Reduction Techniques for Coherent Optical OFDM Transmission Bernhard Goebel, Graduate Student Member, IEEE, Stephan Hellerbrand, Graduate Student Member, IEEE, Norman Haufe, Norbert Hanik, Member, IEEE Institute for Communications Engineering, Technische Universitat Munchen, D-80290 Munich, Germany E-mail: Bernhard. Goebel@tum. de ABSTRACT In coherent optical OFDM systems, the large peak-to-average power ratio (PAPR) gives rise to signal impairments through the nonlinearity of modulator and fiber.
We review the most prominent PAPR reduction techniques that have been proposed for mitigating the impairments with regard to their reduction capability, computational complexity and redundancy. Simulation results are presented for Clipping, Selected Mapping, Active Constellation Extension and Trellis Shaping. Keywords: modulation, OFDM, coherent detection, nonlinear fiber effects, PAPR, coding. 1. INTRODUCTION Orthogonal frequency division multiplexing (OFDM) is considered one of the most promising transmission schemes for future 100 Gigabit Ethernet (100 GbE) networks.
In combination with coherent detection, it offers virtually unlimited electronic compensation of chromatic dispersion and PMD [1] as well as record spectral efficiencies [2]-[3]. One major drawback of OFDM signals is their large peak-to-average power ratio (PAPR) which gives rise to distortions caused by nonlinear devices such as A/D converter, external modulator and transmission fiber [4]. Upon transmission along the fiber, the Kerr effect creates distortions through four-wave mixing (FWM) between OFDM subcarriers; the strength of these FWM products depends on the signal’s PAPR [5].
Various PAPR reduction techniques have been proposed in a wireless communications context [6] and for optical OFDM systems [5], [7]-[10]. In Section 2, we review the most important PAPR reduction methods for coherent optical OFDM systems with respect to their performance, complexity and introduced redundancy. Section 3 presents numerical simulation results, and Section 4 concludes the paper. 2. PAPR REDUCTION TECHNIQUES FOR OPTICAL OFDM SYSTEMS In OFDM, a high-data-rate bit stream is demultiplexed into N lower-rate streams which modulate N equally spaced subcarriers.
The data symbols [X0, X1,…,XN-1], which may be taken e. g. from a QPSK or 16-QAM signal constellation, form a complex OFDM symbol (or data block) of length NT as x(t ) = 1 ?X N n=0 N ? 1 n ? e j 2?? ft , 0 ? t ? NT , (1) where ? f = 1 / NT is the subcarrier spacing [6]. For a sufficiently large N, the real and imaginary part of x(t) follow a Gaussian distribution and the signal power has a central chi-square distribution with two degrees of freedom [6], so that very large power peaks occur with nonzero probability.
When the PAPR is calculated from samples of the continuous signal (1), sampling at a rate of at least four times the Nyquist rate is recommendable to fully capture peaks located in between samples [4], [6]. PAPR reduction methods can be broadly classified into two categories. In one group of methods, the signal is manipulated in a way such that peaks are removed; clipping, active constellation extension (ACE) and precoding are examples for this approach.
In contrast, selected mapping (SLM) and trellis shaping (TS) are schemes which add redundancy to the signal, thereby creating a degree of freedom to reshape the signal or to replace OFDM symbols with a particularly large PAPR. In general, PAPR reduction methods are difficult to compare. For a rough comparison, it is common to use the (complementary) cumulative distribution function (CCDF) of the PAPR depicted in Fig. 1 (right). The aim of PAPR reduction schemes is to shift the CCDF curve as much to the left as possible.
However, the complexity, redundancy and the actual benefit of a method cannot be judged from its CCDF alone. 2. 1 Clipping, Active Constellation Extension and Precoding Clipping all amplitudes that exceed a certain threshold is the simplest PAPR reduction technique. Clipping leads to distortions within as well as out of the signal bandwidth [4]. Filtering the out-of-band clipping noise results in peak re-growth, so that an iterative clipping-and-filtering approach may be necessary [6]. Mach-Zehnder modulators inherently decrease the PAPR through their nonlinear modulation characteristic (“soft clipping”).
Active constellation extension (ACE) reduces the PAPR by moving some of the outer data symbols Xn away from the decision boundaries. This is depicted in Fig. 1 (left) for a QPSK constellation. Each constellation point is allowed to be moved within its respective grey-shaded area. The black dots show the resulting constellation of 978-1-4244-4826-5/09/$25. 00 ©2009 IEEE 1 ICTON 2009 Mo. B2. 4 256 OFDM symbols with 256 subcarriers each. ACE requires no side information at the receiver and the BER is expected to improve initially. However, ACE increases the average signal power; scaling it back to the initial power leads to an SNR decrease.
As seen from Fig. 1 (right), the PAPR reduction capability of ACE decreases for higher constellation orders since only the outermost points can be moved. Determining which data symbols to move in order to reduce the PAPR is a convex optimization problem which is solved by iteratively clipping the amplitude in time domain and re-setting the wrongly moved symbols in frequency domain. Hence, ACE has a complexity of two FFTs per iteration [9]. Figure 1. ACE for QPSK (left) and comparison of CCDFs (right). Results shown are for N=256 subcarriers and QPSK (black) and 16-QAM (green).
The PAPR CCDFs of the original signal, SLM and TS are independent of the modulation format, whereas ACE and precoding degrade with increasing modulation alphabet size. The use of precoding for optical OFDM systems has been proposed in [7], [8]. These schemes reduce the PAPR by decreasing the side lobes of the data symbols’ autocorrelation function, either through multiplication with an appropriate sequence or by a discrete cosine transform (DCT). Both precoding methods have comparable performance, but the DCT is much less complex, especially when the number of subcarriers is large [8].
Precoding is a useful PAPR reduction technique at small constellation sizes. For QPSK modulation, it achieves the lowest PAPR while exhibiting the lowest complexity of all methods (cf. Fig. 1). However, when higher-order modulation such as 16-QAM is used to increase the spectral efficiency, the effect of these schemes is limited. 2. 2 Selected Mapping and Trellis Shaping The idea of selected mapping (SLM) is to generate at the transmitter a set of candidate data blocks, all representing the same data, and to select the block with the lowest PAPR [5].
In practice, this is achieved by predefining a number NSLM random phase sequences of length N. The data symbol vector [X0, X1,…,XN-1] is then element-wise multiplied with each phase vector to obtain the set of candidate data blocks. An IFFT operation is required for each candidate block, so that SLM has a relatively large complexity. To let the receiver know which phase vector was used for encoding, log2(NSLM) bits of side information need to be transmitted along with the payload data. Hence, SLM introduces redundancy and reduces the net rate to Rn = N log2M – log2NSLM bits per OFDM symbol.
The CCDF shown in Fig. 1 was obtained for N = 256, NSLM = 16 and QPSK (i. e. M = 4). In this example, two subcarriers have to be reserved for the side information, so the net rate reduces to 512-4 bits per OFDM symbols. Consequently, to ensure a constant net data rate in bits/s, the symbol rate needs to be increased by 1-254/256 ? 0. 8%. In practice, the transmitted side information requires protection by powerful FEC codes. Therefore, SLM schemes in which no explicit side information is required have been proposed [11].
Trellis shaping (TS) is a coding method that is useful for various signal shaping purposes; its use for PAPR reduction in an optical OFDM context has been proposed in [10]. The required encoder and decoder are depicted in Fig. 2. The input bit sequence for one OFDM symbol is split into vectors s and b. The vector b consists of N(log2M ? 1) bits and the ith group of log2M – 1 bits is used as the least significant bits (LSB) of the M-QAM constellation point in the ith carrier. The remaining N most significant bits (MSB) will be used for shaping.
The MSB of ns consecutive constellation points form one shaping symbol, hence there are N/ns shaping symbols. The input vector s consists of N(ns? 1)/ns bits, which are encoded to vector z by using an (ns? 1)? ns inverse syndrome former matrix (H? 1)T of the convolutional shaping code Cs, i. e. z = s(H? 1)T. Consequently, the vector z consists of N bits, which can be used to select the MSB of the QAM constellation points. The original sequence s can be restored from z by using a syndrome former matrix HT of the code Cs according to s = zHT. The shaping code Cs has a rate R = 1/ns and is defined by the generator matrix G.
The PAPR reduction capability is largely independent of the shaping code that is used. An arbitrary codeword c in Cs can now be added to z, while leaving the restored sequence unchanged, s = ( z ? c ) H T = zH T + 0 , due to cHT = 0. This property implies that no explicit side information is required at the receiver. 2 ICTON 2009 Mo. B2. 4 Figure 2. Encoder (left) and decoder (right) for trellis shaping. The major task that remains to be solved is how to identify the codeword c that results in the lowest PAPR when added to z. One option is to evaluate the resulting PAPR for all possible codewords in Cs.
However, this is a very time-consuming approach. Instead, a Viterbi algorithm based trellis search is used in conjunction with a frequency-domain metric which minimizes the subcarriers’ autocorrelation sidelobes [10]. For large N, the computational complexity of calculating the full metric can become prohibitively large; a sub-optimal metric can be used which minimizes the sidelobes only within a given window [10]. As seen in Section 3. 1, this even leads to lower PAPRs after a certain link distance. The general trellis shaping scheme as depicted in Fig. can be universally applied for multiple purposes by using different metrics. An alternative metric for PAPR reduction was reported in [13]. More importantly, any metric that is directly related to the physical frequency-domain impairments (Kerr-induced FWM subcarrier crosstalk) could be readily applied. Due to the redundancy, which is included in each OFDM symbol, TS reduces the net rate to Rn = N (log2M – 1/ns) bits per OFDM symbol. The reduction of the net rate becomes smaller for increasing constellation size and increasing size ns of the shaping symbol. The CCDF shown in Fig. was obtained using ns = 8, corresponding to rates of Rn = 480 bits per symbol for QPSK and Rn = 992 bits per symbol for 16-QAM, respectively. 3. SIMULATION RESULTS 3. 1 PAPR evolution along the link As shown in Fig. 3 (left), any PAPR reduction is partly undone along propagation as the chromatic dispersion decorrelates the subcarriers’ phases. However, the PAPR remains well below that of an unshaped signal for the entire link irrespective of the data rate, as long as the cyclic prefix (CP) length exceeds the channel memory. For the 56. 3 Gb/s signal (100G PolMux incl. verhead), this is the case only for approximately the first 800 km in the simulated configuration (20% CP). The average PAPR at the transmitter is only partly meaningful; the DCTprecoded signal starts off with the lowest PAPR, but increases rapidly. In Fig. 3 (right), different window sizes of the TS metric are compared for QPSK modulation and 10. 7 Gb/s data rate. It appears that smaller window sizes (corresponding to “local” minimization of autocorrelation side lobes) yield higher PAPRs at the transmitter, but a steadier (and eventually better) PAPR performance along the link.
Figure 3. Average PAPR over link distance for different data rates and PAPR reduction schemes (left) and different widow sizes of the metric used for Trellis shaping (right). 3. 2 Performance comparison For a fair comparison, the various schemes should be compared using the Q-factor or BER at their respective optimum transmit powers for a link length of interest. In our simulation setup, we used N = 256, identical 80-km spans of SSMF (D = 16 ps/nm/km, S = 0. 057 ps/nm2/km, ? = 0. 2 dB/km, ? = 1. 3 /W/km, no PMD), ideal MZM, no DCF, EDFA NF 6 dB.
The net rate is 10. 7 Gb/s, the overhead of SLM and TS was allowed for by an 3 ICTON 2009 Mo. B2. 4 increased total data rate. Fig. 4 shows the results for link lengths of 400 km (left) and 1600 km (right). The depicted effective Q-factor Qeff for 16-QAM was calculated from the BER, which in turn was estimated analytically from the SNR. It can be seen that TS (ns = 8, window size N/8) and SLM (NSLM = 16) can improve the maximum Q by > 1 dB, whereas clipping the signal (at optimum clipping level) does only improve the signal quality at suboptimal power levels.
Because of the non-Gaussian symbol distribution (cf. Fig. 1), calculating a Q-factor for ACE is not sensible. For 16-QAM, a Qeff,ACE can be obtained from the inner (unmoved) constellation points. In our simulations, ACE brought no improvements according to this Qeff,ACE. However, as Qeff,ACE cannot be directly related to the BER, ACE may still bring some gain. To judge this fairly, direct evaluation of the BER is required. Figure 4. Q-factor over input power for 400 km (left) and 1600 km (right) link length and QPSK (black) and 16-QAM (green) subcarrier modulation. . CONCLUSIONS We have introduced and characterized several PAPR reduction schemes proposed for coherent optical OFDM systems. These schemes differ significantly in terms of computational complexity, redundancy and reduction capability. All schemes yield the best performance at high signal power levels. At optimum levels, SLM and Trellis shaping can improve the signal quality by decreasing the nonlinear penalty. The schemes differ considerably with respect to the PAPR evolution along the link.
Hence, a good PAPR reduction scheme should guarantee low PAPR values for the entire link distance of interest.
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