Open Access
Add to BibSonomy
Abstract
In this work, we present a coherent distributed radio frequency (RF) array, discover and quantitatively describe the strong positive correlation between reconstructed signals for the first time. Eight replicable parallel receivers are connected to the phase-locked common trunk link via eight optical couplers spaced 1 km apart. The forward and backward signals at each receiver, extracted from two ports of optical couplers, are recovered to RF signals separately and then mixed to achieve upward frequency conversion. The link delay jitter is counteracted by wavelength-tuning of the optical carrier. With the long-term stability of point-to-multipoint fiber-optic RF dissemination effectively improved, the coherent distributed array is generated, and further the relative frequency stability between signals at different receivers is studied. The proposed correlation coefficient at 103 s is ∼0.8 and shows a slight downward trend with the increase of averaging time based on our experimental results.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Highly coherent radio frequency (RF) arrays are in great demand in various applications, such as deep space network, modern radar systems, and radio telescope imaging [1–4]. Free from considerations of much greater phase noise and periodic deep fades induced by atmospheric turbulence [5], stable frequency dissemination operated over fiber-optic links is recognized as an attractive approach for RF arrays generation. To our knowledge, the most advanced point-to-multipoint fiber-optic RF dissemination scheme has achieved the overlapping Allan deviation (ADEV) of 10−15 order at 1 s and 10−18 order at 104 s [6]. To obtain coherent distributed RF arrays, the essential problem is to eliminate the link delay jitter caused by mechanical vibration or temperature fluctuation in a simple, flexible, and cost-effective way.
In recent years, many ingenious designs based on auxiliary signal sensing and frequency transformation have been reported [7–12]. In these designs, phase conjugation is mainly achieved through frequency mixing, so that the link delay jitter is automatically canceled by pre-compensation. A photonic microwave phase conjugation scheme is proposed to achieve all-optical stable quadruple frequency dissemination, thus avoiding undesired spurious signals from electrical mixers [9]. Despite the fast compensation speed and large compensation range, such a passive phase-noise cancelation scheme is not applicable to the generation of large-scale arrays, for its cumbersome structure as well as high cost. Meanwhile, another phase stabilization scheme using the error signal to feedback control the noise compensation system has emerged, which may include phase-locked loop (PLL) [13], variable delay lines [6,14], and optical combs [15], to name just a few. Suitable topologies are applied to extend to point-to-multipoint ultra-stable frequency dissemination. Specifically, bus topologies [7,13,14] and ring topologies [9,12,15] usually phase-lock the trunk fiber-optic link to stabilize the propagation delay of reconstructed signals at arbitrarily inserted receivers, where optical couplers seem to be indispensable. Tree topologies [6,8,11], also known as branching topologies [10], deploy compensation components at each receiver, and the wavelength-division-multiplexing (WDM) technology is commonly used to distinguish receivers. Almost all of the current schemes are focused on how to distribute RF signals from the local site to multiple receivers with high frequency stability, and independent tests are generally performed on only two receivers. However, little attention is paid to the relative frequency stability between signals at different receivers. It is worth considering because some distributed applications, such as very long baseline interferometry (VLBI) [16], require both good long-term stability of signals received at multiple radio observatories and precise phase alignment between them. Efficient interferometric processing can be carried out when the correlation is precisely known, otherwise it will lead to relatively large coherence loss.
In this manuscript, we demonstrate a bus-topology coherent distributed RF array with ultra-high precision parallel synchronization for multiple receivers and further explore the relative frequency stability between reconstructed signals. The random phase fluctuation in the trunk fiber-optic link is suppressed in a closed-loop, by actively tuning the wavelength of the only laser at the local site, as we have previously proposed in the point-to-point stable RF delivery scheme [17]. After compensation, the coupled forward and backward RF signals at any receiver are mixed and filtered to obtain a sum-frequency signal, which is the phase-stabilized reconstructed signal we expect. Unlike several schemes that also use optical couplers [7,15], there is no need to use additional optical filters due to the use of the same wavelength for two-way transmission, which saves a certain power loss. Modular receivers can be easily inserted or removed by optical couplers at any intermediate position to custom RF arrays. On this basis, we analyze the relative frequency stability between signals at any two receivers and propose a correlation coefficient to provide a quantitative evaluation of the strength and direction of the correlation. In our experiment, a reference RF signal is delivered to eight receivers placed at 1 km intervals. The 30-minute maximum jitter RMS is ∼2 ps and the overlapping ADEV at 103 s is ∼3.73×10−15 at each receiver; The relative overlapping ADEV at 103 s is ∼2.27×10−15 between any two receivers and the correlation coefficient at 103 s is calculated as high as ∼0.8. These results signify good stability and a strong positive correlation of the generated coherent distributed RF array.
2. Principle
The schematic diagram of RF arrays generation is illustrated in Fig. 1, which consists of a local site, a remote site, and two receivers served as examples. At the local site, a reference RF signal can be simply denoted as a cosine function: ${V_0} = \cos ({{w_0}t + {\varphi_0}} )$, where the amplitude is normalized, the angular frequency is ${w_0}$, and the initial phase is regarded as a stable ${\varphi _0}$. A trunk link, composed of optical fiber segments, is laid to connect the local site and remote site. As is well-known, temperature fluctuation ($\Delta T$) is the most important factor affecting the long-term stability of fiber-optic RF propagation delay, which can be determined by $\frac{{\textrm{d}\tau }}{{\textrm{d}T}} = \frac{L}{c}\left( {\frac{n}{L}\frac{{\textrm{d}L}}{{\textrm{d}T}} + \frac{{\textrm{d}n}}{{\textrm{d}T}}} \right) = \frac{L}{c}\xi $, where c is the speed of light in vacuum, $\xi $ is a simplified representation of the rate of change of two temperature-dependent terms, L and n are the length and refractive index of optical fiber, respectively. Note that the length and refractive index vary with environmental perturbation, thus introducing annoying link delay jitter ($\Delta \tau $). The thermal coefficient of delay for standard telecom SMF28 fiber is typically 7 ppm/°C [18], corresponding to 35 ps/km/°C. The round-trip RF signal back to the local site can be expressed as ${V_1} \propto \cos [{{w_0}({t - 2{\tau_p}} )+ {\varphi_0}} ]$, where ${\tau _p}$ is the one-way propagation delay from the local site to the remote site. Here we assume that the propagation delay (including the delay jitter) experienced by the signal transmitted in the forward and backward links is equal. It is reasonable since the use of the same wavelength for two-way transmission guarantees the symmetry of the round-trip link, and the temperature change itself is a slow process. The change in propagation delay with respect to the change in optical wavelength can be written as a product of the dispersion coefficient (${D_\lambda }$) and fiber length ($L$):
where $\lambda $ is the wavelength of the optical carrier modulated by RF signals and ${D_{\lambda = 1550\textrm{nm}}}$ is 17 ps/nm/km. Therefore, the link delay jitter can be greatly eliminated by a closed-loop tunable optical wavelength noise compensation system, according toConsequently, whether the signal delivered to the remote site or returned to the local site is phase-stabilized, expressed as ${\varphi _c} + {\varphi _0}$ and $2{\varphi _c} + {\varphi _0}$, respectively, where ${\varphi _c} ={-} {w_0}{\tau _c}$. Here the stabilized one-way propagation delay is written as ${\tau _c}$, to distinguish it from ${\tau _p}$ in the free-running situation. Note that such an active stabilization scheme is very promising for a very long delivery distance because the tunable delay capacity is synchronously expanded with the increase of fiber length. Referring to our past research in [17], there is no problem with increasing the length to 54 km. Furthermore, the tuning range and granularity of the wavelength play a key role in the compensation effect, as can be seen from Eq. (2).
As the access point of the receiver, two ports of the optical coupler form the link connection, and the remaining two ports are applied to extract forward and backward signals respectively. Assuming that ${\tau _n}({n = 1,2, \ldots } )$ represents the corresponding propagation delay of each optical fiber segment, the propagation delay from the local site to the $k\textrm{ - th}$ receiver (i.e., the forward signal coupled out from one port) can be expressed as $\sum\nolimits_{n = 1}^k {{\tau _n}} $. The backward signal is reflected at the remote site, sent back along the same path, and partially coupled out from the other port. It is observed that the propagation delay experienced by the backward signal is $2{\tau _c} - \sum\nolimits_{n = 1}^k {{\tau _n}} $ in total. Mixing the two recovered RF signals detected by photodetectors (PDs) yields a sum-frequency signal. Since the phase variation of sum-frequency signals at arbitrarily inserted receivers is $2{\varphi _c}$ (setting aside the stable ${\varphi _0}$), approximately equivalent to that of the round-trip signal passing through the trunk link, the phase fluctuation of sum-frequency signals can also be suppressed by wavelength-tuning. The reconstructed signal is calculated as
In this way, the phase stabilization process occurs at the local site, and the parallel synchronized RF array can be generated by implementing upward frequency conversion at each receiver.
Perfect phase stabilization cannot be achieved with any phase-locking technique. Suppose the residual phase noise of reconstructed signals is ${\varphi _A}(t )$ at receiver A and ${\varphi _B}(t )$ at receiver B, then the relative phase noise between them is written as ${\varphi _r}(t )= {\varphi _A}(t )- {\varphi _B}(t )$. By taking the derivative with respect to time, the following relation can be obtained:
where $y(t )= \frac{1}{{\textrm{2}\pi {\nu _0}}}\frac{{\textrm{d}\varphi (t )}}{{\textrm{d}t}}$ is the fractional frequency fluctuation [19]. Since traditional standard variance cannot analyze divergent noise (such as white frequency noise), David Allan proposed the M-sample variance [20]. The more common Allan variance (AVAR), ${\sigma _y}^2(\tau )= \frac{1}{2}\left\langle {{{[{{{\bar{y}}_{i + 1}}(\tau )- {{\bar{y}}_i}(\tau )} ]}^2}} \right\rangle $, is defined as the two-sample variance, where $\tau $ is the observation period and ${\bar{y}_i}(\tau )$ is the $i\textrm{ - th}$ fractional frequency fluctuation average over the observation time $\tau $. It is still the mean square of the difference, just like the standard variance, but the difference is between the averages of two adjacent intervals. The AVAR depicts the degree to which the actual reception of phase or frequency delivery deviates from the RF source, which is usually referred to as “fractional frequency instability”. The ADEV is the square root of the AVAR. Then the relative AVAR is calculated as3. Experiment
3.1 Experimental setup
Our experiment is carried out in the natural environment of the indoor laboratory with the temperature slowly varying and researchers occasionally moving around. The experimental setup of RF arrays generation is shown in Fig. 2. At the local site, a 1.21 GHz reference signal is generated by an analog signal generator (Keysight E8257D) and modulated onto an optical carrier from the wavelength-tunable laser (WTL). The WTL utilized has a wide tuning range covering the whole C-band (1528∼1565 nm), and the tuning granularity reaches 1 pm. For the Mach-Zehnder modulator (MZM), we use a bias control (BC) module to maintain its bias at the quadrature point. Then the modulated signal is guided into the trunk link with the help of an optical circulator (OC). Considering that the increase of receivers will lead to a change in link power allocation, we verify the feasibility of our scheme through the trunk link constituted by eight 1-km single-mode fiber (SMF) segments and eight optical couplers (labeled A∼H) alternately connecting in series. The end of the link is attached to a signal return device, consisting of an optical circulator (OC) and an erbium-doped fiber amplifier (EDFA). To avoid SNR reduction involved with too small signal power in the transmission link, the EDFA at the remote site is necessary to compensate for the power attenuation, and the coupling ratio of 9:1 is chosen for optical couplers. After 16-km round-trip transmission, the signal back to the local site is opto-electronically converted by a photodetector (PD, Menlo Systems FPD310-FC-NIR) and then a clean 1.21 GHz signal is output from a narrowband band-pass filter (BPF). The error signal is obtained by a phase detector and employed as feedback to drive the wavelength-tuning by a PID control algorithm. As a result, the phase stabilization of the trunk link is always attained.
Fig. 2. Experimental setup of RF arrays generation. WTL: wavelength-tunable laser, MZM: Mach-Zehnder modulator, BC: bias control, OC: optical circulator, EDFA: erbium-doped fiber amplifier, PD: photodetector, BPF: band-pass filter, SMF: single-mode fiber.
Optical couplers are served as access points. The structure of parallel receivers and measurement methods are detailed in Fig. 2(b). An optical fiber amplifier module is installed to properly amplify the coupled forward and backward signals, followed by a disseminated RF recovery module, which contains a homemade PD with the gain of 35 dB and the bandwidth of 3 GHz, and a BPF with the center frequency of 1.21 GHz. Consequently, the forward and backward recovered RF signals are obtained. Instead of mixing them directly, a conventional triple-mixing method [23,24] is adopted here. We utilize a 0.99 GHz signal to frequency-convert the two signals upward and downward, and then mix the 2.2 GHz and 0.22 GHz filtered signals to obtain the sum-frequency signal, which greatly restrains second harmonic crosstalk. Moreover, electrical amplifiers are added to compensate for the signal power loss from mixers. The measurement methods are in two parts. For the eye diagram observation, the reconstructed 2.42 GHz signal is fed into the test channel, and the attenuated 1.21 GHz source signal is used as the trigger of the sampling oscilloscope (Keysight 86100C). For the ADEV measurement at each receiver, we are supposed to mix the sum-frequency signal with a 2.41 GHz signal in sync with the source to produce a 10 MHz test signal, due to the relatively low operating frequency of the frequency counter (Keysight 53230A).
3.2 Results and discussion
In the beginning, we observe the time domain waveforms of reconstructed signals by the sampling oscilloscope and intuitively compare the phase fluctuation in the 30-minute recording time. As can be seen in Fig. 3(a), the signal is visibly drifting when the system is free-running, and the maximum jitter RMS is 18.39 ps. By contrast, signals received at eight receivers are all phase-stabilized after compensation, and the maximum jitter RMS is only ∼2 ps. Figure 3(b) shows a stage of wavelength-tuning and phase stabilization. As expected, the phase error is locked to zero as the wavelength is tuned, hence the cancelation of the phase fluctuation induced by environmental perturbation.
Fig. 3. (a) Eye diagrams in the 30-minute recording time of reconstructed signals. (b) A stage of wavelength-tuning and phase stabilization.
To be more explicit about the frequency stability from the local site to multiple receivers, we measure and collect the instantaneous frequency by the frequency counter, and focus on the overlapping ADEV. Taking overlapping samples rather than contiguous samples is to reduce statistical uncertainty and plot more detailed curves. Figure 4(a) shows four free-running situations at receiver A and one stabilized case as a comparison. Note that four curves display different trends in unpredictable environmental perturbation, but one thing in common is that the larger value of the overlapping ADEV (i.e., the worse fractional frequency stability) is presented after the averaging time of 20 s compared with the stabilized case. The results confirm that our scheme has good long-term stability despite a slight deterioration within 10 s (the overlapping ADEV is ∼1.83×10−13 at 1 s). We believe this slight deterioration of short-term stability is caused by the intrinsic RoF link noise (e.g., Rayleigh scattering [25–28]) and the imperfect coefficients of the PID algorithm. Independent tests are conducted at the remaining seven receivers using the same parameter settings and the results are shown in Fig. 4(b). After an averaging time of about 102 s, the stability of the free-running link goes deteriorated significantly, while the wavelength-tuning phase-locked links will lead to an approximately linear decline in ADEV. It is noticed that in this demonstrated bus-topology scheme, wavelength-tuning of the only laser at the local site realizes the parallel synchronization for eight receivers. With compensation enabled, each curve performs similarly and the overlapping ADEV at 103 s is down to ∼3.73×10−15, clearly distinguished from the free-running situation. Referring to our previous experience [29,30], the worsening of the free-running system will be more serious with the extension of timescales (such as 104 s), so that the stabilization effect will be more pronounced. In addition, we measure the noise floor by replacing the generated sum-frequency signal with a 2.42 GHz signal in sync with the source. There is a certain gap between the stabilized result and the noise floor, which may be attributable to the absence of vibration and thermal insulations at the local site and receivers. Phase noise outside the compensation loop is considered as the limit of the frequency stability.
Fig. 4. Overlapping ADEV of (a) the reconstructed signal at receiver A and (b) reconstructed signals at eight receivers.
Next, we concentrate on the relative frequency stability between signals at different receivers. In this part, we take the 10 MHz signal output from one receiver as the time base of the frequency counter to measure the instantaneous frequency of the signal output from the other receiver. Figure 5(a) shows the comparisons of the fractional frequency fluctuation among several groups. Significantly smaller fluctuation is observed in the results between any two receiver signals, which reveals that when the fluctuation at one receiver increases, the fluctuation at the other receiver also tends to increase, in line with the trend of a positive correlation. Likewise, as can be seen in Fig. 5(b), the relative overlapping ADEV of the selected four groups (located in the front, middle, rear of the trunk link and across the trunk link) reaching ∼8.70×10−14 at 1 s and ∼2.27×10−15 at 103 s is roughly lower than the overlapping ADEV at the single receiver. Although the instantaneous frequency data of any two receiver signals is not obtained at the same time, the ADEV value is considered constant when compensation is enabled, so we can calculate the proposed correlation coefficient corresponding to each averaging time, and the result is shown in Fig. 6. Note that the curve shows a slight downward trend, indicating a stronger correlation between different receivers in a short period of time in our experimental scenario. In our scheme, independent fiber segments and optical couplers form a common link in a bus topology and only one set of the compensation component is employed. Thanks to the frequency-mixing process at each receiver, the phase of the reconstructed signal is always approximately equivalent to that of the round-trip signal. A positive correlation is expected when the system is free-running, especially at a longer time scale, because the phase is mainly affected by the trunk link delay jitter, which is equivalent for each receiver. However, when the considerable trunk link delay jitter is greatly suppressed due to the dynamic tuning of the wavelength, the residual phase noise in signal reconstruction at each receiver is more dominant. That is, the correlation is affected by the environmental conditions of each receiver, when the trunk link is phase-locked. Since our experiment is conducted indoors, the receivers are placed under similar environmental conditions, leading to the strong correlation observed. It is worth mentioning that the proposed correlation coefficient visually represents the correlation between the residual phase noise after phase stabilization in the generated coherent distributed RF array, which is a favorable complement to the common Allan variance.
Fig. 5. (a) Comparisons of the fractional frequency fluctuation among several groups. (b) Relative overlapping ADEV of the selected four groups.
4. Conclusion
In summary, we presented a coherent distributed RF array and demonstrated the relative frequency stability between signals at different receivers. Eight receivers were organized in a bus topology and accessed to the trunk link at equal intervals of 1 km by optical couplers. By employing the error signal to drive the wavelength-tuning of the WTL at the local site, the phase fluctuation of sum-frequency signals output from multiple receivers as well as that of the round-trip signal was effectively canceled out. The phase stability is not restricted by the fiber length, and the insertion or removal of receivers is allowed, which make our design suitable for the construction of large-scale coherent distributed RF arrays. Theoretical analysis and experimental verification for relative frequency stability have showed that a positive correlation is expected in the case of free-running, while the correlation is mainly affected by environmental conditions of each receiver when the trunk link is phase-locked. A correlation coefficient is proposed to quantitatively evaluate the correlation, which is valuable especially for some sophisticated applications where the correlation between multiple signals is strictly required.
Funding
National Key Research and Development Program of China (2018YFA0701902); National Natural Science Foundation of China (62071055); National Natural Science Foundation of China (62135014); National Natural Science Foundation of China (61901428).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. Calhoun, S. Huang, and R. L. Tjoelker, “Stable Photonic Links for Frequency and Time Transfer in the Deep-Space Network and Antenna Arrays,” Proc. IEEE 95(10), 1931–1946 (2007). [CrossRef]
2. W. Wang, “GPS-Based Time & Phase Synchronization Processing for Distributed SAR,” IEEE Trans. Aerosp. Electron. Syst. 45(3), 1040–1051 (2009). [CrossRef]
3. S. Wassin and T. B. Gibbon, “A proposed optical frequency synchronization system for the South African MeerKAT radio telescope,” Opt. Fiber Technol. 55, 102164 (2020). [CrossRef]
4. S. Huang and R. L. Tjoelker, “Stabilized Photonic Links for Deep Space Tracking, Navigation, and Radio Science Applications,” in Precise Time and Time Interval Systems and Applications Meeting, (2012),
5. B. P. Dix-Matthews, S. W. Schediwy, D. R. Gozzard, E. Savalle, F.-X. Esnault, T. Lévèque, C. Gravestock, D. D’Mello, S. Karpathakis, M. Tobar, and P. Wolf, “Point-to-point stabilized optical frequency transfer with active optics,” Nat. Commun. 12(1), 515 (2021). [CrossRef]
6. M. Jiang, Y. Chen, N. Cheng, Y. Sun, J. Wang, R. Wu, F. Yang, H. Cai, and Y. Gui, “Multi-Access RF Frequency Dissemination Based on Round-Trip Three-Wavelength Optical Compensation Technique Over Fiber-Optic Link,” IEEE Photonics J. 11(3), 7202808 (2019). [CrossRef]
7. G. Tong, T. Jin, H. Chi, X. Zhu, T. Lai, D. Li, and L. Zuo, “Stable radio frequency dissemination in a multi-access link based on passive phase fluctuation cancellation,” Opt. Commun. 423, 53–56 (2018). [CrossRef]
8. L. Yu, R. Wang, L. Lu, Y. Zhu, J. Zheng, C. Wu, B. Zhang, and P. Wang, “WDM-based radio frequency dissemination in a tree-topology fiber optic network,” Opt. Express 23(15), 19783–19792 (2015). [CrossRef]
9. H. Wang, X. Xue, S. Li, and X. Zheng, “All-Optical Arbitrary-Point Stable Quadruple Frequency Dissemination With Photonic Microwave Phase Conjugation,” IEEE Photonics J. 10(4), 5501508 (2018). [CrossRef]
10. Y. Bai, B. Wang, C. Gao, J. Miao, X. Zhu, and L. Wang, “Fiber-based radio frequency dissemination for branching networks with passive phase-noise cancelation,” Chin. Opt. Lett. 13(6), 061201 (2015). [CrossRef]
11. C. Hu, B. Luo, W. Pan, L. Yan, and X. Zou, “Multipoint stable radio frequency long distance transmission over fiber based on tree topology, with user fairness and deployment flexibility,” Opt. Express 28(16), 23874–23880 (2020). [CrossRef]
12. C. Liu, T. Jiang, M. Chen, S. Yu, R. Wu, J. Shang, J. Duan, and W. Gu, “GVD-insensitive stable radio frequency phase dissemination for arbitrary-access loop link,” Opt. Express 24(20), 23376–23382 (2016). [CrossRef]
13. Y. Bai, B. Wang, C. Gao, W. L. Chen, J. Miao, X. Zhu, and L. J. Wang, “Fiber-Based Multiple-Access Ultrastable Radio and Optical Frequency Dissemination,” in Joint European Frequency and Time Forum & International Frequency Control Symposium (EFTF/IFC), (IEEE, 2013), 1014–1017.
14. P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. Lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(9), 1804–1810 (2013). [CrossRef]
15. J. Shang, T. Jiang, C. Liu, X. Chen, Y. Lu, S. Yu, and H. Guo, “Stable frequency dissemination over multi-access fiber loop link with optical comb,” Opt. Express 26(26), 33888–33894 (2018). [CrossRef]
16. H. Schuh and D. Behrend, “VLBI: A fascinating technique for geodesy and astrometry,” J. Geodyn. 61, 68–80 (2012). [CrossRef]
17. A. Zhang, Y. Dai, F. Yin, T. Ren, K. Xu, J. Li, Y. Ji, J. Lin, and G. Tang, “Stable radio-frequency delivery by λ dispersion-induced optical tunable delay,” Opt. Lett. 38(14), 2419–2421 (2013). [CrossRef]
18. F. Narbonneau, M. Lours, S. Bize, A. Clairon, and G. Santarelli, “High resolution frequency standard dissemination via optical fiber metropolitan network,” Rev. Sci. Instrum. 77(6), 064701 (2006). [CrossRef]
19. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of Frequency Stability,” IEEE Trans. Instrum. Meas. IM-20(2), 105–120 (1971). [CrossRef]
20. D. W. Allan, “Statistics of Atomic Frequency Standards,” Proc. IEEE 54(2), 221–230 (1966). [CrossRef]
21. B. Wang, X. Zhu, C. Gao, Y. Bai, J. W. Dong, and L. J. Wang, “Square Kilometre Array Telescope—Precision Reference Frequency Synchronisation via 1f-2f Dissemination,” Sci. Rep. 5(1), 13851 (2015). [CrossRef]
22. D. J. Rumsey, “How to Interpret a Correlation Coefficient r,” in Statistics For Dummies, 2nd Edition (For Dummies, 2016).
23. H. Li, G. Wu, J. Zhang, J. Shen, and J. Chen, “Multi-access fiber-optic radio frequency transfer with passive phase noise compensation,” Opt. Lett. 41(24), 5672–5675 (2016). [CrossRef]
24. Z. Jiang, F. Yin, Q. Cen, Y. Dai, and K. Xu, “Stable downlink frequency transmission from arbitrary injection point with endless and quick phase error correction,” Opt. Express 28(22), 33690–33698 (2020). [CrossRef]
25. O. Okusaga, J. Cahill, W. Zhou, A. Docherty, G. M. Carter, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Joint Conference of the IEEE International Frequency Control and the European Frequency and Time Forum (FCS), (IEEE, 2011), 1–6.
26. J. Nanni, A. Giovannini, M. Usman Hadi, E. Lenzi, S. Rusticelli, R. Wayth, F. Perini, J. Monari, and G. Tartarini, “Controlling Rayleigh-Backscattering-Induced Distortion in Radio Over Fiber Systems for Radioastronomic Applications,” J. Lightwave Technol. 38(19), 5393–5405 (2020). [CrossRef]
27. J. P. Cahill, O. Okusaga, W. Zhou, C. R. Menyuk, and G. M. Carter, “Superlinear growth of Rayleigh scattering-induced intensity noise in single-mode fibers,” Opt. Express 23(5), 6400–6407 (2015). [CrossRef]
28. Y. Cui, T. Jiang, S. Yu, R. Wu, and W. Gu, “Passive-Compensation-Based Stable RF Phase Dissemination for Multiaccess Trunk Fiber Link With Anti-GVD and Anti-Backscattering Function,” IEEE Photonics J. 9(5), 7203608 (2017). [CrossRef]
29. F. Yin, A. Zhang, Y. Dai, J. Li, Y. Zhou, J. Dai, and K. Xu, “Improved Phase Stabilization of a Radio-OverFiber Link Under Large and Fine Delay Tuning,” IEEE Photonics J. 10(4), 7201706 (2015). [CrossRef]
30. A. Zhang, Y. Dai, F. Yin, T. Ren, K. Xu, J. Li, and G. Tang, “Phase stabilized downlink transmission for wideband radio frequency signal via optical fiber link,” Opt. Express 22(18), 21560–21566 (2014). [CrossRef]
