 dB and 40 dB in Figs. 4(a) and 4(b) respectively. However, the scale factor ∆ is set at a smaller value of 0.001 to compensate for the negative impact of estimating without a training sequence and achieve a similar estimation performance as the LMSA. Moreover, MMSE coefficients are also plotted as the convergence limit. While the system setting is comparable with the LMSA, the convergence time of the CMA tap weights is approximately 30 µs, which is much longer than the LMSA. This phenomenon implies that the absence of the training sequence will induce a negative influence on the adaptive equalization algorithm. Moreover, The coefficients will be less stable and the convergence time will be slightly decreased when the SNR is relatively low. Furthermore, due to the lack of the training sequence, the CMA is not an unbiased estimation algorithm of the 11
coefficients.
5. Numerical Results In this section, we consider a UOWC system with a 514 nm wavelength which is working in harbor water to correspond with Petzold’s measurements [16]. In such channels, the speed of light in water is v = 2.237 × 108 m · s−1
and the IOPs are a = 0.366 m−1 and b = 1.829 m−1 . Moreover, the data rate is set to 1 Gbps and the channel distance is set at 10.93 m and 16.40 m such that the ISI is notable to the system performance. Firstly, the fitting curves of the impulse response under typical channel conditions and system settings are introduced as the preconditions of channels. Secondly, different equalization criteria is discussed and it can be observed from the numerical results that the equalized output is sensitive to channel settings, equalizer types and receiver FOVs. Therefore, it is crucial to carefully choose the system settings in different channels to achieve the optimal performance. Thirdly, we discuss the performance of different adaptive equalizers. By comparing the advantages and disadvantages of LMSA and CMA equalizers, an appropriate algorithm can be chosen to adapt to the given channels and system settings. −7
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Figure 5: Impulse response in harbor water. On-axis. FOV= 20◦ , 40◦ , 180◦ .
Fig. 5 shows the normalized impulse response in different UOWC channels. Relevant data and detailed analysis can be found in [13]. As is seen in Fig. 5, 12
the CEAPF model fits well with the simulation results (RMSE < 1%) with 109 independent transmissions in all typical channel conditions and system settings. Therefore, this model will be applied to our further analysis. 0
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Figure 6: BER performance of different equalization criteria in different transmission lengths and receiver FOVs. Harbor water. w/o=without, Equ=Equalizer.
The performance of different equalization criteria is shown in Fig. 6. Different transmission lengths and receiver FOVs have also been compared in Fig. 6. As is seen in Figs. 6 (a) ∼ (c), when the transmission length is relatively long (L = 16.40 m), there will be significant performance discrepancies among different equalization criteria. Although the Linear (MSE and ZF) equalizers suffer from severe SNR penalties (up to 6 dB at the FEC limit when FOV = 180◦ ), the nonlinear DFE equalizer will achieve an acceptable SNR penalty (< 1.5 dB) without increasing intolerant computational complexity. Moreover, the chan13
nel ISI is so severe that the eye diagram is closed when the FOV is only 20◦ . Therefore, the equalization technology is mandatory in such systems. On the other hand, the multiple scattering effect is stronger when the FOV is larger. Therefore, a higher transmit power is necessary to achieve the same BER performance. It is worth mentioning that the larger FOV will also result in higher receiving power (see Fig. 5), and there will be an inherent tradeoff between the channel ISI and the receiving power. This tradeoff will be discussed in details later. On the other hand, as is seen in Fig. 6 (d), when the transmission length is relatively short (L = 10.93 m), there is almost no performance loss at the feedforward error correction (FEC) limit of 10−3 when applying the MLSE equalizer. And the performance discrepancies among different criteria are less than 1 dB at the FEC limit. Moreover, as the eye diagram is not closed in this system, a desired BER can even be achieved without equalization. But this is at the expense of significant SNR penalty. 76 ZF MSE DFE MLSE
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Figure 7: Transmit power v.s. FOV. BER=10−3 . L=16.40 m. On axis.
As is mentioned earlier, there will be an inherent tradeoff between the channel ISI and the received power. This phenomenon is shown in Fig. 7. SN RT is defined as the ratio of transmitted optical power to background noise. On one 14
hand, the system is mainly restricted by the received power when the receiver FOV is relatively small. when the FOV increases, the total received power will increase by receiving more multiple scattered light, and thus the required transmit power will decrease. On the other hand, when the receiver FOV is relatively large, the system performance is mainly restricted by the ISI which is induced by the multiple scattering effect. When the receiver FOV is decreased, the multiple scattering effect can be suppressed. As a result, the optimal FOV is achieved at approximately 60◦ by balancing the influence of the received power and the multiple scattering effect. It is also worth mentioning that due to the negligible power of the large angle MSE, the SNR performance will be flat and consistent when the FOV is sufficiently large (> 130◦ ). 0
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Figure 8: Comparison of convergence rate for the LMSA and CMA. L=16.40 m. harbor water.
The convergence rate of the LMSA and the CMA is shown in Fig. 8. By 15
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selecting a relatively large scale factor (∆ = 0.015 for LMSA and ∆ = 0.004 for CMA), a relatively fast initial convergence can be achieved. However, this is at the cost of some excess RMSE which is an undesired effect. On the other hand, due to the influence of noise, the convergence time will be shorter and the RMSE will be larger in the small SNR area. Furthermore, by comparing Fig. 8 with Figs. 3 and 4, it can be observed that the convergence rate of the RMSE is consistent with that of the coefficients. As a result, we can conclude from the above discussions that the LMSA is preferred when the channel length is relatively large as it has a faster convergence rate, while the CMA is preferred in a short channel because it is free from a known training sequence.
6. Conclusions In order to compensate the ISI in turbid UOWC channels, the performance of different equalization criteria has been discussed in turbid water. Compared to the previous work with the most preliminary ZF equalizers, we made a comprehensive discussion of ZF, MSE, DFE and MLSE equalization criteria as well as LMSA and CMA adaptive equalizers on different channel and system settings. Furthermore, new results can be obtained and concluded as follows. Firstly, the ISI can be compensated in typical UOWC channels by applying the equalizers. Moreover, there will be significant performance discrepancies when applying different equalization criteria. On the other hand, it is worth mentioning that the DFE equalizer exhibits a preferable tradeoff between the computational complexity and the BER performance by applying the nonlinear algorithm. Secondly, a tradeoff can be made between the ISI and the received power to achieve the optimal system performance. And this is realized by choosing the receiver FOVs. To be specific, the optimal FOV of the given channel is approximately 60◦ and this value may change under different system settings. Thirdly, when refers to the adaptive equalizers, the LMSA equalizer outperforms the CMA equalizer when the ISI is severe due to its relatively fast
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convergence rate. However, the CMA equalizer is preferred in a moderate ISI channel as it is free from training sequence. The above mentioned conclusions can provide guidelines to the UOWC systems. In the next step, the research will be focused on the experimental demonstration of the dispersion compensation technology. And the USWNs in harbor water can be thus considered in the near future.
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