Abstract

Optically multiplexed multi-carrier systems with channel spacing reduced to the symbol rate per carrier are highly susceptible to inter-channel crosstalk, which places stringent requirements for the specifications of system components and hinders the use of high-level formats. In this paper, we investigate the performance benefits of using offset 4-, 16-, and 64-quadrature amplitude modulation (QAM) in coherent wavelength division multiplexing (CoWDM). We compare this system with recently reported Nyquist WDM and no-guard-interval optical coherent orthogonal frequency division multiplexing, and show that the presented system greatly relaxes the requirements for device specifications and enhances the spectral efficiency by enabling the use of high-level QAM. The achieved performance can approach the theoretical limits using practical components.

© 2011 OSA

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  1. X. Zhou, J. Yu, M. F. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, “64Tb/s (640×107Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Optical Fiber Communication Conference (2010), paper PDPB9.
  2. A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, and Y. Takatori, “No-guard-interval coherent optical OFDM for 100Gb/s long-haul WDM transmission,” J. Lightwave Technol. 27(16), 3705–3713 (2009).
    [CrossRef]
  3. S. Chandrasekhar and X. Liu, “Experimental investigation on the performance of closely spaced multi-carrier PDM-QPSK with digital coherent detection,” Opt. Express 17(24), 21350–21361 (2009).
    [CrossRef] [PubMed]
  4. J. Yu, Z. Dong, X. Xiao, Y. Xia, S. Shi, C. Ge, W. Zhou, N. Chi, and Y. Shao, “Generation, transmission and coherent detection of 11.2 Tb/s (112×100Gb/s) single source optical OFDM superchannel,” Optical Fiber Communication Conference (2011), paper PDPA6.
  5. G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
    [CrossRef]
  6. A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005).
    [CrossRef]
  7. J. Zhao and A. D. Ellis, “A novel optical fast OFDM with reduced channel spacing equal to half of the symbol rate per carrier,” Optical Fiber Communication Conference (2010), paper OMR1.
  8. S. Yamamoto, K. Yonenaga, A. Sahara, F. Inuzuka, and A. Takada, “Achievement of sub-channel frequency spacing less than symbol rate and improvement of dispersion tolerance in optical OFDM transmission,” J. Lightwave Technol. 28(1), 157–163 (2010).
    [CrossRef]
  9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. N. Bergono, “High spectral efficiency long-haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. European Conference on Optical Communication (2010), paper We.7.C.4.
  10. R. R. Mosier and R. G. Clabaugh, “Kineplex, a bandwidth-efficient binary transmission system,” AIEE Trans. Commun. 76, 723–728 (1958).
  11. R. W. Chang, “Synthesis of band-limited orthogonal signals fro multi-channel data transmission,” Bell Syst. Tech. J. 45, 1775–1796 (1966).
  12. S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol. Com. 19(5), 628–634 (1971).
    [CrossRef]
  13. J. Zhao and A. D. Ellis, “Electronic impairment mitigation in optically multiplexed multi-carrier systems,” J. Lightwave Technol. 29(3), 278–290 (2011).
    [CrossRef]
  14. G. Gavioli, E. Torrengo, G. Bosco, A. Carena, V. Curri, V. Miot, P. Poggiolini, M. Belmonte, F. Forghieri, C. Muzio, S. Piciaccia, A. Brinciotti, A. L. Porta, C. Lezzi, S. Savory, and S. Abrate, “Investigation of the impact of ultra-narrow carrier spacing on the transmission of a 10-carrier 1Tb/s superchannel,” Optical Fiber Communication Conference (2010), paper OThD3.
  15. D. Hillerkuss, T. Schellinger, R. Schmogrow, M. Winter, T. Vallaitis, R. Bonk, A. Marculescu, J. Li, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moller, M. Huebner, J. Becher, C. Koos, W. Freude, and J. Leuthold, “Single source optical OFDM transmitter and optical FFT receiver demonstrated at line rates of 5.4 and 10.8 Tbit/s,” Optical Fiber Communication Conference (2010), paper PDPC1.
  16. S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
    [CrossRef]
  17. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2000).
  18. B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced groupband data modem using orthogonally multiplexed QAM technique,” IEEE Trans. Commun. 34(6), 587–592 (1986).
    [CrossRef]

2011 (1)

2010 (3)

S. Yamamoto, K. Yonenaga, A. Sahara, F. Inuzuka, and A. Takada, “Achievement of sub-channel frequency spacing less than symbol rate and improvement of dispersion tolerance in optical OFDM transmission,” J. Lightwave Technol. 28(1), 157–163 (2010).
[CrossRef]

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

2009 (2)

2005 (1)

A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005).
[CrossRef]

1986 (1)

B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced groupband data modem using orthogonally multiplexed QAM technique,” IEEE Trans. Commun. 34(6), 587–592 (1986).
[CrossRef]

1971 (1)

S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol. Com. 19(5), 628–634 (1971).
[CrossRef]

1966 (1)

R. W. Chang, “Synthesis of band-limited orthogonal signals fro multi-channel data transmission,” Bell Syst. Tech. J. 45, 1775–1796 (1966).

1958 (1)

R. R. Mosier and R. G. Clabaugh, “Kineplex, a bandwidth-efficient binary transmission system,” AIEE Trans. Commun. 76, 723–728 (1958).

Bosco, G.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

Carena, A.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

Chandrasekhar, S.

Chang, R. W.

R. W. Chang, “Synthesis of band-limited orthogonal signals fro multi-channel data transmission,” Bell Syst. Tech. J. 45, 1775–1796 (1966).

Clabaugh, R. G.

R. R. Mosier and R. G. Clabaugh, “Kineplex, a bandwidth-efficient binary transmission system,” AIEE Trans. Commun. 76, 723–728 (1958).

Curri, V.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

Ebert, P. M.

S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol. Com. 19(5), 628–634 (1971).
[CrossRef]

Ellis, A. D.

J. Zhao and A. D. Ellis, “Electronic impairment mitigation in optically multiplexed multi-carrier systems,” J. Lightwave Technol. 29(3), 278–290 (2011).
[CrossRef]

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005).
[CrossRef]

Forghieri, F.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

Frascella, P.

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

Garcia Gunning, F. C.

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

Gunning, F. C. G.

A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005).
[CrossRef]

Hasegawa, S.

B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced groupband data modem using orthogonally multiplexed QAM technique,” IEEE Trans. Commun. 34(6), 587–592 (1986).
[CrossRef]

Hirosaki, B.

B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced groupband data modem using orthogonally multiplexed QAM technique,” IEEE Trans. Commun. 34(6), 587–592 (1986).
[CrossRef]

Ibrahim, S. K.

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

Inuzuka, F.

Ishihara, K.

Kobayashi, T.

Kudo, R.

Liu, X.

Masuda, H.

Miyamoto, Y.

Mosier, R. R.

R. R. Mosier and R. G. Clabaugh, “Kineplex, a bandwidth-efficient binary transmission system,” AIEE Trans. Commun. 76, 723–728 (1958).

Peters, F. H.

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

Sabato, A.

B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced groupband data modem using orthogonally multiplexed QAM technique,” IEEE Trans. Commun. 34(6), 587–592 (1986).
[CrossRef]

Sahara, A.

Sano, A.

Takada, A.

Takatori, Y.

Weinstein, S. B.

S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol. Com. 19(5), 628–634 (1971).
[CrossRef]

Yamada, E.

Yamamoto, S.

Yamazaki, E.

Yonenaga, K.

Yoshida, E.

Zhao, J.

J. Zhao and A. D. Ellis, “Electronic impairment mitigation in optically multiplexed multi-carrier systems,” J. Lightwave Technol. 29(3), 278–290 (2011).
[CrossRef]

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

AIEE Trans. Commun. (1)

R. R. Mosier and R. G. Clabaugh, “Kineplex, a bandwidth-efficient binary transmission system,” AIEE Trans. Commun. 76, 723–728 (1958).

Bell Syst. Tech. J. (1)

R. W. Chang, “Synthesis of band-limited orthogonal signals fro multi-channel data transmission,” Bell Syst. Tech. J. 45, 1775–1796 (1966).

IEEE Photon. J. (1)

S. K. Ibrahim, J. Zhao, F. C. Garcia Gunning, P. Frascella, F. H. Peters, and A. D. Ellis, “Coherent WDM: analytical, numerical, and experimental studies,” IEEE Photon. J. 2(5), 833–847 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and Co-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010).
[CrossRef]

A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005).
[CrossRef]

IEEE Trans. Commun. (1)

B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced groupband data modem using orthogonally multiplexed QAM technique,” IEEE Trans. Commun. 34(6), 587–592 (1986).
[CrossRef]

IEEE Trans. Commun. Technol. Com. (1)

S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol. Com. 19(5), 628–634 (1971).
[CrossRef]

J. Lightwave Technol. (3)

Opt. Express (1)

Other (7)

X. Zhou, J. Yu, M. F. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, “64Tb/s (640×107Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Optical Fiber Communication Conference (2010), paper PDPB9.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2000).

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, V. Curri, V. Miot, P. Poggiolini, M. Belmonte, F. Forghieri, C. Muzio, S. Piciaccia, A. Brinciotti, A. L. Porta, C. Lezzi, S. Savory, and S. Abrate, “Investigation of the impact of ultra-narrow carrier spacing on the transmission of a 10-carrier 1Tb/s superchannel,” Optical Fiber Communication Conference (2010), paper OThD3.

D. Hillerkuss, T. Schellinger, R. Schmogrow, M. Winter, T. Vallaitis, R. Bonk, A. Marculescu, J. Li, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moller, M. Huebner, J. Becher, C. Koos, W. Freude, and J. Leuthold, “Single source optical OFDM transmitter and optical FFT receiver demonstrated at line rates of 5.4 and 10.8 Tbit/s,” Optical Fiber Communication Conference (2010), paper PDPC1.

J. Zhao and A. D. Ellis, “A novel optical fast OFDM with reduced channel spacing equal to half of the symbol rate per carrier,” Optical Fiber Communication Conference (2010), paper OMR1.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. N. Bergono, “High spectral efficiency long-haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. European Conference on Optical Communication (2010), paper We.7.C.4.

J. Yu, Z. Dong, X. Xiao, Y. Xia, S. Shi, C. Ge, W. Zhou, N. Chi, and Y. Shao, “Generation, transmission and coherent detection of 11.2 Tb/s (112×100Gb/s) single source optical OFDM superchannel,” Optical Fiber Communication Conference (2011), paper PDPA6.

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Figures (9)

Fig. 1
Fig. 1

The model of offset-QAM CoWDM for theoretical analysis. Inset 1: the offset 4-QAM data for modulation.

Fig. 2
Fig. 2

An example to illustrate the crosstalk from the in-phase and quadrature tributaries of the (j + 1)th channel to the in-phase tributary of the j th channel by using a signal pulse of (a) rectangular and (b) raised cosine with the roll-off coefficient of 0.4. The first, second, and fourth columns show the signals before demultiplexing without and with normalized carriers with respect to the j th channel, and after demultiplexing respectively, when the input signal is the in-phase tributary of the j th (the top row) and (j + 1)th (the second row) channels, and the quadrature tributary of the (j + 1)th channel without (the third row) and with (the bottom row) T/2 time offset. The third column shows the impulse response of the demultiplexing filter for the j th channel. In both figures, ϕ j = 0 and ϕ j + 1 = π/2.

Fig. 3
Fig. 3

Simulation setup for systems of offset-QAM CoWDM, no-guard-interval optical OFDM, and N-WDM, with 25Gsym/s per channel and 25GHz channel spacing.

Fig. 4
Fig. 4

(a). Performance versus the received OSNR for offset-QAM CoWDM (solid circles), Nyquist WDM (empty circles), and no-guard-interval optical OFDM (cross circles) using 4-QAM format. (b) Performance versus the received OSNR for CoWDM using offset 16- (solid triangles) and 64-QAM (solid squares), and for 16-QAM N-WDM (empty triangles). Dashed or dotted lines represent the theoretical limits. The phase difference between channels is π/2.

Fig. 5
Fig. 5

Constellation diagrams of (a): 4-QAM no-guard-interval optical OFDM; (b): 4-QAM N-WDM; (c): 16-QAM N-WDM; (d): offset 4-QAM CoWDM; (e): offset 16-QAM CoWDM; (f): offset 64-QAM CoWDM. The system parameters are the same as Fig. 4.

Fig. 6
Fig. 6

(a) Required OSNR (dB) versus the transmitter bandwidth for CoWDM (solid), N-WDM (dashed), and no-guard-interval optical OFDM (dotted). Circles and triangles represent (offset) 4- and 16-QAM respectively (b) Required OSNR (dB) versus the transmitter bandwidth for offset 4- (circles), 16- (triangles), and 64-QAM (squares) CoWDM. The equivalent response of the driving amplifier and the modulator’s electronic interface is 3rd-order Gaussian (solid) or 5th-order Bessel (dashed) shaped. For (a) and (b), the receiver EF bandwidth is optimized.

Fig. 7
Fig. 7

(a) Required OSNR (dB) versus the receiver EF bandwidth for offset 4-QAM CoWDM (solid lines), 4-QAM N-WDM (dashed lines), and 4-QAM no-guard-interval optical OFDM (dotted lines). Solid symbols represent a FIR memory length of 6 for CoWDM and OFDM, or 12 for N-WDM. Empty symbols represent a FIR memory length of 2 for CoWDM, OFDM, and N-WDM. (b) Required OSNR (dB) versus the memory length of the receiver FIR filter. Circles, triangles, and squares represent (offset) 4-, 16-, and 64-QAM respectively. For (a) and (b), the transmitter bandwidth is optimized and the phase difference between channels is π/2.

Fig. 8
Fig. 8

(a) Required OSNR (dB) versus receiver filter bandwidth for offset 4- (circles), 16- (triangles), and 64-QAM (squares) CoWDM when the equivalent response of the driving amplifier and the modulator’s electronic interface is 3rd-order Gaussian (solid) and 5th-order Bessel (dashed). The transmitter bandwidth is optimized. (b) Required OSNR (dB) versus the ADC resolution for offset 4- (circles), 16- (triangles), and 64-QAM (squares) CoWDM under optimized transmitter and receiver bandwidths. In (a) and (b), the receiver FIR filter memory length is 6 and the phase difference between channels is π/2.

Fig. 9
Fig. 9

Performance as a function of phase difference between channels for 3rd-order Gaussian shaped transmitter and receiver EF with optimized bandwidths. Circles, triangles, and squares represent offset 4-, 16-, and 64-QAM respectively.

Tables (4)

Tables Icon

Table 1 The Contributions to the Decoded aj,m (the m th Sample of the j th Channel) in the Conventional System without T/2 Time Offset for the Quadrature Tributary*

Tables Icon

Table 2 The Contributions to the Decoded aj,m (the m th Sample of the j th Channel) in Offset-QAM CoWDM*

Tables Icon

Table 3 The Contributions to the Decoded aj,m (the m th Sample of the j th Channel) in the Conventional System without T/2 Time Offset for the Quadrature Tributary*

Tables Icon

Table 4 The Contributions to the Decoded aj,m (the m th Sample of the j th Channel) in Offset-QAM CoWDM*

Equations (21)

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

E j ( t ) = E 0 k = 1 J n = ( a k , n I k , j ( t n T ) + i b k , n Q k , j ( t n T ) ) e i ( ω k ω j ) n T + i φ k
I k , j ( t ) = + h s ( t τ ) e i ( ω k ω j ) ( t τ ) h D , j ( τ ) e i ω j τ d τ
Q k , j ( t ) = + h s ( t τ T / 2 ) e i ( ω k ω j ) ( t τ ) h D , j ( τ ) e i ω j τ d τ
H i n p h a s e , k , j ( ω ) = H s ( ω ω k + ω j ) H D , j ( ω + ω j )
H q u a d r a t u r e , k , j ( ω ) = H s ( ω ω k + ω j ) e i ( ω ω k + ω j ) T / 2 H D , j ( ω + ω j )
H D , j ( ω + ω j ) = H o p t , j ( ω + ω j ) H e l e , j ( ω + ω j ω l o )
a ' j , m r e a l { ( a j , m I j , j ( 0 ) + i b j , m Q j , j ( 0 ) ) + n m ( a j , n I j , j ( ( m n ) T ) + i b j , n Q j , j ( ( m n ) T ) )              + k j n ( a k , n I k , j ( ( m n ) T ) + i b k , n Q k , j ( ( m n ) T ) ) e i ( φ k φ j ) }
b ' j , m i m a g { ( a j , m I j , j ( 0.5 T ) + i b j , m Q j , j ( 0.5 T ) ) + n m ( a j , n I j , j ( ( m n + 0.5 ) T ) + i b j , n Q j , j ( ( m n + 0.5 ) T ) )             + k j n ( a k , n I k , j ( ( m n + 0.5 ) T ) + i b k , n Q k , j ( ( m n + 0.5 ) T ) ) e i ( φ k φ j ) }
I k , j ( ( m n ) T ) = + h s ( ( m n ) T τ ) e i 2 π ( k j ) ( ( m n ) T τ ) / T h s * ( τ ) d τ
Q k , j ( ( m n ) T ) = + h s ( ( m n ) T τ T / 2 ) e i 2 π ( k j ) ( ( m n ) T τ ) / T h s * ( τ ) d τ
I k , j ( ( m n ) T ) = ( 1 ) ( k j ) ( m n ) + h s ( ( m n ) T / 2 + τ ' ) e i 2 π ( k j ) τ ' / T h s * ( τ ' ( m n ) T / 2 ) d τ '
Q k , j ( ( m n ) T ) = ( 1 ) ( k j ) ( m n 0.5 ) + h s ( ( m n 0.5 ) T / 2 + τ ' ) e i 2 π ( k j ) τ ' / T h s * ( τ ' ( m n 0.5 ) T / 2 ) d τ '
I k , j ( ( m n ) T ) = ( 1 ) ( k j ) ( m n ) + h s ( ( m n ) T / 2 + τ ' ) h s ( τ ' ( m n ) T / 2 ) cos ( 2 π ( k j ) τ ' / T ) d τ '
Q k , j ( ( m n ) T ) = ( 1 ) ( k j ) ( m n 0.5 ) + h s ( ( m n 0.5 ) T / 2 + τ ' ) h s ( τ ' ( m n 0.5 ) T / 2 ) cos ( 2 π ( k j ) τ ' / T ) d τ '
a ' j , m a j , m I j , j ( 0 ) n a j 2 , n I j 2 , j ( ( m n ) T ) n a j + 2 , n I j + 2 , j ( ( m n ) T ) ...
b ' j , m b j , m Q j , j ( 0.5 T ) n b j 2 , n Q j 2 , j ( ( m n + 0.5 ) T ) ) n b j + 2 , n Q j + 2 , j ( ( m n + 0.5 ) T ) ) ...
F H i n p h a s e , k , j ( ω ) = T { I k , j ( 0 ) + I k , j ( T ) e i ω T + I k , j ( T ) e i ω T + I k , j ( 2 T ) e 2 i ω T + I k , j ( 2 T ) e 2 i ω T + ... }
F H q u a d r a t u r e , k , j ( ω ) = T { Q k , j ( 0 ) + Q k , j ( T ) e i ω T + Q k , j ( T ) e i ω T + Q k , j ( 2 T ) e 2 i ω T + Q k , j ( 2 T ) e 2 i ω T + ... }
F H i n p h a s e , k , j ( ω ) = p = + H i n p h a s e , k , j ( ω + 2 π p T )
F H q u a d r a t u r e , k , j ( ω ) = p = + H q u a d r a t u r e , k , j ( ω + 2 π p T )
N o r m a l i z e d    O S N R = T o t a l   S i g n a l   P o w e r 5 × N o i s e   P o w e r   i n   0.1 n m

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