Performance analysis of coherent optical 8-star QAM systems using decision-aided maximum likelihood phase estimation

doi:10.1364/OE.20.009302

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Performance analysis of coherent optical 8-star QAM systems using decision-aided maximum likelihood phase estimation

Hongyu Zhang, Pooi-Yuen Kam, and Changyuan Yu »View Author Affiliations

Optics Express, Vol. 20, Issue 8, pp. 9302-9311 (2012)
http://dx.doi.org/10.1364/OE.20.009302

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Abstract

An approximate bit-error rate (BER) expression of 8-star quadrature amplitude modulation (QAM) in the presence of a phase estimation error is derived. The accuracy of the approximate BER is verified via numerical integration of the conditional BER and the Monte-Carlo (MC) simulations. This approximation allows quick estimation of the BER performance and prediction of laser linewidth tolerance, and also facilitates optimization of the ring ratio.

1. Introduction

Recently, high-order modulation formats with coherent detection have attracted extensive attention, since they can greatly improve the spectral efficiency (SE) of optical systems. One major issue in a coherent optical communication system is to recover carrier phase, which is perturbed by phase noise generated from the laser linewidths of both the transmitter and local oscillator (LO) [1

K. P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).

]. Since the phase estimation (PE) is imperfect, the residual phase estimation error degrades the system performance. The bit-error rate (BER) performance of various modulation formats with phase estimation error has been considered in [2

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol. LT-2, 103–112 (1984).

–7

C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express 18(12), 12088–12103 (2010). [CrossRef] [PubMed]

From previous experience, it is important to optimize the signal constellations in the two-dimensional space [8

Y. Li, S. Xu, and H. Yang, “Design of signal constellations in the presence of phase noise,” in Proc. VTC’08-Fall, (Sept. 21–24, 2008, Calgary, Canada), pp. 1–5.

, 9

L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun. 4(5), 2600–2609 (2005). [CrossRef]

]. In the presence of phase noise, large angular distance between adjacent symbols gives a robust protection over the phase estimation error. Since the angular distance between adjacent symbols in 8-star quadrature amplitude modulation (QAM) is

π / 2,

which is larger than the other possible 8-ary constellations and 16QAM, it makes 8-star QAM more noise-tolerant. Moreover, compared with the 8-star QAM with 45 degrees phase offset in the inner ring, simulation results show that our 8-star QAM has a larger laser linewidth tolerance, and also 8-star QAM is easier to be implemented in experiments. Thus, 8-star QAM is a promising modulation format in coherent optical communication systems. Therefore, in this paper, we evaluate the BER performance of 8-star QAM in the presence of a phase estimation error and additive white Gaussian noise (AWGN). We also obtain a simple and accurate approximate BER, which allows quick estimations of the BER performance and laser linewidth tolerance. Finally, the ring ratio optimization algorithm is derived to optimize the placement of points in 8-star QAM, thus giving the minimum BER. This approach illustrates the procedure in detail how to carry out the analysis for 8-star QAM, and can also be extended to other duo-ring modulation formats to optimize the performance.

2. Exact BER of 8-star QAM

In coherent optical detection, digital-signal-processing (DSP) based PE algorithms, such as M-th power [6

G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express 14(18), 8043–8053 (2006). [CrossRef] [PubMed]

] and Decision-aided (DA) maximum likelihood (ML) [10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

], are preferred at the receiver for carrier phase recovery. The PE algorithm has a residual phase estimation error

Δ θ

which degrades the system performance.

In the presence of a random phase error, the BER

P (e_{B})

of a modulation format can be calculated as

P (e_{B}) = \int_{- π}^{π} P (e_{B} | Δ θ) p (Δ θ) d Δ θ,

(1)

where

P (e_{B} | Δ θ)

represents the BER conditioned on a phase error value

Δ θ,

and

p (Δ θ)

is the probability density function (PDF) of the phase error, which can be assumed as a Gaussian PDF with mean zero and variance

σ_{Δ θ}^{2}

G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express 14(18), 8043–8053 (2006). [CrossRef] [PubMed]

, 10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

, 11

W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall, 1973).

]. Besides laser phase noise, additive white Gaussian noise (AWGN) also exists in the system. The PDF of the additive noise is also Gaussian with mean zero and variance

σ_{n}^{2},

and the variance

σ_{n}^{2}

is assumed to be 1 in our simulations.

The constellation map with Gray coding and decision boundaries is shown in Fig. 1 . Here, in the figure, the dot lines are the decision boundaries.

A_{i}

denote the decision regions. The two hollow points represent the points when the two original signal points “000” and “100” are rotated by a phase error

Δ θ .

Observe that there are two rings in the 8-star QAM constellation. The inner ring radius is defined to be

R_{i n},

and the outer ring radius is

R_{o u t},

so the ring ratio

β

is equal to

R_{o u t} / R_{i n} .

We define

γ_{s}

to be the average energy per symbol, and

γ_{b} = γ_{s} / 3

is the average energy per bit, since there are 3 bits per symbol. Then

R_{i n}

can be written as

R_{i n} = α \times \sqrt{γ_{s}},

where

α

is the coefficient of the inner ring.

R_{o u t}

can be expressed as

R_{o u t} = β \times R_{i n} = β \times α \times \sqrt{γ_{s}} .

Since the mean of energies of the inner ring and outer ring should be equal to the average symbol energy

γ_{s},

therefore,

\frac{R_{i n}^{2} + R_{o u t}^{2}}{2} = \frac{α^{2} \times γ_{s} + β^{2} \times α^{2} \times γ_{s}}{2} = γ_{s} .

(2)

So we have the relation

Fig. 1 Constellation map, decision boundaries and Gray code mapping of 8-star QAM with a phase error

Δ θ .

\frac{α^{2} (β^{2} + 1)}{2} = 1, β >1 and 0< α <1 .

(3)

Due to the symmetry (assuming that the symbols “000” and “100” are transmitted) and equal probability of transmitting each symbol, the average BER conditioned on a phase error

Δ θ

can be written as

P (e_{B} | Δ θ) = 4 \times \frac{1}{8} P (e_{B} | Δ θ, 000) + 4 \times \frac{1}{8} P (e_{B} | Δ θ, 100) = \frac{1}{2} P (e_{B} | Δ θ, 000) + \frac{1}{2} P (e_{B} | Δ θ, 100),

(4)

Supposing that an inner ring symbol “000” is sent, the conditional BER

P (e_{B} | Δ θ, 000)

is defined as the expected number of bit errors made per bit detected [12

A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding (McGraw-Hill, 1979).

], and is given by

\begin{array}{l} P (e_{B} | Δ θ, 000) = \frac{1}{3} [P (A_{1} | Δ θ, 000) + 2 P (A_{2} | Δ θ, 000) + P (A_{3} | Δ θ, 000) + P (A_{4} | Δ θ, 000) \\ + 2 P (A_{5} | Δ θ, 000) + 3 P (A_{6} | Δ θ, 000) + 2 P (A_{7} | Δ θ, 000)], \end{array}

(5)

where

P (A_{i} | Δ θ, 000)

is the probability of the received signal point falling in the decision region

A_{i}

when the symbol “000” is sent. Observe that

U_{1} = A_{1} \cup A_{5} \cup A_{2} \cup A_{6}

and

U_{2} = A_{2} \cup A_{6} \cup A_{3} \cup A_{7}

are half-planes, and thus Eq. (5) can be represented as

P (e_{B} | Δ θ, 000) = \frac{1}{3} [P (U_{1} | Δ θ, 000) + P (U_{2} | Δ θ, 000) + P (A_{4} + A_{5} + A_{6} + A_{7} | Δ θ, 000)] .

(6)

It is easy to show that

P (U_{1} | Δ θ, 000) = P [n > α \cdot \sqrt{γ_{s}} \sin (\frac{π}{4} - Δ θ)] = \frac{1}{2} e r f c [α \cdot \sqrt{γ_{s}} \sin (\frac{π}{4} - Δ θ)],

(7a)

P (U_{2} | Δ θ, 000) = P [n > α \cdot \sqrt{γ_{s}} \cos (\frac{π}{4} - Δ θ)] = \frac{1}{2} e r f c [α \cdot \sqrt{γ_{s}} \cos (\frac{π}{4} - Δ θ)],

(7b)

\begin{array}{l} P (A_{4} + A_{5} + A_{6} + A_{7} | Δ θ, 000) = 1 - P (A_{1} + A_{2} + A_{3} + A_{0} | Δ θ, 000) \\ =1 - P [\begin{array}{l} \sqrt{γ_{s}} \frac{α}{2} (2 \cos Δ θ - β - 1) < n_{1} < \sqrt{γ_{s}} \frac{α}{2} (2 \cos Δ θ + β + 1), \\ - \sqrt{γ_{s}} \frac{α}{2} (2 \sin Δ θ + β + 1) < n_{2} < \sqrt{γ_{s}} \frac{α}{2} (- 2 \sin Δ θ + β + 1) \end{array}] \\ = 1 - \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 \cos Δ θ - β - 1)] \cdot [1 - \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 \cos Δ θ + β + 1)]] \\ \cdot \frac{1}{2} e r f c [- \sqrt{γ_{s}} \frac{α}{2} (2 \sin Δ θ + β + 1)] \cdot [1 - \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (- 2 \sin Δ θ + β + 1)]] . \end{array}

(7c)

Similarly, supposing that signal “100” is transmitted, the conditional BER is given by

P (e_{B} | Δ θ, 100) = \frac{1}{3} [P (U_{3} | Δ θ, 100) + P (U_{4} | Δ θ, 100) + P (A_{1} + A_{2} + A_{3} + A_{0} | Δ θ, 100)],

(8)

where

U_{3} = A_{1} \cup A_{5} \cup A_{2} \cup A_{6}

and

U_{4} = A_{2} \cup A_{6} \cup A_{3} \cup A_{7}

are half-planes. We get

P (U_{3} | Δ θ, 100) = \frac{1}{2} e r f c [α β \cdot \sqrt{γ_{s}} s i n (\frac{π}{4} - Δ θ)],

(9a)

P (U_{4} | Δ θ, 100) = \frac{1}{2} e r f c [α β \cdot \sqrt{γ_{s}} \cos (\frac{π}{4} - Δ θ)],

(9b)

\begin{array}{l} P (A_{1} + A_{2} + A_{3} + A_{0} | Δ θ, 100) = \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 β \cos Δ θ - β - 1)] \cdot [1 - \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 β \cos Δ θ + β + 1)]] \\ \cdot \frac{1}{2} e r f c [- \sqrt{γ_{s}} \frac{α}{2} (2 β \sin Δ θ + β + 1)] \cdot [1 - \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (- 2 β \sin Δ θ + β + 1)]] . \end{array}

(9c)

Combining Eq. (4) through Eq. (9c), the conditional BER in the presence of a random phase estimation error

Δ θ

in 8-star QAM can be obtained. The final expression is omitted for lack of space.

3. Distribution of phase estimation error in DA ML PE

With the conditional BER

P (e_{B} | Δ θ)

of 8-star QAM obtained, we consider the PDF of the phase estimation error

p (Δ θ) .

It is shown in [6

G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express 14(18), 8043–8053 (2006). [CrossRef] [PubMed]

, 10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

, 11

W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall, 1973).

] that the distribution of the phase error

Δ θ

is Gaussian

N (0, σ_{Δ θ}^{2}) .

The relations between phase error variance

σ_{Δ θ}^{2}

and linewidth per laser symbol duration product (

Δ ν \cdot T_{s}

) are also given under different PE methods in [6

G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express 14(18), 8043–8053 (2006). [CrossRef] [PubMed]

] and [10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

]. Here, we only consider the DA ML PE algorithm to recover the carrier phase as an illustration. The laser phase noise is a random-walk process [2

] characterized by a white Gaussian frequency noise with mean zero and variance

σ_{p}^{2} = 2 π (2 Δ ν) T_{s},

where

2 \cdot Δ ν

and

T_{s}

are the beat linewidth of the transmitter and LO and symbol duration, respectively. The phase estimation error variance

σ_{Δ θ}^{2}

of DA ML can be expressed, as described in [10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

], as

σ_{Δ θ}^{2} = \frac{2 L^{2} + 3 L + 1}{6 L} σ_{p}^{2} + \frac{1}{2 L} σ_{n'}^{2},

(10)

where

σ_{n'}^{2} = 1 / γ_{s},

and L is the memory length (number of past symbols used to estimate the carrier phase). The accuracy of this expression is verified by using Monte Carlo (MC) simulations in [10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

]. It shows that the analysis of the phase estimation error variance agrees well with the MC results. The optimal L is also given in [10

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

] as

L_{o p t} = ⌊ 0.25 \sqrt{1 + 24 σ_{n'}^{2} / σ_{p}^{2}} - 0.75 ⌋,

(11)

where

⌊ x ⌋

represents the largest integer less than or equal to

x .

Therefore, the analytical BER of 8-star QAM can be obtained by numerically integrating Eq. (1) for the BER conditioned on a phase estimation error value against the PDF of the phase error. Figure 2 illustrates that the analytical BER expression of 8-star QAM is verified by using MC simulations for different values of phase error variance

σ_{Δ θ}^{2} .

The perfectly coherent curve in Fig. 2 is obtained by setting

Δ θ = 0^{\circ}

in Eq. (7) and Eq. (9). As can be seen in Fig. 2, the analytical BERs agree quite well with the MC simulations.

Fig. 2 The BERs of 8-star QAM from analysis and MC simulations with different phase error variance.

4. Approximate BER in 8-star QAM

For analytical purposes, it is desirable to have a simple and accurate approximate BER with respect to SNR

γ_{s}

and phase error variance

σ_{Δ θ}^{2},

instead of doing numerical integration of the conditional BER. This approximation can be used to quickly estimate the BER performance and laser linewidth tolerance of a system.

It can be observed that in order to get the approximate BER, we need to simplify the four error-functions (erfc) multiplication in Eq. (7c) and Eq. (9c) to a single erfc. Since

1 - \frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 \cos Δ θ - β - 1)]

and

\frac{1}{2} e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 β \cos Δ θ - β - 1)]

are the most dominant terms in Eq. (7c) and Eq. (9c), respectively, therefore, the final conditional BER expression can be approximated as

\begin{array}{l} P (e_{B} | Δ θ) \approx \frac{1}{12} {\begin{cases} e r f c [α \sqrt{γ_{s}} \sin (\frac{π}{4} - Δ θ)] + e r f c [α \sqrt{γ_{s}} \cos (\frac{π}{4} - Δ θ)] \\ + e r f c [\sqrt{γ_{s}} \frac{α}{2} (- 2 \cos Δ θ + β + 1)] \end{cases}} \\ + \frac{1}{12} {\begin{cases} e r f c [α β \sqrt{γ_{s}} \sin (\frac{π}{4} - Δ θ)] + e r f c [α β \sqrt{γ_{s}} \cos (\frac{π}{4} - Δ θ)] \\ + e r f c [\sqrt{γ_{s}} \frac{α}{2} (2 β \cos Δ θ - β - 1)] \end{cases}} . \end{array}

(12)

It can be seen that only some single erfc terms are left in Eq. (12). Putting the 1st erfc term of Eq. (12) into Eq. (1), we obtain

P {(e_{B})}_{1 s t} = \int_{- π}^{π} e r f c [α \sqrt{γ_{s}} \sin (\frac{π}{4} - Δ θ)] \frac{1}{\sqrt{2 π σ_{Δ θ}^{2}}} \exp (\frac{- Δ θ^{2}}{2 σ_{Δ θ}^{2}}) d Δ θ .

(13)

The approximation

e r f c (x) < 1 / \sqrt{π} \exp (- x^{2})

and

\cos (Δ θ) \approx 1 - {(Δ θ)}^{2} / 2

are used in the derivation, so Eq. (13) can be further approximated to

\begin{array}{l} P (_{e_{B}) 1 s t} \approx \frac{1}{\sqrt{2 π^{2} σ_{Δ θ}^{2}}} \exp [- \frac{1}{2} α^{2} γ_{s} + \frac{1}{2} α^{4} {(γ_{s})}^{2} σ_{Δ θ}^{2}] \int_{- π}^{π} \exp [- \frac{1}{2 σ_{Δ θ}^{2}} {(Δ θ - α^{2} γ_{s} σ_{Δ θ}^{2})}^{2}] d Δ θ \\ = \frac{1}{\sqrt{π}} \exp [- \frac{1}{2} α^{2} γ_{s} + \frac{1}{2} α^{4} {(γ_{s})}^{2} σ_{Δ θ}^{2}], \end{array}

(14)

since

\int_{- π}^{π} \exp (- x^{2}) d x = \sqrt{π} .

The details of derivation of the other terms are left to the reader. Finally, the approximate BER in 8-star QAM can be expressed as

P (e_{B}) \approx \frac{1}{12} \sqrt{\frac{1}{π}} {\begin{cases} \exp [\frac{1}{2} α^{2} γ_{s} (α^{2} γ_{s} σ_{Δ θ}^{2} - 1)] + \exp [\frac{1}{2} α^{2} β^{2} γ_{s} (α^{2} β^{2} γ_{s} σ_{Δ θ}^{2} - 1)] \\ + \frac{1}{2} {{[1 + α^{2} (β - 1) σ_{Δ θ}^{2} γ_{s}]}^{- \frac{1}{2}} + {[1 + α^{2} β (β - 1) σ_{Δ θ}^{2} γ_{s}]}^{- \frac{1}{2}}} \cdot \exp [- γ_{s} \cdot \frac{α^{2} {(β - 1)}^{2}}{4}] \end{cases}} .

(15)

The approximation in Eq. (15) is plotted in Fig. 3 for

Δ ν T_{s} = 10^{- 5}

and

10^{- 4},

respectively, which correspond to linewidth per laser

Δ ν

of 100 kHz and 1 MHz in a 10Gsymbol/s system. Exact numerical results and MC simulations are also presented to show that the approximation is accurate. Here, optimum memory length Eq. (11) is used to calculate the phase estimation error variance in Eq. (10). Furthermore, it is also found that the derived approximate BER can quickly estimate the laser linewidth tolerance of 8-star QAM in DA ML PE. The laser linewidth tolerance is examined and plotted in Fig. 4 based on our numerical integration and approximate results. The SNR reference is the SNR/bit (

γ_{b}

) at BER = 10⁻⁴ in the perfectly coherent case, which is 11.7dB. As shown in Fig. 4, by using the approximate BER, the estimated value of

Δ ν \cdot T_{s}

leading to a 1-dB SNR/bit (

γ_{b}

) penalty at BER = 10⁻⁴ is

3 \times 10^{- 4}

for 8-star QAM. The estimated value of laser linewidth tolerance is smaller than the one using exact numerical BER, which is

6.5 \times 10^{- 4} .

This is because the approximate BER is an upper bound of the exact numerical BER. Therefore, it provides a more rigid laser linewidth requirement than the exact BER.

Fig. 3 Comparison between numerical BER, MC simulations, and approximate BER under different laser linewidth with optimum memory length.

Fig. 4 The SNR/bit penalty as a function of linewidth per laser symbol duration product (

Δ ν \cdot T_{s}

) when BER = 10⁻⁴ from numerical integration and approximation.

5. Ring ratio optimization

Generally, varying the radii of the two rings in 8-star QAM would severely affect its performance. Therefore, we need to determine the optimal ring ratio. Based on Eq. (15), an optimal ring ratio

β_{o p t}

can be obtained to give the minimum BER

P (e_{B}) .

From Eq. (15), the term

\exp [\frac{1}{2} α^{2} β^{2} γ_{s} (α^{2} β^{2} γ_{s} σ_{Δ θ}^{2} - 1)]

can be dropped for simplicity because it is much smaller than

\exp [\frac{1}{2} α^{2} γ_{s} (α^{2} γ_{s} σ_{Δ θ}^{2} - 1)]

since the terms in the parentheses are both negative and

β^{2} > > 1.

And also the terms

α^{2} (β - 1) σ_{Δ θ}^{2} γ_{s}

and

α^{2} β (β - 1) σ_{Δ θ}^{2} γ_{s}

can be dropped since they are much less than 1. The dominant terms are left as

\frac{1}{12} \sqrt{\frac{1}{π}} {\exp [\frac{1}{2} α^{2} γ_{s} (α^{2} γ_{s} σ_{Δ θ}^{2} - 1)] + \exp [- γ_{s} \cdot \frac{α^{2} {(β - 1)}^{2}}{4}]} .

(16)

We substitute the relation

β = \sqrt{2 / α^{2} - 1}

into Eq. (16), use the approximation

e^{x} \approx 1 + x,

differentiate it by

α

and make it equal to 0, drop

α^{2}

in the

\sqrt{2 - α^{2}}

term since

α^{2} < < 1,

and arrive at the result

2 \sqrt{2} γ_{s} σ_{Δ θ}^{2} α^{3} - α^{2} - \sqrt{2} α + 1 = 0.

(17)

By solving Eq. (17), we get the solution

A = \frac{\sqrt{2}}{48 {(γ_{s} σ_{Δ θ}^{2})}^{2}} + \frac{\sqrt{2}}{864 {(γ_{s} σ_{Δ θ}^{2})}^{3}} - \frac{\sqrt{2}}{8 γ_{s} σ_{Δ θ}^{2}},

(18a)

B = A^{2} - {[\frac{1}{6 γ_{s} σ_{Δ θ}^{2}} + \frac{1}{72 {(γ_{s} σ_{Δ θ}^{2})}^{2}}]}^{3},

(18b)

α_{o p t} = \frac{1}{6 \sqrt{2} γ_{s} σ_{Δ θ}^{2}} + \frac{- 1 - \sqrt{3} i}{2} \sqrt[3]{A + \sqrt{B}} + \frac{- 1 + \sqrt{3} i}{2} \sqrt[3]{A - \sqrt{B}},

(18c)

β_{o p t} = \sqrt{\frac{2}{α_{o p t}^{2}} - 1} .

(18d)

From Eq. (18a) to (18d), it can be seen that optimal ring ratio

β_{o p t}

is a function of SNR/symbol

γ_{s}

and phase error variance

σ_{Δ θ}^{2} .

The term

β_{o p t}

as a function of

γ_{s} \cdot σ_{Δ θ}^{2}

is plotted in Fig. 5 . We also scan the numerical BER of Eq. (1) as a function of the ring ratio

β

given a fixed average

γ_{b}

= 15dB and

σ_{Δ θ}^{2} = 1 \times 10^{- 3} r a d^{2},

and the result is shown in Fig. 6 . It is shown that given the above condition the optimal ring ratio of 8-star QAM from numerical integration is 2.4. Then we substitute the same parameters into Eq. (18a) to (18d), and obtain the value of 2.45 for the optimal ring ratio from the approximation. The ring ratio shift is only 0.05, which corresponds to an SNR per bit penalty of 0.015dB at BER = 10⁻⁴. We also examine the SNR penalty due to the ring ratio shift under this condition as depicted in Fig. 7 . This figure illustrates that 8-star QAM can tolerate a large ring ratio fluctuation around the optimal ring ratio value. Although numerical integration can be used to determine the optimal ring ratio, it is time consuming. As can be seen in Fig. 6, a large range of values of the ring ratio has to be scanned to locate the optimal value. The ring ratio optimization algorithm gives us an explicit way to see that the optimal ring ratio is not a constant, but depends on the product of SNR/symbol and phase estimation error variance. It also allows engineers to make a quick estimation of the optimal ring ratio under different parameters.

Fig. 5 Optimal ring ratio as a function of SNR/symbol phase error variance product.

Fig. 6 BER as a function of ring ratio

β

from numerical integration given

γ_{b} = 15 d B

and

σ_{Δ θ}^{2} = 1 \times 10^{- 3} r a d^{2} .

Fig. 7 SNR penalty due to ring ratio shift

6. Conclusion

In this paper, the conditional BER expression of 8-star QAM in the presence of a phase estimation error and AWGN is derived. The numerical results can be obtained by numerically integrating Eq. (1) for the BER conditioned on a phase error value against the PDF of the phase error. An approximate BER is also presented analytically through a series of approximations. The approximate BER can be used in any PE algorithms given the PDF of the phase estimation error. It is found that this approximation can quickly estimate the BER performance under different parameters, such as SNR/bit, laser linewidth and symbol rate, and also can predict the lower bound on the laser linewidth requirement. The accuracy of the approximation is confirmed via numerical integration and MC simulation.

To optimize the BER performance, the approximate BER is minimized with respect to the ring ratio. This leads to a new ring ratio optimization algorithm. It is found that the optimal ring ratio is not a constant, but depends on two parameters, namely, SNR/symbol and phase error variance. This algorithm can provide us a quick estimation of the optimal ring ratio for different parameters, instead of using time-consuming numerical integration and scanning over a large range of values of ring ratio. Therefore, our work shows how to place the signal points to optimize the performance in 8-star QAM, and can also be applied for other duo-ring modulation formats.

Acknowledgments

This work was supported by A*STAR SERC PSF 092 101 0054.

References and links

1.	K. P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).
2.	K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol. LT-2, 103–112 (1984).
3.	P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun. 41(7), 1020–1022 (1993). [CrossRef]
4.	Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun. 142(5), 292–296 (1995). [CrossRef]
5.	M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun. 43(2), 1192–1201 (1995). [CrossRef]
6.	G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express 14(18), 8043–8053 (2006). [CrossRef] [PubMed]
7.	C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express 18(12), 12088–12103 (2010). [CrossRef] [PubMed]
8.	Y. Li, S. Xu, and H. Yang, “Design of signal constellations in the presence of phase noise,” in Proc. VTC’08-Fall, (Sept. 21–24, 2008, Calgary, Canada), pp. 1–5.
9.	L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun. 4(5), 2600–2609 (2005). [CrossRef]
10.	S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]
11.	W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall, 1973).
12.	A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding (McGraw-Hill, 1979).

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4080) Fiber optics and optical communications : Modulation

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 15, 2012
Revised Manuscript: April 2, 2012
Manuscript Accepted: April 2, 2012
Published: April 6, 2012

Citation
Hongyu Zhang, Pooi-Yuen Kam, and Changyuan Yu, "Performance analysis of coherent optical 8-star QAM systems using decision-aided maximum likelihood phase estimation," Opt. Express 20, 9302-9311 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-9302

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References

K. P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).
K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).
P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993). [CrossRef]
Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun.142(5), 292–296 (1995). [CrossRef]
M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun.43(2), 1192–1201 (1995). [CrossRef]
G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express14(18), 8043–8053 (2006). [CrossRef] [PubMed]
C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express18(12), 12088–12103 (2010). [CrossRef] [PubMed]
Y. Li, S. Xu, and H. Yang, “Design of signal constellations in the presence of phase noise,” in Proc. VTC’08-Fall, (Sept. 21–24, 2008, Calgary, Canada), pp. 1–5.
L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun.4(5), 2600–2609 (2005). [CrossRef]
S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol.28(11), 1597–1607 (2010). [CrossRef]
W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall, 1973).
A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding (McGraw-Hill, 1979).

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