## Abstract

We propose a maximum likelihood detection (MLD) technique that incorporates beat noise estimation (BNE). MLD can minimize a bit error rate theoretically because a bit pattern with the maximum posteriori probability is selected as the detected signals. Also, BNE can extract a specific beat noise from mixed multiple signals using a correlation. By combining these techniques, the influence of beat noise is reduced and the bit error rate becomes lower in an OCDM-based system. This paper describes the MLD algorithm and the BNE design. And numerical simulation results confirm the validity and performance of this technique.

© 2009 Optical Society of America

## 1. Introduction

An optical code division multiplexing (OCDM) system has attractive characteristics such as scalability, the ability to accommodate a variable bit rate, interference avoidance and high security [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. One issue relates to increasing the number of users and the other is concerned with improving the spectral efficiency. Many approaches have been reported with a view to solving the spectral efficiency problem including orthogonal frequency division multiplexing [5], and polarization division multiplexing techniques. However, as the number of people using these techniques increases, the signal deteriorates, e.g. multiple access interference (MAI), shot noise and beat noise are generated. In particular, receiver sensitivity is degraded by beat noise [6–8], which is generated during photo-electric conversion because of wavelength sharing by some users. And beat 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 decision circuit. Each transmitter (Tx) comprises a multi-frequency light source, a user-specific spectral amplitude encoder, and a modulator (MOD). In each Tx, OCDM signals are generated through the encoder, and generated signals are multiplexed at an optical splitter. In contrast, a receiver (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 employs multi-frequency light sources that are frequency-synchronized between users [9]. Figure 2 shows numerical simulation results obtained using the configuration illustrated in Fig. 1. In this Fig., the horizontal axis is the signal to noise ratio (SNR) for Gaussian noise generated during photo-electric conversion, and the vertical axis is the bit error rate (BER). The deep red plot shows the result when there is no beat noise, and the purple plot is the result with beat noise. Here, polarization adjustment is used for excessive beat noise as the worst case. The BER characteristic with beat noise is worse than that without it. It is recognized that beat noise is the dominant factor as regards BER characteristic degradation because its value remains high when the SNR increases. This is caused by the fact that the beat noise exceeds the threshold of the hard decision technique. Therefore a new detection technique is needed.

In this paper, we propose the maximum likelihood detection (MLD) technique [10, 11] that incorporates beat noise estimation (BNE). First, we describe an MLD algorithm that can minimize the BER theoretically. Then, we describe a BNE design that can extract a specific beat noise from mixed multiple signals using a correlation and estimation procedures. We also consider the correlator conditions. Then, we provide numerical simulation results that represent the BER 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 can minimize BER. Figure 3 shows frequency domain OCDM system diagrams for our proposed MLD technique with BNE.

The transmitter signals *s*
_{i,tx}(*t*) can be expressed as shown in Eq. (1).

where *a _{i}*(

*t*) is the transmitted signal {0} or {1} with probability 1/2,

*A*is the electrical intensity of the carrier wave,

_{i}*c*is an orthogonal code assigned to the

_{ik}*i*th user’s

*k*th wavelength,

*f*is frequency,

_{ik}*ϕ*is the optical initial phase, and

_{ik}*M*is the number of wavelengths. In this case, an orthogonal code such as the Hadamard [5] code is employed. After demultiplexing the signals into each wavelength and direct detection, which denotes square-law detection at PDs, the received signals sk,rx can be expressed as shown by Eq. (2).

where *N* is the number of users, the first term is signal information intensity, and *b _{i jk}* in the second term is the beat noise as shown in Eq. (3).

The beat noise is generated by the *i*th and *j*th users sharing the same *k*th wavelength. And third term *x _{k}* 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 function

*N*(0,

*σ*) as shown in Eq. (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 than two 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 as detected signals. The detected signals for *a _{i}* can be expressed by Eq. (5).

where *b*̂_{i jk} is the estimated beat noise. And by removing the common term, Eq. (5) becomes Eq. (6).

Here, we explain why the technique enables us to reduce the influence of beat noise. Because each signal {0} or {1} is transmitted with the same probability, MLD and maximum posteriori probability detection are equivalent from the Bayes formula [12], and a bit pattern with the maximum posteriori probability is selected as detected signals. In all cases, an estimated beat noise value is assigned to the PDF of the Gaussian noise to make it possible to compare the magnitude relation of posteriori probabilities. If the estimated beat noise *b*̂_{i jk} equals the generated beat noise *b _{i jk}*, the posteriori probability of the bit pattern has the maximum value. So, we apply BNE to MLD as our proposed technique.

Also, we estimate the beat noise and apply the value to our MLD technique to output a bit pattern with the maximum posteriori probability as the detected signals. That is to say, when signals are detected, the threshold for signal detection is changed flexibly by the estimated beat noise. This is shown as a constellation diagram in Fig. 4, where *s*
_{k,rx} are the received signals, black and white circles (∙,∘) denote the signal points of each transmitted bit pattern such as (0, 0), and *θ* is the beat noise phase when *N*=2, *M*=3. Because Eq. (6) means that a signal point with the minimum Euclidean distance from the received signal sk,rx is selected as a detected result, the threshold is set at a point that is equidistant from two signal points. If the beat noise phase is changed, that is to say the signal point (1, 1) is moved as in Fig. 4, the distance between the signal points is changed. In this way, the threshold is optimally set at a point equidistant from two signal points. The proposed method is superior to the hard decision technique 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, overlapping signal 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 signal points. This means that the detection confuses the signal point with another point and results in an error. Therefore to use MLD effectively, we must select a code for the frequency domain OCDM system that avoids any overlap between signal points. In fact, the signal points do not overlap 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 optical signals experience photo-electric conversion simultaneously, a beat noise is generated that depends 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 the multiple beat noises becomes greater and the influence of the correlation between each beat noise increases. Therefore, BNE must meet two conditions. One is that it can estimate multiple beat noises independently that are correlated with each other. The other is that the circuit configuration for BNE is simple and its size is as small as possible because the number of beat noises *b*̂_{ijk} 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 a correlation. Here, we make use of the correlation between each beat noise and the bit patterns to estimate the beat noises. Also, a multiplier and an integrator in combination are employed as a correlator. Because the integrator can be realized using a first-order low pass filter (LPF), this approach can make the size of the circuit small.

Figure 5 shows an example of the proposed configuration. It consists of a time delay, a subtracter, a coefficient table, a multiplier, and an LPF. A combination consisting of a multiplier and an LPF denotes a correlation. In this Fig., the estimated beat noise *b*̂_{ijk}(*t*) can be expressed

as shown by Eq. (7).

where *τ* is the time constant of the LPF. It is assumed that *τ* is sufficiently short for *b _{ijk}*(

*t*) to remain unaltered during

*τ*, this means

*b*(

_{ijk}*t*)≒

*b*(

_{ijk}*t*+

*τ*), and is sufficiently longer than 1 bit time. Under the condition of

*τ, u*(

_{ij}*t*), which is a coefficient multiplied by the noise signals to estimate the specific beat noise

*b*(

_{ijk}*t*), is obtained by solving a simultaneous equation derived from the identity formula so that

*b*̂

_{ijk}≒

*b*.

_{ijk}Next, we explain the derivation of *u _{ij}*(

*t*) for the condition

*b*̂

_{ijk}≒

*b*

_{ijk}. Here, there are three users on the OCDM system. In this case, transmitted signals have eight bit patterns as shown in Table 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 in Table 1, where

*θ*

_{1}=

*ϕ*

_{2}-

*ϕ*

_{1},

*θ*

_{2}=

*ϕ*

_{3}-

*ϕ*

_{1},

*θ*

_{3}=

*ϕ*

_{3}-

*ϕ*

_{2}used in Eq. (1),

*x*is Gaussian noise, and

_{k}*b*is the beat noise generated by the

_{ij}*i*th and

*j*th users sharing the same wavelength. We explain how to estimate the beat noise

*b*

_{12}. In this case, we classified the

*u*(

_{ij}*t*) by strength of correlation, 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

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

For estimating the specific beat noise *b*
_{12} and considering the correlation between each bit pattern, simultaneous Eqs. are derived as shown by Eq. (9).

By solving these Eqs., we could obtain the coefficients *u _{ij}*(

*t*) as shown by Eq. (10).

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

Then, using these coefficients, we describe the procedures for estimating a specific beat noise *b*
_{12} below and pattern diagrams in Fig. 6.

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

(2) In contrast, based on the signal values detected at the MLD, the bit pattern are separated as shown in Table 1 and Fig. 6(a). This depends on the correlation strength with *b*
_{12}. For estimating a specific beat noise *b*
_{12}, the obtained coefficient is outputted: Fig. 6(b). As previously indicated, the coefficient of bit pattern P7, which has only a specific beat noise *b*
_{12} is $\genfrac{}{}{0.1ex}{}{20}{3}$, P8, which has a strong correlation with *b*
_{12} is $\genfrac{}{}{0.1ex}{}{4}{3}$, and the others, which have a weak correlation, are $-\genfrac{}{}{0.1ex}{}{4}{3}$.

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

(4) Finally, the input signals are averaged among appropriate lengths of time at the LPF. If each transmission probability of the bit pattern is the same, eliminating the other noises allows us to estimate the specific beat noise *b*
_{12}: 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 one dominant parameter, the time constant *τ*, which is also known as the cutoff frequency *f _{c}*. This relation is defined as shown by Eq. (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, the number of samples for bit patterns becomes small. And this increases the dispersion of the occurrence rate for each bit pattern. As a result, many estimation errors are generated during BNE. On the other hand, if the averaging time is long, the BNE output exhibits a time lag. In particular, there are many estimation errors when there is a large rate of change in the beat noise over a constant time. In addition, if *τ* is longer, the cutoff frequency *f _{c}* of the LPF decreases from 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 the cutoff frequency should be far higher than the spectrum of the phase noise for outputting the phase noise directly through BNE. Figure 7 is a diagram of the LPF frequency characteristics for

*τ*and the LD phase noise spectrum of as previously described. In this way, because the estimation errors become greater if

*τ*is too long or too short, it is expected that there is an optimum time constant

*τ*.

## 3. Simulation results

First, we consider the optimization of the time constant *τ* of the LPF. Figure 9 shows the time constant characteristic. In this Fig., the horizontal axis is the time constant *τ*, and the vertical axis is the BER. Also there are three users, and the SNR is 18 dB. From the result, the time constant 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 BNE technique with an OCDM system, where there are two users, and the optimum τ of the LPF is 2.22 ns. The dot-dashed line denotes the characteristic of the hard decision technique with and without beat noise as shown in Fig. 2, and the dashed line shows the proposed method without the 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 using empirical data as the beat noise value. The data transmission speed was 50 Gbps. In addition, the Hadamard codes, {1,1,0,0} and {1,0,1,0}, were used. The proposed BER characteristic obtained with our approach is superior to that obtained with the hard decision technique. There is a slight penalty, however we could obtain a lower BER. The simulation results confirmed the validity of the proposed method described in Sec. 2.1.

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

The BER obtained with the empirical beat noise is higher than that obtained with the simulated beat noise, which is shown by the solid line. There is little property degradation caused by the LD phase or amplitude noise, however the penalty is suppressed to within 0.5 dB. This result means that by using an appropriate LPF the beat noise including the phase noise could be correctly estimated as previously described. Moreover, the amplitude noise of the LDs was low and its influence on the BER was also small. The simulation results confirmed the validity of the proposed method described in Sec. 2.2.

Next, Fig. 11 shows the BER characteristic, where the number of users is three, and the optimum *τ* of the LPF is 2.78 ns. The dot-dashed line denotes the hard decision characteristic with and without beat noise, and the dashed line indicates the proposed method without estimate error, 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 using empirical 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 as another dashed line. When the number of users increased from two to three, and the number of 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 signal point for a given bit pattern overlapped another signal point when the beat noise phase was changed as described in Sec. 2.1. This confirms the need to select codes to avoid the signal points overlapping.

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

## 4. Conclusion

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

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