Optical Communication Using Subcarrier Intensity Modulation in ...

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Optical Communication Using Subcarrier Intensity Modulation in Strong Atmospheric Turbulence Xuegui Song, Student Member, IEEE, and Julian Cheng, Member, IEEE

Abstract—Error rate performance of subcarrier intensity modulations is analyzed for optical wireless communications over strong atmospheric turbulence channels. We study the error rate of a subcarrier intensity modulated optical wireless communication system employing -ary phase-shift keying, differential phase-shift keying, and noncoherent frequency-shift keying. Both -distributed turbulence channel (strong) and negative exponential turbulence channel (saturated) are considered. Closed-form error rate expressions are derived using a series expansion of the modified Bessel function. Furthermore, the outage probability expressions are obtained for subcarrier intensity modulated optical -distributed turbuwireless communication systems over the lence and the negative exponential channels. Asymptotic error rate analysis and truncation error analysis are also presented. Our asymptotic analysis shows that differential phase-shift keying suffers a constant signal-to-noise ratio performance loss of 3.92 dB with respect to binary phase-shift keying under strong atmospheric turbulence conditions. The numerical results demonstrate that our series solutions are efficient and highly accurate. Index Terms— -ary phase-shift keying (PSK), optical wireless communication (OWC), strong atmospheric turbulence.

I. INTRODUCTION

O

PTICAL WIRELESS COMMUNICATION (OWC) is well suited for applications requiring low costs, high data rates, enhanced security, and unlicensed bandwidths [1]. When an optical signal is transmitted through free space, it is subject to the corruption of fading induced by the atmospheric turbulence. Fading is a major source of system performance degradation for any outdoor OWC system. Owing to its simplicity and low cost, ON–OFF keying (OOK) with irradiance modulation and direct detection (IM/DD) is commonly used for OWC applications [2], [3]. However, optimal error rate performance of an OOK IM/DD system can only be achieved with adaptive detection threshold. Such a system, if feasible, can be costly to implement and subject to channel estimation errors. In practice, for simplicity, OOK IM/DD systems are often implemented using a predetermined fixed detection threshold [4]. This suboptimal scheme will lead to performance loss with undesirable irreducible error floors, which are more severe under strong turbulence conditions [4], [5]. Manuscript received January 25, 2012; revised August 05, 2012; accepted September 21, 2012. Date of publication September 27, 2012; date of current version October 31, 2012. This work was supported in part by an NSERC Discovery Grant. The authors are with the School of Engineering, The University of British Columbia, Kelowna, BC V1V 1V7, Canada (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/JLT.2012.2220754

To overcome the challenges of OOK IM/DD optical communications, pulse position modulation (PPM) has been proposed as an alternative to the OOK IM/DD modulation. The performance of PPM-based systems has been studied in atmospheric turbulence channels in [6], [7], and the references therein. However, the PPM modulation technique requires complex transceiver design due to tight synchronization requirements and suffers from poor bandwidth efficiency. As another attractive alternative to the OOK IM/DD modulation, subcarrier intensity modulation (SIM) was first proposed for OWC application in [5]. Thereafter, the error rate performance of SIM OWC systems employing various phase-shift keying (PSK) modulations over different atmospheric turbulence channels has been studied extensively. In [5], Huang et al. studied the error rate performance for differential PSK (DPSK) and -ary PSK (MPSK) modulations over the lognormal turbulence channels. Their theoretical analysis was confirmed by experimental results. In [4], Li et al. also studied error rate performance of both uncoded and coded SIM systems employing MPSK modulation over the lognormal turbulence channels. Later, Popoola et al. studied bit error rate (BER) performance of binary PSK (BPSK) scheme assuming lognormal, negative exponential, and Gamma-Gamma channel models for atmospheric turbulence in [8], [9], and [10], respectively. However, none of the aforementioned work provided a closed-form expression for the error rate of subcarrier modulations. More recently, Chatzidiamantis et al. studied adaptive and nonadaptive subcarrier PSK systems over the lognormal and the Gamma-Gamma turbulence channels [11]. For the lognormal turbulence channel, their results are approximations, while for the Gamma-Gamma turbulence channel, the presented results are in terms of Meijer’s G-function, which does not reveal any insights into the SIM system. In [12], Park et al. studied the BER performance of subcarrier BPSK with Alamouti coding using a moment generating function (MGF) approach and a series expansion for the modified Bessel function [13]. However, the authors did not study the truncation error, and they only considered subcarrier BPSK over the Gamma-Gamma turbulence channels. In another work [14], Samimi et al. studied the BER performance of subcarrier BPSK over the negative exponential turbulence channels. Based on the results for the negative exponential turbulence channels, the authors also studied the BER performance of subcarrier BPSK over the -distributed turbulence channels by approximating the probability density function (PDF) of the distribution with a finite sum of weighted negative exponential PDFs. However, their results for the -distributed turbulence channels are only valid for [14] where is a fading parameter in the -distributed turbulence model and is related

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to severity of fading. These results are not useful for practical channel parameter [15] in the -distributed turbulence model. More recently, using a direct integration approach, we studied the error rate of a subcarrier intensity modulated OWC system employing BPSK, quadrature PSK (QPSK), DPSK, and noncoherent frequency-shift keying (NCFSK) over the Gamma-Gamma turbulence channels [16]. However, in that work, we did not obtain a unified error rate solution for MPSK modulation. The truncation error analysis is studied only for DPSK/NCFSK, and the channel consideration is restricted to the Gamma-Gamma turbulence model. In this study, using the same series expansion formula [13], we study the error rate performance and outage probability performance for subcarrier intensity modulated OWC systems employing a variety of digital modulation formats. An MGF approach is used to study the performance of an MPSK SIM system. Both -distributed and negative exponential models are considered for strong atmospheric turbulence. We obtain highly accurate closed-form error rate expression in terms of infinite series for MPSK modulated SIM system over the -distributed turbulence channels. Inspired by the relationship between the negative exponential model and the -distributed model, we also obtain highly accurate error rate expression in terms of infinite series for MPSK modulated SIM system over the negative exponential turbulence channels. The error rate performance of BPSK and QPSK modulations are also obtained as special cases of MPSK. Besides MPSK, both DPSK and NCFSK are also considered. We carry out a detailed study of the truncation error and asymptotic behavior of our series solutions. Our numerical results demonstrate that our series solutions are efficient and highly accurate. The remainder of the paper is organized as follows. In Section II, the SIM system is described. In Section III, the -distributed turbulence model and the negative exponential turbulence model are presented. In this section, we also obtain the MGFs of the squared irradiance for the -distributed model and the negative exponential model. In Section IV, we investigate the performance of subcarrier modulated OWC systems employing a variety of digital modulation schemes over the -distributed turbulence channel and the negative exponential turbulence channel. The truncation error and asymptotic behavior of our series solutions are studied in Section IV as well. Section V presents some numerical results and discussions. Finally, some concluding remarks are provided in Section VI. II. SIM SYSTEM At the transmitter end, the data source is first used to modulate an RF subcarrier signal. After being properly biased, the RF signal is used to modulate the irradiance of a continuous wave optical beam at the laser transmitter. The transmitted optical intensity can be written as (1) where is the average transmitted power and lation index satisfying the condition overmodulation. For simplicity, we assume

is the moduto avoid . Further-

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more, the RF signal at the output of the electrical PSK modulator can be written as (2) where tion,

is the shaping pulse, is the symbol durais the frequency of the RF subcarrier signal, and is the phase of the th transmitted symbol and is the modulation order. At the receiver end, the received optical intensity is converted into an electrical signal through direct detection at the photodetector. When the intensity of the received signal is relatively high and the background radiation (i.e., ambient light) is significant, the noise can be modeled with high accuracy as additive white Gaussian noise (AWGN) [2]. Therefore, for an atmospheric turbulence channel, the photocurrent at the receiver can be written as (3)

where is the photodetector responsivity, is the photodetector area, is the irradiance, and is the AWGN due to thermal and/or background noise. For simplicity, we normalize the power of to be unity. After removing the dc bias and demodulating through an electrical PSK demodulator, the sampled electrical signal at the output of the receiver can be used to recover the transmitted data. The instantaneous signal-to-noise ratio (SNR) at the input of the electrical demodulator of an optical receiver is defined as the ratio of the time-averaged ac photocurrent power to the total noise variance [17], and it can be expressed as (4) where is the background light irradiance, is the electronic charge, denotes the noise equivalent bandwidth of the photodetector, is Boltzmann’s constant, is the temperature in Kelvin, and is the load resistance. The SNR defined in (4) is SNR per symbol, and we have for MPSK modulation with denoting SNR per bit. In (4), the constant is also the average SNR , and therefore, we will use in the ensuing analysis. III. STRONG ATMOSPHERIC TURBULENCE MODELS The statistical models for the intensity fluctuation through OWC channels have been extensively studied over the last two decades [18]–[21]. There are several statistical models that can be used to describe irradiance fluctuation in the literature. As discussed in [2] and [21], it has been well accepted that the lognormal turbulence model describes irradiance fluctuations in weak turbulence conditions. The -distributed turbulence model is another useful turbulence model describing irradiance fluctuations in strong turbulence conditions [20], [22], [23]. The negative exponential model is suitable for describing the limiting case of saturated scintillation (saturated turbulence) [10], [21]. More recently, the Gamma-Gamma model emerges as a useful turbulence model as it has excellent fit with measurement data over a wide range of turbulence conditions

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[21]. Specifically, the Gamma-Gamma model contains the -distributed model (strong) and the negative exponential model (saturated) as special cases [20], [21]. In this section, the -distributed model and negative exponential model will be first introduced. Then, the MGFs of the squared irradiance for both statistical models are obtained to facilitate the ensuing analytical development.

Let denote a channel-dependent RV. Using a series expansion of the modified Bessel function ([13, eq. (6)])

A.

and an integral identity ([24, eq. where 3.478(1)]), we can obtain the MGF of as

-Distributed Model

In the -distributed turbulence channel model, the irradiance can be modeled as a product of two statistically independent random variables (RV’s) and , where follows the exponential distribution and follows the Gamma distribution [14], [19]. Therefore, we have

(12)

(13)

(5) and (6) where is the effective number of discrete refractive scatterers [21] and denotes the Gamma function. By first fixing and writing , we obtain the conditional PDF of a -distributed RV as (7)

where in obtaining (13) we have also used the Euler’s reflection formula . We will use (13) to analyze the error rate performance for subcarrier intensity modulated OWC systems employing MPSK in Section IV. B. Negative Exponential Model In a negative exponential atmospheric turbulence environment, the PDF of the irradiance is given as (14)

where is the conditional mean value of . By averaging (7) over the Gamma distribution, one obtains the unconditional PDF of a -distributed RV as [19], [20]

where is the mean irradiance. Without loss of generality, we normalize and set . Let denote a channel-dependent RV. We can obtain the MGF of as

(8)

(15)

where is the modified Bessel function of the second kind of order . The th moment of can be shown to be [19]

is the complemenwhere tary error function. In obtaining (15), we have used an integral identity ([24, eq. 3.322(2)]). The PDF of Gamma RV, which is given in (6), converges to the Dirac delta function at 1 when the parameter approaches , i.e.,

(9) where denotes the expectation operation. From the definition of the scintillation index [21] (10) and by assuming without loss of generality that can show that

(16) Therefore, when approaches model can be written as

, the PDF of

-distributed

, we (11)

The -distributed model is a reliable turbulence model when the scintillation index SI is confined to the range (2, 3) [15]; therefore, the channel parameter is within (1, 2). The turbulence model is suitable for a distance about 1 km [15]. This model is relevant to commercial OWC products operation. For example, Canobeam DT-130 and DT-150 (manufactured by Canon) can operate up to a maximum distance about 1 km at maximum data rate about 1–1.5 Gb/s.

(17) We can see from the aforementioned derivation that the negative exponential model is a special case of the -distributed model where the channel parameter approaches [19], [21]. This relationship between the negative exponential model and the -distributed model will be useful in obtaining highly accurate approximate error rate expression in terms of infinite series for

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MPSK modulated SIM systems over the negative exponential turbulence channel in Section IV-B.

lation over pressed as

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-distributed turbulence channels can then be ex-

IV. PERFORMANCE ANALYSIS OF SIM In this section, we study the error rate performance for subcarrier MPSK modulation and the outage probability for SIM over the -distributed turbulence channel and the negative exponential turbulence channel using an MGF approach. Performance of DPSK and NCFSK is also studied. We express the instantaneous SNR as where is a channel-dependent RV. The average error rate of subcarrier MPSK modulation can be obtained as [25]

(25) Equation (25) can be simplified for BPSK and QPSK modulations by obtaining alternative representations of for specific values. For the special case of BPSK, we have

(18) (26) We will use (18) to perform the ensuing error rate analysis. A. Subcarrier MPSK Modulation Over Turbulence Channels

-Distributed

Using (13) and (18), we obtain the average error rate for subcarrier MPSK modulation over the -distributed turbulence channel as

where we have used the integral identity ([13, eqs. 3.621(1), 8.384(4)]) and the relationship (27) and the definition of Beta function ([24, eq. 8.384(1)]) (28) For QPSK, the symbol-error rate (SER) can be obtained as (29)

(19) where (20) and (21) and (22) With the definition (23)

(29) where

can be shown to be (30)

with the definition of the Hypergeometric function ([24, eq. (9.111)]) and integration by substitution. A detailed proof of (30) is given in the Appendix. Note that (25), (26), and (29) only involve the Gamma function and the Hypergeometric function, which can be computed easily with a standard scientific software. We comment that the error rate expressions (26) and (29) can also be obtained using the results in [16] by setting the Gamma-Gamma channel parameter to be unity. B. Subcarrier MPSK Modulation Over Negative Exponential Turbulence Channels

we can obtain an integral identity from Mathematica as

The average error rate for subcarrier MPSK modulation over the negative exponential turbulence channel can be obtained by substituting (15) into (18) as (24) is the Hypergeometric function ([24, eq. where (9.111)]). The average error rate for subcarrier MPSK modu-

(31)

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where the parameter has been defined in (21). Generally, the integral described in (31) has no closed-form solution. However, for the special case of BPSK where and ,a closed-form solution has been obtained as [14]

Substituting (8) and (12) into (37) and using an integral identity ([24, eq. 3.326(2)]), we obtain the average BER for DPSK/ NCFSK over the -distributed turbulence channel as

(32) Inspired by the relationship between the negative exponential model and the -distributed model, we can obtain a highly accurate series solution to (31). As discussed in Section III, the negative exponential model is a special case of the -distributed model when the channel parameter approaches . Under this condition, we observe that the second term of the summation in (25) is negligible for all values. Based on this key observation, we can derive the average error rate for subcarrier MPSK modulation over the negative exponential turbulence channel as

(38) Similarly, substituting (14) into (37), we obtain the exact average BER for DPSK/NCFSK over the negative exponential turbulence channel as (39) where in obtaining (39) we have used an integral identity ([24, eq. 3.322(2)]). D. Outage Probability Analysis The outage probability is defined as (40)

(33) For BPSK, we have (34)

is a predefined outage threshold and is the PDF where of the instantaneous SNR. This performance metric can be interpreted as the probability that the instantaneous SNR falls below a predetermined threshold above which the quality of the channel is satisfactory. Using (4), (8), (12), and (40), we obtain the outage probability for a subcarrier intensity modulated system over the -distributed turbulence channels as

and for QPSK, we have (41)

(35) As illustrated by our numerical results in Section V, the proposed error rate expressions in (33)–(35) are efficient and highly accurate. C. Subcarrier DPSK/NCFSK Modulation The conditional BER for subcarrier DPSK and NCFSK is [26]

Similarly, using (4), (14), and (40), we obtain the outage probability for a subcarrier intensity modulated system over the negative exponential channels as (42)

E. Truncation Error Analysis for Subcarrier MPSK Modulation To evaluate the truncation error caused by eliminating the infinite terms after the first terms in (25), we first define the truncation error as

(36) for DPSK and where BER can be obtained as

(43)

for NCFSK. The average where (37)

is defined as (44)

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Using the Taylor series expansion of an exponential function, we can simplify the summation term in (43) and obtain an upper bound of the truncation error as

system over the proximated by

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-distributed turbulence channel can be ap-

(50) (45) To prove that (45) is valid we need to show that the maximum exists. From the definition of value of , we observe that

From (49) and (50) we can derive the SNR penalty factor in decibel for subcarrier DPSK modulated OWC system with respect to subcarrier BPSK modulated OWC system over the -distributed turbulence channel as (51)

(46) is finite for any given (44) can be rewritten as

, and

values. The second term in

(47) After examining the right-hand side of (47), we note that approaches zero when approaches ; therefore, or approaches zero when approaches . Based on the aforementioned analysis, we conclude that the maximum value of exists, and it follows that (45) holds. Thus, the truncation error diminishes with increasing . After investigating (33), we notice that both and approach zero when approaches ; therefore, the truncation error

where is the log function with base 10. We observe from (51) that this SNR penalty factor is a constant. We comment that the asymptotic error rates in (49) and (50) approach the exact error rates faster for greater value. For values close to unity, the asymptotic error rate converges to the exact error rate slowly because the terms and have similar decay rates. Therefore, the leading term of the series in (25) and (38) becomes the dominant term only in large SNR regimes. For the negative exponential channel, we have , and the asymptotic error rate of an subcarrier MPSK modulated OWC system can be obtained from (49) as (52) From (39), we obtain the asymptotic error rate of an subcarrier DPSK/NCFSK modulated OWC system over the negative exponential channel as

(48) diminishes with increasing . From (45) and (48), we notice that the truncation errors and diminish rapidly with increasing average SNR . This suggests that our series solutions are highly accurate in large SNR regimes, and we can therefore use them to perform asymptotic error rate study. F. Asymptotic Error Rate Analysis We now study the asymptotic error rate performance of subcarrier intensity modulated OWC systems over the -distributed turbulence channel and the negative exponential channel. In OWC, we have for -distributed turbulence, so the term decreases faster than the term in (25) for the same value as increases. Consequently, when approaches , the leading term of the series in (25) becomes the dominant term. Therefore, the error rate of MPSK modulation in large SNR regimes for subcarrier modulated OWC system over the -distributed turbulence channels can be approximated by (49) where in obtaining (49) we have used the relationship . Similarly, the error rate of DPSK/NCFSK modulations in large SNR regimes for subcarrier modulated OWC

(53) Similarly, the SNR penalty factor in decibel for subcarrier DPSK modulated OWC system with respect to subcarrier BPSK modulated OWC system over the negative exponential channel can be obtained as (54) We observe from (51) and (54) that the SNR penalty factors for subcarrier DPSK modulated OWC system with respect to subcarrier BPSK modulated OWC system over the -distributed turbulence channel and the negative exponential channel are the same. Furthermore, we comment that the asymptotic error rates in (52) and (53) approach the exact error rates with a moderate SNR value because we have for the negative exponential channel, and therefore, decreases faster than when increasing . V. NUMERICAL RESULTS In this section, we compare the approximate error rate and outage probability with the exact error rate and outage probability to verify our analytical results. The approximate results are obtained by eliminating the infinite terms after the first terms in our series solutions. The exact results are calculated by numerical integration using (18), (37), (31), or (40) when closed-form solution is not available.

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Fig. 1. SERs of subcarrier MPSK modulated OWC systems over the -dis. The exact results are obtained tributed turbulence channel with using numerical integration of (18), while the approximate results are obtained using (25).

Fig. 2. BERs of subcarrier BPSK modulated OWC systems over the -dis. The exact results are obtained using tributed turbulence channel with numerical integration of (18), while the approximate results are obtained using (26).

In Fig. 1, SERs are presented for subcarrier QPSK and 8PSK over the -distributed turbulence channels with different values. The exact results are obtained using numerical integration of (18), while the approximate results are obtained using (25). The results presented in Fig. 1 show excellent agreement between the exact SERs and our series solutions with . In Fig. 2, we present BERs for subcarrier BPSK modulated OWC systems over the -distributed turbulence channels with different values. The exact results are obtained using numerical integration of (18), while the approximate results are obtained using (26). We again observe that our series solutions with have excellent agreement with the exact BERs. From Fig. 2, we also observe that the asymptotic BER approaches the exact BER faster for . For , the convergence of the asymptotic BER is slow because in this case the terms and are similar for small to medium SNR values. Therefore, the leading term in the series

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Fig. 3. BERs of subcarrier intensity modulated OWC systems using NCFSK, DPSK, and BPSK over the -distributed turbulence channel with (or ) and . The exact results are obtained using numerical integration of (18) and (37), while the approximate results are obtained using (26) and (38).

Fig. 4. SERs of subcarrier MPSK intensity modulated OWC systems over the negative exponential turbulence channel with different values. The exact results are obtained using numerical integration of (31), while the approximate results are obtained using (33).

solution (25), which is considered for the asymptotic BER, becomes dominant only at high SNR level. In Fig. 3, we show BERs for subcarrier DPSK, NCFSK, BPSK intensity modulated OWC systems over the -distributed turbulence channel. The exact results are obtained using numerical integration of (18) and (37), while the approximate results are obtained using (26) and (38). We observe that our series solutions (with ) have excellent agreement with the exact BERs. When the BER level is at , the SNR penalty factor for DPSK with respect to BPSK is found to be dB and the SNR penalty factor for NCFSK with respect to DPSK is found to be dB, which agree with the theoretical predictions of 3.92 and 3.0 dB, respectively, from our asymptotic analysis. In Fig. 4, we plot SERs for subcarrier QPSK and 8PSK over the negative exponential turbulence channel with different

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TABLE I REQUIRED TO ACHIEVE A GIVEN TRUNCATION ERROR OF LESS THAN FOR DIFFERENT MODULATION SCHEMES IN THE -DISTRIBUTED TURBULENCE CHANNELS

Fig. 5. BERs of subcarrier intensity modulated OWC systems using NCFSK, DPSK, and BPSK over the negative exponential turbulence channel with . The exact results are obtained using (32) and (39), respectively, while the approximate result is obtained using (34).

values. The exact results are obtained using numerical integration of (31), while the approximate results are obtained using (33). From Fig. 4, we observe that it requires only terms for our series solutions to converge to the exact SERs for both QPSK and 8PSK when SNR is greater than 5 dB. 8PSK requires a larger than QPSK to achieve a perfect agreement between the approximate SER and the exact SER. As expected, the asymptotic SERs converge to the exact SERs rapidly. This is because the equivalent for the negative exponential turbulence channel is and the leading term in the series solution (25) becomes dominant even at medium SNR level. Fig. 5 shows BERs for subcarrier BPSK modulated OWC systems over the negative exponential turbulence channel with . We also plot BERs for subcarrier DPSK and NCFSK modulated OWC systems in this figure. The exact results are obtained using (32) and (39), respectively, while the approximate result is obtained using (34). Again, we observe that it requires only terms for our series solutions to converge to the exact BERs for subcarrier BPSK modulation. The asymptotic BERs converge to the exact BER when SNR is greater than 25 dB. When the BER level is at , the SNR penalty factor for DPSK with respect to BPSK is found to be dB and that for NCFSK with respect to DPSK is found to be dB, which agree with the theoretical predictions of 3.92 and 3.0 dB, respectively, from our asymptotic analysis. In Table I, we present the required number of series terms to achieve a given absolute truncation error of less than for different modulation schemes with various values. The case where refers to the negative exponential turbulence channel. From Table I, we observe that in general it requires less number of terms to achieve the given truncation error level in a stronger turbulence condition. As expected, the required decreases rapidly with increasing average SNR values. At dB, the required is comparatively small for all considered modulation schemes to achieve the given truncation error. For dB, we need no more than six terms to

Fig. 6. Outage probability of subcarrier BPSK modulated OWC systems over and the -distributed turbulence chanthe negative exponential . The exact result for the negative exponential channels is obnels with tained using (42). The exact results for the -distributed channels are obtained using numerical integration of (40), while the approximate results are obtained using (41).

achieve the same truncation error level for all considered modulation schemes. In Fig. 6, outage probabilities of subcarrier BPSK modulated OWC systems over the -distributed turbulence channels with dB are presented for different values. In the same figure, outage probability of the considered system over the negative exponential turbulence channels obtained using (42) is also shown. The exact results for the -distributed channels are obtained using numerical integration of (40), while the approximate results are obtained using (41). Again, we observe excellent agreement between the exact results and our series solutions with for the system over the -distributed turbulence channels. We observe from Fig. 6 that the outage performance of the considered system improves as increases (or SI decreases) because the turbulence gets weaker. However, the performance cannot be improved significantly by increasing the

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value of (or decreasing the value of SI) and it can only approach the performance of the limiting case of the negative exponential channels. From the aforementioned numerical results, we can see that the presented closed-form series solutions for error rate and outage probability can approximate the exact results with high accuracy by adopting only a small number of terms. Therefore, we conclude that the series error rate and outage probability solutions we have developed are highly accurate for an uncoded subcarrier system. They can be used to evaluate the system performance efficiently. VI. CONCLUSION We have developed highly accurate closed-form error rate and outage probability expressions for SIMs. Our truncation error analysis has demonstrated that our series solutions can converge to the exact results rapidly, especially in large SNR regimes. Our asymptotic analysis shows that DPSK suffers a constant performance loss of 3.92 dB with respect to BPSK under the strong atmospheric turbulence conditions. The numerical results demonstrate that our series solutions are efficient and highly accurate for practical system performance estimation. In this study, we have only considered SER of MPSK modulated SIM systems over strong turbulence channels. In practice, the turbulence channel has slow fading and changes from a few milliseconds to a few tens of milliseconds; therefore, a large number of bits experience the same fading, i.e., the channel has memory. A future study is needed to consider the block error probability over a block length that is larger than the coherence time of the turbulence channels. APPENDIX Starting from the definition of

, we have

In obtaining the last equality, we have used the definition of the Hypergeometric function ([24, eq. (9.111)]). Therefore, the integral identity in (30) has been proved.

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SONG AND CHENG: OPTICAL COMMUNICATION USING SUBCARRIER INTENSITY MODULATION

Xuegui Song (S’08) received the B.Eng. degree in electrical and information engineering from the Northwestern Polytechnical University, Xi’an, China in 2003, and the M.A.Sc. degree in electrical engineering from the University of British Columbia, Kelowna, BC, Canada, in 2009, where he is currently working toward the Ph.D. degree in the School of Engineering. His current research interests include digital communications over fading channels, orthogonal frequency division multiplexing, and optical wireless communications.

Julian Cheng (S’96–M’04) received the B.Eng. degree (First Class) in electrical engineering from the University of Victoria, Victoria, BC, Canada in 1995, the M.Sc.Eng. degree in mathematics and engineering from Queen’s University, Kingston, ON, Canada, in 1997, and the Ph.D. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 2003.

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He is currently an Associate Professor in the School of Engineering, The University of British Columbia, Kelowna, BC. From 2005 to 2006, he was an Assistant Professor with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON. He was previously with Bell Northern Research and Northern Telecom (later known as NORTEL Networks). His current research interests include digital communications over fading channels, orthogonal frequency division multiplexing, spread spectrum communications, statistical signal processing for wireless applications, and optical wireless communications. Dr. Cheng was the recipient of numerous scholarships during his undergraduate and graduate studies, which included a President Scholarship from the University of Victoria and a postgraduate scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC). He was also a winner of the 2002 NSERC Postdoctoral Fellowship competition. He is the Co-Chair of the 2011 Canadian Workshop on Information Theory held in Kelowna, BC, and the Chair of the 2012 Wireless Communications held in Banff, AB. He is a registered Professional Engineer in the province of British Columbia, Canada.