低地球軌道（Low-Earth Orbit, LEO）衛星通訊通道具有時變延遲、訊號強度衰減以及都卜勒頻移等特性。由於 LEO 衛星距地球較近，訊號傳輸延遲低，因此特別適合應用於衛星網際網路。然而，低軌道衛星必須以較高的軌道速度繞地運行，致使衛星與地面站之間的距離隨時間快速變化，進而導致訊號傳輸延遲呈現連續時變的特性。為了開發能適應此類通道特性的通訊收發機，需要設計一個能夠真實模擬 LEO 通道效應的通道模擬器，以便有效驗證收發機的性能，並提升其在實際運作環境中的可靠性。基於以上需求，我們首先探討了Farrow 結構的可變分數延遲濾波器的設計。此濾波器是 LEO 通道模擬器中的關鍵元件，可在DSP有效實現連續時變延遲效應。與其他分數延遲濾波器採用頻域設計的方法不同，本論文提出的 Farrow 結構內插器是採用時域上最小ISI效能(RC波形)設計，實現SRRC波形的分數取樣週期的延遲。最後，將設計的Farrow 結構內插器應用於符碼間時序同步(Symbol-Timing Synchronization)與任意符碼率轉換(Arbitray Symbol Rate Converter)上。

Low-Earth Orbit (LEO) satellite communication channels are characterized by time-varying propagation delays, signal attenuation, and Doppler frequency shifts. Owing to the proximity of LEO satellites to the Earth’s surface, they offer high signal transmission speeds and low latency, making them particularly well-suited for applications such as satellite internet. However, LEO satellites must operate at high orbital velocities to maintain their trajectories, causing the distance between the satellite and ground stations to change rapidly over time. This results in continuously time-varying propagation delays.
To enable communication transceivers to operate reliably under such challenging channel conditions, it is crucial to design a channel emulator capable of accurately replicating the effects of a LEO satellite link. Such an emulator allows rigorous validation of transceiver performance and enhances robustness in practical deployment scenarios.
In response to these challenges, this paper investigates the design of a Farrow-structured variable fractional delay filter, a critical component of the LEO channel emulator for efficiently realizing continuously time-varying delays in digital signal processing (DSP). Unlike conventional fractional delay filters designed in the frequency domain, the proposed Farrow-structured interpolator is optimized in the time domain based on the minimum inter-symbol interference (ISI) performance criterion of a raised-cosine (RC) pulse. This approach enables precise fractional-sample delays for signals shaped with square-root raised cosine (SRRC) filtering. Finally, the designed Farrow-structured interpolator is applied to symbol timing synchronization and arbitrary symbol rate conversion in SRRC-based transceivers.

參考文獻
[1] B. Di, L. Song, Y. Li and H. V. Poor, "Ultra-dense LEO: Integration of satellite access networks into 5G and beyond", IEEE Wireless Commun, vol. 26, no. 2, pp. 62-69, Apr. 2019.
[2] T. Darwish, G. K. Kurt, H. Yankomeroglu, M. Bellemare, and G. Lamontagne, “LEO Satellites in 5G and Beyond Networks: A Review From a Standardization Perspective,” IEEE Access, vol. 10, pp. 35040-35060, 2022.
[3] V. M. Baeza, E. Lagunas, H. Al-Hraishawi, and S. Chatzinotas, “An Overview of Channel Models for NGSO Satellites,” IEEE 96th Vehicular Technology Conference, 2022,
[4] D. K. Upadhyay and V. Chaudhary, "Design of Variable Fractional Delay Filter Using DFT Interpolation," 2009 International Conference on Advances in Recent Technologies in Communication and Computing, 2009, pp. 691-693.
[5] T.-B. Deng, "Hybrid Structures for Low-Complexity Variable Fractional-Delay FIR Filters," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, no. 4, pp. 897-910, April 2010.
[6] K. Meena and T. K. Rawat, "Hybrid structure for odd order variable fractional-delay FIR filter," 2014 International Conference on Communication and Signal Processing, 2014, pp. 1273-1278.
[7] T.-B. Deng, "Low-complexity and high-accuracy odd-order variable fractional-delay digital filters," 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012, pp. 1589-1592.
[8] X. Zhang, "Maxflat Fractional Delay IIR Filter Design," in IEEE Transactions on Signal Processing, vol. 57, no. 8, pp. 2950-2956, Aug. 2009.
[9] S.-C. Pei, P.-H. Wang and H.-S. Lin, "Closed-form design of maximally flat FIR fractional delay filters," in IEEE Signal Processing Letters, vol. 13, no. 7, pp. 405-408, July 2006.
[10] H. K. Kwan and A. Jiang, "FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 9, pp. 2064-2074, Sept. 2009.
[11] H. Zhao and J. Yu, "A simple and efficient design of variable fractional delay FIR filters," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 2, pp. 157-160, Feb. 2006.
[12] M. Blok, "On practical aspects of optimal FSD filter design using extracted window method," 2011 20th European Conference on Circuit Theory and Design (ECCTD), 2011, pp. 330-333.
[13] L. Erup, F. M. Gardner and R. A. Harris, "Interpolation in digital modems. II. Implementation and performance," in IEEE Transactions on Communications, vol. 41, no. 6, pp. 998-1008, June 1993.
[14] Digital Video Broadcasting (DVB), Digital Video Broadcasting (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications. ETSI Standard, Part 1: DVB- S2, 2014.
[15] Flexible Advanced Coding and Modulation Scheme for High Rate Telemetry Apllications, The Consultative Committee for Space Data Systems (CCSDS), Feb. 2023.
[16] C. W. Farrow, "A continuously variable digital delay element," 1988., IEEE International Symposium on Circuits and Systems, Espoo, Finland, 1988, pp. 2641-2645.
[17] H. Li, G. Torfs, T. Kazaz, J. Bauwelinck and P. Demeester, "Farrow structured variable fractional delay lagrange filters with improved midpoint response," 2017 40th International Conference on Telecommunications and Signal Processing (TSP), Barcelona, Spain, 2017, pp. 506-509.
[18] H. Meyr, M. Moeneclaey and S. Fechtel, Digital Communication Receivers: Synchronization Channel Estimation and Signal Processing. Hoboken, NJ, USA:Wiley, 1997.
[19] ETSI, Digital Video Broadcasting (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications; Part 2: DVB-S2 Extensions (DVB-S2X), ETSI EN 302 307-2 V1.1.1, Sophia Antipolis, France, Mar. 2015.
[20] Y. -M. Chen and C. -C. Chen, "Design of Farrow Structured Variable Fractional Delay Filter for Time-Varying LEO Communication Channel Emulator With SRRC Communication Waveforms," in IEEE Access, vol. 12, pp. 122229-122238, 2024.
[21] G. Ramirez-Conejo, J. Diaz-Carmona, A. Ramirez-Agundis, A. Padilla-Medina and J. Delgado-Frias, "FPGA Implementation of Adjustable Wideband Fractional Delay FIR Filters," 2010 International Conference on Reconfigurable Computing and FPGAs, Cancun, Mexico, 2010, pp. 406-411
[22] T. Bindima and E. Elias, "Design and Implementation of Low Complexity 2-D Variable Digital FIR Filters Using Single-Parameter-Tunable 2-D Farrow Structure," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 2, pp. 618-627, Feb. 2018
[23] T. C. Jayasree, J. C. Suneina, T. Bindima and M. P. Gilesh, "Design and Implementation of a Low-Complexity Continuously Variable Digital Filter Using a Novel Farrow-Equivalent-Newton Structure-Based Fractional Delay Filter," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 72, no. 7, pp. 3425-3435, July 2025
[24] M. Abbas, O. Gustafsson and H. Johansson, "On the Fixed-Point Implementation of Fractional-Delay Filters Based on the Farrow Structure," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 4, pp. 926-937, April 2013
[25] J. Selva, "An Efficient Structure for the Design of Variable Fractional Delay Filters Based on the Windowing Method," in IEEE Transactions on Signal Processing, vol. 56, no. 8, pp. 3770-3775, Aug. 2008
[26] T. -B. Deng, "Coefficient-Symmetries for Implementing Arbitrary-Order Lagrange-Type Variable Fractional-Delay Digital Filters," in IEEE Transactions on Signal Processing, vol. 55, no. 8, pp. 4078-4090, Aug. 2007
[27] T. -B. Deng, "Symmetric Structures for Odd-Order Maximally Flat and Weighted-Least-Squares Variable Fractional-Delay Filters," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 12, pp. 2718-2732, Dec. 2007
[28] Y. Wan, X. Chen and Zhang Zhonghua, "GPSEM: A SDR-based LEO satellite in-orbit GPS intermediate frequency emulator," Proceedings of 2011 International Conference on Computer Science and Network Technology, Harbin, China, 2011, pp. 407-410
[29] I. R. Petrut, R. Iacoban and C. Balint, "CELEOS - SDR Based Satellite Channel Emulator," 2024 47th International Conference on Telecommunications and Signal Processing (TSP), Prague, Czech Republic, 2024, pp. 119-122
[30] C. Cantore, I. Marasco, D. Monopoli, G. Magno and A. D’Orazio, "Laboratory emulation of LEO downlink optical feeder link employing commercial transceivers," 2024 24th International Conference on Transparent Optical Networks (ICTON), Bari, Italy, 2024, pp. 1-4
[31] Dengwei Fu and A. N. Willson, "Trigonometric polynomial interpolation for timing recovery," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 2, pp. 338-349, Feb. 2005

[32] Ç. Candan, "An Efficient Filtering Structure for Lagrange Interpolation," in IEEE Signal Processing Letters, vol. 14, no. 1, pp. 17-19, Jan. 2007
[33] R. Pulikkoonattu, H. K. Subramanian and S. Laxman, "Least square based piecewise parabolic interpolation for timing synchronization," 2008 IEEE Radio and Wireless Symposium, Orlando, FL, USA, 2008, pp. 155-158
[34] J. Selva, "Interpolation of Bounded Bandlimited Signals and Applications," in IEEE Transactions on Signal Processing, vol. 54, no. 11, pp. 4244-4260, Nov. 2006
[35] R. Hamila, J. Vesma and M. Renfors, "Polynomial-based maximum-likelihood technique for synchronization in digital receivers," in IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 49, no. 8, pp. 567-576, Aug. 2002
[36] K. Mueller and M. Muller, "Timing Recovery in Digital Synchronous Data Receivers," in IEEE Transactions on Communications, vol. 24, no. 5, pp. 516-531, May 1976.
[37] F. Gardner, "A BPSK/QPSK Timing-Error Detector for Sampled Receivers," in IEEE Transactions on Communications, vol. 34, no. 5, pp. 423-429, May 1986.
[38] Q. Yu, Z. Huang and J. Ba, "An Improved Gardner Feedback Timing Synchronization Loop," 2022 10th International Conference on Intelligent Computing and Wireless Optical Communications (ICWOC), Chongqing, China, 2022, pp. 63-67
[39] S. Wensheng and Z. Yuanyuan, "A Frame Synchronization and Symbol Timing Synchronization Algorithm in Burst OFDM Communication Based on IEEE802.11a," 2009 International Forum on Information Technology and Applications, Chengdu, China, 2009, pp. 190-193

[40] L. Zhuo and Z. Tangdi, "A Scheme for Symbol Timing and Synchronization Based on DSP Implementation in TD-LTE System," 2012 International Conference on Industrial Control and Electronics Engineering, Xi'an, China, 2012, pp. 1192-1195,
[41] Z. Yi, L. Jin, F. Wang and G. Zhang, "Symbol Timing Synchronization Scheme of AIS multi-source fusion system based on Improved Zero-Crossing Algorithm," 2025 6th International Conference on Electrical, Electronic Information and Communication Engineering (EEICE), Shenzhen, China, 2025, pp. 1543-1548
[42] M. Cho, Y. Jung and J. Kim, "Symbol timing synchronization for IEEE 802.11n WLAN systems," 2009 First Asian Himalayas International Conference on Internet, Kathmundu, Nepal, 2009, pp. 1-6
[43] T. Kim and S. -C. Park, "A New Symbol Timing and Frequency Synchronization Design for OFDM-based WLAN Systems," The 9th International Conference on Advanced Communication Technology, Gangwon, Korea (South), 2007, pp. 1669-1672
[44] X. Xie and R. Ning, "Improvement of Timing Synchronization Algorithm for FBMC-OQAM System Based on Training Sequence," 2024 6th International Conference on Natural Language Processing (ICNLP), Xi'an, China, 2024, pp. 638-642
[45] T. -L. Kung and K. K. Parhi, "Frequency domain symbol synchronization for OFDM systems," 2011 IEEE INTERNATIONAL CONFERENCE ON ELECTRO/INFORMATION TECHNOLOGY, Mankato, MN, USA, 2011, pp. 1-5
[46] D. Sen, S. Chakrabarti and R. V. Raja Kumar, "Symbol timing synchronization for ultra-wideband (UWB) multi-band OFDM (MB-OFDM) systems," 2008 3rd International Conference on Communication Systems Software and Middleware and Workshops (COMSWARE '08), Bangalore, India, 2008, pp. 200-203
[47] X. Liao and Y. Bai, "Improved symbol timing synchronization algorithms for SC-FDE systems," 2013 3rd International Conference on Consumer Electronics, Communications and Networks, Xianning, China, 2013, pp. 363-366
[48] H. Yi, Q. Shi, Y. Zhu, J. Chen, B. Sun and J. Wei, "Symbol Synchronization Algorithm Based on Delay Multiplication for Non-cooperative Communication Signals," 2024 4th International Conference on Electronic Information Engineering and Computer Communication (EIECC), Wuhan, China, 2024, pp. 802-806
[49] D. -T. Nguyen, H. -N. Le, Q. -K. Trinh, T. -H. -T. Tran and T. -A. Vu, "A Novel Efficient Hardware Implementation of Symbol Timing and Carrier Phase Synchronizer for QPSK Receivers," 2023 12th International Conference on Control, Automation and Information Sciences (ICCAIS), Hanoi, Vietnam, 2023, pp. 39-44
[50] S. Wang, K. Yang, L. Chen, Q. Zhang and P. Ye, "Timing Synchronization Algorithm of Large Range Roll-Down Coefficient of Oscilloscope," 2023 IEEE AUTOTESTCON, National Harbor, MD, USA, 2023, pp. 1-7
[51] N. Surantha, T. Uwai, Y. Nagao, M. Kurosaki and H. Ochi, "Symbol timing synchronization for ISDB-T system in multipath fading channel," 13th International Conference on Advanced Communication Technology (ICACT2011), Gangwon, Korea (South), 2011, pp. 1365-1369
[52] A. Golnari, M. Shabany, A. Nezamalhosseini and G. Gulak, "Design and Implementation of Time and Frequency Synchronization in LTE," in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 23, no. 12, pp. 2970-2982, Dec. 2015
[53] A. J. Al-Dweik, "A novel non-data-aided symbol timing recovery technique for OFDM systems," in IEEE Transactions on Communications, vol. 54, no. 1, pp. 37-40, Jan. 2006
[54] R. Mo, Y. H. Chew, T. T. Tjhung and C. C. Ko, "A New Blind Joint Timing and Frequency Offset Estimator for OFDM Systems Over Multipath Fading Channels," in IEEE Transactions on Vehicular Technology, vol. 57, no. 5, pp. 2947-2957, Sept. 2008
[55] S. J. Lee and N. C. Beaulieu, "Performance Comparison of Non-Data-Aided Symbol Timing-Error Detection for the Raised-Cosine and "Better Than" Raised-Cosine Pulses," in IEEE Transactions on Communications, vol. 57, no. 5, pp. 1261-1264, May 2009
[56] A. -L. Yongwiriyakul and W. Suwansantisuk, "Time and Frequency Synchronization of GFDM Waveforms," in IEEE Access, vol. 12, pp. 61359-61374, 2024
[57] Daeyoung Kim, M. J. Narasimba and D. C. Coc, "Design of optimal interpolation filter for symbol timing recovery," in IEEE Transactions on Communications, vol. 45, no. 7, pp. 877-884, July 1997
[58] Wei-Ping Zhu, Yupeng Yan, M. O. Ahmad and M. N. S. Swamy, "Feedforward symbol timing recovery technique using two samples per symbol," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 11, pp. 2490-2500, Nov. 2005
[59] A. A. D’Amico, G. Colavolpe, T. Foggi and M. Morelli, "Timing Synchronization and Channel Estimation in Free-Space Optical OOK Communication Systems," in IEEE Transactions on Communications, vol. 70, no. 3, pp. 1901-1912, March 2022
[60] F. J. Harris and M. Rice, "Multirate digital filters for symbol timing synchronization in software defined radios," in IEEE Journal on Selected Areas in Communications, vol. 19, no. 12, pp. 2346-2357, Dec. 2001
[61] Yik-Chung Wu, Kun-Wah Yip, Tung-Sang Ng and E. Serpedin, "Maximum-likelihood symbol synchronization for IEEE 802.11a WLANs in unknown frequency-selective fading channels," in IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2751-2763, Nov. 2005
[62] P. Savazzi and P. Gamba, "Iterative symbol timing recovery for short burst transmission schemes," in IEEE Transactions on Communications, vol. 56, no. 10, pp. 1729-1736, October 2008
[63] R. McKilliam, A. Pollok and W. Cowley, "Simultaneous Symbol Timing and Frame Synchronization for Phase Shift Keying," in IEEE Transactions on Communications, vol. 62, no. 3, pp. 1114-1123, March 2014
[64] Dongmin Lim, "A modified Gardner detector for symbol timing recovery of M-PSK signals," in IEEE Transactions on Communications, vol. 52, no. 10, pp. 1643-1647, Oct. 2004
[65] Zhen Lu, Yong Wang, Jianhua Ge and Bo Ai, "A robust timing and frequency synchronization algorithm for DVB-H receiver," in IEEE Transactions on Consumer Electronics, vol. 52, no. 2, pp. 341-346, May 2006
[66] Jian Xiong, Jun Sun and Lei Qin, "Timing synchronization for ATSC DTV receivers using the Nyquist sidebands," in IEEE Transactions on Broadcasting, vol. 51, no. 3, pp. 376-382, Sept. 2005
[67] H. -c. Hsu and J. -h. Wen, "Timing Synchronization in Ultra-Wideband Systems with Delay Line Combination Receivers," in IEEE Communications Letters, vol. 11, no. 3, pp. 264-266, March 2007
[68] N. Kashyap and D. L. Neuhoff, "Data Synchronization With Timing: The Variable-Rate Case," in IEEE Transactions on Information Theory, vol. 55, no. 1, pp. 46-52, Jan. 2009
[69] Y. Yao, X. Dong and N. Tin, "Design and Analysis of Timing Synchronization in Block Transmission UWB Systems," in IEEE Transactions on Communications, vol. 59, no. 6, pp. 1686-1696, June 2011
[70] M. K. Simon and A. Tkacenko, "Noncoherent data transition tracking loops for symbol synchronization in digital communication receivers," in IEEE Transactions on Communications, vol. 54, no. 5, pp. 889-899, May 2006
[71] F. Gong, J. Ge and F. Qian, "Symbol Timing Recovery Algorithm with Near Timing-jitter Free for ATSC DTV Receivers," in IEEE Transactions on Consumer Electronics, vol. 53, no. 2, pp. 313-318, May 2007
[72] N. Kashyap and D. L. Neuhoff, "Data synchronization with timing," in IEEE Transactions on Information Theory, vol. 47, no. 4, pp. 1444-1460, May 2001
[73] T. -K. Kim and M. Min, "Improved Non-Data-Aided Feedforward Symbol Timing Estimator for Low-Rate Sampling Systems," in IEEE Communications Letters, vol. 22, no. 5, pp. 1010-1013, May 2018
[74] S. Boumard and A. Mammela, "Robust and Accurate Frequency and Timing Synchronization Using Chirp Signals," in IEEE Transactions on Broadcasting, vol. 55, no. 1, pp. 115-123, March 2009
[75] J. Armstrong and D. Strickland, "Symbol synchronization using signal samples and interpolation," in IEEE Transactions on Communications, vol. 41, no. 2, pp. 318-321, Feb. 1993
[76] G. Watkins, "Optimal Farrow coefficients for symbol timing recovery," in IEEE Communications Letters, vol. 5, no. 9, pp. 381-383, Sept. 2001
[77] F. Shen, Z. Huang and J. Ba, "Non-Data-Aided Symbol Rate Estimation for a Low Roll-Off Factor Nyquist WDM Signal," in IEEE Photonics Journal, vol. 13, no. 4, pp. 1-10, Aug. 2021, Art no. 7200410
[78] D. Babic and M. Renfors, "Power efficient structure for conversion between arbitrary sampling rates," in IEEE Signal Processing Letters, vol. 12, no. 1, pp. 1-4, Jan. 2005
[79] F. J. Galindo Guarch, P. Baudrenghien and J. M. Moreno Arostegui, "An Architecture for Real-Time Arbitrary and Variable Sampling Rate Conversion With Application to the Processing of Harmonic Signals," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 67, no. 5, pp. 1653-1666, May 2020
[80] N. Bakholdin, V. Chernienko and S. Bakhurin, "Digital Resampler Based on Farrow Filter," 2025 27th International Conference on Digital Signal Processing and its Applications (DSPA), Moscow, Russian Federation, 2025, pp. 1-7