A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery

doi:10.1364/OE.20.025471

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A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery

Luca Barletta, Maurizio Magarini, and Arnaldo Spalvieri »View Author Affiliations

Optics Express, Vol. 20, Issue 23, pp. 25471-25477 (2012)
http://dx.doi.org/10.1364/OE.20.025471

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Abstract

A new lower bound below the information rate transferred through the Additive White Gaussian Noise (AWGN) channel affected by discrete-time multiplicative Wiener’s phase noise is proposed in the paper. The proposed lower bound is based on the Kalman approach to data-aided carrier phase recovery, and is less computationally demanding than known methods based on phase quantization and trellis representation of phase’s memory. Simulation results show that the lower bound is close to the actual channel capacity, especially at low-to-intermediate signal-to-noise ratio.

1. Introduction

Multiplicative phase noise is a major source of impairment in coherent optics. It is recognized from a long time that laser’s phase noise is a Wiener process [1

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988). [CrossRef]

], and the Wiener model has been recently proposed in [2

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,” Opt. Express 23, 22455–22461 (2011). [CrossRef]

] also for the phase noise accumulated during nonlinear propagation, at least for the cases studied in that paper. Several methods have been proposed in the literature to combat the detrimental effects of Wiener phase noise. Among these methods we cite iterative demodulation and decoding techniques of [3

M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 2, 87–95 (2000). [CrossRef]

–5

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. on Commun. 11, 2125–2133 (2007). [CrossRef]

], staged demodulation and decoding [6

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (2012). [CrossRef]

], and the insertion of pilot symbols [7

A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun. 7, 1966–1974 (2011). [CrossRef]

, 8

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. Tech. Lett. 9, 739–741 (2012). [CrossRef]

The capacity of the Additive White Gaussian Noise (AWGN) channel affected by multiplicative memoryless phase noise, that is phase noise with white power spectral density, is studied in [9

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011). [CrossRef]

, 10

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

]. However, Wiener phase noise is a process with memory. Methods for computing the information rate transferred through finite-state channels with memory, as, for instance, the Monte Carlo method proposed in [11

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006). [CrossRef]

], have already demonstrated their usefulness in the context of optical transmission, having been adopted in [12

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005). [CrossRef]

] for calculation of the information rate transferred over optical channels with memory. The Monte Carlo approach has been recently proposed also in [13

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008). [CrossRef]

–16

L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” IEEE J. Lightw. Technol. 30, 1480–1486 (2012). [CrossRef]

] for computing the constrained capacity (i.e. the capacity with a fixed source) of the AWGN plus Wiener phase noise channel, which is a channel with memory and with continuous state.

The new result presented in this paper is a lower bound below the constrained capacity of the AWGN Wiener’s phase noise channel. The bounding technique proposed here relies upon Monte Carlo simulation of a demodulator aided by the past data, where the transfer function of the causal filter used for phase estimation is worked out by the Kalman approach. Compared to methods based on phase quantization and trellis representation of phase’s memory, such as [16

], the method proposed here is less computationally demanding, since a filter is used in place of a trellis. At low-to-intermediate signal-to-noise ratio, it can be seen that the proposed lower bound is so close to the actual channel capacity that we can claim that the new lower bound gives virtually the actual channel capacity.

The outline of the paper is as follows. In Section II the channel model and the source model are introduced. Section III reports the general method behind the bound, while in Section IV the specific bound is presented. Section V gives simulation results, while in Section VI conclusions are drawn.

2. Channel and source model

Let

u_{i}^{k}

indicate the vector (u_i, u_i₊₁, ⋯, u_k) with

u_{i}^{k} \in 𝒰_{i}^{k}

, and let U indicate a stationary and ergodic process, U = (U₀, U₁, ⋯), whose generic realization is the sequence (u₀, u₁, ⋯). When

𝒰_{i}^{k}

is a continuous set,

p (u_{i}^{k})

is used to indicate the multivariate probability density function, while when

𝒰_{i}^{k}

is a discrete set

p (u_{i}^{k})

indicates the multivariate mass probability and |𝒰_i| denotes the number of elements in 𝒰_i.

The k-th output of the channel is

y_{k} = x_{k} e^{j φ_{k}} + w_{k}, k = 1, 2, \dots,

(1)

where j is the imaginary unit, Y is the complex channel output process, X is the channel complex input modulation process, and Φ is the phase noise process which is assumed to be independent of X and W. For concreteness, we assume that the input process X is made of i.i.d. random variables with zero mean and unit variance. Process W is a complex AWGN process with zero mean and variance SNR⁻¹. Process Φ is modelled as a Wiener process:

ϕ_{k} = ϕ_{k - 1} + γ v_{i}, k = 1, 2, \dots

(2)

where the frequency noise V is a i.i.d. sequence of Gaussian random variables with zero mean and unit variance, γ is a scalar, and ϕ₀ is uniformly distributed in [−π, π). The phase evolution given in (2) occurs when the power spectral density of the continuous-time complex exponential e^jϕ(t), whose samples taken at symbol frequency generate the sequence e^jϕ_k, is the Lorentzian function

ℒ (f) = \frac{4 γ^{2} T}{γ^{4} + 16 π^{2} f^{2} T^{2}},

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

γ^{2} = 2 π B_{F W H M} T,

where B_FWHM is the full-width half-maximum bandwidth of the spectral line.

Equations (1) and (2) can be cast in the general framework of state-space approach to modelling dynamic systems. Specifically, Eq. (2) is that of a first-order model where ϕ_k is the state at time k. In this context, the AWGN W is called measurement noise, while the frequency noise V is called process noise. The interested reader is referred to [17

D. Simon, Optimal State Estimation (Wiley, 2006). [CrossRef]

] for a comprehensive book on the subject.

3. The auxiliary probability method

The bound that we are going to present is based on the Kullback-Leibler (KL) divergence. The normalized KL divergence between the multivariate mass probability functions

p (u_{1}^{n})

and

q (u_{1}^{n})

lim_{n \to \infty} \frac{1}{n} E_{p} {{log}_{2} \frac{p (u_{1}^{n})}{q (u_{1}^{n})}} \geq 0,

(3)

where E_p{·} denotes the expectation over

p (u_{0}^{n})

and (n)⁻¹ is the normalization factor. From the normalized KL divergence one has the following upper bound on the entropy rate of process U:

\bar{H} (U) = lim_{n \to \infty} \frac{1}{n} E_{p} {{log}_{2} (\frac{1}{q (u_{1}^{n})})} \geq H (U) .

(4)

Let us regard the auxiliary multivariate probability density function

q (u_{1}^{n})

as an approximation to

p (u_{1}^{n})

. In this perspective, the KL divergence is a measure of the quality of the fit between

p (u_{1}^{n})

and

q (u_{1}^{n})

, and the upper bound is equal to the actual entropy rate when the fit is ideal, that is when

q (u_{1}^{n}) = p (u_{1}^{n})

Assuming that U is ergodic, one can invoke the Shannon-McMillan-Breiman theorem and the chain rule, thus writing for the expectation appearing in (4)

\bar{H} (U) = lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} {log}_{2} (\frac{1}{q (u_{k} | u_{0}^{k - 1})}),

(5)

where

u_{1}^{n}

is generated according to the actual multivariate probability density function

p (u_{1}^{n})

, and the initial condition u₀ is given. The bound can be extended to the conditional entropy rate in a straightforward manner.

4. Lower bound

Assume discrete input alphabet. The lower bound below the information rate is

H (X) - \bar{H} (X | Y) \leq I (X; Y),

where the familiar notation is used for the conditional entropy rate and for the mutual information rate. The upper bound

\bar{H} (X | Y) \geq H (X | Y)

is obtained from (5) as

\bar{H} (X | Y) = lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} {log}_{2} (\frac{1}{q (x_{k} | x_{1}^{k - 1}, y_{1}^{n})}) .

(6)

To obtain a tight bound, one has to work out an auxiliary mass probability that closely approximates the actual mass probability. Aiming to simple yet effective methods, we observe that the portion of joint sequence

(x_{1}^{k - 1}, y_{1}^{k - 1})

can strongly contribute to a data-aided approximation to the wanted probability, while the portion

y_{k}^{n}

, that is of non-data-aided type since the input data

x_{k}^{n}

are not conditioning the observation, gives a weaker contribution. The stronger part of the non-data-aided contribution comes from sample y_k, therefore the auxiliary probability that is hereafter considered is based only on

(x_{1}^{k - 1}, y_{1}^{k})

, while

y_{k + 1}^{n}

is ignored. The auxiliary conditional probability proposed here is

\begin{array}{l} q (x_{k} | y_{1}^{k}, x_{1}^{k - 1}) & = & \int_{- π}^{π} q (ϕ_{k}, x_{k} | y_{1}^{k}, x_{1}^{k - 1}) d ϕ_{k} \\ = & \int_{- π}^{π} p (x_{k} | y_{k}, ϕ_{k}) q (ϕ_{k} | y_{1}^{k}, x_{1}^{k - 1}) d ϕ_{k} \\ \propto & \int_{- π}^{π} q (ϕ_{k} | y_{1}^{k}, x_{1}^{k - 1}) p (y_{k} | x_{k}, ϕ_{k}) p (x_{k}) d ϕ_{k}, \end{array}

(7)

where we have exploited the fact that X_k is conditionally independent of

(Y_{1}^{k - 1}, X_{1}^{k - 1})

given (Y_k, Φ_k). The only distribution in (7) that cannot be computed directly from the channel model is

\begin{array}{l} q (ϕ_{k} | y_{1}^{k}, x_{1}^{k - 1}) \propto q (ϕ_{k} | y_{1}^{k - 1}, x_{1}^{k - 1}) p (y_{k} | ϕ_{k}) \\ = q (ϕ_{k} | y_{1}^{k - 1}, x_{1}^{k - 1}) \cdot \sum_{x_{k} \in 𝒳} p (y_{k} | ϕ_{k}, x_{k}) p (x_{k}), \end{array}

where the auxiliary probability

q (ϕ_{k} | y_{1}^{k - 1}, x_{1}^{k - 1})

adopted here is

q (ϕ_{k} | y_{1}^{k - 1}, x_{1}^{k - 1}) = g ({\hat{ϕ}}_{k}, σ_{k}^{2}; ϕ_{k}),

where g(η, σ²; u) is a Gaussian distribution with mean η and variance σ² over the space spanned by u. The estimate ϕ̂_k appearing in the above equation is worked out by a predictive Kalman filter, that is

{\hat{ϕ}}_{k} = E {ϕ_{k} | y_{1}^{k - 1}, x_{1}^{k - 1}},

and the variance of the estimate is

σ_{k}^{2} = E {{({\hat{ϕ}}_{k} - ϕ_{k})}^{2} | y_{1}^{k - 1}, x_{1}^{k - 1}} .

The mean and the variance can be computed in a recursive manner thanks to the update equations of the Kalman filter. Since the channel model (1) is a nonlinear function of the state, the Kalman filter has to be linearized around the current state estimate [17, Ch. 13.1]. The error that drives the Kalman filter is the one produced by the phase detector of classical data-aided carrier recovery, that is

e_{k} = ℑ {y_{k} x_{k}^{★} e^{- j {\hat{ϕ}}_{k}}},

(8)

where ℑ{·} and the superscript ^★ denote the imaginary part and the complex conjugation, respectively. Assuming that the error is small, it can be linearized as

e_{k} \approx ϕ_{k} - {\hat{ϕ}}_{k} + z_{k},

where z_k is assumed to be white Gaussian noise with zero mean and time-varying variance

σ_{z, k}^{2} = {(2 {| x_{k} |}^{2} SNR)}^{- 1}

. The estimate ϕ̂_k and error’s variance

σ_{k}^{2}

are computed for k ≥ 0 according to the iterative equations

{\hat{ϕ}}_{k + 1} = {\hat{ϕ}}_{k} + K_{k} \cdot e_{k},

(9)

σ_{k + 1}^{2} = \frac{σ_{z, k}^{2} \cdot σ_{k}^{2}}{σ_{z, k}^{2} + σ_{k}^{2}} + γ^{2},

(10)

where K_k is the Kalman gain at time k:

K_{k} = \frac{σ_{k}^{2}}{σ_{k}^{2} + σ_{z, k}^{2}} .

Initial values can be set as ϕ̂₀ = 0 and

σ_{0}^{2} = \infty

. Equations (8) and (9) can be regarded as those of a first-order phase-locked loop (PLL) with time-varying gain, where (8) is the phase detector and (9) describes the loop filter. The analogy between PLL with time-varying gain and Kalman filters has been suggested in [18

A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999). [CrossRef]

5. Simulation results

In this section, the new lower bound is compared to the actual channel capacity worked out by the computationally demanding trellis-based method of [16

]. Specifically, the actual channel capacity has been obtained using a large number of states in the lower bound and in the upper bound of [16

]. The number of states is so large that the upper bound and the lower bound become virtually undistinguishable, leading to the actual channel capacity. Note that, in the trellis based method, N² sums, where N is the number of states in the trellis, should be performed at each step in the trellis, while each step of the Kalman filter requires only the evaluation of (9) and (10). Fig. 1 reports the results obtained with 4-QAM and with two values of γ. Specifically, γ = 0.125 is the largest value obtained in the experimental results of [2

] and can be regarded as a case of strong phase noise in cases of practical interest. Although less realistic, also the huge γ = 0.5 is studied, to show the limits of the proposed method. For γ = 0.125 the lower bound is virtually undistinguishable from the actual capacity in a wide range of information rate, say, below 1.5 bit/2D. This range is the one spanned by codes with rate lower than 0.75, that are codes of large practical interest. For information rate greater than 1.5 bit/2D the bound looses accuracy. This is because we have not exploited the conditioning on

y_{k + 1}^{n}

, which, at high SNR, could potentially bring a non-negligible contribution to the accuracy of the fit between the auxiliary probability and the actual probability. Although being fairly close to the actual capacity, the lower bound is less accurate for γ = 0.5, because, with a so large value of γ, frequent cycle slips affect the performance of Kalman carrier recovery. A similar analysis holds for the results obtained with 16-QAM and reported in Fig. 2. Also in this case, the lower bound virtually gives the actual channel capacity for γ = 0.125 and coding rate below 0.75.

Fig. 1 Actual channel capacity and lower bound for 4-QAM and two values of γ. Capacity of the pure AWGN channel is also reported.

Fig. 2 Actual channel capacity and lower bound for 16-QAM and two values of γ. Capacity of the pure AWGN channel is also reported.

6. Conclusion

In the paper, a new lower bound below the information rate transferred through the Wiener phase noise channel has been presented. The results, compared to the actual channel capacity obtained with the computationally demanding method of [16

], show that the bound is accurate in many cases of practical interest. Before concluding the paper, a remark is in order about phase noise of order higher than one, which is out of the scope of the present paper and that will be subject of future research. Extension of the Kalman-based approach to phase noise with memory of order higher than one, as the second-order case studied in [19

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs , 12, 596–600 (2008).

], is feasible by extending the state space of the Kalman filter. In contrast, quantizing a multidimensional state space according to the approach of [14

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Tech. Lett. 21, 1582–1584 (2011). [CrossRef]

,16

] would lead to an exponential increase of the number of states of the trellis, making computation unfeasible.

References and links

1.	G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988). [CrossRef]
2.	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,” Opt. Express 23, 22455–22461 (2011). [CrossRef]
3.	M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 2, 87–95 (2000). [CrossRef]
4.	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). [CrossRef]
5.	A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. on Commun. 11, 2125–2133 (2007). [CrossRef]
6.	L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (2012). [CrossRef]
7.	A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun. 7, 1966–1974 (2011). [CrossRef]
8.	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. Tech. Lett. 9, 739–741 (2012). [CrossRef]
9.	B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011). [CrossRef]
10.	P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).
11.	D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006). [CrossRef]
12.	I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005). [CrossRef]
13.	J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008). [CrossRef]
14.	L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Tech. Lett. 21, 1582–1584 (2011). [CrossRef]
15.	A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. on Commun. 12, 3223–3228 (2011). [CrossRef]
16.	L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” IEEE J. Lightw. Technol. 30, 1480–1486 (2012). [CrossRef]
17.	D. Simon, Optimal State Estimation (Wiley, 2006). [CrossRef]
18.	A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999). [CrossRef]
19.	A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs , 12, 596–600 (2008).

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4080) Fiber optics and optical communications : Modulation
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 23, 2012
Revised Manuscript: October 5, 2012
Manuscript Accepted: October 8, 2012
Published: October 25, 2012

Citation
Luca Barletta, Maurizio Magarini, and Arnaldo Spalvieri, "A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery," Opt. Express 20, 25471-25477 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25471

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References

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988). [CrossRef]
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,” Opt. Express 23, 22455–22461 (2011). [CrossRef]
M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 2, 87–95 (2000). [CrossRef]
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). [CrossRef]
A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. on Commun. 11, 2125–2133 (2007). [CrossRef]
L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (2012). [CrossRef]
A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun. 7, 1966–1974 (2011). [CrossRef]
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. Tech. Lett. 9, 739–741 (2012). [CrossRef]
B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011). [CrossRef]
P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).
D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006). [CrossRef]
I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005). [CrossRef]
J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008). [CrossRef]
L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Tech. Lett. 21, 1582–1584 (2011). [CrossRef]
A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. on Commun. 12, 3223–3228 (2011). [CrossRef]
L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” IEEE J. Lightw. Technol. 30, 1480–1486 (2012). [CrossRef]
D. Simon, Optimal State Estimation (Wiley, 2006). [CrossRef]
A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999). [CrossRef]
A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, 12, 596–600 (2008).

Arnold, D. M.
Barbieri, A.
Barletta, L.
Belzer, B. J.
Bertolini, M.
Caire, G.
Colavolpe, G.
Dauwels, J.
Djordjevic, I. B.
Essiambre, R.-J.
Fischer, T. R.
Foschini, G. J.
Gabitov, I.
Galan, S.
Gavioli, G.
Goebel, B.
Hanik, N.
Hou, P.
Ivkovic, M.
Kavcic, A.
Kramer, G.
Loeliger, H.-A.
Magarini, M.
Patapoutian, A.
Peleg, M.
Pepe, M.
Pfau, T.
Shamai (Shitz), S.
Simon, D.
Spalvieri, A.
Vacondio, F.
Vannucci, G.
Vasic, B.
Vontobel, P. O.
Winzer, P. J.
Zeng, W.

IEEE Commun. Lett.

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

IEEE J. Lightw. Technol.

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A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery

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Abstract

1. Introduction

2. Channel and source model

3. The auxiliary probability method

4. Lower bound

5. Simulation results

6. Conclusion

References and links

References

IEEE Commun. Lett.

IEEE J. Lightw. Technol.

IEEE J. Sel. Areas Commun.

IEEE Photon. Tech. Lett.

IEEE Trans. Circuits Syst. II, Exp. Briefs

IEEE Trans. Commun.

IEEE Trans. Inf. Theory

IEEE Trans. Inform. Theory

IEEE Trans. on Commun.

Opt. Express

Proc. IEE Commun.

Other

Cited By

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