Two-stage Demodulation and Decoding for thePhase Noise Channel

Luca Barletta, Maurizio Magarini, Arnaldo SpalvieriDipartimento di Elettronica e Informazione

Politecnico di MilanoMilan, Italy

Email: {barletta,magarini,spalvier}@elet.polimi.it

Abstract— Coding for the phase noise channel is investigatedin the paper. Specifically, Wiener’s phase noise, which inducesmemory in the channel, is considered. A general coding principlefor channels with memory is that of interleaving two or morecodes. The interleaved codes are decoded in sequence, using pastdecisions to help future decoding. The paper proposes a methodbased on this principle, and shows its benefits through numericalresults obtained by computer simulation. Analysis of the channelcapacity given by the proposed method is also worked out in thepaper.

Keywords— Coherent communication, Phase noise, Iterativejoint demodulation and decoding, Staged decoding.

I. INTRODUCTION

Coherent demodulation of advanced coded modulation for-mats is a challenging task in classical radio transmissionsystems and in coherent optical transmission systems of newgeneration. Besides the common Additive White GaussianNoise (AWGN), the performance of coherent demodulationcan be strongly impaired by multiplicative phase noise. In thecontext of radio transmission, phase noise is introduced bylocal oscillators used for up-conversion and down-conversion.The impact of phase noise on the performance of OFDMsystems is studied, for instance, in [1]. Also, single carriersystems, especially recent systems based on frequency domainequalization as [2], suffer from phase noise and require specificmitigation techniques [3]. With the advent of coherent optics,the role of phase noise is becoming well recognized also inthe context of optical transmission, where it is known thatlaser’s phase noise is a Wiener process [4], and the Wienermodel has been recently proposed in [5] also for the phasenoise accumulated during nonlinear propagation, at least forthe cases studied in that paper.

Recent papers [6]–[8] address the problem of coherentdemodulation in the presence of Wiener phase noise. Also,Wiener phase noise is adopted in [9]–[11] to assess theperformance of iterative demodulation and decoding, whilethe capacity of the channel affected by Wiener phase noiseis derived in [12] and a simple yet effective lower bound isgiven in [13]. Often, one is lead to introduce pilot symbols toaid carrier recovery in the presence of strong phase noise [9],[14], [15], the capacity of Wiener’s phase noise channel withpilot symbols being studied in [16]. However, pilot symbols

sacrifice spectral efficiency. As an alternative to pilot symbols,one can resort to differential demodulation methods as thoseproposed in [10], [17], [18]. The trellis-based method of [17]in conjunction with iterative differential demodulation anddecoding offers excellent performance at the expense of largecomplexity of signal processing, while the method based onTikhonov parametrization [10] is less demanding, still offeringgood performance.

A general coding principle for channels with memory, asWiener’s phase noise channel is, is that of interleaving two ormore codes. The interleaved codes are decoded in sequenceby the so-called staged decoding technique, which uses pastdecisions to help future decoding. The use of interleaving andstaged decoding aided by past decisions has been proposed in[19] and expanded in [20] for the InterSymbol Interference(ISI) AWGN channel, where the generic stage consists ofequalization and decoding. The general principle has thenfound application to a variety of channels with memory [21],[22].

The scheme proposed here and in the journal paper [23]relies upon interleaving of pilot symbols and coded symbolsfrom two (or more) codes. Channel symbols of the more pow-erful code are demodulated and decoded first, then decisionson first-level coded symbols are used as pilot symbols in thesuccessive demodulation and decoding stage. For instance,with two component block codes one can transmit throughthe channel the sequence

(p, c2,1, c2,2, c2,3, c1,1, c2,4, c2,5, c2,6);(p, c2,7, c2,8, c2,9, c1,2, c2,10, c2,11, c2,12);(p, · · · );· · ·(p, c2,6N−5, c2,6N−4, c2,6N−3, c1,N , c2,6N−2, c2,6N−1, c2,6N ).

(1)

In the above sequence, represented in Fig. 1, p representsone pilot symbol and ci,j is the j-th symbol of the i-th levelcode Ci. In (1), one frame is inserted between parentheses,the entire sequence consists of N frames, the length of codeC2 is 6N while the length of code C1 is N . For the sakeof concreteness, in the following only two-level constructionswill be considered, the extension to multilevel constructions

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Fig. 1. Example of two-stage coded sequence with pilots according toequation (1).

being straightforward.The main novelty of our proposal is the application of the

principle of interleaving and staged decoding to the phasenoise channel, where the individual stage consists of demod-ulation and decoding. Compared to the previous literature oninterleaving and staged decoding, specifically references [19]–[22], other minor novelties that we can claim are the following.

• While in [19] and [21] only one code is considered, herethe use of a set of different channel codes is proposed.

• While in [19] the pattern of pilot symbols is a blockof pilots with block length equal to the memory of thechannel, here blocks of size much smaller than channel’smemory are used. Pilot symbols are not considered in[20] and in [21].

• In the interleaving scheme presented here, a differentnumber of symbols from different codes are interleavedin each frame, while in [20] and [22] the same numberof symbols from different codes is interleaved in eachframe.

The paper is organized as follows. Section II is devoted tothe channel and system model. Section III reports the analysisof channel capacity. In Section IV the results that are obtainedwith the proposed method in contrast with adversary methodsare shown. Finally, in Section V conclusions are drawn.

II. CHANNEL AND SYSTEM MODEL

The k-th received sample yk is

yk = (xk + wk)ejθk ,

where xk is the k-th transmitted symbol, wk is the k-th sampleof Additive White Gaussian Noise (AWGN), and θk is the k-thsample of phase noise. The phase noise is hereafter modelledas a discrete-time Wiener process

θk = θk−1 + γvk, k = 1, 2, · · · , (2)

where γ > 0 is a known parameter, θ0 is uniformly distributedin [0, 2π), and vk is the k-th sample of white Gaussian noisewith zero mean and unit variance. The phase evolution given in(2) occurs when the power spectral density of the continuous-time complex exponential ejθ(t), whose samples at symbolfrequency generate the sequence ejθk , is the Lorentzian func-tion

L(f) =4γ2T

γ4 + 16π2f2T 2,

where T is the symbol repetition interval and f is thefrequency. The parameter γ2 can be expressed as

γ2 = 2πBFWHMT,

where BFWHM is the full-width half-maximum bandwidth ofthe spectral line.

The information rate expressed in bits per channel symbolof a two-level construction with one 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 oneframe, R1 is the information rate of code C1, M2 − 1 isthe number of consecutive symbols of code C2, R2 is theinformation rate of code C2, and M1 ·M2 is the total numberof symbols in one frame. In the example (1), illustrated in Fig.1, one has M1 = 2, M2 = 4.

Iterative demodulation and decoding, as for instance de-scribed in [9], can be used after the first demodulation basedon pilot symbols only. After having decoded the first-levelcode, the transmitted code word is regenerated and its symbolsare used as pilot symbols in the second demodulation anddecoding stage.

III. ANALYSIS OF CHANNEL CAPACITY

Let Xp be the deterministic sequence of pilot symbols andlet X1 and X2 be the random processes of symbols of the first-level code and of the second-level code, respectively. Similarly,the received sequence is divided in three parts called Yp, Y1,Y2, where Yp corresponds to the time instants where pilotsymbols Xp are transmitted, while Yi corresponds to the timeinstants where symbols of 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 betweenY and X is

I(Y;X ) = limn→∞

1n

I(yn1 ;xn

1 ). (3)

By 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 ofeach one of the sub-channels is computed by dividing theinformation between vectors by the number of uses of thecomposite channel, 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,

therefore

I(Y;X ) = I(Y;X1|Xp) + I(Y;X2|X1,Xp).

Invoking the chain rule on Y , the first term in the right sideof the above equation is

I(Y;X1|Xp) = I(Y1,Yp;X1|Xp) + I(Y2;X1|Xp,Y1,Yp),

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leading to

I(Y;X) = I(Y1,Yp;X1|Xp) + I(Y;X2|X1,Xp)+ I(Y2;X1|Xp,Y1,Yp). (4)

The term I(Y2;X1|Xp,Y1,Yp) appearing in the above equa-tion is the contribution coming from the blind processing ofY2 at the first stage. In the numerical results presented in thenext section we renounce to this contribution, which is smallin many cases of practical interest. It can be computed fromequation (4) as

I(Y2;X1|Xp,Y1,Yp) = I(Y;X ) − I(Y1,Yp;X1|Xp)− I(Y;X2|X1,Xp) (5)

where the three terms in the right side of (5) can be computedas in [16]. Specifically, for I(Y;X ) one uses pilot symbols in-serted 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 with period M1 and Wienerphase noise with step

γ√

M2.

(Down-sampling Wiener noise one gets Wiener noise withlarger variance.)

IV. NUMERICAL RESULTS

Numerical results have been derived using low-densityparity-check (LDPC) codes as component codes. Designingthe matrix of the component codes is out of the scope ofthe present paper. To demonstrate the benefits of the proposedscheme we adopt the LDPC family of the popular digital videobroadcasting - satellite (DVB-S2) standard, whose matrices areof public domain, thus making our results easy to reproduce.Fig. 2 reports the performance of the component codes ofthree two-level schemes, the component codes being decodedaccording to [9]. The performance of the second-level code isobtained by assuming no errors from the first-level code. Thisa realistic assumption for capacity-achieving codes, as LDPCcodes are provided that they are operating in the waterfallregion. For capacity-achieving systems the performance ofthe two-level scheme is dominated by the performance of theworst of the two component codes, hence, in a good design,the bit error rate (BER) curves of the two components codesshould be close to each other, as it happens with the codes ofFig. 2. In Fig. 3 and 4 our proposed scheme is compared tothe classical one-level scheme for 4-ary quadrature amplitudemodulation (4-QAM) and 16-QAM, respectively. The numberof iterations of the LDPC decoder of the one-level scheme isthe same as the average number of iterations of the two LDPCdecoders of our scheme. It should be observed that, in theproposed scheme, it turns out to be convenient to make moreiterations at the first level and less iterations at the secondlevel. In Fig. 3 the performance of [17] is also reported,even if it should be said that [17], where demodulation isbased on a trellis, is much more demanding in terms ofcomplexity compared to the adversaries. Moreover, [17] isbased on differential demodulation and it is suited only for

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010

−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. Performance of individual codes with iterative demodulation anddecoding [9]. Solid line: first-level code. Dashed line: second-level code. Thesecond-level code assumes ideal decoding of the first-level code.

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 pilotsymbol per frame and the three two-level codes of Fig. 2. CBC indicatesone-level coding with the algorithm of [9], while soft differential decodingindicates the algorithm of [17]. The performance is evaluated at bit error rateof 10−5 after 24 iterations.

phase shift keying constellations and cannot be applied to16-QAM. From Figs. 3 and 4 it appears the advantage ofour scheme, especially with 16-QAM, where it brings systemperformance closer to the capacity curve of about 0.5 dBcompared to the adversary.

V. CONCLUSIONS

In the paper, a staged demodulation and decoding methodis proposed for channels affected by strong phase noise. Theresults presented in the paper show that the proposed methodoutperforms adversary methods base on conventional one-level demodulation and decoding. We should mention thatsome margin still exists to improve the method, as it can

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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 pilotsymbol per frame and M1 = 7, M2 = 9. CBC indicates one-level codingwith the algorithm of [9]. The performance is evaluated at bit error rate of10−5 after 24 iterations.

−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 figureshows the term I(Y2;X1|Xp,Y1,Yp) of equation (5).

be seen from the results on channel capacity reported inFig. 5. Specifically, Fig. 5 shows the capacity that one loseswhen renounces to help the first-level demodulator by blindprocessing the constellation symbols of the second-level code.The 0.05 bits/2D of capacity loss with 16 QAM at SNR=15 dBcan be converted in decibels by the popular law of 3 dB/bit,leading to a potential margin of 0.15 dB coming from thementioned blind processing. Before concluding the paper, itis worth adding that, although here only results for Wienerphase noise have been presented, the staged demodulation anddecoding scheme can be adopted also to combat phase noiseof higher order, for instance the one proposed in [24], whichcan better fit oscillators used in radio systems.

REFERENCES

[1] A. G. Armada, “Understanding the Effects of Phase Noise in OrthogonalFrequency Division Multiplexing (OFDM),” IEEE Trans. Broadcast. vol.47, no. 2, pp. 153–159, Jun. 2001.

[2] M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation ofthe feedback filter for the hybrid decision feedback equalizer in highlydispersive channels,” IEEE Trans. Wireless Commun., vol. 11, no. 6, pp.2245–2253, June 2012.

[3] M. Asim, M. Ghogho, D. McLemon “Mitigation of phase noise in singlecarrier frequency domain equalization systems,” in Proc. of WirelessCommunications and Networking Conference (WCNC), pp. 920–924,April 2012.

[4] G. J. Foschini, G. Vannucci, “Characterizing filtered light waves cor-rupted by phase noise,” IEEE Trans. Inform. Theory, vol. 34, no. 6, pp.1437–1448, Nov. 1988.

[5] M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G.Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Optics Express, vol. 19, issue 23, pp.22455–22461, Nov. 7, 2011.

[6] M. G. Taylor, “Phase estimation methods for optical coherent detectionusing digital signal processing,” IEEE J. Lightw. Technol., vol. 27, no. 7,pp. 901–914, April 1, 2009.

[7] T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digitalreceiver concept with feedforward carrier recovery for M-QAM constel-lations,” IEEE J. Lightw. Technol., vol. 27, no. 8, pp. 989–999, April15, 2009.

[8] X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrierrecovery algorithm for coherent optical QAM systems,” IEEE J. Lightw.Technol., vol. 29, no. 5, pp. 801–807, March 1, 2011.

[9] G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterativedecoding in the presence of strong phase noise,” IEEE Journal onSelected Areas in Communications, vol. 23, no. 9, pp. 1748–1757, Sept.2005.

[10] A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationallyinvariant codes over channels with phase noise,” IEEE Trans. onCommun., vol. 55, no. 11, pp. 2125–2133, Nov. 2007.

[11] A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channel,” IEEE Trans. onCommun., vol. 59, no. 12, pp. 3223–3228, Dec. 2011.

[12] L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of informationrates of discrete-time first-order Markov phase noise channel,” IEEEPhoton. Technol. Lett., vol 23, no. 21, pp. 1582–1584, Nov.1, 2011.

[13] L. Barletta, M. Magarini, and A. Spalvieri, “A New Lower Bound belowthe Information Rate of Wiener Phase Noise Channel Based on KalmanCarrier Recovery,” Optics Express, accepted for publication.

[14] A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presenceof phase noise,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1966–1974,July 2011.

[15] M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe,M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recov-ery for 100-G PM-QPSK digital coherent receivers,” IEEE PhotonicsTechnol. Letters, vol. 24, no. 9, pp. 739–741, May 01, 2012.

[16] L. Barletta, M. Magarini, and A. Spalvieri, “The information ratetransferred through the discrete-time Wiener’s phase noise channel,”IEEE J. Lightw. Technol., vol. 30, no. 10, pp. 1480–1486, May 15,2012.

[17] M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for codednoncoherent MPSK communications over phase-noisy AWGN channel,”Proc. IEE Commun., vol. 147, pp. 87–95, Apr. 2000.

[18] M. Magarini, L. Barletta, A. Spalvieri, A. Leven, M. Pepe, and G.Gavioli, “Impact of non-ideal phase reference on soft decoding ofdifferentially encoded modulation,” IEEE Photonics Technol. Letters,preprint available for download at IEEEXplore website.

[19] M. V. Eyuboglu, “Detection of coded modulation signals on linearseverely distorted channels using decision-feedback noise prediction andinterleaving,” IEEE Trans. Commun., vol. 36, no. 4, pp. 401–409, April1988.

[20] H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievableinformation rates for finite state ISI channels,” Proc. IEEE Globaltelecommun. Conf., S. Antonio, Texas, Nov. 2001, pp. 2992–2996.

[21] T. Li and O. M. Collins, “A successive decoding strategy for channelswith memory,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 628–646,Feb. 2007.

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[22] S. Das and P. Schniter, “Noncoherent communication over the doublyselective channel via successive decoding and channel re-estimation,” inProc. annu. Allerton Conf. Communications, Control and Computing,Monticello, IL, 2007.

[23] L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation anddecoding,” Optics Express, vol. 20, issue 21, pp. 23728–23734, Oct. 8,2012.

[24] A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-timephase-locked loop with loop delay,” IEEE Trans. Circuits and SystemsII, vol. 55, no. 6, pp. 596–600, June 2008.

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