Abstract

A Quantum Stream Cipher (QSC) using Quadrature Amplitude Modulation (QAM) is presented to greatly increase the secure degree compared with ASK or PSK/QSC. We propose encoding multi-bit data in one symbol with a multi-bit basis state, resulting in QAM/QSC, which employs amplitude and phase encryption of the light beam simultaneously. A 16 QAM/QSC experiment at 10 Gbit/s was successfully carried out over 160 km using a digital coherent optical transmission technique, where 16 QAM data were encrypted in a constellation with 32 × 32~4096 × 4096 symbols. We show experimentally that the Number of Masked Signals (NMS) in the quantum noise ΓQAM for QAM/QSC becomes a square multiple larger than ΓASK for ASK/QSC. ΓQAM exceeds 10,000. This result indicates that the QSC technique is more robust against eavesdroppers than ASK or PSK/QSC.

© 2014 Optical Society of America

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References

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  1. M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds., High Spectral Density Optical Transmission Technologies, Springer (2010).
  2. A. J. Lowery, L. Du, and J. Armstrong, “Orthogonal frequency division multiplexing for adaptive dispersion compensation in long-haul WDM systems,” in Proceedings of OFC2006 (Anaheim, USA) PDP39.
  3. Q. Yang, Y. Ma, and W. Shieh, “107 Gb/s coherent optical OFDM reception using orthogonal band multiplexing,” in Proceedings of OFC2008 (San Diego, USA) PDP7.
  4. Y. Koizumi, K. Toyoda, M. Yoshida, M. Nakazawa, “1024 QAM (60 Gbit/s) single-carrier coherent optical transmission over 150 km,” Opt. Express 20(11), 12508–12514 (2012).
    [CrossRef] [PubMed]
  5. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proc. Int. Conf. Comput., Syst., Signal Process. (Bangalore, India, 2011) 175–179.
    [CrossRef]
  6. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
    [CrossRef] [PubMed]
  7. T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
    [CrossRef] [PubMed]
  8. G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
    [CrossRef] [PubMed]
  9. E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
    [CrossRef]
  10. C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
    [CrossRef]
  11. O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
    [CrossRef]
  12. G. S. Kanter, D. Reilly, N. Smith, “Practical physical-layer encryption: The marriage of optical noise with traditional cryptography,” IEEE Commun. Mag. 47(11), 74–81 (2009).
    [CrossRef]
  13. K. Harasawa, O. Hirota, K. Yamashita, M. Honda, K. Ohhata, S. Akutsu, T. Hosoi, Y. Doi, “Quantum encryption communication over a 192-km 2.5-Gbit/s line with optical transceivers employing Yuen-2000 protocol based on intensity modulation,” J. Lightwave Technol. 29(3), 316–323 (2011).
    [CrossRef]
  14. F. Futami and O. Hirota, “40 Gbit/s (4 x 10 Gbit/s) Y-00 protocol for secure optical communication and its transmission over 120 km,” in Proceedings of OFC2012 (Los Angeles, USA) OTu1H.6.
  15. K. Kato and O. Hirota, “Quantum quadrature amplitude modulation system and its applicability to coherent state quantum cryptography,” SPIE 5893, Quantum Communications and Quantum Imaging III (Bellingham, USA, 2005) 589303.
  16. D. Reilly and G. S. Kanter, “Noise-enhanced encryption for physical layer security in an OFDM radio,” IEEE Radio and Wireless Symposium (RWS ’09), TU2P–28.
    [CrossRef]
  17. K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
    [CrossRef]
  18. H. Nyquist, “Certain topics in telegraph transmission theory,” Transact. Am. Inst. Elec. Eng. 47, 617–644 (1928).
  19. K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
    [CrossRef]
  20. F. Futami and O. Hirota, “Masking of 4096-level intensity modulation signals by noises for secure communication employing Y-00 cipher protocol,” in Proceedings of ECOC2011 (Geneva, Switzerland) Tu.6.C.4.
  21. O. Hirota, “Practical security analysis of a quantum stream cipher by the Yuen 2000 protocol,” Phys. Rev. A 76(3), 032307 (2007).
    [CrossRef]

2012

2011

2009

G. S. Kanter, D. Reilly, N. Smith, “Practical physical-layer encryption: The marriage of optical noise with traditional cryptography,” IEEE Commun. Mag. 47(11), 74–81 (2009).
[CrossRef]

2007

K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
[CrossRef]

O. Hirota, “Practical security analysis of a quantum stream cipher by the Yuen 2000 protocol,” Phys. Rev. A 76(3), 032307 (2007).
[CrossRef]

2006

K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
[CrossRef]

2005

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
[CrossRef]

O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
[CrossRef]

2003

G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
[CrossRef] [PubMed]

2000

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

1991

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[CrossRef] [PubMed]

1928

H. Nyquist, “Certain topics in telegraph transmission theory,” Transact. Am. Inst. Elec. Eng. 47, 617–644 (1928).

Akutsu, S.

Barbosa, G. A.

G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
[CrossRef] [PubMed]

Corndorf, E.

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
[CrossRef]

G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
[CrossRef] [PubMed]

Doi, Y.

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[CrossRef] [PubMed]

Fuse, M.

O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
[CrossRef]

Harasawa, K.

Hirota, O.

K. Harasawa, O. Hirota, K. Yamashita, M. Honda, K. Ohhata, S. Akutsu, T. Hosoi, Y. Doi, “Quantum encryption communication over a 192-km 2.5-Gbit/s line with optical transceivers employing Yuen-2000 protocol based on intensity modulation,” J. Lightwave Technol. 29(3), 316–323 (2011).
[CrossRef]

O. Hirota, “Practical security analysis of a quantum stream cipher by the Yuen 2000 protocol,” Phys. Rev. A 76(3), 032307 (2007).
[CrossRef]

O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
[CrossRef]

Honda, M.

Hongo, J.

K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
[CrossRef]

Hosoi, T.

Jennewein, T.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

Kanter, G. S.

G. S. Kanter, D. Reilly, N. Smith, “Practical physical-layer encryption: The marriage of optical noise with traditional cryptography,” IEEE Commun. Mag. 47(11), 74–81 (2009).
[CrossRef]

C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
[CrossRef]

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

Kasai, K.

K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
[CrossRef]

K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
[CrossRef]

Kato, K.

O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
[CrossRef]

Koizumi, Y.

Kumar, P.

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
[CrossRef]

G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
[CrossRef] [PubMed]

Liang, C.

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
[CrossRef]

Nakazawa, M.

Y. Koizumi, K. Toyoda, M. Yoshida, M. Nakazawa, “1024 QAM (60 Gbit/s) single-carrier coherent optical transmission over 150 km,” Opt. Express 20(11), 12508–12514 (2012).
[CrossRef] [PubMed]

K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
[CrossRef]

K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
[CrossRef]

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” Transact. Am. Inst. Elec. Eng. 47, 617–644 (1928).

Ohhata, K.

Reilly, D.

G. S. Kanter, D. Reilly, N. Smith, “Practical physical-layer encryption: The marriage of optical noise with traditional cryptography,” IEEE Commun. Mag. 47(11), 74–81 (2009).
[CrossRef]

Simon, C.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

Smith, N.

G. S. Kanter, D. Reilly, N. Smith, “Practical physical-layer encryption: The marriage of optical noise with traditional cryptography,” IEEE Commun. Mag. 47(11), 74–81 (2009).
[CrossRef]

Sohma, M.

O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
[CrossRef]

Suzuki, A.

K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
[CrossRef]

Toyoda, K.

Weihs, G.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

Weinfurter, H.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

Yamashita, K.

Yoshida, M.

Y. Koizumi, K. Toyoda, M. Yoshida, M. Nakazawa, “1024 QAM (60 Gbit/s) single-carrier coherent optical transmission over 150 km,” Opt. Express 20(11), 12508–12514 (2012).
[CrossRef] [PubMed]

K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
[CrossRef]

K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
[CrossRef]

Yuen, H. P.

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
[CrossRef] [PubMed]

Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

IEEE Commun. Mag.

G. S. Kanter, D. Reilly, N. Smith, “Practical physical-layer encryption: The marriage of optical noise with traditional cryptography,” IEEE Commun. Mag. 47(11), 74–81 (2009).
[CrossRef]

IEEE Photon. Technol. Lett.

C. Liang, G. S. Kanter, E. Corndorf, P. Kumar, “Quantum noise protected data encryption in a WDM network,” IEEE Photon. Technol. Lett. 17(7), 1573–1575 (2005).
[CrossRef]

IEICE Electron. Express

K. Kasai, A. Suzuki, M. Yoshida, M. Nakazawa, “Performance improvement of an acetylene (C2H2) frequency-stabilized fiber laser,” IEICE Electron. Express 3(22), 487–492 (2006).
[CrossRef]

K. Kasai, J. Hongo, M. Yoshida, M. Nakazawa, “Optical phase-locked loop for coherent transmission over 500 km using heterodyne detection with fiber lasers,” IEICE Electron. Express 4(3), 77–81 (2007).
[CrossRef]

J. Lightwave Technol.

Opt. Express

Phys. Rev. A

O. Hirota, M. Sohma, M. Fuse, K. Kato, “Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme,” Phys. Rev. A 72(2), 022335 (2005).
[CrossRef]

O. Hirota, “Practical security analysis of a quantum stream cipher by the Yuen 2000 protocol,” Phys. Rev. A 76(3), 032307 (2007).
[CrossRef]

E. Corndorf, C. Liang, G. S. Kanter, P. Kumar, H. P. Yuen, “Quantum noise randomized data encryption for wavelength division multiplexed fiber optic network,” Phys. Rev. A 71(6), 062326 (2005).
[CrossRef]

Phys. Rev. Lett.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[CrossRef] [PubMed]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000).
[CrossRef] [PubMed]

G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90(22), 227901 (2003).
[CrossRef] [PubMed]

Transact. Am. Inst. Elec. Eng.

H. Nyquist, “Certain topics in telegraph transmission theory,” Transact. Am. Inst. Elec. Eng. 47, 617–644 (1928).

Other

M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds., High Spectral Density Optical Transmission Technologies, Springer (2010).

A. J. Lowery, L. Du, and J. Armstrong, “Orthogonal frequency division multiplexing for adaptive dispersion compensation in long-haul WDM systems,” in Proceedings of OFC2006 (Anaheim, USA) PDP39.

Q. Yang, Y. Ma, and W. Shieh, “107 Gb/s coherent optical OFDM reception using orthogonal band multiplexing,” in Proceedings of OFC2008 (San Diego, USA) PDP7.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proc. Int. Conf. Comput., Syst., Signal Process. (Bangalore, India, 2011) 175–179.
[CrossRef]

F. Futami and O. Hirota, “40 Gbit/s (4 x 10 Gbit/s) Y-00 protocol for secure optical communication and its transmission over 120 km,” in Proceedings of OFC2012 (Los Angeles, USA) OTu1H.6.

K. Kato and O. Hirota, “Quantum quadrature amplitude modulation system and its applicability to coherent state quantum cryptography,” SPIE 5893, Quantum Communications and Quantum Imaging III (Bellingham, USA, 2005) 589303.

D. Reilly and G. S. Kanter, “Noise-enhanced encryption for physical layer security in an OFDM radio,” IEEE Radio and Wireless Symposium (RWS ’09), TU2P–28.
[CrossRef]

F. Futami and O. Hirota, “Masking of 4096-level intensity modulation signals by noises for secure communication employing Y-00 cipher protocol,” in Proceedings of ECOC2011 (Geneva, Switzerland) Tu.6.C.4.

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Figures (8)

Fig. 1
Fig. 1

Operation principle of 16 QAM/QSC. Here, 16 QAM (each 2 bit for I and Q) data is encrypted by using 64 basis states (3 bits for I and Q, respectively).

Fig. 2
Fig. 2

Generation scheme of encrypted 16 QAM data with a m-bit basis state.

Fig. 3
Fig. 3

Constellations of encrypted electrical data. (a) Original 16 QAM data before encryption, (b) 22(2+m) QAM data after encryption. The blue, yellow, and green squares show three examples of basis state. The QAM signal is moving every time slot.

Fig. 4
Fig. 4

Experimental set-up for 16 QAM/QSC digital coherent transmission over 160 km. The symbol rate is 2.5 Gsymbol/s and the QAM multiplicity is 4 bit/symbol, resulting in a 10 Gbit/s transmission.

Fig. 5
Fig. 5

Comparison of QAM/QSC (case 1) and ASK/QSC (case 2). The normalized minimum decision level, Δ, is defined by Δ = 2/(256−1), where the I and Q levels are normalized to ± 1.

Fig. 6
Fig. 6

ΓQAM (Number of Masked Signals in the quantum noise for QAM/QSC) and ΓASK (Number of Masked Signals in the quantum noise for ASK/QSC) as a function of output power Pout from the transmitter. Here, M (multiplicity of I and Q encrypted signal) is set at 256 levels (8 bits).

Fig. 7
Fig. 7

Experimental results of the demodulation performance for the eavesdropper. Here Detection Failure Probability (DFP) can be described as a function of M.

Fig. 8
Fig. 8

ΓQAM and ΓASK as a function of multiplicity M. ΓASK reported in ref [20]. is also plotted.

Equations (8)

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

σ ¯ 4ASK = 1 4 n=1 4 σ I,n 2 ,
σ ¯ 16QAM = 1 32 n=1 16 ( σ I,n 2 + σ Q,n 2 )
Γ QAM = ( 2 σ ¯ 16QAM /Δ ) 2
Γ ASK =2 σ ¯ 4ASK /Δ
Γ 256×256QAM = ( 2 σ ¯ 16QAM /Δ ) 2 =132.7for P out of 35 dBm,
Γ 256ASK =2 σ ¯ 4ASK /Δ=11.3for P out of 38 dBm.
( Γ ASK ) k/ log 2 M = ( 2 σ ¯ ASK /Δ ) k/ log 2 M = [ σ ¯ ASK (M1)] k/ log 2 M ( σ ¯ ASK M) k/ log 2 M = 2 k σ ¯ ASK k/ log 2 M
( Γ QAM ) k/ log 2 M = ( Γ ASK 2 ) k/ log 2 M = 2 2k σ ¯ ASK 2k/ log 2 M

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