The Bit-Error-Ratio (BER) floor caused by the laser phase noise in the optical fiber communication system with differential quadrature phase shift keying (DQPSK) and coherent detection followed by digital signal processing (DSP) is analytically evaluated. An in-phase and quadrature (I&Q) receiver with a carrier phase recovery using DSP is considered. The carrier phase recovery is based on a phase estimation of a finite sum (block) of the signal samples raised to the power of four and the phase unwrapping at transitions between blocks. It is demonstrated that errors generated at block transitions cause the dominating contribution to the system BER floor when the impact of the additive noise is negligibly small in comparison with the effect of the laser phase noise. Even the BER floor in the case when the phase unwrapping is omitted is analytically derived and applied to emphasize the crucial importance of this signal processing operation. The analytical results are verified by full Monte Carlo simulations. The BER for another type of DQPSK receiver operation, which is based on differential phase detection, is also obtained in the analytical form using the principle of conditional probability. The principle of conditional probability is justified in the case of differential phase detection due to statistical independency of the laser phase noise induced signal phase error and the additive noise contributions. Based on the achieved analytical results the laser linewidth tolerance is calculated for different system cases.
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Coherent detection of optical signals is a very attractive technology for applications in communication systems. It provides potential for superior signal sensitivity and enables efficient compensation for the signal distortions due to the availability of the optical field envelope parameters, including both phase and amplitude, in the electrical domain. The best performance is achieved when the optical signal and the local oscillator (LO) lasers are ideally aligned in frequency and synchronized in phase. In order to have phase synchronization in such an analogue homodyne system a phase locked loop (PLL) needs to be implemented as part of the receiver. The inherent PLL delay in combination with the LO laser phase noise gives a complicated and expensive design, and in practical terms semiconductor external cavity lasers with kHz linewidth are required for stable loop operation [1–3].
Modern optical fiber communication systems target the bit rates beyond 100Gb/s per wavelength channel. Such a high transmission capacity requires the application of advanced modulation formats and coherent detection of optical signals. Coherent detection of the optical signals followed by electronic digital signal processing (DSP) [4, 5] is the step forward in the communication technology which enables higher capacity and at the same time more flexible, cost effective and energy efficient systems. Flexible compensation for the optical signal source and the fiber propagation impairments such as carrier phase recovery [6–10], chromatic and polarization mode dispersion compensation, the optical signal polarization tracking - altogether in the electrical domain [4, 5] using digital filters ; feasibility for high spectral efficiency due to very dense stacking of wavelength channels, higher receiver sensitivity in comparison with direct power detection, and a substantially reduced number or complete elimination of expensive and energy-consuming components or passive components with optical losses such as optical amplifiers, optical polarization controllers/trackers and dispersion compensating fibers are the features of modern communication systems taking advantage of a recent progress in high-speed DSP technology.
In this paper we present novel results on the analytical specification of the system Bit-Error-Ratio (BER) and on the system tolerance to the laser linewidth when the impact of the additive/shot noise is negligibly small and the laser phase noise dominates the system performance. These results are achieved for the system employing quadrature phase shift keying (QPSK) modulation format and an in-phase and quadrature (I&Q) receiver with feed-forward carrier recovery using the so-called block-wise average DSP method . We also present a new analytical BER results for the case when the QPSK receiver is operated in differential phase detection mode.
This paper is organized as follows: Section 2 represents the theoretical outline for the receiver including the impact of the additive/shot and the laser phase noise contributions; Section 3 provides the derivation of the system BER floor of the I&Q receiver with block-wise carrier recovery and presents the comparison of the analytical results with brute-force simulations using the Monte Carlo method; the importance of the phase unwrapping in the I&Q receiver with the carrier recovery is illustrated in Section 4 using analytical derivations; Section 5 summarizes the results on a closed form solution for the system BER with differential phase detection; Section 6 is devoted to discussions of the results presented in the paper as well as of the future work.
2. Theoretical outline
In a system employing coherent detection the optical signal field at the receiver site is mixed with a reference field generated by a LO laser and converted into the electrical domain in such a way that the electrical signal is proportional to the optical field envelope of the signal. In the system with DSP the received signal is sampled by analog-to-digital converters (ADC) and processed by a digital processor in order to compensate for the signal distortions and to extract the encoded data. Let us consider the well know model for the complex amplitude of the QPSK signal at the ADC output of phase diversity receiver
The additive noise is described by a complex uncorrelated Gaussian process (normalized to the signal amplitude) with zero mean and variance , whereis the signal to additive noise ratio per bit. The laser phase noise (white frequency noise) is a random walk process corresponding to the Lorentzian laser line-shape. The phase difference between two time instances is Gaussian distributed with zero mean and the variance proportional to the product of the symbol period and total laser linewidth of the signal and the LO lasers: .
The superior sensitivity of the receiver is achieved when the encoded data in the form of the quadrant number is extracted from the signal based on the value of the complex amplitude - signal quadratures (real and imaginary parts), provided that the impact of the laser phase noise is negligibly small. The effect of the laser phase noise randomly distributes the signal constellation points in the complex plane on a circle and should be eliminated or compensated for in order to minimize the error probability of detecting a wrong quadrant number. In the ideal case, when the random walk of the phase is completely eliminated, the BER of the QPSK system with differential data encoding – differential QPSK (DQPSK) - is well known and specified by the complementary error function – see e.g :
3. I&Q receiver with feed-forward carrier recovery using block-wise averaging
The availability of high-speed electronic components for DSP opens the way to various methods for the laser phase noise compensation and carrier phase recovery. The carrier phase recovery (estimation) can be performed using several feed-forward and decision-feedback algorithms [6–11]. In this paper we limit our consideration to a rather simple feed-forward carrier phase estimation algorithm proposed in Ref , which is well suited for real-time high-speed DSP implementation . By applying this algorithm a common phase value is evaluated for a block of signal samples and subtracted from the received signal prior to making a decision on the data extracted from the signal: . The carrier phase is estimated by raising the signal amplitude to the power of four in order to get rid of the phase modulation due to encoded data and by averaging contributions from a block of signal samples:6, 7]. If the quadrant number difference should be incremented by 1, if it should be decreased by 1, otherwise it should be left unchanged. This correction is essential for an adequate performance of the receiver. However, it compensates only for the leading order contribution to the system errors due to wrapping of the estimated phase into the interval and unphysical phase jumps close to the boundaries [6, 7]. In situations when the physical course of the block phase leads to the phase difference between blocks of the order of (in absolute value) or higher, the correction does not work correctly and leads to the system errors.
In the following we will demonstrate that errors occurring at block transitions are dominating at the limit of large SNR (at the BER floor). Disregarding errors occurring at block transitions, and accounting for errors occurring inside the block of symbols only, leads to an underestimated BER floor and an overly optimistic evaluation of the system performance, especially when the laser phase noise is significant.
Let us now analytically derive the system BER floor. At the BER floor the contribution from the additive noise is negligibly small and therefore the signal phase error can be written (using Eq. (1) and Eq. (3)) in the following form:Eq. (6) as a linear combination of independent, identically distributed Gaussian random variables Eq. (7) in Eq. (6) and taking into consideration Eq. (8) we arrive at the form of , which enables straightforward derivation of the phase error varianceEq. (9):13], in Eq. (10) gives the following expression for the phase error variance:Eq. (11) depends on the symbol position within the block (see Fig. 1 ) and reaches the maximum value of at the first and the last symbols in the block. At the centre of the block the phase error variance is at the minimum value of , when or , when
In order to account for the contribution to the BER due to errors occurring at transitions between blocks one also needs to derive the variance for the estimated phase difference. The variance in this case is estimated accounting for three statistically independent contributions: the phase error at the end of the block, the phase random walk during one symbol period and the phase error at the beginning of the following block:
The dependency of the maximum and minimum values of the phase error variance within the block together with the variance for the estimated phase difference at the block transition given by Eq. (12) versus the block size parameter is illustrated in Fig. 2 . This illustration shows that errors, which occur at the block transitions, cause the dominating contribution to the BER floor. Therefore, the BER floor is specified in the following form (see Appendix):Eq. (13) we performed full Monte Carlo simulations and counted errors among 108 received symbols. The results of the system BER evaluation using the Monte Carlo simulation method (symbols) and the analytical expression given by Eq. (13) (solid lines) are shown in Fig. 3 when the block size parameter is equal to 1, 2, 3, 5, 10 and 15, respectively.
The results shown in Fig. 3 demonstrate that Eq. (13) provides an accurate estimation of the system BER when , i.e. the when the laser phase noise variance is much less than 0.3, 0.2, 0.16, 0.1, 0.05 and 0.03 for the block size parameter of 1, 2, 3, 5, 10 and 15, respectively. The analytically estimated BER floor shown in Fig. 3 is in excellent agreement with the Monte Carlo simulation results down to the level of 10−7 and even lower. Using the analytical estimation leads to the drastic savings in the CPU computing time and makes it possible to consider BER floors below 10−8 whereas this is not feasible using a direct Monte Carlo simulation method.
Using Eq. (13) one can easily estimate the system tolerance to the laser phase noise. The upper limit for the total laser linewidth for the system BER floor at 10−4, 10−6, and 10−9 versus the block size parameter is shown in Fig. 4 at the symbol rate of 10 Gsymbol/s. Such type of data is very useful when selecting the laser sources to be used in the system at the beginning of the system design work: the BER floor level should be well below the forward error correction (FEC) threshold and should be low enough to reserve the system budget for penalties due to additive noise, chromatic dispersion and other signal distortions.
We would like to emphasize here that the results presented in Fig. 4 should not lead to a conclusion that a shorter block length is always preferable for the system performance. These results are obtained when the additive noise contribution is negligibly small. In a more general case the system penalty attributed to the additive noise is in inverse proportion to the block size, because increasing the block size reduces the correspondent part of the signal phase error due to averaging among signal samples. In the case when both the additive and laser phase noise contributions are significant the system BER can be optimized by selecting an optimum value of the block size parameter.
Accounting for the additive noise contribution in Eq. (4) and applying similar approximation as in Eq. (5) leads to a more general expression for the phase error variance: . Unfortunately, we found out that accounting for the additive noise in such a straightforward way does not always lead to a satisfactory agreement of the analytically estimated BER with the results of Monte Carlo simulations. The discussion of this problem is presented in the Section 6 of this paper.
4. I&Q receiver performance without phase unwrapping
In order to emphasize the importance of the estimated phase slips correction for the system performance we also derived the expression for the BER floor when such a correction (phase unwrapping) is not applied (see the Appendix). The BER floor in this case is given by the following expressionEq. (14) when the block size parameter is equal to 1, 2, 3, 5, 10 and 15 is shown in Fig. 5 together with the results of the system simulation using Monte Carlo method with direct error counting. Figure 5 demonstrates excellent agreement between BER floor evaluated using the analytical expression given by Eq. (14) and Monte Carlo simulation results. This figure should be compared to Fig. 3 where the estimated phase unwrapping at the receiver is included in the BER evaluation. It is apparent that the BER results shown in Fig. 5 are several orders of magnitude higher than the corresponding cases shown in Fig. 3. At high count of the block size parameter and small phase noise variance the BER floor when not using phase unwrapping is the following:Equation (15) shows that the total laser linewidth should be not more than 7.4 kHz in order to achieve the BER floor at 10−4 for the block size of 20. This should be compared to the value of 9 MHz when phase unwrapping is used, see Fig. 4. This emphasizes the importance of the estimated phase unwrapping as part of the receiver concept proposed in the references  and .
5. Differential phase detection
It is well known that the data recovery based on the received signal quadratures instead of quadratic product of the signal amplitude (or power) provides potential for better sensitivity in terms of SNR. However, we saw above that the performance of the I&Q receiver with carrier phase recovery is limited by the effect of the laser phase noise, especially, for the large values of the block size parameter. The larger the block size parameter the more random walk drift of the phase is expected that leads to increased probability for the error.
Let us now consider a simpler coherent receiver (in terms of signal processing), which is operated in demodulation mode when the encoded data is recovered by a simple “delay and multiply algorithm” in the electrical domain. In such a case the encoded data is recovered from the received signal based on the phase difference between two consecutive symbols, i.e. the value of the complex decision variable , which using Eq. (1) is given by the following expression14] because the laser phase noise induced distortion to the phase of the decision variable Ψ is statistically independent of the additive noise contribution. First, the BER is evaluated assuming that the phase error in Eq. (16) is a constant (see for example [12,15]) and then the resulted expression is averaged using the phase error distribution:16,17].
Equation (17) provides full information for the system BER in terms of SNR and tolerance to the laser phase noise. The BER for the differential phase receiver versus the SNR is shown in Fig. 6 for the total laser linewidth of 4 MHz, 10 MHz, 20 MHz and 40 MHz as well as for the case of no laser phase noise. The BER for the ideal I&Q receiver estimated using Eq. (2) is also shown in Fig. 6 for comparison.
Please note that in the limit of large SNR the BER given by Eq. (17) for the differential phase receiver is identical to the BER floor of the synchronous I&Q receiver with the carrier phase recovery given by Eq. (13) when the block size parameter. We have also found out, by performing numerical Monte Carlo simulations, that when the BER of both receivers is the same at an arbitrary value of the SNR parameter.
When deriving the BER for the receiver with differential phase detection given by Eq. (17) the principal of conditional probability has been used. It is not straightforward to apply this principle to the I&Q receiver with carrier phase recovery because the phase error in this case is correlated to the additive noise. This correlation occurs because the block phase is estimated by averaging the signal samples distorted by the additive noise, see Eq. (3). However, it has been demonstrated in the reference  that applying the principle of conditional probability when disregarding the correlation between the phase error and the additive noise in the case of I&Q receiver with carrier phase recovery leads to an accurate estimation of the BER when the laser linewidth is below 2 MHz (total linewidth below 4 MHz) in the system operated at 10 Gsymbol/s and when the block size parameter is at the optimum value. In such a case the SNR penalty relative to the BER of ideal I&Q receiver is small (few tenths of a dB) and is due to the phase errors caused by both, the additive and the laser phase noise, see Fig. 6 in the reference . When estimating the BER the authors of the reference  have also assumed that the variance of the phase error is independent of the symbol position in the block. In the range of the system parameters considered in  the BER floor has been negligibly small and is well below the estimated BER values. For instance, for the laser linewidth of 2 MHz (total linewidth of 4 MHz) the BER floor estimated using Eq. (13) derived in this paper is 10−8 for the block size of 14, and the BER is estimated to 2∙10−4 for the SNR parameter used in the reference .
In order to develop a more general theory applicable in a wide range of the laser phase and the additive noise parameters both approaches, published in  and presented in this paper have to be developed further.
The BER floor induced by the laser phase noise is an important system performance parameter especially at increased laser linewidth. In this paper we considered the total laser linewidth up to 100 MHz (or 50 MHz per each laser). Such a linewidth is too high when compared to a typical maximum linewidth of a few MHz for a fixed-wavelength distributed-feedback (DFB) edge-emitting laser used in single-mode fiber communication systems. The reason to consider the laser linewidth range well beyond a few MHz is to include into consideration the laser sources that due to their functionality or cost have an increased amount of the laser phase noise. For instance, it would be of a clear advantage for the flexibility and the reduced operational cost of the system to use the wavelength-tunable distributed Bragg reflector (DBR) lasers. This type of lasers has more complicated cavity design than fixed-wavelength DFBs and is driven by a several current sources enabling the wavelength tunability while maintaining the single-longitudinal-mode operation. Such a laser complexity is very likely to lead to increased laser linewidth. Another example is the vertical-cavity surface-emitting lasers (VCSEL) that are very attractive for the cost-sensitive applications such as access networks. VCSELs are fabricated at a substantially reduced cost; however, they are characterized by the laser linewidth of the order of 50-100 MHz and randomly fluctuating polarization of the output beam, and therefore are at present qualified only for the systems with direct power detection. It is very likely that this type of laser will be developed further in the future for the application in the fiber transmission systems with optical coherent detection.
Finally, in the outlook for the future work it would be interesting to apply the analytical approach for the BER estimation in a system employing new type of feed-forward phase estimators: the Wiener filter based phase estimator and multiplier-free phase estimation algorithm based on the barycentre approach recently proposed in  and , respectively.
In conclusion, we analytically derived an expression for the BER floor induced by the laser phase noise in the DQPSK system with optical coherent I&Q receiver and carrier phase recovery using DSP. The carrier phase recovery algorithm considered in this paper is based on the phase estimation of the sum of a finite number (block) of the signal samples raised to the power of four and the phase unwrapping at block transitions. The system tolerance to the laser phase noise is analytically estimated emphasizing higher requirements for the laser linewidth for a larger block size parameter. The importance of the phase unwrapping is emphasized for adequate operation of the system and the BER floor in the analytical form is derived for the case when the phase unwrapping operation is not applied. The analytically estimated phase noise induced BER floor has been compared to the results of brute-force Monte Carlo simulations and excellent agreement has been found confirming the validity of the analytical approach.
The BER for the QPSK receiver with differential phase detection is presented in a new closed form expression by using the principle of conditional probability and compared to the BER of the I&Q receiver. The BER of both types of receivers influenced by the laser phase noise and additive noise is found to be identical when the block size parameter is equal to one.
Future work on the analytical specification of the BER for the QPSK system with carrier phase recovery should aim at accounting for the additive noise in the model presented in this paper. It should be noted that due to the carrier phase recovery the signal phase error is correlated to the additive noise and therefore, the principle of conditional probability cannot be applied in this case in a straightforward way. The future work should also aim at taking into analytical consideration the new type of feed-forward phase estimators based on Wiener filter and the barycentre approaches.
Appendix: Derivation of BER floor in I&Q receiver with feedforward carrier recovery
It is intuitively understood and justified above (see Eq. (12) and Fig. 2) that the errors that occur at block transitions cause the dominating contribution to the system BER at the limit of high SNR. It is equivalent to the assumption that the system BER floor is proportional to the probability of the estimated phase slips, i.e. to the probability of , where is the estimated phase jump correction:
In the case when the estimated phase jumps are not corrected, i.e. the symbol error rate can be estimated as follows:
where the conditional probability density function (PDF) of the estimated phase is given by a folded Gaussian distribution:
The estimated phases and are statistically independent and wrapped in the modulus range due to the method of carrier phase recovery given by Eq. (3). It is accounted for in Eq. (A2) that the estimated phase is a random walk process and therefore is uniformly distributed in the range from to . The conditional PDF of is given by a folded Gaussian distribution with the mean equal to as illustrated in Fig. A1 (top). Integrating over in Eq. (A2) and accounting only for the leading order contributions gives:
Where now , and originates from the PDF given by Eq. (A3), which is unfolded into a modulus π range due to the phase jump correction, see Fig. A1 (bottom). The correlation between and is accounted for by the unfolding. Finally, performing integration in Eq. (A5) we obtain the solution given by Eq. (13).
The work presented in this paper has been supported by the VINNOVA (The Swedish Governmental Agency for Innovation Systems) and by the European Community's Seventh Framework Programme (FP7) under project 212 352 ALPHA “Architectures for fLexible Photonic Home and Access networks”.
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