须藤元气Simultaneous chromatic dispersion and PMD
compensation by using coded-OFDM and
girth-10 LDPC codes
Ivan B. Djordjevic, Lei Xu*, and Ting Wang*
University of Arizona, Department of Electrical and Computer Engineering, Tucson, AZ 85721, USA
*NEC Laboratories America, Princeton, NJ 08540, USA
ivan@ece.arizona.edu
Abstract: Low-density parity-check (LDPC)-coded orthogonal frequency
division multiplexing (OFDM) is studied as an efficient coded modulation
scheme suitable for simultaneous chromatic dispersion and polarization
mode dispersion (PMD) compensation. We show that, for aggregate rate of
10 Gb/s, accumulated dispersion over 6500km of SMF and differential
group delay of 100ps can be simultaneously compensated with penalty
within 1.5 dB (with respect to the back-to-back configuration) when training
sequence based channel estimation and girth-10 LDPC codes of rate 0.8 are
employed.
©2008 Optical Society of America
OCIS codes: (060.4510) Optical communications; (999.9999) Polarization mode dispersion
(PMD); (999.9999) Chromatic dispersion; (060.4080) Modulation; (060.4230) Multiplexing;
(999.9999) Orthogonal frequency division multiplexing; (999.9999) Low-density parity-check
(LDPC) codes
References and Links
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transmission,” Opt. Express 14, 3767-3775 (2006).
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(2006).
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compensation in long haul WDM systems,” in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.
5.I. B. Djordjevic and B. Vasic, “100 Gb/s transmission using orthogonal frequency-division multiplexing,”
IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
6.I. B. Djordjevic, “PMD compensation in fiber-optic communication systems with direct detection using
LDPC-coded OFDM,” Opt. Express 15, 3692-3701 (2007).
7.W. Shieh, “PMD-supported coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19, 134-136
(2006).
8.S. L. Jansen, I. Morita, N. Takeda, H. Tanaka, “20-Gb/s OFDM transmission over 4,160-km SSMF enabled
by RF-pilot tone phase compensation,” in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP15, March 25-29, 2007, Anaheim, CA, USA.
9. B. J. Schmidt, A. J. Lawery, J. Amstrong, “Experimental demonstration of 20 Gbit/s direct-detection optical
OFDM and 12 Gbit/s with a colorless transmitter,” in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP18, March 25-29, 2007, Anaheim, CA, USA.
10. A. Lowery, “Nonlinearity and its compensation in optical OFDM systems,” presented at ECOC 2007
Worksop 5 (Electronic signal processing for transmission impairment mitigation: future challenges).
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turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163 – 1165 (2007).
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using polarization diversity in coherent optical OFDM systems,” Opt. Express 15, 9936-9947 (2007).
13.I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes
for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
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IEEE Trans. Inform. Theory 50, 1788-1794 (2004).
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communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
#92916 - $15.00 USD Received 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008 (C) 2008 OSA7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10269弓形虫
抗体 16.J. L. Fan, “Array codes as low-density parity-check codes,” in Proc. 2nd Int. Symp. Turbo Codes and
Related Topics, Brest, France, pp. 543-546, Sept. 2000.
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(1981).
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H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft
decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
20.N. Cvijetic, L. Xu, and T. Wang, “Adaptive PMD Compensation using OFDM in Long-Haul 10Gb/s
DWDM Systems,” in Optical Fiber Comm. Conf., Paper OTuA5, Anaheim, CA (2007).
1. Introduction
Orthogonal frequency division multiplexing (OFDM) [1-10] represents a particular multicarrier transmission scheme in which a single information-bearing stream is transmitted over many lower rate sub-channels. OFDM has already been considered for use in fiber-optics communication systems by numerous researchers [2-10]. Because the subcarriers in OFDM are orthogonal the partial overlap of neighboring frequency slots is allowed resulting in improved spectral efficiency as compared to a conventional multicarrier system. By using a sufficiently large number of sub-carriers and cyclic extension principle, the intersymbol interference (ISI) due to chromatic dispersion and polarization mode dispersion (PMD) can be significantly reduced [4-9]. To simultaneously compensate both residual chromatic dispersion and PMD someone may use the turbo equalization [11] or maximum-likelihood sequence detection (MLSD). However, the complexity of such an equalizer grows exponentially as accumulated chromatic dispersion and differential group delay increase. On the other hand, in OFDM, in order to improve the tolerance against chromatic dispersion and PMD someone needs to increase the guard interval only, which does not introduce any complexity compared to the turbo equalization or MLSD.
In most of the publications related to OFDM (see [2-10]), different channel impairments, such as chromatic dispersion, PMD, and intrachannel nonlinearities, were studied by observing scenarios in which each particular impairment dominates, although in practice those impairments act simultaneously. The recent reference [12], in which coherent OFDM transmission is observed, is an exception from this common practice. In this paper we show that LDPC-coded OFDM with direct detection is an excellent candidate for simultaneous chromatic dispersion and PMD compensation. Moreover, we propose a new class of LDPC codes suitable for use in coded-OFDM, which is based on large girth quasi-cyclic LDPC codes (the girth represents the shortest cycle in corresponding bipartite graph representation of a parity-check matrix). To facilitate the implementation at high-speed we prefer the use of structured LDPC codes [13-16] rather than random LPDC codes [17]. To reduce the performance loss of structured LDPC codes, compared to random ones, we use the girth as the optimization parameter, and design the LDPC codes of girth-10, as explained in Section 3. Notice that girth-10 LDPC codes significantly outperform the girth-6 LDPC codes we proposed in [6], and outperform the girth-8 LDPC codes by about 0.5 dB (see Section 3). Given the fact that length of girth-12 LDPC codes approaches 100000 of bits, the girth-10 LDPC codes seem to represent a reasonable FEC option for high-speed implementation.
Because the state-of-the-art fiber-optic communication systems essentially use the intensity modulation/direct detection (IM/DD), we consider the LDPC-coded optical OFDM communications with direct detection only. The coherent optical OFDM systems require the use of an additional local laser, which increases the receiver complexity, and are sensitive to the laser phase noise. In most of OFDM proposals for fiber-optics channels, both in-phase (I) and quadrature (Q) channels are to be transmitted (e.g., [3,4]). In our version of OFDM only the transmission of I-channel is sufficient. In what follows, we will show that by using coded-OFDM with training sequence based channel estimation, equipped with girth-10 LDPC codes, it is possible to simultaneously compensate for the accumulated chromatic dispersion over 6500km of SMF and 100 ps of differential group delay with penalty (against the back-to-back #92916 - $15.00 USD Received 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008 (C) 2008 OSA7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10270
configuration) being less than 1.5 dB, in an optical transmission system of aggregate rate 10 Gb/s.
The paper is organized as follows. The concept of LDPC-coded OFDM transmission is introduced in
Section 2. In Section 3 we describe a class of large-girth LDPC codes suitable for use in coded-OFDM. In Section 4 we provide the numerical results to illustrate the suitability of LDPC-coded OFDM in simultaneous chromatic dispersion and PMD compensation. Finally, in Section 5 some important concluding remarks are given.
2. LDPC-coded optical OFDM Transmission
The transmitter and receiver configurations are shown in Figs. 1(a), and 1(b), respectively. On the transmitter side the information-bearing streams at 10 Gb/s are encoded using identical LDPC codes. The outputs of these LDPC encoders are demultiplexed and parsed into groups of B tot bits corresponding to one OFDM frame. The B tot bits in each OFDM frame are subdivided into N QAM sub-channels with the i th sub-carrier carrying b i bits, QAM tot 1N i i B b ==∑. The b i bits from the i th
sub-channel are mapped into a complex-valued signal from a 2b i -point QAM signal constellation. For example, b i =2 for QPSK and b i =4 for 16-QAM. Notice that different sub-carriers may carry different number of bits. The complex-valued signal points from sub-channels are considered to be the values of the fast Fourier transform (FFT) of a multi-carrier OFDM signal. The
OFDM symbol is generated as follows: N QAM input QAM symbols are zero-padded to obtain N FFT input samples for inverse FFT (IFFT), N G non-zero samples are inserted to create the guard interval, and the OFDM symbol is multiplied by the window function. The purpose of cyclic extension is to preserve the orthogonality among sub-carriers when the neighboring OFDM symbols partially overlap due to chromatic dispersion and PMD, and the role of windowing is to reduce the out-of band spectrum. For efficient chromatic dispersion and PMD compensation, the length of cyclically extended guard interval should be smaller than the total spread due to chromatic dispersion and DGD. The cyclic extension is accomplished by repeating the last N G /2 samples of the effective OFDM symbol part (N FFT samples) as a prefix, and repeating the first N G /2 samples as a suffix. After D/A conversion and RF up-conversion, the RF signal can be converted into the optical domain using one of two possible options: (i) the OFDM signal can directly modulate a distributed-feedback (DFB) laser, or (ii) the OFDM signal can be used as the RF input of a Mach-Zehnder modulator (MZM). A DC bias component is added to the OFDM signal in order to enable recovery of the QAM symbols using direct detection. Because bipolar signals cannot be transmitted over an IM/DD link, the bias component should be sufficiently large so that (when added to the OFDM signal) the resulting signal is non-negative. The main disadvantage of this approach scheme is the poor power efficiency. To improve the OFDM power efficiency two alternative schemes can be used: (i) the “clipp
ed-OFDM” (C-OFDM) scheme, which is based on single-side band (SSB) transmission and clipping of the OFDM signal after the bias addition, and (ii) the “unclipped-OFDM” (U-OFDM) scheme, which is based on SSB transmission using a LiNbO 3 MZM. To avoid distortion due to clipping at the transmitter in the U-OFDM scheme, the information can be imposed by modulating the electrical field of the optical carrier. In this way both positive and negative portions of the electrical OFDM signal can be transmitted up to the photodetector. Distortion introduced by the photodetector, caused by squaring, can be successfully eliminated by proper filtering, as shown later in this Section. It is important to note, however, that the U-OFDM scheme is less power efficient than the C-OFDM scheme. The SSB modulation can be achieved either by appropriate optical filtering the double-side band signal at MZM output [see Fig. 1(a)] or by using the Hilbert transformation of in-phase component of OFDM RF signal. The first version requires the use of only in-phase component of RF OFDM signal, providing that zero-padding is done in the middle of OFDM symbol rather than at the edges. The transmitted OFDM signal is real and can be written as
()()OFDM , (1)s t s t D =+ #92916 - $15.00 USD Received 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008泰拉星球
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10271
where
()()()FFT FFT 2/212OFDM ,/2Re FFT RF i j t kT N T j f t i k k i N w t kT s t X e e ππ⋅−−∞=−∞=−−⎧⎫⎪⎪=⋅∑∑⎨⎬
⎪⎪⎩⎭
is defined for t ∈[kT -T G /2-T win , kT +T FFT +T G /2+T win ]. In the above expression X i ,k denotes the i th subcarrier of the k th OFDM symbol, w (t ) is the window function, and f RF is the RF carrier frequency. T denotes the duration of the OFDM symbol, T FFT denotes the FFT sequence duration, T G is the guard interval duration (the duration of cyclic extension), and T win denotes the windowing interval duration. D denotes the DC bias component, which is introduced to enable the OFDM demodulation using the direct detection.
The PIN photodiode output current can be written as
()()()(){}2PIN OFDM ,(2)i t R s t D h t N t ⎡⎤=+∗+⎣⎦
where s OFDM (t ) denotes the transmitted OFDM signal in RF domain given by (1). D is introduced above, while R PIN denotes the photodiode responsivity. The impulse response of the optical channel is represented by h (t ), with operator * being the convolution operator. The N (t ) represents the amplified spontaneous emission (ASE) noise. The signal after RF down-conversion and appropriate filtering, can be written as ()()()()()cos , (3)RF
RF e r t i t k t h t n t ω⎡⎤=∗+⎢⎥⎣⎦
热敏电阻
where h e (t ) is the impulse response of the low-pass filter, n (t ) is electronic noise in the receiver, and k RF denotes the RF down-conversion coefficient. Finally, after the A/D conversion and cyclic extension removal, the signal is demodulated by using the FFT algorithm. The soft outputs of the FFT demodulator are used to estimate the bit reliabilities that are fed to identical LDPC iterative decoders implemented based on the sum-product algorithm [13].
For the sake of illustration, let us consider the signal waveforms and power-spectral densities (PSDs) at various points in the OFDM system given in Fig. 1. These examples are generated using SSB transmission in a back-to-back configuration. The bandwidth of the OFDM signal is set to B GHz, and the RF carrier to 0.75B . With B we denoted the total symbol transmission rate. The numb
er of OFDM sub-channels is set to 64, the OFDM sequence is zero-padded, and the FFT is calculated using 128 points. The guard interval is obtained by a cyclic extension of 2x16 samples. The average transmitted launch power, in this back-to-back example, is set to 0dBm. The OFDM transmitter parameters are carefully chosen such that RF driver amplifier and MZM, as shown in Figs. 2(a)-2(b), operate in the linear regime. The PSDs of MZM output signal, and the photodetector output signal are shown in Figs. 2(c) and 2(d), respectively. The OFDM term after beating in the photodetector (PD), the
streams from
10-Gb/s data SMF streams
(b)
Fig. 1. LDPC-coded OFDM: (a) transmitter configuration, and (b) receiver configuration.
LDPCE-LDPC encoder, LDPCD-LDPC decoder, S/P-serial-to-parallel converter, MZM-Mach-
Zehnder modulator, SMF-single-mode optical fiber, PD-photodetector, DSB-double-sideband,
SSB-single-sideband. #92916 - $15.00 USD Received 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10272
L D d r i v i
n g s i g n a l , v R F [V ]Time, t [ps]
R
F Time, t [ps]
唐县理想中学
P S D [d B m /H z ]Normalized frequency, (f -f c )/B Normalized frequency, (f-f c )/B
Fig. 2. Waveforms and PSDs of SSB QPSK-OFDM signal at different points during
transmission for electrical SNR (per bit) of 6dB. (f c denotes the optical carrier frequency, LD
denotes the laser diode).
Notice that our proposal requires the use of only I-channel, while both the coherent detection version due to Shieh [3] and the direct detection version due to Lowery [4] require the use of I- and Q-channels. In our recent paper [6] we have shown that coded OFDM is an excellent candidate to be used in PMD compensation. In the same paper we designed the girth-6 LDPC codes suitable for use in PMD compensation by coded-OFDM, because they have small number of cycles of length 6. Moreover, the bit-error rate (BER) performance was evaluated observing the thermal noise dominated scenario. In next Section, we will describe the new class of quasi-cyclic LDPC codes suitable for use in coded-OFDM optical transmission, the girth-10 LDPC codes. Those codes, significantly outperform the girth-6 codes, and provide about 0.5 dB improvement in coding gain over girth-8 LDPC codes and about 1 dB improvement over turbo-product codes. Later, in Section 4, we study the efficiency of coded-OFDM based on girth-10 LDPC codes in simultaneous suppression of chromatic dispersion and PMD, observing the ASE noise dominated scenario.
3. Large girth block-circulant (array) LDPC codes
Now we turn our attention to the design of LDPC codes of large girth. Based on Tanner’s bound for the minimum distance of an LDPC code [18]
()()(2)/4(2)/4(2)/41(1)1,/2212(4)1(1)1(1),/222g g g r r g m
r d r r r g m r −⎢⎥⎣⎦−−⎢⎥⎢⎥⎣⎦⎣⎦⎧+−−=+⎪⎪−≥⎨⎪+−−+−=⎪−⎩
where g and r denote the girth of the code graph and the column weight, respectively, and where d stands for the minimum distance of the code. It follows that large girth leads to an exponential increase in the minimum distance, providing that the column weight is at least 3. (⎣⎦ denotes the largest integer less than or equal to the enclosed quantity.) For example, the minimum distance of girth-10 codes with column weight r =3 is at least 10.
The structured LDPC codes introduced in this Section belong to the class of quasi-cyclic
[14,15] or array [16] codes. Their parity-check matrix can be represented by
#92916 - $15.00 USD Received 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10273
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