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We propose a novel configuration of the finite-impulse-response (FIR) filter adapted by the phase-dependent decision-directed least-mean-square (DD-LMS) algorithm in digital coherent optical receivers. Since fast carrier-phase fluctuations are removed from the error signal which updates tap coefficients of the FIR filter, we can achieve stable adaptation of filter-tap coefficients for higher-order quadrature-amplitude modulation (QAM) signals. Computer simulations show that our proposed scheme is much more tolerant to the phase noise and the frequency offset than the conventional DD-LMS scheme. Such theoretical predictions are also validated experimentally by using a 10-Gsymbol/s dual-polarization 16-QAM signal.
©2012 Optical Society of America
1. Introduction
In digital coherent optical receivers, adaptive finite-impulse-response (FIR) filters can compensate for all of the linear impairments of the optical transmission system as far as the delay-tap length is much longer than the impulse response of the system. For example, we can achieve compensation for chromatic dispersion (CD) and polarization-mode dispersion (PMD) of fibers for transmission [1], sampling-phase adjustment of analog-to-digital conversion (ADC) for timing-jitter cancellation [2], and equalization of optical/electrical filtering impairments. Adaptive FIR filters are thus playing the most important role in >100-Gbit/s coherent optical transmission systems.
Tap coefficients of FIR filters are updated by using tap adaptation algorithms. One of the crucial considerations for such algorithms is whether they can work without any interaction with carrier-phase estimation. Such interaction seriously degrades the performance of FIR filters having long delay taps, when large carrier-phase fluctuations are induced within the delay time by the phase noise of the transmitter laser and the local oscillator (LO) and the frequency mismatch between them.
In the constant-modulus algorithm (CMA) [3], the multiple-modulus algorithm (MMA) [4], and the phase-independent decision-directed least-mean square (DD-LMS) algorithm [5], tap adaptation is done by using the phase-independent error signal, that is, the envelope error between the reference signal and the filter output. Since the error signal does not include the phase information of the filter output, the phase estimator placed after the FIR filter can work without any interaction with the FIR filter. Thus, we can employ arbitrary-long delay taps in the FIR filter. However, such tap-adaptation algorithms suffer from the polarization-singularity problem that two outputs from the polarization-diversity coherent receiver converge on the same polarization tributary due to polarization-dependent loss (PDL), amplified-spontaneous-emission (ASE) noise, and PMD [6]. This is owing to the fact that in the initial stage of tap adaptation, we cannot use the training sequence or can use only the envelope information of the training sequence.
On the other hand, the standard DD-LMS algorithm employs the complex-amplitude error between the reference signal and the filter output [7]. In other words, this algorithm is phase-sensitive. Then, introducing the training sequence, we can use the full information of the reference complex amplitude and avoid the singularity problem perfectly. In addition, its tap-adaptation speed is much higher than that of CMA/MMA/phase-independent DD-LMS because of the full use of complex amplitudes of the training sequence. However, it tracks the carrier phase owing to its phase sensitivity, which results in unstable filtering characteristics. In order to use long-tap FIR filters for signal equalization, we need to modify the FIR-filter structure so that it does not track the carrier phase, while maintaining the advantage of the phase sensitivity.
Considering these contradicting traits of the phase sensitivity of the DD-LMS algorithm, we propose a novel configuration of the FIR filter in this paper: Fast phase fluctuations are removed from the error signal that adapts the FIR filter with long-delay taps, whereas phase estimation is done with a conventional one-tap phase rotator. Using such a scheme, we can simultaneously achieve estimation of fast phase fluctuations and compensation for slowly time-varying CD, PMD, and timing jitter in a stable manner.
The organization of the paper is as follows: Section 2 reviews various tap-adaptation algorithms for FIR filters. Section 3 describes the principle of operation of the proposed FIR-filter configuration. In Sec. 4, we numerically evaluate its tolerance to the phase noise and the frequency offset in 4-, 16-, and 64-ary quadrature-amplitude-modulation (QAM) systems. Section 5 validates such theoretical predictions by back-to-back bit-error ratio (BER) measurements for a 10-Gsymbol/s dual-polarization 16-QAM signal. Finally, we conclude our paper in Sec. 6.
2. Adaptation algorithms of FIR filters
In this section, we review several tap-adaptation algorithms for FIR filters in conjunction with the carrier-phase estimator. For simplicity, we assume the single-polarization system and the symbol-rate sampling for ADC throughout this section.
Let the complex amplitude from the phase-diversity homodyne receiver be E(n), where n denotes the number of the sampled sequence. The column vector E(n) input to the FIR filter is then defined as
where M is the filter order (i.e., the number of taps is given as M + 1) and T stands for the transpose of a matrix. The delay-tap coefficient vector p(n) of the FIR filter is given byIn this case, the FIR-filter output E'(n) can be calculated asThe tap vector of the FIR filter is adapted by the LMS algorithm [8] aswhere μp is the step-size parameter and the superscript * denotes complex conjugation. The definition of the error signal ep(n) depends on tap-adaptation algorithms.Equation (4) implicitly assumes that tap coefficients are updated every symbol. However, using k-fold oversampled data (k: integer), we need to update tap coefficients every k samples. In addition, the tap adaptation rate should be decreased much below the symbol rate when the feedback delay cannot be ignored in real digital circuits.
Figures 1(a) and 1(b) respectively show configurations of the FIR filter adapted by CMA/MMA and the phase-independent DD-LMS algorithm, both of which are followed by phase estimators. Figures 1(c) and 1(d) are those adapted by the standard phase-dependent DD-LMS algorithm. In Fig. 1(c), phase estimation is done by the FIR filter itself, whereas in Fig. 1(d), the phase estimator follows the FIR filter. Details of tap-adaptation algorithms in these schemes are described below.
Fig. 1 Configurations of the FIR filter adapted by (a) CMA/MMA and (b) the phase-independent DD-LMS algorithm, both of which are followed by phase estimators. Figures (c) and (d) are those adapted by the standard phase-dependent DD-LMS algorithm. Figure (c) does not include a phase estimator, whereas in Fig. (d), the phase estimator follows the FIR filter.
2.1 CMA, MMA, and phase-independent DD-LMS algorithm
Figure 1(a) shows the configuration of the FIR filter adapted by CMA and MMA. The error signal eCMA(n) in CMA is given [3] by
where r is the reference radius on the complex IQ plane. Since Eq. (5) does not include the phase error, the FIR filter never tracks phase fluctuations. As a result, we can individually achieve FIR-filter adaptation and phase estimation without any mutual coupling. Although CMA works well for constant-envelope signals such as phase-shift keying (PSK) signals, the demodulation performance is degraded for higher-order QAM signals accompanying amplitude-modulation (AM) components. Such degradation can be reduced by using MMA, in which the reference radius is determined by using radius-decision grids in a symbol-by-symbol manner. For example, the standard square 16-QAM signal has the three-level reference radius. The error signal eMMA(n) in MMA is given [4, 5] aswhere r(n) is the reference radius closest to |E'(n)| on the complex plane.Figure 1(b) shows the configuration of the FIR filter adapted by the phase-independent DD-LMS algorithm. In this scheme, the symbol decision is conducted after phase estimation, and the radius of the decoded signal is used as the reference radius. The error signal ePI-LMS(n) is given [5] by
where d(n) is the decoded signal; namely, |d(n)| stands for the reference radius. Since Euclidean distances among symbols is generally longer than the radius difference among them, the symbol-decision criterion provides better filtering performance compared with the radius-decision criterion used in CMA/MMA. Equation (7) also excludes the phase error.2.2 Phase-dependent DD-LMS algorithm
Figure 1(c) illustrates the FIR filter adapted by the standard DD-LMS algorithm which employs the phase-dependent error signal for tap adaptation. The error signal ePD-LMS(n) is expressed [7] as
Since Eq. (8) includes the phase information in contrast to CMA/MMA and the phase-independent DD-LMS algorithm, the FIR filter tracks phase fluctuations as well as other time-varying linear impairments. In such a case, adaptation of FIR-filter taps is disrupted by the fast phase fluctuation when the filter length is too long. Even if the phase estimator is placed after the FIR filter as shown in Fig. 1(d), mutual coupling occurs between them.3. Novel FIR-filtering scheme adapted by the DD-LMS algorithm
3.1 Proposal of the novel FIR-filter configuration
We propose the novel FIR-filter configuration based on the phase-dependent DD-LMS algorithm, which enables stable operation under fast phase fluctuations. For simplicity, we consider the single polarization case, and the polarization-diversity scheme will be described in 3.4.
Figure 2 shows the proposed configuration of the FIR filter. The phase estimator consists of a one-tap adaptive phase rotator controlled by the DD-LMS algorithm and is placed after the FIR filter. In the proposed configuration, however, the FIR filter is adapted by using the error signal in which the phase fluctuation estimated by the phase estimator is removed. The error signal ep(n) in the proposed configuration is then given by
where d(n) represents either the desired signal in the training mode or the decoded signal in the tracking mode, and f(n) denotes a complex number obtained by the decision-directed phase estimator as shown below. Note that in the conventional DD-LMS schemes shown by Figs. 1 (c) and 1(d), the error signal given by Eq. (8) does not include f(n).Fig. 2 Proposed configuration of the FIR filter followed by the phase estimator. Filter-tap coefficients are adapted by the phase-dependent DD-LMS algorithm. |⋅|/(⋅) denotes |f|/f.
The decision-directed phase estimator consists of a one-tap phase rotator. Adaptation of tap coefficients is done by the normalized LMS algorithm as
where μf is the step-size parameter and ef(n) the error signal controlling the tap coefficient. Using Eq. (10), f(n) is updated in a symbol-by-symbol manner so that ef(n) approaches zero. The small positive offset ε prevents the denominator of the second term of the right-hand side of Eq. (10) becoming zero for any input signal.With the proposed scheme, the FIR filter does not track phase fluctuations faster than its adaptation speed, which is determined from the delay-tap length and the step-size parameter [9]; however, residual phase fluctuations are eliminated by the following phase estimator. In other words, the output of the FIR filter tends to converge on d(n){f(n)/|f(n)|}−1, and the phase estimator rotates the phase of the FIR-filter output by f(n)/|f(n)|; thus, the final output approaches d(n). In such a case, Eq. (9), where fast phase fluctuations are eliminated, gives us a proper error signal for the FIR filter. On the other hand, in the conventional scheme, the error signal given by Eq. (8) is not proper, because the FIR-filter output cannot approach d(n) due to fast phase fluctuations that the FIR filter cannot track. Thus, in our proposed scheme, the FIR filter only compensates for slowly time-varying impairments, whereas the phase estimator tracks fast phase fluctuations. Such interruption-free operation of the FIR filter and the phase estimator improves the filtering performance drastically.
In what follows, we analyze the operation principle of our scheme more specifically. For simplicity, we assume that the number of taps of the FIR filter is one and take only phase fluctuations into account as linear impairments. Then, Eqs. (4) and (9) can be written as
where we use the normalized step-size parameter μ′p = μp|E(n)|2. On the other hand, Eqs. (10) and (11) yieldwhere the offset ε in Eq. (10) is neglected. In Eqs. (12) and (13), the first terms of the right-hand sides represent the contribution of the current tap weight to the next one, which depends on the step-size parameters. On the other hand, the second terms stand for updating values of the tap coefficients. We find that p(n) of the FIR filter tracks the phase left after f(n) of the phase estimator does it, whereas f(n) tracks the phase left after p(n) does it. Even when the adaptation speed for p(n) of the FIR filter is slow owing to the small step-size parameter, f(n) of the phase estimator complements the phase-estimation process. Thus, the FIR filter and the phase estimator work for phase estimation in a mutually complementary manner.In contrast, if we employ the conventional error signal shown by Eq. (8) to adapt the filter-tap coefficients, the FIR filter tries to track fast phase fluctuations disregarding the phase estimator. Consequently, unstable coupling of the FIR filter and the phase estimator occurs when the adaptation speed of the FIR filter is slow.
3.2 Numerical analyses of the proposed configuration
In order to validate the operation principle mentioned in 3.1, we perform numerical analyses under the following conditions: The modulation format is the 10-Gsymbol/s 16-QAM. Effects of shot noise and ASE noise are ignored. The FIR filter consists of symbol-spaced delay taps, and adaptation of filter-tap coefficients is performed after perfect clock recovery.
In Fig. 3 , upper Figs. 3(a), 3(b), and 3(c) show unwrapped phases estimated by the proposed scheme, where the linewidth of the transmitter and the local oscillator is 100 kHz and the frequency mismatch is 0 Hz. On the other hand, lower Figs. 3(d), 3(e), and 3(f) illustrate those obtained when the frequency offset is 10 MHz and the phase noise is ignored. We assume that μp/μf = 1 and M + 1 = 1 in left Figs. 3(a) and 3(d), μp/μf = 1 and M + 1 = 4 in middle Figs. 3(b) and 3(e), and μp/μf = 1/4 and M + 1 = 1 in right Figs. 3(c) and 3(f). The tracking speed of the FIR filter in middle and right figures is slower than that in left figures.
Fig. 3 Unwrapped phases tracked by the proposed configuration. Upper figures (a), (b), and (c) are calculated when the laser linewidth equals to 100 kHz and the offset frequency is 0 Hz. Lower figures (d), (e), and (f) are calculated when the laser linewidth is 0 Hz and the offset frequency is 10 MHz. μp/μf = 1 and M + 1 = 1 in left figures (a) and (d), μp/μf = 1 and M + 1 = 4 in middle figures (b) and (e), and μp/μf = 1/4 and M + 1 = 1 in right figures (c) and (f). Red curves: unwrapped phase φp(n) defined as the phase difference of the signal between the output and input ports of the FIR filter. Blue broken curves: unwrapped phase φf(n) defined as the phase difference of the signal between the output and input ports of the phase estimator. Green broken curves: total phases φp(n) + φf(n). Black curves: sign-inverted values of the actual phase fluctuation -φn(n).
The red curves indicate the unwrapped phase φp(n) defined as the phase difference of the signal between the output and input ports of the FIR filter. The blue broken curves represent the unwrapped phase φf(n) defined as the phase difference of the signal between the output and input ports of the phase estimator. The green broken curves show the total phase φp(n) + φf(n). The actual value of the phase fluctuation is denoted as φn(n), and the black curves represent -φn(n).
In Figs. 3(a) and 3(d), the FIR filter and the phase estimator have the same tracking speed; hence, they equally contribute to tracking of the phase fluctuation, that is, φp(n) = φf(n) and φp(n) + φf(n) = -φn(n). On the other hand, in Figs. 3(b), 3(e), 3(c), and 3(f), the tracking speed of the FIR filter is slower than that of the phase estimator. In such cases, the phase estimator mainly tracks the phase fluctuation, that is, |φp(n + 1)-φp(n)|<<|φf(n + 1)-φf(n)|; however, the total phase always tracks the actual phase fluctuation: φp(n) + φf(n) = - φn(n).
In order to evaluate how the FIR filter and the phase estimator track the phase in a mutually complementary manner, we define the phase-separation ratio α as
where the ratio of differential phases between symbols is averaged over N symbols. Smaller α values mean that the contribution of the FIR filter for phase tracking is less than that of the phase estimator.Figure 4 shows α as a function of the number of taps M + 1. In Fig. 3(a), we include the 100-kHz laser linewidth and ignore the frequency offset, whereas in Fig. 4(b), the 10-MHz frequency offset is taken into account and the laser linewidth is neglected. Squares, dots, and plus marks correspond to μp/μf = 1, 1/4, and 1/16, respectively. As shown by solid curves, α for each μp/μf obeys the relation given by
Noting that the tracking speed of the FIR filter is proportional to the step size parameter and the reciprocal of the number of delay taps [8], we find from Eq. (15) that the ratio of contributions of the FIR filter and the phase estimator is determined from the ratio of their effective tracking speeds. When the FIR filter cannot track the fast phase fluctuation owing to the small step-size parameter and/or the long delay tap, the one-tap phase estimator eliminates the phase fluctuation residual after the FIR filter. Then, we can achieve stable FIR filtering with the long delay tap even under fast phase fluctuations.Fig. 4 Phase-separation ratio α as a function of the number of FIR-filter taps. Figure (a) is calculated when the laser linewidth equals to 100 kHz and the offset frequency is 0 Hz. Figure (b) is calculated when the laser linewidth is 0 Hz and the offset frequency is 10 MHz. Squares, dots, and plus marks correspond to μp/μf = 1, 1/4, and 1/16, respectively. Solid curves show the relation given by Eq. (15).
3.3 FIR-filter configuration enabling frequency-offset cancelation
Generally speaking, since the LMS algorithm works in a causal manner, the estimated value is affected by an adaptation delay [8]. As a result, the output of the phase estimator still has a stationary phase error in the presence of the frequency offset. When the frequency mismatch between the transmitter laser and the local oscillator is large, such a phase error seriously degrades the demodulation performance. However, we can eliminate it by adding another one-tap phase estimator after the first-stage phase estimator as shown in Fig. 5 . The tap coefficient s(n) of the second-stage phase estimator is updated by
where μs is the step-size parameter for the second-stage phase estimator, es(n) is the error signal, and ε stands for the small positive offset. In this case, the error signal for the FIR filter given by Eq. (9) should be modified asThe output from the FIR filter is phase-controlled by f(n) and s(n), and we finally obtain the decoded symbol. With this configuration, the frequency-offset tolerance of the receiver is improved.Fig. 5 FIR-filter configuration adapted by the DD-LMS algorithm, which employ the dual-stage decision-directed phase estimator. |⋅|/(⋅) denotes either |f|/f or |s|/s.
3.4 FIR-filter configuration for dual-polarization systems
We introduce the polarization-diversity scheme into the above configuration [10]. Figure 6 shows the block diagram of the proposed FIR-filter configuration. This configuration consists of two parts. First, FIR filters in the two-by-two butterfly structure mainly conduct compensation for CD and PMD, demultiplexing of polarization tributaries, and phase adjustment of ADC for timing-jitter cancellation. Next, the dual-stage decision-directed carrier-phase estimators eliminate phase fluctuations caused by the laser phase noise and the frequency offset.
Fig. 6 Butterfly-structured FIR filters combined with dual-stage decision-directed phase estimators, which are adapted by the phase-dependent DD-LMS algorithm. |⋅|/(⋅) denotes either |fx,y|/fx,y or |sx,y|/sx,y.
Let Ex,y(n) be the sampled complex amplitude from the x- or y-port of the phase- and polarization-diversity homodyne receiver. The column vector Ex,y(n) given by
is incident on the x or y input port of the butterfly-structured FIR filters. The delay-tap coefficient vector pk,l(n) of the butterfly-structured FIR filter, where k and l are either x or y, is given byEach tap vector is updated in accordance withwhere μp is the step-size parameter. The error signal epx,y(n) is defined bywhere dx,y(n) denotes either the desired signal in the training mode or the decoded signal in the tracking mode. Complex numbers fx,y(n) and sx,y(n) are estimated by the first-stage and the second-stage phase estimator, respectively. The output of the butterfly-structured FIR filters is then given asThese outputs are phase-controlled by the dual-stage decision-directed phase estimators.The tap coefficient of the first-stage phase estimator fx,y(n) is updated by
where μf is the step-size parameter, efx,y(n) the error signal controlling the tap coefficient of the first-stage phase estimator, and ε the small positive offset. On the other hand, the tap coefficient of the second-stage phase estimator sx,y(n) is updated in accordance with where μs is the step-size parameter. Generally, μs should be much smaller than μf, because the laser frequency changes very slowly. The error signal esx,y(n) adapts the tap coefficient of the second-stage phase estimator.When polarization tributaries have the same carrier-phase fluctuation in the polarization-multiplexed system, fx,y(n) can be averaged, by using the carrier-phase correlation between the polarization tributaries, as
Using fave(n) instead of fx,y(n) improves the tolerance against the phase noise and the frequency offset, because the signal-to-noise ratio (SNR) is enhanced by averaging the tap coefficients of both of the polarization tributaries. On the other hand, we should independently use the carrier phase estimated from each polarization tributary, when carrier phases have no correlation between the two polarization tributaries.3.5 Advantage of our scheme for practical implementation
As shown in previous subsections, in our scheme, the long-tap FIR filters and the one-tap phase estimators can operate in a mutually independent manner. This is a great advantage of our scheme for practical implementation in real systems, where the feedback delay in digital circuits is a serious concern.
The tap updating process for log-tap FIR filters usually accompanies large feedback delay; in such a case, we can decrease the tap adaptation rate of our scheme much below the symbol rate, because FIR filters compensate only for slowly time-varying CD, PMD, and timing jitter. On the other hand, the feedback delay for updating one-tap phase rotators is essentially much smaller than that for updating long-tap FIR filters; therefore, the phase rotators can cope with fast carrier phase estimation.
4. Simulations of the tolerance against the phase noise and the frequency offset in our proposed scheme
In order to evaluate robustness of the proposed scheme against the phase noise and the frequency offset, we conduct computer simulations using dual-polarization 4-QAM, 16-QAM, and 64-QAM signals.
In the simulation, δf stands for the linewidth of lasers for the transmitter and LO. The state of polarization of the incoming signal and the frequency offset Δf are assumed to be constant within the calculated symbol-sequence interval. Although the phase singularity of the received signal is avoided by the training sequence, differential encoding is employed to mitigate the impact of cycle slips in the phase-estimation process [11]. After the spectrum of the modulated signal is shaped by the Nyquist filter with the roll-off factor of 1, the signal is incident on the coherent receiver.
In a real system, the clock recovery process is as follows: First, receiver outputs are twofold-oversampled. Second, sampled data are aligned with the half symbol duration through digital interpolation, clock-frequency extraction, and resampling. Third, the sampling phase can be optimally adjusted with the adaptive equalizer [2]. In our simulations below, we assume that the twofold-oversampled data are properly aligned with the half symbol duration; however, since the sampling phase adjustment is done inside our proposed FIR filters, the performance of our scheme is not critically dependent on the accuracy of the extracted clock.
Figure 7(a) shows BER characteristics of the 10-Gsymbol/s dual-polarization 16-QAM signal calculated as a function of the energy per bit to noise spectral density ratio Eb/N0, when the proposed scheme is employed. Phase estimation is done individually for each polarization tributary. Squares, circles, and triangles denote BERs when FIR-filter orders M are 4, 16, and 64, respectively. Red and green curves illustrate BERs when Δf = 0 Hz, and δf = 0 Hz and 100 kHz, respectively. Blue curves are BERs calculated when Δf = 100 MHz and δf = 100 kHz. For comparison, Figs. 7(b) and 7(c) depict BER characteristics when the conventional phase-dependent DD-LMS algorithm is used for filter-tap adaptation. Figure 7(b) corresponds to the scheme shown in Fig. 1(c), and Fig. 7(c) to that shown in Fig. 1(d).
Fig. 7 BER characteristics of 10-Gsymbol/s 16-QAM signals calculated as a function of Eb/N0 (a) with the proposed scheme. Figures (b) and (c) are those with the conventional DD-LMS algorithm. Figure (b) corresponds to the scheme shown in Fig. 1(c), and Fig. (c) to that shown in Fig. 1(d). δf and Δf denote the linewidth of the laser and the frequency offset, respectively. We calculate BERs for δf = 0 Hz and 100 kHz and Δf = 0 Hz and 100 MHz. Filter orders M are 4, 16, and 64.
In the proposed scheme, the step-size parameter μf in Eq. (24) is optimized in the range of 1/2m (m = 0, 1, 2,⋅⋅⋅, 6) so that each bit-error ratio (BER) is minimized. Since we ignore fluctuations of the state of polarization and the laser frequency, μp in Eq. (21) and μs in Eq. (26) are fixed to the smallest value of 1/26. In the scheme shown in Fig. 1(c), the step-size parameter for FIR-filter adaptation is optimized in the range of 1/2m. When the scheme shown in Fig. 1(d) is employed, we search for an optimum combination of the three step-size parameters used for adaptation of the FIR filter and the phase estimator based on the dual-stage phase rotator. In this case, the step-size parameter of the FIR filter is selected from 1/2m. On the other hand, the step-size parameter of the first-stage phase estimator takes any one of 1/2m or zero, and that of the second-stage phase estimator is set to 1/26.
When phase fluctuations are negligible, BER characteristics of the three schemes are identical. However, when the phase noise and the frequency offset are involved, BER characteristics of the conventional two schemes are seriously degraded as the filter order increases. Meanwhile, the BER performance of the proposed scheme is entirely independent of the filter order. Thus, the proposed scheme has much higher tolerance to the phase noise and the frequency offset, even if we use FIR filters with long-delay taps.
Figure 8 illustrates the power penalty of the proposed scheme at BER = 10−3 calculated as a function of the laser linewidth normalized to the symbol rate. As a reference, the laser linewidth in the 10-Gsymbol/s system is shown on the top axis. Red, green, and blue curves correspond to 4-QAM, 16-QAM, and 64-QAM formats, respectively, when M = 64. Solid and broken curves respectively represent power penalties with and without using the phase correlation between the polarization tributaries.
Fig. 8 Phase-noise tolerance of the proposed scheme in 4-QAM, 16-QAM, and 64-QAM systems. Solid curves are calculated by using the phase correlation between the two polarization tributaries, whereas broken curves are calculated without using such correlation.
Figure 9 illustrates the power penalty at BER = 10−3 when the frequency offset is changed and the laser linewidth is ignored. The frequency offset normalized to the symbol rate is shown in the bottom axis, while that in the 10-Gsymbol/s system is shown on the top axis. Definitions of the curves are the same as those of Fig. 8.
Fig. 9 Frequency-offset tolerance of the proposed scheme in 4-QAM, 16-QAM, and 64-QAM systems. Solid curves are calculated by using the phase correlation between the two polarization tributaries, whereas broken curves are calculated without using such correlation.
Tables 1 and 2 summarize laser linewidths and frequency offsets, respectively, required to ensure the power penalty lower than 1 dB at BER of 10−3 in 10-Gsymbol/s QAM systems. For example, when we employ the phase correlation between the polarization tributaries, acceptable values of the laser linewidth and the frequency offset can go up to 400 kHz and 400 MHz, respectively, in the 10-Gsymbol/s 16-QAM systems regardless of the FIR-filter order. Such values increase in proportion to the symbol rate. Since these power penalties are independent of the filter order, we find that these are determined only from the phase noise and the frequency offset.
Table 1. Laser linewidth that ensures the power penalty at BER = 10−3 less than 1 dB in 10-Gsymbol/s QAM systems.
Table 2. Frequency offset that ensures the power penalty at BER = 10−3 less than 1 dB in 10-Gsymbol/s QAM systems.
5. Experiments
We experimentally evaluate the performance of the proposed FIR-filtering scheme in the 10-Gsymbol/s dual-polarization 16-QAM system. The experimental setup is depicted in Fig. 10 .
Fig. 10 Experimental setup for measuring the performance of the proposed FIR-filtering configuration.
A distributed-feedback laser diode (DFB-LD) was used as a transmitter. The wavelength of the DFB-LD was 1552 nm and its linewidth was around 100 kHz. An NRZ 10-Gsymbol/s 16-QAM optical signal was generated by a LiNbO3 optical IQ modulator (IQM). The IQM was driven by two streams of 4-level electrical signals, which were generated from two independent ports of an arbitrary waveform generator (AWG). The 16-QAM signal was differentially-encoded. Polarization multiplexing was conducted by a polarization-beam splitter (PBS) and a polarization-beam combiner (PBC). A fiber delay line, which gave a time delay of around 10 ns, was inserted into one of the split arms to decorrelate bit patterns of the dual-polarization tributaries. In front of the receiver, the average optical power was controlled by a variable optical attenuator (VOA). The signal was then pre-amplified by an erbium-doped fiber amplifier (EDFA) and detected by a phase- and polarization-diversity homodyne receiver, where another DFB-LD having the characteristics same as the transmitter laser was used as a local oscillator. The frequency mismatch between the transmitter laser and the local oscillator was carefully set to 0 Hz or 100 MHz by monitoring their coherently-detected beat frequency with a radio-frequency spectrum analyzer. Output signals from the receiver passed through 8-GHz low-pass filters and were twofold-oversampled asynchronously by a 4-ch ADC with 8-bit resolution. The digitized signal was then sent to the proposed FIR-filter configuration, which performed sampling-phase adjustment, polarization demultiplexing, carrier-phase estimation, frequency estimation, and decoding. Tap coefficients of the phase estimators were updated without using the phase correlation between the polarization tributaries, since their carrier phases were decorrelated through the polarization multiplexing process.
Squares, circles, and triangles in Fig. 11(a) respectively show BERs measured when filter orders M are 4, 16, and 64. Green and blue curves represent BER characteristics when Δf = 0 Hz and 100 MHz, respectively. For comparison, BER performances of the conventional DD-LMS schemes are illustrated in Figs. 11(b) and 11(c). Figure 11(b) is obtained by the scheme shown in Fig. 1(c)), whereas Fig. 11 (c) is measured in the scheme shown in Fig. 1 (d).
Fig. 11 BER characteristics of 10-Gsymbol/s dual-polarization 16-QAM signals measured as a function of the received power (a) with the proposed scheme. Figures (b) and (c) are obtained with the conventional DD-LMS schemes, Fig. 1(c) and Fig. 1(d), respectively. δf and Δf denote the individual laser linewidth and the frequency offset, respectively. We measure BERs when Δf = 0 Hz and 100 MHz, whereas δf is about 100 kHz. Filter orders M are 4, 16, and 64.
From these results, we find that the performances of the conventional schemes are degraded with increase in the filter order. In contrast, the performance of the proposed scheme is independent of the filter order and tolerant to the phase noise and the frequency offset. These experimental results are in good agreement with the simulation results given by Fig. 7.
Figures 12(a) , 12(b), 12(c), and 12(d) illustrate constellation maps of a polarization tributary of the 16-QAM signal before the butterfly-structured filters, after the butterfly-structured filters, after the first-stage phase estimator, and after the second-stage phase estimator, respectively. The frequency mismatch was set to 100 MHz, the filter order was 64, and the received power was −30 dBm. From Figs. 12(a) and 12(b), we can observe that polarization demultiplexing and sampling-phase adjustment of ADC are successfully conducted even in the presence of fast phase fluctuations. Then, the phase noise and the frequency offset are partially estimated by the first-stage phase estimator as shown in Fig. 12(c). Finally, the residual phase error is entirely canceled by the second-stage phase estimator as shown in Fig. 12(d).
Fig. 12 Constellation maps of a tributary of the 10-Gsymbol/s dual-polarization signals (a) before the butterfly-structured FIR filters, (b) after the butterfly-structured FIR filters, (c) after the first-stage phase estimator, and (d) after the second-stage phase estimator. The received power was −30 dBm, the frequency offset 100 MHz, and the filter order 64. The horizontal axis and the vertical axis of each figure denote the real axis and the imaginary axis of the complex plane, respectively.
6. Conclusion
We have proposed a novel configuration of the FIR filter adapted by the DD-LMS algorithm, in which fast phase fluctuations are removed from the error signal used for tap adaptation of FIR filters. With such a scheme, we can employ arbitrary-long delay taps in FIR filters even under the influence of fast phase fluctuations. In addition, our scheme can easily be applied to any multilevel modulation formats, since it employs the DD-LMS algorithm in all of the tap-adaptation processes. Through computer simulations, we show that the proposed FIR-filtering scheme is robust against the phase noise and the frequency offset in QAM systems. Finally, effectiveness of the proposed scheme is also confirmed experimentally by using a 10-Gsymbol/s dual-polarization 16-QAM signal.
Acknowledgments
This work was supported in part by Grant-in-Aid for Scientific Research (A) (22246046), the Ministry of Education, Science, Sports and Culture, Japan.
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