Abstract

In this paper, we propose a simple frequency offset estimation (FOE) algorithm for high-speed coherent optical orthogonal frequency-division multiplexing (CO-OFDM) systems, which avoids the exhaustive search computations on the integer part of FOE by using a merit function. In time domain, the FOE algorithm can estimate all the frequency offset in the range of [-5GHz, +5GHz], for satisfying the practical application requirement, based on only one redesigned training symbol. In addition, we provide the theoretical analysis about the quantitative influences of the residual frequency offset (RFO) on CO-OFDM systems. In order to reduce the final estimation error, a new zero-overhead residual frequency offset estimation (RFOE) algorithm is also proposed in this paper. Finally, the feasibility and effectiveness of the proposed FOE and RFOE algorithms are demonstrated in a 464 Gbit/s polarization-division multiplexed (PDM) 16-ary quadrature amplitude modulation (QAM) CO-OFDM system by the simulation, and numerical results validate the proposed algorithms.

© 2012 OSA

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References

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  1. R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
    [CrossRef]
  2. X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000km of ultra-large-area fiber and five 80 Ghz-grid ROADMs,” J. Lightwave Technol. 29(4), 483–490 (2011).
  3. Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s per channel coherent optical OFDM transmission with subwavelength bandwidth access,” in in National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPC1.
  4. E. Ip, A. P. Lau, D. J. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008).
    [CrossRef] [PubMed]
  5. W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).
  6. Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009).
    [CrossRef]
  7. X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
    [CrossRef]
  8. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
    [CrossRef]
  9. Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
    [CrossRef]
  10. S.-H. Fan, J. Yu, D. Qian, and G.-K. Chang, “A fast and efficient frequency offset correction technique for coherent optical orthogonal frequency division multiplexing,” J. Lightwave Technol. 29(13), 1997–2004 (2011).
    [CrossRef]
  11. Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
    [CrossRef]
  12. S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
    [CrossRef]
  13. F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in 34th European Conference on Optical Communication, 2008. ECOC 2008 (2008), Paper Mo.4.D.4.
  14. Optical Internetworking Forum, “Integrable Tunable Laser Assembly MSA,” OIF-ITLA-MSA-01.1 (Nov. 22, 2005).
  15. W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier, 2011).
  16. B. Aklar, Digital Communications: Fundamentals & Applications (Prentice-Hall, 2001).
  17. X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
    [CrossRef]
  18. S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
    [CrossRef]
  19. K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
    [CrossRef]

2011

2010

R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[CrossRef]

2009

2008

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[CrossRef]

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

E. Ip, A. P. Lau, D. J. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008).
[CrossRef] [PubMed]

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[CrossRef]

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[CrossRef]

2007

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[CrossRef]

2005

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[CrossRef]

1997

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[CrossRef]

Barros, D. J.

Chandrasekhar, S.

Chang, G.-K.

Cox, D. C.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[CrossRef]

Fan, S.-H.

Gao, J.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

Gnauck, A. H.

Ip, E.

Ishihara, K.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Jansen, S. L.

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[CrossRef]

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[CrossRef]

Jiang, W.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

Kahn, J. M.

Kobayashi, T.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Kudo, R.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Lau, A. P.

Liu, X.

Liu, Y.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

Liu, Y. N.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[CrossRef]

Long, K. P.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[CrossRef]

Ma, Y.

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[CrossRef]

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

Masuda, H.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Matsui, M.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Miyamoto, Y.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Mizoguchi, M.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Morita, I.

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[CrossRef]

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[CrossRef]

Peckham, D. W.

Qian, D.

Sano, A.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Schenk, T. C.

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[CrossRef]

Schenk, T. C. W.

Schmidl, T. M.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[CrossRef]

Shieh, W.

Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009).
[CrossRef]

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[CrossRef]

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[CrossRef]

Takatori, Y.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Takeda, N.

Tanaka, H.

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[CrossRef]

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[CrossRef]

Tang, Y.

Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009).
[CrossRef]

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[CrossRef]

Tkach, R.W.

R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[CrossRef]

Winzer, P. J.

Yamada, E.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

Yang, Q.

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

Yi, X. W.

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[CrossRef]

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[CrossRef]

Yu, J.

Zhang, Z.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[CrossRef]

Zhao, M.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[CrossRef]

Zhou, H.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

Zhu, B.

Bell Labs Tech. J.

R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[CrossRef]

Electron. Lett.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[CrossRef]

IEEE Photon. Technol. Lett.

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[CrossRef]

IEEE Trans. Broadcast

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[CrossRef]

IEEE Trans. Commun.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[CrossRef]

IEEE Trans. Wirel. Comm.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. Networking

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[CrossRef]

Opt. Express

Other

Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s per channel coherent optical OFDM transmission with subwavelength bandwidth access,” in in National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPC1.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in 34th European Conference on Optical Communication, 2008. ECOC 2008 (2008), Paper Mo.4.D.4.

Optical Internetworking Forum, “Integrable Tunable Laser Assembly MSA,” OIF-ITLA-MSA-01.1 (Nov. 22, 2005).

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier, 2011).

B. Aklar, Digital Communications: Fundamentals & Applications (Prentice-Hall, 2001).

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Figures (10)

Fig. 1
Fig. 1

(a) The conventional and (b) the redesigned training symbol structures for OFDM symbol synchronization and FOE. CP: cyclic prefix.

Fig. 2
Fig. 2

(a) The ICI influence induced by the RFO, (b) the variation of signal amplitude as a function of the RFO.

Fig. 3
Fig. 3

Schematic illustration of RFO estimation algorithm.

Fig. 4
Fig. 4

(a) Simulation setup of the 464Gbit/s PDM-OFDM-16QAM system, the schematics of the DSP in the transmitter (b) and receiver (c), (d) OFDM frame structure.

Fig. 5
Fig. 5

Timing metric of Schmidl algorithm by using the redesigned training symbol with a frequency offset of 5GHz.

Fig. 6
Fig. 6

Absolute values of normalized FOE error under [-4.9GHz, +4.9GHz] frequency offset with OSNR of 24dB.

Fig. 7
Fig. 7

The output constellations of one polarization after phase recovery at OSNR = 28dB, (a) without RFO, (b) e = 0.01, and (c) e = 0.015.

Fig. 8
Fig. 8

BER performance of 464Gbps PDM-OFDM-16QAM signal as a function of the normalized RFO for OSNR = 24dB, 26dB and 28dB.

Fig. 9
Fig. 9

The absolute normalized estimation error (mean value) versus algorithm parameter m for various e at OSNR = 26dB.

Fig. 10
Fig. 10

(a)The absolute normalized estimation error (mean value), and (b) BER performance as a function of the OSNR.

Equations (27)

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

r ( t ) = exp ( j 2 π Δ f t + j ϕ ( t ) ) ( s ( t ) h ( t ) + w ( t ) )
r ( n ) = exp ( j 2 π Δ f n T s + j ϕ ( n ) ) ( s ( n ) h ( n ) + w ( n ) )
M ( d ) = | P ( d ) | 2 R ( d ) M ( d ) [ 0 , 1 ]
P ( d ) = m = 0 N / 2 1 [ r * ( d + m ) r ( d + m + N / 2 ) ]
R ( d ) = ( m = 0 N / 2 1 | r ( d + m ) | 2 ) ( m = 0 N / 2 1 | r ( d + m + N / 2 ) | 2 )
Δ f F F O = arg ( p ( d ^ ) ) π f c s
| 2 π k ( ± 5 G H z ) T s | π
h = 2 c e i l ( log 2 N max ( k ) )
Δ f e s t = arg ( n = 0 N N h 1 ( r * ( n ) r ( n + N h ) ) ) h f s 2 π N
Δ f e s t = integer ( Δ f e s t f s c ) f c s + Δ f F F O
Δ f e s t = integer ( Δ f e s t Δ f e s t f s c ) f s c + Δ f e s t
r ( n ) = r ( n ) exp ( j 2 π Δ f e s t n T s ) = exp ( j 2 π e f s c n T s + j ϕ ( n ) ) ( s ( n ) h ( n ) + w ( n ) )
r i n = p i exp ( j 2 π e n / N ) ( 1 N k = 0 N 1 C i k H i k exp ( j 2 π k n / N ) + w i n ) p i = exp ( j 2 π e ( i 1 ) ( N + N c p ) / N )
C i k = C i k H i k η 0 p i + I i k + W i k I i k = n = 1 , n k N C i n H i n η n k η m = 1 N sin ( π ( m + e ) ) sin ( π ( m + e ) / N ) exp ( j π ( m + e ) ( 1 1 N ) )
σ I C I 2 = σ C 2 n = 1 , n k N | η | 2 n k
S N R ' = σ c 2 σ I C I 2 + σ w 2 = ( n = 1 , n k N | η | 2 n k + S N R 1 ) 1
B E R = 3 8 e r f c ( S N R 10 )
H ^ i k = C i k C i k = H i k η 0 p i + I i k + W i k C i k
H ^ k = 1 L i = 1 L H ^ i k = 1 L i = 1 L H k η 0 p i + I i k + W i k C i k H k η 0 1 L i = 1 L p i
H ^ k = H k η 0 1 L i = 1 L exp ( j 2 π e ( i 1 ) ( N + N c p ) / N ) = H k η 0 sin ( π e ( N + N c p ) L / N ) L sin ( π e ( N + N c p ) / N ) exp ( j π e ( L 1 ) ( N + N c p ) / N )
C ^ i k = H ^ * k C i k = H k * η 0 * L sin ( π e ( N + N c p ) / N ) sin ( π e ( N + N c p ) L / N ) exp ( j π e ( L 1 ) ( N + N c p ) / N ) × ( C i k H k η 0 p i + W i k ) = α ξ i C i k + W i k
α = L sin ( π e ( N + N c p ) / N ) sin ( π e ( N + N c p ) L / N ) , ξ i = exp ( j 2 π e ( N + N c p ) ( i L ) / N )
α = L sin ( d π e ( N + N c p ) / N ) sin ( d π e ( N + N c p ) L / N ) ξ i = exp ( j 2 π d e ( N + N c p ) ( i L ) / N ) , d = { 1 , without PDM 2 , with PDM
H ^ ( i + m ) k H ^ i k * = exp ( ( j 2 π d e ( m + i 1 ) ( N + N c p ) / N ) ( j 2 π d e ( i 1 ) ( N + N c p ) / N ) + θ n ) = exp ( j 2 π d e m ( N + N c p ) / N + θ n )
P E = k = 1 N H ^ i k H ^ ( i + m ) k * exp ( j 2 π d e m ( N + N c p ) / N )
e ^ f s c = arg ( P E ) 2 π d m ( N + N c p ) T s
- N 2 d m ( N + N c p ) e ^ N 2 d m ( N + N c p )

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