## Synergy of adaptive thresholds and multiple transmitters in free-space optical communication

Optics Express, Vol. 18, Issue 9, pp. 8948-8962 (2010)

http://dx.doi.org/10.1364/OE.18.008948

Acrobat PDF (2398 KB)

### Abstract

Laser propagation through extended turbulence causes severe beam spread and scintillation. Airborne laser communication systems require special considerations in size, complexity, power, and weight. Rather than using bulky, costly, adaptive optics systems, we reduce the variability of the received signal by integrating a two-transmitter system with an adaptive threshold receiver to average out the deleterious effects of turbulence. In contrast to adaptive optics approaches, systems employing multiple transmitters and adaptive thresholds exhibit performance improvements that are unaffected by turbulence strength. Simulations of this system with on-off-keying (OOK) showed that reducing the scintillation variations with multiple transmitters improves the performance of low-frequency adaptive threshold estimators by 1-3 dB. The combination of multiple transmitters and adaptive thresholding provided at least a 10 dB gain over implementing only transmitter pointing and receiver tilt correction for all three high-Rytov number scenarios. The scenario with a spherical-wave Rytov number ^{−5} to 10^{−3}, consistent with the code gain metric. All five scenarios between 0.06 and 0.20 Rytov number improved to within 3 dB of the SNR of the lowest Rytov number scenario.

© 2010 OSA

## 1. Introduction

*can*keep up as they are capable of transmitting at multi-gigabit per second rates [1

1. Staff Writers, “Northrop Grumman Awarded DARPA Contract To Design Hybrid Optical/RF Communications Network,” Spacedaily.com (2008). Dated 5 May 2008, URL http://www.spacedaily.com/reports/Northrop Grumman Awarded DARPA Contract To Design Hybrid Optical RF Communications Network 999.html.

4. R. K. Tyson, “Adaptive optics and ground-to-space laser communications,” Appl. Opt. **35**(19), 3640 (1996). [CrossRef] [PubMed]

5. S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Comm. **21**(8), 1346–1357 (2003). [CrossRef]

7. P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett. **32**(8), 885–887 (2007). [CrossRef] [PubMed]

8. J. A. Louthain, and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express **16**(14), 10,769–10,785 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10769.

## 2. Turbulence conditions

*r*

_{0}, spherical-wave Rytov number

*f*to create the proper phase and amplitude correlation properties. The Rytov number

_{G}*ℛ*is equal to the spherical-wave log-amplitude variance

*σ*

^{2}

_{χ}for weak turbulence and is a common measure of turbulence strength [9]. The parameters were chosen to emulate this air-to-air horizontal scenario with aircraft velocities between 56 and 280 m/s and altitudes between 4 and 15 km. These parameter ranges were chosen based on our previous research that showed a relatively small separation of about 31 cm is required to average atmospheric scintillation effects for a 100-km air-to-air communication link [8

8. J. A. Louthain, and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express **16**(14), 10,769–10,785 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10769.

*r*

_{0},

*f*adequately describe the spatial and temporal turbulence effects, the simulated conditions consist of a one-half fractional factorial design of these three factors. Designing the test in this way enables the determination of the primary driving factors for fades and bit errors. Table 1 summarizes the atmospheric parameters for the simulations. There are five different scenarios with different altitudes, air velocities, sampling times, spherical-wave Rytov numbers

_{G}*f*, and spherical-wave coherence diameters

_{G}*r*

_{0}. Scenarios 3 and 4 are non-physical scenarios and therefore do not have an altitude associated with them. The other three scenarios model horizontal propagation. In this table, the “H” refers to the relatively higher numerical value and the “L” refers to the relatively lower numerical value for each parameter (

*f*, and

_{G}*r*

_{0}). The optical wavelength was

*λ*= 1.55 μm.

### 2.1. Anisoplanatic effects

## 3. Temporal considerations

### 3.1. Frequency of the turbulence effects

*τ*where the turbulence evolves so that the scintillation effects are only slightly different than the previous time slice (time difference for isoplanatic scintillation). These simulations use a conservative estimate of

_{irr}*τ*to ensure they include all potential signal variations. For this case, the log-amplitude structure function value for the time separation used was only 2.2% of the structure function maximum. Using

_{irr}*f*as a reference and varying the temporal sampling frequency of the simulations enables the determination of an adequate sampling rate. The power spectral density (PSD) of signal power was estimated for successively finer temporal resolutions until two sequential estimates were relatively similar from 0 Hz to the frequency at which the PSD is 20 dB below its maximum value. This determination is shown graphically in Fig. 1 . These PSD estimates were consistent for different random realizations. The resulting sampling frequency is

_{G}*f*= 64

_{s}*f*.

_{G}### 3.2. Threshold determination

*H*or

_{1}*H*, respectively. The likelihood ratio test (LRT) determines the optimal decision threshold based upon the probability density function (PDF) of the measured current level

_{0}*i*of the transmission of a ‘1’

_{m}*p*(

*i*|

_{m}*H*

_{1}) and transmission of a ‘0’

*p*(

*i*|

_{m}*H*

_{0}). Using the LRT and the assumption that

*P*(

*H*

_{0}) =

*P*(

*H*

_{1}) (equally likely signaling) leads to the following two relations [15]: ifthe algorithm picks

*H*

_{1}and ifthe algorithm picks

*H*

_{0}. The optimum detection criteria can best be described graphically as the intersection of the PDF of the measurement of the transmission of a ‘1’ and the PDF of the measurement of a ‘0’ transmission. The turbulence conditions vary significantly over time, and thus the receiver performance could benefit from a threshold that varies with the optical signal level [16,17].

#### 3.2.1. Fixed Threshold

*p*(

*s*) due to variations caused by channel conditions. In this case, the channel conditions are dictated by the atmospheric turbulence. As mentioned in the previous section, the optimal threshold depends upon the PDF’s of the measurement of ‘1’ and a ‘0’. The measurement noise of a ‘1’ can be broken into the sum of the thermal, shot, and amplifier noise, defined bywhere

*σ*is the electronic thermal noise,

^{2}_{elec}*σ*

^{2}

*is the shot noise due to the random arrival of photons, and*

_{shot}*σ*is the amplified spontaneous emission (ASE) noise associated with an Erbium-doped fiber amplifier (EDFA) [8

^{2}_{ASE}8. J. A. Louthain, and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express **16**(14), 10,769–10,785 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10769.

*p*(

*s*). To simplify the notation, we let

*p*

_{1}(

*i*) =

_{m}*p*(

*i*|

_{m}*H*

_{1}) and

*p*

_{0}(

*i*) =

_{m}*p*(

*i*|

_{m}*H*

_{0}). Then the LRT for this scenario is [17,19]

*i*(in μA) can be solved for numerically whether the PDF of the turbulence-induced power fluctuations

_{T}*p*(

*s*) is analytic, measured, or calculated from the histogram of the simulated received power before the measurement noise is applied. Since we want to compare the adaptive threshold approaches to the best possible fixed threshold performance, we used the PDF estimate

*p*(

*s*) of the simulated received power. The noise associated with measuring a ‘0’ is primarily due to thermal noise (a.k.a. Johnson noise). The probability of an error

*P*is the probability of a missed detection

_{e}*P*plus the probability of a false alarm

_{md}*P*so thatwhere

_{fa}#### 3.2.2. Adaptive optimal threshold

*ideal*optimal adaptive threshold results in the lowest probability of error for each instant in time [15,16]. Since the threshold is determined for each current level, the PDF of the received signal level

*p*(

*s*) is not required for this calculation. Only the estimates of the means (

*μ*

_{1}and

*μ*

_{0}) and the standard deviations (

*σ*

_{1}and

*σ*

_{0}) of the two conditions are required to set the threshold. Solving for the optimal adaptive threshold current assuming Gaussian distributions for

*p*

_{1}(

*i*) and

_{m}*p*

_{0}(

*i*) yields [17,20

_{m}20. H. Burris, A. Reed, N. Namazi, M. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in *Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’01)*, vol. 4, pp. 2685–2688 (7–11 May 2001).

*μ*

_{0}= 0 (i.e. zero dark current) and

*σ*

_{0}=

*σ*, since

_{elec}*σ*=

_{shot}*σ*= 0 when a ‘0’ is sent. This

_{ASE}*ideal*adaptive threshold system calculates the optimal adaptive threshold for each time slice with the corresponding raw received signal level

*s*in the simulation and implements that threshold to determine whether it is a ‘1’ or a ‘0’. For the adaptive threshold case, the probability of a missed detection and the probability of false alarm now have a threshold that varies with the signal level along with all of the other signal-dependent terms. Accordingly,

*P*becomeswhere the threshold now becomes a function of the received power

_{md}*s*. The

*P*also becomes a function of

_{fa}*s*given by

*realistic*system requires an estimator to determine what threshold

*σ*

_{1}

^{2}is signal-dependent and the signal variation is slow compared to the data rate, the variation in the adaptive threshold is only a function of the signal level for the transmission of a ‘1’. In addition, since the transmission of a ‘1’ or ‘0’ is equally likely, the mean signal level for the transmission of a ‘1’ can be determined by multiplying the mean received signal value

*μ*by two and subtracting the mean signal level of the transmission of a ‘0’ (i.e.

_{rcvd}*i*and the differential signal in the previous two measurements

_{Em-}*i*to determine the estimated signal level. To further refine the estimate, the differential of the measured signal

_{Em–}*i*and

_{Em-}*i*. Figure 2 depicts this process. In these simulations the temperature and bandwidth are constant, so

_{Em–}*σ*

^{2}

*remains constant. The estimated current*

_{elec}*i*is the raw actual estimator value and the noise

_{E-}*n*(

*i*) in the measurement

_{E-}*i*is a zero-mean Gaussian random variable with a variance equal to

_{Em-}*μ*

_{1}in Eq. (9) is set to equal the estimated signal

## 4. Results

### 4.1. Simulation set-up

21. B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE **3125**, 327 (1997). [CrossRef]

22. J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proc. SPIE **3381**, 286–296 (1998). [CrossRef]

*r*

_{0}

*, spherical-wave Rytov number*

_{sph}*ℛ*, and isoplanatic angle

*θ*

_{0}matched within 1% of the full path continuous atmospheric turbulence parameters.

*ρ*= 31 cm. The simulations propagate either one or two collimated Gaussian beams depending upon the Tx configuration with a 1/

_{c}*e*field radius of

*W*

_{0}= 2.5 cm using a split-step Fresnel propagation to a 20 cm diameter receiver aperture. Great care was taken to adequately sample the Fresnel propagation between the screens to avoid aliasing in the beam as well as the quadratic phase term [23]. At the Rx, the light is coupled into a single-mode fiber to be amplified by an EDFA with a spontaneous emission factor of

*n*= 4 and a gain of 30, factoring into the ASE noise σ

_{sp}^{2}

_{ASE}[18]. The simulations modeled the fiber coupling by projecting the field onto the guided mode of a single-mode 4 μm-radius fiber. For this single mode fiber, the efficiency of the coupling was modeled by the LP01 mode field using the Bessel functions of the first and second kind. The mode field diameter of the fiber was 10.5 μm with an index of refraction of

*n*= 1.45 and a V number of 2.405. See Refs [24]. and [25] for a description of the calculation of the coupling of the fundamental guided mode.

_{1}### 4.2. Bit error rate (BER) fade statistics

**16**(14), 10,769–10,785 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10769.

26. J. A. Louthain and J. D. Schmidt, “Integrated approach to airborne laser communication,” Proc. SPIE **7108**(14), (2008). [CrossRef]

^{−3}. Since we are comparing the performance of an adaptive threshold to a fixed threshold, we chose to use this definition to accurately illustrate the advantages of our hybrid technique.

*f*case in Fig. 6. All three adaptive threshold systems performed much better than the fixed threshold system. The higher-fidelity estimator with

_{G}*f*= 64

_{s}*f*performed almost as well as the ideal adaptive threshold system. The low-frequency adaptive threshold estimator performed worse than the high-frequency estimator in almost all cases. The two estimators did perform comparably for an SNR above 20 dB for the two transmitter scenario 4 (HHH) case. Since the scintillation indicated by

_{G}*ℛ*= 0.1979 for this scenario (HHH) was higher than the other four scenarios, the improvement in the low-frequency case was most likely due to the reduction in scintillations when using two transmitters. The peaks in each of the fade rates for Figs. 4–6 can be graphically explained by varying the threshold for a particular signal like the one shown in Fig. 3. As the threshold successively drops, more and more fades are encountered until the fades get so long that they merge to make longer fades, thereby reducing the number of threshold crossings and number of fades. In each case, as the SNR decreases, the mean fade length increases.

### 4.3. Probability density function (PDF) estimates of the received signal

*p*(

*i*) with variations caused by atmospheric turbulence are used to estimate the PDF of the received signal. These PDF’s were compared with their double-transmitter cases for all five scenarios listed in Table 1. If the PDF estimate is heavily weighted to the left, the chances of a missed detection are greater, as it might not reach above the threshold. Figure 7 shows that the PDF’s of the received signal for all of the scenarios shifted to the right when two transmitters were used. Even for the low-

_{s}*ℛ*cases, the PDF's markedly shifted to the right thereby improving performance. This shift to the right reduces the probability of error specifically by reducing the probability of a missed detection. It also shifts the optimal threshold to the right, thereby reducing the probability of a false alarm. These performance improvements are quantified with the BER calculations in the next subsection.

### 4.4. Bit error rate (BER)

*p*(

*i*) over the ensemble of the runs calculated [see Eq. (5)]. There were 10 independent realizations with 1000 time slices for each realization. The time increments were determined by

_{s}*τ*= 1/(64

_{s}*f*) for each of the scenarios. Therefore, each independent realization covered a time frame of over 15 Greenwood time constants, resulting in well over 150 relatively independent realizations per scenario.

_{G}*optimal*fixed threshold case for the particular scenario which used the actual PDF of the received signal to determine the optimal threshold. This

*a priori*knowledge of the turbulence resulted in an optimistic BER for the fixed threshold. In most cases, the fixed threshold is not chosen in such an accurate manner. The double-Tx systems outperformed all other techniques even though improvements due to the adaptive threshold technique were up to 5 dB. As expected, the system with an ideal adaptive threshold and two transmitters performed the best.

*f*= 64

_{s}*f*, and another system with an estimator operating at

_{G}*f*= 16

_{s}*f*. For a single transmitter, the performance for the

_{G}*f*= 16

_{s}*f*estimator was the poorest for the highest-Rytov case in scenario 4 (HHH). For this and all other cases, this lower-sampling-rate estimator performance greatly improved when two transmitters were implemented. The single-transmitter cases have more variability in the received irradiance and require a higher fidelity estimator to keep up with the turbulence. This trickle-down effect indicates multiple transmitters can enable the use of cheaper, lower-sampling-rate estimators.

_{G}## 5. Conclusion

*r*

_{0}was greater than the diameter of the receiver

*D*, therefore changes in the Rytov number had a much larger effect than

*r*

_{0}changes. If

*D*/

*r*

_{0}

*>*1, the phase effects due to the turbulence would likely have had a larger effect on the BER.

*ρ*results in diminishing improvement [8

_{c}**16**(14), 10,769–10,785 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10769.

## Acknowledgements

## References and links

1. | Staff Writers, “Northrop Grumman Awarded DARPA Contract To Design Hybrid Optical/RF Communications Network,” Spacedaily.com (2008). Dated 5 May 2008, URL http://www.spacedaily.com/reports/Northrop Grumman Awarded DARPA Contract To Design Hybrid Optical RF Communications Network 999.html. |

2. | R. K. Tyson, J. S. Tharp, and D. E. Canning, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 2: multichannel configuration, aberration characterization, and closed-loop results,” Opt. Eng. |

3. | A. Belmonte, “Influence of atmospheric phase compensation on optical heterodyne power measurements,” Opt. Express |

4. | R. K. Tyson, “Adaptive optics and ground-to-space laser communications,” Appl. Opt. |

5. | S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Comm. |

6. | E. J. Lee and V. W. S. Chan, “Part 1: Optical Communication Over the Clear Turbulent Atmospheric Channel Using Diversity,” IEEE J. Sel. Areas Comm. |

7. | P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett. |

8. | J. A. Louthain, and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express |

9. | R. J. Sasiela, |

10. | J. A. Louthain, and J. D. Schmidt, “Anisoplanatic Approach to Airborne Laser Communication,” Meeting of the Military Sensing Symposia (MSS) Specialty Group on Active E-O Systems |

11. | D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A |

12. | L. C. Andrews, and R. L. Phillips, |

13. | J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. |

14. | M. C. Roggemann, and B. M. Welsh, |

15. | H. L. VanTrees, |

16. | H. Burris, A. Reed, N. Namazi, W. Scharpf, M. Vicheck, M. Stell, and M. Suite, “Adaptive thresholding for free-space optical communication receivers with multiplicative noise,” in |

17. | P. N. Crabtree, |

18. | S. B. Alexander, |

19. | J. D. Schmidt, |

20. | H. Burris, A. Reed, N. Namazi, M. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in |

21. | B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE |

22. | J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proc. SPIE |

23. | S. Coy, “Choosing Mesh Spacings and Mesh Dimensions for Wave Optics Simulation,” Proc. SPIE |

24. | J. A. Louthain, |

25. | J. A. Buck, |

26. | J. A. Louthain and J. D. Schmidt, “Integrated approach to airborne laser communication,” Proc. SPIE |

27. | J. D. Schmidt, and J. A. Louthain, “Integrated approach to free-space optical communication,” in |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

(060.2605) Fiber optics and optical communications : Free-space optical communication

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: February 18, 2010

Revised Manuscript: March 29, 2010

Manuscript Accepted: April 7, 2010

Published: April 14, 2010

**Citation**

James A. Louthain and Jason D. Schmidt, "Synergy of adaptive thresholds and multiple transmitters in free-space optical communication," Opt. Express **18**, 8948-8962 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-8948

Sort: Year | Journal | Reset

### References

- Staff Writers, “Northrop Grumman Awarded DARPA Contract To Design Hybrid Optical/RF Communications Network,” Spacedaily.com (2008). Dated 5 May 2008, URL http://www.spacedaily.com/reports/Northrop Grumman Awarded DARPA Contract To Design Hybrid Optical RF Communications Network 999.html.
- R. K. Tyson, J. S. Tharp, and D. E. Canning, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 2: multichannel configuration, aberration characterization, and closed-loop results,” Opt. Eng. 44(9), (2005).
- A. Belmonte, “Influence of atmospheric phase compensation on optical heterodyne power measurements,” Opt. Express 16(9), 6756–6767 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-9-6756
- R. K. Tyson, “Adaptive optics and ground-to-space laser communications,” Appl. Opt. 35(19), 3640 (1996). [CrossRef] [PubMed]
- S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003). [CrossRef]
- E. J. Lee and V. W. S. Chan, “Part 1: Optical Communication Over the Clear Turbulent Atmospheric Channel Using Diversity,” IEEE J. Sel. Areas Comm. 22(9), 1896–1906 (2004). [CrossRef]
- P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett. 32(8), 885–887 (2007). [CrossRef] [PubMed]
- J. A. Louthain, and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express 16(14), 10,769–10,785 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10769 .
- R. J. Sasiela, Electromagnetic wave propagation in turbulence. Evaluation and application of Mellin transforms, 2nd Ed. (SPIE Publications, 2007).
- J. A. Louthain, and J. D. Schmidt, “Anisoplanatic Approach to Airborne Laser Communication,” Meeting of the Military Sensing Symposia (MSS) Specialty Group on Active E-O Systems I(AD02), 1–20 (2007).
- D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A 72(1), 52–61 (1982). [CrossRef]
- L. C. Andrews, and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE Optical Engineering Press Bellingham, WA, 2005).
- J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. 46(26), 6561–6571 (2007). [CrossRef] [PubMed]
- M. C. Roggemann, and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
- H. L. VanTrees, Detection, estimation, and modulation theory (Wiley, 2002).
- H. Burris, A. Reed, N. Namazi, W. Scharpf, M. Vicheck, M. Stell, and M. Suite, “Adaptive thresholding for free-space optical communication receivers with multiplicative noise,” in Proc. IEEE Aerospace Conference, vol. 3, pp. 1473–1480 (2002).
- P. N. Crabtree, Dissertation:Performance-Metric Driven Atmospheric Compensation for Robust Free-Space Laser Communication (Air Force Institute of Technology, Wright-Patterson AFB, OH, 2006).
- S. B. Alexander, Optical Communication Receiver Design, SPIE Tutorial Texts in Optical Engineering, vol. TT22; IEE Telecommunications Series, vol. 37 (SPIE Press, Bellingham, WA, 1997).
- J. D. Schmidt, Dissertation: Free-Space Optical Communications Performance Enhancement by Use of a Single Adaptive Optics Correcting Element (University of Dayton, Dayton, OH, 2006).
- H. Burris, A. Reed, N. Namazi, M. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’01), vol. 4, pp. 2685–2688 (7–11 May 2001).
- B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327 (1997). [CrossRef]
- J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proc. SPIE 3381, 286–296 (1998). [CrossRef]
- S. Coy, “Choosing Mesh Spacings and Mesh Dimensions for Wave Optics Simulation,” Proc. SPIE 5894 (2005).
- J. A. Louthain, Dissertation: Integrated approach to airborne laser communication, Air Force Institute of Technology, Wright-Patterson AFB, OH, December 2008.
- J. A. Buck, Fundamentals of Optical Fibers, Wiley-Interscience, 2004.
- J. A. Louthain and J. D. Schmidt, “Integrated approach to airborne laser communication,” Proc. SPIE 7108(14), (2008). [CrossRef]
- J. D. Schmidt, and J. A. Louthain, “Integrated approach to free-space optical communication,” in Proc. SPIE, Optics in Atmospheric Propagation and Adaptive Systems XI, vol. 7200 (SPIE Press, Bellingham, WA, 2009).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.