Abstract

The performances of radio-over-fiber (RoF) link with fixed incident optical power on photodetector (PD) are improved through carrier suppression method. Firstly, a precise analytical model is proposed to quantify the relationship between the improvement of link gain, noise figure (NF), spur-free dynamic range (SFDR) and the carrier suppression ratio x, in which, the modulation index m is fully considered for the first time to our knowledge. Then the optimum optical carrier-to-sideband ratio (CSR) for RoF link performances in both double-sideband and single-sideband modulation is obtained from the optimum x for the link performances. Finally the experiments with the carrier subtraction method realized by Stimulated Brillouin scattering (SBS) are carried out and the experimental results show good agreement with the simulation ones.

© 2010 OSA

1. Introduction

Radio-over-fiber (RoF) link has great potential in wide applications due to its unique advantages of lightweight, low loss, high capacity and high immunity to electromagnetic interferences [1,2]. However, from the view of the key figures of merit for the performances of RoF link, including link gain, noise figure (NF) and spur-free dynamic range (SFDR), there still exist some problems in the commonly used RoF link. For example, the low efficiency of RF-to-optical and optical-to-RF conversion that limits link gain, the additional noises composed of relative intensity noise (RIN) and shot noise that exacerbate NF and SFDR. Since many applications demand for a link with high gain, low NF and high linearity over a wide band [2,3], lots of efforts have been done to improve the RoF links’ performances.

Increasing the optical power incident on the photodetector (PD) can quadratically increase the link gain [4]. However, as shot noise and RIN are proportional to the PD received average optical power linearly and quadratically, NF will be reduced for a shot-noise-dominated link while remains unchanged for an RIN-dominated link when increasing the PD received average optical power [5]. Also, this method will be limited by PD’s power tolerance. The optical carrier power contains no information while takes up most of the total optical power in quadrature-biased modulation link, hence suppressing optical carrier power is considered to be a candidate to improve the RoF link’s performances. Many approaches have been used to suppress the optical carrier power, such as low-biasing Mach-Zehnder modulator (MZM) [58], optical carrier filtering [9], Stimulated Brillouin scattering (SBS) [10]. For the low-biasing scheme, more input optical power is needed to keep the same modulation efficiency [5]. SBS has been demonstrated to be a good carrier filter due to its narrow gain spectrum and immunity to carrier drift [10, 11].

There have been some works quantifying the relationship between carrier suppression and the link performances. For example, the optimum carrier-to-sideband ratio (CSR) for the performances of single-sideband modulated RoF link was thoroughly investigated in [12]. Reference [13] demonstrated that an amount of carrier suppression resulted in an equal amount of RF gain. However, reference [12] just investigated on bit error rate (BER) of RoF link, and the conclusion in [13] was not precise when modulation index is large. In this paper, a more complete theoretical analysis of the carrier subtraction for the performance improvements of RoF link with constant PD incident optical power is investigated by considering the modulation index m. Both double-sideband and single-sideband modulation schemes are included. The optimum carrier suppression ratio and best CSR for the link gain, NF and SFDR are theoretically calculated and experimentally verified through carrier subtraction by SBS. As carrier suppression method will increase even-order distortion; it is only useful in sub-octave bandwidth applications. Therefore the result of the second-order harmonic relative to fundamental component with regards to carrier suppression ratio x is also considered in this paper.

The paper is organized as follows. In Section 2, the theory and simulation results that investigate the impacts of carrier subtraction on the link performances are introduced. The experimental configuration and results are shown in Section 3, including improved results of link gain, NF and SFDR. Section 4 contains a total conclusion of this paper.

2. Theory

The theory below is based on the premises that the incident optical power on PD is constant, and the modulator is a single drive MZM biased at quadrature. The modulation index m in the RoF link can be expressed as

m=π2VπVmod
Here, Vmod is the amplitude of the input RF signal, Vπ is the half-wave voltage of MZM. The output of MZM before the carrier suppression is
Ebefore(t)=Lcos(πVDC2Vπ+mcosωrft)E0(t)=L[cosπVDC2Vπcos(mcosωrft)sinπVDC2Vπsin(mcosωrft)]E0(t)=2L2[cos(mcosωrft)sin(mcosωrft)]E0(t)=2L2[J0(m)+2k=1+(1)kJ2k(m)cos(2kωrft)2k=0+(1)kJ2k+1(m)cos[(2k+1)ωrft]]E0(t)
Pbefore(t)=P0L2[1+cos(πVDCVπ+2mcosωrft)]=P0L2{1+cosπVDCVπ[J0(2m)+2k=1+(1)kJ2k(2m)cos(2kωrft)]sinπVDCVπ[2k=0+(1)kJ2k+1(2m)cos[(2k+1)ωrft]]}
Paverage_before=P0L2[1+cosπVDCVπJ0(2m)]=P0L2
where E0(t) is the electric field input into the MZM, ω0 is the input optical frequency, L is the RoF link total loss, VDC=Vπ2 is the bias voltage of MZM, P0 is the input optical power, and Paverage_before is the average optical power incident on PD before the carrier suppression.

After suppressing the optical carrier, in order to keep PD incident optical power constant, E0(t) is amplified to E1(t) . The output of MZM after carrier subtraction is

Eafter(t)=2L2[J0(m)+2k=1+(1)kJ2k(m)cos(2kωrft)2k=0+(1)kJ2k+1(m)cos[(2k+1)ωrft]xJ0(m)]E1(t)=Lcos(πVDC2Vπ+mcosωrft)E1(t)2L2xJ0(m)E1(t)
Pafter(t)=Eafter(t)Eafter*(t)=P1L2{1+cosπVDCVπ[J0(2m)+2k=1+(1)kJ2k(2m)cos(2kωrft)]sinπVDCVπ[2k=0+(1)kJ2k+1(2m)cos[(2k+1)ωrft]]}P1L22xJ0(m)[J0(m)+2k=1+(1)kJ2k(m)cos(2kωrft)2k=0+(1)kJ2k+1(m)cos[(2k+1)ωrft]]+P1L2x2J0(m)2
Paverage_after=P1L2[1+J0(m)2(x22x)]
Here, P1 is the input optical power after the carrier subtraction, x is the carrier suppression ratio and Paverage_after is the average optical power incident on PD after the carrier suppression. Since PD incident optical power is constant, which means Paverage_after=Paverage_before , the relationship between P0 and P1 can be expressed as

P1P0=11+J0(m)2(x22x)

The RF power gain before and after carrier subtraction can be obtained from Eq. (3) and Eq. (6), respectively.

Gafter={P1L2[2J1(2m)4xJ0(m)J1(m)]}2Gbefore=[P0L22J1(2m)]2
Substituting Eq. (8) into Eq. (9), we can get the relationship between Gafter and Gbefore

GafterGbefore=[11+J0(m)2(x22x)J1(2m)2xJ0(m)J1(m)J1(2m)]2

A conclusion has been drawn in [13] that an amount of carrier suppression results in an equal amount of RF power gain. From the above analysis, we can see that this result is only true when modulation index m is small enough that J1(2m)m , J0(m)1 and J1(m)m/2 . In this case, the RF power gain after carrier subtraction is increased by (1-x)−2 (supposing Gbefore as 1) and the optical carrier power after carrier subtraction is also reduced by (1-x)2. However, when the modulation index is not so small, the above approximation for the Bessel function is not exact, and the result is not precisely true.

Figure 1 shows the simulation results of Gafter with regards to the carrier suppression ratio x under different modulation index m supposing Gbefore as 1. The curve of (1-x)−2 is also shown as curve a. It can be seen that the smaller m is, the more the curve of link gain overlaps with curve a. Besides, the link gain increases with the increasing of carrier suppression ratio x only when m is small enough and x is smaller than a certain value, beyond which the RF gain drops quickly.

 

Fig. 1 The simulation results of normalized RF link gain with regards to the carrier suppression ratio x under different modulation index m. a: the curve of (1-x)−2; b: m = 0.04; c: m = 0.1; d: m = 0.3; e: m = 0.5; f: m = 0.7.

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Figure 1 also shows that the carrier subtraction method for the performance improvement is only useful when m is small. While m>0.5, it has little impact, which can be seen from curve e and f. In order to obtain a simple expression of the optimum x and CSR for this method when m is small, the above unexpanded expressions are simplified by ignoring the third-order and all the higher order sidebands, and leaving only the first item of Bessel function Taylor series. Thus, we can get the approximated expressions as below when m is small.

Ebefore(t)=2P0L2{J0(m)cos(ω0t)J1(m)[cos(ω0tωmodt)+cos(ω0t+ωmodt)]J2(m)[cos(ω0t2ωmodt)+cos(ω0t+2ωmodt)]}
Paverage_before=P0L2{[J0(m)]2+2[J1(m)]2+2[J2(m)]2}P0L2(1+m22)
Eafter(t)=2P1L2{(1x)J0(m)cos(ω0t+φ0)J1(m)[cos(ω0tωmodt)+cos(ω0t+ωmodt)]J2(m)[cos(ω0t2ωmodt)+cos(ω0t+2ωmodt)]}
Paverage_after=P1L2{[(1x)J0(m)]2+2[J1(m)]2+2[J2(m)]2}P1L2[(1x)2+m22]

Based on Eq. (14), the CSR, which is defined as the ratio of the optical power between the carrier and one of the first-order sidebands [12], can be obtained as

CSR=(1x)2m24

And the relationship of link gain before and after carrier subtraction is

Gafter=[(1x)(1+m22)(1x)2+m22]2Gbefore

Therefore, an optimum carrier suppression ratio for the link gain can be obtained from dGafter/dx=0 , as shown in Eq. (17).

xG=1m22

In this case, substituting Eq. (17) into Eq. (15), the optimum CSR is equal to 3 dB for double-sideband modulation scheme. While for the single-sideband modulation with small m, the optimum carrier suppression ratio xG_singleband can be obtained under the same principle, which is shown as Eq. (18). It means the optimum CSR for single-sideband is 0 dB, coincident with [12].

xG_singleband=1m24

Noise figure of RoF link can be expressed as

F=Noise_outGfkT0

In Eq. (19), k is Boltzmann’s constant, T 0 is room temperature (290 K), and Noise_out is the output noise power from PD, which includes three main noise sources, thermal noise, RIN and shot noise [14]. As the average optical power incident on PD is constant, the average photocurrent is constant and the output noise power also keeps unchanged [15]. If link gain is improved by carrier subtraction method, NF can be reduced by this method. Moreover, the optimum CSR to increase the link gain is also the optimum point for reducing NF. Figure 2 is the simulation results of NF with regards to different carrier suppression ratio x under different m, in which the laser-RIN is considered as −165 dB/Hz. The solid lines of curve a and b are the simulation results with m = 0.3 and m = 0.1, respectively, both of which have incident optical power on PD of 2 mW. While curve c and d are the dashed lines with the incident optical power on PD of 5 mW, and m = 0.3 and m = 0.1, respectively. We can see that the parameter m determines the shape of the NF curve, while the incident optical power on PD just determines the NF value at x = 0 with no carrier suppression. Thus, if the modulation index m is determined, the impact of carrier suppression on the improvement of NF and link gain is decided. And the smaller modulation index m is, the greater improvement of NF and link gain can be obtained through the carrier suppression method, just as what is shown in Fig. 1 and Fig. 2

 

Fig. 2 The simulation results of NF with respected to the carrier suppression ratio x under different modulation index m. a: The solid line with m = 0.3 and incident optical power on PD of 2 mW; b: The solid line with m = 0.1 and incident optical power on PD of 2 mW; c: The dashed line with m = 0.3 and incident optical power on PD of 5 mW; d: The dashed line with m = 0.1 and incident optical power on PD of 5 mW.

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As for the third-order SFDR,

SFDR=23(IP3+174NF)=23(IP3Noise+G)[dBHz23]
here, IP 3 is the input RF signal power when the third-order intermodulation output equals to the fundamental output, which only depends on the half-wave voltage of the modulator [7], having nothing to do with the carrier suppression ratio x. The output noise also keeps unchanged when the incident optical power on PD is constant. Therefore, it can be concluded that the increase of link gain leads to an improvement of SFDR. And the best carrier suppression ratio x for link gain is also the best point for SFDR with constant PD incident optical power. In such case, SFDR can be improved through increasing the link gain without lowering the noise level.

The MZM biased at quadrature has no even-order harmonics in principle; however, carrier-suppression increases the even-order distortion. Using the same principle above, we can calculate the second-order harmonic relative to fundamental component with small m from Eq. (13), which is given as

G2f_afterGf_after=[xJ1(m)2(1x)J0(m)]2

With the carrier suppression ratio x increasing, the second-order harmonic component increases. Therefore this method is only limited in sub-octave bandwidth applications.

3. Experiments and results

Figure 3 shows the schematic setup for the experiment. The carrier subtraction is realized through SBS. The output of the laser source is launched into a single drive MZM biased at the quadrature. The modulated optical power is then amplified by an erbium-doped fiber amplifier (EDFA). Two circulators, a coil of dispersion-shifted fiber (DSF) with length of 3 km and a variable optical attenuator (VOA) are combined into a long fiber ring. When the amplified optical carrier power entering into ring reaches the SBS threshold, the Stokes wave propagating in an anticlockwise direction is generated and feeds back into the tail of 3 km DSF to lower the SBS threshold and increase the carrier suppression ratio. Thus, different carrier suppression ratios can be conveniently realized through SBS in this fiber ring. The RF input is provided by a signal generator, and the output is amplified by an RF amplifier and then analyzed by an RF spectrum analyzer. The method of noise power measurement and NF calculation is the same as the gain method in [7].

 

Fig. 3 The experimental configuration of RoF link with SBS as a carrier filter.

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The simulation and experimental results of link gain and NF with respected to carrier suppression ratio x are demonstrated in Fig. 4 , which are shown as solid lines and dots, respectively. The RF input signal is 18 GHz, m = 0.1 and incident optical power on PD is 2 mW. The simulation results of NF in Fig. 4 are much larger than Curve b in Fig. 2 with the same m and the same PD incident optical power. The main reason is that we lack of high-power and low-noise laser source, and the laser RIN in the experiment is much larger than the simulation parameter of −165 dB/Hz in Fig. 2. However, we can still get great improvement of link performances and good agreement between simulation and experiments. From Fig. 4, we can see the optimum carrier suppression ratio for link gain is also the optimum one for NF. When the carrier suppression ratio x is less than 0.93, the value corresponding to CSR of 3 dB for m = 0.1, an amount of carrier suppression results in an equal amount of RF gain, also an equal amount of NF reduction. When x>0.93, link gain drops while NF increases. There is a small deviation between the experimental results and the simulation results after the optimum carrier suppression ratio x = 0.93. The reason lies in that, in order to obtain deep carrier suppression, the Stokes wave feeding back into the tail of DSF will be large enough to induce the second-order stokes, which reduces the useful part in the PD incident optical power and causes the performances worse. It also can be seen from Fig. 4 that there is almost 18 dB improvement of link gain and NF when CSR is 3 dB, which verifies the validity of the carrier subtraction method for the performances improvement of RoF link.

 

Fig. 4 The simulation (solid lines) and experimental results (dots) of link gain and NF with respected to carrier suppression ratio x, in which RF signal is 18 GHz and m = 0.1.

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Figure 5 is another set of experimental results with 18 GHz RF input, m = 0.3 and the incident optical power on PD of 2 mW. Just as the theory predicts, the larger m is, the less improvement of link gain and NF we can get. In Fig. 5, there is less than 10 dB improvement of link gain and NF. However, good agreement between theory and experimental results is also obtained. The optimum carrier suppression ratio for m = 0.3 is x = 0.788 that corresponds to the optimum CSR of 3 dB. The experimental results with 9 GHz RF input and m = 0.5 is shown in Fig. 6 , which only has less than 5 dB improvement. All the results in Fig. 4 to Fig. 6 show that the optimum carrier suppression ratio x for link gain is the same one for NF and the modulation index m is a key parameter for the performances improvement of RoF link with carrier subtraction method.

 

Fig. 5 The simulation (solid lines) and experimental results (dots) of link gain and NF with regards to carrier suppression ratio x, in which RF signal is 18 GHz and m = 0.3.

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Fig. 6 The simulation (solid lines) and experimental results (dots) of link gain and NF with regards to carrier suppression ratio x, in which RF input signal is 9 GHz and m = 0.5.

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The relationship between the second harmonic relative to fundamental component and the carrier suppression ratio x is shown in Fig. 7 , in which RF signal frequency is 9 GHz and m = 0.3. The solid line is the simulation results based on Eq. (21) and the diamond dots are the experimental ones. Good agreement is obtained. Obviously, the carrier subtraction method will increase the second-order harmonic, which should be limited in sub-octave applications.

 

Fig. 7 The results of the second-order harmonic relative to fundamental component with regards to carrier suppression ratio x. Solid line: theoretical curve of Eq. (21) with m = 0.3. Diamonds: experimental results.

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SFDR of RoF link with RF input of 9 GHz is finally measured and the results are shown in Table 1 . The incident optical power on PD remains constant of 2 mW. Group I is biased at quadrature with no carrier suppression; Group II and III are with different carrier suppression via SBS. It can be seen that SFDR is greatly improved by this carrier subtraction method. The RF output power as a function of RF input power for group III is plotted in Fig. 8 . The SFDR is obtained as 94.4 dB·Hz2/3 when EDFA output is 23.3 dBm.

Tables Icon

Table 1. Comparison of SFDR in RoF link with and without carrier subtraction

 

Fig. 8 RF output power versus RF input power for RoF link with carrier subtraction method. The modulation frequency is 9 GHz.

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4. Conclusion

In this paper, a profound analysis is made on the carrier subtraction method for the performances improvement of RoF link that has constant incident optical power on PD. Both theoretical calculation and experimental validation are carried out. The results show that the modulation index m is an important factor to determine whether this method is effective or not to improve the link performances. When m is small enough, all figures of merit including link gain, NF and SFDR can be optimized by choosing a proper carrier suppression ratio. The optimum CSR for double-sideband and single-sideband modulation is 3 dB and 0 dB, respectively. However the carrier subtraction method will increase the second-order harmonic distortion, which may limit its applications. Besides, SBS is an effective way to function as a carrier filter, not only because it can realize carrier subtraction conveniently, but also it is a common nonlinear effect which may need to be avoided in other cases, as it will easily occur when the RoF link is longer and the input power is a bit higher, for example in the low-biasing scheme.

Acknowledgements

This work was supported in part by National Natural Science Foundation of China (grant Nos. 60577028 and 60801003), the Program for New Century Excellent Talents in University (No. NCET-05-518), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant No. 20060335074).

References and links

1. E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993). [CrossRef]  

2. R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008). [CrossRef]  

3. C. H. Cox III, E. I. Ackerman, and J. L. Prince, “What do we need to get great link performance?” Microwave Photonics, International topical meeting (Germany, 1997), pp. 215–218.

4. C. H. Cox III, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006). [CrossRef]  

5. A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007). [CrossRef]  

6. M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993). [CrossRef]  

7. J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

8. W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996). [CrossRef]  

9. M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994). [CrossRef]  

10. K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994). [CrossRef]  

11. Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005). [CrossRef]  

12. C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006). [CrossRef]  

13. R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995). [CrossRef]  

14. C. H. Cox III, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990). [CrossRef]  

15. C. Cox III, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997). [CrossRef]  

References

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  1. E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
    [CrossRef]
  2. R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008).
    [CrossRef]
  3. C. H. Cox III, E. I. Ackerman, and J. L. Prince, “What do we need to get great link performance?” Microwave Photonics, International topical meeting (Germany, 1997), pp. 215–218.
  4. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
    [CrossRef]
  5. A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
    [CrossRef]
  6. M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
    [CrossRef]
  7. J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).
  8. W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
    [CrossRef]
  9. M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
    [CrossRef]
  10. K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
    [CrossRef]
  11. Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
    [CrossRef]
  12. C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
    [CrossRef]
  13. R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
    [CrossRef]
  14. C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
    [CrossRef]
  15. C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
    [CrossRef]

2008

R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008).
[CrossRef]

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

2007

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

2006

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

2005

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

1997

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

1996

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

1995

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

1994

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

1993

M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
[CrossRef]

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

1990

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Ackerman, E.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Ackerman, E. I.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Attygalle, M.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

Betts, G. E.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Burns, W. K.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

Chang, W. S. C.

M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
[CrossRef]

Charezenko, W.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Chen, K. S.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Cox, C.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Cox, C. H.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Daryoush, A. S.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Devenport, J.

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

Esman, R. D.

R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008).
[CrossRef]

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

Farewell, M. L.

M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
[CrossRef]

Gopalakrishnan, G. K.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

Hamilton, M. C.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Helkey, R.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Huber, D. R.

M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
[CrossRef]

Johnson, L. M.

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Karim, A.

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

Kasemset, D.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

LaGasse, M. J.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Lim, C.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

Moeller, R. P.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

Nirmalathas, A.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

Novak, D.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

Prince, J. L.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Samant, N. R.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Shen, Y. C.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Thaniyavarn, S.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Wanuga, S.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Waterhouse, R.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

Williams, K. J.

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

Williamson, R. C.

Zhang, X. M.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Electron. Lett.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

Fiber Integr. Opt.

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

IEEE Photon. Technol. Lett.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
[CrossRef]

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[CrossRef]

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

J. Lightwave Technol.

Opt. Eng.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Other

C. H. Cox III, E. I. Ackerman, and J. L. Prince, “What do we need to get great link performance?” Microwave Photonics, International topical meeting (Germany, 1997), pp. 215–218.

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Figures (8)

Fig. 1
Fig. 1

The simulation results of normalized RF link gain with regards to the carrier suppression ratio x under different modulation index m. a: the curve of (1-x)−2; b: m = 0.04; c: m = 0.1; d: m = 0.3; e: m = 0.5; f: m = 0.7.

Fig. 2
Fig. 2

The simulation results of NF with respected to the carrier suppression ratio x under different modulation index m. a: The solid line with m = 0.3 and incident optical power on PD of 2 mW; b: The solid line with m = 0.1 and incident optical power on PD of 2 mW; c: The dashed line with m = 0.3 and incident optical power on PD of 5 mW; d: The dashed line with m = 0.1 and incident optical power on PD of 5 mW.

Fig. 3
Fig. 3

The experimental configuration of RoF link with SBS as a carrier filter.

Fig. 4
Fig. 4

The simulation (solid lines) and experimental results (dots) of link gain and NF with respected to carrier suppression ratio x, in which RF signal is 18 GHz and m = 0.1.

Fig. 5
Fig. 5

The simulation (solid lines) and experimental results (dots) of link gain and NF with regards to carrier suppression ratio x, in which RF signal is 18 GHz and m = 0.3.

Fig. 6
Fig. 6

The simulation (solid lines) and experimental results (dots) of link gain and NF with regards to carrier suppression ratio x, in which RF input signal is 9 GHz and m = 0.5.

Fig. 7
Fig. 7

The results of the second-order harmonic relative to fundamental component with regards to carrier suppression ratio x. Solid line: theoretical curve of Eq. (21) with m = 0.3. Diamonds: experimental results.

Fig. 8
Fig. 8

RF output power versus RF input power for RoF link with carrier subtraction method. The modulation frequency is 9 GHz.

Tables (1)

Tables Icon

Table 1 Comparison of SFDR in RoF link with and without carrier subtraction

Equations (21)

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

m = π 2 V π V mod
E b e f o r e ( t ) = L cos ( π V D C 2 V π + m cos ω r f t ) E 0 ( t ) = L [ cos π V D C 2 V π cos ( m cos ω r f t ) sin π V D C 2 V π sin ( m cos ω r f t ) ] E 0 ( t ) = 2 L 2 [ cos ( m cos ω r f t ) sin ( m cos ω r f t ) ] E 0 ( t ) = 2 L 2 [ J 0 ( m ) + 2 k = 1 + ( 1 ) k J 2 k ( m ) cos ( 2 k ω r f t ) 2 k = 0 + ( 1 ) k J 2 k + 1 ( m ) cos [ ( 2 k + 1 ) ω r f t ] ] E 0 ( t )
P b e f o r e ( t ) = P 0 L 2 [ 1 + cos ( π V D C V π + 2 m cos ω r f t ) ] = P 0 L 2 { 1 + cos π V D C V π [ J 0 ( 2 m ) + 2 k = 1 + ( 1 ) k J 2 k ( 2 m ) cos ( 2 k ω r f t ) ] sin π V D C V π [ 2 k = 0 + ( 1 ) k J 2 k + 1 ( 2 m ) cos [ ( 2 k + 1 ) ω r f t ] ] }
P a v e r a g e _ b e f o r e = P 0 L 2 [ 1 + cos π V D C V π J 0 ( 2 m ) ] = P 0 L 2
E a f t e r ( t ) = 2 L 2 [ J 0 ( m ) + 2 k = 1 + ( 1 ) k J 2 k ( m ) cos ( 2 k ω r f t ) 2 k = 0 + ( 1 ) k J 2 k + 1 ( m ) cos [ ( 2 k + 1 ) ω r f t ] x J 0 ( m ) ] E 1 ( t ) = L cos ( π V D C 2 V π + m cos ω r f t ) E 1 ( t ) 2 L 2 x J 0 ( m ) E 1 ( t )
P a f t e r ( t ) = E a f t e r ( t ) E a f t e r * ( t ) = P 1 L 2 { 1 + cos π V D C V π [ J 0 ( 2 m ) + 2 k = 1 + ( 1 ) k J 2 k ( 2 m ) cos ( 2 k ω r f t ) ] sin π V D C V π [ 2 k = 0 + ( 1 ) k J 2 k + 1 ( 2 m ) cos [ ( 2 k + 1 ) ω r f t ] ] } P 1 L 2 2 x J 0 ( m ) [ J 0 ( m ) + 2 k = 1 + ( 1 ) k J 2 k ( m ) cos ( 2 k ω r f t ) 2 k = 0 + ( 1 ) k J 2 k + 1 ( m ) cos [ ( 2 k + 1 ) ω r f t ] ] + P 1 L 2 x 2 J 0 ( m ) 2
P a v e r a g e _ a f t e r = P 1 L 2 [ 1 + J 0 ( m ) 2 ( x 2 2 x ) ]
P 1 P 0 = 1 1 + J 0 ( m ) 2 ( x 2 2 x )
G a f t e r = { P 1 L 2 [ 2 J 1 ( 2 m ) 4 x J 0 ( m ) J 1 ( m ) ] } 2 G b e f o r e = [ P 0 L 2 2 J 1 ( 2 m ) ] 2
G a f t e r G b e f o r e = [ 1 1 + J 0 ( m ) 2 ( x 2 2 x ) J 1 ( 2 m ) 2 x J 0 ( m ) J 1 ( m ) J 1 ( 2 m ) ] 2
E b e f o r e ( t ) = 2 P 0 L 2 { J 0 ( m ) cos ( ω 0 t ) J 1 ( m ) [ cos ( ω 0 t ω mod t ) + cos ( ω 0 t + ω mod t ) ] J 2 ( m ) [ cos ( ω 0 t 2 ω mod t ) + cos ( ω 0 t + 2 ω mod t ) ] }
P a v e r a g e _ b e f o r e = P 0 L 2 { [ J 0 ( m ) ] 2 + 2 [ J 1 ( m ) ] 2 + 2 [ J 2 ( m ) ] 2 } P 0 L 2 ( 1 + m 2 2 )
E a f t e r ( t ) = 2 P 1 L 2 { ( 1 x ) J 0 ( m ) cos ( ω 0 t + φ 0 ) J 1 ( m ) [ cos ( ω 0 t ω mod t ) + cos ( ω 0 t + ω mod t ) ] J 2 ( m ) [ cos ( ω 0 t 2 ω mod t ) + cos ( ω 0 t + 2 ω mod t ) ] }
P a v e r a g e _ a f t e r = P 1 L 2 { [ ( 1 x ) J 0 ( m ) ] 2 + 2 [ J 1 ( m ) ] 2 + 2 [ J 2 ( m ) ] 2 } P 1 L 2 [ ( 1 x ) 2 + m 2 2 ]
C S R = ( 1 x ) 2 m 2 4
G a f t e r = [ ( 1 x ) ( 1 + m 2 2 ) ( 1 x ) 2 + m 2 2 ] 2 G b e f o r e
x G = 1 m 2 2
x G _ s i n g l e b a n d = 1 m 2 4
F = N o i s e _ o u t G f k T 0
S F D R = 2 3 ( I P 3 + 174 N F ) = 2 3 ( I P 3 N o i s e + G ) [dB Hz 2 3 ]
G 2 f _ a f t e r G f _ a f t e r = [ x J 1 ( m ) 2 ( 1 x ) J 0 ( m ) ] 2

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