1480 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 10, MAY 15, 2012

The Information Rate Transferred Through theDiscrete-Time Wiener’s Phase Noise Channel

L. Barletta, M. Magarini, Member, IEEE, and A. Spalvieri

Abstract—The information rate transferred through the Addi-tive White Gaussian Noise (AWGN) channel affected by discrete-time multiplicative Wiener’s phase noise is computed in the paper.Upper and lower bounds based on quantization of the phase spaceand trellis representation of phase noisememory are proposed. Theresults presented in the paper show that both the upper bound andthe lower bound converge to the actual information rate as thenumber of quantizer’s bins increases, the convergence of the lowerbound being faster. The analysis presented in the paper allows forpilot symbols being considered. From the one hand, the insertion ofpilot symbols penalizes the net symbol rate, thus subtracting spaceto information transmission, while, from the other hand, they bringinformation on channel state. The balance between what is lost andwhat is gained is computed in the paper.

Index Terms—Information rates, optical communication, phasenoise, Wiener processes.

I. INTRODUCTION

C OHERENT demodulation of advanced modulation for-mats is becoming a hot topic in optical transmission sys-

tems of new generation. It has been recently observed in [1] thatthe probability of cycle slip in carrier synchronization can bemuch higher than what expected, thus emphasizing the centralrole of phase noise in determining system’s performance. Re-cent papers [2]–[4] address the problem of coherent demodula-tion in the presence of Wiener’s phase noise. Also, the Wiener’smodel for phase noise is adopted in [5], [6] to assess the perfor-mance of iterative demodulation and decoding. Actually, it isrecognized from a long time that laser’s phase noise is a Wienerprocess [7], and the Wiener’s model has been recently proposedin [8] also for the phase noise accumulated during nonlinearpropagation, at least for the cases studied in that paper.While the capacity of the Additive White Gaussian Noise

(AWGN) channel with white phase noise is a well-studiedtopic [9]–[11], the capacity of the channel affected by Wiener’sphase noise, which is a channel with memory, is less studied.Methods for computing the information rate transferred throughfinite-state channels with memory, as, for instance, the MonteCarlo method proposed in [12], have already demonstratedtheir usefulness in the context of optical transmission, havingbeen adopted in [13] for calculation of the information ratetransferred over optical channels with memory. However, theAWGN plus Wiener phase noise channel is a channel with

Manuscript received October 01, 2011; revised December 21, 2011, January27, 2012; accepted February 06, 2012. Date of publication February 13, 2012;date of current version April 04, 2012.The authors are with the Dipartimento di Elettronica e Informazione, Po-

litecnico di Milano, Milano 20133, Italy (e-mail: barletta@elet.polimi.it; ma-garini@elet.polimi.it; spalvier@elet.polimi.it).Digital Object Identifier 10.1109/JLT.2012.2187635

memory and with continuous state. To overcome the obstacleposed by the continuous nature of the state, the present authorshave recently proposed a lower bound based on quantizationof the phase space [14]. The same approach has been indepen-dently pursued in [15] to work out a lower bound similar tothe one of [14]. The main new result presented in this paperis an upper bound on the constrained channel capacity, thatis the information rate transferred through the channel with afixed source. Increasing the number of bins, upper and lowerbound become virtually indistinguishable, thus substantiatingour claim of having computed the exact information rate ofthe phase noise channel. By controlling the gap between theupper bound and the lower bound, one controls the numberof bins that are adequate to compute capacity with the desiredaccuracy. Another novelty of the present paper compared to[14], [15] is the consideration of pilot symbols, which play animportant role in the practice of carrier recovery [16], especiallywhen phase noise is present [17].

II. TRELLIS-BASED COMPUTATION OF PROBABILITIES FORFINITE-STATE FIRST-ORDER MARKOV CHANNELS

Let indicate the vector and letindicate a possibly non-stationary

process whose generic realization is the sequence .When the process is continuous, we indicate with themultivariate probability density function, while when theprocess is discrete we indicate with the multivariatemass probability and with the number of elements in .When the process is stationary, is used in place of .In what follows, we consider a first-order, finite-state Markov

process , which is characterized as

(1)

Further, we assume that the channel is memoryless given theinput and the state, that the input process and the state processare independent, and that the input process is a sequence of in-dependent random variables, leading to

(2)

where is the channel continuous output process and is thechannel discrete input process. By the assumption of indepen-dence between the state and the input, the channel probabilitylaw is factored as

0733-8724/$31.00 © 2012 IEEE

BARLETTA et al.: DISCRETE-TIME WIENER’S PHASE NOISE CHANNEL 1481

Using (1) and (2) in the right side of the above equation, onewrites the channel probability law as

(3)

Equation (3) allows for trellis-based recursive computation ofand according to [12]. Specifically, the wanted

probabilities are computed by a forward recursion based ona trellis with states. In the computation of , theproduct

is the branch metric between states and , whereis the channel transition probability and

is the state transition probability. In trellis-basedcomputation of

(4)

is the branch metric between states and resulting froma set of parallel transitions.Besides and , also , which

will be used in the upper bound presented in the following,admits trellis-based computation. Trellis-based computationof , which at least to our best knowledge, seemsto be new, is hereafter detailed. By chain rule, the wantedprobability is

Elaborating upon (3) one writes

therefore

By Bayes rule, the marginal conditional probability is

(5)

where thanks to (3), one can prove that

and is a normalization factor that guaranteesthat

By the Markovian assumption (3), the term ap-pearing in (5) can be computed as

(6)

Substituting

in the numerator of (6), one realizes that can becomputed recursively in the backward direction, that is for

with the initial condition

where is the a priori probability of the initialstate of the backward recursion, often ,meaning that the initial state is uniformly distributed. The term

in the denominator of (6) is a normalizationfactor such that

To sum up, the computation of the wanted probability is basedon a recursion whose building blocks are the same used forcomputation of and , that are the conditionalprobabilities and , with

III. CHANNEL AND SOURCE MODEL

The th output of the channel is

(7)

where is the imaginary unit, is the complex channel outputprocess, is the complex AWGN process, is the channelcomplex input modulation process, and is the phase noiseprocess which is assumed to be independent of and . Weassume that the input process is made of independent randomvariables, and we allow it to be cyclostationary, in the sensethat pilot symbols can be periodically inserted between payloadsymbols. Although not strictly necessary, it is hereafter assumedthat the squared modulus of the pilot symbols is equal to thevariance of the random payload symbols, and, for concreteness,the signal power is normalized to unity and the variance of thecomplex AWGN is denoted . Process is modelled as

(8)

where is a known parameter, is uniformly distributedin , and is white Gaussian noise with zero mean andunit variance. Process is a special case of first-order Markov

1482 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 10, MAY 15, 2012

process, where the conditional density of given dependsonly on the phase difference modulo :

(9)

where is a Gaussian probability density functionwith mean , variance , and argument . The phase evolutiongiven in (8) occurs when the power spectral density of thecontinuous-time complex exponential , whose samples atsymbol frequency generate the sequence , is the Lorentzianfunction

where is the symbol repetition interval and is the frequency.The parameter can be expressed as

where is the full-width half-maximum bandwidth ofthe spectral line.The channel and source model given above are such that

can be factored as in (3), with the continuous stateprocess in place of the discrete state process , and with

where indicates a complex Gaussian probability densityfunction.

IV. THE AUXILIARY PROBABILITY METHOD

The channel and source model given in Section III sharemany similarities with the one given in Section II, but while inSection II the state space is discrete and finite, in the channelmodel of Section III the state space is the phase space, henceit is continuous. As a consequence, the approach of Section IIcannot be applied to the actual phase noise channel. To over-come this difficulty we suggest of renouncing to the exactinformation rate, while computing instead upper and lowerbounds. The bounds that we are going to present are basedon the Kullback-Leibler (KL) divergence. The normalizedKL divergence between the multivariate probability densityfunctions and is

(10)

where denotes the expectation over and is thenormalization factor. From the normalized KL divergence onehas the following upper bound on the differential entropy rateof process :

(11)

Let us regard the auxiliarymultivariate probability density func-tion as an approximation to . In this perspective,the KL divergence is a measure of the quality of the fit be-tween and , and the upper bound is equal to the ac-

Fig. 1. Block diagram representing the actual phase noise and the auxiliaryphase noise that are used in Monte Carlo evaluation of the information rate.

tual differential entropy rate when the fit is ideal, that is when.

The auxiliary channel probability density function

(12)

adopted in this paper relies upon time-invariant scalar uniformquantization of the phase space in the range . Note that,to match the channel law (12) to the finite-state channel lawof (3), only quantization of the phase process is required,while it is not necessary to quantize channel’s output , whichis continuous both in (3) and in (12). In what follows,indicates the centroid of the generic bin,

where

is the bin width, indicates the quantization region whosecentroid is , and, when appears in the argument of an auxiliaryprobability, it is understood that it means .In the section devoted to the simulation results we will show

estimates of the upper bound (11) obtained by the Monte Carlomethod. Specifically, a realization is generated accordingto the actual phase noise model (8) and used in (7) to producechannel’s output. At the same time, regarding the centroids asthe states of the trellis, the discrete sequence obtained bythe scheme of Fig. 1 allows for computing the auxiliary proba-bilities by the trellis-based methods of Section II. Specifically,calculation of entropy rate is performed according to the generalmethod of [12], where probabilities are evaluated in the statesvisited by the discretized phase realization generated by Fig. 1.When pilot symbols are inserted, the probability of constella-tion symbols appearing in (4) is

where is the value taken by the pilot symbols at time instant, being understood that the above equation holds only for thosewhere pilot symbols are inserted.The auxiliary state transition probability density function

hereafter adopted is

(13)

BARLETTA et al.: DISCRETE-TIME WIENER’S PHASE NOISE CHANNEL 1483

where the auxiliary state transition mass probability is

(14)

Note that, although the auxiliary state transition probability(13) is formally a probability density function, actually it isbased on the discrete random variable , thus allowing for afinite-state Markov model. Since depends onlyon the difference , it takes its values in a set ofnumbers. Moreover, the numbers can be pre-computed by(14) before running the Monte Carlo simulation.1 The auxil-iary channel transition probability density function hereafteradopted is

(15)

A comment is in order before concluding this section. Thequantizer in Fig. 1 induces a higher-order Markov model uponobserving the state sequence. In other words, the actual jointprobability cannot be factored as in (3).We hastento point out that this is not a problem, since we are definingthe auxiliary probability, and the accuracy of the fit betweenthe actual channel and the auxiliary channel given by equationsfrom (12) to (15) will affect the tightness of the bound, not itsvalidity.

V. LOWER BOUND

Using in (10) the joint probability density functionsand

where

(16)

and following the steps of [12], one gets the following lowerbound below the mutual information between and :

VI. UPPER BOUND

Since the actual conditional differential entropy of the outputgiven the input is not known, the upper bound of [12] cannot be

1The choice of the auxiliary state transition probability made in [15] is dif-ferent from ours.We have verified that the lower bound obtained with our choiceis tighter than the lower bound obtained with the choice of [15], especially withsmall number of bins. Since the auxiliary state transition probability is pre-com-puted, this benefit comes free.

invoked. This obstacle can be circumvented as follows. Extractfrom

to write

where by the independence between and has beensubstituted in place of . The upper bound

that we propose is

(17)

The four differential entropies appearing in the right side of (17)are computed as follows.The differential entropy is that of the complex

AWGN, which is known to be

For one has

which in view of (9), is independent of , and can benumerically evaluated in a straightforward manner. Due to thefolding represented by the sum in (9), is upper boundedby the entropy of a Gaussian random variable with variance, that is

When , the effect of folding is negligible in the range, therefore is well approximated by the right side

of the above inequality.The upper bound

is obtained by using the auxiliary probability density functionof (16) in the method of [12].

The upper bound

is obtained as

where

and the auxiliary mass probability is recursivelycomputed as detailed in Section II.

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Fig. 2. Information rate versus SNR for 256-QAM. Asterisks: pure AWGN,no pilots. For the remaining curves, pilot symbols with ,are considered. Circles: pure AWGN with pilot symbols. The two families ofcurves are obtained with and . Solid line, from the upperto the lower curve and for the two values of : upper bound with 64 and 128states. Dashed line, from the lower to the upper curve and for the two values of: lower bound with 64 and 128 states.

VII. SIMULATION RESULTS

In what follows we assume that a block of consecutivepilot symbols is periodically inserted after payloadsymbols, and that pilot symbols have constant squared ampli-tude equal to the average signal power. When pilot symbols areperiodically inserted at the transmit side, the source is cyclo-stationary with period equal to the period of insertion of pilotsymbols. However, the combined source and channel model re-mains ergodic, therefore Monte Carlo methods can be used toevaluate the wanted upper and lower bounds. In the practice, anestimate of the actual information is obtained by truncating thelimit for appearing in (11) to some large value of ,which in the cyclostationary case, should be an integer multipleof the period for the estimate to be unbiased.Fig. 2 reports the upper and lower bounds for 256 quadrature

amplitude modulation (QAM) with and ,for and . Values of between 0.066 and0.125 were found adequate to fit the experimental results of [8],therefore one can say that, in the practice, can beused to model a channel affected by strong phase noise, while

can be used to model a channel affected by huge phasenoise. A constellation with high spectral efficiency is intention-ally used to characterize the proposed bounds in a wide rangeof signal-to-noise ratio (SNR). From the figure one realizes that,compared to the pure AWGN channel, the impact ofon the information rate transferred through the channel is smallin the low SNR region, while it becomes larger in the high SNRregion. However, the impact of on the informationrate is not dramatic also at high SNR. For this reason, values oflower than 0.125 are not considered. The figure shows that the

upper and the lower bound converge to each other, hence to theactual information rate, as the number of trellis’ states increases,thus substantiating our claim of having computed the actual in-formation rate. Convergence of the upper and lower bound is an

Fig. 3. Information rate versus SNR for 4-QAM. The three solid lines report, that is the information rate with no pilot symbols. From the upper

to the lower solid line: . The two dash-dottedlines report the information rate of the AWGN channel with pilot symbols. Theupper dash-dotted line is for , the lower dash-dotted line is for .The dashed lines report the actual information rate for andwith pilot symbols arranged in blocks. From the upper to the lower curve of thefamily, blocks of pilots. The dot-marked solidlines report the lower bound of conjecture (18), for and .

expected result, since, in the limit of quantization with vanish-ingly small bin width, the fit between the auxiliary probabilityand the actual probability becomes ideal, thus the upper bound(11) tends to the actual entropy. The number of bins that arenecessary to obtain a close fit between the bounds and the ac-tual information rate depends on the SNR. An intuitive explana-tion of this fact is as follows. In its essence, capacity evaluationis based on an estimate of the phase noise. The estimate is af-fected by the quantization noise and by the additive noise. For afixed number of bins, the accuracy of both bounds is thereforedominated by quantization of the phase space at high SNR, thusmaking the gap between the upper bound and the lower boundmore and more pronounced as SNR increases. The figure showsthat 128 bins are adequate to obtain the information rate withinfew tenths of bit per dimension pair at dB. Also,from the figure one appreciates that the lower bound convergesfaster to the actual information rate. In what follows, only theactual information rate will be reported, having obtained it witha number of bins large enough to make the upper and the lowerbound virtually equal to the actual information rate.The aim of Fig. 3 is to stress the balance between the symbol

rate sacrificed for inserting pilot symbols and the gain due tothe knowledge of the channel brought by pilot symbols. Onecan conjecture that what is sacrificed is only partially balancedby what is gained, that is

(18)

where is the information rate with no pilot symbols.The actual information rate is equal to the upper boundwhen the knowledge of channel state brought by pilot symbolsfully balances the symbol rate sacrificed for pilot symbols, while

BARLETTA et al.: DISCRETE-TIME WIENER’S PHASE NOISE CHANNEL 1485

Fig. 4. Difference between the actual information rate and the conjecturedlower bound of (18) versus SNR for 4-QAM, and .Solid line: . Dashed line: . Dotted line: .

it is equal to the lower bound when pilot symbols do not con-tribute at all to the knowledge of channel state. In Fig. 3, theupper bound is the solid line, the lower bound of con-jecture (18) is the dot-marked solid line, and the actual informa-tion rate is the dashed line. Noticeably, the actual informationrate for with and is much closer tothe upper bound than to the lower bound of (18). This meansthat the information on channel state brought by pilot symbolsbalances almost entirely the symbol rate sacrificed for pilot sym-bols. Fig. 3 also shows that the benefit due to the knowledgeof channel’s state diminishes when pilot symbols are arrangedin blocks. An intuitive explanation is gained by looking to thehigh SNR region, where one pilot symbol allows to measure theexact channel state at the specific time instant where that pilotsymbol is transmitted. What happens is that the benefit due tothe knowledge of channel’s state in one specific time instant ex-tends only to the neighbouring time instants. In time instants thatare sufficiently far from the time instant where channel’s stateis known, channel’s state becomes virtually independent of theknown state, thus the benefit due to the pilot symbol vanishes.Therefore, repeated observations of channel’s state in consec-utive time instants are useless. From the results of Fig. 3 andfrom this reasoning we conclude that the best thing to do is totransmit isolated pilot symbols, thus confirming what observedin [17].Fig. 4 reports the difference between the actual information

rate and the lower bound of (18), that is

(19)

The results of Fig. 4 show that the actual information rate be-comes closer to the lower bound of (18) when the phase noisediminishes. At , which can be still considered a strongphase noise in many cases of practical interest, the difference(19) is in fact negligible, independently of . At high SNRs, thepositive contribution of pilot symbols with vanishes.This can be explained by looking to Fig. 3, where it appears that

the information rate of the phase noise channel withand no pilot symbols is virtually equal to the information rateof the AWGN channel at high SNRs. Therefore, at high SNRs,pilot symbols can only decrease the information rate.

VIII. CONCLUSION

Upper and lower bounds to the information rate transferredthrough the AWGN channel affected by multiplicativeWiener’sphase noise are presented in the paper, the upper bound beingnovel. The upper and the lower bounds, which are based ontrellis representation of channel’s memory, are close to eachother also with trellises of affordable complexity, thus substanti-ating our claim of having computed the actual information rate.The proposed method allows to compute the information ratealso when pilot symbols are inserted. The analysis has shownthat, in the case of huge phase noise, the symbol rate sacrificedfor pilot symbols can be almost entirely balanced by the knowl-edge of channel’s state brought by pilot symbols. Also, the re-sults shown in the paper demonstrate that isolated pilot symbolsshould be preferred to blocks of pilot symbols.

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Luca Barletta received the M.S. degree (cum laude) in telecommunicationsengineering and the Ph.D. degree in information engineering from Politecnicodi Milano, Milano, Italy, in 2007 and 2010, respectively.He currently is a temporary researcher at Politecnico di Milano and he is an

intern at Bell Labs, Alcatel-Lucent, Holmdel, NJ. His research interests includedigital communications, with emphasis on channel coding, synchronization, andanalysis of collision resolution protocols.

Maurizio Magarini (M’09) was born in Milano, Italy, in 1969. He receivedthe M.S. and Ph.D. degrees in electronic engineering from the Politecnico diMilano, Milano, Italy, in 1994 and 1999, respectively.In 1994, he was granted the TELECOM Italia scholarship award for his

Master thesis. From 1999 to 2001 he was a Research Associate in the Di-partimento di Elettronica e Informazione at the Politecnico di Milano wheresince 2001, he has been an Assistant Professor. From August 2008 to January2009 he spent a sabbatical leave at Bell Labs, Alcatel-Lucent, Holmdel, NJ.He has authored and coauthored more than 40 journal and conference papers.His research interests are in the broad area of communication theory. Topicsinclude synchronization, channel estimation, equalization, coding and reducedcomplexity detection schemes for multi-antenna systems.

Arnaldo Spalvieri received his the degree in electronic engineering from theUniversity of Ancona, Italy, in 1985.From 1986 to 1992 he was in the modem laboratory of Telettra (now Al-

catel-Lucent Italia), working at the research and development of mo-demodula-tion systems for high capacity digital radio systems. From 1992 to 1998 he waswith the Dipartimento di Elettronica e Informazione at Politecnico di Milanoas an Assistant Professor. In 1998 he was appointed Associate Professor at Po-litecnico di Milano, where he now holds the course digital communication. Hisresearch is focused on channel coding, channel equalization, synchronization.In 2006 he co-founded the company Binary Core, a spinoff of Politecnico diMilano active in the area of design on FPGA of mo-demodulation systems forpoint-to-point terrestrial radio and for digital video broadcasting.