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1
Spectral Efficiency of Coded Phase-Shift Keying for Fiber-Optic Communication Gerhard Kramer, Alexei Ashikhmin, Adriaan J. van Wijngaarden, and Xing Wei Abstract— Several optical modulation and detection schemes are compared by computing their spectral efficiencies over additive white Gaussian noise channels. The bandwidth savings of differential quadrature phase shift keying (D-QPSK) over both direct-detection on-off keying (DD-OOK) and differential binary phase-shift keying (D-BPSK) suggest that D-QPSK can improve the reach and efficiency of wavelength-division multiplexing (WDM) systems. To test the theory, Reed-Solomon and low-density paritycheck forward error correction (FEC) codes are designed and evaluated. The codes generally behave as expected, except that for D-QPSK the gains are hampered by the differential detector. It is further shown that neither multiple-symbol differential detection nor decision-feedback detection is attractive when using strong codes. Keywords— error correcting coding, optical signal detection, optical transmitters, phase shift keying
TABLE I C OMPARISON OF MODULATIONS FOR 40 G BIT / S TRANSMISSION
Modulation DD-OOK D-BPSK
D-QPSK
Code Rate 15/16 15/16 15/32 1/2 2/3 3/4 7/8 15/16
Symbol Rate [GHz] 42.7 42.7 42.7 40.0 30.0 26.7 22.9 21.3
I. I NTRODUCTION Differential binary phase shift keying (D-BPSK) has recently attracted renewed attention for use in ultra-long haul, optical wavelength-division multiplexed (WDM) systems [1]. The main reason is that D-BPSK gives a substantial ( dB) detection energy gain over direct-detection on-off keying (DD-OOK) when using a delay interferometer and a balanced receiver for differential detection. Furthermore, D-BPSK tends to suppress fiber non-linear penalties such as those due to cross-phase modulation better than OOK [1]–[4]. Although D-BPSK is an old technique (see, e.g., [5, p. 527]), the importance of its combined linear and non-linear gains has been recognized only recently. This paper investigates the performance of several coded modulation and detection schemes that are suited for high speed, fiber optic communication. For instance, Table I compares the spectral efficiencies of DD-OOK, D-BPSK, and differential quadrature PSK (D-QPSK) on an additive white Gaussian noise (AWGN) channel. All schemes have a data rate of 40 Gbit/s. Also, all schemes use non-coherent receivers having only one decision threshold per detector arm, also known as hard detection. Consider the first, second, and seventh rows of numbers in Table I. The DD-OOK and D-BPSK schemes use a 6.7% overhead code at a symbol rate of 42.7 GHz. The D-QPSK scheme uses a 14.3% overhead code at a symbol rate of 22.9 GHz. The “energy gain” refers to how much energy is saved over either DD-OOK or D-BPSK when capacity-achieving codes are used, and a 3dB gain means that half the energy is required. We see that D-QPSK requires dB less energy than DD-OOK, and a mere dB more energy than D-BPSK, but with 46% less bandwidth. This suggests that D-QPSK can improve the efficiency of WDM systems. There are many practical issues to resolve before one can realize the gains of Table I. For example, the gains are for extremely long capacity-achieving forward error correction (FEC)
The authors are with Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974 USA (e-mail:
[email protected],
[email protected],
[email protected],
[email protected]).
Energy Gain [dB] vs.
DD-OOK
D-BPSK
2.46 4.43 4.37 3.83 3.40 2.41 1.52
1.97 1.91 1.37 0.94 0.05 0.94
codes. We address this issue in Section V and show that the gains remain intact for practical codes. The main exception is that the D-QPSK differential detector causes additional losses if the code is not close to optimal. A second important issue is that our analysis does not include non-linear fiber effects. Thus, our conclusions will apply primarily to transmission in a linear regime, e.g., low power or short distances. We concentrate on PSK to keep the uniform envelope advantages of D-BPSK. This paper is organized as follows. In Section II we derive and compute the spectral efficiencies of several modulation and detection schemes. This section organizes known results and should be considered a review of communications theory for optical high-speed transmission where one uses noncoherent and hard detection (see, e.g., [6, pp. 282–284, 380– 387]). This theoretical work was done in conjunction with the experiments of [1]. Section III discusses the behavior of spectral efficiency at low signal-to-noise ratios. Section IV presents results on multiple-symbol differential detectors and decisionfeedback detectors, and suggests that these are not so useful when combined with strong codes. Section V considers the performance of two code families used in practice: Reed-Solomon (RS) codes and low-density parity-check (LDPC) codes. The latter are decoded with a suboptimal but high-speed and lowpower decoder. Section VI summarizes our results. II. M ODULATION /D ETECTION S PECTRAL E FFICIENCY The spectral efficiency of a modulation set is the number of bits per second per hertz that the set can support. A first order approach is to compute spectral efficiency without FEC coding [7]. However, in practice codes are a necessity so that one should take them into account. To define spectral efficiency, we follow an approach similar to [8, Sec. VIII] and [5, Sec. 7.2] and make the following idealizations. i) The channel passes frequencies
(measured in hertz) in the
2
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range
A. Computing Spectral Efficiency
where the center frequency is much larger than . The equivalent baseband representations of the channel input and output signals are therefore complex and band-limited to . We say that the channel bandwidth is . ii) The channel is used for a period of seconds. We describe the channel input and output signals by complex samples spaced seconds apart, i.e., we can reconstruct the baseband input signal as
Suppose we have chosen a modulation set and a detector, and we use the channel bcd times. For large the system capacity is
"!$# & % '% () % ' % (
e
-/.
021
*,+
We will consider primarily PSK for which 0 for all ( . iii) The channel is modeled as adding complex Gaussian noise of energy 3 to each output input sample, i.e., the ( th channel
6 where 6 768 :9;6=< and 6=8 sample is 4 5 and 6 < are independent Gaussian random variables each having variance 3 . Usually 3 increases proportionally with , and 3 where 3 is the noise power per hertz. we write 35 We remark that the above setup considers transmission using only one polarization of light. A model that includes both polarizations can be found, e.g., in [9]. We define the modulation rate in bits as >@?ACBEDGF 0 &H where H is the number of values that can take on. For instance, 3-phase PSK has > ? IBEDGF 0 &J bits and QPSK has >K?L bits. We assume that FEC is done with a binary code of rate >KM so the overall rate is >5 N>@MO>K? bits. The energy *QP * + > and we define the inforsent per information bit is *KPO mation bit signal-to-noise ratio as P 3 . We write the bit-error probability out of the decoder as R . It is now natural to define the spectral efficiency of the modulation as
P S * P T 3 GUOR VW>YMX >Q?
bits/s/Hz
e G
(2)
xfy kml$g nai o
qO4T bits/s/Hz p
(3)
where and 4 are the input and output symbols for one use of the channel. We will often drop the maximization and simply assume that Rhrz is a uniform distribution. * + 3 , i.e., we The mutual information in (3) is a function of have
e
*,+{
m
3|
* P
}&>
3Ya"
(4)
where* m is some non-decreasing function, and where wee have *QP + ~> 3 3 . Note further that > used and , so e G e that the best is the largest X satisfying e X
e
}
*QP
G X
3
(5) e
* P
This equation gives the ultimate P 3Y in terms of X e * the ultimate X in terms of 3 . We thus have Y * P S Q 3
e
`
X
bits/s/Hz
, or (6)
* P
Some properties of S 3Ta are developed in Section III. * P We illustrate how to compute S 3Ta by considering the case where arbitrary complex are permitted. Since >? is in principle infinite, one now maximizes the overall rate > without separating it into > ? and > M . Shannon showed that for this case the capacity is [11, Sec. 25] e
(1)
where > MX is the maximum code rate for which one can achieve P a bit-error probability of when the informationP bit signalR * P to-noise ratio is 3Y . For optical transmission R can be required to be [Z]\^ or less. We will in fact consider only the case P`_ R because the spectral efficiency hardly changes from that P when R P [Z]\^ . We denote the resulting spectral efficiency * by S 3Ta . Note that (1) defines spectral efficiency without taking into account the spectral guard bands that are necessary in practice. The reason for doing this is that the amount of guard band needed, in proportion to the bandwidth, will depend primarily on the pulse shape, and less on the number of points in the signal set. We will not consider pulse shaping here.
q"4 bits/s
/p
where is the length–b random vector of channel inputs, 4 is the random vector of channel outputs, Rhr is of , and q"4 s the probability distribution p Rhr U"t B$D F 0Gu Rhr Ut Rhr Rht wv is the mutual information between and 4 [10, Sec. 2.3]. In many cases of interest the channel is memoryless and the best Rhr is the one that chooses every symbol of r independently. For such cases, we can rewrite (2) as
where is a complex number whose squared amplitude and phase are the respective energy and phase of the ( th input sample. The average energy per sample is *,+
/fh kml$gjn i o
*Q+
We set Sec. VIII]
A>
* P
BEDGF 0
, 3
* +
S
3
bits/s
(7)
3Y , and use (7) to obtain [8,
*,P 3T
(8)
We plot the S which satisfy (8) with equality as _ the curve labeled “Gauss, soft” in Fig. 1. Observe that as S we have _ *,P G # 3 B J which is dB. This ultimate enP R increases but it remains at ergy limit will decrease slowly as P essentially dB for R Zz^ . The other curves in Fig. 1 are those for various modulation and detection schemes. We review how to compute these curves
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3
2
3
Gauss soft
1.5
QPSK hard
[b/s/Hz]
Gauss soft
2.5
@
D−QPSK hard
8−PSK, soft
2
QPSK, soft
[b/s/Hz] 1.5
1
3−PSK, soft
1 0.5
BPSK hard 0 −2
0
D−BPSK hard
4
2
0.5
DD−OOK
BPSK, soft
6
8
10
0 −2
0
OOK, soft
DD−OOK
ABCD4E F G HI
2
6
8
10
Fig. 1. Spectral efficiencies of various modulations.
Fig. 2. Spectral efficiencies for soft coherent detection.
in the following sections. We are interested mainly in*,PSK mod+ 0 ulation sets, i.e, takes on the values r U U UOH where h . We will consider three types of detection schemes: 1) soft coherent detection, meaning the detector passes 4 to the decoder, 2) hard coherent detection, meaning the detector decides which of the H phases was sent and passes this decision to the decoder, and 3) hard differential detection, meaning differential detection is used and the detector passes one of H decisions to the decoder. We begin, however, with the commonly used direct-detection OOK method for purposes of comparison.
satisfies (5). The resulting curve is depicted in Fig. 1 as the curve labeled DD-OOK. Observe the sharp rise of this curve from S to approximately S . This means that it makes little sense to use a code with rate less than 1/2 with DD-OOK. We discuss this behavior in more detail in Section III. Consider next OOK with soft coherent detection. The spectral efficiency is easily seen to be 3dB worse than BPSK with soft coherent detection, so we proceed to consider PSK.
B. Direct-detection OOK
On-off keying involves transmitting either or *,+ *K+ so that the average energy is . Direct detection means that the decoder sees either 4 or 4 depending on whether 7 6 is smaller or larger, respectively, than some threshold . The capacity will be a function of the probabilities that the detector puts out 4~ when d , and 4~ when *Q+ . These probabilities are the respective (see [6, p. A! 47])
Z%$& ( ' ) % * , & 1 * &.-0/ t +
R"$# Q
"$# \ R
3
5
p243
t
(9)
* +
3
687
t
m
* +
3
:9
TR
"
bits
C. PSK with Soft Coherent Detection The modulation capacity of PSK with soft coherent detection is q"4} , where takes on one of H equally-likely phases. p We have q"4}KJ4T LJ4 | , where J&4} is the p entropy of 4 and J4 | is the conditional entropy of 4 given . We further have J&4 'K BEDGF 0 % 3 bits/s and (see [6, p. 92], [11, Sec. 20, 25])
J4
(11)
where m is the same function as in (4), and where 9
3
8
bits/s/Hz
(20)
As another example, if H and a Gray mapping from the coded bits to the phases is used, i.e., r , r , r 0 , r9 represent \ the coded bits , , , , respectively, then the channel is equivalent to two uses of a BSC with crossover probability R "K * P "h 6 > 3 . The normalized capacity is thus
e
;:
:
9
*QP 6
9
" "
6
ential detector puts out
, where
0 Z0 Y 3 "!E# 2 / ,h /;V iVU.1 3 (19) "!E# 0 2 3 % S 4 7 s Of course, R OY P R t r j is simply XW Z]\ \ R OQ P R t0 r . For example, for H the channel becomes a BSC with * P crossover probability R "5"h76 > 3| . The normalized e
Suppose we employ differential encoding so that K *,+ K where y and carries the information on K Z \ ) for (Y . A differone of H phases (with U U"J U
>
* P
3KM 3
=
bits/s/Hz
(21)
L
4
4X
K
:6
2
:6
(22) "X Z]\ Z]\ Z]\ Z \ where 4 is the ( th channel output, and 4 X is the complexZ \ conjugate of 4 . Suppose the receiver uses a soft detector Z]\ with 4 L . The receiver can now recover 4 recursively via 0 4 4 4 U UOJ[U for ( . Thus, a soft differenZ]\ M Z \
tial detector can in principle perform as well as a soft coherent detector (see, e.g., [17] and the references therein). Consider next a hard differential detector which puts out one L K of H decisions 4 basedL on . The channel from to 4 is not memoryless because is a function of successive channel outputs 4 and 4 . Thus, one should use (2) rather than (3) Z \ to compute S . However, for practical reasons such as encoding and decoding complexity one usually treats the channel as memoryless. This approach will be accurate if one places a long (random-like) interleaver between the encoder and modulator, and the appropriate de-interleaver between the detector and decoder. The de-interleaver breaks the statistical dependence between successive symbols. We will adopt this approach and will K K rather than q 4 . therefore compute q 4} p
p
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5
K
We compute q 4 to write J\4} by using symmetry y p K B$D F 0 &H bits/s, and write JX4 as in (15). However, instead of (16) or (18) we now use (see [15, eq. (11)] or [16, eq. (5)])
O P t0 R
)
for h
" ,
r V-,"
U
3
% Y
H
" 3
%
,
H
6
S
Z0
i
U
* +
0 3 "!$# /; {D / 2D 3
Y
1
s
9 3
Z
/
('
7 4
bits/s/Hz
6
3
0.5
L OW
AT
*QP
*QP
3
3 An important parameter is the smallest for which S *,P 3 . Let mrz denote the derivative of m at r . The paper [12, eq. (30)] shows that we can transmit at
m
DD−OOK
D−BPSK hard
0 −2
0
2
4
6
8
10
Fig. 4. Spectral efficiencies for hard (solid lines) and soft (dashed lines except for DD-OOK) differential detection.
coherent BPSK and QPSK to obtain *,P
(25)
III. S PECTRAL E FFICIENCY
D−3PSK, hard
(24)
D−QPSK, hard
1.5
The spectral efficiency for general H is again by nue computed * P G 3Y and merically finding the relation between S X in (5). The results for H U"J U U] are plotted in Fig. 4 together with the soft detection curves from Fig. 2 as dashed lines. Observe that with hard differential detection one cannot trans*QP 3 mit below an of about 4.8dB for H , and about 4.5dB for H . Observe also the sharp rise J and H in all the DPSK curves. We further remark that D-QPSK can outperform D-BPSK both in terms of bandwidth and energy efficiency by choosing appropriate code rates > M . For example, if * P 3TQ then one could improve the bandwidth efficiency one half the D-BPSK spectral efficiency, by choosing7 >@M from which is > M , to one half the D-QPSK spectral ]. efficiency, which is > M Finally, note that hard differential detection is much less efficient than soft coherent detection and, as discussed above, the soft detection curves can be approached by using soft differential detection. This suggests that large energy gains are possible once high-speed A/D converters with better than 1-bit resolution become available.
D−8PSK hard
2
[b/s/Hz]
Again, R OQ P t r j is simply XW Z]\ \ R OQ P t r . the detector error probability is For example, for H *,+) R"| i U 3 (see [5, p. 527], [6, p. 276]). The normalizedS capacity of D-BPSK with hard detection (and ignoring the channel memory) is thus
1
, where
%
e
Gauss soft
2.5
(23)
UH
/
6
3
3
B#
%
(28)
T i.e., one loses =BEDGF % dB using hard rather than \ soft detection (see, e.g., [18, p. 155]). Consider next hard coherent 3-PSK, DD-OOK and DPSK. For these schemes m is not concave so that (26) does not give *@PO 3 the right answer [12, Sec. V.E]. Instead, the minimum is found by using the following graphical procedure: 1) plot mr against r and 2) draw a line through the origin and tangent The to m so that all points of m lie on or below the*Kline. P 3 inverse of the slope of this line gives the minimum we * P are seeking. Furthermore, the size of the S jump at this 3Y is simply mr X , where r X is the point where the line is tangent to m . Observe that if m is concave this graphical procedure is equivalent to using (26). The aboveP discussion implies that for non-concave m the _ _ * *K+{ minimum or . 3T is not characterized by S 3 This is the reason for the sharp rises in the hard 3-PSK, DDOOK and DPSK spectral efficiency curves. The consequence of the abrupt behavior is that if one lowers > M beyond some threshold, then one pays more in the code’s =BEDGF >@M energy \ loss than one can gain back with error control capabilities. Thus, it makes little sense to choose > M less than this threshold. For instance, it makes little sense to use strong codes with hard DBPSK and >@M .
IV. OTHER D IFFERENTIAL D ETECTORS
We discuss two techniques for improving a differential detector. The first is multiple-symbol differential detection [21, : = 3 (26) Fig. L3],[22], i.e., the hard decisions 4 are functions of more than just x4 4 X . For example, [22, eq. (20)] uses the detector as long as and m is concave. For example, the soft Z]K\ L K K K X 4 X coherent Gaussian curve of (7) has 4 4 X 4 X 0 X
4 4 X 0 X (29) *QP
* P
and
B$D F
3Y
1
7
7
B # "K
r
B$D F 0
G
rz
Z \
n
Z]\
W] 4
xB #
K
(27)
dB. One gets the same result of dB for soft coherent PSK. We can also apply (26) to hard \
Z \
Z \
Z
Z \
Z
Z \ K
is a decision about the ( th phase change . Observe where that (29) requires two types of differential outputs: 4 4 X and Z \ 4 4 X 0 . These outputs could be realized with two delay interferZ ometers: one having a delay of one symbol and the other having a delay of two symbols.
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According to [22], equation (29) achieves a detector gain of *KP 3 about 1dB for D-QPSK and for large (see also [21, Table 1]). However, [22, Fig. 5] shows that this gain disappears at low * P 3T where R" Z 0 . This means that using (29) will not increase the spectral efficiency by much at rates where >TM 9 . We do emphasize, however, that this statement is valid only if one uses strong codes. When using weak codes, or even no coding at all, one will in fact get an energy gain of 1dB over uncoded D-QPSK because one must operate at large * P 3T . A second approach related to multiple-symbol differential detection is decision-feedback detection (see [21] and [23]). However, as we proceed to show, the advantages of this technique are also small if one uses strong codes. The simplest decisionfeedback structure of [21] puts out the decision statistic
L
K
4
4
4
Z]\
L
K
Z 0
Z]\
X
(30)
K
where is computed from only. In other words, Z]\ Z]\ Z]\ th hard-decision is the ( that is fed back to provide a better K K phase estimate than 4 alone. Observe that if Z]\ Z]\ Z]\ then the term in brackets in (30) is
Z]\
K x
6 0 : 0 Z]\ Z Z \ K Z
6
6 0 Z]\ Z \ Z Z \ ]
: 6
(31)
Note that (31) has twice the signal-to-noise ratio of 4 alone. Z \ K Thus, as long as is correct, (30) will give a less noisy esti Z \ K K mate of . However, if is incorrect one will have a poor Z]\ the term in brackets in phase reference. For instance, if H (30) becomes
:6 Z]\ 6 Z \
Z]\ 6
x K
Z 0
Z]\
:6 Z 0 U
2 Z 0
K
Z]\
(32)
K
i.e., the phase reference is completely noisy. In this case will be in error exactly one half the time. For H the error analysis is similar, although somewhat more complicated. K We continue to consider the case H . Let R "K K be W the raw bit-error rate out of the detector, and let K K K K . We compute
R
"
Z \ K
K
K
Z \ K
K
K
Z]\ Z \ ] Z \ Z \ (33) ~ R "
R " We thus have R "Q and, ignoring the channel mem-
9
ory as in (25), we have e
Y
bits/s/Hz.
(34)
We can compute using the techniques of*,[15] or [16].*,More +2 +2 precisely, we use [16, eq. (5)] with
3 , 0 3 \ and / x% to obtain
%
) Z
0
0 S
i
U.1
J,
* +
3| !E# 3 4
7
3
(35)
1
0.8
0.6
[b/s/Hz]
perfect DFB
0.4
D−BPSK hard
0.2
0 3
4
D−BPSK DFB
DD−OOK
6 7
5
8
9
10
Fig. 5. Spectral efficiencies for decision-feedback (DFB) hard differential detection.
The numerical results are plotted in Fig. 5. We see that the decision-feedback scheme performs virtually the same as DBPSK. Fig. 5 also shows the hypothetical curve if all the feedback decisions are correct. The large loss of imperfect feedback is not surprising because at low rates the noise energy is large and the decisions are incorrect with high probability. In summary, the decision-feedback detector (30) holds little promise if one uses strong codes. At the same time, we emphasize that when using weak codes, or no coding, it might very well make sense to use (30) rather than the usual DPSK detector. V. C OMPARISON
OF
C ODES
The spectral efficiencies of Fig. 1 4 are ultimate limits on performance if one uses capacity-approaching codes. Although practical codes cannot exactly reach these limits, the relative comparisons between the modulation and detection schemes will hopefully be accurate when one uses a particular code or code family. We consider two families of codes: ReedSolomon (RS) codes [24] and low-density parity-check (LDPC) codes [25], [26]. The RS codes have parameters =U [ where is the number of 8-bit symbols in a code word and is the number of 8-bit symbols containing information bits. We consider RS . U J , U J and U codes with parameters The code words thus have 2040 bits and the respective rates are and ] . We assume that all error patterns with J , ] , } "! or fewer symbol errors are corrected and that no miscorrections occur. The last assumption is optimistic, but pessimistic bounds show the approximation to be accurate. The LDPC codes have parameters =U [ where is now the number of bits in a code word and is the number of information bits. We consider three length 32,000 LDPC codes whose rates are , ] and . The codes are iteratively decoded and we determine their performance by a combination of simulation and analysis. P Fig. 6 plots the decoder output bit-error rate R versus the decoder input bit-error rateP R " . Table II lists the corresponding operating points when R Z \ ^ . For example, uncoded DBPSK and D-QPSK lose 0.29dB and 2.33dB, respectively, ver-
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10
7
0
2
21.3 GHz 22.9 GHz LDPC
10
uncoded
−5
RS
D−QPSK
[b/s/Hz] 1
10 10
RS
1.5
40 GHz 42.7 GHz
−10
D−BPSK
LDPC
RS
−15
10
LDPC
−12
10 −16 10
LDPC
−4
RS
−3
10
0.5
D−QPSK hard 0 3
−2
=BEDGF
1
\
4
dB
(36)
Inserting H and H into (36), we find that the respective asymptotic losses are 0dB and 2.32dB. However, these losses degrade to the respective 1.16dB and 2.41dB at the R " operating point of the rate LDPC code. In fact, uncoded D-QPSK outperforms uncoded D-BPSK for very high R" . This is why D-QPSK is superior to D-BPSK at low rates.
A. Code Spectral Efficiencies We plot the operating points of our codes in Fig. 7. There are six sets of code curves corresponding to the two code families and the three modulations D-BPSK, D-QPSK, DD-OOK. We have also included some of the ultimate operating points from Table I. Observe that these points (even the LDPC points) are rather far from the ultimate operating curves. The main reason is that we cannot use optimal message passing iterative decoders for high decoder speeds such as 10 Gbit/s and 40 Gbit/s. In the following we will compare a few of the operating points of Fig. 7. Many further comparisons can be made using Fig. 6 and Table II. Suppose we choose as our “standard” the > M LDPC code with DD-OOK. WeP see from Fig. 7 and Table II that this combination achieves R Z \ ^ at P a symbol rate of 42.7 GHz and a signal-to-noise ratio of * G dB. Alternatively, one could use D-BPSK to 3T *QPO 3 ] operate at dB, thereby saving 2.57dB in energy (this does not include the observed fiber non-linearity gains of D-BPSK). Suppose next that one decides to use D-QPSK with the >YM ] LDPC code. One can now send at a symbol rate of 22.9 *,P G 3 ] dB, i.e., one uses 46% less bandwidth GHz and than our standard and still gains 2.10dB in energy. Moreover, the bandwidth savings over D-BPSK are also 46% but one sacrifices 0.47dB in energy. This 0.47dB loss is 0.42dB greater than the 0.05dB loss of Table I. The main factor contributing
RS
7 !8#"
6
9
LDPC
10
RS
11
12
Fig. 7. Code operating points for $&%('*),+.-0/21 and a data rate of 40Gbit/s. The point labels refer to the symbol rates.
sus coherent BPSK. These numbers are close to the asymptotic degradations given in [27, p. 253]: "!E# 0 % H "!$# 0 % H
5
10
Fig. 6. Code bit-error probability transfer curves.
4
LDPC
D−BPSK hard
−1
10
DD−OOK
to the 0.42dB is a 0.36dB loss because a D-QPSK detector per*@PO 3 forms progressively worse than a D-BPSK detector as increases. This means that a better code will achieve a double gain as compared to the current code: the coding gain and a detector gain. Finally, suppose one decides to use D-QPSK with *K theP >YM 3 LDPC code. The symbol rate is now 40 GHz and dB. Thus, one saves a small amount of bandwidth and still saves 3.03dB in energy over our standard DD-OOK scheme. However, the energy savings over D-BPSK are only 0.46dB which is much less than the 1.97dB expected from Table I. The main factor contributing to the large degradation is that, for hard LDPC code gains only 1.88dB detection BPSK, our rate rather than the expected 3.25dB over our rate LDPC code. This suggests that one can design a better rate LDPC code.
VI. S UMMARY This paper has investigated the performance of several combinations of codes, modulations and detectors. We have concentrated on PSK because of its linear gains and its potential fiber non-linearity gains. The main points we wish to make are: 1) D-BPSK outperforms DD-OOK by about 2.5dB in energy efficiency with strong codes. 2) D-QPSK with low-rate codes outperforms D-BPSK with high-rate codes, both in terms of bandwidth and energy. The bandwidth gains are particularly large. 3) Real codes can realize the gains predicted by the information-theoretic analysis, even for high-speed systems where optimal decoding is not possible. 4) Soft differential detection will give further large gains. However, this requires A/D converters with multi-bit resolution at high speeds such as 10 Gbit/s and 40 Gbit/s. ACKNOWLEDGMENTS The LDPC code performance analysis was based in part on unpublished work by T. J. Richardson and R. L. Urbanke. The authors also wish to thank S. Hunsche, J. E. Mazo, J. Salz and R. E. Slusher for their support of this work.
8
APPEARED IN THE IEEE/OSA JOURNAL OF LIGHTWAVE TECHNOLOGY, PP. 2438–2445, OCTOBER 2003
TABLE II C ODE O PERATING P OINTS : $ AND (%
Code uncoded LDPC
RS
> M
1 1/2 7/8 15/16 0.498 0.875 0.937
R
"
[Z]\^
0.0363 0.00362 0.00114 0.0114 5.84 [Z 6.35 Z
[2]
[3]
[4]
[5] [6] [7]
[8]
[9]
[10] [11]
[12] [13] [14]
[15]
[16] [17]
[18]
3
.
[dB]
BPSK
D-BPSK
D-QPSK
DD-OOK
15.29 5.08 6.15 6.96 7.16 7.80 8.94
15.58 7.20 7.51 8.12 8.80 8.88 9.81
17.62 7.66 8.59 9.37 9.64 10.20 11.32
18.47 9.49 10.02 10.69 11.23 11.47 12.49
R EFERENCES [1]
$ %('*),+ -#/21
FOR
*QPO
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