Coding and Error Correction in Optical Fiber Communications Systems

Chapter 3

Coding and Error Correction in Optical Fiber Communications Systems

Vincent W. S. Chan Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Massachusetts

3.1 Introduction During the development of lightwave communications with fiber since the mid-1970s, the major emphasis for research has been on the technology of lightwave devices and subsystems. Fiber is one of the most predictable and stable communications channels ever developed and characterized. In lightwave systems, the modulation and demodulation subsystems are typically designed and manufactured with very low bit error probabilities. In fact, the extremely competent device designers and fabricators may have spoiled the system engineers, often creating pristine devices and subsystems with outstanding properties and tolerances. Examples are low-chirp laser diodes to prevent pulse spreading due to dispersion in the fiber, laser diodes with ultralow mode partition noise for good pulse detection performance, and highly sensitive positive-intrinsic-negative field-effect transistor (PINFET) and avalanche photodiode (APD) receivers for good receiver sensitivities. Although the lightwave systems available to date have detection sensitivities at least 10 dB away from theoretical limits, there has never been a compelling reason, economic or otherwise, to improve receiver performance. For a few years in the 1980s, coherent detection was explored to gain a few decibels of receiver performance, especially for undersea cables. With the advent of erbium-doped fiber amplifiers (EDFAs), coherentsystem-like performance became possible with a preamplified direct detection receiver. Indeed, near-quantum-limit detection performance at 10 Gbps was demonstrated in 1996 [3.1]. Thus, it is not surprising that 42 OPTICAL FIBER TELECOMMUNICATIONS, VOLUME IliA

Copyright © 1997 by Lucent Technologies. All rights of reproduction in any form reserved. ISBN: 0-12-395170-4

3. Coding and Error Correction

43

lightwave communications systems have not yet exploited what other communications systems have for years - forward error-correction coding and decoding to improve channel reliability. As the field of telecommunications moves into the 21st century, lightwave systems will be more than point-to-point links with a highly controlled and predictable environment. For instance, the use of EDFAs will reduce the need for frequent regenerators, wavelength-division multiplexing (WDM) will increase single fiber capacities many fold, the use of optical routing and switching will introduce a new way of building optical networks, and the interconnection of many types of optical and nonoptical networks into a global network will result in a mix of networks of different designs and different generations interacting by means of an open network architecture. In these scenarios, applications of forward error-correction coding and decoding may substantively affect device fabrications, system designs, and their economics. Thus, it is worthy to consider how this technique can be profitably used in lightwave systems of the future. The technique of forward error-correction coding has its theoretical foundations on the celebrated papers by Claude E. Shannon in the late 1940s [3.2, 3.3] and has seen many years of highly sophisticated mathematical development in the field of Information Theory. In the following brief discussion of the Noisy Channel Coding Theorem, the spirit of the main enabling result of those works is captured. For a more rigorous treatment of the subject, the reader should refer to a text on Information and Coding Theory [3.4-3.6]. 3.1.1

NOISY CHANNEL CODING THEOREM

For a channel with known degradations (e.g., noise power and statistics) and restrictions on resources (e.g., signal power, bandwidth), a quantity called the capacity of the channel (usually expressed in bits per second) can be established such that (1) For information transmission rates less than the capacity, there exists a forward error-correcting code that can be used to transmit data over the channel with arbitrary accuracies. (2) For information transmission rates exceeding the capacity, no matter what the code may be, there will always be an unavoidable and nonzero error probability. This result may seem counterintuitive at first glance, but realizing how it is practically implemented can greatly help understanding. Figure 3.1

44

Vincent W. S. Chan Data in

-1

Encoder

~

Modulator

Data out

Fig. 3.1 Basic model of a digital communication system with forward error correction.

illustrates the essential building blocks of a coded digital communication system. The encoder introduces redundancy to the input data before transmissions. The decoder uses this redundancy to reconstitute the input sequence even in the presence of transmission errors, provided the frequency of errors is less than a threshold defined as the power of the code. The codes used in current systems are typically well designed to minimize the overhead (redundancy) required and usually have efficient decoding algorithms so that they can be implemented in modest hardware or software modules. For example, a code with a 25% increase in redundancy can be effective enough to substantively shift operating points of lightwave systems (e.g., from 10- 12 to 10- 6 channel bit error rates) and affect device specifications. This chapter provides some insight into the process of using forward error correction in noisy lightwave channels, and, in particular, indicates how lightwave system designs can be more efficient with the addition of coding. Some of these impacts can be significant. For example, WDM systems can pack more wavelengths into the EDFA band for more capacity, longer transmission distances can be attained with more cascaded EDFAs before signal regeneration, and optical devices such as routers, frequency translators, and amplifiers can function well with less stringent tolerance on cross talk and the amount of additive noise.

3.2 Common Modulation Formats for Fiber Systems Direct Detection and Coherent Systems Although there are many types of waveforms being used in current communication systems, the discussions in this chapter are restricted to the waveforms that are most commonly used in lightwave systems. For the purpose

3. Coding and Error Correction

45

of understanding the fundamental limits of detection of these signals, the signals can be divided into three classes: on-offsignaling, orthogonal signaling, and antipodal signaling. Most modulation formats used in lightwave systems are binary (i.e., each symbol will carry 1 bit of information). Higher order modulations (those that carry multiple bits per symbol) tend to be more power efficient but do so at the expense of more bandwidth required per bit of information transmitted. Because of the cost of high-speed optoelectronics, these schemes are seldom used in high-speed lightwave systems. On-off signaling is the simple intensity modulation of the optical field. The presence of a pusle indicates the symbol one and the absence zero. This type of signaling is used in most lightwave systems because of its bandwidth efficiency and ease of implementation, such as direct laser current modulation or electrooptic waveguide modulation by a Mach-Zehnder interferometer or with an electroabsorption modulator. Soliton systems almost exclusively use this format because of its simplicity and because it is easier to implement at very high speeds. Orthogonal signaling formats use modulation waveforms that are orthogonal, such as frequency shift keying (FSK), where different orthogonal frequencies denote different symbols, and pulse position modulation (PPM), where orthogonal time slots denote different symbols. Binary PPM is sometimes used in lightwave systems. These signaling formats are more commonly known as Manchester Coding (not to be confused with errorcorrection coding) in the lightwave community. Orthogonal signaling is common and well understood in classical radio frequency communications, but it is a misnomer with regard to lightwave communications. The reason is that although the classical waveforms of the optical fields corresponding to different symbols are orthogonal, at high receiver sensitivities where quantum effects are significant the quantum state representations of these signals are not orthogonal. In fact, unlike classical RF communications, in the absence of additive noise the detection error probability of orthogonal optical signals (in the classical sense) is not zero but a well-defined amount dictated by quantum mechanics. Antipodal signals are almost never used in lightwave systems, although one was used in 1996 to achieve high detection sensitivity [3.1]. Because these are the quantum optimum signal sets for binary signaling, they are included here for reference. The simplest form of antipodal signaling is phase shift keying (PSK), where different phase shifts of the optical carrier denote different symbols. In binary phase shift keying (BPSK), the signals one and zero are separated by 180 in optical phase of the carrier. 0

46

Vincent W. S. Chan

Two classes of receiver structures are used to detect optical signals: direct (incoherent) detection and coherent (heterodyne or homodyne) detection. Optically preamplified direct detection is equivalent to heterodyne detection in terms of its fundamental quantum model and detection performance. In direct detection, the received optical field is energy detected by means of a photodetector that usually provides gain. Examples are APDs or PMTs (photomultiplier tubes). Modulation schemes for direct detection systems are limited to intensity modulations such as on-off signaling and Manchester Coding. In coherent detection, an optical local oscillator field is added to the received optical field, and the sum is detected by a photodetector. The resulting signal is further processed at base band (homodyne detection) or at an intermediate frequency (heterodyne detection). Phase and/or frequency tracking of the signal field by the local oscillator laser is required. The mixing of the weak signal field and the strong local oscillator field at the front-end of a coherent receiver provides linear amplification and converts the optical signal into an electrical output with gain (usually tens of decibels), raising the signal level well above the noise level of subsequent electronics. This is why a detector with gain is not required. Coherent detection can be used on any of the three classes of modulations discussed. However, because antipodal signaling is a class of signals that rely on phase modulation, it can be detected only by coherent detection, not by direct detection.

3.3 Fundamental Detection Performances and Deviations of Currently Available Systems Often, sensitive lightwave receivers can be at or a few decibels from the quantum detection limit. Thus, understanding where the quantum limit lies is important to the system designer. For, if the performance of the designer's hardware is close, the design will be of diminishing return to try to improve system performance, whereas if the design is far from the quantum limit, there is a chance of significant payoffs if he or she spends more effort to improve the design. The output of a laser well above threshold can be represented by a special quantum state called the coherent state [3.7]. A coherent state I ex) is labeled by a complex number lX, where the modulus corresponds to the amplitude of the optical field, and the phase of lX corresponds to the phase

3. Coding and Error Correction

47

of the field. When direct detection is performed on such a state, the photon count is Poisson distributed with intensity (also known as the rate parameter) I a I 2, and when coherent detection is used, the output has the complex amplitude a, with added white Gaussian noise, which is actually a manifestation of quantum detection noise. Suppose that a signal with two possible states, a and fJ, representing the symbol zero and the symbol one, respectively, is received by an optical receiver. The quantum optimum receiver can be found by optimizing over the set of all possible receivers, consistent with the laws of quantum mechanics. This set of receivers is usually known as quantum observables. The mathematical structure of the quantum optimum receiver and its performance were found in 1970 by Helstrom, Liu, and Gordon [3.8]. The probability of error can be found exactly: Pr[e]

==

!(1 -

VI -

I (a I fJ)

2 1

(3.1)

),

where the inner product of the two coherent states I a) and I fJ) is given by (a I fJ). The magnitude of the inner product between the two coherent states is I

(a I fJ) 1 2 == exp (-

I

a - fJ I 2).

(3.2)

The average number of photons in the field represented by the state I a) is simply given by N, == (a I a). For small errors, the inner product is also small and the probability of error can be simplified to Pr[e] ~

i (a fJ) 1

1

1

2

==

iexp (- a - fJ I

1

2

)

==

ie-o,

(3.3)

where (J == I a - fJ I 2 is the exponential parameter that describes the behavior of the system at low error rates. It is trivial to note that for on-off signaling, fJ == 0, (J == a 2 == N, == 2Ns ' where N, is the average number of photons received per bit. It is also easy to see that under an energy constraint per bit, the signal set that minimizes the error probability is antipodal signaling, which maximizes the error exponent (J at 4Ns . Often the quantum optimum receiver is unrealizable with known techniques, or its implementation, even if it is known, is extremely complicated. Thus, simple receiver realizations are used as near-optimum compromises. For example, direct, or incoherent, detection is most commonly used in lightwave systems (note that the quantum model for preamplified direct detection is the same as that for heterodyne detection and should be viewed as equivalent to that of heterodyne detection rather than that of standard direct detection). The ideal form of direct detection is a photon-counting I

1

48

Vincent W. S. Chan

receiver (i.e., a receiver with enough electrical gain per photoelectron emitted by the detector surface such that individual photo events are detected and counted by subsequent electronics). APD and PIN-FET receivers, however, are generally at least 10 dB less sensitive. As stated before, modulations for such detection schemes are limited to intensity modulations. In lightwave systems with high-performance single-mode lasers, the detected photo events can be well characterized by Poisson statistics. Given that one chooses binary signaling and direct detection, on-off signaling is the optimum signal set. The optimum receiver is a counting receiver with a threshold. When the count exceeds the threshold, a one is considered sent, with a zero otherwise. The probability of detection error can be well approximated by the bound (3.4)

where N, is the average number of photons detected per symbol. It can be shown that the exponent (J == 2Ns gives the tightest exponential bound (sometimes known as the Chernoff Bound) for the error probability and for all practical purposes (at low error rates) can be used as an excellent approximation of the actual error probability. If Manchester Coding or binary PPM is used, the exponent (J == N s , which means that PPM is 3 dB less efficient than on-off signaling with ideal direct detection. When the local oscillator field intensity of a coherent receiver is high enough (which can be achieved in lightwave systems), the detector output is quantum noise limited. The output noise process can be modeled as an additive white Gaussian noise with power spectral density No/2 == ! for heterodyning and i for homodyning (normalized to the energy of a single photon and assuming unity quantum efficiency detectors). The communication performance of the detection of signals in additive white Gaussian noise is well known. Given that one chooses binary signaling, binary antipodal signaling (e.g., PSK) is the optimum signal set. The channel error probability for homodyne detection is given by (3.5)

where Q is the Gaussian error function and N, is again the average received number of photons per symbol. On-off signaling and binary PPM are 3 dB less efficient with the exponent (J == N; Heterodyne detection is again 3 dB less efficient than homodyne detection because the upper and lower

3. Coding and Error Correction

49

sidebands present twice the amount of noise to the receiver, compared with that of the base-band signal of the homo dyne receiver. The performance of the various common signal sets and detection schemes is summarized in Table 3.1. The most common signaling format and detection scheme currently used in lightwave communications is on-off signaling and direct detection. The direct detection photon-counting receiver achieves quantum optimum performance with a sensitivity of 10- 12 at 28 detected photons per bit. Current state-of-the-art commercial APD or PIN-FET receivers are ~0-15 dB away. The degradation of performance at modest data rates (100 Mb/s to 1 Gb/s) is mostly due to the noisy avalanche process of an APD or the front-end noise of electronics after detection. This type of noise increases as the bandwidth of the receiver increases. Thus, higher rate receivers are noisier, with greater degradations in performance. At very high rates (10-100 Gb/s), in addition to electronics noise, there are the effects of bandwidth limitations of electronics and timing system jitters giving rise to intersymbol interference and perhaps also cross talk. With a suitable low-noise optically pre amplified direct detection receiver such as an EDFA, several of the lost decibels can be recovered. As alluded to before, this receiver curiously has the same quantum mechanical model and detection limit as those of a heterodyne receiver. This limit has been approached within a factor of 2 [3.1] in an EDFA receiver at 10 Gb/s. In general, it would be difficult and/or costly to improve transmitter and receiver performance enough to approach quantum limited performance. With the advent of high-density and high-speed applications-specific Table 3.1

Receiver Performance Comparison: Probability of Detection Error, Pr[ E], for Binary Signalinga,b

Direct Detection

Signal Set

Heterodyne Detection

Homodyne Detection

Quantum Optimum

On -off signaling

2N s

N s/2

Ns

2N s

Orthogonal signaling (PPM, FSK)C

Ns

N s/2

Ns

2N s

Antipodal signaling (PSK)d

Not applicable

Ns

2N s

4N s

Exponent 8 of tightest exponential bound, Pr[e] :::; e-o. N, == average number of detected photons per bit. e PPM, pulse position modulation; FSK, frequency shift keying. d PSK, phase shift keying. a

b

50

Vincent W. S. Chan

integrated circuits (ASICs), the use of error-correcting codes is an attractive alternative to the development of highly accurate and pristine devices. The next section presents some of these opportunities.

3.4 Potential Role of Forward Error-Correcting Codes in Fiber Systems and Its Beneficial Ripple Effects on System and Hardware Designs Forward error correction can be implemented simply in a lightwave system by encoding the information symbols into code words by means of an encoder (Fig. 3.1). At the receiver, two major types of decoding, hard and soft decisions decoding, can be used to recover the information bits. With hard decisions decoding, the receiver first makes tentative decisions on the channel symbols and then passes these decisions to the decoder, where errors are corrected. With soft decisions decoding, the receiver, in principle, would pass on to the decoder the analog signal at the output of the demodulator. The decoder would then make use of the information in the analog signal (rather than the hard decisions) to re-create the transmitted information bits. Soft decision decoding thus is always better performing than hard decisions decoding, usually by a couple of decibels. With hard decision decoding, most lightwave channels can be modeled as binary symmetrical channels (BSCs) (Fig. 3.2). A BSC will make an error with the same probability p for inputs zero and one. The parameter p can be measured experimentally or derived using a model of the receiver via a similar process that leads to the expressions in Table 3.1. The capacity of the BSC is well known [3.4-3.6]: Chard ==

1 + P log2P + (1 - p) log, (1 - p). Output alphabet

Input alphabet

1- P o ...",.-------------;:::::"'" o

p p 1- P

Fig. 3.2 Binary symmetrical channel.

(3.6)

3. Coding and Error Correction

51

The capacity Chard is for each use of the channel (per channel bit transmitted for the BSC). The error probabilities given in Table 3.1 can be used to find Chard for the various modulation and detection schemes. Often it is difficult for implementable systems to work near capacity. A second quantity, R com p, the computation cutoff rate of the channel, is used as a convenient measure of the performance of the overall communication channel. R comp is usually less than the capacity of a channel and represents a soft upper limit of information rates for which moderate-size decoders are readily implementable. The R comp for hard decisions decoding is Rcomp,hard

== 1 - log- [1 + 2yp(1 - p)].

(3.7)

is a realistically achievable performance to expect of a coded system with present-day electroncis technology. Later in this chapter, examples of practical coders and decoders are given. To achieve the ultimate capacity of most communication systems, it can be shown that soft decisions decoding must be used. Soft decisions decoding is currently done for only modest-rate communication systems (e.g., 10 Mb/s) and is unlikely to be used soon in typically high-rate lightwave systems. For an appreciation of the potential gains, the capacity Cso ft and the computation cutoff rate Rcomp,soft for binary PPM signaling (Manchester Coding) are given next: Rcomp,hard

Rcomp,soft

== 1 - 10g2 (1 + e- N s ) .

(3.8)

Figure 3.3 depicts a plot of the capacity and the cutoff rate in bits per use of the binary PPM channel for hard and soft decisions decoding. In most interesting regions of operations, Rcomp,hard is within 3-6 dB of C soft. The added complexity of a soft decisions decoder makes it difficult to implement soft decisions decoding at high rates (> 100 Mb/s). Thus, only hard decisions decoding is used in the examples given subsequently. The ultimate performance limit of lightwave systems lies in nonbinary systems, where the signaling alphabet can be much larger than 2. The capacities and cutoff rates of direct and coherent detection systems are included in Table 3.2 for reference. Derivations can be found in Ref. 3.8, or from Eqs. (3.6) to (3.8). Note that the capacity for the direct detection channel with no additive noise and only quantum detection noise, given in Table 3.2, is infinite. This may sound counterintuitive at first, but this performance occurs in an unrealistic scenario, when the PPM signaling

52

Vincent W. S. Chan

0.8 Q5

c c

~

o

0.6-

a

<. R


en

:::::>

Q5

0.4

comp, hard

a. ~

en

0.2

Rcomp, soft

2 3 4 5 Average Number of Signal Photons in Each Channel Bits

Fig. 3.3 C sof t, Chard, Rcomp,sojt, and Rcomp,hard versus the average number of photons per channel bit for quantum limited performance.

symbol size and the energy in the pulse both approach infinity. As is evident, current lightwave systems are very far away (>20 dB) from these limits. Even for binary systems, current lightwave systems are about 10 dB away from the theoretical limits. To recover a few decibels of performance will require better optical devices and electronics, which can be expensive. Another way of recovering a few decibels (e.g., 5) is the use of forward error correction. Not only can error-correcting codes provide a few decibels of power efficiency, but they can also shift the operating point of a link from virtually error free for the uncoded channel to frequent errors for a coded channel. For example, a code with a modest coding gain of 3 dB Table 3.2

Receiver Performance Comparison: Computation Cutoff Rate R o and Capacity, C, of Coded Systems"

Detection Scheme

Direct Detection

Computation cutoff rate R o

1 nat/photon

1 nat/photon

Capacity, C

00

2 nat/photon

a

1 nat = log-e bits.

Homodyne Detection

3. Coding and Error Correction

53

(i.e., it can transmit at the performance of the uncoded channel with a factor of 2 better in power efficiency) can operate at a raw link bit error rate of 10- 6 but yields a delivered information bit error rate after decoding of 10- 12 . The next section introduces some practical codes and a little more insight into the technique.

3.5 Some Practical Codes The simplest code to understand is the repeat code - repeating the information bits several times and letting the receiver correct for errors by using majority logic (i.e., when there are disagreements on the binary bit sent among replicas, the symbol with more replicas is selected). This primitive error-correction code, although easy to implement, is inefficient, requiring a significant increase of bandwidth, and seldom provides any coding gain. A more efficient code called the parity check code is illustrated in Fig. 3.4. Its operation is easy to understand also. The data sequence is first broken into blocks of bits. Each block of information bits is arranged in a twodimensional array with one parity check (exclusive-or, XOR) for each row and one for each column. The parity bit in the lower right-hand corner is a check on the parity bits. If a single error occurs during transmission, it can be detected and corrected by cross-checking the corresponding row and column parity bits. Generally, multiple errors cannot be corrected unless they occur at certain places. Although this code is effective for the occasional isolated error, it is especially weak against short bursts of errors. Because such bursts of errors can occur in lightwave systems (such as those caused by power dropouts due to laser partition noise), the use of this code

a Information bits

b

100 1 0 0 1 1 1 1 001 0 1

Horizontal checks

100 1 0 1 001 0 1

o 1 @] 1

1 0 1 1 1

1 0 1 1 1

1 1 1

011 1 1

o1

Vertical checks

Horizontal checks

Vertical checks

Fig. 3.4 Horizontal and vertical parity check code. (a) Coded message. (b) A single channel error (boxed bit) can be located and corrected.

54

Vincent W. S. Chan

is undesirable. However, the basic principle of the two-dimensional parity check code can be readily translated into more general parity check codes. A more powerful set of codes operating on the same principle as the parity code is linear block codes. For binary input data, a linear code is constructed from a set of modulo-2linear combinations of the data symbols. Again, the binary information sequence is segmented into message blocks of length k, denoted by u. There are 2k possible distinct messages. The encoder uses either a codebook or an encoding algorithm to map each input message u into another binary block v of length n > k (see Fig. 3.5). Thus, there are n - k binary bits added as redundancy. the set of 2k ntuple code words is called a block code (usually labeled as an [n, k] block code to indicate the amount of redundancy). For a block code to be useful, the code length n should be large. The decoder looks at each block of length n and finds in the codebook the closest code word to the received block. The inverse mapping to an information block of length k recovers the information sequence. The ratio kin is sometimes known as the code rate of the code and is an indication of the degree of redundancy in the code. Usually, the smaller the code rate, the more error-correction capability of the code, albeit at the expense of more channel bits transmitted per input bit. Unless a block code has a special mathematical structure, it is difficult to encode and decode a long code. Almost all block codes currently used are linear block codes (just as parity check codes). Some commonly used block codes are the Reed-Solomon Code and the BCH (Bose-Chaudhuri-

Information sequence x ... 0001000010

Transmitted coded sequence Y n-tuple v ... 10010010011100000001 ...

Decoded message sequence ~ ... 00010000101011101101 ...

Fig. 3.5 Block coding and decoding with an (n, k) code.

3. Coding and Error Correction

55

Hocquenghem) Code. Coding gains of 3-5 dB can be attained. Of significant note is that the best linear block codes can perform about as well as the best block codes (without the linearity constraint) with the same lengths and code rates. The encoding and decoding processes of these codes would require extensive mathematical development before they could be presented here. Thus, instead, the reader should refer to Refs. 3.4 through 3.6. The most common experiment on the use of error-correction code in lightwave systems to date is the application to the extension of long-haul repeaterless transmission with EDFAs. When multiple EDFAs are cascaded to extend repeaterless long-haul transmission, noise accumulation will ultimately limit the number of EDFAs that can be used in series before data regeneration. Reed-Solomon Codes have been used for such experiments [3.9-3.12]. When no forward error correction is used, the noise accumulated from EDFAs imposes a floor on the bit error rate performance. When error correction is used, the raw link can operate at much higher error rates, which renders the error rate floor inconsequential. As we see later in this chapter, not only can more EDFAs be used in cascade, but EDFA spacings can also be increased. This fact has significant implications on the design and costs of long-haul, especially transoceanic, systems. Because the decoding algorithm of Reed-Solomon Codes requires significant processing, these experiments have been realized with stream rates of less than 1 Gb/s and multiplexed up to higher rates through interleaving. Recent advances in GaAs and InP high-speed electronics promise decoders at base rates of a few to tens of gigabits per second in the future. Convolutional Codes [3.4-3.6] are another set of commonly used errorcorrection codes. They differ from the codes discussed before in that they are not block codes. The continuous convolutional structure of the encoder has a property that facilitates hard and soft decision decoding and improves coding performance; thus it is the most popular coding scheme currently in use. Figure 3.6 shows a binary convolutional encoder with a K-stage register. The information sequence enters the encoder 1 bit at a time, and every bit in the register shifts right one position, with the last one on the extreme right discarded. The taps feeding into the modulo-2 (XOR) adders are selected for good coding performance. There can be v of such adders, so v channel bits are generated for each information bit, which makes the code rate l/v. There are a number of ways to decode convolutional codes with various levels of decoding complexities. Threshold and sequential decoding schemes are easy to implement, providing modest but not best coding gains

56

Vincent W. S. Chan Stage 1

Stage 2

Stage K x-register

~ Commutator

("'0

y

•

Fig. 3.6 Binary convolutional encoder with a K-stage shift register. XOR,

exclusive-or.

(---3 dB). They are ideal for very high-speed lightwave systems [3.13]. A much more powerful scheme is called Viterbi decoding, especially the soft decision version. Viterbi decoding is the optimum decoding scheme, and coder-decoder chips are available commercially at speeds up to tens of Mb/s. All the coding schemes discussed thus far use hard decisions (i.e., tentative decisions on the channel bits are made before further processing by the decoder). Viterbi decoding has a natural way to accept analog (usually quantized to 3 bits for ease of operation) outputs from the matched filters and processes these signals to recover the information sequence. Additional coding gains of up to 2 dB can usually be realized. A common convolutional code using Viterbi decoding is a rate !, constraint length 7 code. Unfortunately, this decoding scheme is complicated, requiring many computations, and will not be useful for lightwave systems until ultrafast electronics technology improves drastically. A recent example of a high-rate code that has been implemented in a lightwave system is a convolutional code at 2 Gb/s [3.13, 3.14]. It is a rate ! code with a constraint length (K) of 332. The rate! code requires only 25% bandwidth expansion (i.e., requires 25% more transmitted bits). This is an important consideration for high-rate lightwave systems, not because fiber bandwidth is precious, but because high-speed electronics are expensive and generally have poorer performance. Therefore, requiring a lot of bandwidth expansion is the wrong direction to take for high-rate systems. The encoder and decoder are implemented as a single, custom, silicon bipolar integrated circuit gate array. Threshold decoding was chosen as the decoding algorithm because of its simple decoder design, which permits high-speed operations. Figure 3.7 is a graph of the decoded bit error rate

3. Coding and Error Correction

57

1 0-2 Q)

~ 1 O~

e

w 10"

"0 Q)

"0

8 Q)

o

10"

1 0-10~-""-~.&....I..-I~L-~--L-~..........u.----L~....L-L...J..U..LJ 0.0001 0.001 0.01 0.1 Channel Error Rate

Fig. 3.7 Decoded bit error rate versus channel error rate for on-off keying (OOK) and frequency shift keying (FSK) experiments. WDM, wavelength-division multiplexed.

plotted against the channel error rate. Note that one phenomenon is common to all coded systems: the presence of a threshold raw channel bit error rate beyond which the code starts to take effect. In this particular example, the transition point is at a 4 X 10- 2 bit error rate, below which there is a coding gain of approximately 3 dB (i.e., the exponent of the decoded bit error rate is approximately twice as large as the exponent of the raw channel). The raw channel uses NRZ (non-return-to-zero) signaling, achieving a system sensitivity of 87 received photons per bit at 10- 12 • The coded system achieves the same performance at 37 received photons per information bit, realizing a gain of 3.7 dB (see Fig. 3.8). -2.........- - - - - - - - - - - - - ,

CD-3

ro

a:

e -4

~

o --.-

Ideal PA-OOK Uncoded Coded Model

-5

@. -6

(!J

o -7

....J

-8 -9 -1 O...._ .......................,.-_....................~ 100 1000 10 Received Photons/Info Bit

Fig. 3.8 Measured bit error rate performance of the codec at a 1.0-Gb/s data rate on a single OOK channel. Coded sensitivity is 37 photons per bit. PA-OOK, pre amplified on-off keying.

58

3.6

Vincent W. S. Chan

Applications of Coding to Future Lightwave Systems Such as WDM Systems and Optical Networks

Gaining a few decibels of receiver sensitivity is not very important for present-day lightwave systems, especially at 1.5-JLm wavelengths, where an EDFA preamplifier can readily improve receiver sensitivity. However, the application of forward error correction does have some significant implications in the design of future lightwave systems. Some of these implications are discussed next.

3.6.1

DECREASED SENSITIVITY TO DEVICE AND SUBSYSTEM FLA WS

A coded system can and will operate the raw lightwave channel at much higher error rates. Minute device and subsystem flaws do not usually become apparent until the error rate is extremely low. Hence, a coded system barely sees the difference between pristine and not-so-perfect devices. This system feature allows the use of less expensive and more mass-producible optoelectronics, which has significant implications on system costs. In addition, a system's lifetime can be extended as devices slowly degrade. Thus, a lightwave system using a laser with appreciable mode partition noise may never be able to achieve 10- 12 raw channel error performance but will with the use of forward error correction. A detailed theoretical analysis and simulation of coded lightwave systems using a direct detection APD receiver can be found in Ref. 3.15. Performance gains up to 9 dB are predicted. Similarly, when significant laser phase noise is present, a coherent system will exhibit an error rate floor but can have perfectly acceptable performance when an error-correcting code is used [3.16]. To first order, in the limit of high signalto-detection-noise ratio, the laser phase noise behaves like additive white Gaussian noise in a phase- or frequency-modulated system (see Ref. 3.16). The exponent (J, of the bit error rate floor e- 8 due to laser phase noise, increases linearly with the laser linewidth. Thus, the specification on laser phase noise in a coherent system can be significantly relaxed, by approximately the coding gain of the code used (e.g., if the coding gain is 3 dB, (J and hence the laser linewidth can increase by a factor of 2). Finally, when installed fiber plants exhibit reflections from fiber discontinuities, which occur more frequently than expected because of variations in the quality of work in the field, the raw channel may see error rate floors that a coded system may ride through and provide acceptable performance.

3. Coding and Error Correction

3.6.2

59

INCREASED LINK DISTANCES FOR OPTICALLY AMPLIFIED FIBER SYSTEMS

As alluded to before, forward error correction can be used to extend the link distances of optically amplified fiber systems. To quantify the potential gain, assume that the gain of each of the in-line optical amplifiers exactly equals the loss over the fiber between amplifiers [3.17]. Thus, it is easy to see that the amplifier noise accumulated over N amplifiers is N.Nsp , where N sp is the spontaneous emission noise added by each amplifier. A good estimate of the maximum number of amplifiers, N max, that can be used in series before regeneration can be derived from the condition that the accumulated amplifier noise, N.Nsp , cannot exceed the receiver detection noise, N det • Obviously, under such conditions, when the receiver sensitivity is increased with the use of forward error correction, the maximum number of amplifiers increases proportionally. Because coding gains of 3-6 dB are achievable at lightwave system rates, the number of in-line amplifiers and thus the link distance before regeneration can be increased by a factor of 2-4. Also, one can see that the input optical power, Pin, into the fiber should be as large as possible without paying a severe penalty resulting from fiber nonlinearities [3.18]. In the next subsection, we see that the use of forward error correction can also alleviate degradations due to fiber nonlinearities. Hence, further gains in the ability to cascade links and in link distance can be realized. Because the explicit gains are complicated functions of the operating points of the system and fiber nonlinearity parameters, the exercise is left to the reader. 3.6.3 INCREASED PACKING DENSITY OF WA VELENGTHS IN WDM ALL-OPTICAL NETWORKS WDM all-optical networks are becoming a reality as a result of rapid research and development worldwide. The use of optical amplifiers and wavelength multiplexing promises higher throughputs and more economical networks. The erbium amplifier bandwidth is of the order of 20 nm. There is much incentive to pack as many wavelengths into a fiber as possible. In addition to a larger capacity, an increased number of wavelengths provides increased connectivity for an all-optical network. When wavelengths are packed close together, there will be degradations due to spectrum spillover from one signal to an adjacent channel. This problem is particularly limiting when passive or active multiplexers-demultiplexers, routers, and switches

60

Vincent W. S. Chan

soo . - - - - - - - - - - - - -....t--..

0'>

I

o

o

Ii a::

Theory Measured: o Uncoded • Coded

400

w

co

@ 300 +-'

ffi

6 C

2.5X Improvement

200

~

c 100

o

(5 ..c

c...

0

o

5

10

15

20

Channel Spacing/Channel Rate

Fig. 3.9 Receiver sensitivity versus channel spacing. Solid lines are calculated from a model and indicate an expected improvement in channel spacing of approximately 2.5X. The measured improvement was 2X. BER, bit error rate.

are used. If, in addition, dispersion-shifted fiber is used for long-haul transmissions, fiber nonlinearities will also result in severe dgradations. The dominant nonlinear fiber effect is four photon mixing [3.18], the magnitude of which increases as the wavelengths move closer. Thus, both these effects limit wavelength packing density. The use of forward error correction shifts the operating point of the raw channel and allows more cross talk and

nonlinear products. To first order, because the bit error rate curve is exponential in the signal-to-noise ratio, the gain in terms of allowable increase of these interference effects equals the coding gain. In Ref. 3.19, an increase of channel packing density of a factor of 2.5 has been demonstrated, using the convolutional code described in Section 3.5, which has a coding gain of 3.7 dB (see Fig. 3.9).

3.6.4 RANDOM ACCESS OF A SHARED FIBER SYSTEM VIA CODE-DIVISION MULTIPLEXING When a transport medium is used as a shared broadcast medium, such as a star network, the benefit is that every user can hear the same information. The drawback is that every user signal acts as interference to other users' signals. There are several standard techniques employed to work around the interference problem. Time-division and frequency-division techniques are commonly used. Code-division multiplexing, first used in defense communications and more recently in cellular communications, is a potential candidate for lightwave networks. In this scheme [3.20], each user encodes

3. Coding and Error Correction

61

his or her messages using a unique signature code and broadcasts the resulting signal into the medium. The receiver uses a decoder to sort out the intended user signal, treating all other user signals as noise. This random access scheme is particularly attractive when time synchronization is difficult, as in the case of a sizable all-optical network. Generally, there is a significant bandwidth expansion of the message rate (as much as the number of users sharing the medium), to accommodate many users in the network; thus, this method is less attractive for high-rate lightwave systems except when it is being used in the low-rate signaling channel for network management and diagnostics. The ability to work without time synchronization is an attractive feature for network management because of its ease of operation, particularly during network cold starts.

3.7

Summary

Although forward error correction has been used for decades in many communication systems, it did not proliferate into lightwave systems because of the advanced state of development of fiber and optoelectronics and the lack of high-rate coder-decoder technology. However, high-rate electronics technology has now finally caught up with the speed of lightwave systems, and single-chip coders-decoders are available. This development, coupled with novel new architectures, such as long-haul repeaterless transmissions with the use ofEDFAs and WDM all-optical networks, will create new opportunities for the insertion of this powerful technique to enhance even more the performance of future lightwave systems and reduce costs significantly.

References [3.1]

[3.2]

[3.3] [3.4]

Livas, J. 1996. High sensitivity optically pre amplified 10 Gb/s receivers. In OFC'96, San Jose, CA, February. Postdeadline paper 4. Washington, DC: Optical Society of America. Shannon, C. E. 1948. A mathematical theory of communications. Bell Syst. Tech. J. 27:379-423, 623-656. Reprinted in 1949 in book form, with a postscript by W. Weaver, by the University of Illinois Press, Urbana. Shannon, C. E. 1948. Communications in the presence of noise. Proc. IRE 37:10-21. Gallager, R. G. 1968. Information theory and reliable communications. New York: Wiley.

62 [3.5] [3.6] [3.7] [3.8] [3.9]

[3.10]

[3.11]

[3.12]

[3.13]

[3.14] [3.15]

[3.16]

[3.17] [3.18]

[3.19]

[3.20]

Vincent W. S. Chan Lin, S., and D. Costello. 1983. Error control coding: Fundamentals and applications. Englewood Cliffs, NJ: Prentice-Hall. Viterbi, A., and J. Omura. 1979. Principles of digital communication and coding. New York: McGraw-Hill. Glauber, R. J. 1963. Coherent and incoherent states of the radiation field. Phys. Rev. 131:2766-2788. Helstrom, C. W., J. Liu, and J. Gordon. 1970. Quantum mechanical communication theory. Proc. IEEE 58:1578-1598. Yamamoto, S., H. Takahira, and M. Tanaka. 1994. 5Gbit/s optical transmission terminal equipment using forward error correcting code and optical amplifier. Electron. Lett. 30(3)254-255. Pamart, J. L., E. Lefranc, S. Morin, G. Balland, Y. C. Chen, T. M. Kissell, and J. W. Miller. 1994. Forward error correction in a 5 Gbit/s 6400 km EDFA based system. Electron. Lett. 30(4). Pamart, J. L., E. Lefranc, S. Borin, G. Ballard, Y. C. Chen, T. M. Kissell, and J. W. Miller. 1994. Forward error correction in a 5 Gbit/s 6400 km EDFA based system. In OFC'94, San Jose, CA, February. Postdeadline paper PD18-1. Washington, DC: Optical Society of America. Yamamoto, S., H. Takahira, E. Shibano, M. Tanaka, and Y. C. Chen. 1994. BER performance improvement by forward error correcting code in 5 Gbit/s 9000 km EDFA transmission system. Electron. Lett. 30(8). Castagnozzi, D. M., J. C. Livas, E. A. Bucher, L. L. Jeromin, and J. W. Miller. 1994. Performance of a 1 Gbit/s optically pre amplified communication system with error correcting coding. Electron. Lett. 30(1):65-66. Castagnozzi, D. M., J. C. Livas, E. A. Bucher, L. L. Jeromin, and J. W. Miller. 1994. High data rate error correcting coding. SPIE 2123. Jeromin, L. L., and V. W. S. Chan. 1983. Performance of a coded optical communication system using an APD direct detection receiver. In Proceedings ofthe IEEE International Conference on Communications, June, Boston, Massachusetts. New York: IEEE. Jeromin, L. L., and V. W. S. Chan. 1986. M-ary FSK performance for coherent optical communication systems using semiconductor lasers. IEEE Trans. Commun. COM-34(4). Olsson, N. A. 1989. Lightwave systems with optical amplifiers. J. Lightwave Tech. 7(7). Tkach, R. W., A. R. Chraplyvy, F. Forghieri, A. H. Gnauk, and R. M. Derosier. 1995. Four-photon mixing and high-speed WDM systems. J. Lightwave Tech. 13(5). Livas, J. C., D. M. Castagnossi, L. L. Jeromin, E. A. Swanson, and E. A. Bucher. 1994. Forward error correction in a 1 Gbit/s/channel wavelengthdivision multiplexed system. In IEEE/LEOS summer topical meeting on optical networks and their enabling technologies, Lake Tahoe, Nevada, 11 July. Chan, V. W. S. 1979. Multiple-user random access optical communication system. In ICC'79.