Abstract

Based on optical frequency combs (OFC), we propose an efficient and flexible multi-band frequency conversion scheme for satellite repeater applications. The underlying principle is to mix dual coherent OFCs with one of which carrying the input signal. By optically channelizing the mixed OFCs, the converted signal in different bands can be obtained in different channels. Alternatively, the scheme can be configured to generate multi-band local oscillators (LO) for widely distribution. Moreover, the scheme realizes simultaneous inter- and intra-band frequency conversion just in a single structure and needs only three frequency-fixed microwave sources. We carry out a proof of concept experiment in which multiple LOs with 2 GHz, 10 GHz, 18 GHz, and 26 GHz are generated. A C-band signal of 6.1 GHz input to the proposed scheme is successfully converted to 4.1 GHz (C band), 3.9 GHz (C band) and 11.9 GHz (X band), etc. Compared with the back-to-back (B2B) case measured at 0 dBm input power, the proposed scheme shows a 9.3% error vector magnitude (EVM) degradation at each output channel. Furthermore, all channels satisfy the EVM limit in a very wide input power range.

© 2014 Optical Society of America

1. Introduction

The demands on great capacity and large instantaneous bandwidth have driven today’s RF systems to operate at higher frequencies and more bands [1]. The RF repeaters in communication satellites, for example, are now converting signals among C, Ku, Ka and even V bands to mitigate the frequency spectrum congestion and orbital resource depletion problems while fulfilling the increasing requirements on the throughput [2]. In addition, frequency reuse technique enabled by multi-spot coverage with tens of narrow beam antennas is also used to further improve the satellite capacity [3].

However, such a multi-port, multi-band and high-frequency RF system as a satellite repeater encounters numerous new challenges in terms of cost, weight, complexity, efficiency, bandwidth and flexibility. For instance, in a multi-band RF system, frequency sources are commonly required for transmitters, receivers, reference generators and processors [4]. Equipping each unit with an individual local oscillator (LO) could be costly and bulky. Therefore, one demand is a centralized LOs generation and widely distribution architecture. This can make it more cost-effective, power-efficient and space-saving in satellite.

For future multi-band communication satellite repeaters, flexible band-to-band, beam-to-beam cross-connect will be in strong demand to build agile links among different areas and users [5]. To achieve these functions, frequency converters and path switches are used in satellites. However, conventional satellite repeaters can only support intra-band downconversion of the incoming signals. For inter-band conversion, advanced repeaters use multiple stage conversion scheme to avoid deleterious spurs, where incoming RF signals within different bands are first down-converted to intermediate frequencies (IFs) and then up-converted again to the desired outcoming bands after executing processing like switching, amplification, etc. However, multiple stage conversion would severely reduce the system dynamic range and conversion efficiency [6]. Hence, another demand is band-to-band direct conversion and high frequency handling ability. This can bring flexibility and high performance for satellite.

Aiming at the two main demands mentioned above, microwave photonic technology is probably the very candidate to bring them into reality. For decades, photonic techniques have contributed much to extending the bandwidth of RF system and signal distribution distance due to its unique characteristics of extremely high bandwidth, low loss, parallel processing capability, etc [7]. In the meantime, photonic approaches offer additional benefits such as transparency to signal formats, high RF isolation, dramatic mass saving as well as immunity to electro-magnetic interference (EMI) [8, 9]. Therefore, microwave photonic techniques are well suited to multi-band high-frequency RF systems.

In the area of photonic microwave generation and frequency conversion, a wide variety of techniques have been reviewed in [7, 1012]. Recently, some of these techniques have been implemented in practical large projects to process multi-band microwave signals. For example, In the European Space Agency’s SAT’N LIGHT project [13], an advanced satellite payload concept was proposed and demonstrated with simultaneous multi-band downconversion capability and enhanced flexibility. In the Atacama Large Millimeter Array (ALMA) [14] project collaborated by the United States, Canada, Europe, and Japan, multi-band (27-124 GHz) LO reference signals were optically generated and distributed to the remote antennas for downconversion of the received signals at 31-950 GHz. These tasks can hardly be completed by their electrical counterparts.

Alternatively, optical frequency combs (OFCs) [1517] or mode-locked lasers (MLLs) [12, 18, 19] have been applied to perform frequency downconversion due to the fact that the comb itself consists a large set of coherent oscillators. However, the general idea of these techniques, from a frequency-domain perspective, is making the signal spectrum overlap each other to get narrower spaces between the beating tones. Therefore, each channel is dependent with other channels and aliasing effect (spectra overlapping) can easily take place.

Recently, the dual-comb technique is proposed for the application of spectroscopy with multi-heterodyne method, which takes advantage of the broadband, coherent and high-resolution nature of the dual OFCs [20, 21]. In this work, we incorporate the dual OFCs and electro-optic mixing as well as the channelization method to implement flexible photonic microwave conversion. The input RF signal is replicated besides each combline of one OFC, which is then coherently heterodyne with another OFC with different center frequency and mode spacing. Thus, each optical channel performs one frequency band conversion and all channels can be easily separated by periodic optical filtering. A proof of concept experiment is also carried out to verify the proposed function with good performance.

It is worth noting that by introducing the dual-comb technique and channelization method, the proposed technique in this paper is quite different from the prior arts. Firstly, dual OFCs with different mode spacing can generate LOs with more flexible frequencies than those generated by single OFC. Secondly, with the periodic optical filter, the heterodyne takes place in separated channels. The processes are in parallel and independent with each other. Finally yet importantly, the scheme is capable of simultaneous inter- and intra-band conversion (including up-conversion).

2. Principle

2.1 Multi-band LOs generation

Besides for frequency conversion, multi-band LOs can also be used as shared reference sources by various instruments distributed in multi-band RF systems or networks. Considering the OFCs having both the low phase noise characteristic and the parallel processing capability, we propose an OFC-based multi-band LOs generation and distribution scheme which can generate more than five bands LOs with only three frequency-fixed microwave sources.

As illustrated in Figs. 1(a) and 1(b), the schematic diagram of multiple LOs generation consists of dual coherent OFCs generation part, channelization part and heterodyne detection part. In the first part, two OFCs (OFC1 and OFC2) are generated with their mode spacing and center frequency offset controlled by two different RF frequencies δ1, δ2 and a third RF frequency δs respectively. The coherence of the dual OFCs is ensured by using a shared CW laser. The frequency of each combline in OFC1 can be expressed as

fOFC1(i)=fc+iδ1.
where the index i (-n≤i≤n) denotes the order of the combline (or sideband) relative to the center carrier fc, n is the maximum order of the available combline.

 figure: Fig. 1

Fig. 1 (a) Schematic diagram of multiple LOs generation (without mixing in the dashed line box) and inter & intra-band frequency conversion. (b) Schematic illustration of multiple LOs generation through channelizing dual coherent OFCs. (c) Signal multicasting for frequency conversion and the instantaneous bandwidth illustration.

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Similarly, the frequency of each combline in OFC2 can be given as

fOFC2(i)=fc+δs+iδ2.

The values of δ1, δ2 and δs are carefully assigned that every combline pair which is composed of the comblines from the two OFCs with the same index falls in each passband of a periodic optical filter such like wavelength division multiplexer (WDM). By heterodyne detection of each pair of these comblines, multiple microwave LOs can be achieved with a serials of frequencies

fLO(i)=δs+i|δ1δ2|.

Specifically, to successfully extract these combline pairs, the values of δ1, δ2 and δs should satisfy the following relations,

nδs|δ1δ2|andnBWδs|δ1δ2|.
where BW denotes passband width of the WDM filter. According to Eq. (4), more LOs can be achieved with less difference between δ1 and δ2. The total available channels or obtained number of LOs will be 2n + 1. Besides, the highest frequency of the LOs is only limited by the BW of the WDM filter.

2.2 Inter- and intra-band frequency conversion

Based on the works above, inter- and intra-band frequency conversion can be simply implemented by inserting an electro-optical (E/O) mixing module in one branch of the multi-band LOs generation structure, as shown in the dashed line box of Fig. 1(a). An input signal fin within a supported bandwidth is replicated besides each combline to form a signal OFC as shown in Fig. 1(c). To efficiently make use of the available bandwidth and avoid residual sideband crosstalk, carrier suppressed single sideband modulation (CS-SSB) is preferred. Alternatively, carriers can be further removed by aligning their frequencies at the gaps between the WDM passbands. The signal OFC is then combined with the other OFC, which can be called Local OFC correspondingly. After channelization and heterodyne detection process to the combined lights, the converted signals within different bands are obtained,

fout(i)=|fLO(i)fin|.

Theoretically, since the signal’s carriers are removed by CS-SSB modulation or the filter gaps, only one pair of signal-LO comblines are left in separated channels for heterodyne. A heterodyne with only two optical frequencies can be considered with ultra-wide Nyquist zone (hundreds of THz) as illustrated in Fig. 1(c). However, the Nyquist criterion implies that the 2nd and 3rd Nyquist zones should not be used as instantaneous bandwidth simultaneously [15]. Otherwise, aliasing effect will happen. If we choose the LO combline’s left side space (within the 2nd Nyquisit zone) as operation bandwidth, considering the LO comblines locate at different positions in the filter passbands, the supported instantaneous bandwidth in the ith channel can be given by

IBW(i)=δs+i|δ1δ2|.

Thus, to implement this technique in practical applications, the minimum combline offset and the space increment between the combline pairs should be tuned to accommodate the required operation bandwidth. Otherwise, if the input signal have a bandwidth between IBW(-n) and IBW(-n + 1), the signal’s frequency content will cross the local combline in a certain channel, leading to the spectral aliasing problem. But the rest channels are still available. At worst, if the input signal’s bandwidth exceeds IBW(n), spectral aliasing will appear in all channels.

3. Experiment and results

3.1 Multi-band LOs generation and analysis

The experimental setup is shown in Fig. 2. In order to generate two coherent seed lights with different central frequencies, a continuous wave (CW) light centered at fc = 193.55 THz with a power of 16 dBm is divided into two branches by a polarization maintained tunable optical coupler (PM-TOC). The PM-TOC also keeps the power between the two branches balanced. The light in the lower branch is modulated by a δs = 18 GHz sinusoidal wave to generate a double sideband (DSB) spectrum. Using a narrow optical filter centered at 193.568 THz with bandwidth of 10 GHz, the right sideband of the DSB spectrum is then extracted out with the carrier more than 40 dB suppressed. This sideband serves as a new carrier for the following OFC generation stage. The evolution of the spectrum can be illustrated at point D and E in Fig. 2.

 figure: Fig. 2

Fig. 2 Experimental setup of the inter- and intra-band frequency conversion. CW Laser: continuous wave laser; PM-TOC: polarization maintained tunable optical coupler; PolM: polarization modulator; Pol: polarizer; MZM: Mach Zehnder modulator; DPMZM-dual-parallel MZM; OC: optical coupler; EDFA: erbium doped fiber amplifier; PD: photo detector; EA: electrical amplifier.

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As a proof of concept experiment, a simple OFC generation method [22] which generates 5 flat comblines with only one polarization modulator (PolM) and a polarizer is chosen to validate our proposal. Flatness of the OFC determines the power uniformity of the output microwave signals in each channel. Two steps can be followed to generate a very flat comb with 5 lines based on PolM. First, for a carrier modulated by a sinusoidal wave, the amplitude of the modulated carrier and each sideband can be calculated using Bessel functions of the first kind. We can make the amplitude of the 2nd order sideband equals to that of the carrier by applying a particular modulation index β,

J0(β)=J2(β)β1.84,5.33,.
where J0, J2 are Bessel functions of the first kind with 0th and 2nd order, respectively.

Second, for a linear polarized CW lightwave modulated by a sinusoidal wave through a PolM with an angle of 45° relative to the PolM’s principle axis, the resulted odd and even sidebands will have orthogonal states of polarization. A polarizer placed behind the PolM can balance the power between the two orthogonal sidebands, namely the ± 1st sidebands and the carrier or the ± 2nd sidebands, with a particular polarization angle α relative to the principle axis of PolM,

sin(2α)cosϕ=J12(β)J02(β)J12(β)+J02(β).
where ϕ is a static phase term between principle axes of the PolM. α can be adjusted by a polarization controller (PC) before the polarizer.

By executing the two steps in turn, we can make the amplitude of the carrier, the ± 1st sidebands and the ± 2nd sidebands of the resulted OFC identical.

In the experiment, two sinusoidal waves with frequencies δ1 = 38 GHz and δ2 = 30 GHz are chosen to drive the PolMs through a 30 dB amplifier to get a large enough β. One of the PolM as shown in the upper branch of Fig. 2 is customized that the light is aligned to have an angle to the principle axis of the PolM, so there is no PC before it. Under a particular state of polarization controlled by the PC, dual coherent OFCs with combline spacing corresponding to their driven frequencies are achieved at point B and F in the setup. As shown in Fig. 3, when the power of the two branches is balanced, the 5-line power variation of both OFCs is less than 0.9 dB and unwanted-mode suppression ratio is greater than 14.7 dB.

 figure: Fig. 3

Fig. 3 Dual coherent OFCs with 18 GHz center frequency shift and 38 GHz, 30 GHz mode spacing respectively.

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The resulted dual coherent OFCs are combined together via an optical coupler (OC) and then amplified and channelized by an EDFA and a periodic optical filter respectively. A WaveShaper (Finisar 4000S) is used to obtain a tunable channelized bandwidth. But for practical applications, either the OFCs or the WDM can be customized to match the FSR (Free Spectral Range) of each other. The WaveShaper is programmed to have 4 passbands output ports uniformly centered at 193.639 THz, 193.601 THz, 193.563 THz and 193.525 THz respectively. Each passband has a bandwidth of 33 GHz. The channelized signals composed of a pair of comblines in each channel are then heterodyne detected using a 40 GHz photodetecter (u2t) and measured by a signal analyzer (Agilent N9030A). The resulted frequency spectra of each channel are given in Fig. 4.

 figure: Fig. 4

Fig. 4 Multiple LOs within different bands of (a) 2 GHz, (b) 10 GHz, (c) 18 GHz and (d) 26 GHz.

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It can be seen that different bands of frequency components at 2 GHz, 10 GHz, 18 GHz and 26 GHz, which can be taken as multi-band LOs, are generated. The signal to noise ratio (SNR) of the multi-band LOs are between 31 dB to 52 dB in 2 GHz bandwidth under a resolution bandwidth (RBW) of 20 kHz measurement condition. Unfortunately, the fifth frequency component which can be predicted at 34 GHz was unable to be measured due to the limited measurement range of signal analyzer.

3.2 Inter- and intra-band frequency conversion and analysis

The E/O mixing of the conversion process is implemented by a dual parallel Mach Zehnder modulator (DPMZM) placed behind OFC1 generator. Under carrier suppressed single sideband (CS-SSB) modulation, A 6.1 GHz test signal generated from a vector signal generator (Agilent E8267D) is replicated besides each combline of OFC1 to form a signal OFC as the black line shows in Fig. 5(a).It can be seen that the carrier extinction ratio is greater than 21.4 dB. Correspondingly, OFC2 can be considered as local OFC. The spectra of the two OFCs are shown jointly in Fig. 5(a). It can also be obviously observed that the newly formed combline pairs, which are composed of two beating frequencies each, have frequency differences of 4.1 GHz, 3.9 GHz, 11.9 GHz, 19.9 GHz, 27.9 GHz, as Eq. (5) predicts.

 figure: Fig. 5

Fig. 5 Coherent signal OFC and local OFC (a) before and (b) after channelization.

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The channelization and heterodyne detection processes are similar to those in the previous section except that the four channels of the WaveShaper are changed to uniformly center at 193.642 THz, 193.604 THz, 193.566 THz and 193.528 THz for more effective filtering. The 5th channel centered at 193.490 THz is also measured separately. Due to the dense passbands of the filters which are determined by the OFCs’ mode spacing, the combline’s power in each channel will leak into the adjacent channels as shown in Fig. 5(b). The measured maximum crosstalk in the 1st to the 5th channel is −17.1 dB, −15.6 dB, −24.3 dB, −15.1 dB and −21.9 dB, respectively.

It is worth noting that, the leaked comblines from other channels will beat with the signal combline or LO combline in the channel of interest, generating undesired RF signals in electrical domain. Since the 33 GHz channel bandwidth is beyond the measurement range of the instrument, we calculate the power ratios of the undesired RF signals’ power to the desired RF signals’ power according to the optical spectra. The ratios are listed in Table 1 versus the frequencies.

Tables Icon

Table 1. Power ratios of the undesired signals to the desired signal

As shown in Table 1, the nearest undesired signal with a −13.6 dB power ratio (in Ch4) is 1.8 GHz away from the desired signal. It can be further removed by placing a RF filter behind the photodetector. Alternatively, the crosstalk can be greatly reduced by using OFCs with much wider mode spacing. Figure 6 presents the electrical spectrum of the converted signals at 4.1 GHz, 3.9 GHz (C band) and 11.9 GHz (X band) from the first three channels in 2 GHz bandwidth. Note that the spurious frequency components in Figs. 6(b) and 6(c) come from non-ideal CS-SSB modulation. The spurious products suppression ratios of all the three channels are between 30 dB and 46.7 dB under 20 kHz RBW measurement condition. To evaluate the performance of the converter, EVM is tested by modulating a 10 M Symbol/s QPSK data on the 6.1 GHz test signal. As shown in Fig. 6(d), the third channel, also the worst one of the three, has an EVM degradation of 9.3% compared with the back-to-back (B2B) case at 0 dBm input RF power. However, all the results of three channels are far below the maximum EVM limit of 31.6% for QPSK signal in a very wide power range from −19 dBm to 0 dBm.

 figure: Fig. 6

Fig. 6 The converted signal from the first three channel at (a) 4.1 GHz, (b) 3.9 GHz, (c) 11.9 GHz and the EVM test results of the corresponding channel compared with back-to-back (B2B) case.

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4. Conclusion

In conclusion, we propose a single photonic-assisted system for multi-band microwave frequency conversion based on dual coherent OFCs. The system has potential applications in satellite repeaters. The supported band number can be far greater than the count of the microwave sources used. A proof of concept experiment is carried out in which multiple microwave LOs with 2 GHz, 10 GHz, 18 GHz, 26 GHz are generated. An input C band frequency of 6.1 GHz is successfully converted to 4.1 GHz, 3.9 GHz, 11.9 GHz, etc. in separate channels. System impairments are evaluated that about 9.3% EVM degradation compared with B2B case is observed at 0 dBm input power. Furthermore, all the results are far below the maximum EVM limit of 31.6% for QPSK signal in a very wide power range.

Acknowledgments

This work was supported in part by National 973 Program (2012CB315705), NSFC Program (61302086, 61302016, and 61302060), NCET-13-0682, and China Postdoctoral Science Foundation (2013M540891).

References and links

1. G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich Jr, J. W. de Graaf, W. Habicht II, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005). [CrossRef]  

2. A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004). [CrossRef]  

3. S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.

4. L. A. Mallette, “Atomic and quartz clock hardware for communication and navigation satellites,” in Proceedings of the 39th Annual Precise Time and Time Interval Meeting (Long Beach, CA, 2007), pp. 45–58.

5. G. Jean-Didier, “Evolution of telecommunication payloads: the necessity of new technologies,” in Conference on 20th AIAA International Communication Satellite Systems Conference and Exhibit (American Institute of Aeronautics and Astronautics, Montreal, Quebec, Canada, 2002), paper AIAA-2002–1848.

6. J. P. Dunsmore, Handbook of Microwave Component Measurements: with Advanced VNA Techniques (Wiley, 2012), Chap. 7.

7. J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009). [CrossRef]  

8. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007). [CrossRef]  

9. R. Won, “Microwave photonics shines,” Nat. Photonics 5(12), 736 (2011). [CrossRef]  

10. W. S. C. Chang, RF Photonic Technology in Optical Fiber Links (Cambridge Univ. Press, 2002), Chap. 10.

11. G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993). [CrossRef]  

12. P. Juodawlkis, J. Hargreaves, R. Younger, G. Titi, and J. Twichell, “Optical Down-Sampling of Wide-Band Microwave Signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003). [CrossRef]  

13. M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.

14. J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the atacama large millimeter array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006). [CrossRef]  

15. V. Torres-Company, D. E. Leaird, and A. M. Weiner, “Simultaneous broadband microwave downconversion and programmable complex filtering by optical frequency comb shaping,” Opt. Lett. 37(19), 3993–3995 (2012). [CrossRef]   [PubMed]  

16. A. O. Wiberg, L. Liu, Z. Tong, E. Myslivets, V. Ataie, B. P. Kuo, N. Alic, and S. Radic, “Photonic preprocessor for analog-to-digital-converter using a cavity-less pulse source,” Opt. Express 20(26), B419–B427 (2012). [CrossRef]   [PubMed]  

17. Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012). [CrossRef]  

18. C. H. von Helmolt, U. Krüger, and K. Krüger, “Microwave fiber-optic downconverter,” in Proceeding of Conference on Optical Fiber Communications (Dallas, TX, 1997), pp. 357–358. [CrossRef]  

19. P. Ghelfi, G. Serafino, F. Scotti, F. Laghezza, and A. Bogoni, “Flexible receiver for multiband orthogonal frequency division multiplexing signals at the millimeter waveband based on optical downconversion,” Opt. Lett. 37(18), 3924–3926 (2012). [CrossRef]   [PubMed]  

20. S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. 27(9), 766–768 (2002). [CrossRef]   [PubMed]  

21. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008). [CrossRef]   [PubMed]  

22. C. He, S. L. Pan, R. H. Guo, Y. J. Zhao, and M. H. Pan, “Ultraflat optical frequency comb generated based on cascaded polarization modulators,” Opt. Lett. 37(18), 3834–3836 (2012). [CrossRef]   [PubMed]  

References

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  • |

  1. G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
    [Crossref]
  2. A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004).
    [Crossref]
  3. S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.
  4. L. A. Mallette, “Atomic and quartz clock hardware for communication and navigation satellites,” in Proceedings of the 39th Annual Precise Time and Time Interval Meeting (Long Beach, CA, 2007), pp. 45–58.
  5. G. Jean-Didier, “Evolution of telecommunication payloads: the necessity of new technologies,” in Conference on 20th AIAA International Communication Satellite Systems Conference and Exhibit (American Institute of Aeronautics and Astronautics, Montreal, Quebec, Canada, 2002), paper AIAA-2002–1848.
  6. J. P. Dunsmore, Handbook of Microwave Component Measurements: with Advanced VNA Techniques (Wiley, 2012), Chap. 7.
  7. J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  8. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [Crossref]
  9. R. Won, “Microwave photonics shines,” Nat. Photonics 5(12), 736 (2011).
    [Crossref]
  10. W. S. C. Chang, RF Photonic Technology in Optical Fiber Links (Cambridge Univ. Press, 2002), Chap. 10.
  11. G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993).
    [Crossref]
  12. P. Juodawlkis, J. Hargreaves, R. Younger, G. Titi, and J. Twichell, “Optical Down-Sampling of Wide-Band Microwave Signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003).
    [Crossref]
  13. M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.
  14. J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the atacama large millimeter array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006).
    [Crossref]
  15. V. Torres-Company, D. E. Leaird, and A. M. Weiner, “Simultaneous broadband microwave downconversion and programmable complex filtering by optical frequency comb shaping,” Opt. Lett. 37(19), 3993–3995 (2012).
    [Crossref] [PubMed]
  16. A. O. Wiberg, L. Liu, Z. Tong, E. Myslivets, V. Ataie, B. P. Kuo, N. Alic, and S. Radic, “Photonic preprocessor for analog-to-digital-converter using a cavity-less pulse source,” Opt. Express 20(26), B419–B427 (2012).
    [Crossref] [PubMed]
  17. Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
    [Crossref]
  18. C. H. von Helmolt, U. Krüger, and K. Krüger, “Microwave fiber-optic downconverter,” in Proceeding of Conference on Optical Fiber Communications (Dallas, TX, 1997), pp. 357–358.
    [Crossref]
  19. P. Ghelfi, G. Serafino, F. Scotti, F. Laghezza, and A. Bogoni, “Flexible receiver for multiband orthogonal frequency division multiplexing signals at the millimeter waveband based on optical downconversion,” Opt. Lett. 37(18), 3924–3926 (2012).
    [Crossref] [PubMed]
  20. S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. 27(9), 766–768 (2002).
    [Crossref] [PubMed]
  21. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
    [Crossref] [PubMed]
  22. C. He, S. L. Pan, R. H. Guo, Y. J. Zhao, and M. H. Pan, “Ultraflat optical frequency comb generated based on cascaded polarization modulators,” Opt. Lett. 37(18), 3834–3836 (2012).
    [Crossref] [PubMed]

2012 (5)

2011 (1)

R. Won, “Microwave photonics shines,” Nat. Photonics 5(12), 736 (2011).
[Crossref]

2009 (1)

2008 (1)

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

2007 (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

2006 (1)

J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the atacama large millimeter array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006).
[Crossref]

2005 (1)

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

2004 (1)

A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004).
[Crossref]

2003 (1)

2002 (1)

1993 (1)

G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993).
[Crossref]

Alic, N.

Alter, J. J.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Arapoglou, P. D. M.

A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004).
[Crossref]

Ataie, V.

Benazet, B.

M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.

Bogoni, A.

Bulmer, C. H.

G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993).
[Crossref]

Burns, W. K.

G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993).
[Crossref]

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Cliche, J. F.

J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the atacama large millimeter array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006).
[Crossref]

Coddington, I.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Cottis, P. G.

A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004).
[Crossref]

Crnkovich, J. G.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

de Graaf, J. W.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Deng, L.

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Evins, J. B.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Ghelfi, P.

Gopalakrishnan, G. K.

G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993).
[Crossref]

Guo, R. H.

Habicht, W.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Hagewood, S. M.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Hargreaves, J.

He, C.

Hilterbrick, C. L.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Hrin, G. P.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Hsu, C.

S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.

Juodawlkis, P.

Krüger, K.

C. H. von Helmolt, U. Krüger, and K. Krüger, “Microwave fiber-optic downconverter,” in Proceeding of Conference on Optical Fiber Communications (Dallas, TX, 1997), pp. 357–358.
[Crossref]

Krüger, U.

C. H. von Helmolt, U. Krüger, and K. Krüger, “Microwave fiber-optic downconverter,” in Proceeding of Conference on Optical Fiber Communications (Dallas, TX, 1997), pp. 357–358.
[Crossref]

Kuo, B. P.

Laghezza, F.

Le Kernec, A.

M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.

Leaird, D. E.

Lessin, S. A.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Liu, L.

Maignan, M.

M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.

Mallette, L. A.

L. A. Mallette, “Atomic and quartz clock hardware for communication and navigation satellites,” in Proceedings of the 39th Annual Precise Time and Time Interval Meeting (Long Beach, CA, 2007), pp. 45–58.

Minh, T.

S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.

Monroy, I. T.

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Myslivets, E.

Newbury, N. R.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Pan, M. H.

Pan, S. L.

Panagopoulos, A. D.

A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004).
[Crossref]

Pang, X. D.

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Radic, S.

Rao, S.

S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.

Schiller, S.

Scotti, F.

Serafino, G.

Shillue, B.

J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the atacama large millimeter array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006).
[Crossref]

Sotom, M.

M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.

Swann, W. C.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Tavik, G. C.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Titi, G.

Tong, Z.

Torres-Company, V.

Twichell, J.

von Helmolt, C. H.

C. H. von Helmolt, U. Krüger, and K. Krüger, “Microwave fiber-optic downconverter,” in Proceeding of Conference on Optical Fiber Communications (Dallas, TX, 1997), pp. 357–358.
[Crossref]

Wang, J.

S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.

Weiner, A. M.

Wiberg, A. O.

Won, R.

R. Won, “Microwave photonics shines,” Nat. Photonics 5(12), 736 (2011).
[Crossref]

Wu, D. C.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Yao, J. P.

Younger, R.

Yu, X. B.

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Zhao, Y.

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Zhao, Y. J.

Zheng, X. P.

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

IEEE Commun. Surveys Tuts. (1)

A. D. Panagopoulos, P. D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: propagation impairments and mitigation techniques,” IEEE Commun. Surveys Tuts. 6(3), 2–14 (2004).
[Crossref]

IEEE Contr. Syst. Mag. (1)

J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the atacama large millimeter array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

G. K. Gopalakrishnan, W. K. Burns, and C. H. Bulmer, “Microwave-optical mixing in LiNbO3 modulators,” IEEE Trans. Microw. Theory Tech. 41(12), 2383–2391 (1993).
[Crossref]

J. Lightwave Technol. (2)

Nat. Photonics (2)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

R. Won, “Microwave photonics shines,” Nat. Photonics 5(12), 736 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Other (7)

M. Sotom, B. Benazet, A. Le Kernec, and M. Maignan, “Microwave photonic technologies for flexible satellite telecom payloads,” in Proceedings of 35th European Conference on Optical Communication (Vienna, Austria, 2009), pp. 1–4.

C. H. von Helmolt, U. Krüger, and K. Krüger, “Microwave fiber-optic downconverter,” in Proceeding of Conference on Optical Fiber Communications (Dallas, TX, 1997), pp. 357–358.
[Crossref]

W. S. C. Chang, RF Photonic Technology in Optical Fiber Links (Cambridge Univ. Press, 2002), Chap. 10.

S. Rao, T. Minh, C. Hsu, and J. Wang, “Advanced antenna technologies for satellite communication payloads,” in Proceedings of First European Conference on Antennas and Propagation (Nice, France, 2006), pp. 1–6.

L. A. Mallette, “Atomic and quartz clock hardware for communication and navigation satellites,” in Proceedings of the 39th Annual Precise Time and Time Interval Meeting (Long Beach, CA, 2007), pp. 45–58.

G. Jean-Didier, “Evolution of telecommunication payloads: the necessity of new technologies,” in Conference on 20th AIAA International Communication Satellite Systems Conference and Exhibit (American Institute of Aeronautics and Astronautics, Montreal, Quebec, Canada, 2002), paper AIAA-2002–1848.

J. P. Dunsmore, Handbook of Microwave Component Measurements: with Advanced VNA Techniques (Wiley, 2012), Chap. 7.

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram of multiple LOs generation (without mixing in the dashed line box) and inter & intra-band frequency conversion. (b) Schematic illustration of multiple LOs generation through channelizing dual coherent OFCs. (c) Signal multicasting for frequency conversion and the instantaneous bandwidth illustration.
Fig. 2
Fig. 2 Experimental setup of the inter- and intra-band frequency conversion. CW Laser: continuous wave laser; PM-TOC: polarization maintained tunable optical coupler; PolM: polarization modulator; Pol: polarizer; MZM: Mach Zehnder modulator; DPMZM-dual-parallel MZM; OC: optical coupler; EDFA: erbium doped fiber amplifier; PD: photo detector; EA: electrical amplifier.
Fig. 3
Fig. 3 Dual coherent OFCs with 18 GHz center frequency shift and 38 GHz, 30 GHz mode spacing respectively.
Fig. 4
Fig. 4 Multiple LOs within different bands of (a) 2 GHz, (b) 10 GHz, (c) 18 GHz and (d) 26 GHz.
Fig. 5
Fig. 5 Coherent signal OFC and local OFC (a) before and (b) after channelization.
Fig. 6
Fig. 6 The converted signal from the first three channel at (a) 4.1 GHz, (b) 3.9 GHz, (c) 11.9 GHz and the EVM test results of the corresponding channel compared with back-to-back (B2B) case.

Tables (1)

Tables Icon

Table 1 Power ratios of the undesired signals to the desired signal

Equations (8)

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

f O F C 1 ( i ) = f c + i δ 1 .
f OFC2 ( i )= f c + δ s +i δ 2 .
f LO ( i )= δ s +i| δ 1 δ 2 |.
n δ s | δ 1 δ 2 | andn BW δ s | δ 1 δ 2 | .
f out ( i )=| f LO ( i ) f in |.
IBW( i )= δ s +i| δ 1 δ 2 |.
J 0 ( β )= J 2 ( β )β1.84,5.33,.
sin( 2α )cosϕ= J 1 2 ( β ) J 0 2 ( β ) J 1 2 ( β )+ J 0 2 ( β ) .

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