Abstract

A semi-analytical method to evaluate the bit error ratio (BER) in direct-detection (DD) optical fibre transmission systems employing orthogonal frequency division multiplexing (OFDM) and optically preamplified receivers is proposed. The method considers a Gaussian approach for the signal at the equalizer output and allows evaluating accurately the BER of each OFDM subcarrier for the receiver structure considered and for practical optical and electrical filters shapes, being a powerful tool to perform the optimization of these systems. The results obtained by the proposed method have shown excellent agreement with Monte Carlo estimates for DD-OFDM ultra-wideband radio signals and for two different DD-OFDM signals proposed for long-haul systems.

© 2009 Optical Society of America

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References

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  1. J. Armstrong, "OFDM for optical communications", J. Lightwave Technol. 27, 189-204 (2009).
    [CrossRef]
  2. R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
    [CrossRef]
  3. C. Lin, Y. Lin, J. Chen, S. Dai, P. Shih, P. Peng and S. Chi, "Optical direct-detection OFDM signal generation for radio-over-fibre link using frequency doubling scheme with carrier supression," Opt. Express 16, 6056-6063 (2008).
    [CrossRef]
  4. J. Tang, P. Lane and K. Shore, "30 Gbit/s transmission over 40 km directly modulated DFB laser-based SMF links without optical amplification and dispersion compensation for VSR and metro applications," in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper JThB8.
    [CrossRef]
  5. B. Schmidt, A. Lowery and J. Armstrong, "Experimental demonstration of 20 Gbit/s direct-detection optical OFDM and 12 Gbit/s with a colorless transmitter", in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18.
  6. M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
    [CrossRef]
  7. T. Alves and A. Cartaxo, "Performance degradation due to OFDM-UWB radio signal transmission along dispersive single-mode fiber," Photon. Technol. Lett. 21, 158-160 (2009).
    [CrossRef]
  8. F. Buchali and R. Dischler, "Optimized sensitivity direct detection O-OFDM with multi level subcarrier modulation," in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper OMU5.
  9. A. Lowery and J. Armstrong, "Orthogonal-frequency-division multiplexing for dispersion compensation of longhaul optical systems," Opt. Express 14, 2079-2084 (2006).
    [CrossRef] [PubMed]
  10. W. Peng, K. Feng, S. Chi and A. Willner, "Bit error rate calculation for a single sideband OFDM signal with direct detection optically pre-amplified receivers," in Conference on Lasers and Electro-Optics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper CWN3.
  11. A. Lowery, "Amplified-spontaneous noise limit of optical OFDM lightwave systems," Opt. Express 16, 860-865 (2008).
    [CrossRef] [PubMed]
  12. J. Rebola and A. Cartaxo, "Gaussian approach for performance evaluation of optically preamplified receivers with arbitrary optical and electrical filters," IEE Proc.-Optoelectronics 148, 135-142 (2001).
    [CrossRef]
  13. High Rate UltraWideband PHY and MAC Standard, 2nd ed. Geneve, Switzerland: ECMA Int. (2007).
  14. W. Peng, X. Wu, V. Arbab, K. Feng, B. Shamee, L. Christen, J. Yang, A. Willner and S. Chi, "Theoretical and experimental investigations of direct-detected RF-tone-assisted optical OFDM systems," J. Lightwave Technol. 27, 1332-1339 (2009).
  15. B. Schmidt, A. Lowery and L. Du, "Low sample rate transmitter for direct-detection optical OFDM", in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper OWM4.
  16. S. Jansen, I. Morita and H. Tanaka, "Carrier-to-signal power ratio in fiber-optic SSB-OFDM transmission systems," in IEICE General Conference, Nagoya, Japan, (Institute of Electronics, Information and Communication Engineers, 2007), paper B-10-24.
  17. A. Carlson, P. Crilly and J. Rutledge, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, (McGraw-Hill, New York, 2002).

2009 (3)

2008 (4)

A. Lowery, "Amplified-spontaneous noise limit of optical OFDM lightwave systems," Opt. Express 16, 860-865 (2008).
[CrossRef] [PubMed]

C. Lin, Y. Lin, J. Chen, S. Dai, P. Shih, P. Peng and S. Chi, "Optical direct-detection OFDM signal generation for radio-over-fibre link using frequency doubling scheme with carrier supression," Opt. Express 16, 6056-6063 (2008).
[CrossRef]

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

2006 (1)

2001 (1)

J. Rebola and A. Cartaxo, "Gaussian approach for performance evaluation of optically preamplified receivers with arbitrary optical and electrical filters," IEE Proc.-Optoelectronics 148, 135-142 (2001).
[CrossRef]

Alves, T.

T. Alves and A. Cartaxo, "Performance degradation due to OFDM-UWB radio signal transmission along dispersive single-mode fiber," Photon. Technol. Lett. 21, 158-160 (2009).
[CrossRef]

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

Arbab, V.

Armstrong, J.

Beltran, M.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

Breyer, F.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Bunge, A.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Cartaxo, A.

T. Alves and A. Cartaxo, "Performance degradation due to OFDM-UWB radio signal transmission along dispersive single-mode fiber," Photon. Technol. Lett. 21, 158-160 (2009).
[CrossRef]

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

J. Rebola and A. Cartaxo, "Gaussian approach for performance evaluation of optically preamplified receivers with arbitrary optical and electrical filters," IEE Proc.-Optoelectronics 148, 135-142 (2001).
[CrossRef]

Chen, J.

Chi, S.

Christen, L.

Dai, S.

Feng, K.

Lee, S.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Lin, C.

Lin, Y.

Llorente, R.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

Lowery, A.

Marti, J.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

Morant, M.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

Peng, P.

Peng, W.

Perez, J.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

Petermann, K.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Randel, S.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Rebola, J.

J. Rebola and A. Cartaxo, "Gaussian approach for performance evaluation of optically preamplified receivers with arbitrary optical and electrical filters," IEE Proc.-Optoelectronics 148, 135-142 (2001).
[CrossRef]

Schuster, M.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Shamee, B.

Shih, P.

Spinnler, B.

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

Willner, A.

Wu, X.

Yang, J.

IEE Proc.-Optoelectronics (1)

J. Rebola and A. Cartaxo, "Gaussian approach for performance evaluation of optically preamplified receivers with arbitrary optical and electrical filters," IEE Proc.-Optoelectronics 148, 135-142 (2001).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (3)

Photon. Technol. Lett. (3)

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo and J. Marti, "Ultra-wideband radio signals distribution in FTTH networks", Photon. Technol. Lett. 20, 945-947 (2008).
[CrossRef]

M. Schuster, S. Randel, A. Bunge, S. Lee, F. Breyer, B. Spinnler and K. Petermann, "Spectrally efficient compatible single-sideband modulation for OFDM transmission with direct detection," Photon. Technol. Lett. 20, 670-672 (2008).
[CrossRef]

T. Alves and A. Cartaxo, "Performance degradation due to OFDM-UWB radio signal transmission along dispersive single-mode fiber," Photon. Technol. Lett. 21, 158-160 (2009).
[CrossRef]

Other (8)

F. Buchali and R. Dischler, "Optimized sensitivity direct detection O-OFDM with multi level subcarrier modulation," in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper OMU5.

J. Tang, P. Lane and K. Shore, "30 Gbit/s transmission over 40 km directly modulated DFB laser-based SMF links without optical amplification and dispersion compensation for VSR and metro applications," in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper JThB8.
[CrossRef]

B. Schmidt, A. Lowery and J. Armstrong, "Experimental demonstration of 20 Gbit/s direct-detection optical OFDM and 12 Gbit/s with a colorless transmitter", in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18.

High Rate UltraWideband PHY and MAC Standard, 2nd ed. Geneve, Switzerland: ECMA Int. (2007).

B. Schmidt, A. Lowery and L. Du, "Low sample rate transmitter for direct-detection optical OFDM", in Optical Fibre Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper OWM4.

S. Jansen, I. Morita and H. Tanaka, "Carrier-to-signal power ratio in fiber-optic SSB-OFDM transmission systems," in IEICE General Conference, Nagoya, Japan, (Institute of Electronics, Information and Communication Engineers, 2007), paper B-10-24.

A. Carlson, P. Crilly and J. Rutledge, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, (McGraw-Hill, New York, 2002).

W. Peng, K. Feng, S. Chi and A. Willner, "Bit error rate calculation for a single sideband OFDM signal with direct detection optically pre-amplified receivers," in Conference on Lasers and Electro-Optics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper CWN3.

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Figures (6)

Fig. 1.
Fig. 1.

Block diagram of the optically pre-amplified DD-OFDM receiver.

Fig. 2.
Fig. 2.

a) Spectra of the OFDM-UWB radio signal applied to MZM arms and at the MZM output, with ν0 the optical carrier frequency. PDF of the Q component of two subcarriers of the signal at the equalizer output for a) a modulation index of 19% and b) a modulation index of 64%. PDF obtained by using the GA (lines) and MC simulation (marks).

Fig. 3.
Fig. 3.

a) Mean of the Q component of the first OFDM symbol at equalizer output as a function of the transmitted information subcarrier. Results obtained by using the GA (×) and MC simulation (circles). b) Variance contributions to the total variance of the Q component of the first OFDM symbol at the equalizer output as a function of the transmitted information subcarrier. Results obtained by using the GA (lines) and MC simulation (marks).

Fig. 4.
Fig. 4.

Variance of the Q component of the first OFDM symbol at the equalizer output as a function of the transmitted information subcarrier for a) different types of LPF and modulation indexes values and b) different types of optical filters. Variance obtained by the GA (lines) and the MC simulation (marks).

Fig. 5.
Fig. 5.

BER at the equalizer output as a function of a) the information subcarriers and b) the modulation index. Results obtained by using the GA (lines) and MC simulation (marks).

Fig. 6.
Fig. 6.

a) Spectra of the OFDM signal at the IQ modulator output (I and III) and at the PIN output (II and IV) considering the A-OFDM and the B-OFDM transmitters. b) BER at the equalizer output as a function of CSPR. Results obtained by using the GA (lines) and MC simulation (marks), with the approach i) of OFDM-A (diamond marks and dashed lines) and the approach ii) of OFDM-A (square marks and dashed-dotted lines).

Equations (67)

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e1(t)=[sl,(t)+nl,I,(t)+jnl,Q,(t)]u+[nl,I,(t)+jnl,Q,(t)] u
me,I(γ)[k]={E[se(γ)[k]]}me,Q(γ)[k]={E[se(γ)[k]]}
(σFFT,(I,Q)(γ)[k])2=y=14(σsASE,(I,Q),y(γ)[k])2+(σASEASE,,(I,Q),y(γ)[k])2 + (σASEASE,,(I,Q),y(γ)[k])2
(σsASE,(I,Q),y(γ)[k])2 = Rλ24 +Sy(f1) { exp(j2πfRFt0(γ))2 H1,m,(I,Q)(γ) (f1+fRF,k) +
+exp(j2πfRFt0(γ))2H2,m,(I,Q)(γ)(f1fFR,k)}{exp(j2πfRFt0(γ))2H2,p,(I,Q)(γ)(f1fRF,k)+
+exp(j2πfRFt0(γ))2H1,p,(I,Q)(γ)(f1+fRF,k)}*df1
(σASEASE,(I,Q),y(γ)[k])2=18 +[Sy(f1)*Sy(f1)]Heq,(I,Q)(γ)(f1,k)2df1
(σe,(I,Q)(γ)[k])2=(Ae(γ)[k])2 (σFFT,(I,Q)(γ)[k]))2
BER(I,Q)[k]=1Ns[γa(I,Q)(γ)=0=1NsQ(F(I,Q)[k]me,(I,Q),0(γ)[k]σe,(I,Q)(γ)[k])+γa(I,Q)(γ)=1=1NsQ(me,(I,Q),1(γ)[k]F(I,Q)[k])σe,(I,Q)(γ)[k])]
BER= k=1(kΩ)N × BER[k]Ni
sPIN(t)=Rλel(t)·el*(t)=Rλ { s(t)2 + 2sI (t) nI, (t)+2sQ (t) nQ, (t) + nI,2 (t) + nQ,2 (t)+
+ nI,2 (t) + nQ,2 (t) }
sLPF,I(t)=[sPIN(t)2cos(2πfRFt+ϕ1)] * hr (t) = Rλ2 +[s(τ)2×+2sI(τ)nI,(τ) +
+ 2sQ (τ) nQ, (τ) + nI,2 (τ) + nQ,2 (τ)+nI,2(τ) + nQ,2 (τ) ] cos(2πfRFτ+ϕ1)hr(t-τ)dτ
sLPF,Q (t) = [sPIN(t)2sin(2πfRFt+ϕ2)] * hr (t)=Rλ2 +[s(τ)2+2sI(τ)nI,(τ)+
+ 2sQ (τ)nQ,(τ)+nI,2(τ)+nQ,2(τ)+nI,2 (τ) + nQ,2 (τ)]sin(2πfRFτ+ϕ2)hr(t-τ)dτ
sLPF (t0(γ)+(i1)Tc)=+sPIN(τ)2[cos(2πfRFτ+ϕ1)jsin(2πfRFτ+ϕ2)]×
×hr(t0(γ)+(i1)Tcτ)
sFFT(γ) [k] = sFFT,I(γ) [k]+jsFFT,Q(γ)[k]=i=1NsLPF(γ)[i]exp[jWN(k,i)]
sFFT,(I,Q)(γ) [k]=12i=1N+sPIN(τ)f(I,Q)(τ,k,i)hr(t0(γ)+(i1)Tcτ)
fI (τ,k,i)=cos(2πfRFτ+ϕ1)cos[WN(k,i)]sin(2πfRFτ+ϕ2)sin[WN(k,i)]
fQ (τ,k,i)=cos(2πfRFτ+ϕ1)sin[WN(k,i)]sin(2πfRFτ+ϕ2)cos[WN(k,i)]
E [sFFT(γ)[k]] = Rλ2 i=1N{{s(t0(γ)+(i1)Tc)2[cos(2πfRF(t0(γ)+(i1)Tc)+ϕ1)
j sin (2πfRF(t0(γ)+(i1)Tc)+ϕ2)]}*hr(t0(γ)+(i1)Tc)+
+ Pn [{Hr(fRF)exp[j2πfRF[t0(γ)+(i1)Tc]]exp(1)}{Hr(fRF)×
×exp[j2πfRF[t0(γ)+(i1)Tc]]exp(2)}]}exp[jWN(k,i)]
E [se(γ)[k]] = E [sFFT(γ)[k]] Ae(γ) [k]exp(jΦe(γ)[k])
me,I(γ) [k]={E[se(γ)[k]]}me,Q(γ)[k]={E[se(γ)[k]]}
(σFFT,(I,Q)(γ)[k])2 = E [(sFFT,(I,Q)(γ)[k])2] {E[sFFT,(I,Q)(γ)[k]]}2
E [sFFT,(I,Q)(γ)[k]] = 12 i=1N+E[sPIN(τ)]f(I,Q)(τ,k,i)hr(t0(γ)+(i1)Tcτ)
E = [sPIN(τ)]=Rλ{s(τ)2+E[nI,2(τ)]+E[nQ,2(τ)]+E[nI,2(τ)]+E[nQ,2(τ)]}
(σFFT,(I,Q)(γ)[k])2 = Rλ2 i=1Nn=1N++{+[sI(τ1)sI(τ2)+SnI,nI,(f1)+sI(τ1)×
× sQ (τ2)SnI,nQ,(f1)+sQ(τ1)sI(τ2)SnQ,nI,(f1)+sQ(τ1)sQ(τ2)SnQ,nQ,(f1)]×
×exp[j2πf1(τ1τ2)]df1+12++[SnI,nI,(f2)SnI,nI,(f3)+SnI,nQ,(f2)SnI,nQ,(f3)+
+ SnQ,nI, (f2)SnQ,nI,(f3)+SnQ,nQ,(f2)SnQ,nQ,(f3)+SnI,nI,(f2)SnI,nI,(f3)+
+ SnI,nQ, (f2)SnI,nQ,(f3)+SnQ,nI,(f2)SnQ,nI,(f3)+SnQ,nQ,(f2)SnQ,nQ,(f3)]×
×exp[j2π(f2+f3)(τ1τ2)]df2df3}g(I,Q)(τ1,τ2,k,i,n)dτ1dτ2
g(I,Q) (τ1,τ2,k,i,n)=f(I,Q)(τ1,k,i)f(I,Q)(τ2,k,n)hr(t0(γ)+(i1)Tcτ1)×
×hr(t0(γ)+(n1)Tcτ2)
(σsASE,(I,Q),y(γ)[k])2 = Rλ2 i=1Nn=1N+++sm(τ1)sp(τ2)Sy(f1)exp[j2πf1(τ1τ2)]df1×
×g(I,Q)(τ1,τ2,k,i,n)dτ1dτ2
sm (τ1)={sI(τ1)ify=1ory=2sQ(τ1)ify=3ory=4 sp(τ2)={sI(τ2)ify=1ory=3sQ(τ2)ify=2ory=4
Sy (f1)={SnI,nI,(f1)ify=1SnI,nQ,(f1)ify=2SnQ,nI,(f1)ify=3SnQ,nQ,(f1)ify=4
(σsASE,(I,Q),y(γ)[k])2 =Rλ2i=1Nn=1N++Sy(f1)Sm(f2)F(I,Q)(f3,k,i)Hr(f4)Sp(f5)×
×F(I,Q)(f6,k,n)Hr(f7)exp[j2πf4(t0(γ)+(i1)Tc)]exp[j2πf7(t0(γ)+(n1)Tc)]×
×exp[j2π(f1+f2+f3f4)τ1]exp[j2π(f1+f5+f6f7)τ2]df1df7dτ1dτ2
FI (f,k,i)=𝓕{f1(f,k,i)}=12[A(k,i)exp(1)+jB(k,i)exp(2)]δ(ffRF)+
+ 12 [A(k,i)exp(1)jB(k,i)exp(2)]δ(f+fRF)
FQ (f,k,i)=𝓕{fQ(f,k,i)}12[B(k,i)exp(1)+jA(k,i)exp(2)]δ(ffRF)+
+ 12 [B(k,i)exp(1)jA(k,i)exp(2)]δ(f+fRF)
(σsASE,(I,Q),y(γ)[k])2 Rλ24 +Sy(f1){Rl,m,(I,Q)(γ)(f1fRF,k)+R2,m,(I,Q)(γ)(f1+fRF,k)}×
×{R2,p,(I,Q)(γ)(f1+fRF,k)+R1,p,(I,Q)(γ)(f1fRF,k)}*df1
R1,m,(I,Q)(γ) (f1fRF,k)=exp(j2πfRFt0(γ))2 H1,m,(I,Q)(γ) (f1+fRF,k)
R2,m,(I,Q)(γ) (f1+fRF,k)=exp(j2πfRFt0(γ))2 H2,m,(I,Q)(γ) (f1fRF,k)
Hi,m,(I,Q)(γ) (f,k)=𝓕 {νm(γ)(t)gi,(I,Q)(γ)(t,k)}gi,(I,Q)(γ)(t,k)=𝓕1{Gi,(I,Q)(γ)(f,k)}
νm(γ) = 𝓕1 {Vm(γ)(f)}=𝓕1{exp(j2πft0(γ))sm(f)}
G(I,Q)(γ) = [G1,(I,Q)(γ)(f,k)G2,(I,Q)(γ)(f,k)] = Hr (f)P(I,Q)T(γ)
T1(γ) (f,k)=exp{j2πN12[fTc+(k1)N]}[sin{2π[fTc+(k1)N]N2}sin{π[fTc+(k1)N]}]
T2(γ) (f,k)=exp{j2πN12[fTc(k1)N]}[sin{2π[fTc(k1)N]N2}sin{π[fTc(k1)N]}]
PI = [exp(1)+exp(2)exp(1)exp(2)exp(1)exp(2)exp(1)+exp(2)]
PQ = j [exp(1)+exp(2)exp(1)+exp(2)exp(1)exp(2)exp(1)exp(2)]
(σASEASE,(I,Q),y(γ)[k])2 = Rλ22 i=1Nn=1N++++Sy(f2)Sy(f3)×
×exp[j2π(f2+f3)(τ1τ2)]df2df3g(I,Q)(τ1,τ2,k,i,n)dτ1dτ2
(σASEASE,(I,Q),y(γ)[k])=18+[Sy(f1)*Sy(f1)]Heq,(I,Q)(γ)(f1,k)2df1
[Heq,1,(I,Q)(γ)(f,k)Heq,2,(I,Q)(γ)(f,k)]=Hr(f)2exp[j2πft0(γ)]P(I,Q)T(γ)
se,I(γ) [k]=Ae(γ)[k]cos(Φe(γ)[k])sFFT,I(γ)[k]+Ae(γ)[k]sin(Φe(γ)[k])sFFT,Q(γ)[k]
se,Q(γ) [k]=Ae(γ)[k]cos(Φe(γ)[k])sFFT,Q(γ)[k]Ae(γ)[k]sin(Φe(γ)[k])sFFT,I(γ)[k]

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