Abstract

This paper extends our prior coherent MSDD Carrier Recovery system from QPSK to QAM operation and also characterizes for the first time the Carrier Frequency Offset (CFO) mitigation capabilities of the novel MSDD for QAM systems. We introduce and numerically investigate the performance of an improved MSDD carrier recovery system (differing from the one disclosed in our MSDD for QPSK prior paper), automatically adapting to the channel statistics for optimal phase-noise mitigation. Remarkably, we do not require a separate structure to estimate and mitigate CFO, but the same adaptive structure originally intended for phase noise mitigation is shown to also automatically provide frequency offset estimation and recovery functionality. The CFO capture range of our system is in principle infinite, whereas prior CFO mitigation systems have CFO capture ranges limited to a small a fraction of the baud-rate. When used for 16-QAM with coherent-grade lasers of 100 KHz linewidth, our MSDD system attains the best tradeoffs between performance and complexity, relative to other carrier recovery systems combining blind-phase-search with maximum likelihood detection. We also present additional MSDD phase-noise recovery system variants whereby substantially reduced complexity is traded off for slightly degraded performance. Our MSDD system is able to switch “on-the-fly” to various m-QAM constellation sizes, e.g. seamlessly transition between 16-QAM and QPSK, which may be useful for dynamically adaptive optical networks.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
    [CrossRef]
  2. E. Ip and J. M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J. Lightwave Technol. 23(12), 4110–4124 (2005).
    [CrossRef]
  3. R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
    [CrossRef]
  4. M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital processor,” in, ECOC’05 European Conf. of Optical Communication, Tu 4.2.6 (2005).
  5. E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
    [CrossRef]
  6. S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
    [CrossRef]
  7. T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
    [CrossRef]
  8. M. G. Taylor, “Algorithms for coherent detection,” in OFC/NFOEC’10, Conf. on Optical Fiber Communication, OThL4 (2010).
  9. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
    [CrossRef]
  10. C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express 18(12), 12088–12103 (2010).
    [CrossRef] [PubMed]
  11. S. Zhang, P. yuen Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010).
    [CrossRef]
  12. X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
    [CrossRef]
  13. N. Kikuchi, “Chromatic dispersion-tolerant higher-order multilevel transmission with optical delay detection,” in Signal Processing in Photonic Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper SPMA2.
  14. Y. Gao, A. P. T. Lau, S. Yan, and C. Lu, “Low-complexity and phase noise tolerant carrier phase estimation for dual-polarization 16-QAM systems,” Opt. Express 19(22), 21717–21729 (2011).
    [CrossRef] [PubMed]
  15. N. Sigron, I. Tselniker, and M. Nazarathy, “Carrier phase estimation for optically coherent QPSK based on Wiener-optimal and adaptive Multi-Symbol Delay Detection (MSDD),” Opt. Express 20(3), 1981–2003 (2012).
    [CrossRef] [PubMed]
  16. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
    [CrossRef]
  17. J. C. Tao, Z. Zhang, H. Isomura, A. Li, L. Hoshida, and T. Rasmussen, “Simple, robust, and wide-range frequency offset monitor for automatic frequency control in digital coherent receivers,” in ECOC’07 European Conference of Optical Communication (2007).
  18. L. Li, Z. Tao, S. Oda, T. Hoshida, and J. C. Rasmussen, “Wide-range, accurate and simple digital frequency offset compensator for optical coherent receivers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OWT4.
  19. M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in ECOC’09 European Conference of Optical Communication, P3.08 (2009).
  20. T. Nakagawa, K. Ishihara, T. Kobayashi, R. Kudo, and M. Matsui, “Wide-range and fast-tracking frequency offset estimator for optical coherent receivers,” in ECOC’10 European Conference of Optical Communication (2010).
  21. S. Zhang, L. Xu, J. Yu, M.-F. Huang, P. Y. Kam, C. Yu, and T. Wang, “Novel ultra wide-range frequency offset estimation for digital coherent optical receiver,” in OFC/NFOEC’10 - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, OWV3 (2010).
  22. T. Nakagawa, M. Matsui, T. Kobayashi, K. Ishihara, and R. Kudo, “Non-data-aided wide-range frequency offset estimator for QAM optical coherent receivers,” in OFC/NFOEC’11 - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, OMJ1 (2011).
  23. J. C. R. F. Diniz, J. C. M. Rosa, E. S. Ribeiro, V. B. da Silva, R. Silva, E. P. Herbster, A. F. Bordonalli, and A. C. de Oliveira, “Simple feed-forward wide-range frequency offset estimator for optical coherent receivers,” in ECOC’11 European Conference of Optical Communication (2011).
  24. N. Kikuchi and S. Sasaki, “Highly Sensitive Optical Multilevel Transmission of Arbitrary Quadrature-Amplitude Modulation (QAM) Signals With Direct Detection,” J. Lightwave Technol. 28(1), 123–130 (2010).
    [CrossRef]
  25. T. Adali and S. Haykin, Adaptive signal processing - next generation solutions (John Wiley, 2010).

2012 (1)

2011 (1)

2010 (4)

2009 (2)

2008 (1)

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

2007 (2)

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[CrossRef]

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
[CrossRef]

2005 (2)

E. Ip and J. M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J. Lightwave Technol. 23(12), 4110–4124 (2005).
[CrossRef]

R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
[CrossRef]

1983 (1)

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

Adamczyk, O.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Bhandare, S.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Chen, J.

Chen, Y.-K.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[CrossRef]

Gao, Y.

Hoffmann, S.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[CrossRef]

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Ip, E.

Kahn, J. M.

Kam, P. Y.

Kaneda, N.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[CrossRef]

Kikuchi, N.

Koc, U.-V.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[CrossRef]

Lau, A. P. T.

Leven, A.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[CrossRef]

Lu, C.

Nazarathy, M.

Noe, R.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[CrossRef]

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Noé, R.

R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
[CrossRef]

Peveling, R.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Pfau, T.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[CrossRef]

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Porrmann, M.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Sasaki, S.

Sigron, N.

Taylor, M. G.

Tselniker, I.

Viterbi, A.

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

Wordehoff, C.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Yan, S.

Yu, C.

yuen Kam, P.

Zhang, S.

Zhou, X.

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
[CrossRef]

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[CrossRef]

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

J. Lightwave Technol. (6)

Opt. Express (3)

Other (11)

M. G. Taylor, “Algorithms for coherent detection,” in OFC/NFOEC’10, Conf. on Optical Fiber Communication, OThL4 (2010).

M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital processor,” in, ECOC’05 European Conf. of Optical Communication, Tu 4.2.6 (2005).

J. C. Tao, Z. Zhang, H. Isomura, A. Li, L. Hoshida, and T. Rasmussen, “Simple, robust, and wide-range frequency offset monitor for automatic frequency control in digital coherent receivers,” in ECOC’07 European Conference of Optical Communication (2007).

L. Li, Z. Tao, S. Oda, T. Hoshida, and J. C. Rasmussen, “Wide-range, accurate and simple digital frequency offset compensator for optical coherent receivers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OWT4.

M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in ECOC’09 European Conference of Optical Communication, P3.08 (2009).

T. Nakagawa, K. Ishihara, T. Kobayashi, R. Kudo, and M. Matsui, “Wide-range and fast-tracking frequency offset estimator for optical coherent receivers,” in ECOC’10 European Conference of Optical Communication (2010).

S. Zhang, L. Xu, J. Yu, M.-F. Huang, P. Y. Kam, C. Yu, and T. Wang, “Novel ultra wide-range frequency offset estimation for digital coherent optical receiver,” in OFC/NFOEC’10 - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, OWV3 (2010).

T. Nakagawa, M. Matsui, T. Kobayashi, K. Ishihara, and R. Kudo, “Non-data-aided wide-range frequency offset estimator for QAM optical coherent receivers,” in OFC/NFOEC’11 - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, OMJ1 (2011).

J. C. R. F. Diniz, J. C. M. Rosa, E. S. Ribeiro, V. B. da Silva, R. Silva, E. P. Herbster, A. F. Bordonalli, and A. C. de Oliveira, “Simple feed-forward wide-range frequency offset estimator for optical coherent receivers,” in ECOC’11 European Conference of Optical Communication (2011).

T. Adali and S. Haykin, Adaptive signal processing - next generation solutions (John Wiley, 2010).

N. Kikuchi, “Chromatic dispersion-tolerant higher-order multilevel transmission with optical delay detection,” in Signal Processing in Photonic Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper SPMA2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(top): QAM Tx with modulus-preserving Differential Precoder (DP) as in [15] [25]. (bottom): QAM receiver block diagram detailing the novel “not U-U” MSDD carrier recovery efficient hardware structure with the Uop-1 normalization applied onto the improved reference but not to the partial references. This MSDD realization for QAM differs from the “U-notU” MSDD structure of Fig. 6 in [15] (which is not reproduced here) in the positioning of the Uop-1 normalization. This scheme may also seamlessly “switch-on-the-fly” from QAM to QPSK operation. The coefficients control block implementing an LMS adaptive algorithm is detailed in Fig. 2.

Fig. 2
Fig. 2

Coherent QAM/QPSK receiver with novel “notU-U” adaptive MSDD, including full detail on the novel LMS coefficients adaptation mechanism introduced in this paper, differing from that of Fig. 8 of [15] (providing 0.4dB performance advantage – see section 7). Notice the presence of two Uop modules, however the one within the slicer does not incur extra complexity, as it may be implemented as a look-up table. The delay line with multipliers at the top of the figure incurs negligible complexity, as multiplication with QAM constellation elements may be implemented as lookup tables.

Fig. 3
Fig. 3

The notU-U adaptive MSDD in the presence of Carrier Frequency Offset (CFO). (a): First stage of the derivation, propagating the CFO-affected received signal through the delay line. (b): Second stage of the derivation propagating the linear phase factor e jθk all the way to the demodulator, where it gets cancelled. The resulting system is equivalent to one with no CFO but with coefficients modified by a linear phase taper. A third stage of the derivation, not shown, makes the phase-tapered coefficients, c i e jθi , equal to the optimal coefficients, c i o (by setting c i = c i o e jθi , i.e. applying an inverse phase taper to the optimal coefficients in the absence of CFO), resulting in cancelling out the CFO.

Fig. 4
Fig. 4

Simulated complex taps in adaptive cancellation of CFO using the notU-U MSDD CR for 16-QAM at 14 GBaud for an L = 8 window with parallelization factor P = 8, 100 KHz linewidths for the Tx and LO lasers and OSNR = 18.6 dB. The frequency offset is CFO = (0.2154 + 14 n) GHz, where n is any integer. (a). The tap values ci in the complex (I-Q) plane. The numerals are the indexes i. It is apparent that as the tap index, i, increases, the magnitude | ci | diminishes while the phase of phase of ci follows a monotonically increasing progression, which is shown in the figure on the right to be linear. (b): The taps phases as a function of the tap index i. The phases are seen to be linear in i. In both plots ‘LMS taps’ (red) means the coefficients ci as determined at the end of the training sequence, whereas ‘OPT taps’ (blue) means optimal taps evaluated offline assuming a U-notU MSDD system (the optimal coefficients are not analytically known for the actual notU-U MSDD system used here, but the differences in coefficient values are minute). The match between the phases of the LMS and OPT taps (red and blue) is seen to be nearly perfect.

Fig. 5
Fig. 5

Low complexity notU-U MSDD for QAM reception with uniform taps and the common multiplier adaptively adjusted for AGC capability.

Fig. 6
Fig. 6

Ultra-low complexity notU-U MSDD for QAM reception with uniform taps and fixed common multiplier c0 = 1/L, which may be discarded, absorbed within the slicer scaling.Thus, the CP estimation is performed multiplier-free, and just a single full-fledged complex multiplier, the one used for demodulation, is required– all the other multipliers at the top, marked green, are trivial (multiplications by QAM constellation points are realized as lookup tables).

Fig. 7
Fig. 7

Comparison of BER vs. OSNR performance for three carrier recovery systems, our MSDD or BPS [9] and BPS + 2ML [12]. The last two conventional systems correspond to the lowest two curves. The poor performance top curve is a naïve delay detector (corresponding to an MSDD with an L = 1 window). From the top down we generally progress through increasingly larger window sizes, L, for the MSDD, selecting either notU-U vs. notU-U structures, and uniform fixed or AGC-ed taps, vs. adaptive taps. Key conclusions are that the notU-U variant generally performs better than the U-notU variant. Our best system is an adaptive notU-U MSDD with L = 8, performing only 0.3 dB worse than the BPS + 2ML, but being less complex. MSDD complexity may be significantly further reduced by using a uniform taps structure with fixed taps, falling just 0.15 dB behind our MSDD adaptive “leader”.

Fig. 8
Fig. 8

Comparison of BER vs. OSNR performance for three CR systems, adopting as CP E&R either our QAM MSDD or BPS [7] or BPS + 2ML [12] with the prior-art systems either preceded or not by the CFO E&R of [20], for either 16-QAM or 64-QAM; Tx and LO laser are fixed at LW = 100KHz in (b,c,d); the baud-rates are 14 GBaud and the parallelization factors are P = 8 in (a,c,d). (a): BER vs. normalized linewidth (LW/baudrate) at OSNR = 19 dB and P = 1: MSDD and BPS + 2ML closely track, BPS has best performance but has prohibitive complexity. Generally, due to the hardware parallelization, the effective LW of MSDD is enhanced by P, whereas the LW of the feedforward BPS( + 2ML) is not. For low-baudrate applications, such as PON, P is small, hence the parallelization penalty is negligible. (b): BER vs. OSNR for a Coherent PON 16-QAM system: At the PON lower baudrate, P is as low as 2. The BPS + 2ML and MSDD performances are then virtually identical (but the MSDD is less complex (Table 1)). (c,d): BER vs. OSNR for long-haul 16/64-QAM without and with CFO E&R: At the higher baudrate, P is increased to 8 and MSDD lags BPS + 2ML by 0.3 dB for 16-QAM and by 2 dB for 64-QAM. However, this is for an ideal situation with no CFO E&R; For 3 GHz CFO, when the BPS( + 2ML) CP E&R are preceded by a practical CFO E&R such as the coarse CFO mitigation stage of [19], due to enhanced phase noise and residual CFO from the CFO E&R stage, the performance of the CP E&R stage [7] [12] is severely degraded. In contrast, our MSDD is virtually unaffected by CFO in the range 0...8GHz (and even for arbitrarily large CFO values, tested but not shown). Thus, for 16-QAM, on an end-to-end system basis the MSDD CP + CFO E&R significantly outperforms BPS + 2ML in performance and has lower complexity.

Tables (1)

Tables Icon

Table 1 Complexity Comparisons. CM = complex multiplier; PR = Phase Rotator (may be simpler to realize than standard CM using the CORDIC algorithm); CMeff is an effective CM effectively representing both CP and real multipliers (RM): 3RM = 1CMeff; RAeff = real-adder (effective, corrected due to representing real mult. as CMeff); LUT = lookup table; SLICER = decision unit; CMP = comparators used to find min/max; MUX = multiplexer. Assumed parameters: MSDD: L = 8; BPS + 2ML [12]: N = 6; B = 5(14) for 16(64)QAM, 10bit numbers representation.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

r ˜ k = A ˜ k e j ϕ k LPN + n ˜ k = A ˜ k p ˜ k ; p ˜ k ( 1+ η ˜ k ) e j ϕ k LPN ; η ˜ k e j ϕ k LPN n ˜ k / A ˜ k
R ˜ k1 = c 1 r ˜ k1 R ˜ k1 (1) + c 2 s ˜ ^ k1 r ˜ k2 R ˜ k1 (2) + c 3 s ˜ ^ k1 s ˜ ^ k2 r ˜ k3 R ˜ k1 (3) + c 4 s ˜ ^ k1 s ˜ ^ k2 s ˜ ^ k3 r ˜ k4 R ˜ k1 (4) +... s ˜ k (i)U-notU = r ˜ k R ˜ k1 (i)* s ˜ k U-notU r ˜ k R ˜ k1 * = r ˜ k i=1 L c ¯ i R ˜ k1 (i)* = i=1 L c ¯ i r ˜ k R ˜ k1 (i)* = i=1 L c ¯ i s ˜ k (i)U-notU
R ˜ k1 = c 1 r ˜ k1 R ˜ k1 (1) + c 2 s ˜ ^ k1 r ˜ k2 R ˜ k1 (2) + c 3 s ˜ ^ k1 s ˜ ^ k2 r ˜ k3 R ˜ k1 (3) + c 4 s ˜ ^ k1 s ˜ ^ k2 s ˜ ^ k3 r ˜ k4 R ˜ k1 (4) +... R ˜ k1 =U{ R ˜ k1 } s ˜ k notU-U r ˜ k R ˜ * k1
R ˜ k1 R ˜ k1 /| R ˜ k1 |= R ˜ k1 / R ˜ k1 R ˜ ¯ k1 = R ˜ k1 / R ˜ ¯ k1 ; R ˜ k1 * = R ˜ ¯ k1 / R ˜ k1
s ˜ k notU-U r ˜ k R ˜ * k1 = r ˜ k R ˜ ¯ k1 / R ˜ k1
| ε ˜ k | 2 = ε ˜ k ε ˜ ¯ k =( s ˜ k s ˜ k notU-U ) ( s ˜ k s ˜ k notU-U ) * = | s ˜ k | 2 s ˜ k r ˜ ¯ k R ˜ k1 / R ˜ ¯ k1 s ˜ ¯ k r ˜ k R ˜ ¯ k1 / R ˜ k1 + | r ˜ k | 2
[ c | ε ˜ k | 2 ] i =2( ρ ˜ k1 (i)* / R ˜ k1 * )jIm{ s ˜ k s ˜ k * } =2j ( ρ ˜ k1 (i) / R ˜ k1 ) * | s ˜ k || s ˜ k notU-U |sin[ s ˜ k notU-U s ˜ k ]
[ c | ε ˜ k | 2 ] i =0 s ˜ k notU-U s ˜ k =0mod2π s ˜ k notU-U = s ˜ k mod2π
| ε ˜ k | 2 = | s ˜ k | 2 2Re{ s ˜ k notU-U s ˜ ¯ k }+ | s ˜ k notU-U | 2 = | s ˜ k | 2 2| s ˜ k || s ˜ k |cos( s ˜ k s ˜ k )+ | r ˜ k | 2
U i [k]= μ 2 [ c | ε ˜ k | 2 ] i =μ( R ˜ k1 (i)* / R ˜ k1 * )jIm{ s ˜ k s ˜ k * }
c i [k+1]= c i [k]+μ( R ˜ k1 (i)* / R ˜ k1 * )jIm{ s ˜ k s ˜ k * }.
θ=2πΔνTmod2π
R ˜ k1 CFO = i=1 L c i e jθ(ki) r ˜ ki s ˜ ^ ki+1 s ˜ ^ ki+2 .. s ˜ ^ k1 = i=1 L c i e jθi R ˜ k1 (i) e jθk = R ˜ k1 noCFOeff e jθk where R ˜ k1 noCFOeff i=1 L c i e jθi R ˜ k1 (i) = i=1 L c i eff R ˜ k1 (i) ; c i eff c i e jθi
s ˜ k CFO = r ˜ k U{ R ˜ k1 CFO } ¯ =( r ˜ k o e jθk ) U{ R ˜ k1 noCFOeff e jθk } ¯ =( r ˜ k o e jθk ) U{ R ˜ k1 noCFOeff } e jθk ¯ = r ˜ k o U{ R ˜ k1 noCFOeff } ¯ = r ˜ k o U{ i=1 L c i e jθi R ˜ k1 (i) } ¯ = r ˜ k o U{ i=1 L c i CFOeff R ˜ k1 (i) } ¯
s ˜ k CFO = r ˜ k o U{ i=1 L c i CFOeff R ˜ k1 (i) } ¯ = r ˜ k o U{ i=1 L c i o R ˜ k1 (i) } ¯
c i CFOeff = c i o e jθi C CFOeff ( e jω )= C o ( e j( ωθ ) )
| ε ˜ k | 2 = | s ˜ k | 2 s ˜ k r ˜ ¯ k R ˜ k1 / R ˜ ¯ k1 s ˜ ¯ k r ˜ k R ˜ ¯ k1 / R ˜ k1 + | r ˜ k | 2
[ c | ε ˜ k | 2 ] i =2 c ¯ i | ε ˜ k | 2 =2 c ¯ i ( s ˜ k r ˜ ¯ k R ˜ k1 / R ˜ ¯ k1 + s ˜ ¯ k r ˜ k R ˜ ¯ k1 / R ˜ k1 ) =2( c ¯ i R ˜ ¯ k1 ) R ˜ ¯ k1 ( s ˜ k r ˜ ¯ k R ˜ k1 / R ˜ ¯ k1 + s ˜ ¯ k r ˜ k R ˜ ¯ k1 / R ˜ k1 ) = ρ ˜ ¯ k1 (i) R ˜ ¯ k1 1 ( s ˜ k r ˜ ¯ k R ˜ k1 / R ˜ ¯ k1 s ˜ ¯ k r ˜ k / R ˜ ¯ k1 / R ˜ k1 ) =2j ρ ˜ ¯ k1 (i) R ˜ ¯ k1 1 Im{ s ˜ k r ˜ ¯ k R ˜ k1 / R ˜ ¯ k1 }=2j ρ ˜ ¯ k1 (i) R ˜ ¯ k1 1 Im{ s ˜ k r ˜ ¯ k e j R ˜ k1 } =2j ρ ˜ ¯ k1 (i) R ˜ ¯ k1 1 Im{ s ˜ ¯ k r ˜ k R ˜ k1 * }=2j ( ρ ˜ k1 (i)* / R ˜ k1 * ) Im{ s ˜ k * s ˜ k }
c ¯ i R ˜ ¯ k1 = c ¯ i j=1 L c ¯ j ρ ˜ ¯ k1 (j) = ρ ˜ ¯ k1 (i)
s ˜ k CFO = r ˜ k o ( i=1 L c i e jθi R ˜ k1 (i) ) * = i=1 L c ¯ i e jθi r ˜ k o R ˜ k1 (i)* = i=1 L c ¯ i e jθi s ˜ k (i)o
MS E CFO [ { c i };θ ]= | s ˜ k s ˜ k CFO | 2 = | s ˜ k i=1 L c ¯ i e jθi s ˜ k (i)o | 2
MS E noCFO [ { c i o } ]= | s ˜ k i=1 L c ¯ i s ˜ k (i)o | 2 =MS E min noCFO
MS E CFO [ { c i o e jθi };θ ]= | s ˜ k i=1 L c i o e jθi ¯ e jθi s ˜ k (i)o | 2 = | s ˜ k i=1 L c i o ¯ s ˜ k (i)o | 2

Metrics