A novel beat-noise-reducing en/decoding technology for a coherent 2-D OCDMA system

Jilin Zheng; Rong Wang; Tao Pu; Lin Lu; Tao Fang; Yun Cheng; Xiangfei Chen

doi:10.1364/OE.17.019264

1. Introduction

Coherent optical code-division multiple-access (OCDMA) systems are expected to provide excellent performances superior to those of incoherent schemes in terms of the correlation property, code cardinality and so on, since the coding operations are performed in a bipolar (−1, + 1) manner in the optical domain [1–12]. However, due to the square law photodetector (PD) characteristic, beat noise (BN) that arises from the mixing of the desired data signal and the unwanted interferer signals will severely deteriorate coherent OCDMA systems performances when the two signals are in the same wavelength [3,13–18]. Several solutions have been proposed to mitigate the effect of BN through improving the code design, receiver design or the format of signal. For code design, more wavelengths or longer code length are required [3,15], which may increase the hardware cost, decrease wavelength efficiency or limit the transmission data rate. For improving the receiver, extra devices such as optical thresholding [19–23], time gating [24] and optical hard limiter(OHL) [25] etc. may be necessary, which will increase the complexity of the receiver. For the approach to improve the format of the signal, pulse position modulation (PPM) [25] or differential phase-shift keying (DPSK) combined with balanced detection [26] are usually used, making the system more complex. Among the coherent OCDMA systems, coherent two-dimensional (2-D) wavelength-time (WT) OCDMA system is attractive because of the increased capacity, enhanced security and improved flexibility [12,15,18,27–29]. However, such a system nonexceptionally suffers from the degradation caused by BN [13–15,18]. How to solve such a problem is a difficult issue.

In this paper, to the best of our knowledge, a novel inherent beat-noise-reducing FBG based en/decoder for coherent 2-D OCDMA system, namely chirp-coded en/decoder is proposed and investigated for the first time. Profiting from the introduction of chirp in spectral chip, BN can be mostly suppressed. Furthermore, combined with the reconstruction-equivalent-chirp (REC) technology [30], such an en/decoder comprising several segments of various chirped sub-gratings can be flexibly designed and fabricated using a single uniform phase mask with submicron precision control. The present paper is organized as follows: the principle of chirp-en/decoding is introduced in section 2, and the en/decoder’s characteristics are studied both in the simulation and experiment. The system performance is numerically analyzed in section 3 and the conclusions are drawn in section 4.

2. En/decoding principle and demonstrations

2.1 Principle

For comparison, Fig. 1 shows a detailed schematic of the conventional 2-D encoder and the proposed FBG-based chirp-coded 2-D encoder, both of which include four wavelength chips and four time chips for simplicity. For the traditional 2-D encoder, as shown in Fig. 1(a), each wavelength chip contains only a wavelength “point”. However, in the proposed chirp-coded 2-D scheme, as shown in Fig. 1(b), the single wavelength “point” is chirped and becomes a wavelength “band” $Δ λ$ . In other words, it’s a chirped chip-pulse centered at the original wavelength “point” that fills in each time chip, which will determine the inherent capability of BN suppression illustrated in the following analysis. To further improve the performance, the distribution of the chips’ chirp polarity (positive or negative chirp), follows any bipolar code sequence that has good correlation properties such as m-sequence, Gold sequence and so on. In such a chirp-coded 2-D code, there is an additional orthogonality related to the distribution of the chirp polarity, for example, a pulse reflected by a positive chirped sub-grating can be restored by an inversely chirped sub-grating but further dispersed by the same positive chirped sub-grating. Such an orthogonality helps to guarantee the same good correlation properties as those in conventional 2-D code system. It should be noted that: 1) The optical source should be coherent and non-chirped; 2)the wavelength band $Δ λ$ in each chip should not be wider than the interval between the two adjacent wavelengths in order to avoid inter-chip overlapping in wavelength domain.

Fig. 1 Illustration of 2-D en/decoder: (a) Conventional structure; (b) Proposed chirp-coded structure.

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In terms of the analysis shown in [3], although in coherent regime the phase noise of the interfering chip is a small constant within the integral duration, it varies from bit to bit, which results in the arising of BN. In incoherent regime, the distribution of the phase noise is random over [-π,π] during the integral duration, so the BN can be ignored when averaging phase nosie signal. Because the chirp-coded chip can disperse the pulse over a relative wide wavelength band, the en/decoding process in chirp-coded system is similar with that in incoherent regime in conventional 2-D OCDMA system. The reason is also illustrated in Fig. 2 . For conventional 2-D OCDMA systems, as shown in the upper part of Fig. 2, when data chips forming the peak of auto-correlation mix with different interfering chips at the same wavelength but with various original phase, the output photocurrent after PD fluctuates wildly, it means that BN has arisen due to the square law of PD. However, for chirp-coded 2-D OCDMA systems, as shown in the lower part of Fig. 2, there is only a tiny fluctuation in the photocurrent. Such an improvement of BN suppression results from the existence of chirp in the interfering chip. Because the frequency of the interferer chip is varying throughout the chip time, the phase difference between the two chips is no longer a small constant but a random variable during the integral duration. That is to say, similar to incoherent regime, the phase matching condition required for beating between the two chips has been destroyed. It should be noted that, for the correctly decoded data signals, the total chirp is zero and the auto-correlation peak can be formed.

Fig. 2 Illustration of beating between the data chip and the interferer chip (with different phase noise) for the conventional (upper part) and the novel (bottom part) scheme.

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In chirp-coded 2-D OCDMA systems, various chirped sub-gratings with different chirp are required and various chirped phase masks are needed in fabrication, which will lead to increased complexity and cost Here, REC technology [30] is used to avoid such a problem and only a single uniform phase mask can do well in the coder design and fabrication. That is to say, equivalent chirp is used to replace the conventional true chirp in grating pitch. For example, the four different colors in the bottom part of Fig. 1(b) represent four segments of a sampling-chirp sampled grating with four different center sampling periods.

2.2 Simulation demonstration

In this part, both the conventional and the chirp-coded 2-D OCDMA en/decoding performance is simulated for comparison. Based on a frequency hopping m-sequence with four wavelengths and 15 time chips [12], two types of en/decoders are designed both using REC technology. The codes listed in Table 1 in [12] are used for conventional scheme, and the codes used for chirp-coded scheme are shown in Table 1 . The center wavelengths of the four chips are chosen to be 1551.85, 1552.4, 1552.95 and 1553.5nm, respectively. The total length of each coder is 110mm. Figure 3 shows the wavelength spectra of the two kinds of encoders. Gauss pulse with a FWHM (full width at half maximum) of 2ps is used as an input optical pulse. The encoded waveforms of desired users and the frequency distribution in each chip are shown in Fig. 4 . In contrast with the conventional 2-D scheme, the novel method exhibits “+” or “-” variations of frequency chirp which reflect the address code shown in the bottom part. The basic decoded performance is examined at a data rate of 5Gb/s as shown in Fig. 5 . Since the length of encoder is 110mm, there will be 11 inter-symbol interference (ISI) bits for each decoded signal, which may result in a large number of inter-symbol BN [19]. To further examine the performance of the proposed method, interferer users are added to investigate the cross-correlation property simultaneously. For the conventional 2-D OCDMA system using a 30-GHz PIN receiver, there are a lot of sub-peaks in the autocorrelation, which arise from spectral interference [3,12,24]. Although such mutual spectral interferences could be eliminated after a 10-GHz PIN receiver since the wavelength spacing between neighboring channels is about 0.55 nm (about 70 GHz), the received electrical eye diagram is half closed due to the ISI and inter-symbol BN. However, for the chirp-coded 2-D OCDMA system, there are few sub-peaks in the auto-correlation, the eye diagram after a 30-GHz or 10-GHz PIN receiver is quite open. It indicates that the BN arising from mutual spectral interference and inter-symbol overlapping has been mostly suppressed. It should be noted that since the ISI noise has the same effect as the multiple-access interference (MAI) [19,31], the BN arising from MAI can also be suppressed. This conclusion can also be confirmed by adding some interferer users. We can see that the eye diagram of conventional scheme is fully closed whereas the chirp-coded scheme is quite open when 3 interferer users are active. The residual pedestal and upper eyelid in the eye diagram arise mainly from ISI, partly from cross-correlation and the remnants of BN. These simulation results indicate that the proposed method is effective in the suppression of BN.

Table 1. The codes used for chirp-coded scheme. The order of numbers follows its corresponding time chips, numbers 1~4 represent four wavelength chips, and the minus punctuation “-” represents inverse chirp polarity.

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Fig. 3 Wavelength spectra of the two kinds of encoders: the conventional (left) and the novel (right)

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Fig. 4 Encoded waveforms and frequency distribution corresponding to code word for the conventional (left) and the novel (right) scheme.

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Fig. 5 Eye diagrams after 30-GHz PIN and 10-GHz PIN with different interferer users for the conventional (upper part) and the novel (lower part) scheme.

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2.3 Experiment demonstration

In our experiments, chirp-coded en/decoders with the same parameters as those in the above simulation are fabricated. The coders are fabricated by inscribing gratings on hydrogen-loaded fibers using a ultraviolet 244-nm laser and a uniform phase mask with grating pitch of 1070 nm. The measured wavelength spectrum of one encoder is shown in the left part of Fig. 6 . A preliminary system test-bed is set up in order to examine the performance of the novel en/decoders, as shown in the right part of Fig. 6. The mode-locked laser diode(MLLD) generates ~2-ps optical pulses at a repetition rate of 10 GHz with a central wavelength of 1552.5 nm. This optical pulse train is converted to a 2⁷-1 pseudorandom bit sequence (PRBS) by a LiNbO3 intensity modulator (LN-IM). The data rate is set to 2.5, 5, 10Gb/s respectively by manually setting the PRBS data patterns to investigate the impact of ISI and inter-symbol BN with different inter-symbol overlapping. An unmatched encoder is added to compare the decoded waveforms between with and without cross-correlation interference. The optical bandwidth of optical sampling oscilloscope(OSO) is 30 GHz.

Fig. 6 Measured wavelength spectrum of encoder (left) and experiment setup (right)

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The measured eye diagrams of decoded signals for one user and two users with different data rate are shown in Fig. 7 . For comparison purpose, the simulated eye diagrams with the same parameters are shown in the lower part. It can be found that these measured eye diagrams agree well with the simulation results which have few sub-peaks profiting from the suppression of mutual spectral interference BN. It can be seen that the adding of one interference user doesn’t obviously deteriorate the quality of eye diagrams. The opening of eye diagram becomes narrower with more inter-symbol overlapping, which is mainly caused by ISI. However, when system operates at even 10Gb/s, i.e., there are up to 23 ISI bits for each decoded signal, the eye diagrams are still open due to the inter-symbol BN suppression. It should be noted that the en/decoders used in simulation and experiment are purposely designed to be 11cm long in order to magnify the effect of ISI and inter-symbol BN. However, we can expect better decoding performance if shorter en/decoders are used.

Fig. 7 Measured (upper) and simulated (lower) eye diagrams of decoded signals for one user and two users with different data rate

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3. Numerical analysis of the impact of BN

3.1 Performance evaluation

For simplicity, two assumptions are applied. First, chip synchronous case is assumed, which will result in an upper bound with respect to the asynchronous system performance [15]. Second, since the ISI has the same effect as the MAI, inter-symbol overlapping is not included and only MAI and the corresponding BN are considered. Consider “K” optical waveforms representing the data and interfering signals. The received optical field at the photodetector of the target user is

E (t) = E_{d} (t) + E_{i} (t)

Then

E (t) \propto \vec{P_{d}} \sum_{l = 1}^{w} \sqrt{P_{d} (t)} \exp {j (ω_{d, l} t + ϕ_{d, l} (t))} + \vec{P_{i}} \sum_{l = 1}^{w} \sum_{i = 1}^{K - 1} \sqrt{P_{i} (t)} \exp {j (ω_{c, l, i} (t - τ_{l, i}) + ϕ_{c, l, i} (t - τ_{l, i}))}

where

{\vec{P}}_{d}

and

{\vec{P}}_{i}

are the unit polarization vectors for the data and interferers, and

P_{d} (t)

and

P_{i} (t)

are the instantaneous optical chip powers for data and interferers at the photodetector, respectively. w represents the code weight.

ω_{d, l}

is the frequency of the intended data chip corresponding to wavelength

λ_{l}

.

ω_{c, l, i}

is the frequency of the interferer signal corresponding to a chip of wavelength

λ_{l}

originating from the ith interferer.

φ_{d, l} (t)

and

φ_{c, l, i} (t)

represent the phase noise for the data and ith interferer signals corresponding to wavelength

λ_{l}

, which are uniformly distributed over [-π,π].

τ_{l, i}

is the propagation transit delay of the ith interferer relative to the data pulse at

λ_{l}

. We assume that the data and the interferer are of the same polarization, i.e.,

{\vec{P}}_{d} • {\vec{P}}_{i} = 1

, which results in the worst case scenario.

For a receiver employing chip-rate square-law photodetector, the output signal Z from the integrator is

\begin{array}{l} Z = \int_{0}^{T_{c}} ℜ • (E • E^{*}) d t + \int_{0}^{T_{c}} n_{0} (t) d t \\ = \underset{\begin{array}{c} Data \end{array}}{\underline{T_{c} ℜ • w • P_{d}}} + \underset{\overset{}{Interference}}{\underline{T_{c} ℜ • \sum_{i = 1}^{K - 1} P_{i}}} \\ + \underset{\overset{}{Data-interference beat terms}}{\underline{2 ℜ \sqrt{P_{d}} \sum_{l = 1}^{w} \sum_{i = 1}^{K - 1} \sqrt{P_{i}} \int_{0}^{T_{c}} \cos (ϕ_{c, l, i} (t - τ_{l, i}) - ω_{d, l} τ_{l, i} - ϕ_{d, l} + Δ ω_{i} (t) (t - τ_{l, i})) d t}} \\ + \underset{}{\underline{2 ℜ \sum_{l = 1}^{w} \sum_{i = 1}^{K - 1} \sum_{g = i + 1}^{K - 2} \sqrt{P_{i}} \sqrt{P_{g}} \int_{0}^{T_{c}} \cos (ϕ_{c, l, i} (t - τ_{l, i}) - ω_{d, l} τ_{l, i} + ω_{d, l} τ_{l, g} - ϕ_{c, l, g} (t - τ_{l, g})}} \\ \underset{\begin{array}{c} Interference-interference beat terms \end{array}}{\underline{+ (Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}) - Δ ω_{g} (t) τ_{i, g}) d t}} + \underset{\overset{}{Receiver noise}}{\underline{\int_{0}^{T_{c}} n_{0} (t) d t}} \end{array}

where ℜ is the responsivity of the photodetector,

T_{c}

is the chip duration, and

n_{0}

denotes the receiver noise current. The data chip forming the auto-correlation peak is chirp-free, while the frequency difference between the data chip and the ith interferer chip is

Δ ω_{i} (t)

that can be described by

Δ ω_{i} (t) = {\begin{array}{c} 0 Probability = \frac{1}{2} \\ - \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t Probability = \frac{1}{4} \\ \frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t Probability = \frac{1}{4} \end{array}

where

Δ ω

is one half of the bandwidth

Δ λ

in each wavelength chip, and the probability values are given based on the fact that the probability of appearance of “-1” and “1” in any bipolar code is approximately the same. Different from conventional 2-D scheme,

Δ ω_{i} (t)

here is a nonconstant variable during the integral duration, which will help to reduce the data-interferer beat noise. The frequency difference between the ith interferer chip and the gth interferer chip is

Δ ω_{i} (t) - Δ ω_{g} (t)

, which is described by

Δ ω_{i} (t) - Δ ω_{g} (t) = {\begin{array}{c} \begin{array}{c} 0 Probability = \frac{3}{8} \\ - \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t Probability = \frac{1}{4} \end{array} \\ \frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t Probability = \frac{1}{4} \\ 2 • (- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) Probability = \frac{1}{16} \\ 2 • (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t) Probability = \frac{1}{16} \end{array}

From expression (5), we can find that similar to expression (4), different from conventional 2-D scheme,

Δ ω_{i} (t) - Δ ω_{g} (t)

here is a nonconstant variable during the integral duration. This characteristic will help to reduce the interferer-interferer beat noise.

The instantaneous optical chip power for the ith interferer is described by

P_{i} = {\begin{array}{c} ξ P_{d} if Δ ω_{i} (t) = 0 \\ \frac{ξ P_{d}}{2} if Δ ω_{i} (t) = (- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t) \end{array}

where ξ is the crosstalk level parameter, which is defined as the ratio of the average optical intensity of the decoded signal of interferers’ chips to the desired data per chip [15,25]. Note that the expression (6) indicates the decline of interference intensity from the viewpoint of statistics, since the interferer chip power will decrease to half of its original value when this chip is double-dispersed.

For simplicity, we introduce two parameters

Φ_{1} = ϕ_{c, l, i} (t - τ_{l, i}) - ω_{d, l} τ_{l, i} - ϕ_{d, l}

Φ_{2} = ϕ_{c, l, i} (t - τ_{l, i}) - ω_{d, l} τ_{l, i} + ω_{d, l} τ_{l, g} - ϕ_{c, l, g} (t - τ_{l, g})

Usually

Φ_{1}

and

Φ_{2}

are small constants within the integral duration [3], then the third and fourth terms of expression (3) related to the beat noise can be expressed as

\begin{array}{l} Z_{B N} = 2 ℜ \sqrt{P_{d}} \sum_{l = 1}^{w} \sum_{i = 1}^{K - 1} \sqrt{P_{i}} [\cos (Φ_{1}) \int_{0}^{T_{c}} \cos (Δ ω_{i} (t) (t - τ_{l, i})) d t - \sin (Φ_{1}) \int_{0}^{T_{c}} \sin (Δ ω_{i} (t) (t - τ_{l, i})) d t] \\ + 2 ℜ \sum_{l = 1}^{w} \sum_{i = 1}^{K - 1} \sum_{g = i + 1}^{K - 2} \sqrt{P_{i}} \sqrt{P_{g}} [\cos (Φ_{2}) \int_{0}^{T_{c}} \cos ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}) - Δ ω_{g} (t) τ_{i, g}) d t \\ - \sin (Φ_{2}) \int_{0}^{T_{c}} \sin ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}) - Δ ω_{g} (t) τ_{i, g}) d t] \end{array}

The last term can be further expressed as

\begin{array}{l} 2 ℜ \sum_{l = 1}^{w} \sum_{i = 1}^{K - 1} \sum_{g = i + 1}^{K - 2} \sqrt{P_{i}} \sqrt{P_{g}} [\cos (Φ_{2}) (\int_{0}^{T_{}} \cos ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}))) d t • \int_{0}^{T_{c}} \cos (Δ ω_{g} (t) τ_{i, g}) d t \\ - \int_{0}^{T_{c}} \sin ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}))) d t • \int_{0}^{T_{c}} \sin (Δ ω_{g} (t) τ_{i, g}) d t) \\ - \sin (Φ_{2}) (\int_{0}^{T_{}} \sin ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}))) d t • \int_{0}^{T_{c}} \cos (Δ ω_{g} (t) τ_{i, g}) d t \\ - \int_{0}^{T_{}} \cos ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i}))) d t • \int_{0}^{T_{c}} \sin (Δ ω_{g} (t) τ_{i, g}) d t)] \end{array}

All the integrals in expression (7) and (8) are random variables relating to τ (i.e., τ_l,i, τ_i,g or τ_l,g). To figure out the second-order moments of these variables are necessary for calculating the signal-to-noise ratio (SNR). Set reasonable values for the two definite parameters:

T_{c} = 20 ps

,

Δ ω = 2 π * 35 GHz

, we have

E ({(\frac{1}{T_{c}} • \int_{0}^{T_{c}} \cos (Δ ω_{i} (t) (t - τ_{l, i})) d t)}^{2}) = 1 {.1523e}^{-4}

E ({(\frac{1}{T_{c}} • \int_{0}^{T_{c}} \sin (Δ ω_{i} (t) (t - τ_{l, i})) d t)}^{2}) = 2 {.7612e}^{-5}

E ({(\frac{1}{T_{c}} • \int_{0}^{T_{c}} \cos ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i})) d t)}^{2}) = 4 {.6128e}^{-5}

E ({(\frac{1}{T_{c}} • \int_{0}^{T_{c}} \sin ((Δ ω_{i} (t) - Δ ω_{g} (t)) (t - τ_{l, i})) d t)}^{2}) = 3 {.1280e}^{-6}

E ({(\frac{1}{T_{c}} • \int_{0}^{T_{c}} \cos (Δ ω_{i} (t) τ_{i, g}) d t)}^{2}) = 0.0012

E ({(\frac{1}{T_{c}} • \int_{0}^{T_{c}} \sin (Δ ω_{i} (t) τ_{i, g}) d t)}^{2}) = 0

where

E (•)

represents the mathematical expectation operator. It should be noted that (9)~(12) may only be reckoned by the help of computing software such as MATLAB, since they have no analytical expressions.

The probability distribution of the Φ-related terms in (7) and (8) follows an arcsine, “two-pronged” density function [15,16], and it can be modeled as a Gaussian distribution following the central limit theorem if the number of interferer is equal or greater than “5” [3,15]. In this paper, the total noise source consist of thermal noise, shot noise, relative intensity noise (RIN). After the average received signal is scaled by $\frac{1}{T_{c} ℜ}$ , the SNR is then analyzed.

SNR for signal “1” is:

S N R_{1} = \frac{P_{1}}{σ_{t h}^{2} + σ_{s h_{1}}^{2} + σ_{R I N_{1}}^{2} + σ_{s i g - int}^{2} + σ_{int - int}^{2}}

where the

σ_{}^{2}

represents the variance. The power of the useful signal can be expressed as:

P_{1} = {(ω P_{d} + k' ξ P_{d} • P (P_{i} = ξ P_{d}) + k' \frac{ξ P_{d}}{2} • P (P_{i} = \frac{ξ P_{d}}{2}) - ω P_{d} D)}^{2}

where

k'

represents the total number of interferers’ pulses that overlap (“hit”) the active wavelengths in the desired codeword [15]. D is the threshold level.

P (•)

denotes the probability. The power of the noise is shown in the following:

σ_{t h}^{2} = \frac{(4 • K_{B} • T • B_{e})}{R_{L}}

σ_{s h_{1}}^{2} = 2 • q • (ω P_{d} + k' ξ P_{d} • P (P_{i} = ξ P_{d}) + k' \frac{ξ P_{d}}{2} • P (P_{i} = \frac{ξ P_{d}}{2})) • B_{e}

σ_{R I N_{1}}^{2} = R I N • {(ω P_{d} + k' ξ P_{d} • P (P_{i} = ξ P_{d}) + k' \frac{ξ P_{d}}{2} • P (P_{i} = \frac{ξ P_{d}}{2}))}^{2} • B_{e}

\begin{array}{l} σ_{s i g - int}^{2} = V a r (data - interference / Δ ω_{i} (t) = 0) • P (Δ ω_{i} (t) = 0) \\ + V a r (data - interference / Δ ω_{i} (t) = - \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) • P (Δ ω_{i} (t) = - \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \\ + V a r (data - interference / Δ ω_{i} (t) = \frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t) • P (Δ ω_{i} (t) = \frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t) \\ = \frac{1}{2} • 2 • ξ • P_{d}^{2} • k' + \frac{1}{2} • 4 • P_{d} • \frac{ξ P_{d}}{2} • k' [V a r (\frac{1}{T_{c}} \cos (Φ_{1}) \int_{0}^{T_{c}} \cos (Δ ω_{i} (t) (t - τ_{l, i})) d t) \\ + V a r (\frac{1}{T_{c}} \sin (Φ_{1}) \int_{0}^{T_{c}} \sin (Δ ω_{i} (t) (t - τ_{l, i})) d t)] \end{array}

\begin{array}{l} σ_{int - int}^{2} = V a r i a n c e_{0 - 0} • P (x = 0, y = 0) + V a r i a n c e_{single - 0} • P (x = single, y = 0) \\ + V a r i a n c e_{0-single} • P (x = 0, y = single) + V a r i a n c e_{double - single} • P (x = double, y = single) \\ + V a r i a n c e_{single - single} • P (x = single, y = single) \end{array}

where, the subscripts of Variance_x-y, i.e., x, y, represent the values of

Δ ω_{i} (t) - Δ ω_{g} (t)

and

Δ ω_{g} (t)

respectively. The “single” means

(- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)

, and the “double” represents

2 • (- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup 2 • (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)

.

P (x,y)

denotes the joint probability.

The joint probability distribution of (x,y) is given in Table 2.

Table 2. The joint probability distribution of (x,y)

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SNR for signal “0” is:

S N R_{0} = \frac{P_{0}}{σ_{t h}^{2} + σ_{s h_{0}}^{2} + σ_{R I N_{0}}^{2} + σ_{int - int}^{2}}

The power of the useful signal can be expressed as:

P_{0} = {(ω P_{d} D - (k' ξ P_{d} • P (P_{i} = ξ P_{d}) + k' \frac{ξ P_{d}}{2} • P (P_{i} = \frac{ξ P_{d}}{2})))}^{2}

The power of the noise is shown in the following:

σ_{t h}^{2} = \frac{(4 • K_{B} • T • B_{e})}{R_{L}}

σ_{s h_{0}}^{2} = 2 • q • (k' ξ P_{d} • P (P_{i} = ξ P_{d}) + k' \frac{ξ P_{d}}{2} • P (P_{i} = \frac{ξ P_{d}}{2})) • B_{e}

σ_{R I N_{0}}^{2} = R I N • {(k' ξ P_{d} • P (P_{i} = ξ P_{d}) + k' \frac{ξ P_{d}}{2} • P (P_{i} = \frac{ξ P_{d}}{2}))}^{2} • B_{e}

\begin{array}{l} σ_{int - int}^{2} = V a r i a n c e_{0 - 0} • P (x = 0, y = 0) + V a r i a n c e_{single - 0} • P (x = single, y = 0) \\ + V a r i a n c e_{0-single} • P (x = 0, y = single) + V a r i a n c e_{double - single} • P (x = double, y = single) \\ + V a r i a n c e_{single - single} • P (x = single, y = single) \end{array}

Finally, the total probability of error is then given by

P_{e} = \sum_{i = 1}^{K - 1} (\begin{array}{c} K - 1 \\ i \end{array}) • 2^{- (K - 1)} • \sum_{j = 1}^{i} (\begin{array}{c} i \\ j \end{array}) • {(h_{a v})}^{j} • {(1 - h_{a v})}^{i - j} • \frac{1}{2} • {Q (s q r t (S N R_{1}) + Q (s q r t (S N R_{0}))}

where

Q (x) = \frac{1}{2 π} \int_{x}^{\infty} \exp (\frac{- y^{2}}{2}) d y

and

h_{a v}

is the average probability of a “hit” between two users, which is listed in [15].

3.2 Results and discussions

System performance is evaluated for both symmetric (S) code and asymmetric (AS) code for generality. In this paper, prime hop (PH) code (w = 17) and asymmetric prime hop (APH) code (λ_p = 41, w = 13) are selected for S and AS types respectively. The RIN is assumed to be −150dB/Hz. The probability of error versus the number of simultaneous users for the conventional (with and without BN) and the novel (with BN) schemes is plotted in Fig. 8 . Here the useful received optical power w⋅P _d, unity crosstalk level ξ and threshold level D are chosen to be −14dBm, 1 and 1/2, respectively. It is clear that for both PH and APH, the BERs of the chirp-coded schemes are much lower than those of the conventional schemes with BN and even comparable to the conventional schemes without BN. Although this novel scheme is inferior to conventional case without BN when K<15 because of the remnants of BN, it will outperform the latter when K>15 due to the relatively weaker interferer chip power indicated by expression (6).

Fig. 8 Comparison of BER performance versus the number of simultaneous users between the conventional and novel scheme for PH (left) and APH (rigth)

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Several BERs between different chip bandwidths have been performed in order to study the effect of different amount of chirp on BN suppression. The result for K = 20 is shown in Fig. 9 . Here, the chip bandwidth is scaled by $\frac{1}{35 GHz}$ which is a typical value used above. In general, it can be seen that the BER performance is degraded with the decrease of chirp, which will tend towards the case of conventional with BN. However, it is clear that the property almost keeps well when the normalized chip bandwidth changes from 0.01 to 10 and the large scale offers us a great freedom during our design.

Fig. 9 Influence of the different amount of chirp on the BER performance for PH (left) and APH (rigth)

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The above results are calculated with a fixed threshold level D = 1/2, however, it may be not the optimal threshold value for different simultaneous users. Figure 10 illustrates the BER performance at different threshold values for K = 5~40. It is clear that the optimal threshold value increases with the adding of more interfering users. Therefore, the receiver with dynamic threshold adjustment may be of certain benefit to the improvement of such system.

Fig. 10 BER performance versus different threshold values for the number of simultaneous users K = 5~40, step = 5. PH (left) and APH (rigth)

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4. Conclusion

A novel FBG-based en/decoder with inherent capability of BN suppression for coherent 2-D OCDMA system is proposed and investigated in this paper. The comparison between the conventional and the proposed en/decoding scheme is performed by simulation, which indicates that the BN can be suppressed based on the method proposed by us. The corresponding experiments at a data rate of 2.5Gb/s, 5Gb/s and 10 Gb/s with two users are carried out to validate the simulation results, in which open eye diagrams are observed even under serious inter-symbol overlapping. The impact of BN is numerically analyzed. The results show that in BER performance, the proposed method outperforms the conventional method. Profiting from the BN suppression and weaker interferer chip power, it even rivals the conventional method without BN. The discussions about the relationship between the BN suppression and chirp reveal that there is a wide range of chirp for BN suppression without sacrificing system performance. The effect of the threshold level is also investigated, which indicates that optimal threshold value is needed to further improve the system performance. Such a chirp-coded en/decoding technology with the inherent capability of BN suppression may play an important role in improving the performance of coherent OCDMA systems.

Acknowledgments

This work is partly supported by the National Science Foundation of Jiangsu Province (BK2007501) and the National “863” Project of China (No. 2007AA01Z274). The authors would like to thank Prof. Shousheng Liu for the technical support in the numerical deduction.

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Desired user	1 2 3 −2 2 −1 −4 3 4 −4 1 −3 −2 −3
Interferer user 1	2 1 3 −1 1 −2 −3 4 3 −3 2 - 4 −1 4 - 4
Interferer user 2	3 4 2 −4 4 −3 - 2 1 2 −2 3 −1 −4 −1
Interferer user 3	4 3 1 −3 3 −4 −1 2 1 −1 4 −2 −3 −2

Desired user	1 2 3 −2 2 −1 −4 3 4 −4 1 −3 −2 −3
Interferer user 1	2 1 3 −1 1 −2 −3 4 3 −3 2 - 4 −1 4 - 4
Interferer user 2	3 4 2 −4 4 −3 - 2 1 2 −2 3 −1 −4 −1
Interferer user 3	4 3 1 −3 3 −4 −1 2 1 −1 4 −2 −3 −2

A novel beat-noise-reducing en/decoding technology for a coherent 2-D OCDMA system

Abstract

1. Introduction

2. En/decoding principle and demonstrations

2.1 Principle

2.2 Simulation demonstration

2.3 Experiment demonstration

3. Numerical analysis of the impact of BN

3.1 Performance evaluation

3.2 Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Tables (2)

Equations (31)

Optics Express

$Δ ω_{i} (t) - Δ ω_{g} (t)$	$Δ ω_{g} (t)$	Probab-ility
0	0	1/4
$(- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)$	0	1/4
0	$(- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)$	1/8
$2 • (- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup 2 • (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)$	$(- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)$	1/8
$(- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)$	$(- \frac{Δ ω}{2} + \frac{Δ ω}{T_{c}} • t) \cup (\frac{Δ ω}{2} - \frac{Δ ω}{T_{c}} • t)$	1/4