Abstract

Efficient algorithms for timing, carrier frequency and phase recovery of Nyquist and OFDM signals are introduced and experimentally verified. The algorithms exploit the statistical properties of the received signals to efficiently derive the optimum sampling time, the carrier frequency offset, and the carrier phase. Among the proposed methods, the mean modulus algorithm (MMA) shows a very robust performance at reduced computational complexity. This is especially important for optical communications where data rates can exceed 100 Gbit/s per wavelength. All proposed algorithms are verified by simulations and by experiments using optical M-ary QAM Nyquist and OFDM signals with data rates up to 84 Gbit/s.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  6. F. M. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
    [Crossref]
  7. H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing (Wiley, 1997).
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    [Crossref]
  9. M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988).
    [Crossref]
  10. M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital clock recovery algorithm for Nyquist signal,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTu2I.7.
    [Crossref]
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    [Crossref]
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    [Crossref]
  13. R. Bouziane, R. Schmogrow, D. Hillerkuss, P. A. Milder, C. Koos, W. Freude, J. Leuthold, P. Bayvel, and R. I. Killey, “Generation and transmission of 85.4 Gb/s real-time 16QAM coherent optical OFDM signals over 400 km SSMF with preamble-less reception,” Opt. Express 20(19), 21612–21617 (2012).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2012 (2)

2011 (1)

2010 (1)

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

2008 (1)

1998 (1)

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

1997 (1)

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

1988 (1)

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

1986 (1)

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

1985 (1)

N. D’Andrea and U. Mengali, “A simulation study of clock recovery in QPSK and 9QPRS systems,” IEEE Trans. Commun. 33(10), 1139–1142 (1985).
[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]

1973 (1)

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).
[Crossref]

1959 (1)

J. E. Volder, “The CORDIC Trigonometric Computing Technique,” IRE Trans. Electron. Comput. EC-8(3), 330–334 (1959).
[Crossref]

Baeuerle, B.

Bao, H.

Bayvel, P.

Becker, J.

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express 20(1), 317–337 (2012).
[Crossref] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

Behm, J. D.

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

Ben-Ezra, S.

Bosco, G.

Bouziane, R.

Brown, D. R.

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

Carena, A.

Casas, R. A.

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

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]

Curri, V.

D’Andrea, N.

N. D’Andrea and U. Mengali, “A simulation study of clock recovery in QPSK and 9QPRS systems,” IEEE Trans. Commun. 33(10), 1139–1142 (1985).
[Crossref]

Dreschmann, M.

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express 20(1), 317–337 (2012).
[Crossref] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

Endres, T. J.

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

Forghieri, F.

Freude, W.

Gardner, F. M.

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

Hillerkuss, D.

Huebner, M.

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express 20(1), 317–337 (2012).
[Crossref] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

Johnson, C. R.

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

Killey, R. I.

Koos, C.

Leuthold, J.

Ludwig, A.

Mengali, U.

N. D’Andrea and U. Mengali, “A simulation study of clock recovery in QPSK and 9QPRS systems,” IEEE Trans. Commun. 33(10), 1139–1142 (1985).
[Crossref]

Meyer, J.

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express 20(1), 317–337 (2012).
[Crossref] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

Meyer, M.

Meyr, H.

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

Milder, P. A.

Nebendahl, B.

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express 20(1), 317–337 (2012).
[Crossref] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

Oerder, M.

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

Personick, S. D.

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).
[Crossref]

Poggiolini, P.

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]

Schmogrow, R.

Schniter, P.

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

Shieh, W.

Tang, Y.

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]

Volder, J. E.

J. E. Volder, “The CORDIC Trigonometric Computing Technique,” IRE Trans. Electron. Comput. EC-8(3), 330–334 (1959).
[Crossref]

Winter, M.

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express 20(1), 317–337 (2012).
[Crossref] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

Wolf, S.

Bell Syst. Tech. J. (1)

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).
[Crossref]

IEEE Photon. Technol. Lett. (1)

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. 22(21), 1601–1603 (2010).
[Crossref]

IEEE Trans. Commun. (4)

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

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

N. D’Andrea and U. Mengali, “A simulation study of clock recovery in QPSK and 9QPRS systems,” IEEE Trans. Commun. 33(10), 1139–1142 (1985).
[Crossref]

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

IRE Trans. Electron. Comput. (1)

J. E. Volder, “The CORDIC Trigonometric Computing Technique,” IRE Trans. Electron. Comput. EC-8(3), 330–334 (1959).
[Crossref]

J. Lightwave Technol. (1)

Opt. Express (3)

Proc. IEEE (1)

C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE 86(10), 1927–1950 (1998).
[Crossref]

Other (3)

J. Leuthold and W. Freude, “Optical OFDM and Nyquist multiplexing,” in Optical Fiber Telecommunications Volume VIB: Systems and Networks (Optics and Photonics), I. Kaminow, T. Li, A.E. Willner, eds. (Academic Press, 2013).

M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital clock recovery algorithm for Nyquist signal,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTu2I.7.
[Crossref]

H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing (Wiley, 1997).

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Figures (14)

Fig. 1
Fig. 1 Color-coded eye-diagrams and mean values (black dashed lines) as a function of time (Ts: symbol period). The plots are obtained from amplitudes (upper row) and powers (lower row) of binary phase shift keyed (BPSK) signals with (a) a sinc-like pulse shape with raised-cosine spectrum and (b) a sinc-pulse shape. While the mean power of signals with “raised-cosine” pulses (a) does reveal timing information, the signals comprising sinc-shaped pulses (b) produce large overshoots leading to a constant mean power for all t.
Fig. 2
Fig. 2 Sinc-shaped pulses in time (Nyquist) or frequency domain (OFDM) as defined by the sum terms of Eq. (3) for R = 0 and Q = 2 neighbors. Pulses are separated by multiples k of the symbol period Z. The offset from the ideal sampling position mZ is named ζ.
Fig. 3
Fig. 3 For finding the ideal Nyquist sampling time or OFDM carrier frequency, the ensemble average of the power S(ζ) has been calculated (lines), simulated (solid squares), and plotted as a function of the sampling offset ζ normalized to the symbol period Z. The ideal sampling time or carrier frequency is given by ζ = 0. Sinc-shaped functions with low roll-off factors β close to β = 0 only have a maximum at ζ = 0 if a small number Q of neighbor symbols/spectra is included in the averaging process. For sincl-functions (having a raised-cosine spectrum) with β close to 1, the number of neighbor symbols Q is less important, because sincl-functions with larger β in Eq. (2) decay much faster than sinc-functions with β = 0.
Fig. 4
Fig. 4 Signal with a rect-shaped pulse envelope (green) showing either an OFDM signal in time domain or a Nyquist sinc-pulse signal in frequency domain (Eq. (6) for R = 0). The black curves show three superimposed “neighbors” (i.e., for Q = 2). The coefficients cik = c0k can be extracted by applying an FFT (for OFDM) or an IFFT (for sinc-pulses). An offset from the ideal I/FFT-window position is defined as ζ.
Fig. 5
Fig. 5 Rectangular symbols and average power in receiving window as a function of the offset ζ. Signal comprising rect-shaped pulse envelopes (green) either in the time domain (OFDM) or in the frequency domain (Nyquist) separated by a guard interval zg. The coefficients c ik are extracted within a rectangular window with an I/FFT.
Fig. 6
Fig. 6 The ideal OFDM time or the ideal Nyquist frequency window position, respectively, may be found by calculating the ensemble average of the power S(ζ) within the window as a function of the time/frequency offset ζ. Signals with a guard interval in time or frequency, i.e. with zg > 0, have a distinct maximum exactly at the ideal timing or carrier frequency ζ = 0. Therefore, time window or carrier frequency offsets can only be found for signals with a guard interval. The solid lines show the analytically obtained (Eq. (10) and (11)) results and the solid squares show results obtained by simulations.
Fig. 7
Fig. 7 Performance of MMA and the CMA for simulated QPSK (top) and 16QAM (bottom) OFDM signals with noise loading. (a) RMS of error made by (a) timing or (b) frequency recovery.
Fig. 8
Fig. 8 Performance of the MMA and the CMA for simulated QPSK (top) and 16QAM (bottom) Nyquist signals. (a) RMS of error made by (a) timing or (b) frequency recovery.
Fig. 9
Fig. 9 Experimental setup for transmitting Nyquist or OFDM signals. Two synchronized field programmable gate arrays (FPGA) store pre-computed waveforms and drive attached digital-to-analog converters (DAC). The generated complex waveform is then encoded on an external cavity laser (ECL) with an optical I/Q-modulator. After amplification with an erbium doped fiber amplifier (EDFA) the signal passes a 1 nm-wide bandpass, is coherently received by an optical modulation analyzer (OMA), and processed offline.
Fig. 10
Fig. 10 Experiment showing the performance of four cost functions for OFDM carrier frequency recovery for either QPSK (black), 16QAM (blue), or 64QAM (red) modulated SCs. (a) The MPA does not provide an extremum and thus cannot be used to recover the frequency. (b) The mean modulus algorithm (MMA) of the extracted coefficients cik yields a maximum at the frequency offset ν = 0. (c) The CMA and (d) the CPA provide a minimum for ν = 0.
Fig. 11
Fig. 11 Demonstration of the performance of four cost functions for OFDM timing recovery. (a) MPA cost function for various offset times for simulated OFDM symbols with a temporal guard interval τg in-between symbols. (b) – (d) show the result of applying the cost functions to experimentally obtained OFDM signals using either (b) the MMA, (c) the CMA or (d) the CPA algorithms. These algorithms provide an extremum at the ideal time even if τg = 0.
Fig. 12
Fig. 12 The performance of four cost functions for Nyquist carrier frequency recovery. (a) MPA cost function as a function of frequency offsets applied to a simulated Nyquist signal with spectral guard interval νg in-between channels. (b) – (d) Results of cost functions obtained by applying the operations on experimentally received Nyquist signals without a guard interval using either (b) the MMA, (c) the CMA or (d) the CPA algorithms.
Fig. 13
Fig. 13 Performance of four cost functions to recover the timing offset for Nyquist signals with different roll-off factors β. We evaluate the MPA (upper left), the MMA (upper right), the CMA (lower left), and the CPA (lower right) for (a) QPSK and (b) 64QAM encoded Nyquist signals and as a function of the temporal sampling offset τ. The MPA only yields a good measure for β > 0.1. Another limitation arises for Nyquist signals with large β where the CMA and CPA algorithms do not provide a distinct minimum for τ = 0 (e.g. for 64QAM). (c) Constellation diagrams for QPSK and β = 0 (top) and for 64QAM and β = 1 (bottom). Sampling with τ = 0 yields the red symbols whereas sampling with τ ≠ 0 results in the black inter-symbol transitions.
Fig. 14
Fig. 14 Carrier phase estimation by applying the MMA to the real and imaginary values of the signal: Theory (lines, Eq. (17)) and simulations (squares) agree well. Typically, an offset φ between signal carrier phase and LO phase is observed. Constellation diagrams for different φ are shown as insets. (a) QPSK. (b) 16QAM. (c) 32QAM. (d) 64QAM.

Equations (17)

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s Nyquist ( t )= i=0 N1 k= + c ik exp( +j2π it T s )sincl( t T s k ) , s OFDM ( f )= 1 F s i= + k=0 N1 c ik exp( j2π if F s )sinc( f F s k ) .
sincl( z Z )=sinc( z Z ) cos( πβz /Z ) 1 4 β 2 z 2 / Z 2 sinc( z Z )={ 1 for z=0 sin( πz/Z ) πz/Z else
s( z )= k=Q/2 +Q /2 c k ( z )sincl( z Z k ) , c k ( z )= i=R/2 +R/2 c ik e j2π iz Z .
0.5<ζ/Z 0.5.
S( ζ )= | s( ζ ) | 2 = k, k =Q/2 Q/2 c k c k sincl( ζ Z k )sincl( ζ Z k ) , S( ζ ) ¯ = k, k =Q/2 Q/2 c k c k ¯ sincl( ζ Z k )sincl( ζ Z k ) | c k | 2 ¯ k=Q/2 Q/2 sin cl 2 ( ζ Z k ) .
s OFDM ( t )= i= + k=0 N1 c ik exp( +j2π kt T s )rect( t T s i ) , s Nyquist ( f )= 1 F s i=0 N1 k= + c ik exp( j2π kf F s )rect( f F s i ) .
rect( z Z )={ 1 for | z | < Z/2 0 for | z | > Z/2
s( z )= i=R/2 +R/2 c i (z)rect( z Z i ) , c i ( z )= k=Q/2 +Q/2 c ik exp( j2π kz Z ) .
S( ζ )= 1 Z ζZ/2 ζ+Z/2 | s( z ) | 2 dz , S( ζ ) ¯ = 1 Z i, i = R /2 +R /2 ζZ/2 ζ+Z/2 c i ( z ) c i ( z )rect( z Z i )rect( z Z i )dz ¯ = 1 Z i= R /2 +R /2 ζZ/2 ζ+Z/2 | c i ( z ) | 2 rect( z Z i )dz ¯ .
S( ζ ) ¯ = k, k c ik c i k ¯ 1 Z ζZ/2 ζ+Z/2 exp( j2π ( k k )z Z )dz = k | c ik | 2 ¯ .
S( ζ ) ¯ = k | c ik | 2 ¯ ( 1 | ζ | Z )forR=0 (no neighbors).
f( S( ζ ) ) ¯ = 1 L l=1 L f( S( ζ ) ) | l .
S( ζ ) ¯ = 1 L l=1 L S( ζ ) | l ( MPA )
S( ζ ) ¯ = 1 L l=1 L S( ζ ) | l ( MMA )
σ S( ζ ) 2 = ( S( ζ ) S( ζ ) ¯ ) 2 ¯ = S( ζ ) ¯ S( ζ ) ¯ 2 (CMA).
σ S( ζ ) 2 = ( S( ζ ) S( ζ ) ¯ ) 2 ¯ = S 2 ( ζ ) ¯ S( ζ ) ¯ 2 (CPA).
| { s( φ ) } |+| { s( φ ) } | ¯ = 1 M m=1 M [ | { c m e jφ } |+| { c m e jφ } | ] , c m =| c m | e j θ m , { c m e jφ }=| c m |[ sin( θ m )sin( φ )cos( θ m )cos( φ ) ], { c m e jφ }=| c m |[ sin( θ m )cos( φ )+cos( θ m )sin( φ ) ].

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