Performance Bounds of Optical Receivers With Electronic Detection and Decoding Tobias Rankl, Christian Kurz, and Joachim Speidel, Member, IEEE

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1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, Performance Bounds of Optical Receivers With Electronic Detection Decoding Tobias Rankl, Christian Kurz, Joachim Speidel, Member, IEEE Abstract In this paper, we investigate lower bounds of optical-signal-to-noise ratio (OSNR) for three types of electronic digital-signal-processing (DSP) receivers, which are used for an intensity-modulated direct-photo-detected optical transmission link. Target is to determine the performance bound of this transmission link if equalization /or forward error correction (FEC) are used. The investigated DSP methods are hard-input FEC decoding, soft-input FEC decoding, sequence detection. The impact of chromatic polarization mode dispersion, as well as the effect of code rate on the bounds, is shown. Signal statistics are computed analytically. Several implementations of coding detection schemes, such as Reed Solomon (RS) low-density parity-check (LDPC) decoding as well as Turbo equalization, are investigated, we show the distance of their performances to the derived bounds in quite some detail. Index Terms Achievable information rates, direct photo detection, electronic signal processing, fiber optical communications, system capacities, turbo equalization. I. INTRODUCTION T HE increasing dem for high bit rates in optical communications requires sophisticated coding detection methods to cope with noise intersymbol interference (ISI) caused by chromatic (CD) polarization mode dispersion (PMD) of the fiber. The high performance speed of modern digital signal processing devices (DSPs) offer significant chances to implement coding detection methods electronically at the transmitter the receiver [1] [9]. In this paper, we investigate the performance bounds of such methods from the perspective of information theory answer the question what can be achieved theoretically. As engineers we are also interested in real-world solutions, therefore we consider various dedicated methods, show how close their performance are to the predicted bounds. Throughout this paper, binary intensity modulated (IM) transmission with direct optical photo-detection (DD) is assumed. The information bitrate is 10 Gb/s. As depicted in Fig. 1(a), we focus on a communication link with forward error correction. The net information bit rate is fixed the code rate is. Thus, the symbol rate on the fiber is Manuscript received November 10, 2008; revised May 08, First published June 12, 2009; current version published July 24, The authors are with the Institute of Telecommunications, University of Stuttgart, Stuttgart 70569, Germany ( rankl@inue.uni-stuttgart.de). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JLT (1) After encoding for forward error correction (FEC) of the information bits, the binary symbols are externally modulated by a chirp-free electroabsorption modulator onto the output signal of a continuous-wave (CW) laser with mean power carrier frequency. The signal then propagates along the stard single-mode fiber (SSMF). At the receiver, the optical signal is amplified by an erbium-doped fiber amplifier (EDFA) that generates amplified spontaneous emission (ASE) noise is then optically filtered. After optical-to-electrical conversion by a photodiode the signal is filtered again the electrical receive signal is obtained. Subsequent sampling of is done at, yielding the sequence with. is the symbol interval,. If, the spacing of the samples is one sample per symbol is obtained. If, the spacing is two samples per are obtained. is necessary for synchronization to adjust the sampling phase. We distinguish between sampling at the maximal inner eye opening (with respect to the dispersion free case) later. In the following, these sampling instances are referred to as, respectively. The relation between discrete time can be expressed as with. Finally, the samples are used as input to a DSP are equalized FEC decoded according to the capabilities of the DSP. The following three electronic detection/decoding methods at the receiver are considered in quite some detail. 1) Hard-input decoding : This means that a threshold decision is performed on to obtain a binary sequence, which is used as input to a single FEC decoder. 2) Soft-input decoding : This means that no threshold decision on takes place a soft-input FEC decoder is used. Implementations of this method are softdecision Reed Solomon (RS) or low-density parity-check (LDPC) codes [10]. 3) Sequence detection ( arbitrary): It means that no threshold decision on takes place, detection is on the basis of consecutive samples, i.e., a soft-input equalizer a subsequent FEC decoder are used in the DSP. The concatenation of Viterbi equalizer a hard-input FEC decoder as well as the Bahl Cocke Jelinek Raviv (BCJR) equalizer [4] with a subsequent soft-input FEC decoder belong to this category. Equalizer FEC decoder may also be embedded in an iterative loop resulting in the wellknown Turbo equalizer [1] [4]. Compared to 1) 2), method 3) is also capable of reducing ISI. (2) /$ IEEE

2 3568 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, 2009 Fig. 1. Fiber-optic transmission with transmitter (TX), stard singe-mode fiber (SSMF), receiver (RX). (a) Block diagram. (b) Equivalent low-pass system model. The target of this paper is to investigate bounds of the signal-to-noise ratios (SNRs) for the three methods 1), 2), 3) to achieve data transmission without bit errors or with a predefined bit error probability (BER). We use fundamentals of Shannon s information rate-distortion theory, investigate the mutual information between channel input channel output. Therefore, the results are quite general are not subject to a dedicated code or equalizer implementation. is also called information rate (IR). To compute the mutual information, the probability density functions (pdfs) of the samples are required. Compared to [11], we calculate these pdfs analytically, no approximations are used. The computation is based on a Karhunen Loève series expansion (KLSE) a moment generation function (MGF) [12] [15]. This is done in the Appendix. Furthermore, we investigate the impact of PMD, CD, code rate on the SNR bounds. This is important, because an FEC code with a smaller code rate can provide a more powerful error correction. On the other h, according to (1), the symbol rate is increased, which causes higher ISIs. As already indicated, we also consider dedicated coding detection schemes, reflect their performance with respect to the derived bounds. In detail, we consider advanced coding detection schemes like soft iterative LDPC RS decoding as well as Viterbi equalization turbo equalization with one two samples per branch metric. As pointed out in [11], several papers appeared recently, which deal with the capacity of optical transmission links that use DSP in the receiver. The capacity is most often determined (as, for example, in [11], [16], [17]) by applying the method of Arnold [18], which we also follow. An alternative is introduced in [19], which uses the perturbation theory. However, no paper investigates compares the limits for detection decoding for various code rates channel distortions by use of analytically calculated channel statistics. Furthermore, the comparison of the performance bounds to turbo equalization with samples per metric, to iterative LDPC decoding was also not considered before. This paper is organized as follows. In Section II, the system model is introduced. Section III derives detection decoding bounds, in Section IV, some basic properties of signal noise are discussed. Section V discusses numerical examples, in Section VI, a comparison of the bounds with the performance of dedicated coding detection schemes is presented. Finally, Section VII concludes the paper. The Appendix contains some further discussions of the signal noise properties as well as the determination of the pdfs that are necessary for the derivation of the capacity limits. II. SYSTEM MODEL The equivalent low-pass (elp) system model of the considered IM-DD optical transmission link is shown in Fig. 1(b). The low-pass representation of the transmit signal is where with is the polarization Jones vector that characterizes the polarization of the CW laser output. are the transmit symbols, which include the extinction ratio of the modulator is the impulse response of the pulse shaper [20]. The signal exhibits a 2-D field distribution, 1 can be written as. After transmission over the fiber, the signal at the fiber output is superimposed by the ASE noise of the EDFA, is the input of the optical RX filter. The ASE noise is modeled as a complex-valued zero mean white Gaussian noise process with two polarization directions the spectral density per complex polarization component. The operator describes the signal propagation in the SSMF is modeled by a pair of nonlinearly coupled nonlinear Schrödinger equations [4], [21]. The electrical receive signal is 1 Vectorized signals in the remaining part of the paper do always deploy this property. (3) (4) (5)

3 RANKL et al.: PERFORMANCE BOUNDS OF OPTICAL RECEIVERS WITH ELECTRONIC DETECTION AND DECODING 3569 where is the optoelectronic conversion factor of the photodiode, are the impulse responses of the optical 2 electrical filter, respectively. Factor 1/2 comes from the derivation of the equivalent low-pass model, due to a squaring of the carrier, by the photodiode, where the high-frequency component is at 384 THz, for 1550 nm. However, this frequency component is filtered out by the photodiode. Consequently, the equivalent low-pass representation, with the factor 1/2, shows the same input/output relation as the bpass model. As stated in [11], all effects of memory within section in Fig. 1 can be modeled by a sequence of adjacent transmit (TX) symbols. Each results in dedicated statistical parameters for. As is known, for square law photodetection, these statistical properties are of chi-square type are signal dependent. In order to be able to treat this signal dependency accurately, a state-input description is introduced, where the state variable the are combined to, which is equivalent to. As is binary, states exist. Using the state-input representation, the statistical properties of a sample can be given by the pdf To simplify notation, sometimes instead of is used. The SNR is defined as where equally likely transmit symbols are assumed. is the energy per bit, is the symbol energy, is the total noise spectral density, is the impulse shaper duty cycle according to [20]. The OSNR can be given as is the measurement bwidth that is mostly chosen as 12.5 GHz. For 10 Gb/s, db. III. SNR BOUNDS FOR DETECTION AND DECODING Shannon derived the channel capacity of communication channels by maximizing the IR between input process output process over the set of channel input distributions [22]. For the considered optical channel, the input values are binary thus the input distribution exhibits a binary Bernoulli pdf. Because of this fact, the maximization cannot be performed to derive the channel capacity. As an alternative, we consider of a given system configuration. In the following, the system capacities are derived discussed for 2 The two upper bars of h (t) indicate that this impulse response is characterized by the matrix, because of the 2-D input h (t) 0 0 h (t) signal. (6) (7) (8) Fig. 2. Discrete representation of the optical transmission link with decision device. the three detection/decoding methods 1), 2), 3), defined in Section I. Fig. 2 can be used as a principle block diagram for the theoretical considerations. A discrete-time representation of a fiber optic transmission link with binary input, continuous output samples, binary output is shown. The rom variables (RVs) are,,, respectively. The section of Fig. 1 is also shown. Since the optical communication channel is dispersive thus belongs to the class of Markov channels, in each sample, the information of several adjacent transmit symbols is superimposed each is spread over a number of adjacent. Therefore, sequences of transmit symbols receive samples are of interest for sophisticated detection/decoding methods, such as method 3). Consequently, between the binary fiber input sequence the output sequence has to be determined [11], [18], [23]. Since hard-input soft-input decoding algorithms do not use knowledge about the ISI, the sequence information can be neglected have to be computed for detection/decoding methods 1) 2), respectively. Both do not take advantage out of the Markov property of the transmission channel. Using,,, SNRs that fulfill the condition for error-free data communication the rate-distortion limits can be determined. A. Shannon s Channel Coding Theorem As is well known, Shannon stated in his coding theorem [22] that for any code rate reliable communication with an arbitrary small error probability can be achieved by use of proper FEC codes, possibly of infinite length. Conversely, for code rates, is bounded away from zero. Since in the case at h the code rate has a strong impact on ISI, Shannon s channel coding theorem is used to find the SNR that fulfills the condition. This SNR is the bound of reliable data transmission for the respective channel system settings is referred to as system capacity limit; it is For (9) reliable (i.e., bit error free) transmission is possible; for is bounded away from zero. Using this, FEC codes with code rate can be found that enables error-free transmission for SNRs that are larger than.

4 3570 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, 2009 B. System Capacity SNR Bound To calculate the system capacity bound, has to be found. This can be done numerically by evaluating for various SNR values until the condition is fulfilled. Section III-D derives the IRs,,, which are necessary to determine for the three detection/decoding methods. In Section V, the system capacities are evaluated numerically are discussed. C. Rate-Distortion Limit Unlike is assumed in Section III-A, in optical communications, the acceptable bit-error rates (BERs) are in range of to not zero. Since for BERs larger than zero (i.e., ) lower SNRs can be achieved than that determined by (9), another SNR bound has to be used. This SNR bound is based on the rate-distortion theory is called the rate-distortion limit ( -limit) [24]. The respective SNR bound can be determined by modifying the code rate in (9) to (10) where is the binary entropy function [24], is the probability of occurrence of, is the acceptable distortion. Since, holds for equally probable, we get. Instead of (9) (11) has to be solved. For, the -limit (11) converges to the SNR system capacity limit (9). D. The IRs for the Three Detection/Decoding Methods In this section, the IRs for the three detection/decoding methods 1), 2), 3) are derived. In the above discussion of system capacity SNR -limit, the was used as placeholder. Now it has to be replaced by for method 1), for method 2), for method 3). Equally probable TX symbols with are used in the following. 1) Detection/Decoding Method 3) Sequence Detection : can be calculated by solving (12) where is the source entropy, is the receive (RX) entropy, is the joint entropy of the TX RX sequence. is the conditional entropy of TX sequence under the condition that the RX sequence is known. For equally probable, the entropy. can be determined by use of the Shannon McMillan Breiman theorem 3 [25], [26] as Using these equations as (13) (14) as well, the equation (15) results. can be computed by Monte Carlo simulations using the forward recursion of a BCJR algorithm [27] as well as by the use of the pdf (34). The SNR bounds based on with are found in [11] to be the fundamental limits for optical transmission with direct photodetection. 2) Detection/Decoding Method 2) Soft-Input Decoding : In case of soft-input decoding, the communication channel exhibits a binary-input continuous-output property. In the following, equiprobable signal states with are assumed. For soft-input decoding instead of the has to be used to determine the SNR bound. Then (16) holds, can be determined according to [28] by use of the IR formula for binary-input continuous-output channels, If (17) as well as are applied to (17), the expression (18) results. The pdf is again given by (34). 3) Detection/Decoding Method 1) Hard-Input Decoding : If this decoding technique is applied, a binary decision device as shown in Fig. 2 is used in order to quantize the continuous output of the transmission link to binary samples. Then, the communication exhibits the binary-input binary-output property as shown in Fig. 3. Since the error probabilities (a transmitted 0 is received as 1) (a transmitted 1 is received as 3 Neither ergodicity nor stationarity is necessary if this theorem is used. However, N -stationarity (cyclostationarity) of signal noise is required [25], [26]. The N -stationarity assumption exhibits N cases of stationarity that repeat depending on the transmit sequence. In the case at h, N =2.

5 RANKL et al.: PERFORMANCE BOUNDS OF OPTICAL RECEIVERS WITH ELECTRONIC DETECTION AND DECODING 3571 Fig. 3. Binary nonsymmetric channel model with error probabilities P for transition 0! 1 P for transition 1! 0. 0) are different, the transmission link can be modeled as a binary nonsymmetric channel. In this case, has to be used can be determined as (19) After applying Bayes rule some further elementary rules as well as is obtained. The probability terms in (20) can be written as (20) (21) (22) is given later in (35),, is the optimal decision threshold value that delivers the lowest BER at the output of the decision device. Further, the probabilities for receiving correct bits can be determined by if. IV. SOME STATISTICAL PROPERTIES OF SIGNAL AND NOISE In the Appendix, we derive some important statistical properties of. As an example as a result of (34), Fig. 4. The pdfs of q with =0 R = 10 Gb/s, R = 0.937, 1 = 60 ps, R = 0 ps/nm. 2 markers indicate the pdfs of transmitted zeros markers of transmitted ones. The optimal decision threshold q is also shown. the pdfs of are shown in Fig. 4. These pdfs are necessary to characterize the receive samples for the given channel conditions. A similar result holds for. Another important measure to characterize the stochastic signal is the correlation of samples with different. While the pdf describes the statistic of single RX samples, the autocorrelation function (ACF) gives the relation of two samples. According to the theoretical analysis in the Appendix, is composed of a data signal noise components. The data signal contains the digital transmit data the ISI, which is a kind of correlation. This correlation is introduced by the TX impulse shaper, the fiber, the photodiode, the receiver filters. It is described by the state-input description of the transmission link, which is introduced in Section II. Since, in front of the optical filter, the ASE noise is superimposed with this data signal, the correlation of the noise components in is introduced by the transfer characteristic of the receiver. As can be seen in (23), signal noise interact with each other in the receiver. Therefore, the correlation of the noise in also depends on the data signal. Since in a practical system is the only available signal, a good way to analyze the correlation of the noise components is to determine the ACF of to derive the autocovariance function (ACOV) that exclusively contains the correlation of the noise components of. In order to do this, the ACOV of, which is defined as with mean values of is determined. The ACOVs of with are shown in Fig. 5 for, i.e., eight different ; further system parameters are given in Section V. The impact of the data signal in on the noise can be recognized by the various shapes in Fig. 5 that are all related to different symbol patterns of. The solid line shows, the markers indicate, which is the signal-dependent noise correlation of for. The case can be obtained by using every second marker starting at. As can be seen in the diagram, for, the correlation of adjacent samples is huge, however the correlation for

6 3572 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, 2009 Fig. 5. Autocovariance functions of q (t) (solid) sampled at mt the autocovariance functions of q ( markers) for l =2 l =1(if every second is chosen starting at t =0). The functions are plotted for the 8-bit patterns (a ;a ;a ) for the case 1 = 60 ps R = 400 ps/nm. is below 10% compared to the ACOVs at. In summary, the correlation of the noise components noise noise the signal noise in is quite significant. A. The Correlation of the Noise the IR Computation To incorporate the correlation properties of the noise in in the mathematics of Section III, correctly, is a challenging task. Since we determine bounds of existing DSP methods not fundamental limits as [11], we can ignore this correlation for the computation of the bounds, because most state-of-the-art detection/decoding algorithms also do not consider this correlation. Therefore, bounds algorithms are based on the same statistical mismatch thus the determined bounds are the lower bounds of the investigated algorithms. Some of the considered algorithms are BCJR detection [27], maximum-likelihood sequence estimation (MLSE, Viterbi equalizer) [29], [30], sum product algorithm (SPA) LDPC decoding [31], decoding of RS codes [32], turbo equalization [2], [4], others. None of them consider the noise correlation. V. NUMERICAL ANALYSIS The numerical analysis in this section is based on a 10-Gb/s optical transmission scheme with binary input symbols. The impulse shaper exhibits a cos-square waveform with rolloff factor duty cycle [20]. The electroabsorption modulation is chirp free has a signal extinction ratio of 13 db. The CW laser power is 3 dbm. The SSMF is assumed to be lossless. The RX filters are of Gaussian shape exhibit 14 GHz optical filter bwidth 7 GHz electrical filter cutoff frequency. The optoelectronic conversion factor of the photodiode is. The SNR definition is given in (7), CD PMD are assumed to be the main distortions. (i.e., the length of ) is always chosen to cover the system memory perfectly. Franceschini et al. [11] intensively investigates the effects of sampling rate, sampling phase, quantization on for CD. In contrast, this paper focuses on the comparison of the system capacity bounds that are based on,,, as well as on impacts of code rate, PMD, CD on the bounds. We restrict our investigations to because for no significant performance improvement can be observed [11]. Furthermore, the results for are almost independent of the sampling phase. Therefore, variations in the sampling phase are only considered for. Quantization of with 3 bits provides nearly no degradation of performance. Therefore, quantization effects are also not considered. Fig. 6(a) (b) shows a comparison of the SNR bounds for the three detection/decoding methods 1), 2), 3) of Section I as a function of the differential group delay of the PMD the residual chromatic dispersion. It turns out that hardinput decoding exhibits a pole at a certain, where decoding beyond this point is impossible. Interestingly, the soft-input decoding develops the same pole but decoding beyond this pole is possible. Soft-input decoding further allows transmission with a significantly lower SNR. Sequence detection with exhibits the lowest bound marks also the fundamental limit for electronic receivers for IM-DD optical transmission [11]. As expected, the sampling phase has an impact on the SNR bounds with. The SNR bounds for are also shown. Our results at zero dispersion are about 0.5 db worse compared to that shown in [11, Fig. 6]. This gap is probably caused by differences in the transmission parameters. Figs show similar SNR bounds as Fig. 6(a) (b), however as a function of separately for the three methods 1), 2), 3), for various code rates. The given corresponds to a coding overhead of 5%, 6.7%, 10%, 12.5%, 15%. The hard-input decoding SNR bounds in Fig. 7(a) (b) show rather strong variations, if code rate is changed. For low, FEC codes with low (i.e., strong codes) perform better than codes with high (i.e., weak codes). If increase, the SNR bounds enter into an intersection exhibit opposite performances as before. The reason is quite obvious, because at low, there is almost no ISI the increase of ISI due to a lower code rate (higher symbol rate ) is not significant. For large, the high ISI is further increased by lowering the code rate the performance drops. Further, there is a dispersion bound (decoding pole) for hard-input decoding at, where hard-input decoding is no longer possible. as it was chosen in the ITU-T G.975 G recommendations [33], [34] seems to be a good compromise. The properties of soft-input decoding in Fig. 8(a) (b) are similar to Fig. 7(a) (b). However, soft-input decoding beyond the decoding pole is possible a coding gain compared to hard-input decoding is present. Interestingly, these findings do not hold for detection/decoding method 3) sequence detection, as is shown in Figs No intersection is observed a decreasing code rate improves the system performance. This can be seen best in Fig. 10(a) (b) where samples per symbol interval are processed by the receiver. Fig. 11 shows bounds of achievable BER over SNR for the three detection/decoding methods 1), 2), 3) for 0 ps 0 ps/nm as well as 60 ps 400 ps/nm. The curves represent the BER bound that can be achieved at the respective SNR. They are obtained on the basis of rate distortion -limits, are plotted for a code rate. For low BER, i.e.,, the characteristics

7 RANKL et al.: PERFORMANCE BOUNDS OF OPTICAL RECEIVERS WITH ELECTRONIC DETECTION AND DECODING 3573 Fig. 6. System capacity SNR bound for R =0:937 (O = 6.7%), as a function of (a) differential group delay 1 (b) residual dispersion R various detection/decoding methods. As SNR the E =N is given on the left-h side axis of the diagram the OSNR on the right-h side axis. Fig. 7. System capacity SNR bounds for detection/decoding method 1) hard-input decoding as a function of (a) differential group delay 1 (b) residual dispersion R for various code rates R (the same axis labeling as in Fig. 6). are equivalent to the SNR system capacity bounds, however, for large BER, i.e.,, the -limit is represented by the curves. Even though the -limits predict lower SNR bounds than the limits discussed before, for BER distortions of or, as required in optical communications, the -limits do generally not result in a nameable gain. This holds, because the -limits already reach the system capacities at BER of about. As shown in Fig. 11, in the dispersion-free case, detection with samples is worse than detection with. We believe that this property is caused by the strong correlation of, which leads to a strong mismatch of signal statistic statistical assumption in the receivers in the theoretical analysis (see Section IV-A). Because of that, receivers with the better matching statistical assumption (as, for example, in the case of ) have better performance. However, if ISI dominates, the improved equalization capabilities of the detection dominate, thus, the detector again outperforms the detector. The area that can be achieved by electronic detection/decoding is indicated in Figs by achievable region. This region is bounded by the system capacity SNR curves. Hardware realizations of electronic receivers exhibit a performance on this area. VI. PERFORMANCE OF REAL DETECTION/DECODING ALGORITHMS Turbo equalization [2], [4], iterative LDPC decoding [31], RS decoding [32] are examples of the considered electronic detection/decoding methods 1), 2), 3). They can be used to approach the system capacity bounds that were derived discussed before. As is well known, due to technological restrictions, Shannon s capacity bounds cannot be reached exactly. However, some of the mentioned detection decoding methods are able to approach the system capacity bounds quite close with a small gap that is below 1 db of SNR. In the following, turbo equalization, iterative LDPC decoding, RS decoding, serially concatenated Viterbi detection, RS decoding as well as a serial concatenation of BCJR detection iterative LDPC decoding are investigated. The RS code RS(255,239) of the ITU-T recommendation G.975 [33] is applied. Its code rate is. Furthermore,

8 3574 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, 2009 Fig. 8. System capacity SNR bounds for detection/decoding method 2) soft-input decoding as a function of (a) differential group delay 1 (b) residual dispersion R for various code rates R (the same axis labeling as in Fig. 6). Fig. 9. System capacity SNR bounds for detection/decoding method 3) sequence detection as a function of (a) differential group delay 1 (b) residual dispersion R for l =1sample per symbol interval sampling phase =0(solid) as well as sampling phase = T =2 (dashed) various code rates R (the same axis labeling as in Fig. 6). we consider an Euclidean geometry LDPC code of type 1 [31] with parameters, which is 16 times column extended. Details about the applied LDPC code, i.e., about its generation method, the code parameters can be found in [31]. Further, the code is of size, with bit codeword length bit information word length; the code rate is also. Fig. 12 shows the BER versus SNR for different FECs receivers. The fiber link has a differential group delay of 60 ps a residual chromatic dispersion of 400 ps/nm. It turns out that a turbo equalizer based on LDPC code BCJR soft-output equalizer samples per metric performs best. The SNR difference to the rate-distortion limit is only about 0.5 db. This is possible because this method is able to use not only or. The BCJR equalizer combined with LDPC decoding, but without turbo iterations, shows a small performance loss compared to the turbo equalizer. Soft-input decoding of LDPC codes with an SPA decoder performs worse with a larger SNR gap of about 2.5 db to the turbo equalizer with 1 db to its rate-distortion limit. Furthermore, hard-input RS decoding is shown. As predicted by Fig. 11, the BER performance of this receiver is much worse with a gap of 3.5 db to its -limit almost 7 db to the turbo equalizer with. Additionally, the performance of a hard-output Viterbi equalizer (, ) concatenated with an RS decoder is shown. In order to allow the evaluation of the BERs at, in this section, the respective SNRs were estimated by an extrapolation of the turbo cliff. To prevent error floors, an additional outer FEC code with low redundancy may be used as suggested in [2] [35]. In Fig. 13, the required SNR versus differential group delay [Fig. 13(a)] residual chromatic dispersion [Fig. 13(b)] is plotted for the receivers of Fig. 12. Since sequence detection methods with perform better than with, the investigations are focused on. It turns out that the sequence detection methods perform as predicted by their bounds, of course, with a small gap. For a receiver with concatenated equalizer LDPC decoder, the difference to the SNR bound of is not larger than 2 db for all much smaller for all. Its performance is very similar to the turbo equalizer, however, turbo equalization with samples per metric is able to further reduce the SNR gap to

9 RANKL et al.: PERFORMANCE BOUNDS OF OPTICAL RECEIVERS WITH ELECTRONIC DETECTION AND DECODING 3575 Fig. 10. System capacity SNR bounds for detection/decoding method 3) sequence detection as a function of (a) differential group delay 1 (b) residual dispersion R for l =2samples per symbol interval sampling at =0 = T =2 various code rates R (the same axis labeling as in Fig. 6). Fig. 11. BER as a function of SNR for R(D)-limits at 1 = 0 ps R = 0 ps/nm (solid) 1 = 60 ps R = 400 ps/nm (dashed) for the detection/ decoding methods 1), 2), 3) l =1; =0, l =2. the system capacity bound to about 1 db for 100 ps to 200 ps. In case of CD, no significant improvement of the turbo equalizer compared to the concatenation of BCJR, soft-input LDPC decoder without turbo iterations can be observed; both are within about 0.5 db to the respective bound. Soft-input decoding of LDPC codes has a distance of about 1 db to its bound. It develops a decoding pole at 93.7 ps. Interestingly, this method is able to decode also beyond this value. As the soft-input values of the decoder are computed similarly to the soft bits at the output of a soft demapper [36], the decoding beyond 93.7 ps can be understood. Hard-input decoding with RS codes develops a gap to its SNR bound of about 3.5 db. A pole is also present at 93.7 ps. The pole for CD is not shown in Fig. 13(b) because it builds up beyond 1800 ps/nm. Further, some Viterbi-equalizer-based receivers are shown. The performances of these receivers are similar to the other sequence detection receivers, however with a much larger gap to the bounds. VII. CONCLUSION We have investigated fiber-optic transmission with FEC direct optical photodetection. The net information bit rate for all numerical investigations is 10 Gb/s. ISI in the received signal is caused by the polarization mode dispersion the residual chromatic dispersion. Furthermore, the signal is corrupted by signal-dependent noise that was initially generated by an optical amplifier. SNR bounds for three digital signal processing methods, which can provide error-free transmission, are investigated on basis of Shannon s information theory on the rate-distortion theory. The methods under study are 1) hard-input decoding, 2) soft-input decoding, 3) sequence detection. As real-world solutions of these methods, we consider LDPC RS codes for FEC. Further, BCJR Viterbi equalization are considered as sequence detectors. Numerical investigations of the bounds were performed for FEC code rates in the range of It turned out that for all investigated code rates sequence detection receivers with two receive samples per transmit symbol exhibit the lowest bounds. In comparison, soft-input FEC decoders exhibit almost the same bounds at low ISI. However, their SNR performance bound increase rapidly if the ISI increases finally the bounds approach infinity, i.e., the bounds approach a pole where FEC decoding is impossible. Hard-input FEC decoders exhibit a very similar performance as the soft-input decoders, however, the bounds are about 1 db worse. Surprisingly, soft-input FEC decoding can be performed also beyond the decoding pole, at higher ISI, however with a worse performance. In the second part of this paper, we consider implementations of receivers, which are able to approach the performance bounds. We present soft-input LDPC decoding, hard-input RS decoding, concatenations of Viterbi BCJR equalizer with FEC decoders. As already shown during the analysis of the bounds, it is again shown that FEC coding as a single element is outperformed by far by concatenated equalization FEC decoding receivers. We show that a BCJR equalizer concatenated with an LDPC decoder gets within about 2 db to the SNR bound. This gap can be further reduced by applying the turbo principle to less than 1 db. The receiver with the best performance is a

10 3576 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, 2009 Fig. 12. BER versus E =N for different FECs receivers. R = 10 Gb/s net bit rate. 1 = 60 ps R = 400 ps/nm. turbo equalizer with LDPC code, which operates with two samples per transmit symbol at its input. In summary, electronic receivers with advanced detection decoding methods show a large potential to improve optical transmission systems. APPENDIX SIGNAL AND NOISE PROPERTIES The noise in the samples exhibits some special properties that differ significantly from additive white Gaussian noise channels. In order to discuss this, is given as (23) where the signal in front of the photodiode is represented as with the signal components the filtered Gaussian noise. The ACFs of the noise with polarization are denoted as, respectively. These ACFs can be easily determined by use of the system parameters, given in Section II. As can be seen in (23), the received signal is a superposition of three components in each polarization. is the signal signal, is the noise noise, is the signal noise component of -polarization, similarly for -polarization. Because of, the noise in is strongly signal dependent. Due to the fact that the transmitted binary data sequence is contained in in (23), depends on the process is -stationary [26]. Further, because of the electrical filter, the correlation of the noisy samples depends on the data signal, i.e., on. The eye diagram of the receive signal is shown in Fig. 14. The bold lines indicate a noise-free signal. As mentioned, we consider two sampling phases for to get the samples :. Also, the threshold for hard-input decoding is shown., however, varies for different channel conditions like dispersion or noise. After sampling of at, results. In order to use these samples in probabilistic detection decoding methods, their statistic has to be known. In the following section, the pdfs of the samples are derived. The correlation of signal noise is determined in Section IV. A. Derivation of pdf cdf of the RX Samples To derive the pdf as well as the cumulative distribution function (cdf) of, which are both necessary to compute the IRs of Section III, a three-step method based on the Karhunen Loève series expansion (KLSE) a moment generation function (MGF) is applied to, with according to (2). The applied method is similar to that published in [15],

11 RANKL et al.: PERFORMANCE BOUNDS OF OPTICAL RECEIVERS WITH ELECTRONIC DETECTION AND DECODING 3577 Fig. 13. Required SNR versus dispersion of various detection decoding methods the respective system capacity bounds: (a) PMD (b) CD. The receivers operate at a BER of 10 at R = 10 Gb/s net bit rate (the same axis labeling as in Fig. 6). (26) with the sets of pairwise orthogonal coefficients (i.e., ) the sets of pairwise orthonormal base function [see (28a) (28b)]. In order to derive the noise series expansions (25), the Fredholm integral equations of second kind (27a) Fig. 14. Eye diagram of q (t) for R = 10 Gb/s, R = 0.937, 1 = 60 ps, R = 0 ps/nm. The noise-free signal is bold, the noisy signal for E =N = 20 db is in the background. however, we extended the method to two polarization dimensions, i.e., the the component of the received signal. From (5), we obtain have to be solved under the orthonormality conditions (27b) (28a) (28b) (24) The coefficients can then be determined by (29a) The three-step method can be subdivided into 1) the expansion of the noise signal parts in (24) into Karhunen Loève (KL) series, 2) a discretization of (24) by the use of the KL series, 3) the computation of the pdf cdf of by use of an MGF. Expansion of noise signal yields (25) (29b) The variances of the coefficients are, respectively. The sets of signal expansion coefficients can be determined by use of (30a) (30b)

12 3578 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 16, AUGUST 15, 2009 After applying (25), (26), (28) to (24), the discretized expression (31) is obtained. Since the variances tend to zero for large index values, the first coefficients can provide a reasonable approximation of (31) [15]. Thus, combining,, in a similar way for,,, it follows,,,. Using the property of mean values, the MGF of can be expressed as (32) Since the sets of coefficients are pairwise orthogonal the underlying noise processes of are independent Gaussian, the sets consist of independent therefore uncorrelated Gaussian RVs with variances, respectively. As a consequence, the pdf is a product of 1-D complex-valued Gaussian pdfs, namely,. The MGF its region of convergence (ROC) is then given by the pdf cdf can be determined by (33) (34) (35) In this context, is used as complex variable of the MGF should not be mixed with the state variable. REFERENCES [1] H. F. Haunstein, W. Sauer-Greff, A. Dittrich, K. Sticht, R. 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13 RANKL et al.: PERFORMANCE BOUNDS OF OPTICAL RECEIVERS WITH ELECTRONIC DETECTION AND DECODING 3579 [28] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw- Hill, [29] G. D. Forney Jr., Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference, IEEE Trans. Inf. Theory, vol. IT-18, no. 3, pp , May [30] O. E. Agazzi, M. R. Hueda, H. S. Carrer, D. Crivelli, Maximum-likelihood sequence estimation in dispersive optical channels, J. Lightw. Technol., vol. 23, no. 2, pp , Feb [31] Y. Kou, S. Lin, M. P. C. Fossorier, Low-density parity-check codes based on finite geometries: A rediscovery new results, IEEE Trans. Inf. Theory, vol. 47, no. 6, pp , Nov [32] S. Lin, J. Daniel, J. Costello, Error Control Coding, 2nd ed. New York: Prentice-Hall, [33] International Telecommunications Union (ITU), Telecommunications Stardization Sector of the ITU (ITU-T), ITU-T Recommendation G.975 Forward Error Correction for Submarine Systems, Geneva, Switzerl, [34] International Telecommunications Union (ITU), Telecommunications Stardization Sector of the ITU (ITU-T), ITU-T Recommendation G Forward Error Correction for High Bit-Rate DWDM Submarine Systems, Geneva, Switzerl, [35] Y. Miyata, W. Matsumoto, H. Yoshida, T. Mizuochi, Efficient FEC for optical communications using concatenated codes to combat error-floor, in Proc. Opt. Fiber Commun./Nat. Fiber Opt. Eng. Conf., 2008, OtuE4. [36] S. ten Brink, J. Speidel, R. H. Yan, Iterative demapping decoding for multilevel modulation, in Proc. Int. Conf. Global Commun., Sydney, Australia, Nov. 1998, pp Tobias Rankl received the Dipl.-Ing. degree in electrical engineering information technology from the University of Stuttgart, Stuttgart, Germany, in 2003, where he is currently working towards the Dr.-Ing. degree. He is a Teaching Research Staff Member at the Institute of Telecommunications, University of Stuttgart. In 2008, he visited the Keio University, Yokohama, Japan, as a Japan Society for the Promotion of Science (JSPS) Research Fellow. His research interests include information theory, channel coding, equalization, optical communications. He is was involved in research projects with the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), Agilent Technologies, the Agilent Technologies Foundation, Alcatel R&I (now Alcatel-Lucent Bell Laboratories), the Keio University, the JSPS. Christian Kurz was born in Aalen, Germany, in He received the Dipl.-Ing. degree in electrical engineering information technology from the University of Stuttgart, Stuttgart, Germany, in 2008, where he is currently working towards the Ph.D. degree at the Fraunhofer Institute for Biomedical Engineering. His research interests include cellular biosensors, electrical monitoring of living cells, biohybrid microdevices for biomedical biotechnological applications. Joachim Speidel (M 94) received the Dipl.-Ing. Dr.-Ing. degrees (both summa cum laude) from the University of Stuttgart, Stuttgart, Germany, in , respectively. From 1980 to 1992, he was with Philips Communications (now Lucent Technologies Bell Labs Innovations), where he worked in the broad field of digital communications. He has held various positions in research development as a Member of a Technical Staff, Laboratory Head, finally as Vice President. Since Autumn 1992, he has been a Full Professor at the University of Stuttgart Director of the Institute of Telecommunications. His research areas are digital communications in mobile, optical, electrical networks, with an emphasis on transmission, modulation, source, channel coding.