## Ergodic capacity comparison of optical wireless communications using adaptive transmissions |

Optics Express, Vol. 21, Issue 17, pp. 20346-20362 (2013)

http://dx.doi.org/10.1364/OE.21.020346

Acrobat PDF (915 KB)

### Abstract

Ergodic capacity is investigated for the optical wireless communications employing subcarrier intensity modulation with direct detection, and coherent systems with and without polarization multiplexing over the Gamma-Gamma turbulence channels. We consider three different adaptive transmission schemes: (i) variable-power, variable-rate adaptive transmission, (ii) complete channel inversion with fixed rate, and (iii) truncated channel inversion with fixed rate. For the considered systems, highly accurate series expressions for ergodic capacity are derived using a series expansion of the modified Bessel function and the Mellin transformation of the Gamma-Gamma random variable. Our asymptotic analysis reveals that the high SNR ergodic capacities of coherent, subcarrier intensity modulated, and polarization multiplexing systems gain 0.33 bits/s/Hz, 0.66 bits/s/Hz, and 0.66 bits/s/Hz respectively with 1 dB increase of average transmitted optical power. Numerical results indicate that a polarization control error less than 10° has little influence on the capacity performance of polarization multiplexing systems.

© 2013 OSA

## 1. Introduction

1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. **24**, 4750–4762 (2006). [CrossRef]

2. N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun. **59**, 1368–1377 (2011). [CrossRef]

3. Md. Z. Hassan, Md. J. Hossain, and J. Cheng, “Performance of non-adaptive and adaptive subcarrier intensity modulations in Gamma-Gamma turbulence,” IEEE. Trans Commun. **61**, 2946–2957 (2013). [CrossRef]

16. A. Laourine, M.-S. Alouini, S. Affes, and A. Stéphenne, “On the capacity of generalized-K fading channels,” IEEE Trans. Wireless Commun. **7**, 2441–2445 (2008). [CrossRef]

34. A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Info. Theory **43**, 1986–1992 (1997). [CrossRef]

34. A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Info. Theory **43**, 1986–1992 (1997). [CrossRef]

8. C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over Gamma-Gamma turbulence channels with pointing errors,” Electron. Lett. **46**, 851–853 (2010). [CrossRef]

12. K. P. Peppas, A. N. Stassinakis, G. K. Topalis, H. E. Nistazakis, and G. S. Tombras, “Average capacity of optical wireless communication systems over I− K atmospheric turbulence channels,” J. Opt. Commun. Netw. **4**, 1026–1032 (2012). [CrossRef]

20. 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. Netw. **4**, 586–601 (2005). [CrossRef]

21. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “On the capacity of FSO links over Gamma-Gamma atmospheric turbulence channels using OOK signaling,” EURASIP Journal on Wireless Communications and Networking **2010**. Article ID 127657, 9 pages, 2010. doi: [CrossRef]

## 2. Channel and system models

### 2.1. Atmospheric turbulence models

### 2.2. Coherent OWC system

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

27. M. Niu, J. Schlenker, J. Cheng, J. F. Holzman, and R. Schober, “Coherent wireless optical communications with predetection and postdetection EGC over Gamma-Gamma atmospheric turbulence channels,” J. Opt. Commun. Netw. , **3**, 860–869 (2011). [CrossRef]

27. M. Niu, J. Schlenker, J. Cheng, J. F. Holzman, and R. Schober, “Coherent wireless optical communications with predetection and postdetection EGC over Gamma-Gamma atmospheric turbulence channels,” J. Opt. Commun. Netw. , **3**, 860–869 (2011). [CrossRef]

18. N. Cvijetic, D. Y. Qian, J. J. Yu, Y. K. Huang, and T. Wang, “Polarisation-multiplexed optical wireless transmission with coherent detection,” J. Lightwave Technol. **28**, 1218–1227 (2010). [CrossRef]

29. E. J. Lee and V. W. S. Chan, “Diversity coherent and incoherent receivers for free-space optical communication in the presence and absence of interference,” J. Opt. Commun. Netw. **1**, 463–483 (2009). [CrossRef]

*P*is the received optical signal power,

_{s}*P*is the LO power,

_{LO}*ϕ*is the phase information associated with the modulation order, and

*ω*=

_{IF}*ω*−

_{c}*ω*is the intermediate frequency, where

_{LO}*ω*and

_{c}*ω*denote the carrier frequency and the LO frequency, respectively. The photocurrent generated by the photodetector can be written as

_{LO}*i*

_{r}_{,}

*(*

_{c}*t*) =

*i*+

_{DC}*i*(

_{AC}*t*) +

*n*(

_{c}*t*) where

*i*=

_{DC}*R*(

*P*+

_{s}*P*) and

_{LO}*R*is the responsivity of phtotodetector, and

*n*(

_{c}*t*) is a shot noise limited AWGN process with variance

*P*≫

_{LO}*P*, and hence, the DC term of the photocurrent is dominated by the term

_{s}*RP*. The shot noise generated by the LO is dominant compared to the background irradiance generated shot noise and/or receiver thermal noise. The shot noise variance can be written as

_{LO}*q*is the electronic charge, and Δ

*f*is the noise equivalent bandwidth of photodetector. The signal-to-noise ratio (SNR) at the input of an electrical demodulator can be written as the ratio of the time-averaged AC photocurrent power to the total noise variance. The received optical power

*P*can be written as

_{s}*P*=

_{s}*AI*where

*A*is the photodetector area, and

*I*is the received optical irradiance. Assuming the mean of optical irradiance

*I*is unity, the instantaneous SNR at the input of a electrical demodulator can be obtained as [33

33. G. P. Agrawal, *Fiber-Optical Communication Systems* (Wiley, 2002). [CrossRef]

*g*is a constant path-loss factor, and

### 2.3. POLMUX OWC system with coherent detection

30. X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarisation shift-keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw. **1**, 307–312 (2009). [CrossRef]

19. G. D. Xie, F. X. Wang, A. Dang, and H. Guo, “A novel polarisation-multiplexing system for free-space optical links,” IEEE Photon. Technol. Lett. **23**, 1484–1486 (2011). [CrossRef]

**E**

*(*

_{r}*t*) = [

**E**

_{x}_{,}

*(*

_{r}*t*),

**E**

_{y}_{,}

*(*

_{r}*t*)]

*where the subscripts*

^{T}*x*and

*y*denote the two orthogonal channels, and where At the receiver, a polarization controller (PC) is employed to adjust the SOP of the received optical beam. The adjusted optical beam is splitted into two orthogonal channels using a second PBS. The output of the second PBS can be written as [19

19. G. D. Xie, F. X. Wang, A. Dang, and H. Guo, “A novel polarisation-multiplexing system for free-space optical links,” IEEE Photon. Technol. Lett. **23**, 1484–1486 (2011). [CrossRef]

*ε*is the polarization control error which describes the SOP difference between the light after PBC at the transmitter and the light after PBS at the receiver. Following [18

18. N. Cvijetic, D. Y. Qian, J. J. Yu, Y. K. Huang, and T. Wang, “Polarisation-multiplexed optical wireless transmission with coherent detection,” J. Lightwave Technol. **28**, 1218–1227 (2010). [CrossRef]

18. N. Cvijetic, D. Y. Qian, J. J. Yu, Y. K. Huang, and T. Wang, “Polarisation-multiplexed optical wireless transmission with coherent detection,” J. Lightwave Technol. **28**, 1218–1227 (2010). [CrossRef]

*λ*

_{1}= cos

*ε*+ sin

*ε*, and

*λ*

_{2}= cos

*ε*− sin

*ε*. For a system with fully equalized cross-polarization interference, we have

*ε*= 0° or

*λ*

_{1}=

*λ*

_{2}= 1. The POLMUX systems with coherent detection are widely used for the fiber-optic applications since the need of significant high data rate predominates the implementation cost and complexity. The major impairments of the fiber-optics POLMUX systems are the polarization mode dispersion (PMD), polarization dependent loss (PDL), and depolarization [31

31. I. B. Djordjevic, L. Xu, and T. Wang, “Beyond 100 Gb/s optical transmission based on polarization multiplexed coded-OFDM with coherent detection,” J. Opt. Commun. Netw. **1**, 50–56 (2009). [CrossRef]

**28**, 1218–1227 (2010). [CrossRef]

### 2.4. OWC system with subcarrier intensity modulation/direct detection

*s*(

*t*), pre-modulated with data source, is used to modulate the irradiance of a continuous wave optical beam at the laser transmitter after being properly biased. For an atmospheric turbulence channel, the received photocurrent after direct detection using photodetector can be expressed as where

*ξ*is the modulation index satisfying the condition −1 ≤

*ξs*(

*t*) ≤ 1 in order to avoid overmodulation,

*I*(

*t*) is assumed to be a stationary random process for the received irradiance fluctuation caused by atmospheric turbulence, and

*n*(

*t*) is the noise term caused by background radiation (i.e., ambient light) and/or thermal noise, and it is modeled as an AWGN process with variance

*I*=

*I*(

*t*)|

_{t=t0}at a time instant

*t*=

*t*

_{0}gives the RV

*I*. Normalizing the power of

*s*(

*t*) to unity, the instantaneous SNR at the input of electrical demodulator can be written as [33

33. G. P. Agrawal, *Fiber-Optical Communication Systems* (Wiley, 2002). [CrossRef]

*I*is the background light irradiance,

_{b}*k*is Boltzman constant,

_{b}*T*is the temperature in Kelvin,

_{k}*R*is the load resistance, and

_{L}*E*[

*I*] = 1, and using (2), we define the average SNR of the subcarrier IM/DD OWC system as

*γ*by The average SNR can also be expressed as

_{s}## 3. Ergodic capacity of coherent OWC system

### 3.1. Variable-power, variable-rate adaptive transmission

*γ*. For a VPVR adaptive scheme, the ergodic capacity (in bits/s/Hz) of a fading channel can be calculated by [34

_{o}34. A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Info. Theory **43**, 1986–1992 (1997). [CrossRef]

*γ*must satisfy [34

_{o}**43**, 1986–1992 (1997). [CrossRef]

*f*

_{γc}(

*γ*) is the PDF of the SNR

_{c}*γ*. Using a series expansion of the modified Bessel function of second kind [25

_{c}25. Md. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw. **6**, 522–532 (2012). [CrossRef]

*γ*, respectively, as and where

_{c}#### 3.1.1. Computation of Cutoff SNR *γ*_{o}

_{o}

*r*

_{1,}

*(*

_{c}*α*,

*β*,

*E*[

*I*

^{−1}] is the first negative integer moment of the Gamma-Gamma RV

*I*. The Mellin transformation of a positive RV

*X*provides all moments of

*X*including positive integer, negative integer, and fractional moments [35]. Using the Mellin transformation of the Gamma-Gamma RV (see Appendix), we obtain

*r*

_{1,}

*(*

_{c}*α*,

*β*,

*r*

_{2,}

*(*

_{c}*α*,

*β*,

*γ*,

_{o}*γ*. For min{

_{o}*α*,

*β*} > 1, such expression can be numerically solved along with Eq. (14) in order to compute the cutoff SNR

*γ*for a given average SNR. From Eqs. (16), (19) and (20), when

_{o}*F*

_{γc}(·) → 0,

*r*

_{1,}

*(*

_{c}*α*,

*β*,

*r*

_{2,}

*(*

_{c}*α*,

*β*,

*γ*,

_{o}*γ*approaches unity. Therefore, for the Gamma-Gamma turbulence channels, the value of the cutoff SNR,

_{o}*γ*, is restricted to [0, 1].

_{o}#### 3.1.2. Computation of ergodic capacity

*g*

_{1,}

*(*

_{c}*α*,

*β*,

*γ*, i.e., E[ln

_{c}*γ*]. In order to estimate the expected value of ln

_{c}*γ*, we first define

_{c}*Z*= ln

*γ*. The moment generating function of

_{c}*Z*is The expected value of ln

*γ*can be obtained as It can be easily shown that if

_{c}*k*-th moment of

*γ*is

_{c}*I*] is the

^{k}*k*-th moment of RV

*I*and it is given by [24

24. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma-Gamma atmospheric turbulence model,” Opt. Express **18**, 12824–12831 (2010). [CrossRef] [PubMed]

36. The Wolframe Function Site. [Online]: http://functions.wolfram.com.

*g*

_{1,}

*(*

_{c}*α*,

*β*,

*ψ*(·) is the Euler’s digamma function [23, Eq. 8.360(1)]. In order to evaluate

*g*

_{2,}

*(*

_{c}*α*,

*β*,

*γ*,

_{o}

_{z}_{→0}

*z*ln

^{a}*z*= 0 for any real non-negative

*z*and

*a*. Substituting Eq. (15) into the second integral of Eq. (21) and using [23, Eq. 2.723(1)], we obtain

*g*

_{2,}

*(*

_{c}*α*,

*β*,

*γ*,

_{o}*K*+ 1 terms. Consequently, a truncation error is introduced due to elimination of infinite terms after the first

*K*+ 1 terms. Using a similar method described in [25

25. Md. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw. **6**, 522–532 (2012). [CrossRef]

*K*values and/or average SNR

#### 3.1.3. Asymptotic high SNR capacity

*γ*approaches unity, and consequently, log

_{o}_{2}(

*γ*) approaches zero. Also, it can be shown that in large SNR regimes

_{o}### 3.2. Complete channel inversion with fixed rate

**43**, 1986–1992 (1997). [CrossRef]

*K*fading channel given by [16

16. A. Laourine, M.-S. Alouini, S. Affes, and A. Stéphenne, “On the capacity of generalized-K fading channels,” IEEE Trans. Wireless Commun. **7**, 2441–2445 (2008). [CrossRef]

16. A. Laourine, M.-S. Alouini, S. Affes, and A. Stéphenne, “On the capacity of generalized-K fading channels,” IEEE Trans. Wireless Commun. **7**, 2441–2445 (2008). [CrossRef]

*β*is an integer. The asymptotic capacity in large SNR regimes can be obtained as

### 3.3. Truncated channel inversion with fixed rate

*γ*

_{coh}. The channel will not be used if the received SNR falls below

*γ*

_{coh}. For this adaptive transmission policy, the ergodic capacity (in bits/s/Hz) is given by [34

**43**, 1986–1992 (1997). [CrossRef]

*r*

_{1,}

*(·, ·, ·) and*

_{c}*r*

_{2,}

*(·, ·, ·, ·) are obtained from Eqs. (18) and (20), respectively. Here, the cutoff level*

_{c}*γ*

_{coh}is selected in order to maximize the channel capacity in Eq. (31) for a given average SNR. Our numerical result shows that the cutoff SNR that maximizes the channel capacity of a coherent TCIFR system increases with an increase of average SNR. Recall that the cutoff SNR of a coherent VPVR system is restricted to the range [0, 1]. Hence, due to increasing cutoff SNR values, coherent TCIFR systems exhibit higher outage probability compared to the coherent VPVR systems. Also note that, for coherent systems, Eqs. (30) and (32) are valid when min{

*α*,

*β*} > 1, which is typically satisfied for the Gamma-Gamma turbulence channels [26].

## 4. Ergodic capacity of coherent POLMUX OWC system

### 4.1. Variable-power, variable-rate adaptive transmission

*f*

_{γx}(

*γ*) and

_{x}*f*

_{γx}(

*γ*) are respectively PDFs of

_{x}*γ*and

_{x}*γ*, and

_{y}*γ*and

_{a}*γ*are the cutoff SNRs for the two orthogonal channels satisfying and An asymptotic expansion of Eqs. (34) and (35) reveals that both

_{b}*γ*and

_{a}*γ*are confined to the range [0, 1]. Using a similar method applied to the coherent OWC system, it can be shown and where

_{b}*F*

_{γx}(·) and

*F*

_{γy}(·) are respectively the CDFs of

*γ*and

_{x}*γ*, and they can be obtained by substituting

_{y}### 4.2. Complete channel inversion with fixed rate

### 4.3. Truncated channel inversion with fixed rate

*γ*

_{pol,x}and

*γ*

_{pol,y}are selected in order to maximize

*F*

_{γi}(

*γ*

_{pol,i})) where

*i*=

*x*,

*y*. Similar to the coherent OWC, Eqs. (39) and (41) are valid for min{

*α*,

*β*} > 1.

## 5. Ergodic capacity of subcarrier IM/DD OWC system

### 5.1. Variable-power, variable-rate adaptive transmission

*γ*is the cutoff SNR satisfying the condition In Eqs. (42) and (43),

_{e}*f*

_{γs}(

*γ*) and

_{s}*F*

_{γs}(·) are respectively the PDF and CDF of the SNR of the subcarrier IM/DD OWC systems, and they are given by and where in Eqs. (44) and (45)

*γ*is also confined to the range [0, 1]. Using a similar approach in deriving Eq. (25), we can evaluate

_{e}*g*

_{1,}

*as Following Eq. (26), we obtain*

_{s}*g*

_{2,}

*as where*

_{s}### 5.2. Channel inversion with fixed rate

*γ*

_{th}is the cutoff SNR below which no power adaption is accomplished, and

*α*,

*β*} > 2.

## 6. Numerical results

*α*= 2.04 and

*β*= 1.10) with 4.35 dB/Km path loss; 2) a 900 m light smoke optical channel (in moderate turbulence with

*α*= 2.50 and

*β*= 2.06) with 9.56 dB/Km path loss; 3) a 700 m light fog optical channel (in weak turbulence

*α*= 4.43 and

*β*= 4.39) with 11.5 dB/Km path loss. In order to generate the numerical plots of capacity versus the average transmitted optical power, we also make the following assumptions [26]: the modulation index

*ε*= 0.85, the photodetector responsivity

*R*= 0.75 A/W, load resistance

*R*= 50 Ω, the bit duration

_{L}*T*= 1ns with approximate transmission bandwidth 1 GHz, thermal noise variance 3.3 × 10

^{−13}Amp

^{2}, and the background noise variance 10

^{−15}Amp

^{2}. Assuming a typical LO power 10

^{−2}W [37

37. J. M. Hunt, F. Holmes, and F. Amizajerdian, “Optimum local oscialltor levels for coherent detection using photodetectors,” Appl. Opt. **27**, 3135–3141 (1988). [CrossRef] [PubMed]

^{−12}Amp

^{2}. Also, in our numerical results we have selected the range of the average transmitted optical power from −12 dBm to 6 dBm so that the assumption

*P*

_{LO}≫

*P*remains valid.

_{s}*ε*= 10° has no noticeable influence on the ergodic capacity performance. Asymptotic channel capacities for the VPVR and CCIFR schemes are also shown in both figures, and they converge to the exact capacity at large SNR values, as expected.

*α*= 2.50 and

*β*= 2.06. This result agrees with the predicted SNR gap of 3.06 dB between the VPVR and CCIFR schemes from our asymptotic ergodic capacity analysis. Asymptotic capacities are also shown in this figure, and they converge to the exact ergodic capacity when

*ε*= 0° and

*ε*= 30°, respectively. On the other hand, for the same average transmitted optical power, the coherent and subcarrier IM/DD VPVR schemes attain channel capacity of 14.19 bits/s/Hz and 5.03 bits/s/Hz, respectively. From Fig. 2(b), at an average transmitted optical power of −4 dBm coherent POLMUX CCIFR scheme achieves 32.74 bits/s/Hz and 31.83 bits/s/Hz channel capacity with polarization control error

*ε*= 0° and

*ε*= 30°, respectively. On the other hand, for the same average transmitted optical power, the coherent and subcarrier IM/DD CCIFR schemes obtain channel capacity of 17.42 bits/s/Hz and 9.738 bits/s/Hz, respectively. Both Figs. 2(a) and 2(b) illustrate that the performance gap between the coherent and the subcarrier IM/DD systems gets narrowed with an increase of transmitted optical power. However, the performance gap between coherent POLMUX and subcarrier IM/DD systems remains constant with an increase of transmitted optical power. This is because both coherent POLMUX and subcarrier IM/DD systems have the same slope of 0.66 bits/s/Hz whereas the coherent systems have a slope of 0.33 bits/s/Hz with respect to the dB values of average transmitted optical power.

## 7. Conclusion

## Appendix

*I*is defined as

*I*is a Gamma-Gamma RV, one can show

*I*=

*XY*, where

*X*and

*Y*are two independent Gamma RVs with PDFs respectively given by

*M*(

_{I}*z*) =

*M*(

_{X}*z*)

*M*(

_{Y}*z*). Using an integral identity [23, Eq. 3.326(2)], we obtain the Mellin transformations of RVs

*X*and

*Y*, respectively, as

*I*becomes It follows that

*E*[

*I*

^{−1}] and

*E*[

*I*

^{−2}], which are required by Eqs. (18) and (54), can be obtained by letting

*z*= 0 and

*z*= −1 in Eq. (56) [35].

## References and links

1. | V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. |

2. | N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun. |

3. | Md. Z. Hassan, Md. J. Hossain, and J. Cheng, “Performance of non-adaptive and adaptive subcarrier intensity modulations in Gamma-Gamma turbulence,” IEEE. Trans Commun. |

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8. | C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over Gamma-Gamma turbulence channels with pointing errors,” Electron. Lett. |

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36. | The Wolframe Function Site. [Online]: http://functions.wolfram.com. |

37. | J. M. Hunt, F. Holmes, and F. Amizajerdian, “Optimum local oscialltor levels for coherent detection using photodetectors,” Appl. Opt. |

38. | P. Galambos and I. Simonelli, |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 14, 2013

Revised Manuscript: July 26, 2013

Manuscript Accepted: August 2, 2013

Published: August 22, 2013

**Citation**

Md. Zoheb Hassan, Md. Jahangir Hossain, and Julian Cheng, "Ergodic capacity comparison of optical wireless communications using adaptive transmissions," Opt. Express **21**, 20346-20362 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-20346

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