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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23728–23734
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Staged demodulation and decoding

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


Optics Express, Vol. 20, Issue 21, pp. 23728-23734 (2012)
http://dx.doi.org/10.1364/OE.20.023728


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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 the interleaving of two or more codes. The interleaved codes are decoded in sequence, using past decisions to help future decoding. The paper proposes a method based on this principle, and shows its benefits through numerical results obtained by computer simulation. Analysis of the channel capacity given by the proposed method is also worked out in the paper.

© 2012 OSA

1. Introduction

Coherent demodulation of advanced coded modulation formats is a hot topic in new generation optical transmission systems. Besides the common additive white Gaussian noise (AWGN), the performance 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

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

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

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. The impact of phase noise on the performance of coherent optical transmission systems is discussed in [3

3. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

, 4

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.

]. Basically, phase noise afflicts the accuracy of carrier recovery, which becomes a critical task of the receiver. In the presence of strong phase noise, carrier recovery is so bad that cycle slips do appear [4

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.

], leading to a lack of coherency of the demodulator that definitely compromises system’s performance. Recent papers [5

5. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol. 7, 901–914 (2009). [CrossRef]

7

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). [CrossRef]

] address the problem of coherent demodulation in the presence of Wiener phase noise. Also, Wiener phase noise is adopted in [8

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). [CrossRef]

10

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). [CrossRef]

] to assess the performance of iterative demodulation and decoding, while the capacity of the channel affected by Wiener phase noise is derived in [11

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). [CrossRef]

]. Often, one is lead to introduce pilot symbols to aid carrier recovery in the presence of strong phase noise [8

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). [CrossRef]

, 12

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

, 13

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). [CrossRef]

], and the capacity of Wiener’s phase noise channel with pilot symbols is studied in [14

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

]. However, pilot symbols sacrifice spectral efficiency. As an alternative to pilot symbols, one can resort to differential demodulation methods as those proposed in [9

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). [CrossRef]

, 15

15. 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]

]. The trellis-based method of [15

15. 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]

] in conjunction with iterative differential demodulation and decoding offers an excellent performance at the expense of large complexity of signal processing, while the less demanding method based on Tikhonov parametrization [9

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). [CrossRef]

] still offers a good performance.

A staged demodulation and decoding method is proposed in this paper. The method relies upon interleaving of pilot symbols and coded symbols from two (or more) codes. Channel symbols 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 decoding stage. For instance, with two channel block codes one can transmit through the 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,c2,6N5,c2,6N4,c2,6N3,c1,N,c2,6N2,c2,6N1,c2,6N).
(1)
In the above sequence, represented in Fig. 1, p represents one pilot symbol and ci,j is the j-th symbol of the i-th level code 𝒞i. In Eq. (1), one frame is inserted between parentheses, the entire sequence consists of N frames, and the length of code 𝒞2 is 6N while the length of code 𝒞1 is N. For the sake of correctness, only two-level constructions will be studied, and the extension to multilevel constructions becomes straightforward.

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

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 obtained with the proposed method in contrast with adversary methods are shown. Finally, in section V conclusions are drawn.

2. 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 sample of AWGN, and θk is the k-th sample of phase noise. The phase noise is hereafter modeled as a discrete-time Wiener process
θk=[θk1+γνk]mod2π,k=1,2,,
(2)
where γ > 0 is a known parameter, θ0 is uniformly distributed in [0,2π), and vk is the k-th sample 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 e (t), whose samples at symbol frequency generate the sequence ek, is the Lorentzian function
(f)=4γ2Tγ4+16π2f2T2,
where T is the symbol repetition interval and f is the frequency. The parameter γ2 can be expressed as
γ2=2πBFWHMT,
where BFWHM is the full-width half-maximum bandwidth of the spectral line.

The information rate expressed in bits per channel symbol of a two-level construction with one pilot symbol per frame is
R=R1(M11)+R2M1(M21)M1M2,
where M1 1 is the number of symbols of code 𝒞1 in one frame, R1 is the information rate of code 𝒞1, M2 1 is the number of consecutive symbols of code 𝒞2, R2 is the information rate of 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

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). [CrossRef]

], can be used after the first demodulation based on pilot symbols only. After having decoded the first-level code, the transmitted code word is regenerated and its symbols are used as pilot symbols in the second demodulation 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 processes of symbols of the first-level and of the second-level, respectively. Similarly, the received sequence is divided in three parts called Yp, Y1, Y2, where Yp corresponds to the time instants where pilot symbols Xp are transmitted, while Yi corresponds to the time instants where symbols of level i are transmitted.

Let x1n and y1n 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)=limn1nI(y1n;x1n).
(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 computed by dividing the information between vectors by the number of uses of the composite channel, e.g.
I(Y;Xp)=limn1nI(y1n;xp,1n),
where xp,1n 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 side of the above equation is
I(Y;X1|Xp)=I(Y1,Yp;X1|Xp)+I(Y2;X1|Xp,Y1,Yp),
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 equation is the contribution coming from the blind processing of Y2 at the first stage. In our proposal we suggest to renounce to this contribution. It can be computed from Eq. (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 Eq. (5) can be computed as in [14

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

]. Specifically, for I(Y;X) one uses pilot symbols inserted with period M1 · M2, for I(Y;X2|X1,Xp) one uses pilot symbols inserted with period M2, while for I(Y1,Yp;X1|Xp) one uses pilot symbols inserted with period M1 and Wiener phase noise with step
γM2.

4. Numerical results

Fig. 2 4-QAM, γ = 0.125, LDPC codes of length 64800 from the DVB-S2 standard. Performance 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 decoding of the first-level code.

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, operating in the waterfall region. For these systems the performance of the two-level scheme is dominated by the performance of the worst of the two component codes, hence, in a good design, 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 staged decoding is compared to one-level schemes for 4-ary quadrature amplitude modulation (QAM) and 16-QAM, respectively.

Fig. 3 4-QAM, γ = 0.125. The two-stage scheme is based on one pilot symbol per frame and the three two-level codes of Fig. 2. CBC indicates one-level coding with the algorithm of [8], while soft differential decoding indicates the algorithm of [15]. The performance is evaluated at bit error rate of 10−5 after 24 iterations.
Fig. 4 16-QAM, γ = 0.125. The two-stage scheme is based on one pilot symbol per frame and M1 = 7, M2 = 9. CBC indicates one-level coding with the algorithm of [8]. The performance is evaluated at bit error rate of 10−5 after 24 iterations.

The number of iterations of the LDPC code for the one-level code is the same as the average number of iterations of the two LDPC codes of our method, where it turns out to be convenient to make more iterations at the first level and less iterations at the second level. In Fig. 3 the performance of [15

15. 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]

] is also reported, even if it should be said that [15

15. 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]

], where demodulation is based on a trellis, is much more demanding in terms of complexity compared to the adversaries. Moreover, [15

15. 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]

] is based on differential demodulation and it is suited only for phase shift keying (PSK)-type constellations therefore it cannot be applied to 16-QAM. From Figs. 3 and 4, the advantage of our method appears, especially with 16-QAM, where our method brings system performance closer to the capacity curve of about 0.5 dB compared to the adversary.

Fig. 5 4-QAM and 16-QAM, M1 = 7, M2 = 9, γ = 0.125. The figure shows the term I(Y2;X1|Xp,Y1,Yp) of Eq. (5).

5. Conclusions

In this work, a staged demodulation and decoding method is proposed for channels affected by strong phase noise. The results presented in the paper show that the proposed method outperforms adversary methods based on conventional one-level demodulation and decoding.

Before concluding the paper, it is worth adding that, although here only results for Wiener phase noise have been presented, the general principle of staged demodulation and decoding can be adopted also to combat phase noise of higher order [20

20. A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I 49, 1782–1791 (2002). [CrossRef]

], for instance the second-order phase noise studied in [21

21. A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II 55, 596–600 (2008). [CrossRef]

].

References and links

1.

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. 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.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

12.

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

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). [CrossRef]

14.

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

15.

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]

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). [CrossRef]

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). [CrossRef]

19.

S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decoding and channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).

20.

A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I 49, 1782–1791 (2002). [CrossRef]

21.

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II 55, 596–600 (2008). [CrossRef]

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 21, 2012
Revised Manuscript: September 10, 2012
Manuscript Accepted: September 10, 2012
Published: October 1, 2012

Citation
Luca Barletta, Maurizio Magarini, and Arnaldo Spalvieri, "Staged demodulation and decoding," Opt. Express 20, 23728-23734 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23728


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References

  1. G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 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. Express23, 22455–22461 (2011). [CrossRef]
  3. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28, 662–701 (2010). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  12. A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.7, 1966–1974 (2011). [CrossRef]
  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). [CrossRef]
  14. L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol.30, 1480–1486 (2012). [CrossRef]
  15. 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]
  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). [CrossRef]
  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. Theory2, 628–646 (2007). [CrossRef]
  19. S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decoding and channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).
  20. A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I49, 1782–1791 (2002). [CrossRef]
  21. A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II55, 596–600 (2008). [CrossRef]

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