## Efficiency of complex modulation methods in coherent free-space optical links

Optics Express, Vol. 18, Issue 4, pp. 3928-3937 (2010)

http://dx.doi.org/10.1364/OE.18.003928

Acrobat PDF (303 KB)

### Abstract

We study the performance of various binary and nonbinary modulation methods applied to coherent laser communication through the turbulent atmosphere. We compare the spectral efficiencies and SNR efficiencies of complex modulations, and consider options for atmospheric compensation, including phase correction and diversity combining techniques. Our analysis shows that high communication rates require receivers with good sensitivity along with some technique to mitigate the effect of atmospheric fading.

© 2010 OSA

## 1. Introduction

*M*-ary modulation techniques with coherent detection, including phase-shift keying (PSK), quadrature-amplitude modulation (QAM), and pulse-amplitude modulation (PAM). The parameter

*M*is the number of points in the signal constellation and, consequently, log

_{2}

*M*describes the number of coded bits per symbol. We also consider binary differential phase-shift keying (2-DPSK or DBPSK) with differentially coherent detection. As we are primarily interested in the high-spectral-efficiency regime, we do not consider orthogonal modulation formats, such as pulse-position modulation (PPM) or frequency-shift keying (FSK).

## 2. Statistical model of optical heterodyne detection

*P*and noise power spectral density

*N*

_{0}/2,

*γ*

_{0}

*=P/N*

_{0}

*B*is the SNR per symbol per unit bandwidth

*B*. The SNR per symbol

*γ*

_{0}can be interpreted as the detected number of photons (photocounts) per symbol when 1/

*B*is the symbol period. Coherently detected signals are modeled as narrowband RF signals with additive white Gaussian noise. In free-space optical communication through the turbulent atmosphere, we must consider fading channels, which are a class of channels with multiplicative noise. In the fading AWGN channel, we let

*α*

^{2}denote the atmospheric channel power fading and (

*P/N*

_{0}

*B*)

*α*

^{2}=

*γ*

_{0}

*α*

^{2}denote the instantaneous received SNR per symbol. Now, the SNR can be taken as the number of signal photons detected on the receiver aperture

*γ*

_{0}multiplied by a heterodyne mixing efficiency

*α*

^{2}. When, due to the impact of atmospheric turbulence-induced phase and amplitude fluctuations, the spatial field of the received signal does not properly matches that of the local oscillator, the contributions to the current signal from different parts of the receiver aperture can interfere destructively and result in a reduced instantaneous mixing efficiency, causing fading.

*α*

^{2}[5

5. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**(18), 14151–14162 (2008). [CrossRef] [PubMed]

6. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express **17**(15), 12601–12611 (2009). [CrossRef] [PubMed]

*σ*

_{β}^{2}=0 in all results presented in the following paragraphs.

*γ=γ*

_{0}

*α*

^{2}. The statistical properties of the atmospheric random channel fading

*α*

^{2}provide a statistical description of the SNR

*γ*. We have previously modeled the impact of atmospheric turbulence-induced phase and amplitude fluctuations on free-space optical links using synchronous detection and found that the SNR γ at the output of a perfect

*L*-element diversity coherent combiner in the atmosphere would be described by a noncentral chi-square distribution with 2

*L*degrees of freedom [5

5. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**(18), 14151–14162 (2008). [CrossRef] [PubMed]

6. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express **17**(15), 12601–12611 (2009). [CrossRef] [PubMed]

*I*

_{L}_{−1}(∙) is the (

*L*–1)-order modified Bessel function of the first kind. The average SNR (or average detected photocounts) per symbol

*/r*describe turbulence effects. Both

*/r*are described in terms of the amplitude and phase variances

*σ*

_{χ}^{2}and

*σ*

_{f}^{2}in Eq. (1) [5

5. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**(18), 14151–14162 (2008). [CrossRef] [PubMed]

*N*different coherent regions within one aperture. In this model, the signal is characterized as the sum of a constant (coherent) term and a random (incoherent) residual halo. The contrast parameter 1

*/r*is a measure of the strength of the residual halo relative to the coherent component. The parameter

*r*ranges between 0 and ∞. It can be shown that when the constant term is very weak (

*r*→0), turbulence fading causes the SNR to become gamma-distributed, just as in a speckle pattern [7]. Likewise, when the dominant term is very strong (

*r*→∞), the density function becomes highly peaked around the mean value

*L*independent branch signals and equal average branch SNR per symbol

*L*diversity branches by the complex conjugates of their respective fading gains and sum them. By setting

*L*= 1, the PDF in Eq. (2) describes the SNR

*γ*for a single receiving branch and corresponds to a noncentral chi-square probability with two degrees of freedom [1

1. J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. **10**(1), 59–70 (1975). [CrossRef]

1. J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. **10**(1), 59–70 (1975). [CrossRef]

*γ*, which it is defined by as the expected value of

*p*(

_{γ}*γ*) with the argument

*γ*reversed in sign.

## 3. Spectral and power efficiencies of coherent modulation formats

*γ*may be approximated by [9]:Here,

*a*and

*b*depend on the modulation type. In this approach,

*a*describes the number of nearest neighbors to a constellation point at the minimum distance, and

*b*is a constant that relates minimum distance to average symbol energy. In Table 1 , we summarize the specific values of

*a*and

*b*for the

*M*-ary coherent modulations PSK, QAM, and PAM considered in this analysis. A slightly different formulation should be used with differential PSK [9].

*γ*varies randomly and, consequently, the SER

*p*(

_{s}*E|γ*) conditioned on the SNR

*γ*is also random. The performance metrics depends on the rate of change of the fading and on the average SER. The unconditional SER

*p*(

_{s}*E*) of an ideal coherent receiver in the presence of fading must be obtained by averaging the AWGN conditional SER

*p*(

_{s}*E|γ*) in Eq. (4) where

*p*(

_{γ}*γ*) is the PDF of the instantaneous fading SNR in Eq. (2). Our goal is to evaluate the various coherent modulation schemes described earlier for our atmospheric fading channel. An MGF-based approach is quite useful in simplifying the analysis [10

10. M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis of digital communications over generalized fading channels,” IEEE Proc. **86**(9), 1860–1877 (1998). [CrossRef]

*M*(

*s*) is defined by Eq. (3). Although this result cannot be put in a closed form, we are able to carry out the integration in Eq. (6) using a simple Gaussian-Legendre quadrature formula, which yields high accuracy. The MGF approach can also be applied to

*M*-ary differential PSK through a somewhat more complicated formulation [9, 11]. Note that performance specifications are generally more concerned with the bit-error ratio (BER)

*p*(

_{s}*E*)/log

_{2}

*M*as a function of the photons-per-bit (SNR per bit) γ/log

_{2}

*M*. Here, for

*M*-ary signaling, we make the typical assumption that the symbol energy is divided equally among all bits, and that Gray encoding is used so that, at reasonable SNRs, one symbol error corresponds to exactly one bit error.

*M*=2, PAM and PSK are identical modulation formats. As it should be expected, in AWGN the BER decreases exponentially with increasing photons per bit. However, under atmospheric fading, when no compensation techniques are considered, the BER decreases more gradually with increasing average photons per bit. We see that it requires approximately 4 photons per bit (6 dB SNR) to maintain a 10

^{−3}bit error rate in AWGN while it requires more than 1000 photons per bit (larger than 30 dB SNR) to maintain the same error rate in fading when no atmospheric compensation techniques are considered. It is clear from these plots that to maintain good receiver sensitivity requires some technique to mitigate the effect of atmospheric fading. In these plots, turbulence is characterized by a moderate phase coherence length

*r*

_{0}such that

*D*/

*r*

_{0}=4. When phase correction is applied in Fig. 1(a), the compensation of just 6 modes (phase aberrations are compensated up to astigmatism) brings the number of photons required to maintain the 10

^{−3}BER down to 9 (9.5 dB SNR). When we consider MRC diversity combining of the received signal, using 8 independent branches (Fig. 1(b)) also reduces the power to acceptable levels (about 12 photons per bit, less than 11 dB SNR).

*M*=2, 4, 8, 16; note that theoretically 2- and 4-PSK have similar BER performance). For modulations QAM PAM, and DPSK, just the high-order 16-ary formats have been considered in the figure. Also, note that 4-QAM is equivalent to 4-PSK. In all modulations analyzed, we have assumed that either turbulence-induced phase aberrations have been compensated up to

*J*=20 (5th-order aberrations) (Fig. 2(a)), or a 12-branch MRC combining has been used (Fig. 2b). The plots illustrate the photon efficiency for the various binary and nonbinary modulation formats using coherent detection.

*B*, and the number of points in the signal constellation is

*M*, the spectral efficiency is defined aswhere

*R*is the bit rate in bits/s,

_{b}*R*is the symbol rate in symbols/s, and unitless parameter

_{s}*R*≤1 is the rate of an error-correction encoder that is used to add redundancy to the signal in order to improve the photon efficiency. The uncoded modulations considered in this analysis correspond to

_{c}*R*=1. In all situations, prevention of intersymbol interference requires

_{c}*R*≤

_{s}*B*. Without a loss of generality, we assume the ideal case where

*R*=

_{s}*B*(1/

*B*is the symbol period), and use the number of coded bits per symbol log

_{2}

*M*as the figure of merit for spectral efficiency. Figure 3 compares spectral efficiency log

_{2}

*M*and photon per bit requirements for 10

^{−9}BER. It clearly illustrates the tradeoff between spectral efficiency and photon efficiency of various binary and nonbinary modulation formats applied to coherent laser communication through the turbulent atmosphere. A similar spectral efficiency analysis was done for fiber communications in DWDM transmission systems [12

12. J. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. **10**(2), 259–272 (2004). [CrossRef]

*S*=log

_{2}

*M*below 1 bit/s/Hz, 2-PAM and 2-DPSK are attractive techniques. They are simple to implement and lead to the best photon efficiencies in terms of photons per bit required for 10

^{−9}BER. Between 1 and 2 bits/s/Hz, 4-DPSK and 4-PSK are perhaps the most interesting modulation formats. At spectral efficiencies above 2 bits/s/Hz, 8-PSK and 16-QAM becomes the most appealing modulations. In general, the performance of 8-QAM and 8-PSK are very similar because the mean energy of the constellation is just slightly different in both modulations. The complicating factor is that the 8-QAM points are no longer all the same amplitude and so the demodulator must now correctly detect both phase and amplitude, rather than just phase, and consequently 8-PSK is a better choice. For data-rates beyond those of 8-PSK and 8-QAM, it is better to consider QAM since it attains a greater distance between contiguous points in the I-Q plane by distributing the points more squarely. In Fig. 3(a), the 150 photon-per-bit required for 16-QAM with phase compensating up to astigmatism (

*J*=6) is better than the near 400-photons-per-bit requirement for the equivalent 16-PSK, and is even better than the near 200-photons-per-bit required for 16-PSK with a higher phase compensation (

*J*=10). With

*J*=6, 16-QAM offers a benefit of more than 4 dB over 16-PSK.

^{−9}BER as a function of data-rate for various turbulence conditions and several degrees of atmospheric compensation. Heterodyne detection of BPSK provides among the best theoretical receiving sensitivity, it is spectrally efficient, and it is easily demodulated by using a balanced receiver. In Fig. 4, lines of constant power, representing notional laser communications link budgets, have been estimated for a 1550-nm working wavelength.

*D*/

*r*

_{0}=2, and for a nominal –50 dBm power budget, a high-rate 1-Gbit/s link with ~3 dB margin can be achieved using an 8-branch (

*L*=8) MRC combiner. Receiver sensitivity, just 3 dB over the AWGN limit of 18-photons-per-bit at 10

^{−9}BER, is high. The excess margin could be used to deal with the stronger turbulence condition considered in the figure, where

*D*/

*r*

_{0}=4, or used to increase the data rate up to 2 Gbit/s with less than 1 dB margin. Under the same initial turbulence conditions

*D*/

*r*

_{0}=2, a receiver phase-compensated up to astigmatism (a moderate

*J*=6) can achieve a 1-Gbit/s link with ~5 dB margin. A high receiver sensitivity just ~1.5 dB over the AWGN limit is expected. Now, the excess margin can be traded either for stronger turbulence conditions

*D*/

*r*

_{0}=4, while retaining a ~2 dB margin, or for an increase of the data rate up to 3 Gbit/s.

*L*=8 to

*L*=4, or phase compensation is limited to a modal expansion up to tilt (

*J*=3, instead of the previous

*J*=6), only low-data-rate links are possible. This can be easily understood in Fig. 4, as increasing the number of combiner branches

*L*from 4 to 8 grants a diversity gain of ~9 dB. Similarly, increasing the order

*J*of the phase compensation from 3 to 6 provides a net modal gain of ~7 dB. Now, for the same nominal –50 dB power link budget, and under turbulence conditions such as

*D*/

*r*

_{0}=2, the MRC 4-branch combiner can only support a medium-rate 200-Mbit/s link with ~1 dB margin. (If stronger turbulence conditions, such as those described by

*D*/

*r*

_{0}=4, are considered, the BPSK 4-branches MRC combiner suffers and only a low-rate 30-Mbit/s link with similar ~1 dB margin is supported). Note that these small 1-dB excess margins may become necessary to compensate for possible scintillation fading: Although in most situations considered, phase distortion becomes the dominant effect on the coherent performance of MRC combiners, the penalty due to amplitude fluctuations may still be relevant. For example, as seen in Fig. 4, for the BPSK 4-branches MRC combiner, atmospheric conditions leading to a scintillation index σ

_{β}

^{2}= 1 yield a performance scintillation penalty of roughly 1.3 dB.

## 4. Conclusion

*L*degrees of freedom, which we introduced recently as a model for atmospheric fading in a single coherent receiver affected by amplitude and phase fluctuations, is used to study the performance of various binary and nonbinary modulation methods for coherent laser communication through the turbulent atmosphere. We have compared the spectral efficiencies and SNR requirements of complex modulations in the presence of fading noise from atmospheric turbulence and local oscillator shot noise.

*M*-ary QAM is not just spectrally efficient, but also can significantly improve photon efficiency in atmospheric fading as compared to high-order PSK modulation.

## Acknowledgments

## References and links

1. | J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. |

2. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

3. | M. Born, and E. Wolf, |

4. | D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. |

5. | A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express |

6. | A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express |

7. | J. W. Goodman, |

8. | J. D. Parsons, “Diversity techniques in communications receivers,” in |

9. | J. G. Proakis, and M. Salehi, |

10. | M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis of digital communications over generalized fading channels,” IEEE Proc. |

11. | A. Goldsmith, |

12. | J. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. |

**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: November 11, 2009

Revised Manuscript: February 1, 2010

Manuscript Accepted: February 1, 2010

Published: February 12, 2010

**Citation**

Aniceto Belmonte and Joseph M. Kahn, "Efficiency of complex modulation methods in coherent free-space optical links," Opt. Express **18**, 3928-3937 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3928

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### References

- J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975). [CrossRef]
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]
- M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. 3(6), 213–221 (1967). [CrossRef]
- A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16(18), 14151–14162 (2008). [CrossRef] [PubMed]
- A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express 17(15), 12601–12611 (2009). [CrossRef] [PubMed]
- J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Ben Roberts & Company, 2007).
- J. D. Parsons, “Diversity techniques in communications receivers,” in Advanced Signal Processing, D. A. Creasey, ed. (Peregrinus, 1985), Chap. 6.
- J. G. Proakis, and M. Salehi, Digital Communications, (Mc Graw-Hill, 2007).
- M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis of digital communications over generalized fading channels,” IEEE Proc. 86(9), 1860–1877 (1998). [CrossRef]
- A. Goldsmith, Wireless Communications (Cambridge University Press, 2005)
- J. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004). [CrossRef]

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