## LDPC-coded MIMO optical communication over the atmospheric turbulence channel using *Q*-ary pulse-position modulation

Optics Express, Vol. 15, Issue 16, pp. 10026-10032 (2007)

http://dx.doi.org/10.1364/OE.15.010026

Acrobat PDF (136 KB)

### Abstract

We describe a coded power-efficient transmission scheme based on *repetition* MIMO principle suitable for communication over the atmospheric turbulence channel, and determine its *channel capacity*. The proposed scheme employs the Q-ary pulse-position modulation. We further study how to approach the channel capacity limits using low-density parity-check (LDPC) codes. Component LDPC codes are designed using the concept of pairwise-balanced designs. Contrary to the several recent publications, bit-error rates and channel capacities are reported assuming that p.i.n. photodetectors are used instead of ideal photon-counting receivers. The atmospheric turbulence channel is modeled using the Gamma-Gamma distribution function due to Al-Habash *et al*. Excellent bit-error rate performance improvement, over uncoded case, is found.

© 2007 Optical Society of America

## 1. Introduction

*multi-laser multi-detector*(MLMD) concept [2

2. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. **53**, 1402–1412 (2005). [CrossRef]

3. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. **23**, 1901–1910 (2005). [CrossRef]

6. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. **18**, 1491–1493 (2006). [CrossRef]

2. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. **53**, 1402–1412 (2005). [CrossRef]

3. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. **23**, 1901–1910 (2005). [CrossRef]

6. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. **18**, 1491–1493 (2006). [CrossRef]

*ideal photon-counting receiver*. The recent paper due to Cvijetic

*et al*. [7], is an exception form this common practice. The simulation results reported in [5,6

6. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. **18**, 1491–1493 (2006). [CrossRef]

*non-ideal photodetection*based on p.i.n. photodiodes?” The purpose of the paper is twofold: (i) to determine the channel capacity of a coded MIMO FSO system in the presence of atmospheric turbulence, and (ii) to see how much this limit can be approached using the best known coding techniques.

8. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory **28**, 55–67 (1982). [CrossRef]

9. I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. **12**, 555–562 (2006). [CrossRef]

10. J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking **4**, 586–601 (2005). [CrossRef]

12. G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory **44**, 927–946 (1998). [CrossRef]

**18**, 1491–1493 (2006). [CrossRef]

13. H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory **IT-23**, 371–377 (1977). [CrossRef]

4. M. A. Al-Habash, L.C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. **40**, 1554–1562 (2001). [CrossRef]

## 2. MIMO concept, channel description, and channel capacity

*M*laser sources and

*N*photodetectors are employed [2

2. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. **53**, 1402–1412 (2005). [CrossRef]

3. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. **23**, 1901–1910 (2005). [CrossRef]

**18**, 1491–1493 (2006). [CrossRef]

14. K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. **4**, 35–39 (2005). [CrossRef]

15. V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory **45**, 1456–1467 (1999). [CrossRef]

*incoherent*superposition of the transmitted signals.

**18**, 1491–1493 (2006). [CrossRef]

**18**, 1491–1493 (2006). [CrossRef]

*L*RF/microwave sources are multiplexed together and encoded using an (

*n*,

*k*) LDPC code of code rate

*r*=

*k*/

*n*(

*k*-the number of information bits,

*n*-the codeword length). The

*m*×

*n*block-interleaver, collects

*m*code-words written row-wise. The mapper accepts

*m*bits at a time from the interleaver column-wise and determines the corresponding slot for Q-ary (

*Q*=2

*) PPM signaling using a*

^{m}*Gray mapping*rule. With this BICM scheme, the neighboring information bits from the same source are allocated into different PPM symbols. In each signaling interval

*T*a pulse of light of duration

_{s}*T*=

*T*/

_{s}*Q*is transmitted by a laser. (The signaling interval

*T*is subdivided into

_{s}*Q*slots of duration

*T*.) The total transmitted power

*P*is fixed and independent of the number of lasers so that emitted power per laser is

_{tot}*P*/

_{tot}*M*. This technique improves the tolerance to atmospheric turbulence, because different Q-ary PPM symbols experience different atmospheric turbulence conditions. The

*i*th (

*i*=1,2,…,

*M*) laser modulated beam is projected toward the

*j*th (

*j*=1,2,…,

*N*) receiver using the expanding telescope, and the receiver is implemented based on a p.i.n. photodetector in a trans-impedance amplifier (TA) configuration. Notice that the MLC scheme employs different (

*n*,

*k*) LDPC codes (

_{i}*k*-dimensionality of

_{i}*i*th component code), and it is able to carry Σ

*k*/

_{i}*n*bits per symbol, which is generally smaller than

*m*. Therefore the spectral efficiency (expressed in bits/symbol) of the BICM scheme is higher. Moreover, the BICM scheme employs only one LDPC code for all RF/microwave users, which simplifies the implementation. The use of only one LDPC code allows iterating between the

*a posteriori*probability (APP) demapper and the LDPC decoder (we will call this step the

*outer iteration*), further improving the BER performance.

*N*receivers in response to symbol

*q*, denoted as

*Z*

_{n,q}(

*n*=1,2,…,

*N*;

*q*=1,2,…,

*Q*), are processed to determine the symbol reliabilities λ(

*q*) (

*q*=1,2,…,

*Q*) by

*I*is the signal intensity,

*Γ*(·) is the gamma function, and

*K*

_{α-β}(·) is the modified Bessel function of the second kind and order

*α*-

*β*.

*α*and

*β*are PDF parameters describing the scintillation experienced by plane waves, and in the case of zero-inner scale are given by [4

4. M. A. Al-Habash, L.C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. **40**, 1554–1562 (2001). [CrossRef]

*σ*

_{R}^{2}is the Rytov variance given by

*k*=2

*π*/λ is the optical wave number,

*L*is propagation distance, and

*C*

_{n}^{2}is the refractive index structure parameter, which we assume to be constant for horizontal paths.

*L*(

*c*), (

_{j}*j*=1,2,…,

*m*) (

*c*is the

_{j}*j*th bit in observed symbol

*q*binary representation

*c*=(

*c*

_{1},

*c*

_{2},…,

*c*)) are determined from symbol reliabilities by

_{m}8. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory **28**, 55–67 (1982). [CrossRef]

**=(**

*Z**Z*

_{n,q})

_{NxQ}(

*Z*

_{n,q}-the nth receiver response to

*q*th symbol) the channel capacity can be determined by

*q*

_{0}is an arbitrary symbol, and with E[·] we denoted the operator of ensemble averaging (other parameters are introduced earlier). Because we assumed equally probable transmission (Pr(

*q*)=1/

*Q*,

*q*=1,..,

*Q*), the averaging over different symbols in (6) will not affect the result. Notice that ensemble averaging is to be done for different channel conditions (

*I*̄

*) and for different thermal noise realizations (*

_{n}**|**

*Z**I*̄

*) by using Monte Carlo simulations, and taking into account the fact that conditional probability density function*

_{n}*p*(

*Z*

_{n,q}|

*I*̄

*) is Gaussian*

_{n}## 3. PBD based LDPC codes

*v*,

*K*,{0,1,…,λ}) is a collection of subsets (blocks) of a

*v*-set

*V*with a size of each block

*k*∊

_{i}*K*(

*k*≤

_{i}*v*), so that each pair of elements occurs in

*at most*λ of the blocks. (Notice that we have relaxed the constraint in definition of PBD from [11] by replacing the word

*exact*with

*at most*.) For example, the following blocks: {1,6,9}, {2,7,10}, {3,8,11}, {4,12}, {5,13}, {1,7,11}, {2,8,12},{3,13}, {1,8,13}, {2,9}, {3,10}, {4,6,11}, {5, 7,12}, {1,10}, {2,11}, {3,6,12}, {4,7,13}, {5,8,9}, {1,12}, {2,6,13}, {3,7,9}, {4,8,10}, and {5,11} create an PBD(13,{2,3},{0,1}) (having 9 blocks of size 2, and 14 blocks of size 3), with parameter λ=1. By considering elements of blocks as position of ones in corresponding element-block incidence matrix, a parity-check matrix of an equivalent

*irregular*LDPC code of girth-6 is obtained.

## 4. Numerical results

*σ*

_{R}=3.0,

*α*=5.485,

*β*=1.1156) for different number of lasers, photodetectors, and number of slots are given in Fig. 2. The significant spectral efficiency improvement is possible by using the multi-level schemes, such as

*Q*-ary pulse position modulation.

*σ*

_{R}=3.0,

*α*=5.485,

*β*=1.1156) are shown in Fig. 3, for different number of lasers, photodetectors and number of slots, by employing an (6419,4794) irregular girth-6 LDPC code of rate 0.747 designed using the concept of PBD introduced in Section 3. The MLC scheme with spectral efficiency of 2.241 bits/symbol combined with MLMD scheme employing 2 lasers and 4 photodetectors provides about 21dB improvement over LDPC coded binary PPM employing one laser and one photodetector. Corresponding BICM scheme of higher spectral efficiency (3bits/symbol) provides about 20dB improvement. The number of inner iteration in message-passing LDPC decoder is set to 25 in both schemes, while the number of outer iterations in BICM scheme is set to 10. MLC employs a parallel-independent LDPC decoding as explained in [6

**18**, 1491–1493 (2006). [CrossRef]

14. K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. **4**, 35–39 (2005). [CrossRef]

15. V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory **45**, 1456–1467 (1999). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and Links

1. | H. Willebrand and B.S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks: Sams Publishing, 2002. |

2. | S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. |

3. | S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. |

4. | M. A. Al-Habash, L.C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. |

5. | I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel,” in Proc. OFC 2007, Paper no. JThA46, March 25–29, 2007, Anaheim, CA, USA. |

6. | I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. |

7. | N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. |

8. | G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory |

9. | I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. |

10. | J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking |

11. | I. Anderson, |

12. | G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory |

13. | H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory |

14. | K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. |

15. | V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory |

16. | I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express |

17. | I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007. |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 21, 2007

Revised Manuscript: July 20, 2007

Manuscript Accepted: July 20, 2007

Published: July 25, 2007

**Citation**

Ivan B. Djordjevic, "LDPC-coded MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation," Opt. Express **15**, 10026-10032 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10026

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### References

- H. Willebrand, and B.S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks: Sams Publishing, 2002.
- S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402-1412 (2005). [CrossRef]
- S. G. Wilson, M. Brandt-Pearce, Q. Cao, M. Baedke, "Optical repetition MIMO transmission with multipulse PPM," IEEE Selected Areas Comm. 23, 1901-1910 (2005). [CrossRef]
- M. A. Al-Habash, L.C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001). [CrossRef]
- I. B. Djordjevic, B. Vasic, and M. A. Neifeld, "Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel," in Proc. OFC 2007, Paper no. JThA46, March 25-29, 2007, Anaheim, CA, USA.
- I. B. Djordjevic, B. Vasic, M. A. Neifeld, "Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel," IEEE Photon. Tehnol. Lett. 18, 1491-1493 (2006). [CrossRef]
- N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, "Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM," IEEE Photon. Tehnol. Lett. 19, 1491-1493 (2007).
- G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE. Trans Inf. Theory 28, 55-67 (1982). [CrossRef]
- I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006). [CrossRef]
- J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, "Shannon capacities and error-correction codes for optical atmospheric turbulent channels," J. Opt. Networking 4, 586-601 (2005). [CrossRef]
- I. Anderson, Combinatorial Designs and Tournaments: Oxford University Press, 1997.
- G. Caire, G. Tarrico, E. Biglieri, "Bit-interleaved coded modulation," IEEE. Trans Inf. Theory 44, 927-946 (1998). [CrossRef]
- H. Imai, and S. Hirakawa, "A new multilevel coding method using error correcting codes," IEEE. Trans Inf. Theory IT-23, 371-377 (1977). [CrossRef]
- K. Simon, and V.A Vilnrotter, "Alamouti-type space-time coding for free-space optical communication with direct detection," IEEE Trans. Wireless Comm. 4, 35 - 39 (2005). [CrossRef]
- V. Tarokh, H. Jafarkani, and A. R. Calderbank, "Space-time block codes from orthogonal designs," IEEE. Trans Inf. Theory 45, 1456-1467 (1999). [CrossRef]
- I. B. Djordjevic, B. Vasic, and M. A. Neifeld, "LDPC coded OFDM over the atmospheric turbulence channel," Opt. Express 15, 6332-6346 (2007).
- I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, "LDPC-Coded MIMO optical communication over the atmospheric turbulence channel," accepted for presentation at Globecom 2007.

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