Abstract: Coding for the phase noise channel is investigated in the paper.Specifically, Wiener’s phase noise, which induces memory in the channel,is considered. A general coding principle for channels with memory is theinterleaving of two or more codes. The interleaved codes are decoded insequence, using past decisions to help future decoding. The paper proposesa method based on this principle, and shows its benefits through numericalresults obtained by computer simulation. Analysis of the channel capacitygiven by the proposed method is also worked out in the paper.
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lation of phase noise in long-haul coherent optical systems,” Opt. Express 23, 22455–22461 (2011).3. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber net-
works,” J. Lightwave Technol. 28, 662–701 (2010).4. T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in
Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.5. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-
wave Technol. 7, 901–914 (2009).6. T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier
recovery for M-QAM constellations,” J. Lightwave Technol. 8, 989–999 (2009).7. X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical
QAM systems,” J. Lightwave Technol. 5, 801–807 (2011).8. G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,”
IEEE J. Sel. Areas Commun. 9, 1748–1757 (2005).9. A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase
noise,” IEEE Trans. Commun. 11, 2125–2133 (2007).10. A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise
channels,” IEEE Trans. Commun. 12, 3223–3228 (2011).11. L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov
phase noise channels,” IEEE Photon. Technol. Lett. 21, 1582–1584 (2011).12. A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.
7, 1966–1974 (2011).13. M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-
symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol.Lett. 9, 739–741 (2012).
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15. M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communicationsover phase-noisy AWGN channel,” Proc. IEE Commun. 2, 87–95 (2000).
16. M. V. Eyuboglu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback noise prediction and interleaving,” IEEE Trans. Commun. 4, 401–409 (1988).
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23728
17. H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievable information rates for finite state ISI channels,”in Proc. of IEEE Globecom (2001).
18. T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory 2,628–646 (2007).
19. S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decodingand channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).
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21. A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEETrans. Circuits Syst. II 55, 596–600 (2008).
1. Introduction
Coherent demodulation of advanced coded modulation formats is a hot topic in new generationoptical transmission systems. Besides the common additive white Gaussian noise (AWGN), theperformance of coherent demodulation can be strongly impaired by multiplicative phase noise.It is recognized for a long time that laser’s phase noise is a Wiener process [1], and the Wienermodel has been recently proposed in [2] also for the phase noise accumulated during nonlinearpropagation, at least for the cases studied in that paper. The impact of phase noise on the per-formance of coherent optical transmission systems is discussed in [3,4]. Basically, phase noiseafflicts the accuracy of carrier recovery, which becomes a critical task of the receiver. In thepresence of strong phase noise, carrier recovery is so bad that cycle slips do appear [4], leadingto a lack of coherency of the demodulator that definitely compromises system’s performance.Recent papers [5–7] address the problem of coherent demodulation in the presence of Wienerphase noise. Also, Wiener phase noise is adopted in [8–10] to assess the performance of iter-ative demodulation and decoding, while the capacity of the channel affected by Wiener phasenoise is derived in [11]. Often, one is lead to introduce pilot symbols to aid carrier recovery inthe presence of strong phase noise [8,12,13], and the capacity of Wiener’s phase noise channelwith pilot symbols is studied in [14]. However, pilot symbols sacrifice spectral efficiency. Asan alternative to pilot symbols, one can resort to differential demodulation methods as thoseproposed in [9, 15]. The trellis-based method of [15] in conjunction with iterative differentialdemodulation and decoding offers an excellent performance at the expense of large complexityof signal processing, while the less demanding method based on Tikhonov parametrization [9]still offers a good performance.
A staged demodulation and decoding method is proposed in this paper. The method reliesupon interleaving of pilot symbols and coded symbols from two (or more) codes. Channelsymbols of the more powerful code are demodulated and decoded first, then decisions on first-level coded symbols are used as pilot symbols in the successive demodulation and decodingstage. For instance, with two channel block codes one can transmit through the channel thesequence
In the above sequence, represented in Fig. 1, p represents one pilot symbol and ci, j is the j-thsymbol of the i-th level code Ci. In Eq. (1), one frame is inserted between parentheses, the entiresequence consists of N frames, and the length of code C2 is 6N while the length of code C1 isN. For the sake of correctness, only two-level constructions will be studied, and the extensionto multilevel constructions becomes straightforward.
The use of interleaving and staged decoding aided by past decisions has been proposedin [16] and expanded in [17] for the intersymbol interference (ISI) AWGN channel, wherethe generic stage consists of equalization and decoding. The general principle has then found
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23729
Fig. 1. Example of two-stage coded sequence with pilots according to Eq. (1).
application in a variety of channels with memory [18, 19]. The main novelty of our proposal isthe application of the principle of interleaving and staged decoding to the phase noise channel,where the individual stage consists of demodulation and decoding. Compared to the previ-ous literature on interleaving and staged decoding, specifically references [16–19], other minornovelties that we can claim are the following:
• While in [16] and [18] only one code is considered, here the use of a set of differentchannel codes is proposed.
• While in [16] the pattern of pilot symbols is a block of pilots with block length equal tothe memory of the channel, here blocks of size much smaller than channel’s memory areused. Pilot symbols are not considered in [17] and [18].
• In the interleaving scheme presented here, a different number of symbols from differentcodes are interleaved in each frame, while in [17] and [19] the same number of symbolsfrom different codes is interleaved in each frame.
The paper is organized as follows. Section II is devoted to the channel and system model.Section III reports the analysis of channel capacity. In section IV the results that are obtainedwith the proposed method in contrast with adversary methods are shown. Finally, in section Vconclusions are drawn.
2. Channel and system model
The k-th received sample yk isyk = (xk +wk)e
jθk ,
where xk is the k-th transmitted symbol, wk is the k-th sample of AWGN, and θk is the k-thsample of phase noise. The phase noise is hereafter modeled as a discrete-time Wiener process
θk = [θk−1 + γvk] mod 2π , k = 1,2, · · · , (2)
where γ > 0 is a known parameter, θ0 is uniformly distributed in [0,2π), and vk is the k-th sam-ple of white Gaussian noise with zero mean and unit variance. The phase evolution given in Eq.(2) occurs when the power spectral density of the continuous-time complex exponential ejθ(t),whose samples at symbol frequency generate the sequence e jθk , is the Lorentzian function
L ( f ) =4γ2T
γ4 +16π2 f 2T 2 ,
where T is the symbol repetition interval and f is the frequency. The parameter γ2 can beexpressed as
γ2 = 2πBFWHMT,
where BFWHM is the full-width half-maximum bandwidth of the spectral line.
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23730
The information rate expressed in bits per channel symbol of a two-level construction withone pilot symbol per frame is
R =R1 · (M1 −1)+R2 ·M1 · (M2 −1)
M1 ·M2,
where M1 −1 is the number of symbols of code C1 in one frame, R1 is the information rate ofcode C1, M2 − 1 is the number of consecutive symbols of code C2, R2 is the information rateof code C2, and M1 ·M2 is the total number of symbols in one frame. In the example (Eq. (1)),illustrated in Fig. 1, M1 = 2, M2 = 4.
Iterative demodulation and decoding, as for instance described in [8], can be used after thefirst demodulation based on pilot symbols only. After having decoded the first-level code, thetransmitted code word is regenerated and its symbols are used as pilot symbols in the seconddemodulation and decoding stage.
3. Analysis of channel capacity
Let Xp be the deterministic sequence of pilot symbols and let X1 and X2 be the random pro-cesses of symbols of the first-level and of the second-level, respectively. Similarly, the receivedsequence is divided in three parts called Yp, Y1, Y2, where Yp corresponds to the time instantswhere pilot symbols Xp are transmitted, while Yi corresponds to the time instants where symbolsof level i are transmitted.
Let xn1 and yn
1 denote the channel input vector (x1,x2, · · · ,xn) and the channel output vector(y1,y2, · · · ,yn), respectively. The information rate between Y and X is
I(Y ;X) = limn→∞
1n
I(yn1;xn
1). (3)
By the chain rule on X one writes
I(Y ;X) = I(Y ;Xp)+ I(Y ;X1|Xp)+ I(Y ;X2|X1,Xp),
where, here and in what follows, the information rate of each one of the sub-channels is com-puted by dividing the information between vectors by the number of uses of the compositechannel, e.g.
I(Y ;Xp) = limn→∞
1n
I(yn1;xn
p,1),
where xnp,1 is the vector of pilot symbols where zeros are inserted in the positions occupied by
first-level and second-level coded symbols. Note that, since Xp is a known sequence,
I(Y ;Xp) = 0,
thereforeI(Y ;X) = I(Y ;X1|Xp)+ I(Y ;X2|X1,Xp).
Invoking the chain rule on Y , the first term in the right side of the above equation is
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23731
The term I(Y2;X1|Xp,Y1,Yp) appearing in the above equation is the contribution coming fromthe blind processing of Y2 at the first stage. In our proposal we suggest to renounce to thiscontribution. It can be computed from Eq. (4) as
where the three terms in the right side of Eq. (5) can be computed as in [14]. Specifically, forI(Y ;X) one uses pilot symbols inserted with period M1 ·M2, for I(Y ;X2|X1,Xp) one uses pilotsymbols inserted with period M2, while for I(Y1,Yp;X1|Xp) one uses pilot symbols inserted withperiod M1 and Wiener phase noise with step
γ√
M2.
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
SNR [dB]
BE
R
R1=1/2, M
1=6
R2=3/5, M
2=9
R1=3/5, M
1=7
R2=5/6, M
2=9
R1=3/4, M
1=5
R2=9/10, M
2=9
Fig. 2. 4-QAM, γ = 0.125, LDPC codes of length 64800 from the DVB-S2 standard. Per-formance of individual codes with iterative demodulation and decoding [8]. Solid line:first-level code. Dashed line: second-level code. The second-level code assumes ideal de-coding of the first-level code.
4. Numerical results
Numerical results have been derived using low-density parity-check (LDPC) codes from thepopular digital video broadcasting—satellite (DVB-S2) standard. Figure 2 reports the perfor-mance of the component codes of three two-level schemes, the component codes being decodedaccording to [8].
The performance of the second-level code is obtained by assuming no errors from the first-level code. This a realistic assumption for capacity-achieving codes, as LDPC codes are, op-erating in the waterfall region. For these systems the performance of the two-level scheme isdominated by the performance of the worst of the two component codes, hence, in a good de-sign, the bit error rate (BER) curves of the two components codes should be close to each other,as it happens with the codes of Fig. 2. In Figs. 3 and 4 the two-level coding scheme with stageddecoding is compared to one-level schemes for 4-ary quadrature amplitude modulation (QAM)and 16-QAM, respectively.
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23732
1 2 3 4 5 6 7 8 9
1
1.2
1.4
1.6
1.8
2
SNR [dB]
Spe
ctra
l Effi
cien
cy [b
it/2D
]
AWGN capacityCapacity with phase noiseTwo−stage scheme with pilotsCBC with pilotsSoft Differential Decoding
Fig. 3. 4-QAM, γ = 0.125. The two-stage scheme is based on one pilot symbol per frameand the three two-level codes of Fig. 2. CBC indicates one-level coding with the algorithmof [8], while soft differential decoding indicates the algorithm of [15]. The performance isevaluated at bit error rate of 10−5 after 24 iterations.
10 11 12 13 14 15 162.8
3
3.2
3.4
3.6
3.8
4
SNR [dB]
Spe
ctra
l Effi
cien
cy [b
it/2D
]
AWGN capacityCapacity with phase noiseTwo−stages scheme with pilotsCBC with pilots
Fig. 4. 16-QAM, γ = 0.125. The two-stage scheme is based on one pilot symbol per frameand M1 = 7, M2 = 9. CBC indicates one-level coding with the algorithm of [8]. The per-formance is evaluated at bit error rate of 10−5 after 24 iterations.
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23733
The number of iterations of the LDPC code for the one-level code is the same as the averagenumber of iterations of the two LDPC codes of our method, where it turns out to be convenientto make more iterations at the first level and less iterations at the second level. In Fig. 3 theperformance of [15] is also reported, even if it should be said that [15], where demodulation isbased on a trellis, is much more demanding in terms of complexity compared to the adversaries.Moreover, [15] is based on differential demodulation and it is suited only for phase shift keying
−5 0 5 10 15 20 25−0.01
0
0.01
0.02
0.03
0.04
0.05
SNR [dB]
I(Y
2;X1|X
p,Y1,Y
p) [b
it/2D
]
16−QAM, γ=0.125, M1=7, M
2=9
4−QAM, γ=0.125, M1=7, M
2=9
Fig. 5. 4-QAM and 16-QAM, M1 = 7, M2 = 9, γ = 0.125. The figure shows the termI(Y2;X1|Xp,Y1,Yp) of Eq. (5).
(PSK)-type constellations therefore it cannot be applied to 16-QAM. From Figs. 3 and 4, theadvantage of our method appears, especially with 16-QAM, where our method brings systemperformance closer to the capacity curve of about 0.5 dB compared to the adversary.
We should mention that some margin still exists to improve the method, as it can be seen fromthe results on channel capacity reported in Fig. 5. Specifically, Fig. 5 shows the capacity that oneloses when one does not help the first-level demodulator by blind processing the constellationsymbols of the second-level code. The 0.05 bits/2D of capacity loss with 16 QAM at SNR=15dB can be converted in decibels by the popular law of 3 dB/bit, leading to a potential margin of0.15 dB coming from the mentioned blind processing.
5. Conclusions
In this work, a staged demodulation and decoding method is proposed for channels affected bystrong phase noise. The results presented in the paper show that the proposed method outper-forms adversary methods based on conventional one-level demodulation and decoding.
Before concluding the paper, it is worth adding that, although here only results for Wienerphase noise have been presented, the general principle of staged demodulation and decodingcan be adopted also to combat phase noise of higher order [20], for instance the second-orderphase noise studied in [21].
#174752 - $15.00 USD Received 21 Aug 2012; revised 10 Sep 2012; accepted 10 Sep 2012; published 1 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 23734
