Abstract

Intermodulation distortions are generated by Mach-Zehnder modulators when they are driven by signals with certain bandwidth in microwave (MW) and millimeter-wave (MMW) radio-over-fiber (ROF) links. The optical spectral structure of the distorted optical signal is investigated. A strategy to improve the dynamic range of MW and MMW ROF links directly in the optical domain is proposed and experimentally demonstrated. Based on optical spectrum processing, the third-order intermodulation distortions (IMD3s) of the generated signals are suppressed. A 107.2dB∙Hz2/3 spurious-free dynamic range (SFDR) of the MW/MMW ROF link is obtained, which is improved more than 20dB. A 16QAM signal is transmitted in the system and the error vector magnitude (EVM) is measured with and without the proposed technique. The influence of the nonlinearity of modulators on EVM is almost completely eliminated.

© 2012 OSA

1. Introduction

In the past two decades, many institutions have focused on the development of the wireless systems operating at much higher carrier frequencies in the microwave (MW) and millimeter-wave (MMW) range where more bandwidth is available [1]. With the inherent high-propagation-loss characteristics of MW/MMW signals, it is essential to deploy picocellular or microcellular architectures to provide efficient geographical coverage. To accommodate such architecture, a large number of base stations (BSs) have to be deployed to optimize the coverage. Radio over fiber (ROF) has been considered as a technique of potential due to its capability of reducing the complexity of the BS by optical remote distribution. In a ROF system, intensity modulation direct detection (IM-DD) is the most commonly used technique, and the downlink MMW signal is generated by optical up-conversion at the center station (CS) [2]. However, this technique is susceptible to the nonlinear characteristics of external Mach-Zehnder modulators (MZMs), which limit the SFDR of the overall system [3]. In a MW/MMW ROF link, the third-order intermodulation distortions (IMD3s) dominate the dynamic range, since the center frequency of the signal is much higher than its bandwidth. A number of approaches have been reported to eliminate the IMD3s and improve the dynamic range of the system. These approaches use or demonstrate complex modulators, such as dual-parallel MZM (DPMZM) [4], dual-electrode MZM (DEMZM) [5], mixed polarization DEMZM [6], and electro-optic polymeric DPMZM [7]. However, these approaches depend on the development of the modulators and need to be improved to be applicable to MMW ROF links. In WDM system, there is a post linearization technique presented in [8] to suppress the four wave mixing between the WDM channels. Unfortunately, little attention has been paid to the post-compensation technique for the conventional MZM in a MW/MMW ROF link.

In this paper, we analyze the output spectrum of a conventional MZM in a MMW ROF link with optical up-conversion and design the optical spectrum. Two frequency bands with the same spectral structure are simultaneously generated when the optical signal is detected by the photo detector. A compensation strategy based on optical spectrum processing is proposed to improve the dynamic range of the system simultaneously in MW band and MMW band. A 107.2dB∙Hz2/3 SFDR is obtained experimentally, which is improved more than 20dB. To demonstrate the influence on signals, a 15Msymbols/s 16-QAM data is transmitted in the ROF link. The experimental result shows that the influence of MZM nonlinearity on EVM is almost completely eliminated.

2. Optical spectral structure and operational principle

Figure 1 shows the block diagram of a classic MMW ROF link with optical up-conversion. The optical input of the MZM is a coherent dual-wavelength optical source with the corresponding angular frequencies of ω1 and ω2. Assume that the two wavelengths have the same phase and amplitude. The MZM is biased at the quadrature point to perform a double sideband (DSB) modulation. A two-tone RF signal at the angular frequencies of Ω1 and Ω2 is used to drive the MZM. The output of the MZM can be expressed as:

Eout(t)=12E0i=12exp(jωit){exp[jπ4+jπVin2Vπ(sinΩ1t+sinΩ2t)]+exp[jπ4jπVin2Vπ(sinΩ1t+sinΩ2t)]}
where E0 is the amplitude of each wavelength, Vin is the amplitude of each RF tone, and Vπ is the half-wave voltage of the MZM. Using Bessel expansion, Eq. (1) can be further developed as:
Eout(t)=12E0i=12p=q=[exp(jπ4)+(1)p+qexp(jπ4)]Jp(m)Jq(m)exp(jωit+jpΩ1t+jqΩ2t)
where m = πVin/2Vπ is the modulation index, Jp(m) and Jq(m) is the Bessel function of the first kind of order p and q. To simplify the analysis, only terms of order |p| < 3 and |q| < 3 are considered, because terms of higher order are negligible. Thus, there are altogether thirty-eight terms of different order. Each term represents an optical frequency component. All these frequencies compose ten optical sidebands.

 

Fig. 1 Block diagram of classic MMW ROF link with optical up-conversion. PD: photo detector.

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Figure 2(a) shows the output ten optical sidebands of MZM. They are divided into two groups. Each group generated by one optical wavelength is composed of five optical sidebands. We name these sidebands optical carrier band of ω1 (ω1-OCB), 1st-order optical upper/lower sideband of ω1 (ω1-1-OUSB/ ω1-1-OLSB), 2nd-order optical upper/lower sideband of ω1 (ω1-2-OUSB/ ω1-2-OLSB), optical carrier band of ω2 (ω2-OCB), 1st-order optical upper/lower sideband of ω2 (ω2-1-OUSB/ ω2-1-OLSB), 2nd-order optical upper/lower sideband of ω2 (ω2-2-OUSB/ ω2-2-OLSB), respectively. Only four sidebands are reserved for a linearized MW/MMW ROF link. They are ω2-2-OUSB, ω2-1-OUSB, ω1-1-OUSB, and ω1-OCB, as shown in Fig. 2(b). Figure 2(c) shows the detailed frequency components of the desired optical sidebands according to Eq. (2).

 

Fig. 2 (a) Output optical spectrum of MZM; (b) 4 desired sidebands in linearized MW/MMW ROF link; (c) frequency components of the desired sidebands.

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When the optical signal is detected by a photo detector (PD), all the reserved optical sidebands will beat with each other and generate fundamental frequencies and IMD3s in electrical domain. In our system, signals in microwave band and millimeter-wave band are generated simultaneously with the same spectral structure. The generated electrical signal can be expressed as:

Ielec(t)=I0+(I101+I112)(sinΩ1t+sinΩ2t)+(I'101+I'112)[sin(ω2ω1+Ω1)t+sin(ω2ω1+Ω2)t]+(I301+I312){sin[(2Ω1Ω2)t]+sin[(2Ω2Ω1)t]}+(I'301+I'312){sin[(ω2ω1+2Ω1Ω2)t]+sin[(ω2ω1+2Ω2Ω1)t]}
where I1-01 and I3-01 are coefficients generated by ω1-OCB and ω1-1-OUSB, I1-12 and I3-12 are generated by ω2-1-OUSB and ω2-2-OUSB, I’1-01 and I’3-01 are generated by ω1-OCB and ω2-1-OUSB, I’1-12 and I’3-12 are generated by ω1-1-OUSB and ω2-2-OUSB. We can easily figure out that I1-01 = I’1-01, I1-12 = I’1-12, I3-01 = I’3-01, and I3-12 = I’3-12. Thus, Eq. (3) can be further simplified to:
Ielec(t)=I0+(I101+I112)[sinΩ1t+sinΩ2t+sin(ω2ω1+Ω1)t+sin(ω2ω1+Ω2)t]+(I301+I312){sin[(2Ω1Ω2)t]+sin[(2Ω2Ω1)t]+sin[(ω2ω1+2Ω1Ω2)t]+sin[(ω2ω1+2Ω2Ω1)t]}.
I1-01, I1-12, I3-01, and I3-12 can be positive or negative. Ω1, Ω2, ω2-ω1 + Ω1 and ω2-ω1 + Ω2 are fundamental frequencies, while 2Ω12, 2Ω21, ω2-ω1 + 2Ω12, and ω2-ω1 + 2Ω21 are IMD3s. By attenuating the amplitude of ω1-OCB to certain value and shifting the phase of ω2-2-OUSB to its opposite, the IMD3s are eliminated with I3-01 = -I3-12. Figure 3 shows the spectrum evolution as described above.

 

Fig. 3 Spectrum evolution without (a) and with (b) optical spectrum processing.

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3. Experimental setup and results

Figure 4 shows the experimental setup of the MW/MMW ROF link with optical up-conversion. A 40GHz MZM (AVANEX AM40, MZM1) is biased at its optical carrier-suppression (OCS) point to generate a coherent dual-wavelength light. The driving frequency of MZM1 is 27GHz, resulting in a 54GHz optical frequency interval. The light is amplified by an erbium-doped fiber amplifier (EDFA) to supply enough power before it is fed into the next stage. The output spectrum of the coherent dual wavelength source (CDWS) is shown in Fig. 4. The light is fed into another 40GHz MZM (AVANEX AM40, MZM2) to perform a double sideband (DSB) modulation. To investigate the SFDR of the system, a two-tone RF signal at the frequency of 13.46GHz and 13.54GHz is supplied to the modulator. Thus, the desired fundamental frequencies in MMW band are 67.46GHz and 67.54GHz, and the frequencies in MW band are 13.46GHz and 13.54GHz.

 

Fig. 4 Schematic diagram of the experimental setup.

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An optical spectrum processor is placed after the MZM2. There are two functions on this processor: one is to filter the four desired optical sidebands, the other is to manipulate the amplitude and phase of the optical frequency to compensate the nonlinearity of the MZM2. The schematic diagram of the processor is shown in Fig. 4. It is based on a reflected optical pulse shaper [9]. The input light is decomposed into its constituent optical spectrum by a spectral disperser (such as a grating with 1200 grooves/mm) which is placed on the front focal plane of a focus element. The focus element (such as a lens) makes each light beam with different optical frequency converge to the different position of the focal plane. A flat totally reflecting mirror is placed on the back focal plane to return the light which is then recombined by the focus element and the disperser. A programmable liquid crystal spatial light modulator (LCSLM, 128 pixels, 100 μm pixel pitch, 2 μm inter-pixel gap) is placed just before the mirror to modulate the amplitude and phase of the spatially dispersed optical spectrum. The effective optical bandwidth for each LCSLM pixel is about 3 GHz. The optical signal from the processor is amplified by an EDFA to maintain the optical power before PD at about 5dBm. The detected signal is analyzed by an electrical spectrum analyzer (ESA, Agilent E4446A) directly.

Figure 5 shows the measured SFDR of the ROF link. The measured noise floor is −150dBm/Hz, which is limited by the noise floor of the ESA and higher than the calculated result −160dBm/Hz dominated by the thermal noise and shot noise. By optical spectrum processing, the SFDR is improved from 87dB∙Hz2/3 to 107.2dB∙Hz2/3. An improvement of more than 20dB is obtained. To investigate the influence on vector signals, a 15Msymbols/s 16-QAM signal centered at 13.5GHz generated by a vector signal generator (VSG, Agilent E8267D) is supplied to the MZM2. We measure the EVM performance of the output of the VSG, the ROF link without and with processing. At the input power of 21dBm, the EVM is decreased from 12.8% to 5.3%, as shown in Fig. 6 . The experimental results indicate that the influence of MZM nonlinearity on EVM is almost completely eliminated using the proposed processing, and the EVM is limited only by the dynamic range of the VSG.

 

Fig. 5 Measured SFDR without (dashed line) and with (solid line) optical spectrum processing.

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Fig. 6 Measured EVM performance of the VSG (dash-dot line), ROF link without processing (dashed line), and ROF link with processing (solid line) and constellation diagram for 16-QAM at the input power of 21dBm.

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4. Conclusion

The nonlinear performance of the MW/MMW ROF link with optical frequency up-conversion is analyzed. A dynamic range improvement strategy based on optical spectrum processing is proposed and experimentally demonstrated for this MMW ROF link. As a result, a SFDR of 107.2dB∙Hz2/3 is measured, which is improved more than 20dB. The influence of the nonlinearity of MZM on EVM is almost completely eliminated using the proposed processing.

Acknowledgments

This work was supported in part by National Key Basic Research Program of China under grant No 2012CB315603 and 2012CB315604, National Nature Science Foundation of China (NSFC) under grant No. 60736003, 61025004, 61032005, Foundation of the Key State Lab of Integrated Optoelectronics under grand No. 2010KFB007, and the Ph.D. Programs Foundation of Ministry of Education of China, under grand No. 20100002110039, China Postdoctoral Science Foundation under grant No. 20110490426, 2012M510442.

References and links

1. A. M. J. Koonen and M. G. Í. Larrodé, “Radio-Over-MMF Techniques-Part II: Microwave to Millimeter-Wave Systems,” J. Lightwave Technol. 26(15), 2396–2408 (2008). [CrossRef]  

2. M. J. Fice, E. Rouvalis, F. van Dijk, A. Accard, F. Lelarge, C. C. Renaud, G. Carpintero, and A. J. Seeds, “146-GHz millimeter-wave radio-over-fiber photonic wireless transmission system,” Opt. Express 20(2), 1769–1774 (2012). [CrossRef]   [PubMed]  

3. J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010). [CrossRef]  

4. S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010). [CrossRef]  

5. C. Lim, A. T. Nirmalathas, K.-L. Lee, D. Novak, and R. Waterhouse, “Intermodulation Distortion Improvement for Fiber–Radio Applications Incorporating OSSB+C Modulation in an Optical Integrated-Access Environment,” J. Lightwave Technol. 25(6), 1602–1612 (2007). [CrossRef]  

6. B. Masella, B. Hraimel, and X. Zhang, “Enhanced Spurious-Free Dynamic Range Using Mixed Polarization in Optical Single Sideband Mach-Zehnder Modulator,” J. Lightwave Technol. 27(15), 3034–3041 (2009). [CrossRef]  

7. S. K. Kim, W. Liu, Q. Pei, L. R. Dalton, and H. R. Fetterman, “Nonlinear intermodulation distortion suppression in coherent analog fiber optic link using electro-optic polymeric dual parallel Mach-Zehnder modulator,” Opt. Express 19(8), 7865–7871 (2011). [CrossRef]   [PubMed]  

8. J. Chou, O. Boyraz, and B. Jalali, “Adaptive optical post distortion linearization,” Opt. Express 13(15), 5711–5718 (2005). [CrossRef]   [PubMed]  

9. A. M. Weiner, Ultrafast Optics (Wiley, 2009).

References

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  1. A. M. J. Koonen and M. G. Í. Larrodé, “Radio-Over-MMF Techniques-Part II: Microwave to Millimeter-Wave Systems,” J. Lightwave Technol. 26(15), 2396–2408 (2008).
    [CrossRef]
  2. M. J. Fice, E. Rouvalis, F. van Dijk, A. Accard, F. Lelarge, C. C. Renaud, G. Carpintero, and A. J. Seeds, “146-GHz millimeter-wave radio-over-fiber photonic wireless transmission system,” Opt. Express 20(2), 1769–1774 (2012).
    [CrossRef] [PubMed]
  3. J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
    [CrossRef]
  4. S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
    [CrossRef]
  5. C. Lim, A. T. Nirmalathas, K.-L. Lee, D. Novak, and R. Waterhouse, “Intermodulation Distortion Improvement for Fiber–Radio Applications Incorporating OSSB+C Modulation in an Optical Integrated-Access Environment,” J. Lightwave Technol. 25(6), 1602–1612 (2007).
    [CrossRef]
  6. B. Masella, B. Hraimel, and X. Zhang, “Enhanced Spurious-Free Dynamic Range Using Mixed Polarization in Optical Single Sideband Mach-Zehnder Modulator,” J. Lightwave Technol. 27(15), 3034–3041 (2009).
    [CrossRef]
  7. S. K. Kim, W. Liu, Q. Pei, L. R. Dalton, and H. R. Fetterman, “Nonlinear intermodulation distortion suppression in coherent analog fiber optic link using electro-optic polymeric dual parallel Mach-Zehnder modulator,” Opt. Express 19(8), 7865–7871 (2011).
    [CrossRef] [PubMed]
  8. J. Chou, O. Boyraz, and B. Jalali, “Adaptive optical post distortion linearization,” Opt. Express 13(15), 5711–5718 (2005).
    [CrossRef] [PubMed]
  9. A. M. Weiner, Ultrafast Optics (Wiley, 2009).

2012 (1)

2011 (1)

2010 (2)

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
[CrossRef]

2009 (1)

2008 (1)

2007 (1)

2005 (1)

Accard, A.

Boyraz, O.

Carpintero, G.

Chou, J.

Dalton, L. R.

Fetterman, H. R.

Fice, M. J.

Gomes, N. J.

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

Hraimel, B.

Jalali, B.

James, J.

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

Kim, S. K.

Koonen, A. M. J.

Larrodé, M. G. Í.

Lee, K.-L.

Lelarge, F.

Li, S.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
[CrossRef]

Liang, X.

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

Lim, C.

Liu, W.

Masella, B.

Nirmalathas, A. T.

Nkansah, A.

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

Novak, D.

Pei, Q.

Renaud, C. C.

Rouvalis, E.

Seeds, A. J.

Shen, P.

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

van Dijk, F.

Waterhouse, R.

Zhang, H.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
[CrossRef]

Zhang, X.

Zheng, X.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
[CrossRef]

Zhou, B.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly Linear Radio-Over-Fiber System Incorporating a Single-Drive Dual-Parallel Mach-Zehnder Modulator,” IEEE Photon. Technol. Lett. 22(24), 1775–1777 (2010).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

J. James, P. Shen, A. Nkansah, X. Liang, and N. J. Gomes, “Nonlinearity and Noise Effects in Multi-Level Signal Millimeter-Wave Over Fiber Transmission Using Single and Dual Wavelength Modulation,” IEEE Trans. Microw. Theory Tech. 58(11), 3189–3198 (2010).
[CrossRef]

J. Lightwave Technol. (3)

Opt. Express (3)

Other (1)

A. M. Weiner, Ultrafast Optics (Wiley, 2009).

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Figures (6)

Fig. 1
Fig. 1

Block diagram of classic MMW ROF link with optical up-conversion. PD: photo detector.

Fig. 2
Fig. 2

(a) Output optical spectrum of MZM; (b) 4 desired sidebands in linearized MW/MMW ROF link; (c) frequency components of the desired sidebands.

Fig. 3
Fig. 3

Spectrum evolution without (a) and with (b) optical spectrum processing.

Fig. 4
Fig. 4

Schematic diagram of the experimental setup.

Fig. 5
Fig. 5

Measured SFDR without (dashed line) and with (solid line) optical spectrum processing.

Fig. 6
Fig. 6

Measured EVM performance of the VSG (dash-dot line), ROF link without processing (dashed line), and ROF link with processing (solid line) and constellation diagram for 16-QAM at the input power of 21dBm.

Equations (4)

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E out (t)= 1 2 E 0 i=1 2 exp(j ω i t) {exp[j π 4 +j π V in 2 V π (sin Ω 1 t+sin Ω 2 t)] +exp[j π 4 j π V in 2 V π (sin Ω 1 t+sin Ω 2 t)]}
E out (t)= 1 2 E 0 i=1 2 p= q= [exp(j π 4 )+ (1) p+q exp(j π 4 )] J p (m) J q (m)exp(j ω i t+jp Ω 1 t+jq Ω 2 t)
I elec (t)= I 0 +( I 101 + I 112 )(sin Ω 1 t+sin Ω 2 t) +(I ' 101 +I ' 112 )[sin( ω 2 ω 1 + Ω 1 )t+sin( ω 2 ω 1 + Ω 2 )t] +( I 301 + I 312 ){sin[(2 Ω 1 Ω 2 )t]+sin[(2 Ω 2 Ω 1 )t]} +(I ' 301 +I ' 312 ){sin[( ω 2 ω 1 +2 Ω 1 Ω 2 )t] +sin[( ω 2 ω 1 +2 Ω 2 Ω 1 )t]}
I elec (t)= I 0 +( I 101 + I 112 )[sin Ω 1 t+sin Ω 2 t +sin( ω 2 ω 1 + Ω 1 )t+sin( ω 2 ω 1 + Ω 2 )t] +( I 301 + I 312 ){sin[(2 Ω 1 Ω 2 )t]+sin[(2 Ω 2 Ω 1 )t] +sin[( ω 2 ω 1 +2 Ω 1 Ω 2 )t]+sin[( ω 2 ω 1 +2 Ω 2 Ω 1 )t]}.

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