Adaptive threshold decision for on-off keying transmission systems in atmospheric turbulence

Shengli Ding; Jiankun Zhang; Anhong Dang

doi:10.1364/OE.25.024425

1. Introduction

Optical wireless communication (OWC) has attracted wide interests in the recent decades as its high data rate, cost-effectiveness, easy installation, anti-electromagnetic interference and license-free property [1, 2]. Due to these properties, OWC system is an attractive solution for large capacity communication, temporary communication, emergency communication and last-mile problem [3]. Because of the high system complexity of phase or frequency modulation, current OWC systems usually adopt intensity modulation and direct detection (IM/DD) with on-off keying (OOK) format [4, 5]. However, the OWC link through atmospheric turbulence channel is susceptible to the atmospheric conditions, which leads to the intensity fluctuation of the IM/DD OOK signals.

Due to the turbulence-induced intensity fluctuation, fixed decision threshold would not be effective [6]. In the literature, many researches on the optimization of decision threshold have been carried out to improve the IM/DD performance in OWC systems. The optimum decision utilizes the instantaneous signal-to-noise ratio (SNR) to decide the signal symbol-by-symbol [7]. However, the optimum decision needs perfect instantaneous channel state information (CSI), which is unavailable in practical systems. In [8], maximum likelihood (ML) symbol-by-symbol detection and ML sequence detection techniques were introduced, where the receiver is assumed to have knowledge of the channel model instead of instantaneous CSI. The simplified version of ML sequence detection was developed in [9] as fast multiple symbol detection. However, the accurate channel model is also non-attainable in practice. In order to accommodate this problem, pilot symbol assisted modulation (PSAM) was proposed to approach the system performance, where the pilot symbol is used to recover the CSI [10]. However, the frequency insertion of pilot symbol will lead to loss of bandwidth efficiency. Blind detection (BD) is proposed in [11] to detect optical signals without the aid of CSI or pilot symbol. Nevertheless, the bit error rate (BER) performance of BD is floored. Although the error floor is mitigated in [12], the effectiveness is obvious only when the SNR is very high. A generalized likelihood ratio test (GLRT) receiver is developed in [13], which uses the GLRT metric to detect OOK sequence with Viterbi algorithm. Excellent performance will be achieved, however, pilot symbols are needed at the beginning of data transmission. In [14], it was demonstrated that IM/DD systems could use the correlation in turbulence-induced scintillation at closely spaced wavelength channels, and one of the wavelengths is used as reference channel to recover the instantaneous CSI. Obviously, the improved performance is at the cost of decreased spectrum effectiveness. To overcome this shortcoming, the theory of OOK with source information transformation was proposed in [15], where at least one laser transmits bit ‘1’ during each symbol duration, so as to extract explicit turbulence fading references for decision without a specialized reference channel. However, the hardware setup is rather complicated, for its employment of several lasers.

In this study, we propose an adaptive threshold decision (ATD) scheme without knowledge of CSI. According to the results of researches on plane wave model and spherical wave model, there exists a sharp cutoff frequency in the turbulence spectrum, and the spectrum beyond this cutoff frequency is dramatically attenuated. The temporal power spectrum of intensity fluctuation is studied in [16–18] and found to be low-pass, with cutoff frequency smaller than 1 kHz. Based on the low-pass characteristic of turbulence channel, the received signal is divided into two branches with one straight forward as the signal to be decided, while the other is low-pass filtered to extract the approximate instantaneous CSI. By decomposing the filtered output, we find that the filtered output consists of half the instantaneous intensity fluctuation and some perturbation terms. Thus, the filtered output can be directly used as the decision threshold, and the decision error depends on the perturbation terms which are determined by ratio of the data rate and filter bandwidth. Theoretical analyses and Monte Carlo simulations show that performance of the proposed scheme is very close to the lower bound given by decision with perfect CSI, when ratio of the data rate and filter bandwidth exceeds a critical value depending on the channel condition. An indoor experiment is carried out with 2.5 Gbps data rate, where a specially customized phase screen is employed to simulate the turbulence channel. The experimental results match well with the theoretical predictions. As the processing procedure shows high real-time capability without loss of power and bandwidth efficiency, ATD is particularly suitable for high-speed transmissions.

2. Scheme of adaptive threshold decision

In this section, we first present a detailed description of the proposed ATD scheme. Property of the filtered signal is analyzed, and decision criterion is given. Then, the BER performance is deduced followed by a numerical analysis.

2.1 Scheme description

Figure 1 shows the system block diagram of the proposed ATD scheme. Here, we consider a point-to-point OWC system adopting IM/DD OOK modulation over turbulence channel. At the transmitter, the laser carrier is sent into the turbulence channel after being OOK modulated by data sequence. Suppose the signal amplitude is unit, then light intensity of the transmitted signal is $s (t) = \sum_{k = - \infty}^{\infty} d [k] g (t - k T),$ where $d [k] \in {0, 1}$ denotes the data sequence, g(t) represents the square-wave shaping pulse and T is the symbol period [15]. In the turbulence channel, the OOK modulated signal is affected by the turbulence-induced intensity fluctuation. At the receiver, the received optical signal is converted to electrical current under the influence of noise generated in the detecting process, which consists of thermal noise, dark current and shot noise. In OWC systems, the detecting process is usually considered to be dominated by the thermal noise, which can be modeled as additive white Gaussian noise (AWGN) [9]. Then the analog electrical current r(t) is converted into digital signal through analog to digital conversion (ADC). In k-th bit interval, the digital received signal r[k] is obtained through sampling the electrical current at time kT, and is [12]

r [k] = I [k] s [k] + n [k],

where I[k] is the intensity fluctuation and n[k] denotes the AWGN. Obviously, there always exists the relationship of s[k] = d[k], while physical significances of the two variables are distinctly different. d[k] represents the bit information, while s[k] is the sampling value of the baseband signal s(t). As the received signal amplitude varies with time, the real-time SNR is also time-varying. In order to uniformly demonstrate the signal quality, normalized SNR is defined as

γ = E {s {[k]}^{2}} / E {n {[k]}^{2}} = 1 / 2 σ_{n}^{2}

, where E{∙} denotes the expected average operation and

σ_{n}^{2}

is the noise power. The advantage of employing normalized SNR rather than real-time SNR is that, the normalized SNR is irrelevant to either the time-varying intensity fluctuation or turbulence strength, so that we can analyze the impact of SNR and intensity fluctuation separately.

Fig. 1 System block diagram of ATD. TA/RA, transmitting/receiving antenna; ADC, analog-to-digital conversion; LPF, low-pass filter.

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After the sampling process, the received signal r[k] is divided into two branches, and the upper branch is straightly fed into the decision module with proper delay to maintain synchronization with the lower branch as shown in Fig. 1. At the same time, the lower branch passes through a low-pass filter (LPF), and the filtering output φ[k] is employed as the decision threshold and the decision output is $\hat{d} [k]$ . In the following, we will demonstrate the property of φ[k] and why it can be used as decision threshold.

Suppose the filter bandwidth is B_f, and obviously it should satisfy the relationship of $B_{c} < B_{f} < < B_{s}$ , where B_c denotes cut-off frequency of the intensity fluctuation and the B_s is the signal bandwidth. As low-pass filtering is a kind of linear transformation, the filtering output in the k-th bit interval can be formulated as

φ [k] = I [k] s_{f} [k] + n_{f} [k],

where s_f[k] represents the filtering output of baseband signal s[k] individually passing through the LPF and n_f[k] is that of the AWGN n[k]. Hence n_f[k] is band-limited Gaussian white noise with zero mean, and its variance is

σ_{n f}^{2} = σ_{n}^{2} B_{f} / B_{s}

. As

B_{f} < < B_{s}

,

σ_{n f}^{2}

is usually of small value and

σ_{n f}^{2} < < σ_{n}^{2}

. Furthermore, the one-sided power spectrum density (PSD) of square-wave baseband signal s[k] is [19]

Ψ_{s} (f) = \frac{1}{2 B_{s}} \sin c^{2} (\frac{f}{B_{s}}) + \frac{1}{4} δ (f), f \geq 0,

where sinc(x) = sin(πx)/πx for x≠0 and sinc(0) = 1, and δ(f) is the Dirac delta function. The first term represents PSD of the alternating current (AC) component and the second term is PSD of the direct current (DC) component. Therefore, s_f[k] also contains DC and AC component, i.e. s_f[k] = s_fD[k] + s_fA[k], where s_fD[k] = 1/2 is the DC component and s_fA[k] denotes the AC component. Obviously, s_fA[k] is zero-mean and its variance can be obtained from Eq. (3) as

σ_{s, f A}^{2} = \frac{1}{2 B_{s}} \int_{0}^{B_{f}} \sin c^{2} (\frac{f}{B_{s}}) d f \approx \frac{B_{f}}{2 B_{s}},

where the approximation holds true as

B_{f} < < B_{s}

. Obviously, there exists

σ_{s, f A}^{2} < < 1

. Based on the above analysis, Eq. (2) can be rewritten as

φ [k] = \frac{I [k]}{2} + I [k] s_{f A} [k] + n_{f} [k] .

The first term I[k]/2 is exactly the threshold of decision with perfect CSI [11]. The second and third terms are perturbation terms as $B_{f} < < B_{s}$ . Therefore, φ[k] can be employed as the decision threshold and the decision criterion is

r [k] ≷_{\hat{d} [k] = 0}^{\hat{d} [k] = 1} φ [k] .

2.2 BER performance

According to the above proposed decision criterion, the pairwise error probability of bit ‘1’ and bit ‘0’ respectively are

\begin{array}{l} P (0 | 1) = \Pr {I [k] + n [k] < φ [k]} \\ =Pr {n [k] - n_{f} [k] > I [k] (\frac{1}{2} - s_{f A} [k])}, \end{array}

\begin{array}{l} P (1 | 0) = \Pr {n [k] > φ [k]} \\ =Pr {n [k] - n_{f} [k] > I [k] (\frac{1}{2} + s_{f A} [k])}, \end{array}

where Pr{∙} represents the probability operation, P(0|1) is the conditional probability that if a bit ‘1’ is transmitted, the decision output is a bit ‘0’, and P(1|0) is just the opposite. As

σ_{n f}^{2} < < σ_{n}^{2}

, the impact of n_f[k] in the BER expression can be neglected. Given I and s_fA, the conditioned BER can be formulated as

\begin{array}{l} P_{e} (I, s_{f A}) = P (1) P (0 | 1) + P (0) P (1 | 0) \\ = P (1) Q [I \sqrt{2 γ} (\frac{1}{2} - s_{f A})] + P (0) Q [I \sqrt{2 γ} (\frac{1}{2} + s_{f A})], \end{array}

where P(1) = P(0) = 0.5 are the priori probabilities, and Q(∙) represents the Gaussian-Q function. Taking the probability distribution function (PDF) of I and s_fA into account, we can get the unconditioned BER as

P_{e} = \int_{0}^{\infty} \int_{- \infty}^{\infty} P_{e} (I, s_{f A}) f_{I} (I) p_{s, f A} (s_{f A}) d s_{f A} d I,

where p_s,fA(s_fA) denotes the PDF of s_fA and f_I(I) is the PDF of I. Under the condition of

B_{f} < < B_{s}

, s_fA is proved to approximately follow the Gaussian distribution with zero mean and variance of

σ_{s, f A}^{2}

in the Appendix. As to the PDF of I, Gamma-Gamma distribution is considered to fit the turbulence from weak turbulence to strong turbulence, and can be expressed as [20]

\begin{matrix} f_{I} (I) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} I^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β I}), & I > 0 \end{matrix},

where Γ(∙) is Gamma function, and K_v(∙) is the vth-order modified Bessel function of the second kind, α and β are the parameters determined by the Rytov variance. The Rytov variance is a measure of turbulence strength and defined as

σ_{R}^{2} = 1.23 C_{n}^{2} κ^{7 / 6} L^{11 / 6}

[20], where

C_{n}^{2}

is the structure constant of atmosphere, κ is the wave number and L is the transmission distance.

In order to manifest the scheme performance, BER of decision with perfect CSI is used as the comparison object, whose expression is [21]

P_{e}^{c s i} = \int_{0}^{\infty} f_{I} (I) Q (\sqrt{\frac{γ}{2}} I) d I .

In fact, P_e in Eq. (10) is convergent to $P_{e}^{c s i}$ in Eq. (12) with $σ_{s, f A}^{2} \to 0$ . Under the influence of s_fA, P_e is certainly greater than $P_{e}^{c s i}$ and the difference reflects the performance loss of ATD. The above analysis has already implied that the performance loss is related to s_fA. As s_fA is a zero-mean Gaussian random process and its variance $σ_{s, f A}^{2}$ rests with B_s/B_f, the performance loss depends on B_s/B_f. Figure 2 shows BER comparison of ATD and decision with perfect CSI under several typical SNR and turbulence conditions, where the BER of ATD is obtained from Eq. (10). In each case, when B_s/B_f is relatively small, the BER difference is significant. With the increase of B_s/B_f, BER of ATD gradually gets close to that of decision with perfect CSI. When B_s/B_f exceeds a critical value, the BER difference can hardly be distinguished. From Fig. 2, this critical value varies with SNR and turbulence condition. When SNR is lower or turbulence gets stronger, the critical value shows smaller, and the reverse is also true.

Fig. 2 Theoretical BER comparison of ATD and decision with perfect CSI. The abbreviations “ATD” and “CSI” denote the case of ATD and decision with perfect CSI, respectively.

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3. Monte Carlo Simulation

In this section, we first demonstrate simulation to verify the model of s_fA. Then complete system simulation is presented to show the scheme performance.

3.1 Model validation

From the above analysis, s_fA performs a crucial role in the proposed scheme. In order to validate the analytical model, it is essential to demonstrate the statistical property of s_fA by using Monte Carlo simulation. With the help of MATLAB, random square-wave pulse sequence is generated with magnitude of either 0 or 1, and then filtered through digital low-pass Butterworth filter. The square-wave pulse sequence is realized through quadruple oversampling operation on random bit sequence. Then the statistical property of s_fA can be analyzed from the filtering output, where the simulated variance and distribution are mainly concerned. Figure 3(a) shows comparison between the simulated and theoretical variances of s_fA, where the latter is calculated by using Eq. (4). As anticipated, the simulated variance takes very small value with $B_{s} / B_{f} > > 1$ and decreases with the increase of B_s/B_f. In a wide range of B_s/B_f, the simulated variance accords very well with the theoretical variance. Nevertheless, when B_s/B_f>10⁵, the simulated variance becomes smaller compared with theoretical variance. The reason is that, the design of digital filter becomes inaccurate when B_s/B_f tends to be very large.

Fig. 3 Simulation results of s_fA. (a) Comparison of simulated and theoretical variances of s_fA. (b) Correlation coefficient of simulated and theoretical distributions of s_fA.

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As to the distribution of s_fA, correlation coefficient between the simulated and theoretical distributions determines the validity of the Gaussian model of s_fA. The correlation coefficient between two arbitrary zero-mean random processed u and v can be formulated as $ρ = {\sum_{k} u_{k} v_{k} / (\sum_{k} u_{k}^{2} \sum_{k} v_{k}^{2})}^{1 / 2},$ which will also be used in the following discussions. Figure 3(b) shows correlation coefficients between the simulated and theoretical distributions varying as B_s/B_f. When B_s/B_f<10, s_fA does not meet the condition of following Gaussian distribution as expected, and the simulated distribution is shown in the left inset in Fig. 3(b). In the range of 10<B_s/B_f<10⁵, the correlation coefficient is higher than 0.98, which means the distribution of s_fA fits very well with Gaussian distribution. When B_s/B_f>10⁵, due to the inaccurate filter design as aforementioned, s_fA no longer follows Gaussian distribution. In fact, in the range of B_s/B_f>10⁵, ATD still shows good performance as the simulated variance of s_fA is very small, but the BER expressed in Eq. (10) can hardly be figured out due to the lose efficacy of the Gaussian model of s_fA. Therefore, application range of the analytical model is 10<B_s/B_f<10⁵ in our simulations.

3.2 System simulation

In the system simulation, the data rate is set as B_s = 1 Mbps which is subjected to the computing power, and the digital frequencies are B_c = 1.2 × 10⁻⁴π and B_f = 2 × 10⁻⁴π, which correspond to 60 Hz and 100 Hz analog frequency respectively. Thus B_s/B_f = 10⁴. The reason that B_f is set slightly larger than B_c is taking the non-ideal frequency spectral characteristic of intensity fluctuation and LPF.

As previously discussed, ATD is based on the premise that CSI can be approximately extracted through low-pass filtering operation. Thus accuracy of the extracted CSI is most concerned. In the simulation, the real intensity fluctuation (RIF) is generated with software. From Eq. (5), the extracted intensity fluctuation (EIF) is $\hat{I} = 2 φ$ . Figure 4 shows the waveform comparison of EIF and RIF under several typical SNR and turbulence conditions. Waveforms of EIF and RIF nearly overlap with each other in each case. Figure 5 demonstrates the correlation coefficients between EIF and RIF. From weak turbulence ( $σ_{R}^{2} = 0.05$ ) to strong turbulence ( $σ_{R}^{2} = 0.8$ ), with relatively low SNR (γ = 3dB) or high SNR (γ = 23dB), the correlation coefficients are all greater than 0.995. Therefore, the difference between EIF and RIF is very small, which verifies the above analysis.

Fig. 4 Waveform comparison of EIF and RIF in simulation.

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Fig. 5 Correlation coefficients between EIF and RIF in simulation.

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Figure 6 demonstrates the BER comparison of ATD and decision with perfect CSI in Monte Carlo simulation, where the theoretical results obtain from Eq. (10) and Eq. (12) are also presented as the lower bounds. In all instances, the performance of ATD is very close to that of decision with perfect CSI. For $σ_{R}^{2} = 0.05$ , the SNR loss of ATD compared to decision with CSI is 0.0221dB when SNR is 21dB (marked position A), while the SNR loss is 0.0071dB when SNR is 11dB (marked position C). For another side, given the same SNR as γ = 21dB, the SNR loss is 0.0221dB when $σ_{R}^{2} = 0.05$ (marked position A) while the SNR loss is 0.0076dB when $σ_{R}^{2} = 0.4$ (marked position B). Hence, ATD can achieve high performance in a wide range of SNR and turbulence strength. Furthermore, ATD performs better when the signal quality is worse, which confirms the conclusion of Fig. 2.

Fig. 6 BER comparison of ATD and decision with perfect CSI in Monte Carlo simulation. The abbreviations “ATD” and “CSI” denote the case of ATD and decision with perfect CSI, respectively. “sim.” and “the.” denote the case of simulation results and theoretical results, respectively.

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4. Experimental study

In order to verify the scheme performance in practical OWC system, an indoor experiment is implemented. Figure 7 shows the block diagram and photograph of the experimental setup. At the transmitter, a 1550nm high power fiber laser (NP Photonics Scorpio Laser Sources) is used to generate the continuous wave beam. The carrier laser is fed into a Mach-Zehnder modulator (MZM) and modulated by data sequence, which is generated by a high-speed signal quality analyzer (Anritsu MP1800A) with 2.5 Gbps data rate. Before fed into MZM, the data sequence is amplified to a certain electrical level (6 V) with a microwave amplifier (MWA), to match the input voltage of the MZM. Then the modulated light is amplified by an erbium doped fiber amplifier (EDFA) before sent into the OWC link through a collimator.

Fig. 7 Experimental setup. (a) Block diagram of the experimental setup. (b) Photograph of the experimental setup. The DC (direct current) electrical source is used to provide the DC bias for the MZM. MWA, microwave amplifier; MZM, Mach-Zehnder modulator; EDFA, erbium doped fiber amplifier; VOA, variable optical attenuator; DSP, digital signal processing.

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In OWC link, the phase screen used in [22, 23] is adopted to simulate the turbulence channel. The phase screen is a glass plate with elaborated refractive index distribution and fixed on a rotating stage. By changing position of the phase screen in the OWC link, turbulence strength can be adjusted continuously and flexibly. The simulated turbulence shows good low-pass characteristic with cut-off frequency less than 1 kHz. Figures 8(a)-8(d) show measured distributions of EIF in experiment fitted with Gamma-Gamma distribution and the Rytov variance takes 0.0427, 0.0977, 0.2030 and 0.4073, respectively. Correlation coefficients between the measured distributions and the corresponding Gamma-Gamma distributions are 0.9846, 0.9951, 0.9931 and 0.9945, which means the phase screen can simulate the turbulence-induced intensity fluctuation quite well. Besides, a variable optical attenuator (VOA) is placed in the OWC link to adjust SNR flexibly.

Fig. 8 Experimental data of intensity fluctuation. (a)-(d) Measured distributions of EIF in experiment fitted with Gamma-Gamma distributions. The green histograms show the measured distributions, and the red curves are the fitting Gamma-Gamma distributions. (e) Extracted Rytov variances from received signals in experiment. The marked curves show the extracted Rytov variances, and the dash horizontal lines with the same color present the corresponding real Rytov variance.

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At the receiver, received light beam is collected and coupled into the fiber link by another collimator, and then detected by a high-speed photo-detector (Thorlabs PDA8GS) with 8 GHz bandwidth. The detector output is converted into digital signal and stored through a high-speed oscilloscope (Agilent DSA-X91604A). Finally, off-line digital signal processing (DSP) program executes the filtering and decision operations.

In off-line DSP, we utilize a low-pass Butterworth filter with 1.2 × 10⁻⁴π digital frequency (corresponding to 150 kHz analog frequency for the data of 2.5 Gbps) to extract the intensity fluctuation and thus B_s/B_f = 1.67 × 10⁴. As the real-time intensity fluctuation can hardly be obtained, Rytov variance is selected to manifest the accuracy of EIF in experiment. Figure 8(e) shows extracted Rytov variances in experiment. From this figure, the extracted Rytov variances are in good agreement with real Rytov variances with minor errors when SNR is relatively high. Nevertheless, when SNR decreases, the extracted Rytov variances tend to get larger than the real Rytov variances, which is caused by non-ideal characteristic of the photo-detector.

Figure 9 shows the BER performance of ATD in experiment. The theoretical BERs are obtained from Eq. (10) by using numerical calculations. This figure shows the experimental results can match the theoretical performance relatively well with inevitable experimental error. The experimental error is mainly caused by three factors. The first is the non-ideal characteristic of the photo-detector as aforementioned. The second is minor deviation between the phase screen simulated channel and theoretical Gamma-Gamma channel (shown in Figs. 8(a)-8(d)). The last factor is the non-ergodic of the intensity fluctuation in the limited volume experimental data.

Fig. 9 Experimental BER performance of ATD. Abbreviations follow the comma “The.” and “Exp.” denote the theoretical BER and the experimental BER, respectively.

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5. Summary

In this paper, an adaptive threshold decision scheme without knowledge of CSI in OWC systems using OOK modulation was investigated theoretically and experimentally. Based on the low-pass characteristic of atmospheric turbulence, the low-pass filtering is operated on the received signal to extract the approximate instantaneous intensity fluctuation. Detailed theoretical analysis and Monte Carlo simulation showed that data rate and filter bandwidth are the two factors which will influence the scheme performance. With the designed filter, the proposed scheme can achieve performance very close to that with perfect CSI, especially for high-speed transmissions. An elaborate designed experiment with data rate of 2.5 Gbps was implemented, and the results showed the excellent performance of the proposed scheme without CSI.

Appendix

In this appendix, we will prove that s_fA approximately follows the Gaussian distribution when B_s/B_f tends to be very large. The time-domain response of an ideal rectangular low-pass filter with cut-off frequency of B_f is $h (t) = 2 B_{f} \sin c (2 B_{f} t)$ [19]. When the square wave random pulse s(t) passes through the filter, the output is the convolution of s(t) and h(t)

\begin{matrix} s_{f} (t) = \int_{- \infty}^{\infty} h (τ) s (t - τ) d τ, \\ = 2 B_{f} \int_{- \infty}^{\infty} \sin c (2 B_{f} τ) \sum_{k = - \infty}^{\infty} d [k] g (t - τ - k T) d τ, \\ = 2 B_{f} \sum_{k = - \infty}^{\infty} d [k] \int_{- \infty}^{\infty} \sin c (2 B_{f} τ) g (t - τ - k T) d τ . \end{matrix}

The square wave g(t) values unit in range of $t \in [- T / 2, T / 2]$ , and values zero otherwise. Hence, the above integral can be reduced and Eq. (A1) is converted to

s_{f} (t) = 2 B_{f} \sum_{k = - \infty}^{\infty} d [k] \int_{t - k T - \frac{T}{2}}^{t - k T + \frac{T}{2}} \sin c (2 B_{f} τ) d τ .

As $T = 1 / B_{s} < < 1 / B_{f}$ , the value of $sinc (2 B_{f} τ)$ almost holds the same in the range of $τ \in [t - k T - T / 2, t - k T + T / 2]$ . Therefore the integral in Eq. (A2) can be approximately as the product of $sinc [2 B_{f} (t - k T)]$ and the integral interval T, and Eq. (A2) becomes

s_{f} (t) = 2 B_{f} T \sum_{k = - \infty}^{\infty} d [k] \sin c [2 B_{f} (t - k T)] .

As $d [k] \in {0, 1}$ and P(0) = P(1) = 0.5, let $ξ [k] = d [k] \sin c [2 B_{f} (t - k T)]$ , then $ξ [k]$ is a random variable follows the binomial distribution, and values in ${0, \sin c [2 B_{f} (t - k T)]}$ with equal probability. The mean value and variance of $ξ [k]$ are $μ_{ξ k} = \sin c [2 B_{f} (t - k T)] / 2$ and $σ_{ξ k}^{2} = \sin c^{2} [2 B_{f} (t - k T)] / 4$ . Furthermore, if we suppose $ψ = \sum_{k = - \infty}^{\infty} ξ [k]$ , then ψ is the sum of independent non-identically distributed random variables of infinite number. Define $M = \sum_{k = - \infty}^{\infty} μ_{ξ k}$ and $D^{2} = \sum_{k = - \infty}^{\infty} σ_{ξ k}^{2}$ , then according to the property of Sinc function [24], there are

M = \frac{1}{2} \sum_{k = - \infty}^{\infty} \sin c [2 B_{f} (t - k T)] = \frac{1}{4 B_{f} T},

D^{2} = \frac{1}{4} \sum_{k = - \infty}^{\infty} \sin c^{2} [2 B_{f} (t - k T)] = \frac{1}{8 B_{f} T} .

When $B_{s} / B_{f} \to \infty$ , there are $M \to \infty$ and $D^{2} \to \infty$ . Meanwhile, for each k, there is $| ξ_{k} | \leq 1$ and $σ_{ξ k}^{2} < 1$ . Therefore, the Lindeberg condition is satisfied, which is the condition of sum of independent non-identically distributed random variables following the Gaussian distribution [25], and can be expressed as

\frac{1}{D^{2}} \sum_{k = - \infty}^{\infty} \int_{| x - μ_{ξ k} | > ε D} {(x - μ_{ξ k})}^{2} f_{ξ k} (x) d x = 0,

where ε is an arbitrary positive number,

f_{ξ k}

is the PDF of

ξ_{k}

which is mentioned as binomial distribution above. In the above analysis, the value range of k is [-∞, ∞]. Hence, the limit problem of k in the Lindeberg condition is unnecessary and naturally omitted. Consequently, ψ follows the Gaussian distribution with mean value of M and variance of D² when

B_{s} / B_{f} \to \infty

. As

s_{f} (t) = 2 B_{f} T ψ

, then s_f(t) will also follows the Gaussian distribution, and the mean value and variance are

μ_{s f} = 2 B_{f} T M = \frac{1}{2}

σ_{s f}^{2} = {(2 B_{f} T)}^{2} D^{2} = \frac{B_{f}}{2 B_{s}}

As s_fA(t) = s_f(t)-1/2, then s_fA follows Gaussian distribution with zero mean and variance of B_f/2B_s with $B_{s} / B_{f} \to \infty$ . The resulted variance of s_fA is consistent with the result in Eq. (4) which is obtained from integral over the PSD. Nevertheless, Bs/B_f is sufficiently large but not infinite in practice. Hence, s_fA approximately follows Gaussian distribution.

Funding

National Key Basic Research Program of China (2013CB329205); National Natural Science Foundation of China (NSFC) (60572002).

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Adaptive threshold decision for on-off keying transmission systems in atmospheric turbulence

Abstract

1. Introduction

2. Scheme of adaptive threshold decision

2.1 Scheme description

2.2 BER performance

3. Monte Carlo Simulation

3.1 Model validation

3.2 System simulation

4. Experimental study

5. Summary

Appendix

Funding

References and links

Cited By

Figures (9)

Equations (20)

Optics Express