Subcarrier Phase-Shift Keying Systems With Phase ... - OSA Publishing

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 33, NO. 9, MAY 1, 2015

Subcarrier Phase-Shift Keying Systems With Phase Errors in Lognormal Turbulence Channels Xuegui Song, Student Member, IEEE, Fan Yang, Julian Cheng, Senior Member, IEEE, Naofal Al-Dhahir, Fellow, IEEE, and Zhengyuan Xu, Senior Member, IEEE

Abstract—We study the bit-error rate performance of different subcarrier phase-shift keying systems with carrier phase errors (CPE) in lognormal turbulence channels where the CPE is modeled as a Tikhonov random variable. The CPE induced asymptotic noise reference losses for the studied systems are quantified analytically by introducing the lognormal-Nakagami fading as an auxiliary channel model. The auxiliary channel method used in this study can be potentially applied to other performance analysis problems involving the lognormal channels. Index Terms—Atmospheric turbulence, communication, freespace optical, lognormal fading, subcarrier intensity modulation.

I. INTRODUCTION REE-SPACE optical (FSO) communication is an attractive technology having the advantages of lower cost, larger available bandwidth, better security, greater deployment flexibility, and reduced time-to-market [1]–[3]. These benefits make FSO a promising solution to the “first-mile” and “last-mile” problems [1], [2]. Among different types of FSO systems, the subcarrier intensity modulated FSO system has gained increasing attention because it can offer better performance than an on-off keying based system, and has less complex transceiver design than a pulse position modulation based system [4]. Another advantage of the subcarrier FSO system is that such a system can employ a variety of modulation schemes (coherent modulation, noncoherent modulation, and differentially coherent modulation) at the electrical modulator. It is well-known that a coherent modulation scheme is superior to a noncoherent or differentially coherent modulation scheme in terms of error rate performance; therefore, the research community has mainly focused on subcarrier FSO systems using coherent modulation schemes. The error rate performance of such systems over different atmospheric turbulence channels has been studied extensively (refer to [5]–[7], and the references therein). However,

F

Manuscript received October 21, 2014; revised January 28, 2015; accepted January 29, 2015. Date of publication February 1, 2015; date of current version March 13, 2015. This work was supported by the National Key Basic Research Program of China under Grant 2013CB329201 and by the National Natural Science Foundation of China under Grant 61171066. This work was also supported by the Natural Sciences and Engineering Research Council of Canada. X. Song, F. Yang, and J. Cheng are with the School of Engineering, The University of British Columbia, Kelowna, BC V1V 1V7, Canada (e-mail: [email protected]; [email protected]; [email protected]). N. Al-Dhahir is with the University of Texas at Dallas, Richardson, TX 75080 USA (e-mail: [email protected]). Z. Xu is with the Key Laboratory of Wireless-Optical Communications, Chinese Academy of Sciences, School of Information Science and Technology, University of Science and Technology of China, Hefei 230027, China (e-mail: [email protected]). Digital Object Identifier 10.1109/JLT.2015.2398847

one challenge of using coherent modulation schemes is that such a system requires carrier phase recovery for the subcarrier signal in order to achieve its optimal performance [8]. The required carrier phase cannot always be tracked accurately. In such scenarios, noncoherent modulation or differentially coherent modulation schemes can be employed with inferior system performance. The error rate performance of subcarrier FSO systems using noncoherent or differentially coherent modulation schemes has also been studied [5], [9]–[12]. However, to the authors’ best knowledge, no prior work has studied the effect of carrier phase synchronization error on performance of subcarrier phase-shift keying systems in atmospheric turbulence channels. In this work, we study the bit-error rate (BER) performance of subcarrier binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK) systems with carrier phase errors (CPE) in the lognormal turbulence channels. The asymptotic analysis is then extended to a subcarrier M -ary phase-shift keying (MPSK) system based on a highly accurate approximate BER expression of MPSK in additive white Gaussian noise (AWGN) channel with CPE. The contributions of this paper are as follows: 1) We introduce an auxiliary random variable (RV) approach to study the asymptotic performance of wireless communication systems over the lognormal channels where the traditional asymptotic analysis techniques cannot be applied directly. This novel analytical approach can be potentially applied to other performance analysis problems involving the lognormal channels. 2) We derive exact asymptotic noise reference loss expressions for subcarrier BPSK and QPSK systems with CPE over the lognormal channels. Such asymptotic noise reference loss results can quantify the performance degradation introduced by the CPE. Our analysis reveals that the performance degradation introduced by the CPE for a BPSK system over the lognormal channels is negligible. 3) The asymptotic noise reference loss analysis is extended to a subcarrier MPSK system based on a highly accurate approximate BER expression of MPSK in AWGN channel with CPE. The obtained asymptotic noise reference loss expression is general, including the BPSK and QPSK results as special cases. II. SYSTEM MODEL For a subcarrier FSO system, the instantaneous receiver signal-to-noise ratio (SNR) is defined as the ratio of the

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SONG et al.: SUBCARRIER PHASE-SHIFT KEYING SYSTEMS WITH PHASE ERRORS IN LOGNORMAL TURBULENCE CHANNELS

time-averaged ac photocurrent power to the total noise variance [13], and it can be expressed as [5], [6] γ=

(P Rξ)2 2 I = γI 2 σn2

(1)

where P is the average transmitter power, R is the photodetector responsivity, ξ is the modulation index, σn2 is the AWGN variance, I is the turbulence induced channel gain, and γ is the electrical SNR assuming normalized mean of channel gain [14]. The SNR γ defined in (1) is SNR per symbol. For an M ary modulation scheme, we have γ = κγb and γ = κγ b where κ = log2 M and γb and γ b denote the instantaneous SNR per bit and the electrical SNR per bit, respectively.

fΘ (θ|γb ) =

exp (Cγb cos θ) . 2πI0 (Cγb )

1897

(5)

With a specific fading model, we can average the conditional pdf of the CPE over the pdf of SNR per bit γb and obtain the pdf of Θ as ∞ exp (Cγb cos θ) fΘ (θ) = fγ b (γb )dγb (6) 2πI0 (Cγb ) 0 where fγ b (γb ) is the pdf of the instantaneous SNR per bit. It is important to note that the condition ρ >> 1 holds in practice; therefore, the variance of the CPE σθ2 can be approximated by σθ2 ≈ 1/ρ [18], [20], [21]. We will make this assumption in our analysis.

A. The Lognormal Tubulence Channel Several statistical models have been proposed to describe irradiance fluctuations in FSO systems. It has been well accepted that the lognormal model describes irradiance fluctuations under weak turbulence conditions where the link length is several hundred meters in clear sky [15]. In a lognormal turbulence environment, the channel gain I is I = exp(X) where X is a Gaussian RV having mean μ and variance σ 2 . To facilitate our analysis, we normalize the mean of I and obtain its probability density function (pdf) as (ln I + σ 2 /2)2 1 exp − , I > 0 (2) fLN (I) = √ 2σ 2 2πσI where σ is the scintillation level whose typical value is less than 0.5 for FSO applications [14], [16].

In this work, the CPE is assumed to be derived from the pilot tone using a first-order phase-locked loop (PLL) in the presence of AWGN and fading. Thus, the CPE follows a Tikhonov distribution conditioned on the channel gain [17], [18]. The PLL SNR ρ can be expressed as ρ = Pc /(N0 BL ) where Pc is the power allocated to the carrier phase recovery pilot, N0 is the noise spectral density, and BL is the loop bandwidth. We further assume that a fixed fraction (ς) of the available total power Pt is allocated to the pilot (i.e., Pc = ςPt ) and the remaining fraction is for data detection. Therefore, we have (1 − ς)Pt = Eb /Tb where Eb is the energy per bit and Tb is the bit interval. In a fading environment, it is straightforward to obtain [18] 1 I 2 Eb 1 ς ς = γb . 1 − ς BL Tb N0 1 − ς BL Tb

(3)

From (3), we observe that the PLL SNR ρ is proportional to the instantaneous receiver SNR γb , i.e., ρ = Cγb where C is a constant defined as C = ς/[(1 − ς)BL Tb ] with typical value around 10 [18], [19]. The pdf of the CPE conditioned on the fading is given by [17] fΘ (θ|ρ) =

exp (ρ cos θ) , −π ≤ θ ≤ π 2πI0 (ρ)

A. Subcarrier BPSK System For nonideal coherent detection of BPSK, the conditional BER is Pb,BPSK (θ, γb ) =

(4)

where I0 (·) is the first kind modified Bessel function of order zero. Substituting ρ = Cγb into (4), we obtain

1 √ erfc ( γb cos θ) 2

(7)

where θ denotes the CPE and erfc(x) complementary error ∞ is the 2 √ function defined as erfc(x) = x 2e−t / πdt. Thus, the average BER of a subcarrier BPSK system over turbulence channels can be obtained as ∞ π Pb,BPSK = Pb,BPSK (θ, γb )fΘ (θ|γb )fγ b (γb )dθdγb 0

=

B. Phase Error Distributions

ρ=

III. ASYMPTOTIC NOISE REFERENCE LOSS ANALYSIS

1 4π

−π

∞ 0

π

−π

exp (Cγb cos θ) I0 (Cγb )

√ × erfc ( γb cos θ) fγ b (γb )dθdγb .

(8)

For a specific turbulence channel, eq. (8) can be used to evaluate the average BER of a subcarrier BPSK system. If we plot Pb,BPSK from (8) for a fixed PLL parameter C and the BER for an ideal coherent detection in the same figure, the relative performance gap between the two curves reveals the amount of CPE induced degradation in this performance measure. The noisy reference loss SNRL ,2 , defined as the amount of additional SNR (compared with the ideal coherent detection) required to achieve a specific BER in the presence of CPE [18], can be used to quantify this CPE introduced performance degradation. This noisy reference loss at a specific BER level depends on the value of the given BER or the corresponding average SNR [18]. The calculation of such noisy reference loss for turbulence channels is challenging at finite SNR. On the other hand, we can obtain the asymptotic noisy reference loss, denoted by SNR∞ L ,2 , when the SNR approaches ∞. This asymptotic measurement is independent of the average SNR. Most importantly, it can reveal some interesting insights into the system. In a lognormal turbulence environment, the receiver SNR γb is another lognormal RV having pdf 1 (ln γb − ln γ b + σ 2 )2 exp − fγ b ,LN (γb ) = √ (9) 8σ 2 2 2πσγb

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where ln γb is a Gaussian RV with mean σ 2 − ln γ b and variance 4σ 2 . Using (8) and (9), we can evaluate the BER of subacrrier BPSK systems in the lognormal turbulence channels. However, it is well-known that the asymptotic BER analysis cannot be performed on a lognormal channel directly because the diversity order of such a channel is undefined. In order to investigate the asymptotic noisy reference loss of subcarrier BPSK in the lognormal channels, we introduce the lognormal-Nakagami fading as an auxiliary channel model where the receiver SNR follows a lognormal-Gamma distribution. Since the lognormal-Gamma distribution approaches a lognormal distribution as the channel parameter m approaches infinity, the results obtained for the lognormal-Nakagami channels will reveal the performance characteristics of the lognormal channels when m → ∞. In a lognormal-Nakagami fading environment, the receiver SNR γb follows a lognormal-Gamma distribution with pdf ∞ m m −1 mγ m γb 1 b exp − fγ b ,LG (γb ) = √ Ω 2 2πσΩ 0 Ωm Γ(m) (ln Ω − ln γ b + σ 2 )2 × exp − dΩ (10) 8σ 2 where m is the Nakagami-m fading parameter and Ω is the second moment of a Nakagami-m RV. When m → ∞, we can show that [22] lim fγ b ,LG (γb ) =

m →∞

1 √ 2 2πσγb (ln γb − ln γ b + σ 2 )2 × exp − (11) 8σ 2

which, as expected, is the lognormal pdf given in (9). Substituting u = Ω/γ b into (10), we rewrite the lognormalGamma pdf as ∞ mm γ m −1 γ −m 1 mγb b exp − fγ b ,LG (γb ) = √ b uγ b 2 2πσΓ(m) 0 um +1 (ln u + σ 2 )2 × exp − du. (12) 8σ 2 Substituting (12) into (8), we obtain the average BER of subcarrier BPSK over the lognormal-Nakagami channels as Pb,BPSK,LG

∞ π ∞ m −1 mm γ −m γb exp (Cγb cos θ) √ b m u +1 I0 (Cγb ) 8π 2πσΓ(m) 0 −π 0 mγb (ln u + σ 2 )2 × exp − − uγ b 8σ 2 √ × erfc ( γb cos θ) dudθdγb . (13)

=

When γ b approaches ∞, we obtain from (13) the asymptotic BER of subcarrier BPSK in the lognormal-Nakagami channels as ∞ Pb,BPSK,LG =

gc G1 (m)γ −m b 4π

(14)

where mm gc = √ 2 2πσΓ(m)

∞

1 um +1

0

and G1 (x) is defined as ∞ G1 (x) =

π

−π

0

(ln u + σ 2 )2 exp − du 8σ 2 (15)

exp (Cγb cos θ) γb x−1 I0 (Cγb )

√ × erfc ( γb cos θ) dθdγb .

(16)

It is straightforward to obtain the asymptotic BER of subcarrier BPSK in the lognormal-Nakagami channels with perfect phase compensation as ∞ Pb,BPSK,LG =

gc Γ(m + 12 ) −m √ γb . 2 πm

(17)

From (14) and (17), we can derive the exact asymptotic noisy reference loss for the lognormal-Nakagami channels as mG1 (m) 1 10 log SNR∞ (m) = dB. (18) √ L ,2,LG m 2 πΓ(m + 12 ) It can be shown analytically that the asymptotic noisy reference loss in (18) for the lognormal-Nakagami channels approaches 0 dB when m → ∞, i.e., ∞ SNR∞ L ,2,LN = lim SNRL ,2,LG (m) = 0 dB m →∞

(19)

indicating that the CPE introduced exact asymptotic noisy reference loss of subcarrier BPSK is 0 dB in the lognormal channels. A detailed proof of (19) is given in Appendix A. B. Subcarrier QPSK System The BER of QPSK conditioned on the CPE and fading is [18] 1 √ erfc ( γb (cos θ + sin θ)) 4 1 √ + erfc ( γb (cos θ − sin θ)) . (20) 4 Thus, the average BER of a subcarrier QPSK system over turbulence channels can be obtained as ∞ π exp (γb cos θ) 1 Pb,QPSK = 8π 0 −π I0 (Cγb ) √ × [erfc ( γb (cos θ + sin θ)) √ + erfc ( γb (cos θ − sin θ))] fγ b (γb )dθdγb . (21) Pb,QPSK (θ, γb ) =

In order to investigate the asymptotic noisy reference loss of subcarrier QPSK in the lognormal channels, we will study the asymptotic noisy reference loss of subcarrier QPSK over the lognormal-Nakagami fading first. Substituting (12) into (21), we obtain the average BER of subcarrier QPSK over the lognormalNakagami channels as ∞ π ∞ mm γ −m mγb b √ exp − Pb,QPSK,LG = uγ b 16π 2πσΓ(m) 0 −π 0 2 2 γbm −1 exp (Cγb cos θ) exp − (ln u8σ+σ2 ) × um +1 I0 (Cγb )


√ × [erfc ( γb (cos θ + sin θ)) √ + erfc ( γb (cos θ − sin θ))] dudθdγb .

(22)

When γ b approaches ∞, we obtain from (22) the asymptotic BER of subcarrier QPSK in the lognormal-Nakagami channels as ∞ = Pb,QPSK,LG

gc G2 (m) −m γb 8π

(23)

where G2 (x) is defined as ∞ π exp (Cγb cos θ) γb x−1 G2 (x) = I0 (Cγb ) 0 −π √ × [erfc ( γb (cos θ + sin θ)) √ +erfc ( γb (cos θ − sin θ))] dθdγb .

(24)

gc Γ(m + 12 ) −m √ γb . 2 πm

The CPE induced asymptotic noisy reference loss of subcarrier QPSK in the lognormal channels can be obtained by letting m approach ∞ in (26) as

= 10 log

ˆ − C cos θˆ C + 1 + sin (2θ)

−1

M 2

G3 (x) =

∞

π

exp (Cγb cos θ) γb x−1 I0 (Cγb ) −π n =0 0 (2n + 1)π √ +θ dθdγb . ×erfc κγb sin M (33)

The asymptotic BER of subcarrier MPSK over the lognormalNakagami channels with perfect phase compensation has been presented in [24] as Pb,∞M PSK,LG =

1

When γ b approaches ∞, we obtain from (31) the asymptotic BER of subcarrier MPSK in the lognormal-Nakagami channels as gc G3 (m) −m ∞ γb (32) Pb,M PSK,LG = 4πκ where G3 (x) is defined as

(25)

From (23) and (25), we derive the exact asymptotic noisy reference loss for the lognormal-Nakagami channels as mG2 (m) 1 10 log SNR∞ (m) = dB. (26) √ L ,4,LG m 4 πΓ(m + 12 )

∞ SNR∞ L ,4,LN = lim SNRL ,4,LG (m) m →∞

Substituting (5), (12), and (29) into (30), we obtain the average BER of subcarrier MPSK over the lognormal-Nakagami channels as M 2 −1 ∞ π ∞

mm γ −m γbm −1 b √ Pb,M PSK,LG = 8π 2πσΓ(m)κ n =0 0 −π 0 um +1 (ln u +σ 2 ) 2 γb exp (Cγb cos θ) exp − m uγ b − 8σ 2 × I0 (Cγb ) (2n + 1)π √ +θ dudθdγb . κγb sin ×erfc M (31)

It is straightforward to obtain the asymptotic BER of subcarrier QPSK in the lognormal-Nakagami channels with perfect phase compensation as ∞ = Pb,QPSK,LG

1899

(27)

×

gc Γ (m) γ −m b 2πκ M −1

dk (Pk∞ (mk , Ak ) − Pk∞ (nk , Bk ))

k =1

where

θˆ = sin−1

C−

√

C 2 + 32 8

(34) .

(28)

Pk∞ (mk , Ak ) =

A detailed derivation of (27) is given in Appendix B. C. Subcarrier MPSK System A highly accurate approximate BER expression of MPSK conditioned on the CPE and fading has been presented in [23] as M 2

Pb,M PSK (θ, γb ) =

−1

1

√ erfc κγb sin 2κ n =0

(2n + 1)π +θ M

.

(29)

Therefore, the average BER of a subcarrier MPSK system over turbulence channels can be obtained as ∞ π Pb,M PSK = Pb,M PSK (θ, γb )fΘ (θ|γb )fγ b (γb )dθdγb . 0

−π

where

(30)

m k π 2 m sin θ dθ 0 . (κAk )m

In (34), the parameter dk is

κ

k

k

k

− k − + 2 dk = 2

i i M M 2 2 i=2

(35)

(36)

where |x| takes the absolute value of x and x rounds x to the nearest integer. In (35), we have mk = (M − 2k + 1)/M , nk = (M − 2k − 1)/M , Ak = sin2 ((2k − 1)π/M ), Bk = sin2 ((2k + 1)π/M ). From (32) and (34), we derive the asymptotic noisy reference loss for the lognormal-Nakagami channels as 1 10 log SNR∞ L ,M ,LG (m) = m G3 (m)/ (2Γ (m)) × M −1 dB. (37) ∞ ∞ k =1 dk (Pk (mk , Ak ) − Pk (nk , Bk ))

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Fig. 1. BERs of subcarrier BPSK over the lognormal channels under different turbulence conditions with the PLL parameter C = 10.

Fig. 2. BERs of subcarrier QPSK over the lognormal channels under different turbulence conditions with the PLL parameter C = 10.

The CPE induced asymptotic noisy reference loss of subcarrier MPSK in the lognormal channels can be obtained by letting m approach ∞ in (37). Following the detailed derivation in Appendix B and that in the Appendix of [24], we can obtain this asymptotic noisy reference loss as ∞ SNR∞ L ,M ,LN = lim SNRL ,M ,LG (m) m →∞ ⎛ ⎞ π κ sin2 M ⎠ dB = 10 log ⎝ π C − C cos θ˜ + κ sin2 M + θ˜

(38) where

π +θ . (39) θ˜ = argmin C − C cos θ + κ sin2 θ M The asymptotic noisy reference loss expression for subcarrier MPSK in (38) will specialize to that for subcarrier QPSK in (27) when M = 4. It will become 0 dB when M = 2 for a subcarrier BPSK system. IV. NUMERICAL RESULTS Fig. 1 presents BER curves of subcarrier BPSK with CPE over the lognormal channels under different turbulence conditions. The BER curves of subcarrier BPSK with perfect phase compensation and subcarrier differential binary phase-shift keying (DPSK) are also presented for comparison. We observe from Fig. 1 that the noise reference loss introduced by the CPE is negligible for a subcarrier BPSK system in the lognormal channels. The performance advantage of subcarrier BPSK over subcarrier DPSK can also be observed from Fig. 1. This interesting observation agrees with the 0 dB prediction from (19), and provides us strong motivation to use subcarrier BPSK because no tight phase synchronization is required. Fig. 2 presents BER curves of subcarrier QPSK with CPE over the lognormal channels under different turbulence conditions.

Fig. 3. BERs of subcarrier 8PSK over the lognormal channels under different turbulence conditions with the PLL parameter C = 10.

The BERs of subcarrier QPSK with perfect phase compensation and subcarrier differential quadrature phase-shift keying (DQPSK) are also presented for comparison. We observe from Fig. 2 that the noisy reference loss is around 0.9 dB for C = 10 at the BER level of 10−10 under the considered turbulence conditions, which agrees with the exact asymptotic prediction of 0.92 dB from (27). The performance advantage of subcarrier QPSK over subcarrier DQPSK can also be observed from Fig. 2. These observations suggest that phase compensation is critical in a subcarrier QPSK system. The BER curves are presented in Fig. 3 for a subcarrier 8PSK system over the lognormal channels under different turbulence conditions. We also present BERs of a subcarrier 8PSK system with perfect phase compensation for comparison. We observe from Fig. 3 that the noisy reference loss is around 1.9 dB for C = 10 at the BER level of 10−8 under the considered turbulence


Fig. 4. Exact asymptotic noisy reference loss of subcarrier BPSK system over the lognormal-Nakagami channel with different PLL parameter values.

conditions, which agrees with the exact asymptotic prediction of 1.88 dB from (38). These observations suggest that phase compensation is critical for a subcarrier 8PSK system. Based on the observed noisy reference loss values from Fig. 1–3, we conclude that the impact of CPE becomes stronger as the modulation order increases. This observation can also be made by studying the asymptotic noisy reference loss expression presented in (38) and verifying that it is an increasing function of the modulation order M . In Fig. 4, we present the exact asymptotic noisy reference loss of subcarrier BPSK system over the lognormal-Nakagami channel with different PLL parameter values. This asymptotic noisy reference loss is obtained from (18). We observe from Fig. 4 that the exact asymptotic noisy reference loss is a decreasing function of the channel parameter m, and it approaches 0 dB as m increases. This observation agrees with the theoretical prediction from (19) that the exact asymptotic noisy reference loss of subcarrier BPSK system over the lognormal channels is 0 dB. The exact asymptotic noisy reference losses of subcarrier QPSK system over the lognormal-Nakagami channel, which is obtained from (26), and that for the lognormal channels, which is obtained from (27), are presented in Fig. 5. We observe from Fig. 5 that the exact asymptotic noisy reference loss for the lognormal-Nakagami channel is a decreasing function of the channel parameter m, and it approaches that for the lognormal channels when m is sufficiently large. V. CONCLUSION In this work, we studied the BER performance of subcarrier BPSK and QPSK systems with CPE in the lognormal turbulence channels where the CPE is modeled as a Tikhonov RV. We quantified the CPE induced asymptotic noise reference loss over the lognormal channels analytically. This asymptotic noise reference loss analysis is extended to a subcarrier MPSK system based on a highly accurate approximate BER expression of MPSK in AWGN channel with CPE. The auxiliary channel method we used in this work can be potentially applied to other

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Fig. 5. Exact asymptotic noisy reference loss of subcarrier QPSK system over the lognormal-Nakagami channel and the lognormal channels with different PLL parameter values.

performance analysis problems involving the lognormal channels. Our derived asymptotic noise reference loss expressions can be used to predict the performance degradation introduced by CPE. Both analytical results and numerical results showed that the CPE induced performance degradation for a subcarrier BPSK system over the lognormal channel is negligible. As the modulation order increases, the impact of CPE becomes stronger and phase compensation becomes critical. APPENDIX A In order to prove (19), we investigate the function G1 (m) first. According to (16), we have G1 (m) π ∞ = −π

π 2

= − π2

π 2

=

− π2

0

γbm −1 exp(Cγb cos θ) √ erfc( γb cos θ)dγb dθ I0 (Cγb )

γbm −1 √ [exp(Cγb cos θ)erfc( γb cos θ) I (Cγ ) 0 b 0 √ + exp(−Cγb cos θ)erfc(− γb cos θ)] dγb dθ ∞ m −1 γb exp(Cγb cos θ) √ erfc( γb cos θ)dγb I0 (Cγb ) 0 ∞

γbm −1 exp(−Cγb cos θ) √ erfc( γb cos θ)dγb I (Cγ ) 0 b 0 ∞ m −1 2γb exp(−Cγb cos θ) dγb dθ. + I0 (Cγb ) 0

−

∞

(40)

Using the large argument approximations1 erfc(x) ≈

exp(−x2 ) x

(41)

1 These approximations are justified because the practical value of C is around 10, and lim γbm = 0 for 0 < γb < 1. m →∞

1902


and

Using (18), (19), (48), and (49), we obtain (50). exp(x) I0 (x) ≈ √ 2πx

(42)

and assuming an arbitrarily small quantity τ , we rewrite G1 (m) from (40) as G1 (m) π −τ 2 ≈ − π2

+τ

⎛ ∞

0

√ m−1 2 2πCγb 2 exp(−Cγb (1 + cos θ))dγb

√ 2πCγbm −1 exp −γb (C cos θ + C + cos2 θ) dγb − cos θ 0 ∞√ 2πCγbm −1 exp γb (C cos θ − C − cos2 θ) dγb dθ. + cos θ 0

∞

(43) Using the identity ∞

we express (43) as G1 (m) ≈

+ 10 log ⎝ lim

m →∞

tn exp(−kt)dt =

Γ(n + 1) k n +1

(44)

−τ

For large m values, we have √ √ 2 2πCΓ(m + 12 ) 2πCΓ(m) 1 cos θ(C − C cos θ + cos2 θ)m [C(1 + cos θ)]m + 2

(46)

√ 2πCΓ(m) 2πCΓ(m) . 2 m (C − C cos θ + cos θ) (C + C cos θ + cos2 θ)m (47) Therefore, when m → ∞ we can simplify (45) further to √ π −τ 2 2πCΓ (m) G1 (m) ∼ dθ. (48) 2 m − π2 +τ (C − C cos θ + cos θ) cos θ

lim

m →∞

(49)

m1 ⎞ ⎠

(50)

CΓ (m + 1) √ 2 cos θ0 Γ m + 12

m1 =1

(51)

and π 2

−τ

lim

m →∞

− π2 +τ

m1 1 dθ (C − C cos θ + cos2 θ)m 1 = max (52) C − C cos θ + cos2 θ

where θ ∈ [−π/2 + τ, π/2 − τ ]. For C > 2, we have π π 1 + τ, − τ . = 1, θ ∈ − max C − C cos θ + cos2 θ 2 2 (53) Therefore, SNR∞ L ,2,LN = 10 log (1) + 10 log (1) = 0 dB.

(54)

APPENDIX B In order to obtain (27), we investigate the function G2 (m) first. According to (24), we have

√

Since 1/cos θ is a continuous function when θ is within the range of [−π/2 + τ, π/2 − τ ], we can employ the first mean value theorem to show that there exists a θ value (say, θ0 ) within the range of [−π/2 + τ, π/2 − τ ] such that √ π −τ 2 2πCΓ(m) dθ π (C − C cos θ + cos2 θ)m cos θ − 2 +τ √ π −τ 2 2πCΓ(m) 1 = dθ. π cos θ0 − 2 +τ (C − C cos θ + cos2 θ)m

CΓ (m + 1) √ 2 cos θ0 Γ m + 12

√

√ 2πCΓ (m) π (C − C cos θ + cos2 θ)m cos θ − 2 +τ √ 2πCΓ (m) − (C + C cos θ + cos2 θ)m cos θ √ 2 2πCΓ m + 12 dθ. (45) + 1 [C(1 + cos θ)]m + 2 π 2

√

It is straightforward to show that

0

and

∞ SNR∞ L ,2,LN = lim SNRL ,2,LG (m) m →∞ ⎛ π m1 ⎞ 2 −τ 1 ⎠ = 10 log ⎝ lim dθ 2 m m →∞ − π2 +τ (C − C cos θ + cos θ)

G2 (m) = 0

∞

π

−π

exp (Cγb cos θ) γb m −1 I0 (Cγb )

√ × [erfc ( γb (cos θ + sin θ)) √ +erfc ( γb (cos θ − sin θ))] dθdγb .

(55)

Using the large argument approximations (41) and (42), we rewrite G2 (m) from (55) as (56) at the top of next page. Using the identity in (44), we express (56) as G2 (m) π ≈

√ 2πCΓ(m) (cos θ + sin θ)[C − C cos θ + (cos θ + sin θ)2 ]m −π √ 2πCΓ(m) + dθ. (cos θ − sin θ)[C − C cos θ + (cos θ − sin θ)2 ]m (57)


√

2πCγbm −1 exp γb C cos θ − C − (cos θ + sin θ)2 dγb G2 (m) ≈ cos θ + sin θ −π 0 ∞√ 2πCγbm −1 exp γb C cos θ − C − (cos θ − sin θ)2 dγb dθ + cos θ − sin θ 0

π

∞

It is straightforward to show that √ π 2πCΓ(m) dθ 2 m −π (cos θ + sin θ)[C − C cos θ + (cos θ + sin θ) ] √ π 2πCΓ(m) dθ = 2 m −π (cos θ − sin θ)[C − C cos θ + (cos θ − sin θ) ] √ π 2πCΓ(m) dθ = m −π (cos θ + sin θ)[C − C cos θ + 1 + sin(2θ)] (58) so that we can rewrite (57) as G2 (m) π ≈ −π

√ 2 2πCΓ(m) dθ. (cos θ + sin θ)[C − C cos θ + 1 + sin(2θ)]m (59)

Using (26), (27), and (59), we obtain (60). SNR∞ L ,4,LN = lim SNR∞ L , 4 , L G (m) m →∞

⎡

⎛

= ⎣10 log ⎝ lim

m →∞

+10 log

−π π

lim

m →∞

π

−π

√

CΓ(m + 1) √ dθ 2(cos θ + sin θ)Γ m + 12

1 dθ [C − C cos θ + 1 + sin(2θ)]m

m1 ⎞ ⎠

m1

dB (60)

Following the analysis in Appendix A, we can show that m1 √ π CΓ(m + 1) √ dθ = 1 (61) lim m →∞ 2(cos θ + sin θ)Γ m + 12 −π and m1 1 dθ m −π [C − C cos θ + 1 + sin(2θ)] 1 = max , θ ∈ [−π, π] C − C cos θ + 1 + sin(2θ)

lim

m →∞

π

=

1

(62)

ˆ C − C cos θˆ + 1 + sin(2θ)

where

θˆ = sin−1

C−

√

C 2 + 32 8

1903

.

(63)

(56)

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1904

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