Abstract

We demonstrate a novel low-cost photonic-layer secured communication system that incorporates intensity modulation direct detection (IM/DD) scheme with a 4-level pulse amplitude modulation (PAM-4). In the proposed system, the signal is buried under an amplifier’s spontaneous emission (ASE) noise and coded with a spectral phase mask. As a result, the signal is stealthy and encrypted in both frequency and time domains. We analyze the reception performance of the secured signal under direct detection, and analytically compare its SNR with a conventional PAM-4 system, demonstrating that only an eligible receiver achieves a decryption of the stealthy and encrypted signal. Furthermore, we experimentally validate these findings showing a significant processing gain, which allows an error-free reception, despite a negative optical-SNR.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. Sadot, G. Dorman, A. Gorshtein, E. Sonkin, and O. Vidal, “Single channel 112Gbit/sec PAM4 at 56Gbaud with digital signal processing for data centers applications,” Opt. Express 23(2), 991–997 (2015).
    [Crossref] [PubMed]
  2. L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
    [Crossref]
  3. B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
    [Crossref]
  4. J. A. Salehi, “Code division multiple-access techniques in optical fiber networks. I. fundamental principles,” IEEE Trans. Commun. 37(8), 824–833 (1989).
    [Crossref]
  5. T. Yeminy, D. Sadot, and Z. Zalevsky, “Spectral and temporal stealthy fiber-optic communication using sampling and phase encoding,” Opt. Express 19(21), 20182–20198 (2011).
    [Crossref] [PubMed]
  6. T. Yeminy, D. Sadot, and Z. Zalevsky, “Sampling impairments influence over fiber-optic signal decryption,” Opt. Commun. 291(15), 193–201 (2013).
    [Crossref]
  7. E. Wohlgemuth, T. Yeminy, Z. Zalevsky, and D. Sadot, “Experimental demonstration of encryption and steganography in optical fiber communications,” in 43th European Conference on Optical Communication (ECOC 2017), paper P1.SC3.43.
    [Crossref]
  8. E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Demonstration of coherent stealthy and encrypted transmission for data center interconnection,” Opt. Express 26(6), 7638–7645 (2018).
    [Crossref] [PubMed]
  9. E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Low cost PAM-4 IM/DD photonic-layer secured communication for DCI based on phase mask,” in 44th European Conference on Optical Communication (ECOC 2018), paper We1F.6.
  10. M. Chagnon, S. Lessard, and D. V. Plant, “336 Gb/s in direct detection below KP4 FEC threshold for intra data center applications,” IEEE Photonics Technol. Lett. 28(20), 2233–2236 (2016).
    [Crossref]
  11. N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
    [Crossref]

2018 (1)

2016 (1)

M. Chagnon, S. Lessard, and D. V. Plant, “336 Gb/s in direct detection below KP4 FEC threshold for intra data center applications,” IEEE Photonics Technol. Lett. 28(20), 2233–2236 (2016).
[Crossref]

2015 (2)

D. Sadot, G. Dorman, A. Gorshtein, E. Sonkin, and O. Vidal, “Single channel 112Gbit/sec PAM4 at 56Gbaud with digital signal processing for data centers applications,” Opt. Express 23(2), 991–997 (2015).
[Crossref] [PubMed]

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

2014 (1)

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

2013 (1)

T. Yeminy, D. Sadot, and Z. Zalevsky, “Sampling impairments influence over fiber-optic signal decryption,” Opt. Commun. 291(15), 193–201 (2013).
[Crossref]

2011 (1)

1989 (1)

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks. I. fundamental principles,” IEEE Trans. Commun. 37(8), 824–833 (1989).
[Crossref]

Chagnon, M.

M. Chagnon, S. Lessard, and D. V. Plant, “336 Gb/s in direct detection below KP4 FEC threshold for intra data center applications,” IEEE Photonics Technol. Lett. 28(20), 2233–2236 (2016).
[Crossref]

Dochhan, A.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Dorman, G.

Eiselt, M.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Eiselt, N.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Elbers, J.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Gorshtein, A.

Griesser, H.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Jiang, C.

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

Lessard, S.

M. Chagnon, S. Lessard, and D. V. Plant, “336 Gb/s in direct detection below KP4 FEC threshold for intra data center applications,” IEEE Photonics Technol. Lett. 28(20), 2233–2236 (2016).
[Crossref]

Mittal, P.

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

Monroy, I. T.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Olmos, J. J. V.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Plant, D. V.

M. Chagnon, S. Lessard, and D. V. Plant, “336 Gb/s in direct detection below KP4 FEC threshold for intra data center applications,” IEEE Photonics Technol. Lett. 28(20), 2233–2236 (2016).
[Crossref]

Prucnal, P. R.

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

Ren, Y.

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

Sadot, D.

Salehi, J. A.

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks. I. fundamental principles,” IEEE Trans. Commun. 37(8), 824–833 (1989).
[Crossref]

Shastri, B. J.

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

Sonkin, E.

Tait, A. N.

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

Vidal, O.

Wang, J.

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

Wei, J.

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

Wohlgemuth, E.

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Demonstration of coherent stealthy and encrypted transmission for data center interconnection,” Opt. Express 26(6), 7638–7645 (2018).
[Crossref] [PubMed]

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Low cost PAM-4 IM/DD photonic-layer secured communication for DCI based on phase mask,” in 44th European Conference on Optical Communication (ECOC 2018), paper We1F.6.

Wu, B.

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

Xu, L.

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

Yeminy, T.

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Demonstration of coherent stealthy and encrypted transmission for data center interconnection,” Opt. Express 26(6), 7638–7645 (2018).
[Crossref] [PubMed]

T. Yeminy, D. Sadot, and Z. Zalevsky, “Sampling impairments influence over fiber-optic signal decryption,” Opt. Commun. 291(15), 193–201 (2013).
[Crossref]

T. Yeminy, D. Sadot, and Z. Zalevsky, “Spectral and temporal stealthy fiber-optic communication using sampling and phase encoding,” Opt. Express 19(21), 20182–20198 (2011).
[Crossref] [PubMed]

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Low cost PAM-4 IM/DD photonic-layer secured communication for DCI based on phase mask,” in 44th European Conference on Optical Communication (ECOC 2018), paper We1F.6.

Yoffe, Y.

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Demonstration of coherent stealthy and encrypted transmission for data center interconnection,” Opt. Express 26(6), 7638–7645 (2018).
[Crossref] [PubMed]

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Low cost PAM-4 IM/DD photonic-layer secured communication for DCI based on phase mask,” in 44th European Conference on Optical Communication (ECOC 2018), paper We1F.6.

Yuan, J.

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

Zalevsky, Z.

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Demonstration of coherent stealthy and encrypted transmission for data center interconnection,” Opt. Express 26(6), 7638–7645 (2018).
[Crossref] [PubMed]

T. Yeminy, D. Sadot, and Z. Zalevsky, “Sampling impairments influence over fiber-optic signal decryption,” Opt. Commun. 291(15), 193–201 (2013).
[Crossref]

T. Yeminy, D. Sadot, and Z. Zalevsky, “Spectral and temporal stealthy fiber-optic communication using sampling and phase encoding,” Opt. Express 19(21), 20182–20198 (2011).
[Crossref] [PubMed]

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Low cost PAM-4 IM/DD photonic-layer secured communication for DCI based on phase mask,” in 44th European Conference on Optical Communication (ECOC 2018), paper We1F.6.

IEEE Access (1)

L. Xu, C. Jiang, J. Wang, J. Yuan, and Y. Ren, “Information security in big data: privacy and data mining,” IEEE Access 2, 1149–1176 (2014).
[Crossref]

IEEE J. Sel. Top. Signal Process. (1)

B. Wu, B. J. Shastri, P. Mittal, A. N. Tait, and P. R. Prucnal, “Optical signal processing and stealth transmission for privacy,” IEEE J. Sel. Top. Signal Process. 9(7), 1185–1194 (2015).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Chagnon, S. Lessard, and D. V. Plant, “336 Gb/s in direct detection below KP4 FEC threshold for intra data center applications,” IEEE Photonics Technol. Lett. 28(20), 2233–2236 (2016).
[Crossref]

IEEE Trans. Commun. (1)

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks. I. fundamental principles,” IEEE Trans. Commun. 37(8), 824–833 (1989).
[Crossref]

Opt. Commun. (1)

T. Yeminy, D. Sadot, and Z. Zalevsky, “Sampling impairments influence over fiber-optic signal decryption,” Opt. Commun. 291(15), 193–201 (2013).
[Crossref]

Opt. Express (3)

Other (3)

N. Eiselt, H. Griesser, J. Wei, A. Dochhan, M. Eiselt, J. Elbers, J. J. V. Olmos, and I. T. Monroy, “Real-time evaluation of 26-GBaud PAM-4 intensity modulation and direct detection systems for data-center interconnects,” in Optical Fiber Communication Conference, (Optical Society of America,2016), paper Th1G.3.
[Crossref]

E. Wohlgemuth, Y. Yoffe, T. Yeminy, Z. Zalevsky, and D. Sadot, “Low cost PAM-4 IM/DD photonic-layer secured communication for DCI based on phase mask,” in 44th European Conference on Optical Communication (ECOC 2018), paper We1F.6.

E. Wohlgemuth, T. Yeminy, Z. Zalevsky, and D. Sadot, “Experimental demonstration of encryption and steganography in optical fiber communications,” in 43th European Conference on Optical Communication (ECOC 2017), paper P1.SC3.43.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Experimental setup for encrypted and stealthy optical system. The following abbreviation were used: ECL - external cavity laser, SSMF - standard single mode fiber; VOA - variable optical attenuator; BPF - (optical) bandpass filter; EDFA - Erbium-doped fiber amplifier; MZM - Mach-Zehnder modulator; PD -photodiode.
Fig. 2
Fig. 2 SNR versus received OSNR after executing the decryption process (solid lines) and before (dashed lines), measured for a varying number of replicas, which is inversely proportional to the information bandwidth f m . Horizontal dashed line indicates the pre-FEC BER threshold.
Fig. 3
Fig. 3 PG versus received OSNR, for varying number of f m . Solid lines represent the simulation results.
Fig. 4
Fig. 4 Histogram of the received signal before and after spectral phase decoding (SPD), for OSNR of 8 dB/0.1nm.
Fig. 5
Fig. 5 Authorized user BER versus received OSNR, for various information bandwidth cases f m , corresponding to a different number of spectral replicas. A dashed black line represents the Pre-FEC BER threshold.
Fig. 6
Fig. 6 Received SNR penalty as a result of residual chromatic dispersion.

Tables (1)

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Table 1 Experimental setup hardware parameters

Equations (26)

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S tx ( f ) = ( l=L/2 L/2 S( fl f m ) )Ψ( f ) H ps ( f ),
Ψ( f ) = Ψ * ( f ).
E tx ( t ) = E t cos( s tx (t) V π + 3 4 π ).
i Pre-proc. ( t ) = RG( | E tx (t) | 2 E[ | E tx (t) | 2 ] ) i sig ( t ) + R( | n( t ) | 2 E[ | n( t ) | 2 ] ) i sp-sp ( t ) + 2R G ( E tx (t)Re{ n(t) }E[ E tx (t)Re{ n(t) } ] ) i sig-sp ( t ) ,
γ Pre-proc. = σ sig 2 σ sp-sp 2 + σ sig-sp 2 ,
I Post-proc. (f) = l=L/2 L/2 I Pre-proc. ( fl f m ) Ψ * ( fl f m ) H PS * ( fl f m ) .
γ s Post-proc. = E[ | l=L/2 L/2 I sig ( fl f m ) Ψ * ( fl f m ) H PS * ( fl f m ) | 2 ] E[ | l=L/2 L/2 I sp-sp ( fl f m )+ I sp-sig ( fl f m ) Ψ * ( fl f m ) H PS * ( fl f m ) | 2 ] .
PG= γ s Post-proc. / γ s Pre-proc. =L.
i Pre-proc. ( t ) = RG( | E tx (t) | 2 E[ | E tx (t) | 2 ] ) i sig ( t ) + R( | n( t ) | 2 E[ | n( t ) | 2 ] ) i sp-sp ( t ) + 2R G ( E tx (t)Re{ n(t) }E[ E tx (t)Re{ n(t) } ] ) i sig-sp ( t ) ,
σ sig 2  =  R 2 G 2 E[ ( E tx 2 ( t )E[ E tx 2 ( t ) ] ) 2 ] =  R 2 G 2 ( E t 4 / V π 4 ) σ s 2 , σ sig-sp 2  = 4 R 2 GE[ ( E tx ( t )Re{ n( t ) }E[ E tx ( t )Re{ n( t ) } ] ) 2 ] =  4 R 2 GE[ E tx 2 ( t ) ]E[ ( Re{ n( t ) } ) 2 ] =  R 2 G E t 2 N 0 2 BW, σ sp-sp 2  =  R 2 E[ ( | n 2 ( t ) |E[ | n 2 ( t ) | ] ) 2 ] = 2 R 2 ( N 0 2 ) 2 BW,
γ s Pre-proc. = σ sig 2 σ sp-sp 2 + σ sig-sp 2 = G 2 ( E t 2 / V π 2 ) σ s 2 2 ( N 0 2 ) 2 +G E t 2 N 0 2 .
z Pre-proc. ( t ) = i sp-sp ( t )+ i sig-sp ( t ),
Z Pre-proc. ( f ) = I sp-sp ( f )+ I sig-sp ( f ).
Z Post-proc. ( f ) = l=L/2 L/2 ( I sp-sp (fl f m )+ I sig-sp (fl f m ) ) Ψ * ( fl f m ) H PS * ( fl f m ) .
Var{ z Post-proc. (t) }=E[ | z Post-proc. (t) | 2 ]= E[ Z Post-proc. ( f 1 ) e j2π f 1 t d f 1 ( Z Post-proc. ( f 2 ) ) * e j2π f 2 t d f 2 ]= E[ Z Post-proc. ( f 1 ) ( Z Post-proc. ( f 2 ) ) * ] e j2π( f 1 f 2 )t d f 1 df.
E[ Z Post-proc. ( f 1 ) ( Z Post-proc. ( f 2 ) ) * ] =  l=L/2 L/2 l'=L/2 L/2 { E[ ( I sp-sp ( f 1 l f m )+ I sig-sp ( f 1 l f m ) ) ( I * sp-sp ( f 2 l' f m )+ I * sig-sp ( f 2 l' f m ) ) ] Ψ( f 2 l' f m ) Ψ * ( f 1 l f m ) H PS ( f 2 l' f m ) H PS * ( f 1 l f m ) }  .
E[( I sp-sp ( f 1 l f m )+ I sig-sp ( f 1 l f m ) )( I sp-sp * ( f 2 l f m )+ I sig-sp * ( f 2 l f m ) )] =        (a) E[ I sp-sp ( f ˜ 1 ) I sp-sp * ( f ˜ 2 )] +   (b) E[ I sp-sp ( f ˜ 1 ) I sig-sp * ( f ˜ 2 )] +        (c) E[ I sig-sp ( f ˜ 1 ) I sp-sp * ( f ˜ 2 )] +   (d) E[ I sig-sp ( f ˜ 1 ) I sig-sp * ( f ˜ 2 )],
(a) E[ I sp-sp ( f ˜ 1 ) I sp-sp * ( f ˜ 2 )] = 2 R 2 ( N 0 2 ) 2  δ( f 1 f 2 +( l l ) f m ).
(b) E[ I sp-sp ( f ˜ 1 ) I sig-sp * ( f ˜ 2 ) ] =  E[ R( | n( τ ) | 2 E[ | n( τ ) | 2 ] ) e j2π f ˜ 1 τ dτ 2R G E tx ( λ )Re{ n( λ ) } e j2π f ˜ 2 λ dλ ] =  2 G R 2 E[ E tx ( λ ) ]E[ Re{ n( λ ) }( | n( τ ) | 2 E[ | n( τ ) | 2 ] ) ] e j2π f ˜ 1 τ e j2π f ˜ 2 λ dλdτ.
(d) E[ I sig-sp ( f ˜ 1 ) I sig-sp * ( f ˜ 2 ) ] =  E[ 2R G E tx ( τ )Re{ n( τ ) } e j2π f ˜ 1 τ dτ 2R G E tx ( λ )Re{ n( λ ) } e j2π f ˜ 2 λ dλ ] =  4G R 2 E[ Re{ n( τ ) }Re{ n( λ ) } ]E[ E tx ( τ ) E tx ( λ ) ] e j2π f ˜ 1 τ e j2π f ˜ 2 λ dτdλ.
(d) E[ I sig-sp ( f ˜ 1 ) I sig-sp * ( f ˜ 2 ) ] =  R 2 G N 0 E[ E tx 2 ( τ ) ] e j2πτ( f ˜ 1 f ˜ 2 ) dτ R 2 G N 0 E t 2 1 2 e j2πτ( f ˜ 1 f ˜ 2 ) dτ =  R 2 G E t 2 N 0 2 δ( f ˜ 1 f ˜ 2 ).
(d) E[ I sig-sp ( f 1 l f m ) I sig-sp * ( f 2 l f m )] = G R 2 N 0 E t 2 2  δ( f 1 f 2 +( l l ) f m ).
E[ Z Post-proc. ( f 1 ) ( Z Post-proc. ( f 2 ) ) * ] =  l=L/2 L/2 { ( 2 R 2 ( N 0 2 ) 2  δ( f 1 f 2 )+ R 2 G E t 2 N 0 2 δ( f 1 f 2 ) ) Ψ( f 2 l f m ) Ψ * ( f 1 l f m ) H PS ( f 2 l f m ) H PS * ( f 1 l f m ) } ,
Var{ z Post-proc. ( t ) } =    l=L/2 L/2 { ( 2 R 2 ( N 0 2 ) 2 + R 2 G E t 2 N 0 2 ) Ψ( f 1 l f m ) Ψ * ( f 1 l f m ) H PS ( f 1 l f m ) H PS * ( f 1 l f m ) } d f 1   = ( 2 R 2 ( N 0 2 ) 2 + R 2 G E t 2 N 0 2 ) | H PS ( f 1 l f m ) | 2 d f 1  = L( 2 R 2 ( N 0 2 ) 2 + R 2 G E t 2 N 0 2 )BW.
Var{ z Post-proc. ( t ) } = L Var{ z Pre-proc. ( t ) }.
Var{ i  Sig, Post-proc. ( t ) } =  L 2 Var{ i  Sig, Pre-proc. ( t ) }.

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