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We analyze the clock-recovery process based on adaptive finite-impulse-response (FIR) filtering in digital coherent optical receivers. When the clock frequency is synchronized between the transmitter and the receiver, only five taps in half-symbol-spaced FIR filters can adjust the sampling phase of analog-to-digital conversion optimally, enabling bit-error rate performance independent of the initial sampling phase. Even if the clock frequency is not synchronized between them, the clock-frequency misalignment can be adjusted within an appropriate block interval; thus, we can achieve an asynchronous clock mode of operation of digital coherent receivers with block processing of the symbol sequence.
© 2011 Optical Society of America
1. Introduction
The digital coherent optical receiver can compensate for various impairments in the optical transmission system by using adaptive finite-impulse-response (FIR) filters [1]. Impairments stemming from narrow-band filtering, group-velocity dispersion (GVD), polarization-mode dispersion (PMD), and polarization-dependent loss (PDL) in the optical transmission system are equalized in an adaptive manner, which enables large-capacity optical transmission based on multilevel modulation, dense wavelength-division multiplexing, and polarization-division multiplexing [2, 3]. Algorithms for adaptive equalization have been well established; the constant-modulus algorithm (CMA) [4] and the decision-directed least-mean square (DD-LMS) algorithm [5] are most commonly employed in the digital coherent receiver.
A unique function of such adaptive FIR filters is variable time delay with resolution much higher than the sampling time interval. If the number of filter taps is large enough, we can generate continuously-variable time delay for the received signal even by using discrete time-delay elements in the FIR filters [6, 7, 8, 9]. Such function has successfully been applied to compensation for differential group delay (DGD) between two principal states of polarization (PSP). Another possible application of it may be optimal adjustment of the sampling phase of AD conversion. Owing to the continuously-variable time delay and the adaptive equalization algorithm, the signal waveform can be delayed so that the time best for symbol discrimination coincides with the sampling instant. In fact, this process is nothing but clock recovery. Although such ability of clock recovery using adaptive FIR filters has been reported in the radio-communication field [6, 7, 8], it has not been well recognized in the optical-communication field.
So far, the clock recovery in the digital coherent receiver has been done in the following way [2]: First, data sampled by AD converters are up-sampled. Then, the clock is extracted from the up-sampled data through discrete Fourier transform (DFT). Next, by using the clock, the up-sampled data are re-sampled so as to keep one sample within one symbol interval. On the other hand, in this paper, we analyze the clock-recovery process based on adaptive FIR filtering. When the clock frequency is synchronized between the transmitter and the receiver, as is the case in SONET/SDH networks, only five taps in half-symbol-spaced FIR filters can adjust the sampling phase optimally, and we obtain good bit-error rate (BER) performance independent of the initial sampling phase for both polarization tributaries. Even if the clock frequency is not synchronized, as is the case in Ethernet-based networks, the clock-frequency offset can be adjusted within an appropriate block interval. We show the possibility of an asynchronous clock mode of operation of digital coherent receivers based on such an analysis.
The organization of this paper is as follows: Section 2 summarizes principle of operation of equalization and clock recovery using adaptive FIR filters. Section 3 deals with the method of computer simulations. In Sec.4, we discuss characteristics of sampling-phase adjustment by adaptive FIR filters when the clock frequency is synchronized between the transmitter and the receiver. In Sec.5, we propose and analyze an asynchronous clock mode of operation of digital coherent receivers, where synchronization of the clock frequency is not required between the transmitter and the receiver. Finally, Sec.6 concludes this paper.
2. Principle of operation of equalization and clock recovery using adaptive FIR filters
2.1. FIR-filter-based equalization
If the input power launched on a transmission link is low enough to operate in the linear region, the transfer function of the link can be modeled as a concatenation of the scalar GVD element D(ω), the two-by-two unitary matrix for PMD U(ω), the two-by-two Hermite matrix for polarization-dependent loss (PDL) T, and the unitary Jones matrix expressing the birefringence K [10]:
where ω is the angular frequency of the optical carrier. The transfer matrix of the equalizer in the digital coherent receiver Heq (ω) should be the inverse matrix of H (ω) as The input vector for the n-th sampled signal is expressed as [x(n),y(n)]T, where x(n) denotes the complex amplitude of the electric field from the x port of the homodyne receiver comprising phase and polarization diversities, and y(n) that from the y port. The output vector is written as [X(n),Y(n)]T, where X(n) denotes the complex amplitude of the x-polarized tributary, and Y(n) that of the y-polarized tributary.Each element of the matrix hp (ω) (p = (xx,xy,yx,yy)) can be realized by an FIR filter; then, the matrix given by Eq.(2) is implemented in two-by-two butterfly-structured FIR filters. Let the number of taps be k, and the delay-time interval be given as T/m, where T denotes the symbol interval and m is an integer showing the oversampling ratio of AD conversion. Input column vectors x⃗(n) and y⃗(n) for the FIR filter are defined as
Tap-coefficient column vectors h⃗p(n) are written as Filter outputs are then given asUpdate of filter-tap coefficients of each FIR filter is done by using CMA or the DD-LMS algorithm, and we can perform signal equalization and polarization demultiplexing simultaneously. To avoid the singularity problem to polarization demultiplexing inherent in blind-equalization techniques [11], we use the DD-LMS algorithm with the training mode in this paper. On the basis of the DD-LMS algorithm, we update tap coefficients as follows:
In these equations, μ is a step-size parameter, and eX,Y (n) is an error signal. On the other hand, dX,Y (n) is either a training symbol in the training mode or a decoded symbol in the tracking mode for each polarization tributary. It is any one of , , , and for the quadrature phase-shift-keying (QPSK) modulation format. After updating tap coefficients in the training mode, we switch it into the tracking mode. In the case of ×2 oversampling (m = 2), update of tap coefficients is done per every two samples.2.2. FIR-filter-based clock recovery
The Nyquist bandwidth of the optical signal with a symbol duration of T is given as B = 1/T. Then, outputs from in-phase and quadrature (IQ) ports of the phase-diversity homodyne receiver have the bandwidth of B/2 = 1/(2T), and the Nyquist sampling rate R for these signals is R = 1/T. When the clock is extracted by an external circuit, we can control the sampling phase for AD converters using the extracted clock; in such a synchronous sampling case, we can employ the Nyquist-rate sampling at R, because the sampling phase has already been fixed at an optimal value. The signal sampled at the symbol rate is sent to signal processing circuits such as adaptive FIR filters for equalization and polarization demultiplexing, carrier-phase estimators, and decoders.
On the other hand, when the sampling is done asynchronously, we usually need to use ×2 oversampling to remove the aliasing effect [12]. In this case, the clock recovery has been performed as follows: First, data sampled by AD converters are interpolated in the time domain (i.e., up-sampling). Then, the clock is extracted from the up-sampled data through DFT, and it re-samples the up-sampled data so as to keep one sample within one symbol interval. The re-sampled data are sent to the DSP circuit for further signal processing including adaptive FIR filters with one-symbol-spaced delay taps.
In this paper, we analyze the clock-recovery process based on adaptive FIR filtering as an alternative approach. The FIR filter can generate an arbitrary transfer function as far as the number of delay taps is large enough. When the transfer function is exp (jωτ), the FIR filter gives the signal the time delay of τ, which is continuously variable by optimal setting of tap coefficients, even through we use discrete time-delay elements in FIR filters. Together with the adaptive equalization algorithm, the signal can be delayed so that the time best for symbol discrimination coincides with the sampling instant. This function is equivalent to clock recovery, and only using adaptive FIR filters, we can achieve such function; thus, the DSP circuit of the receiver is simplified by this clock-recovery scheme.
3. Simulation method
The method of numerical analyses of clock-recovery characteristics is as follows: Electric fields for x and y polarizations are independently modulated in the QPSK format, where each symbol is differentially encoded. The non-return-to-zero (NRZ) waveform includes 23 samples in one symbol duration. White Gaussian noise is loaded to these signals, and such noise-loaded signals are filtered with Nyquist filters having a roll-off parameter of α. Its transfer function is given as
We change the energy per bit to noise spectral-power-density ratio, Eb/N0, to estimate BER characteristics. After the x- and y-polarization components are multiplexed, the state of polarization of the multiplexed signal is scrambled. At the receiver, outputs are sampled at the Nyquist rate R or at twice the Nyquist rate 2R. The sampled data are sent to two-by-two butterfly-structured FIR filters. For adaptive filter-tap control, we employ the DD-LMS algorithm with the training mode. After the signals are processed by FIR filters, they are decoded and the number of bit errors is counted for each polarization tributary. In our calculations, we introduce none of CD, PMD, and PDL of the transmission system; and FM noise of lasers for simplicity.4. Sampling-phase adjustment with FIR filters
In this section, we analyze the property of sampling-phase adjustment with FIR filters when the clock frequency is locked between the transmitter and the receiver, as is usually the case in SONET/SDH networks, but the initial sampling phase at AD conversion is uncontrolled.
First, we consider the case that the signal is filtered by an ideal Nyquist filter with α = 0. In this case, even by using the asynchronous Nyquist-rate sampling at R = 1/T, we can eliminate the aliasing effect, because the signal bandwidth at the baseband is restricted within 1/(2T). The initial sampling instant is set to t = nT/8 (n = 0, 1, 2, ⋯,7) within one symbol interval. The step-size parameter μ in the DD-LMS algorithm is 2−8. The number of training symbols is 210, and the total number of symbols is 214.
Figure 1 shows bit-error rates calculated as a function of Eb/N0 for eight initial sampling phases mentioned above. Figures (a), (b), and (c) correspond to tap numbers of 5, 17, and 65, respectively, and red and black curves are bit-error rates for x- and y-polarization tributaries, respectively. When the number of taps is as large as 65, the sampling phase is quasi-continuously adjusted with FIR filters, resulting in the BER performance relatively insensitive to the initial sampling phase. However, the tap length is shortened to 17 or less, the BER performance is dependent on the initial sampling phase significantly. On the other hand, Fig. 3(d) shows BER characteristics when α = 0.2 and the tap length is 65. In this case, BER is very sensitive to the initial sampling phase due to the aliasing effect, even if we use a long tap length of 65. Thus, when we employ symbol-rate asynchronous sampling, we need ideal Nyquist filtering and relatively long delay taps in FIR filters.
Fig. 1 Bit-error rates calculated as a function of Eb/N0 for eight initial sampling phases when we apply Nyquist-rate sampling. Numbers of taps are 5 in (a), 17 in (b), and 65 in (c) and (d). In (a), (b), and (c), the roll-off parameter of the Nyquist filter α is 0, whereas in (d), α = 0.2. Red and black curves correspond to the x-polarization tributary and y-polarization tributary, respectively.
Fig. 3 Data structure in the asynchronous clock mode of operation. The symbol sequence consists of blocks containing training sequences at the head and null sequences at the tail.
These difficulties in symbol-rate asynchronous sampling are overcome by ×2 oversampling as shown in what follows. The initial sampling instant is set to (t = 0, T/2), (t = T/8,5T/8), (t = T/4, 3T/4), or (3T/8, 7T/8) within one symbol interval. We assume the use of Nyquist filters having α = 0.2. The step-size parameter μ in the DD-LMS algorithm is 2−6. The number of training symbols is 27, and the total number of symbols is 214.
Figure 2 shows bit-error rates calculated as a function of Eb/N0 for four initial sampling phases mentioned above. Figures (a), (b), and (c) correspond to tap numbers of 1, 3, and 5, respectively. The one-tap FIR filter does not have any ability of giving time delay to the signal; therefore, BER curves are strongly dependent on the initial sampling phase. On the other hand, those for the three-tap and five-tap filters become insensitive to the initial sampling phase. We find that the FIR filter with five-delay taps can entirely remove the dependence of BER on the initial sampling phase, achieving perfect clock recovery. Compared with symbol-rate asynchronous sampling, the required tap length is greatly reduced. This is because the continuous time delay is more easily generated with fractionally-spaced FIR filters.
Fig. 2 Bit-error rates calculated as a function of Eb/N0 for four initial sampling phases when we apply ×2 oversampling. Numbers of taps are 1 in (a), 3 in (b), and 5 in (c). In all cases, the roll-off parameter of the Nyquist filter α is 0.2. Red and black curves correspond to the x-polarization tributary and y-polarization tributary, respectively.
We introduce static shifts of the sampling phase ranging from 0 to 2π into our calculations; however, we can cope with dynamic timing jitter in the spectral range from 0 to μ/T owing to the tap updating process at the symbol rate. The maximum permissible jitter is defined in ITU-T G.8251 [13] and can be accommodated by using FIR filters with sufficient tap length.
5. Asynchronous clock mode of operation of digital coherent receivers
In this section, we assume that the clock frequency is not locked between the transmitter and the receiver, as is often the case in Ethernet-based networks. Figure 3 shows the symbol sequence used in the asynchronous clock mode of operation, which consists of blocks containing training sequences at the head and null sequences at the tail. At t = t1 in Fig.3, the DD-LMS starts to operate in the training mode. Then, it is switched to the tracking mode at t = t2. The null sequence is added to avoid data overlapping between two blocks due to the dispersive effect during signal transmission through a fiber. At t = t4, all tap coefficients are reset, and the DD-LMS begins to operate in the training mode again. Time instances of the block head are recovered by using regular repetition of null sequences.
We use ×2 oversampling and half-symbol-spaced FIR filters. The detuning of the clock frequency between the transmitter and the receiver is set to 2−12 = 2.44 × 10−4 (244 ppm) by taking the Ethernet standard into account.
Figure 4(a) shows BER characteristics, whereas Fig. 4(b) represents the magnitude of error in the DD-LMS algorithm, which is |eX,Y (n)|2 (Eqs.(10) and (13)) relative to the magnitude of the symbol. In the calculation, the block length ntot is 213 symbols, the tap length of FIR filters 9, the step-size parameter μ 2−6, the number of training symbols ntr 27, the number of null symbols n0 26, and the number of blocks 3. We take the moving average of the error magnitude over (1 + 1/μ) symbols. In this case, the misalignment between the symbol sequence and the receiver clock occurring in one block interval is 2 symbols. Since the maximum time delay given by T/2-spaced 9-tap FIR filters is ±2 symbols, FIR filters can absorb this misalignment; therefore, we can obtain good BER characteristics over 3 block sequences.
Fig. 4 Successful asynchronous clock mode of operation when the clock-frequency detuning is 2−12, the block length 213 symbols, and the tap length of FIR filters 9. (a): Bit-error rates as a function of Eb/N0. (b): The magnitude of error in the DD-LMS algorithm as a function of the symbol number. Red and black curves correspond to the x-polarization tributary and y-polarization tributary, respectively.
If the number of symbols in a block is increased or the number of taps is decreased, however, we cannot compensate for the misalignment between the symbol sequence and the receiver clock any more. Figure 5 shows an example in such case, where the number of symbols in a block is increased to 214 and the number of taps is kept at 9. As shown in Fig. 5(b), the magnitude of error does not converge because FIR filters with 9 delay taps cannot absorb the misalignment of 4 symbols in one block interval. As a result, the BER performance is seriously degraded as shown in Fig.5(a).
Fig. 5 Example of failed asynchronous clock mode of operation when the clock-frequency detuning is 2−12, the block length is increased to 214 symbols, and the tap length of FIR filters is kept at 9. (a): Bit-error rates as a function of Eb/N0. (b): The magnitude of error in the DD-LMS algorithm as a function of the symbol number. Red and black curves correspond to the x-polarization tributary and y-polarization tributary, respectively.
6. Conclusion
We have analyzed characteristics of sampling-phase adjustment based on adaptive of FIR filters in digital coherent receivers. The half-symbol-spaced FIR filter with 5 taps generates quasi-continuous time delay, which can adjust the sampling phase of AD conversion optimally. Using this function, we can achieve clock recovery without adding any special signal-processing unit. Even when the clock frequency is not locked between the transmitter and the receiver, the clock-frequency misalignment is absorbed in the delay taps of FIR filters by block processing of the symbol sequence; thus, the asynchronous clock mode of operation of digital coherent receivers is made possible.
Acknowledgments
The author thanks K. Igarashi and Md. S. Faruk of The University of Tokyo for fruitful discussions. This work was supported in part by Strategic Information and Communications R&D Promotion Programme (SCOPE) ( 081503001), the Ministry of Internal Affairs and Communications, JapanCand Grant-in-Aid for Scientific Research (A) ( 22246046), the Ministry of Education, Science, Sports and Culture, Japan.
References and links
1. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008). [CrossRef] [PubMed]
2. K. Kikuchi, High Spectral Density Optical Communication Technologies, edited by M. Nakazawa, K. Kikuchi, and T. Miyazaki (Springer, 2010), Chap. 2.
3. K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation of with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12, 563–570 (2006). [CrossRef]
4. D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980). [CrossRef]
5. S. U. H. Qureshi, “Adaptive equalization,” IEEE Proc . 73, 1349–1387 (1985). [CrossRef]
6. G. Ungerboeck, “Fractional tap-spacing equalizer and consequences for clock recovery in data modems,” IEEE Trans. Commun. 24, 856–864 (1976). [CrossRef]
7. C. W. Farrow, “A continuously variable digital delay element,” in IEEE International Symposium on Circuits and Systems, Espoo, Finland, 2641–2645 (1988).
8. H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing, 2nd ed. (Wiley-Interscience, 1997).
9. H. M. Nguyen, K. Igarashi, K. Katoh, and K. Kikuchi, “Continuously-tunable optical delay line using PLC-based optical FIR filter,” in Conference on Lasers and Electro-Optics (CLEO 2010), San Jose, CA, USA, CFE2 (2010).
10. Md. S. Faruk, Y. Mori, C. Zhang, K. Igarashi, and K. Kikuchi, “Multi-impairment monitoring from adaptive finite-impulse-response filters in a digital coherent receiver,” Opt. Express 18, 26929–26936 (2010). [CrossRef]
11. K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in IEEE LEOS Summer Topical Meeting, Acapulco, Mexico, MC2.2 (2008).
12. G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22, 1129–1131 (2010). [CrossRef]
13. ITU-T Recommendation G.8251, “The control of jitter and wander within the optical transport network (OTN).”
