Abstract

The performances of various blind timing phase estimators (TPE) for digital coherent receiver are analyzed. The equivalence among four TPE algorithms is analytically presented, showing that two TPE algorithms applying squaring pre-filters are in fact identical. Three TPE algorithms applicable to Nyquist signals are proposed based on the equivalence analysis. In addition, the impact of receiver bandwidths, spectrum weighting bandwidths and signal timing phases on TPE performance are investigated. The definition of sampling diversity and the analysis of sampling diversity gain for four pulse shapes are presented. The effect of sampling diversity is observed and verified via both simulations and experiments.

© 2014 Optical Society of America

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  1. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
    [CrossRef] [PubMed]
  2. M. Kuschnerov, F. Hauske, K. Piyawanno, B. Spinnler, M. Alfiad, A. Napoli, B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27(16), 3614–3622 (2009).
    [CrossRef]
  3. K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express 19(6), 5611–5619 (2011).
    [CrossRef] [PubMed]
  4. H. Sun and K. Wu, “Clock recovery and jitter sources in coherent transmission systems,” in Proceedings OFC/NFOEC, Los Angeles, CA (2012), Paper OTh4C.1.
    [CrossRef]
  5. T. Tanimura, T. Hoshida, S. Oda, H. Nakashima, M. Yuki, Z. Tao, L. Liu, and J. C. Rasmussen, “Digital clock recovery algorithm for optical coherent receivers operating independent of laser frequency offset,” in Proceeding ECOC, Brussels, Belgium (2008), Paper Mo.3.D.2.
    [CrossRef]
  6. N. Stojanovic, F. N. Hauske, C. Xie, and M. Chen, “Clock recovery in coherent optical receivers,” in Proceeding Photonic Networks 12.ITG Symposium, Leipzig, Germany (2011), Paper 9.
  7. M. Oerder, H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988).
    [CrossRef]
  8. F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
    [CrossRef]
  9. D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978).
    [CrossRef]
  10. S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002).
    [CrossRef]
  11. K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004).
    [CrossRef]
  12. Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003).
    [CrossRef]
  13. E. Panayirci, E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996).
    [CrossRef]
  14. T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991).
    [CrossRef]
  15. M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997).
    [CrossRef]
  16. H. Sun, K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5.
    [CrossRef]
  17. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice Hall, 1999).
  18. M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7.
    [CrossRef]
  19. N. Stojanovic, N. G. Gonzalez, C. Xie, Y. Zhao, B. Mao, J. Qi, and L. M. Binh, “Timing recovery in nyquist coherent optical systems,” in Proceeding 20th Telecommunications Forum (TELFOR), Serbia, Belgrade (2012), pp. 895–898.
    [CrossRef]
  20. J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)
  21. N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994).
    [CrossRef]
  22. L. Franks, J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974).
    [CrossRef]
  23. N. A. D’Andrea, M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
    [CrossRef]
  24. R. Kudo, T. Kobayashi, K. Ishihara, Y. Takatori, A. Sano, Y. Miyamoto, “Coherent optical single carrier transmission using overlap frequency domain equalization for long-haul optical systems,” J. Lightwave Technol. 27(16), 3721–3728 (2009).
    [CrossRef]

2011 (1)

2009 (2)

2008 (1)

2004 (1)

K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004).
[CrossRef]

2003 (1)

Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003).
[CrossRef]

2002 (1)

S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002).
[CrossRef]

1997 (1)

M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997).
[CrossRef]

1996 (1)

E. Panayirci, E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996).
[CrossRef]

1994 (1)

N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994).
[CrossRef]

1993 (1)

N. A. D’Andrea, M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

1991 (1)

T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991).
[CrossRef]

1988 (1)

M. Oerder, H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988).
[CrossRef]

1986 (1)

F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[CrossRef]

1978 (1)

D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978).
[CrossRef]

1974 (1)

L. Franks, J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974).
[CrossRef]

Alfiad, M.

Bar-Ness, E. K.

E. Panayirci, E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996).
[CrossRef]

Bubrouski, J.

L. Franks, J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974).
[CrossRef]

Ciblat, P.

Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003).
[CrossRef]

D’Andrea, A. N.

M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997).
[CrossRef]

D’Andrea, N. A.

N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994).
[CrossRef]

N. A. D’Andrea, M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

Fang, T. T.

T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991).
[CrossRef]

Franks, L.

L. Franks, J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974).
[CrossRef]

Gardner, F.

F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[CrossRef]

Godard, D.

D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978).
[CrossRef]

Hauske, F.

Ishihara, K.

Kikuchi, K.

Kobayashi, T.

Kudo, R.

Kuschnerov, M.

Lankl, B.

Lee, S.

S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002).
[CrossRef]

Luise, M.

N. A. D’Andrea, M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

Mengali, U.

M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997).
[CrossRef]

N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994).
[CrossRef]

Meyr, H.

M. Oerder, H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988).
[CrossRef]

Miyamoto, Y.

Morelli, M.

M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997).
[CrossRef]

Napoli, A.

Oerder, M.

M. Oerder, H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988).
[CrossRef]

Panayirci, E.

E. Panayirci, E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996).
[CrossRef]

Piyawanno, K.

Reggiannini, R.

N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994).
[CrossRef]

Sano, A.

Savory, S. J.

Serpedin, E.

K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004).
[CrossRef]

Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003).
[CrossRef]

Shi, K.

K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004).
[CrossRef]

Spinnler, B.

Sun, H.

H. Sun, K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5.
[CrossRef]

Takatori, Y.

Wang, Y.

K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004).
[CrossRef]

Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003).
[CrossRef]

Wu, K.

H. Sun, K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5.
[CrossRef]

IEEE Commun. Lett. (2)

S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002).
[CrossRef]

M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997).
[CrossRef]

IEEE Trans. Commun. (10)

N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994).
[CrossRef]

L. Franks, J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974).
[CrossRef]

N. A. D’Andrea, M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004).
[CrossRef]

Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003).
[CrossRef]

E. Panayirci, E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996).
[CrossRef]

T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991).
[CrossRef]

M. Oerder, H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988).
[CrossRef]

F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[CrossRef]

D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (2)

Other (8)

H. Sun and K. Wu, “Clock recovery and jitter sources in coherent transmission systems,” in Proceedings OFC/NFOEC, Los Angeles, CA (2012), Paper OTh4C.1.
[CrossRef]

T. Tanimura, T. Hoshida, S. Oda, H. Nakashima, M. Yuki, Z. Tao, L. Liu, and J. C. Rasmussen, “Digital clock recovery algorithm for optical coherent receivers operating independent of laser frequency offset,” in Proceeding ECOC, Brussels, Belgium (2008), Paper Mo.3.D.2.
[CrossRef]

N. Stojanovic, F. N. Hauske, C. Xie, and M. Chen, “Clock recovery in coherent optical receivers,” in Proceeding Photonic Networks 12.ITG Symposium, Leipzig, Germany (2011), Paper 9.

H. Sun, K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5.
[CrossRef]

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7.
[CrossRef]

N. Stojanovic, N. G. Gonzalez, C. Xie, Y. Zhao, B. Mao, J. Qi, and L. M. Binh, “Timing recovery in nyquist coherent optical systems,” in Proceeding 20th Telecommunications Forum (TELFOR), Serbia, Belgrade (2012), pp. 895–898.
[CrossRef]

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)

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Figures (14)

Fig. 1
Fig. 1

Block diagram of the (a) feedback and (b) feedforward timing phase recovery (TPR) for an asynchronous coherent optical receiver.

Fig. 2
Fig. 2

(a) Illustration of the different weighting of spectra auto-correlations for Gardner’s and Godard’s original TPE. (b) S-curves of Godard’s, the SLN, Gardner’s and Lee’s TPE.

Fig. 3
Fig. 3

S-curves and zero-crossing phase histograms of (a) Gardner’s, (b) Godard’s, (c) SLN and (d) Lee’s TPE.

Fig. 4
Fig. 4

Normalized jitter variance vs OSNR (in 0.1nm) for NRZ-QPSK, 16-QAM and 64QAM signals.

Fig. 5
Fig. 5

Normalized timing jitter variance as function of (a) receiver bandwidth and (b) weighting window bandwidth at various OSNR levels.

Fig. 6
Fig. 6

Evolution of timing phase estimation for (a) feedback timing recovery using Gardner’s TPE and (b) feedforward timing recovery using Godard’s TPE.

Fig. 7
Fig. 7

OSNR penalty with respect to sampling phase for 28GBaud/s (a) QPSK, (b) 16QAM and (c) 64QAM system.

Fig. 8
Fig. 8

(a) The NRZ waveform with 0.75 × baudrate bandwidth (solid line) and infinite bandwidth (dash line) containing 3 symbols (−1,1,1) and (b) the calculated signal power, noise power and SNR with normalized scale. Samples at sampling phase of 0.25 (square) and the recovered signal using linear interpolation (circle) are marked.

Fig. 9
Fig. 9

OSNR penalty with respect to sampling phase for 28GBaud/s (a) NRZ, (b) RZ 33%, (c) RZ 50%, and (d) RZ 67% QPSK system.

Fig. 10
Fig. 10

(a) The RZ33% waveform with 0.75 × baudrate bandwidth (solid line) and infinite bandwidth (dash line) containing 3 symbols (−1,1,1) and (b) the calculated signal power, noise power and SNR with normalized scale. Samples at sampling phase of 0.25 (square) and the recovered signal using linear interpolation (circle) are marked.

Fig. 11
Fig. 11

(a) Normalized timing jitter variance and (b) error vector magnitude (EVM) with respect to fiber launch power over 13*80 km spans of SSMF transmission.

Fig. 12
Fig. 12

Block diagram of the experiment setup.

Fig. 13
Fig. 13

(a) S-curves of (i) Gardner’s, (ii) Godard’s, (iii) SLN and (iv) Lee’s TEDs; (b) normalized jitter variance with respect to OSNR at 0.1nm resolution

Fig. 14
Fig. 14

(a) BER versus OSNR and (b) Required OSNR with respect to sampling phase for the four TPE algorithms.

Tables (1)

Tables Icon

Table 1 Algorithms of timing phase estimator

Equations (24)

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n=0 N1 x n y n = 1 N k=0 N1 X k Y k
DFT[ x 2n ]= X k + X k+N/2 2 , n,k=0,1,...,N/21
DFT[ x 2n+1 ]= e j 2πk /N X k X k+N/2 2 , n,k=0,1,...,N/21
ε ^ SLN = 1 2π arg{ n=0 2N1 | x n | 2 e jπn/2 } = 1 2π arg{ n=0 2N1 x n ( x n e j2πn( N/2 )/( 2N ) ) * } substituting Eq.( 1 ) = 1 2π arg{ k=0 2N1 X k ( X kN/2 ) * } ,
X k|k=0,1,...,N/21 ={ X k|k=0,1,...,N/21 , k=0,1,...,N/21 0, k=N/2,N/2+1,...,3N/21 X k|k=N/2,N/2+1,...,N1 , k=3N/2,3N/2+1,...2N1.
ε ^ SLN = 1 2π arg{ k=0 N/2 1 X k ( X k+N/2 ) * }= ε ^ Godard .
ε ^ Gardner =Re{ n=0 N/21 ( x 2n x 2n+2 ) x 2n+1 } =Re{ n=0 N/21 ( x 2n x 2n+1 x 2n+2 x 2n+1 ) } substituting Eq.( 1 ),( 2 ) and ( 3 ) = 2 N k=0 N/21 { sin( 2π N k )Im( X k X k+N/2 * ) }
ε ^ Lee = 1 2π arg{ n=0 N1 | x n | 2 e jnπ + n=0 N1 Re{ x n x n+1 } e j( n0.5 )π } = 1 2π arg{ n=0 N1 | x n | 2 ( 1 ) n +j n=0 N1 Re{ x n x n+1 } ( 1 ) n } = 1 2π arg{ n=0 N/21 [ | x 2n | 2 | x 2n+1 | 2 ] +j n=0 N/21 [ Re{ x 2n x 2n+1 }Re{ x 2n+1 x 2n+2 } ] } = 1 2π arg{ ε ^ D +j ε ^ Gardner },
ε ^ D = n=0 N/21 [ | x 2n | 2 | x 2n+1 | 2 ] = n=0 N/21 | x 2n | 2 n=0 N/21 | x 2n+1 | 2 substituting Eq.( 1 ),( 2 ) and ( 3 ) = 2 N Re{ k=0 N/21 X k X k+N/2 }.
ε ^ Lee = 1 2π arg{ k=0 N/21 [ Re( X k X k+N/2 )+jsin( 2π N k )Im( X k X k+N/2 * ) ] }.
ε ^ P = n=0 N/21 p 2n ( p 2n+1 p 2n1 )
ε ^ P = 2 N k=0 N/21 { sin( 2π N k )Im( P k P k+N/2 * ) } ,
ε ^ 4PPD =Im{ k=0 N/21 P k P k+N/2 * }
ε ^ Godard,P = 1 2π arg{ k=0 N/21 P k P k+N/2 * }
ε ^ SLN,P = 1 2π arg{ n=0 2N1 | x n | 4 e jnπ/2 }
ε ^ Lee,P = 1 2π arg{ n=0 N1 | x n | 4 e jnπ + n=0 N1 | x n | 2 | x n+1 | 2 e j( n0.5 )π }
SNR= E [ ( 1μ ) s 1 +μ s 2 ] 2 ( 1μ ) 2 n 1 2 + μ 2 n 2 2 .
SNR= E [ ( 1μ ) s 1 +μ s 2 ] 2 ( 12μ+2 μ 2 ) n 2 .
s NRZ ( t )= P m C m rect(tmT)
rect(t)={ 1, 0, 0.5Tt0.5T otherwise
SN R NRZ = P n 2 1 ( 12| ε | ) 2 + ( 2| ε | ) 2
SN R RZ33 = P n 2 [ ( 12| ε | )cos( π 2 sin( π| ε | ) )+2| ε |cos( π 2 sin( π( 0.5| ε | ) ) ) ] 2 ( 12| ε | ) 2 + ( 2| ε | ) 2
SN R RZ50 = P n 2 [ ( 12| ε | )sin[ π 4 cos( 2π| ε | )+ π 4 ]+2| ε |sin[ π 4 cos( 2π( 0.5| ε | ) )+ π 4 ] ] 2 ( 12| ε | ) 2 + ( 2| ε | ) 2
SN R RZ67 == P n 2 [ ( 12| ε | )sin( π 2 cos( π| ε | ) )+2| ε |sin( π 2 cos( π( | ε |+0.5 ) ) ) ] 2 ( 12| ε | ) 2 + ( 2| ε | ) 2

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