IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 19, NO. 13, JULY 1, 2007
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Chromatic Dispersion Compensation Using Digital IIR Filtering With Coherent Detection Gilad Goldfarb and Guifang Li
Abstract—Digital infinite impulse response (IIR) filtering is proposed as a means for compensating chromatic dispersion in homodyne-detected optical transmission systems with subsequent digital signal processing. Compared to finite impulse response (FIR) filtering, IIR filtering achieves dispersion compensation (DC) using a significantly smaller number of taps. DC of 80 and 160 km in a 10-Gb/s binary phase-shift-keying is experimentally compared for the two filtering schemes. IIR filtering can achieve performance similar to the FIR filtering scheme. Index Terms—Chromatic dispersion, coherent detection, digital signal processing (DSP), optical fiber communication.
experimental results for a 10-Gb/s binary phase-shift-keying (BPSK) transmission system employing the suggested IIR filtering scheme, and in Section V, a discussion of the suggested IIR filtering approach is presented. II. THEORY The effect of dispersion may be modeled as a linear filtering process given by
(1)
I. INTRODUCTION IGH-SPEED digital signal processing (DSP) has recently been suggested for use in conjunction with coherent detection to allow demodulation of various modulation formats. One major advantage of using DSP after sampling of the outputs from a phase-diversity receiver is that hardware optical phase locking can be avoided and only digital phase-tracking is needed [1], [2]. DSP algorithms can also be used to mitigate degrading effects of optical fiber such as chromatic dispersion and polarization-mode dispersion. As suggested in several papers [1], [3], tap delay finite impulse response for a symbol rate of , a (FIR) filter may be used to reverse the effect of (first order) fiber chromatic dispersion. The number of FIR taps required grows linearly with increasing dispersion [3]. As reported in [4], the number of (complex) taps required to compensate 1280 ps/nm of dispersion is roughly 5.8. At long propagation distances, the added power consumption required for this task becomes significant. Moreover, a longer FIR filter introduces a longer delay and requires more area on a DSP chip. Alternatively, an infinite impulse response (IIR) filter may achieve similar performance with a substantially reduced number of operations. This leads to lower power consumption and a smaller device footprint. This letter describes, for the first time to the best of our knowledge, the use of IIR filtering for dispersion compensation (DC). In contrast to optical DC which processes bandpass signals, digital FIR or IIR DC processes baseband signals [5]. Section II introduces the theory pertaining to IIR filtering for DC. Simulation results comparing FIR and IIR filtering are presented in Section III. Section IV contains
H
Manuscript received October 25, 2006; revised March 6, 2007. This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) under Contract DAAD1702C0097. The authors are with the College of Optics and Photonics, CREOL & FPCE, University of Central Florida, Orlando, FL 32816-2700 USA (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LPT.2007.898819
where , , and are the received, transmitted signals, and the fiber impulse response, respectively, and denotes convolution. designate the real and imaginary parts for each Subscripts term. The all-pass transfer function of a standard single-mode fiber (SSMF) is (2) where , , , and are baseband radial frequency, transmitter (and local oscillator, LO) wavelength, fiber dispersion parameter at wavelength , and the propagation distance, respectively. It can be readily shown that and , where and are the real and imaginary parts of (2), respectively. The Fourier transform (FT) of (1) , where an asterisk denotes comyields plex conjugation. By separating into real and imaginary parts, it is obtained that (3) , , and are the FTs of the real and imaginary where parts of the transmitted and received signals, respectively. Noting that the phase response of the fiber is even, a stable all-pass IIR filter (of real or complex coefficients) cannot be designed to match the fiber’s response [6]. It is necessary then in (3) using monotonous phase response to express matrix (where functions only. Defining is the transfer function of the Hilbert transform), matrix can be rewritten as (4) In SSMF at , ps/km/nm. This imincreases monotonously, and plies that the phase response of the response of monotonously decreases. Hence, the group
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 19, NO. 13, JULY 1, 2007
Fig. 1. Block diagram for DC of the real part of a transmitted signal.
delay of , defined as , is always positive, making causal. A real-coefficients IIR filter having a response matching (as much as possible) to may be designed using several methods [7]. may be chosen using the following The IIR filter’s order argument. Each order of the IIR filter contributes a phase shift in the IIR filter phase response. At the band edge (half the samradians is accumulated [6]. To pling rate), a total phase of , where is the allow DC, the required sampling rate is symbol rate. To find the filter order which best matches a given dispersion, (2) may be used where the phase at the band edge is considered (5) Using (5), it is noted that every additional filter order (i.e., two taps) compensates for approximately 78 km (using ps/km/nm, GHz) of SSMF transmission. , having an inverse phase response compared The filter to , is noncausal and unstable. It is, therefore, necessary to implement an equivalent filtering process to using . By in (3), the FT of the real part of the inserting (4) instead of transmitted signal is given by (6) in (6) is easily obtained since The term containing is stable and causal. To obtain the term containing , an equivalent expression may be considered: . Noting that a complex conjugate of a signal’s FT is equivalent to a time-reversal operation, the required filtering process can be implemented using a time reversal device. The time-reversal technique was previously studied for noncausal filtering achieving linear phase response using IIR filters [8]. A block diagram of the required filtering is presented in Fig. 1. This scheme which allows DC for approach can be extended to obtain the imaginary part of the transmitted signal . III. SIMULATION Simulation of a 10-Gb/s BPSK system with ps/km/nm is considered. To isolate the effect of dispersion, laser linewidth and nonlinearities were not considered. IIR filtering was achieved using a sixth-order IIR Hilbert-transformer
Fig. 2. ECP as a function of total dispersion/IIR filter order.
and an IIR filter of appropriate order that matches the dispersion of the fiber, as obtained in (5). Eye closure penalty (ECP) is defined as shown in the equation at the bottom of the page. The ECP for the IIR filtering scheme as a function of total dispersion (and IIR filter order) is presented in Fig. 2. The FIR filter orders which achieve the same ECP as the IIR filter are presented for selected total dispersion values. For the first simulated point, 11 complex FIR taps are required. For FIR filtering, approximately 5.8 additional (complex) taps are required to compensate each additional 1248 ps/nm of total dispersion. IIR filtering requires only one additional filter order (two additional real taps). At larger total dispersion values, the IIR filter ECP grows slightly. This may be explained by the fact that in designing the IIR filter, a lower degree of freedom is available compared to the FIR design because of the reduced filter order. A better IIR filter design algorithm may reduce this problem. To compare the actual benefit of IIR versus FIR DC, the number of operations in each case is considered. At large dispersion values, the overhead incurred by the Hilbert transformer in the IIR case becomes negligible and the ratio between the number of operations needed for FIR versus IIR filtering is given , where and are the FIR and IIR by filter orders, respectively. A reduction of 2.5 times or more in the number of operations required by IIR compared to FIR is achieved starting at 15 000 ps/nm. IV. BPSK EXPERIMENT To verify the effectiveness of the proposed IIR filtering scheme, a transmission experiment was conducted. A 10-Gb/s BPSK transmission system was set up as shown in Fig. 3. A Mach–Zehnder modulator is biased (at zero transmission) ) to achieve BPSK modulation. Pseudoand driven (at was used. Back-to-back, random bit sequence of length 80- and 160-km transmission distances were considered. SSMF used had a dispersion parameter and attenuation coefficient of ps/km/nm and dB/km, respectively. Launched
GOLDFARB AND LI: CHROMATIC DC USING DIGITAL IIR FILTERING WITH COHERENT DETECTION
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the IIR filter (two taps). In the FIR case, six complex taps were necessary. V. DISCUSSION
Fig. 3. Experimental setup for BPSK transmitter/phase diversity receiver.
Fig. 4. Eye diagrams with and without IIR DC, 80 and 160 km.
optical power into each span was 2.5 dBm. An optical amplifier (erbium-doped fiber amplifier) with appropriate amplified spontaneous emission filter was used at each span to compensate for attenuation. The LO wavelength and polarization were carefully tuned to match that of the transmitter laser. A phase diversity receiver and subsequent sampling using a 40-GSamples/s real-time oscilloscope were used for detection. The samples were resampled to 20 GSamples/s and processed offline. DC was implemented using the IIR filtering technique described above. A sixth-order IIR Hilbert-transformer was used. Phase estimation was achieved as in [1] and the best back-to-back -factor of 17.17 dB was measured. DC for 80 and 160 km was achieved using first- and second-order IIR filters, respectively. The eye diagrams for 80 and 160 km obtained using IIR filtering are presented in Fig. 4. When IIR filtering was implemented the -factor improved from 9 to 13.44 dB and from 3.64 to 10.16 dB, for the 80and 160-km transmission distances, respectively. In the FIR filtering case, the resulting -factors after DC were 11.91 (using 14 taps) and 9 dB (using 20 taps), for 80- and 160-km transmission distances, respectively. It can be concluded that an additional 80-km span requires an addition of only one order to
As seen from both the simulation and experimental results, the IIR filtering approach may be used to mitigate chromatic dispersion for optical transmission systems employing a phase diversity receiver with subsequent sampling and DSP. The benefit obtained by this approach is the reduced tap count required by the IIR filter to achieve DC, which originates from the inherent feedback process in IIR filtering. This advantage becomes more significant at longer transmission distances since the overhead incurred by the Hilbert transformer is constant. The use of an iterative process to obtain the IIR filter coefficients does not guarantee perfect matching between the actual IIR filter response and the desired response [conjugate of (2)]. This leads to a small penalty for IIR filtering compared to FIR filtering with larger dispersion values. The penalty may be reduced by cascading the IIR filter with a low tap-count (complex) FIR filter to compensate for the small discrepancy between the desired and obtained IIR responses. It is important to note that the IIR scheme involves time-reversal operations. The benefit from the reduced number of operations required by IIR filtering is to be weighed against the added complexity of the time reversal operation and the effect on real-time implementation through parallelization. ACKNOWLEDGMENT The authors wish to thank M. G. Taylor for the use of his Matlab code for phase estimation and -factor calculation in the experimental section. REFERENCES [1] M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” IEEE Photon. Technol. Lett., vol. 16, no. 2, pp. 674–676, Feb. 2004. [2] R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photon. Technol. Lett., vol. 17, no. 4, pp. 887–889, Apr. 2005. [3] J. H. Winters, “Equalization in coherent lightwave systems using a fractionally spaced equalizer,” J. Lightw. Technol., vol. 8, no. 10, pp. 1487–1491, Oct. 1990. [4] S. J. Savory, A. D. Stewart, S. Wood, G. Gavioli, M. G. Taylor, R. I. Killey, and P. Bayvel, “Digital equalisation of 40 Gbit/s per wavelength transmission over 2480 km of standard fibre without optical dispersion compensation,” in ECOC’06, Cannes, France, Sep. 2006, Paper Th2.5.5. [5] C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach. New York: Wiley, 1999. [6] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Ciffs, NJ: Prentice-Hall, 1993, pp. 71–77. [7] C. C. Tseng, “Design of IIR digital all-pass filters using least pth phase error criterion,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 50, no. 9, pp. 653–656, Sep. 2003. [8] S. R. Powell and P. M. Chau, “A technique for realizing linear phase IIR filters,” IEEE Trans. Signal Process., vol. 39, no. 11, pp. 2425–2435, Nov. 1991.
