Theoretical and experimental studies of turbo product code with time diversity in free space optical communication

Yaoqiang Han; Anhong Dang; Yongxiong Ren; Junxiong Tang; Hong Guo

doi:10.1364/OE.18.026978

1. Introduction

Free space optical communication (FSOC) has attracted a lot of attention recently due to its advantages over radio frequency technology, such as high data rate, interference immunity, and fast deployment [1,2]. FSOC is widely applied in the fields of earth observation, broadband wireless access and so on [3–6].

FSOC system is often a line-of-sight point-to-point system, and the main challenge for practical systems is the atmospheric turbulence. Inhomogeneities in the pressure and temperature of the atmosphere lead to variations of the refractive index along the transmission path, which causes fluctuations in both the amplitude and the phase of the laser beam. These fluctuations will lead to an increase in the link error probability, which is the so-called channel fading [7]. To focus on mitigating channel fading, we assume that the transmitter and the receiver are perfectly aligned, and do not consider effects such as pointing error [8].

One possible approach to mitigate channel fading is to increase transmit power, however, it is not feasible and safe in many practical systems. Diversity techniques are often used in wireless communication systems for mitigating channel fading, and they can be divided into two categories: temporal-domain techniques and spatial-domain techniques. In the temporal-domain techniques, interleaving is commonly used to cope with burst errors caused by channel fading, and it helps to improve the channel coding performance [9]. In the spatial-domain techniques, aperture averaging is often used to mitigate the turbulence-induced power scintillation. For better performance, multiple receivers can be employed to collect the signal at different positions or from different spatial angles, and they should be placed as far apart as possible to maximize the diversity reception gain [10]. In summary, channel coding with diversity techniques are suitable for practical systems due to their good performance and easy implementation.

There have been a lot of papers concerning the performance of various coding schemes applied in FSOC system [9–12]. However, due to the complexity of the atmospheric channel, different channel models and approximate treatments are used. For example, the atmospheric channel is modeled as uncorrelated ergodic in [11], which means that the correlation time of channel fading is considered to correspond to single bit duration. In fact, the correlation time is much greater than single bit duration, thus it should be modeled as a block-fading channel [7]. Interleaving is often used in block-fading channel to combat burst errors [13]. Although some papers have introduced the performance of interleaving applied in FSOC system, they only present brief simulation results [9, 14]. In fact, the selection of the interleaver pattern and its parameters significantly influence the interleaving performance.

In this paper, we first introduce the outdoor experiment we did and determine the parameters of the block-fading channel model. Based on this model, we propose to use turbo product code (TPC) as the coding scheme due to its good resistance to burst errors [15,16]. Then a performance comparison of TPC, recursive systematic convolutional (RSC) code and low-density parity-check (LDPC) code is presented. Simulation results indicate that TPC shows significant advantages over RSC codes, and even provide better performance than some regular LDPC codes. In order to cope with burst errors caused by channel fading, b $σ_{I}^{2}$ block interleaving is used. We investigate its efficiency and determine the optimum interleaving depth for TPC. Finally, an experimental study of TPC with interleaving is presented, and experimental results also indicate that TPC with interleaving can significantly mitigate channel fading effects. Since such studies in FSOC system have not been reported, it is meaningful for engineering design of practical FSOC systems.

The structure of the paper is arranged as follows: the experimental study of the atmospheric channel is presented in Section 2. In Section 3, a brief introduction to TPC, RSC and LDPC is given, and a performance comparison of the above coding schemes is also demonstrated in this section. The analysis of interleaving is presented in Section 4. The outdoor experimental study of TPC with interleaving is presented in Section 5. The last section concludes this paper.

2. Channel fading

In FSOC systems, the laser beam will travel through atmosphere before it arrives at the receiver, and atmospheric turbulence will significantly influence the beam properties. Due to the complexity associated with phase or frequency modulation, current FSOC systems typically use intensity modulation with direct detection (IM/DD) [17], therefore, we focus on the irradiance fluctuation of the laser beam. In this section, the distribution of irradiance fluctuation and the correlation time of channel fading will be introduced.

In order to quantify the irradiance fluctuation, scintillation index (SI) is introduced, which is defined as below [18]:

σ_{I}^{2} = \frac{< I^{2} >}{< I >^{2}} - 1,

where Idenotes the received laser beam irradiance.

σ_{I}^{2}

is related to the weather condition, propagation distance, and the diameter of the receiver, etc. Since the distribution of irradiance fluctuation is complicated, several models have been proposed, such as gamma-gamma model and lognormal (LN) model [7]. Gamma-gamma model can be used to describe both moderate and weak scintillation, while LN model is often used in the case of weak scintillation. In practical systems, aperture averaging is often used to lower down the irradiance fluctuation of the received laser beam [19], therefore LN model is used in this paper due to its simplicity, and its expression is given as below:

p (I) = \frac{1}{I \sqrt{2 π σ_{I}^{2}}} \exp {- \frac{{[\ln (I / I_{0}) + \frac{1}{2} σ_{I}^{2}]}^{2}}{2 σ_{I}^{2}}},

where

I_{0}

denotes the mean value of the received laser beam irradiance.

In order to examine the accuracy of the model in practical system, an outdoor experiment has been done. Since the experiment setup is only a part of that of the experiment which we will introduce in Section 5, we will not introduce it in this section. The PDF of I is shown in Fig. 1 , and SI takes 0.05, 0.1, 0.25, 0.5 and 1 respectively. In this figure, the outdoor experiment data is also presented, which is marked by the abbreviation “Exp”. It can be seen that the experiment data and the corresponding curve match well, thus LN model is accurate enough in our study.

Fig. 1 Simulation curves and experiment data of the distribution of irradiance fluctuation for different scintillation index (SI).

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Another important parameter is the correlation time of channel fading $τ_{c}$ , and its value is related to the correlation length of irradiance fluctuation $d_{0}$ .When the propagation path lengthLsatisfies the below inequality [18]:

l_{0} < \sqrt{λ L} < L_{0},

d_{0}

can be estimated as below:

d_{0} \approx \sqrt{λ L},

where λ is the wavelength of the laser beam, and

l_{0}

,

L_{0}

are inner and outer scales of the atmospheric turbulence, respectively. In our experiment, λ is 780nm, L is 500m, therefore, Eq. (4) can easily be satisfied, and

d_{0}

is approximately 20mm.

According to Taylor’s frozen-in hypothesis [7], the refractive index fluctuation along the direction of propagation is well-averaged and weaker than those along the direction transverse to propagation. Therefore only the component of the wind velocity vector perpendicular to the propagation direction $V_{⊥}$ is taken into account, and the correlation time $τ_{c}$ can be expressed as:

τ_{c} = \frac{d_{0}}{V_{⊥}} .

According to the values of

d_{0}

and the wind speed,

τ_{c}

ranges from a few milliseconds to tens of milliseconds.

Based on the above discussion, the atmospheric channel can be modeled as a block-fading channel. The distribution of the irradiance fluctuation satisfies the LN model, and the correlation time $τ_{c}$ ranges from a few milliseconds to tens of milliseconds. For simplicity, we assume that the channel fade remains constant during a fading block which corresponds to the interval of $τ_{c}$ , and it is considered to be independent and identically distributed.

3. Theoretical analysis of TPC

To improve the FSOC system performance, channel coding is always used. In this section, we first introduce three typical channel coding schemes applied in FSOC systems, and then a performance comparison will be given. Since RSC code and LDPC code are well known and widely applied, we only present a brief review of them [20,21].

3.1 Introduction of channel coding schemes

Recursive systematic convolutional (RSC) code is a widely-applied powerful coding technique. Its performance in FSOC system has been presented in [9], and simulation results show that RSC code can provide significant coding gain over uncoded system. In this paper, for a fair comparison, the code rate of RSC code should be the same as that of the other two coding schemes, therefore punctured RSC code is used in this paper [21,22].

LDPC code as one of the most advanced channel coding techniques has received a lot of attention in the past decade, and I. B. Djordjevic has presented the performance of LDPC in FSOC system in [11, 22]. However, in [23], J. Li points out that LDPC codes are sensitive to channel characteristics and require specific design and optimization to achieve near-capacity performance. The optimization for LDPC codes will make the coding process complicated. Without loss of generality, regular LDPC code is used in this paper.

TPC is also a kind of powerful coding techniques which can provide near-capacity performance [15]. We consider two linear block codes $C_{1}$ with parameters $(n_{1}, k_{1})$ and $C_{2}$ with parameters $(n_{2}, k_{2})$ , where n and k stand for codeword length and the number of information bits, respectively. As shown in Fig. 2(a) , the product code $P = C_{1} \otimes C_{2}$ is obtained by 1) placing $(k_{1} \times k_{2})$ information bits in an array of $k_{2}$ rows and $k_{1}$ columns; 2) coding the $k_{2}$ rows using code $C_{1}$ ; 3) coding the $k_{2}$ rows using code $C_{1}$ . The parameters of the product code P are $n = n_{1} \times n_{2}$ , $k = k_{1} \times k_{2}$ , and the code rate $R = R_{1} \times R_{2}$ , where $R_{1}$ and $R_{2}$ are the code rates of $C_{1}$ and $C_{2}$ respectively. In this paper, we use extended Hamming code (64,57) and (32,26) to construct the product code, which are denoted by TPC (64,57)² and TPC (32,26)² respectively. Because the minimum Hamming distance of TPC is high, there is no error floor for TPC. Moreover, data scrambling in the coding process makes TPC having good resistance to burst errors [24]. Therefore TPC is suitable for block-fading channel.

Fig. 2 Encoding and decoding scheme of TPC. (a) Construction of TPC, (b) block diagram of elementary TPC decoder (row or column).

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Chase algorithm and iterative decoding are used for TPC decoding [15], and the structure of the elementary decoder of the rows (or columns) is shown in Fig. 2(b).

R (m) = R + a (m) \cdot W (m),

where m indicates the m^th iterative decoding process, R(m) denotes the soft-input data of the m^th iterative decoding, R denotes the soft-input data of the received signal, a(m) denotes the scaling factor which is related to the difference between R and W, W(m) denotes the extrinsic information,

β (m)

denotes the reliability factor. W(m + 1) is obtained by performing Chase algorithm with R(m) and

β (m)

as input, and is sent to the next decoding process. One iterative decoding process contains the decoding of the rows and the columns, and more iterative decoding will provide more coding gain. However, the coding gain will not increase significantly when iterative decoding process exceeds 5 times. Therefore, the iterative time takes 5 in this paper. In this paper, we use OOK (on-off keying) as the modulation scheme, and the channel log-likelihood ratio (LLR) for the FSO is given as below:

L L R = \frac{2 \hat{I} r_{s} - {\hat{I}}^{2}}{2 σ_{0}^{2}},

where

\hat{I}

is the estimated value of the instantaneous channel fading. In order to obtain the channel state information (CSI), training bits are inserted into the frame.

3.2 Performance comparison

In this section, we present simulation results to compare the performance of RSC, LDPC and TPC. Some research has been done to compare the performance of TPC and LDPC in AWGN channel [25], however, to the best of our knowledge, there is no similar study in FSOC system. The coding performance is evaluated in terms of BER as a function of SNR. For a fair comparison, the code rates of the three coding schemes are set approximately equal. As the RSC code, we consider RSC (1,133/171) of constraint length K = 7, where 133 and 171 represent the code polynomial generators in octal, and we use the punctured RSC (PRSC) code of code rate 3/4. As the LDPC code, we consider LDPC (4320,3242) of code rate 0.75, which is also widely used in optical fiber communication systems. As the TPC code, we consider TPC (64,57)² of code rate 0.79 and TPC (32,26)² of code rate 0.66.

As for the atmospheric channel, we set SI to 0.1, and the channel correlation time $τ_{c}$ is 3 ms, which are typical values in practical FSOC systems [19]. For the FSOC system, we set the data rate to 1Gbps that corresponds to the bit duration $T_{b} = 1 n s$ . In order to scramble the burst errors, interleaving is used and the interleaver depth is set to $2 \times 10^{7}$ bits, which corresponds to the length of seven channel fades.

From Fig. 3 , it can be seen that: 1) all the three coding schemes provide significant coding gain over uncoded system. For instance, at BER = $10^{- 5}$ , compared to the uncoded case, we obtain a coding gain of 5db, 7.8db, 8db and 8.9db by using RSC, LDPC, TPC (64,57)² and TPC (32,26)², respectively. 2) TPC (64,57)² shows better performance than the other coding schemes of approximately equal code rate and there is no error floor, while LDPC (4320,3240) has an error floor when BER is lower than $10^{- 5}$ . Although it is not a complete comparison, we can learn that TPC is suitable for FSOC system, and it can even provide better performance than some regular LDPC codes.

Fig. 3 Performance comparison of the three coding schemes, SI = 0.1, $τ_{c} = 3 m s$ , CSI assisted.

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4. Interleaving gain analysis

Channel fading causes burst errors, and only channel coding cannot cope with these errors. In general, a random or S-random interleaver has excellent performance for combating burst errors. However, it requires the storage of the entire interleaver pattern, which makes it infeasible for FSOC system due to the large amounts of errors. Algebraic interleaver is preferred in some block-fading channels, which can be generated on-the-fly using a few parameters. However, when the interleaver length N is long, searching for a primitive element in the Galois field GF (N) is difficult [26]. Considering the amount of burst errors and implementation difficulty, block interleaving is used in our system. In fact, some successful long distance FSOC systems also use block interleaving in their systems [27].

At the transmitter, data is written row-wise into the interleaver as shown in Fig. 4 . At the receiver, the data is read out column-wise, which is the inverse process of interleaving. In order to quantify the time diversity order, the term interleaving depth fading ratio (IDFR) is introduced, whose definition is:

I D F R = \frac{N_{i n t}}{burst errors} = \frac{M \times N}{R_{b i t} \cdot τ_{c}},

where

N_{i n t}

is the total bit number of the interleaver, N is the number of columns of the interleaver, M is the number of rows, and

R_{b i t}

is the data rate of the system. If N is the same as the codeword length (N = n), then the deinterleaver should spread burst errors to at least t columns, where t is the number of correctable bits. When M is above the value

M_{t h}

, the probability of having more than t errors in a codeword will be small and the estimated value of

M_{t h}

is:

M_{t h} = ⌈ \frac{R_{b i t} \cdot τ_{c}}{t} ⌉,

where

⌈ \cdot ⌉

is the top integral function. According to Eq. (8) and Eq. (9), the estimated IDFR_th value is 9 (for TPC(32,26)²) or 11(for TPC(64,57)²). When IDFR value is above IDFR_th, the interleaving gain is close to that of perfect interleaving. However, interleaving also leads to latency, a trade-off between interleaving depth and system performance should be made in practical system. Figure 5 presents the relation between the interleaving gain (at BER = 10⁻⁶) and IDFR using block interleaver.

Fig. 4 Concept of channel fading and the structure of block interleaver.

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Fig. 5 The relation between interleaving gain and IDFR using block interleaver, $τ_{c} = 3 m s$ .

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From Fig. 5, it can be seen that the interleaving gain increases with the increase of IDFR, and it can approximately be divided into three stages: 1) When IDFR is below 4, due to the limited interleaving depth, the number of error bits still exceeds the error correcting ability of TPC, interleaving gain increases slowly. 2) When IDFR is between 4 and 12 (for TPC(32,26)²) or 15 (for TPC(64,57)²), interleaving gain increases rapidly with the increase of IDFR, because more and more error bits can be corrected. 3) When IDFR is above 12 (for TPC(32,26)²) or 15 (for TPC(64,57)²), burst errors are spread relatively uncorrelated, interleaving gain increases slowly with the increase of IDFR and it approaches the value corresponding to perfect interleaving. 4) Since interleaving provides more interleaving gain for the coding scheme with higher code rate, the interleaving gain for TPC(64,57)² exceeds that of TPC(32,26)² when IDFR is above the IDFR_th of TPC(32,26)². 5) Since the laser beam irradiance distribution satisfies the LN model, and it can be seen that more fades are below the mean value of the laser beam irradiance, more errors are introduced than that in AWGN channel. Therefore, IDFR_th values in Fig. 5 are bigger than the estimated values of IDFR_th. In summary, the optimum IDFR_th value for TPC(32,26)² is 12 and for TPC(64,57)² is 15, it makes a balance between interleaving depth and interleaving gain.

5. Experimental study of TPC

5.1 Experiment setup

In order to study the TPC performance in practical system, an experimental study of TPC with different code rates and interleaving depths under different turbulence conditions is demonstrated in this section. Since the experiment we did is to evaluate the performance of long distance FSOC system which we will deploy in the future, the parameters of the atmospheric channel are taken the same as that of the long distance FSOC system. In the stratospheric optical payload experiment (STROPEX) conducted by DLR (German Aerospace Center) in August, 2005, the distance between terminals is 61km, SI is kept under 0.1 by using a antenna whose diameter is 40cm, and above 0.25 without the antenna [19]. Therefore, in our experiment, SI takes 0.1 and 0.25, which also correspond to cases of with and without aperture averaging in our future long distance FSOC system. The experiment setup is demonstrated in Fig. 6 , and the distance between the terminals in our experiment is 500m.

Fig. 6 Outdoor experiment setup.

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At the transmitting end, since high speed TPC encoding and decoding chip (or IP core) is expensive, we chose the off-the-shelf chip AHA4524 in our experiment, which is produced by AHA Corporation. TPC (64,57)² and TPC (32,26)² are chosen in our experiment, and the data rate is only 32Mbps by using AHA4524. High data rate (>600Mbps) TPC encoding and decoding module will be implemented in our future FSOC system. From Fig. 5, it can be seen that the interleaving gain increases with the increase of IDFR, however, big interleaver needs large memory. Due to the limited memory on the board, the maximum IDFR the board can provide is approximately 7. The laser diode HL7851G is used as transmitter, which is produced by HITACHI Inc..

At the receiving end, the diameter of the telescope is 40mm, which is bigger than the correlation length 20mm, therefore aperture averaging will lower down power scintillation. However, due to the limited distance, the telescope is not used when we want to conduct experiments in cases of large SI. The receiver placed after the telescope is PDA10A purchased from Thorlabs Inc., whose peak response is near 780nm. After de-interleaving and decoding, the decoded bits are sent into the on board bit error rate tester (BERT) to obtain BER performance of the FSOC system, and then the result is displayed on the PC.

5.2 Experimental results

The experimental results of TPC performance in the cases of SI equals to 0.1 and 0.25 are presented in Fig. 7 and Fig. 8 respectively, where abbreviations follow the dash “NoInt”, “Int”, “Exp” and “Sim” denote the cases of “no interleaving”, “interleaving”, “experiment data” and “simulation curve” respectively.

Fig. 7 Experimental results of TPC performance. SI = 0.1, IDFR = 7, CSI assisted.

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Fig. 8 Experimental results of TPC performance. SI = 0.25, IDFR = 7, CSI assisted.

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In Fig. 7, the curves of BER versus SNR for TPC (64,57)² and TPC (32,26)² with and without interleaving are presented. It can be seen that: 1) when IDFR takes 7, interleaving provides about 2db interleaving gain for TPC (64,57)² and TPC (32,26)², which approximately matches the simulation result presented in Fig. 5. 2) Interleaving provide more interleaving gain for TPC (32,26)² than TPC (64,57)² due to the reduction of code rate from 0.79 to 0.66, therefore a tradeoff between code rate and system performance should be made when channel fading significantly influences the system performance. 3) Experiment data is a little worse than simulation result, because the correlation time is time varying and it will degrade the interleaving performance.

Similar to Fig. 7, Fig. 8 presents the TPC performance in the case of SI equals to 0.25, and it can be seen that: 1) interleaving and the reduction of code rate will provide much more coding gain than that in the case of SI = 0.1, and it can be up to 5db. That means that channel coding and interleaving can significantly improve the system performance when SI is big. 2) From the comparison of Fig. 7 and Fig. 8, it is obviously that when SI decreases from 0.25 to 0.1, the SNR requirement at BER = 10⁻⁶ decreases by 5 to 6 db, which means that aperture averaging also contributes a lot to mitigate atmospheric turbulence effects, especially in long distance FSOC systems. 3) Experiment data is also worse than the simulation result for the same reason, and the difference between them is larger than that in the case of SI = 0.1.

From the experimental results, we can conclude that TPC combined with interleaving can significantly improve the FSOC system performance. Though the IDFR value in our experiment only takes 7, interleaving also contributes a lot to combat burst errors.

6. Conclusion

In this paper, we first introduce the outdoor experiment for measuring atmospheric channel, and then the channel model and its parameters are also determined. Based on the channel characteristic, we propose to use TPC as the channel coding scheme, which features good resistance to burst errors and no error floor. Further more, a comparison of RSC, LDPC and TPC has been conducted, and simulation results show that TPC can even provide better performance than some regular LDPC codes. Only channel coding cannot cope with burst errors, interleaving has to be used. Considering the characteristic of the atmospheric channel and the implementation difficulty, block interleaving is used, and the relation between interleaving gain and IDFR using block interleaver is also presented. Finally, an outdoor experimental study of TPC with interleaving is presented, and experimental results also show the excellent performance of TPC combined with interleaving.

Acknowledgement

This work is supported by the National Nature Science Foundation of China (60572002 and 60837004).

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Theoretical and experimental studies of turbo product code with time diversity in free space optical communication

Abstract

1. Introduction

2. Channel fading

3. Theoretical analysis of TPC

3.1 Introduction of channel coding schemes

3.2 Performance comparison

4. Interleaving gain analysis

5. Experimental study of TPC

5.1 Experiment setup

5.2 Experimental results

6. Conclusion

Acknowledgement

References and links

Cited By

Figures (8)

Equations (9)

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