Abstract

The energy efficiency of pulse-amplitude-modulated discrete multitone modulation (PAM-DMT) decreases as the modulation order of M-PAM modulation increases. Enhanced PAM-DMT (ePAM-DMT) was proposed as a solution to the reduced energy efficiency of PAM-DMT. This was achieved by allowing multiple streams of PAM-DMT to be superimposed and successively demodulated at the receiver side. In order to maintain a distortion-free unipolar ePAM-DMT system, the multiple time-domain PAM-DMT streams are required to be aligned. However, aligning the antisymmetry in ePAM-DMT is complex and results in efficiency losses. In this paper, a novel simplified method to apply the superposition modulation on M-PAM modulated discrete multitone (DMT) is introduced. Contrary to ePAM-DMT, the signal generation of the proposed system, termed augmented spectral efficiency discrete multitone (ASE-DMT), occurs in the frequency domain. This results in an improved spectral and energy efficiency. The analytical bit error rate (BER) performance bound of the proposed system is derived and compared with Monte-Carlo simulations. The system performance is shown to offer significant electrical and optical energy savings compared with ePAM-DMT and DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM).

© 2016 Optical Society of America

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  1. Cisco Visual Networking Index, “The Zettabyte era: trends and analysis,” White Paper, Cisco (2015), http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/VNI_Hyperconnectivity_WP.pdf .
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    [Crossref]
  3. D. Tsonev, S. Videv, and H. Haas, “Towards a 100 Gb/s visible light wireless access network,” Opt. Express 23, 1627–1637 (2015).
    [Crossref] [PubMed]
  4. J. Armstrong and A.J. Lowery, “Power efficient optical OFDM,” Elect. Lett. 42, 370–372 (2006).
    [Crossref]
  5. S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
    [Crossref]
  6. D. Tsonev, S. Sinanovic, and H. Haas, “Novel unipolar orthogonal frequency division multiplexing (U-OFDM) for optical wireless,” in Proceedings of IEEE Vehicular Technology Conference (VTC Spring), (IEEE, 2012). pp. 1–5.
  7. N. Fernando, Y. Hong, and E. Viterbo, “Flip-OFDM for Unipolar Communication Systems,” IEEE Tran. on Commun. 60, 3726–3733 (2012).
    [Crossref]
  8. D. Tsonev, S. Videv, and H. Haas, “Unlocking spectral efficiency in intensity modulation and direct detection systems,” IEEE J. Sel. Areas Commun. 33, 1758–1770 (2015).
    [Crossref]
  9. M. Islim, D. Tsonev, and H. Haas, “Spectrally enhanced PAM-DMT for IM/DD optical wireless communications,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2015), pp. 877–882.
  10. M. Islim, D. Tsonev, and H. Haas, “On the superposition modulation for OFDM-based optical wireless Communication,” in Proceedings of IEEE Global Signal and Information Processing conference, (IEEE, 2015). pp. 1022–1026.
  11. H. Elgala and T. Little, “SEE-OFDM: spectral and energy efficient OFDM for optical IM/DD systems,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2014), pp. 851–855.
  12. Q. Wang, C. Qian, X. Guo, Z. Wang, D. G. Cunningham, and I. H. White, “Layered ACO-OFDM for intensity-modulated direct-detection optical wireless transmission,” Opt. Express 23, 12382–12393 (2015).
    [Crossref] [PubMed]
  13. A.J. Lowery, “Enhanced asymmetrically-clipped optical OFDM,” Opt. Express 24, 3950–3966 (2016).
    [Crossref] [PubMed]
  14. D. Tsonev, S. Sinanovic, and H. Haas, “Complete modelling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightw. Technol. 31, 3064–3076 (2013).
    [Crossref]
  15. M. Islim, D. Tsonev, and H. Haas, “A generalized solution to the spectral efficiency loss in unipolar optical OFDM-based systems,” in Proceedings of IEEE International Conference on Communications (ICC), (IEEE, 2015). pp. 5126.
  16. F. Xiong, Digital Modulation Techniques, (Artech House Publishers, 2006), 2nd ed.
  17. J. Proakis and G. Manolakis, Digital Signal Processing (Pearson, 2013), Chap. 8.
  18. H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse fourier transform,” in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, (ACM, 2012), pp. 563–578.
  19. J. Armstrong and B. J. C. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12, 343–345 (2008).
    [Crossref]

2016 (1)

2015 (3)

2013 (1)

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modelling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightw. Technol. 31, 3064–3076 (2013).
[Crossref]

2012 (1)

N. Fernando, Y. Hong, and E. Viterbo, “Flip-OFDM for Unipolar Communication Systems,” IEEE Tran. on Commun. 60, 3726–3733 (2012).
[Crossref]

2009 (1)

S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
[Crossref]

2008 (1)

J. Armstrong and B. J. C. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12, 343–345 (2008).
[Crossref]

2006 (1)

J. Armstrong and A.J. Lowery, “Power efficient optical OFDM,” Elect. Lett. 42, 370–372 (2006).
[Crossref]

Armstrong, J.

J. Armstrong and B. J. C. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12, 343–345 (2008).
[Crossref]

J. Armstrong and A.J. Lowery, “Power efficient optical OFDM,” Elect. Lett. 42, 370–372 (2006).
[Crossref]

Breyer, F.

S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
[Crossref]

Cunningham, D. G.

Dimitrov, S.

S. Dimitrov and H. Haas, Principles of LED Light Communications: Towards Networked Li-Fi, (Cambridge University, 2015).
[Crossref]

Elgala, H.

H. Elgala and T. Little, “SEE-OFDM: spectral and energy efficient OFDM for optical IM/DD systems,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2014), pp. 851–855.

Fernando, N.

N. Fernando, Y. Hong, and E. Viterbo, “Flip-OFDM for Unipolar Communication Systems,” IEEE Tran. on Commun. 60, 3726–3733 (2012).
[Crossref]

Guo, X.

Haas, H.

D. Tsonev, S. Videv, and H. Haas, “Towards a 100 Gb/s visible light wireless access network,” Opt. Express 23, 1627–1637 (2015).
[Crossref] [PubMed]

D. Tsonev, S. Videv, and H. Haas, “Unlocking spectral efficiency in intensity modulation and direct detection systems,” IEEE J. Sel. Areas Commun. 33, 1758–1770 (2015).
[Crossref]

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modelling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightw. Technol. 31, 3064–3076 (2013).
[Crossref]

M. Islim, D. Tsonev, and H. Haas, “A generalized solution to the spectral efficiency loss in unipolar optical OFDM-based systems,” in Proceedings of IEEE International Conference on Communications (ICC), (IEEE, 2015). pp. 5126.

M. Islim, D. Tsonev, and H. Haas, “On the superposition modulation for OFDM-based optical wireless Communication,” in Proceedings of IEEE Global Signal and Information Processing conference, (IEEE, 2015). pp. 1022–1026.

M. Islim, D. Tsonev, and H. Haas, “Spectrally enhanced PAM-DMT for IM/DD optical wireless communications,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2015), pp. 877–882.

D. Tsonev, S. Sinanovic, and H. Haas, “Novel unipolar orthogonal frequency division multiplexing (U-OFDM) for optical wireless,” in Proceedings of IEEE Vehicular Technology Conference (VTC Spring), (IEEE, 2012). pp. 1–5.

S. Dimitrov and H. Haas, Principles of LED Light Communications: Towards Networked Li-Fi, (Cambridge University, 2015).
[Crossref]

Hassanieh, H.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse fourier transform,” in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, (ACM, 2012), pp. 563–578.

Hong, Y.

N. Fernando, Y. Hong, and E. Viterbo, “Flip-OFDM for Unipolar Communication Systems,” IEEE Tran. on Commun. 60, 3726–3733 (2012).
[Crossref]

Indyk, P.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse fourier transform,” in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, (ACM, 2012), pp. 563–578.

Islim, M.

M. Islim, D. Tsonev, and H. Haas, “A generalized solution to the spectral efficiency loss in unipolar optical OFDM-based systems,” in Proceedings of IEEE International Conference on Communications (ICC), (IEEE, 2015). pp. 5126.

M. Islim, D. Tsonev, and H. Haas, “On the superposition modulation for OFDM-based optical wireless Communication,” in Proceedings of IEEE Global Signal and Information Processing conference, (IEEE, 2015). pp. 1022–1026.

M. Islim, D. Tsonev, and H. Haas, “Spectrally enhanced PAM-DMT for IM/DD optical wireless communications,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2015), pp. 877–882.

Katabi, D.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse fourier transform,” in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, (ACM, 2012), pp. 563–578.

Koonen, A. M. J.

S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
[Crossref]

Lee, S. C. J.

S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
[Crossref]

Little, T.

H. Elgala and T. Little, “SEE-OFDM: spectral and energy efficient OFDM for optical IM/DD systems,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2014), pp. 851–855.

Lowery, A.J.

A.J. Lowery, “Enhanced asymmetrically-clipped optical OFDM,” Opt. Express 24, 3950–3966 (2016).
[Crossref] [PubMed]

J. Armstrong and A.J. Lowery, “Power efficient optical OFDM,” Elect. Lett. 42, 370–372 (2006).
[Crossref]

Manolakis, G.

J. Proakis and G. Manolakis, Digital Signal Processing (Pearson, 2013), Chap. 8.

Price, E.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse fourier transform,” in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, (ACM, 2012), pp. 563–578.

Proakis, J.

J. Proakis and G. Manolakis, Digital Signal Processing (Pearson, 2013), Chap. 8.

Qian, C.

Randel, S.

S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
[Crossref]

Schmidt, B. J. C.

J. Armstrong and B. J. C. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12, 343–345 (2008).
[Crossref]

Sinanovic, S.

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modelling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightw. Technol. 31, 3064–3076 (2013).
[Crossref]

D. Tsonev, S. Sinanovic, and H. Haas, “Novel unipolar orthogonal frequency division multiplexing (U-OFDM) for optical wireless,” in Proceedings of IEEE Vehicular Technology Conference (VTC Spring), (IEEE, 2012). pp. 1–5.

Tsonev, D.

D. Tsonev, S. Videv, and H. Haas, “Unlocking spectral efficiency in intensity modulation and direct detection systems,” IEEE J. Sel. Areas Commun. 33, 1758–1770 (2015).
[Crossref]

D. Tsonev, S. Videv, and H. Haas, “Towards a 100 Gb/s visible light wireless access network,” Opt. Express 23, 1627–1637 (2015).
[Crossref] [PubMed]

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modelling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightw. Technol. 31, 3064–3076 (2013).
[Crossref]

M. Islim, D. Tsonev, and H. Haas, “A generalized solution to the spectral efficiency loss in unipolar optical OFDM-based systems,” in Proceedings of IEEE International Conference on Communications (ICC), (IEEE, 2015). pp. 5126.

M. Islim, D. Tsonev, and H. Haas, “Spectrally enhanced PAM-DMT for IM/DD optical wireless communications,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2015), pp. 877–882.

M. Islim, D. Tsonev, and H. Haas, “On the superposition modulation for OFDM-based optical wireless Communication,” in Proceedings of IEEE Global Signal and Information Processing conference, (IEEE, 2015). pp. 1022–1026.

D. Tsonev, S. Sinanovic, and H. Haas, “Novel unipolar orthogonal frequency division multiplexing (U-OFDM) for optical wireless,” in Proceedings of IEEE Vehicular Technology Conference (VTC Spring), (IEEE, 2012). pp. 1–5.

Videv, S.

D. Tsonev, S. Videv, and H. Haas, “Unlocking spectral efficiency in intensity modulation and direct detection systems,” IEEE J. Sel. Areas Commun. 33, 1758–1770 (2015).
[Crossref]

D. Tsonev, S. Videv, and H. Haas, “Towards a 100 Gb/s visible light wireless access network,” Opt. Express 23, 1627–1637 (2015).
[Crossref] [PubMed]

Viterbo, E.

N. Fernando, Y. Hong, and E. Viterbo, “Flip-OFDM for Unipolar Communication Systems,” IEEE Tran. on Commun. 60, 3726–3733 (2012).
[Crossref]

Wang, Q.

Wang, Z.

White, I. H.

Xiong, F.

F. Xiong, Digital Modulation Techniques, (Artech House Publishers, 2006), 2nd ed.

Elect. Lett. (1)

J. Armstrong and A.J. Lowery, “Power efficient optical OFDM,” Elect. Lett. 42, 370–372 (2006).
[Crossref]

IEEE Commun. Lett. (1)

J. Armstrong and B. J. C. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12, 343–345 (2008).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

D. Tsonev, S. Videv, and H. Haas, “Unlocking spectral efficiency in intensity modulation and direct detection systems,” IEEE J. Sel. Areas Commun. 33, 1758–1770 (2015).
[Crossref]

IEEE Photonics Technology Letters (1)

S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, “PAM-DMT for intensity-modulated and direct-detection optical communication systems,” IEEE Photonics Technology Letters 21, 1749–1751 (2009).
[Crossref]

IEEE Tran. on Commun. (1)

N. Fernando, Y. Hong, and E. Viterbo, “Flip-OFDM for Unipolar Communication Systems,” IEEE Tran. on Commun. 60, 3726–3733 (2012).
[Crossref]

J. Lightw. Technol. (1)

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modelling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightw. Technol. 31, 3064–3076 (2013).
[Crossref]

Opt. Express (3)

Other (10)

M. Islim, D. Tsonev, and H. Haas, “A generalized solution to the spectral efficiency loss in unipolar optical OFDM-based systems,” in Proceedings of IEEE International Conference on Communications (ICC), (IEEE, 2015). pp. 5126.

F. Xiong, Digital Modulation Techniques, (Artech House Publishers, 2006), 2nd ed.

J. Proakis and G. Manolakis, Digital Signal Processing (Pearson, 2013), Chap. 8.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse fourier transform,” in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, (ACM, 2012), pp. 563–578.

D. Tsonev, S. Sinanovic, and H. Haas, “Novel unipolar orthogonal frequency division multiplexing (U-OFDM) for optical wireless,” in Proceedings of IEEE Vehicular Technology Conference (VTC Spring), (IEEE, 2012). pp. 1–5.

Cisco Visual Networking Index, “The Zettabyte era: trends and analysis,” White Paper, Cisco (2015), http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/VNI_Hyperconnectivity_WP.pdf .

S. Dimitrov and H. Haas, Principles of LED Light Communications: Towards Networked Li-Fi, (Cambridge University, 2015).
[Crossref]

M. Islim, D. Tsonev, and H. Haas, “Spectrally enhanced PAM-DMT for IM/DD optical wireless communications,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2015), pp. 877–882.

M. Islim, D. Tsonev, and H. Haas, “On the superposition modulation for OFDM-based optical wireless Communication,” in Proceedings of IEEE Global Signal and Information Processing conference, (IEEE, 2015). pp. 1022–1026.

H. Elgala and T. Little, “SEE-OFDM: spectral and energy efficient OFDM for optical IM/DD systems,” in Proceedings of IEEE Personal, Indoor, and Mobile Radio Communication, (IEEE, 2014), pp. 851–855.

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Figures (8)

Fig. 1
Fig. 1 ASE-DMT transmitter block diagram. Xd[k] refers to the kth subcarrier at depth d; S/P denotes for serial to parallel; DAC denotes for digital to analogue conversion; and CP refers to the cyclic prefixing.

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Fig. 2
Fig. 2 An illustration of the frequency domain subcarrier loading at three depths ASE-DMT and the effects of zero clipping. (a) and (c) and (e) shows the imaginary components of the subcarriers before and after zero level time-domain clipping, ℑ[Xd[k]] and ℑ [ X d c [ k ] ], respectively. (b) and (d) and (f) shows the real components of the subcarriers before and after zero level time-domain clipping, ℜ[Xd[k]] and ℜ [ X d c [ k ] ], respectively.

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Fig. 3
Fig. 3 The spectral efficiency of ASE-DMT, eU-OFDM, and ePAM-DMT compared to the spectral efficiency of DCO-OFDM for different FFT and CP lengths at D = 3 and D = 6.

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Fig. 4
Fig. 4 The BER performance of 16-QAM ASE-DMT depths with a total number of depths D = 5. The BER of DCO-OFDM is only shown for comparison purposes.

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Fig. 5
Fig. 5 The relative computation complexity of ASE-DMT and ePAM-DMT in comparison with the computation complexity of DCO-OFDM as a function of the total number of depths D, and the cyclic prefix percentage of the frame size NCP/N.

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Fig. 6
Fig. 6 The peak to average power ratio of ASE-DMT (depths and overall), ePAM-DMT and DCO-OFDM.

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Fig. 7
Fig. 7 The BER performance comparison of ASE-DMT, ePAM-DMT, and DCO-OFDM for different spectral efficiencies in an AWGN channel as a function of: (a) electrical SNR, and (b) optical SNR. The DC biasing levels for DCO-OFDM at M = {4, 64, 1024} are estimated through Monte Carlo simulations at respectively 6 dB, 9.5 dB, and 13 dB as described in (32).

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Fig. 8
Fig. 8 The BER performance comparison of ASE-DMT, ePAM-DMT, and DCO-OFDM for different spectral efficiencies in an AWGN channel as a function of: (a) electrical SNR, and (b) optical SNR. The spectral effiency η is given in [bits/s/Hz]. The DC biasing levels for DCO-OFDM at η = {1.5, 3, 4.5} are estimated through Monte Carlo simulations at respectively 7 dB, 9.5 dB, and 12 dB as described in (32).

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Tables (3)

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Table 1 Computational complexity of the Transmitter and receiver of DCO-OFDM, ePAM-DMT and ASE-DMT.

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Table 2 Energy efficiency gains of ASE-DMT over DCO-OFDM at a BER of 10−4.

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Table 3 The optimal combination of constellation sizes and scaling factors for ASE-DMT and the associated electrical and optical gains over DCO-OFDM at a BER of 10−4, where Md and γd denote the constellation size and the scaling factor for the modulation depth-d, respectively.

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Equations (36)

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x 1 [ n ] = 1 N ∑ k = 0 N − 1 X 1 [ k ] e j 2 π k n N = − 2 N ∑ k = 1 N / 2 − 1 B [ k ] sin 2 π k n N .
x d c [ n ] = x d [ n ] + | x d [ n ] | 2 ,
X d c [ k ] = X d [ k ] + FFT { | x d [ n ] | } 2 ,
FFT { | x 1 [ n ] | } = 1 N ∑ n = 0 N − 1 | x 1 [ n ] | e − j 2 π k n N = 2 N ∑ n = 1 N / 2 − 1 | x 1 [ n ] | cos 2 π k n N .
X 2 [ k ′ ] = { A 2 [ k ′ ] , if     k ′ = 2 k + 1 0 , Otherwise ,
FFT { | x 2 [ n ] | } = 1 N ∑ n = 0 N / 2 − 1 | x 2 [ k ] | e − j 2 π k n N ( 1 + e − j π k ) ,
X d [ k ′ ] = { A d [ k ′ ] , if     k ′ = 2 d − 2 ( 2 k + 1 ) 0 , Otherwise ,
x d [ n ] = − x d [ n + N / 2 d − 1 ]                     ∀ d > 1 .
X d [ k ] = 1 N ∑ n = 0 N / 2 d − 1 − 1 x d [ k ] e − j 2 π k n N κ ( 1 − e − j π k 2 D − 2 ) ,
FFT { | x d [ n ] | } = 1 N ∑ n = 0 N / 2 d − 1 − 1 | x d [ k ] | e − j 2 π k n N κ ( 1 + e − j π k 2 D − 2 ) ,
κ = ∏ d = 2 D − 1 ( 1 + e − j π k 2 d − 2 ) .
x T [ n ] = ∑ d = 1 D x d c [ n ] .
X T [ k ] = j B 1 [ k ] + ∑ d = 2 D A d [ k ] + ∑ d = 1 D FFT { | X d [ n ] | } 2 .
η PAM ( 1 ) = log 2 ( M 1 ) ( N − 2 ) 2 ( N + N CP )                           bits / s / Hz ,
η PAM ( d ) = log 2 ( M d ) N 2 d ( N + N CP )                           bits / s / Hz ,
η ASE ( D ) = ∑ d = 1 D η PAM ( d )                       bits / s / Hz = log 2 ( M 1 ) ( N − 2 ) + ∑ d = 2 D log 2 ( M d ) ( N ) 2 d − 1 2 ( N + N CP ) .
α η ( D , d ) = η ASE ( D ) η PAM ( d ) 2 d − 1 .
P Ele avg ( D , γ _ ) = E [ x T 2 ( t ) ] = E [ ( ∑ d = 1 D x d ( t ) ) 2 ] = σ s 2 ( ∑ d = 1 D γ d − 1 2 d + 2 ϕ 2 ( 0 ) ∑ d 1 = 1 D ∑ d 2 = 1 d 1 ≠ d 2 D ( γ d 1 γ d 2 ) − 1 2 d 1 + d 2 ) ,
P Opt avg ( D , γ _ ) = ∑ d = 1 D E [ s d ( t ) ] = ϕ ( 0 ) σ s ∑ d = 1 D γ d − 1 2 d − 1 ,
α Ele P ( D , γ _ ) = P Ele avg ( D , γ _ ) P Ele , d avg ( γ d ) .
y = Hx + w ,
H = F * Λ F ,
α Ele ( D , d , γ _ ) = α Ele P ( D , γ _ ) α η ( D , d ) .
BER ( D , d , γ _ ) ≅ 2 log 2 ( M d ) ( 1 − 1 M d ) × ∑ l = 1 2 ∑ k = 1 N Q ( ( 2 l − 1 ) 3 | Λ k | 2 E b , elec log 2 ( M d ) N o α Ele ( D , d , γ _ ) ( M d 2 − 1 ) ) ,
C DCO = 2 𝒪 ( 3 N / 2 log 2 ( N ) ) log 2 ( M DCO ) ( N − 2 ) ,
C ASE Tx = 𝒪 ( N / 2 log 2 ( N ) ) + ∑ d = 2 D 𝒪 ( N / 2 d log 2 ( N ) ) ≈ 𝒪 ( N log 2 ( N ) ) ,
C ASE Rx = 𝒪 ( N / 2 log 2 ( N ) ) + ∑ d = 2 D 𝒪 ( N / 2 d log 2 ( N ) ) + ∑ d = 2 D − 1 𝒪 ( N / 2 d log 2 ( N ) ) ≈ 𝒪 ( N log 2 ( N ) ) .
C ASE = 2 ( C ASE Tx + C ASE Rx ) log 2 ( M d ) ( N − 2 ) + ∑ d = 2 D log 2 ( M d ) N / 2 d − 1 ,
C ePAM Tx = ∑ d = 1 D 2 d − D 𝒪 ( N / 2 log 2 ( N ) ) ≈ 𝒪 ( N log 2 ( N ) ) .
C ePAM Rx = 𝒪 ( N / 2 log 2 ( N ) ) + 2 ∑ d = 2 D 2 d − 1 𝒪 ( N / 2 log 2 ( N ) ) 2 D − 1 ≈ 𝒪 ( 2 N log 2 ( N ) ) ,
C ePAM = 2 ( C ePAM Tx + C ePAM Rx ) log 2 ( M ) ( N − 2 ) + ∑ d = 2 D log 2 ( M ) N d / 2 d − 1 ,
B DC dB = 10 log 10 ( k M DCO 2 + 1 ) .
N CP ePAM , max = ⌊ N / ( 2 D − 2 ) − 1 ⌋ .
log 2 ( M DCO ) = 2 ∑ d = 1 D log 2 ( M d ) 2 d .
P Ele avg ( D , γ _ ) ≤ P Ele avg ( D , 1 1 × D ) , P Old avg ( D , γ _ ) ≤ P Opt avg ( D , 1 1 × D ) .
BER ≅ ∑ d = 1 D ( BER ( D , d , γ _ ) α η ( D , d ) ) .

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