maximum likelihood detection with beat noise estimation for minimizing bit error rate in...

Maximum likelihood detection with beatnoise estimation for minimizing bit error

rate in OCDM-based system

Takahito Kirihara, Noriki Miki, Shin Kaneko, Hideaki Kimuraand Kiyomi Kumozaki

NTT Access Network Service Systems Laboratories, NTT Corporation1-6 Nakase, Mihama-ku, Chiba-shi, Chiba 261-0023 Japan

[email protected]

Abstract: We propose a maximum likelihood detection (MLD) techniquethat incorporates beat noise estimation (BNE). MLD can minimize a biterror rate theoretically because a bit pattern with the maximum posterioriprobability is selected as the detected signals. Also, BNE can extracta specific beat noise from mixed multiple signals using a correlation.By combining these techniques, the influence of beat noise is reducedand the bit error rate becomes lower in an OCDM-based system. Thispaper describes the MLD algorithm and the BNE design. And numericalsimulation results confirm the validity and performance of this technique.

© 2009 Optical Society of AmericaOCIS codes: (060.4510) Optical communications; (060.4250) Networks; (070.4550) Correla-tors.

References and links1. A. Stok and E. H. Sargent, “The Role of Optical CDMA in Access Networks,” IEEE Commun. Mag. 40, 83-87

(2002).2. D. Zaccarin and M. Kavehrad, “An Optical CDMA System Based on Spectral Encoding of LED,” IEEE Photon.

Technol. Lett. 4, 479-482 (1993).3. V. J. Hernandez, W. Cong, J. Hu, C. Yang, N. K. Fontaine, R. P. Scott, Z. Ding, B. H. Kolner, J. P. Heritage,

and S. J. B. Yoo, “A 320-Gb/s Capacity SPECTS O-CDMA Network Testbed With Enhanced Spectral EfficiencyThrough Forward Error Correction,” IEEE J. Lightwave Technol. 25, 79-86 (2007).

4. S. Huang, K. Kitayama, K. Baba, and M. Murata, “Impact of MAI Noise Cycle Attack on OCDMA-basedOptical Networks and its Diagnostic/Mitigation Algorithm,” in Proceedings of IEEE Global CommunicationsConference (GLOBECOM) (IEEE, 2007) pp. 2412-2416.

5. S. Kaneko, H. Suzuki, N. Miki, H. Kimura, and M. Tsubokawa, “1-bit/s/Hz Spectral Efficiency OCDM TechniqueBased on Multi-frequency Homodyne Detection and Optical OFDM,” European Conference and Exhibition onOptical Communication (ECOC) 5, PS-P088, 203-204 (2007).

6. E. D. J. Smith, P. T. Gough, and D. P. Taylor, “Noise limits of optical spectral-encoding CDMA systems,” Elec-tron. Lett. 31, 1469-1470 (1995).

7. L. Tancevski and L. A. Rusch, “Impact of the Beat Noise on the Performance of 2-D Optical CDMA Systems,”IEEE Commun. Lett. 4, 264-266 (2000).

8. M. Fujiwara, J. Kani, H. Suzuki, and K. Iwatsuki, “Impact of Backreflection on Upstream Transmission in WDMSingle-Fiber Loopback Access Networks,” IEEE J. Lightwave Technol. 24, 740-746 (2006).

9. W. Lee, H. Izadpanah, R. Menendez, S. Etemad, and P. J. Delfyett, “Synchronized Mode-Locked SemiconductorLasers and Applications in Coherent Communications,” IEEE J. Lightwave Technol. 26, 908-921 (2008).

10. J. H. Winters and S. Kasturia, “Constrained Maximum-Likelihood Detection for High-Speed Fiber-Optic Sys-tems,” in Proceedings of IEEE Global Communications Conference (GLOBECOM) (IEEE, 1991) pp. 1574-1579.

11. A. Farbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J. -P. Elbers, H. Wernz, H.Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Like-lihood Sequence Estimation,” European Conference and Exhibition on Optical Communication (ECOC) PD-Th.4.1.5 (2004).

#106098 - $15.00 USD Received 7 Jan 2009; revised 4 Jun 2009; accepted 7 Jun 2009; published 7 Jul 2009

(C) 2009 OSA 20 July 2009 / Vol. 17, No. 15 / OPTICS EXPRESS 12433

12. P. G. Hoel, Elementary Statistics, 2nd edition (John Wiley & Sons Inc., New York, 1966)

1. Introduction

An optical code division multiplexing (OCDM) system has attractive characteristics such asscalability, the ability to accommodate a variable bit rate, interference avoidance and high se-curity [1]. Recently various frequency domain based OCDM methods have been reported [2–4].However, there are two main issues related to OCDM technologies that must be addressed. Oneissue relates to increasing the number of users and the other is concerned with improving thespectral efficiency. Many approaches have been reported with a view to solving the spectralefficiency problem including orthogonal frequency division multiplexing [5], and polarizationdivision multiplexing techniques. However, as the number of people using these techniquesincreases, the signal deteriorates, e.g. multiple access interference (MAI), shot noise and beatnoise are generated. In particular, receiver sensitivity is degraded by beat noise [6–8], which isgenerated during photo-electric conversion because of wavelength sharing by some users. Andbeat noise is a dominant factor limiting the number of users.

Figure 1 is a conceptual diagram of a frequency domain OCDM that has a hard decisioncircuit. Each transmitter (Tx) comprises a multi-frequency light source, a user-specific spec-

Multi-frequency

light source

Encoder #1

{1,1,0,0}

Decoder #1

{1,1,0,0}

Decoder #2

{1,0,1,0}

Data #1Data #1

Data #2

Tx #1

Multi-frequency

light source

Encoder #2

{1,0,1,0}

Data #2 Tx #2

f0 f1 f2 f3f0 f1 f2 f3

AWG

MOD

PD1

PD2

MOD

f0 f1 f2 f3Rx

Fig. 1. Example OCDM system.

Hard decision with beat noise for two users.

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

Hard decision w/o beat noise.

Degradation caused

by beat noise.

Fig. 2. Influence of beat noise on BER characteristics.



Multi-frequency

light source

Encoder #1

{1,1,0,0}

Data #1Data #1

Data #2

Tx #1

Multi-frequency

light source

Encoder #2

{1,0,1,0}

Data #2 Tx #2

f0 f1 f2 f3f0 f1 f2 f3

Rx

AWG

MOD

PD1

PD2

MOD

f0 f1 f2 f3

Decoding & Decision

MLD

BNE

Maximum Likelihood

Detection

Beat Noise Estimation

**

*

*

**

Fig. 3. Proposed detector.

tral amplitude encoder, and a modulator (MOD). In each Tx, OCDM signals are generatedthrough the encoder, and generated signals are multiplexed at an optical splitter. In contrast, areceiver (Rx) comprises a demultiplexer such as an arrayed waveguide grating (AWG), PDs,user-specific decoders, and detectors, which are signal hard decision circuits. This system em-ploys multi-frequency light sources that are frequency-synchronized between users [9]. Figure2 shows numerical simulation results obtained using the configuration illustrated in Fig. 1. Inthis Fig., the horizontal axis is the signal to noise ratio (SNR) for Gaussian noise generatedduring photo-electric conversion, and the vertical axis is the bit error rate (BER). The deep redplot shows the result when there is no beat noise, and the purple plot is the result with beatnoise. Here, polarization adjustment is used for excessive beat noise as the worst case. TheBER characteristic with beat noise is worse than that without it. It is recognized that beat noiseis the dominant factor as regards BER characteristic degradation because its value remains highwhen the SNR increases. This is caused by the fact that the beat noise exceeds the threshold ofthe hard decision technique. Therefore a new detection technique is needed.

In this paper, we propose the maximum likelihood detection (MLD) technique [10, 11] thatincorporates beat noise estimation (BNE). First, we describe an MLD algorithm that can min-imize the BER theoretically. Then, we describe a BNE design that can extract a specific beatnoise from mixed multiple signals using a correlation and estimation procedures. We also con-sider the correlator conditions. Then, we provide numerical simulation results that represent theBER characteristics. And we confirm the validity and performance of the proposed technique.Finally, we conclude this paper.

2. Maximum likelihood detection with beat noise estimation

2.1. MLD algorithm

We first describe an algorithm for a maximum likelihood detection (MLD) technique that canminimize BER. Figure 3 shows frequency domain OCDM system diagrams for our proposedMLD technique with BNE.

The transmitter signals si,tx(t) can be expressed as shown in Eq. (1).

si,tx(t) = ai(t)Ai

M∑

k=1

cik · cos(2π fikt+φik) (1)

where ai(t) is the transmitted signal {0} or {1} with probability 1/2, Ai is the electrical intensityof the carrier wave, cik is an orthogonal code assigned to the ith user’s kth wavelength, fik is



frequency, φik is the optical initial phase, and M is the number of wavelengths. In this case,an orthogonal code such as the Hadamard [5] code is employed. After demultiplexing the sig-nals into each wavelength and direct detection, which denotes square-law detection at PDs, thereceived signals sk,rx can be expressed as shown by Eq. (2).

sk,rx =

N∑

i=1

c2ika

2i A2

i

2+

N−1∑

i=1

N∑

j=i+1

cikaiAic jka jA jbi jk + xk (2)

where N is the number of users, the first term is signal information intensity, and bi jk in thesecond term is the beat noise as shown in Eq. (3).

bi jk = cos(2π( fik − f jk)t+ (φik −φ jk)) (3)

The beat noise is generated by the ith and jth users sharing the same kth wavelength. And thirdterm xk is thermal noise and denotes Gaussian noise generated during photo-electric conversion.The probability density function (PDF) of this thermal noise depends on the Gaussian functionN(0,σ) as shown in Eq. (4).

Pr(x) =1√2πσ

exp

(− x2

2σ2

)(4)

where σ is a standard deviation of the Gaussian distribution.From Eq. (2), if there are some users who have transmitted signal {1} at a certain wavelength,

beat noise is generated because of users sharing the same wavelength. And if there are more thantwo users, multiple beat noises are generated, so the impact of the beat noise becomes greater.Therefore, each beat noise has to be estimated independently. And with estimated beat noises,a bit pattern that denotes combined users’ signals with the maximum likelihood is defined asdetected signals. The detected signals for ai can be expressed by Eq. (5).

argmax{ai}

⎧⎪⎪⎪⎨⎪⎪⎪⎩∏

k

Pr(sk,rx−N∑

i=1

cikai

2−N−1∑

i=1

N∑

j=i+1

cikaic jka jbi jk)

⎫⎪⎪⎪⎬⎪⎪⎪⎭(5)

where bi jk is the estimated beat noise. And by removing the common term, Eq. (5) becomesEq. (6).

argmin{ai}

⎧⎪⎪⎪⎨⎪⎪⎪⎩∑

k

(sk,rx−N∑

i=1

cikai

2−N−1∑

i=1

N∑

j=i+1

cikaic jka jbi jk)2

⎫⎪⎪⎪⎬⎪⎪⎪⎭(6)

Here, we explain why the technique enables us to reduce the influence of beat noise. Becauseeach signal {0} or {1} is transmitted with the same probability, MLD and maximum posterioriprobability detection are equivalent from the Bayes formula [12], and a bit pattern with themaximum posteriori probability is selected as detected signals. In all cases, an estimated beatnoise value is assigned to the PDF of the Gaussian noise to make it possible to compare themagnitude relation of posteriori probabilities. If the estimated beat noise bi jk equals the gener-ated beat noise bi jk, the posteriori probability of the bit pattern has the maximum value. So, weapply BNE to MLD as our proposed technique.

Also, we estimate the beat noise and apply the value to our MLD technique to output a bitpattern with the maximum posteriori probability as the detected signals. That is to say, whensignals are detected, the threshold for signal detection is changed flexibly by the estimatedbeat noise. This is shown as a constellation diagram in Fig. 4, where sk,rx are the receivedsignals, black and white circles (•,◦) denote the signal points of each transmitted bit patternsuch as (0,0), and θ is the beat noise phase when N =2, M =3. Because Eq. (6) means that a



(0,0)(0,0)

(0,1)(0,1)

(1,0)(1,0)

(1,1)(1,1)

θθ == 00

θθ == ππ θθ ==

ππ

22θθ ==

ππ

22

ss1,1,rxrx

ss3,3,rxrx

ss2,2,rxrx

Fig. 4. Changes of signal points for bit pattern for two users.

signal point with the minimum Euclidean distance from the received signal sk,rx is selected asa detected result, the threshold is set at a point that is equidistant from two signal points. Ifthe beat noise phase is changed, that is to say the signal point (1,1) is moved as in Fig. 4, thedistance between the signal points is changed. In this way, the threshold is optimally set at apoint equidistant from two signal points. The proposed method is superior to the hard decisiontechnique in this respect, and it enables us to realize minimum BER.

However, it is necessary to consider that even if the threshold is set optimally, overlappingsignal points lead to incorrect detection with a probability of 1/2. If a beat noise is not generated,signal points do not overlap under an orthogonal condition such as a Hadamard code. However,if a beat noise is generated, changing the phase of the beat noise may cause overlapping signalpoints. This means that the detection confuses the signal point with another point and resultsin an error. Therefore to use MLD effectively, we must select a code for the frequency domainOCDM system that avoids any overlap between signal points. In fact, the signal points do notoverlap with the code used in Fig. 4.

2.2. BNE design

Here, we describe the BNE design and how to estimate a specific beat noise. If some opticalsignals experience photo-electric conversion simultaneously, a beat noise is generated that de-pends on the frequency spacing, and it fluctuates periodically as a sine waveform. In addition,if there are more than two users, a number of beat noises are generated, so the impact of themultiple beat noises becomes greater and the influence of the correlation between each beatnoise increases. Therefore, BNE must meet two conditions. One is that it can estimate multi-ple beat noises independently that are correlated with each other. The other is that the circuitconfiguration for BNE is simple and its size is as small as possible because the number of beatnoises bi jk increases in proportion to the cube of the number of users.

The most widely used way to extract a specific signal from mixed multiple signals is to use acorrelation. Here, we make use of the correlation between each beat noise and the bit patternsto estimate the beat noises. Also, a multiplier and an integrator in combination are employed asa correlator. Because the integrator can be realized using a first-order low pass filter (LPF), thisapproach can make the size of the circuit small.

Figure 5 shows an example of the proposed configuration. It consists of a time delay, asubtracter, a coefficient table, a multiplier, and an LPF. A combination consisting of a multiplierand an LPF denotes a correlation. In this Fig., the estimated beat noise bi jk(t) can be expressed



Table 1. Relationship between bit patterns and received signals including beat and Gaussian noises.

P1

P2

P3

P4

P5

P6

P7

P8

User Signals

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

User1 User2 User3

Received SignalsPattern

No.s1

1

2

1x+

1321coscoscos

2

3x++++ θθθ

2

2

1x+

4

2

1x+

1x

1

2

1x+

1

2

1x+

13cos1 x++ θ

12cos1 x++ θ

11cos1 x++ θ

s2

s3

s4

3

2

1x+

2

2

1x+

4x

3

2

1x+

2

2

1x+

4

2

1x+

3x

2

2

1x+

4x3

x

2x

4

2

1x+

3

2

1x+

2x

4x

3

2

1x+

2x

4

2

1x+

3x

2x

4x

3x

1231312xbbb +++

1x

123xb +

113xb +

112xb +

Beat noise &

Gaussian noise

1x

1x

1x

ＭＬＤO/E

（Estimated beat noises）

ＢＮＥ

Beat noises and Gaussian noises

Delay

( )tsk rx,

( )( )tasi

ˆ

est

( )tbijk

∧

Correlation

LPF

Ｃｏｅｆｆｉｃｉｅｎｔ

ｔａｂｌｅ( )tu

ij

( )tai

ˆ

Fig. 5. BNE configuration.

as shown by Eq. (7).

bi jk(t) =1τ

∫ t

−∞e−

t−t′τ

{sk,rx(t′)− sest(t

′)}ui j(t

′)dt′ (7)

where τ is the time constant of the LPF. It is assumed that τ is sufficiently short for bi jk(t) toremain unaltered during τ, this means bi jk(t) � bi jk(t+ τ), and is sufficiently longer than 1 bittime. Under the condition of τ, ui j(t), which is a coefficient multiplied by the noise signals toestimate the specific beat noise bi jk(t), is obtained by solving a simultaneous equation derivedfrom the identity formula so that bi jk � bi jk.

Next, we explain the derivation of ui j(t) for the condition bi jk � bi jk. Here, there are threeusers on the OCDM system. In this case, transmitted signals have eight bit patterns as shown inTable 1. The received signals including the beat and Gaussian noises generated during photo-electric conversion and the coefficients for transmitted bit patterns are expressed as shown inTable 1, where θ1 = φ2 − φ1, θ2 = φ3 − φ1, θ3 = φ3 − φ2 used in Eq. (1), xk is Gaussian noise,and bi j is the beat noise generated by the ith and jth users sharing the same wavelength. Weexplain how to estimate the beat noise b12. In this case, we classified the ui j(t) by strength ofcorrelation, for example, the coefficient of a bit pattern with a specific beat noise (P7) is α,another with a strong correlation with a specific beat noise (P8) is β, and the others with a weak



Gaussian x1

b23

b13

b12

P1 P2 P3 P4 P5 P6 P7 P8

(a) Beat and Gaussian noises for bit pattern. (b) Coefficient for bit pattern.

(c) LPF input :multiplier signals. (d) LPF output.

Fig. 6. How to estimate specific beat noise b12.

correlation (P1-P6) are γ, as shown by Eq. (8).

ui j(t) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

α , aia jal = 1

β , aia jal = 1 (l � i, l � j)

γ , others(8)

For estimating the specific beat noise b12 and considering the correlation between each bitpattern, simultaneous Eqs. are derived as shown by Eq. (9).

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

18 (α+β) = 1 , for b1218 (β+γ) = 0 , for b13 or b2318 (α+β+6γ) = 0 , for x1

(9)

By solving these Eqs., we could obtain the coefficients ui j(t) as shown by Eq. (10).

ui j(t) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

α = 203 , aia jal = 1

β = 43 , aia jal = 1 (l � i, l � j)

γ = − 43 , others

(10)

In this way, the design of BNE is considered. Even when the number of users is more thanthree, the coefficients can be calculated by the same approaches.

Then, using these coefficients, we describe the procedures for estimating a specific beat noiseb12 below and pattern diagrams in Fig. 6.

(1) First, the beat noise estimator subtracts signal values detected at the MLD from receivedsignals with a time delay. Therefore, we can obtain the noise signals that denote beat andGaussian noise as multiplier inputs: Fig. 6(a).



(2) In contrast, based on the signal values detected at the MLD, the bit pattern are separatedas shown in Table 1 and Fig. 6(a). This depends on the correlation strength with b12.For estimating a specific beat noise b12, the obtained coefficient is outputted: Fig. 6(b).As previously indicated, the coefficient of bit pattern P7, which has only a specific beatnoise b12 is 20

3 , P8, which has a strong correlation with b12 is 43 , and the others, which

have a weak correlation, are − 43 .

(3) Next, the beat and Gaussian noise signals and the coefficient are multiplied, and we obtainthe product as the LPF input: Fig. 6(c).

(4) Finally, the input signals are averaged among appropriate lengths of time at the LPF. Ifeach transmission probability of the bit pattern is the same, eliminating the other noisesallows us to estimate the specific beat noise b12: Fig. 6(d).

2.3. Optimum time constant of LPF

Here, the characteristics and conditions of the LPF are described. As previously described,an LPF is used as an integrator for averaging signals in the correlator. An LPF has only onedominant parameter, the time constant τ, which is also known as the cutoff frequency fc. Thisrelation is defined as shown by Eq. (11).

fc =1

2πτ(11)

The time constant τ is thought of as having an optimum value for the two reasons given below.One is that if the averaging time, which is defined as about double the τ value, is short, thenumber of samples for bit patterns becomes small. And this increases the dispersion of theoccurrence rate for each bit pattern. As a result, many estimation errors are generated duringBNE. On the other hand, if the averaging time is long, the BNE output exhibits a time lag. Inparticular, there are many estimation errors when there is a large rate of change in the beat noiseover a constant time. In addition, if τ is longer, the cutoff frequency fc of the LPF decreasesfrom Eq. (11). In this case, because the high frequency content of the beat noise is eliminated,the estimation errors increase. For example, the beat noise includes LD phase noise, so thecutoff frequency should be far higher than the spectrum of the phase noise for outputting thephase noise directly through BNE. Figure 7 is a diagram of the LPF frequency characteristicsfor τ and the LD phase noise spectrum of as previously described. In this way, because theestimation errors become greater if τ is too long or too short, it is expected that there is anoptimum time constant τ.

3. Simulation results

First, we consider the optimization of the time constant τ of the LPF. Figure 9 shows the timeconstant characteristic. In this Fig., the horizontal axis is the time constant τ, and the verticalaxis is the BER. Also there are three users, and the SNR is 18 dB. From the result, the timeconstant was at its optimum value, in this case τ = 2.78 ns, as previously described in Sec. 2.3,and this is confirmed.

Figure 10 shows the BER characteristic of the proposed MLD method incorporating the BNEtechnique with an OCDM system, where there are two users, and the optimum τ of the LPF is2.22 ns. The dot-dashed line denotes the characteristic of the hard decision technique with andwithout beat noise as shown in Fig. 2, and the dashed line shows the proposed method withoutthe estimation error, instead substituting the well-known beat noise as the estimated beat noise.The solid line shows the proposed method and the diamond (�) also shows the proposed method



f [Hz]

Gain [a.u.]

fc

fc

’ fc

’’

LD phase noise

τ

longer shorter

3dB

0

-3

Fig. 7. Diagram of LPF characteristics for τ.

ＰＤ

LD

LD

PCSpectrum

AnalyzerATT

ATT

Power Meter

LD

ATT

OSC

Coupler

: Polarization Control

: Oscilloscope

PC

OSC

: Laser Diode,

: Attenuator,

Fig. 8. Experimental system.

Time constant τ [ns]

BER

1 20

10

10

10

10

-4

-3

-2

-1

10-5

10-6

Time constant τ [ns]

BER

1 20

10

10

10

10

-4

-3

-2

-1

10-5

10-6

102.78

Fig. 9. BER characteristics for τ for three users, SNR = 18 dB.

using empirical data as the beat noise value. The data transmission speed was 50 Gbps. In addi-tion, the Hadamard codes, {1,1,0,0} and {1,0,1,0}, were used. The proposed BER characteristicobtained with our approach is superior to that obtained with the hard decision technique. Thereis a slight penalty, however we could obtain a lower BER. The simulation results confirmed thevalidity of the proposed method described in Sec. 2.1.

Next, we consider the empirical beat noise data. The purpose of using the empirical datais to take account of the noise (e.g. phase and amplitude noise) exhibited by LDs. Figure 8shows our experimental system. It consists of LDs, attenuators (ATT), a polarization controller(PC), couplers, PDs, and electrical and optical measuring instruments. Here we used distributedfeedback laser diodes (DFB-LDs) with a spectral width of 80 kHz. To allow us to focus on thefrequency spacing and phase fluctuation, the carrier waves were not modulated. In addition, thepolarization and power values of two signals were set using the PC and ATT, and mixed at thecoupler. At the receiver, the signal was photo-electric converted, and the received signal wasobserved with a digital sampling oscilloscope (OSC). The transmission consisted of continuouswave (CW) light at 1.55 μm. The OSC parameter was 50 giga samples per second (50 GS/s)for an observation time of 10 μs.

The BER obtained with the empirical beat noise is higher than that obtained with the simu-lated beat noise, which is shown by the solid line. There is little property degradation caused



SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

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10

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-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

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10

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-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

for two users.

Hard decisionw/o beat noise.

Hard decision.

w/o estimate error.

Proposed.

using empirical beat noise data.

(1,1,0)

(1,0,1)

- used code

Fig. 10. BER characteristics for two users, τ = 2.22 ns.

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

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10

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10

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-2

-1

0

SNR [dB]

BER

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-3

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-1

0

SNR [dB]

BER

0 5 10 15 20 2510

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10

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-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

SNR [dB]

BER

0 5 10 15 20 2510

10

10

10

10

-4

-3

-2

-1

0

w/o estimate error.

for three users.

(1,1,0,0)

(1,0,1,0)

(0,1,1,0)

- used codeProposed.

(1,1,0,0)

(1,0,1,0)

(1,0,0,1)

Hard decisionw/o beat noise.

Hard decision.

using empirical beat noise data.

- used codeProposed.

Fig. 11. BER characteristics for three users, τ = 2.78 ns.

by the LD phase or amplitude noise, however the penalty is suppressed to within 0.5 dB. Thisresult means that by using an appropriate LPF the beat noise including the phase noise couldbe correctly estimated as previously described. Moreover, the amplitude noise of the LDs waslow and its influence on the BER was also small. The simulation results confirmed the validityof the proposed method described in Sec. 2.2.

Next, Fig. 11 shows the BER characteristic, where the number of users is three, and theoptimum τ of the LPF is 2.78 ns. The dot-dashed line denotes the hard decision characteristicwith and without beat noise, and the dashed line indicates the proposed method without estimateerror, instead substituting the well-known beat noise as the estimated beat noise. The solid line



shows the proposed method, and the diamonds (�) also show the proposed method when usingempirical data as the beat noise value. In addition, the Hadamard codes, {1,1,0,0}, {1,0,1,0},and {1,0,0,1}, were used. And here, the result using the length of the 3 codes is shown asanother dashed line. When the number of users increased from two to three, and the numberof wavelengths (the length of the codes) remained 3, an error floor was observed. In this case,the Hadamard codes, {1,1,0,0}, {1,0,1,0}, and {0,1,1,0}, were assigned. This is because a signalpoint for a given bit pattern overlapped another signal point when the beat noise phase waschanged as described in Sec. 2.1. This confirms the need to select codes to avoid the signalpoints overlapping.

From the simulation results, the proposed method provides a better BER characteristic thanthe hard decision method. The penalty that was caused by the phase and amplitude noises ofthe LDs could be suppressed to less than 1 dB. This, the validity of proposed technique isconfirmed.

4. Conclusion

In this paper, we proposed a maximum likelihood detection method with a beat noise estimationtechnique for a frequency domain OCDM system. By selecting a bit pattern with the maximumposteriori probability as the detected signals, the BER decreased because the influence of beatnoise was reduced. This was because, by using the correlation between each beat noise and thebit patterns, beat noises were estimated that included the LD phase noise. Simulation resultsconfirmed the validity and the performance of our proposed technique.



maximum likelihood detection with beat noise estimation for minimizing bit error rate in...

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