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Pulsars, a small portion of celestial sources that emit radiation at varying intensity, provide new possible navigation algorithms which are different from steady point sources. Time-delay estimation is one of the key aspects of pulsar-based navigation technology. Previous work for pulse phase estimation uses a maximum likelihood estimator (MLE) for the phase-in time domain, which is seen as one of the most useful phase estimators. However, the analytic solution for phase cannot be found using MLE. As a result, a brute-force method involving nested, iterative grid-searches is needed to solve this MLE issue, which leads to lots of computations. In order to solve this problem, a multiple harmonic X-ray pulsar signal phase estimation (MHSPE) method is proposed. This method uses the times of arrivals (TOAs) measured pulsar signal to estimate the time-delay in the frequency domain. In this paper, firstly we use the arrival time to derive the maximum-likehood (ML) estimation of phase-delay by fundamental frequency, then an analytic expression for the fundamental frequency phase is obtained. The MHSPE method based on the fundamental frequency phase equation, calculates different harmonic phases by generalizing the analytic expression of fundamental frequency phase, and the normalized amplitude of each harmonic in the spectrum is used as the weight of each harmonic phase. Finally, the weighted average of harmonic phases, which is given by the final analytic expression, is used as the estimation of the moment. To evaluate the MHSPE method, the error and variance equations are calculated and the MHSPE method is demonstrated to be unbiased and consistent. Moreover, by comparing with the ML estimation of the first harmonic, if the amplitudes of harmonic in the spectrum are almost the same, the signal-to-noise ratio (SNR) of MHSPE improves m1/2 times when the number of harmonic waves is m. Compared with the Cramer-Rao bound of pulsar time-delay estimation, the derivative of pulsar signal in the time domain reflects the number of harmonic waves in the frequency domain. Hence, the MHSPE can greatly approximate to the Cramer-Rao bound for the estimation of pulsar signal timedelay when the harmonic number increases. Finally, we utilize the TOAs of Crab pulsar, observed by Rossi X-ray timing explorer (RXTE) spacecraft, to verify the performance of MHSPE. The results show that for low SNR of pulsar signal, MHSPE can obtain high precision phase estimations. When the amplitude of the harmonic in the spectrum is larger, the estimation variance of the harmonic phase tends to be smaller. The projection orbit determined by MHSPE method can match the projection of RXTE in the direction of Crab pulsar, with the observed time increasing, the estimation accuracy converges rapidly to Cramer-Rao bound.
[1] Shuai P, Li M, Chen S L, Huang Z 2009 The principle and method the X-ray pulsar-Based navigation system (Beijing: China Astronautic Publishing House) p20 (in Chinese) [帅平, 李明, 陈绍龙, 黄震 2009 X射线脉冲星导航系统原理与方法(北京: 中国宇航出版社) 第20页]
[2] Zhou Q Y, Ji J F, Wang T 2013 Acta Phys. Sin. 62 019701 (in Chinese) [周庆勇, 姬剑锋, 任红飞 2013 62 019701]
[3] Zhu J, Ji P Y 2008 Chin. Phys. B 17 356
[4] Hewish A, Bell S J, Pilkington J D H, Scoptt P F, Collins R A 1968 Nature 217 709
[5] Taylor J H 1992 Philos. Trans. Roy. Soc. 341 117
[6] Liao S L, Vernon R J 2009 IEEE transactions on Antennas and Propagation 57 2068
[7] Pines D J 2004 DAR-PA/TTO 571 4339
[8] Xie Z H, Xu L P, Ni G R 2008 Acta Phys. Sin. 57 6683 (in Chinese) [谢振华, 许录平, 倪广仁 2008 57 6683]
[9] Su Z, Xu L P, Wang T 2011 Journal of Astronautics. 32 1256 (in Chinese) [苏哲, 许录平, 王婷 2011 宇航学报 32 1256]
[10] Emadzadeh A A, Speyer J L, Golshan A 2009 AIAA Guidance, Navigation, and Control Conferenc,Chicago, August 10–13, 2009 p5974
[11] Sun H F, Bao W M, Fang H Y, Li X P 2014 Acta Phys. Sin. 63 069701 (in Chinese) [孙海峰, 包为民, 方海燕, 李小平 2014 63 069701]
[12] Emadzadeh A A, Speyer J L 2010 IEEE Transactions on Signal Processing 58 4484
[13] Jahoda K, Swank J H, Cues A B, Stark M J, Strohmayer T, Zhang W 1996 Proc. SPIE 2808, EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy VII, United States, August 4, 1996 p59
[14] Kay S M (translated by Huang J G) 1994 Modern Spectral Estimation: Theory and Ap-plication (Beijing: Science Press) pp326-331 (in Chinese) [Key S M 著 (黄建国 译) 1994 现代谱估计:原理与应用(北京: 科学出版社) 第326–331页]
[15] Rawley L, Stinebring D, Taylor J 1986 Proceedings of the Eighteenth Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Washington, December 1986 p453
[16] Emadzadeh A A 2009 Ph. D. Dissertation (Los Angeles: University of California)
[17] Li J X, Ke X Z 2011 Chinese Astronomy and Astrophysics 35 19
[18] Nguyen D T, Renaux A, Boyer R 2014 IEEE Transactions on Aerospace and Electronic Systems 50 786
[19] Golshan A R, Sheikh S I Proceedings of the 63rd Annual Meeting of the Institute of Navigation Cambridge, April 23-25, 2007, p413
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[1] Shuai P, Li M, Chen S L, Huang Z 2009 The principle and method the X-ray pulsar-Based navigation system (Beijing: China Astronautic Publishing House) p20 (in Chinese) [帅平, 李明, 陈绍龙, 黄震 2009 X射线脉冲星导航系统原理与方法(北京: 中国宇航出版社) 第20页]
[2] Zhou Q Y, Ji J F, Wang T 2013 Acta Phys. Sin. 62 019701 (in Chinese) [周庆勇, 姬剑锋, 任红飞 2013 62 019701]
[3] Zhu J, Ji P Y 2008 Chin. Phys. B 17 356
[4] Hewish A, Bell S J, Pilkington J D H, Scoptt P F, Collins R A 1968 Nature 217 709
[5] Taylor J H 1992 Philos. Trans. Roy. Soc. 341 117
[6] Liao S L, Vernon R J 2009 IEEE transactions on Antennas and Propagation 57 2068
[7] Pines D J 2004 DAR-PA/TTO 571 4339
[8] Xie Z H, Xu L P, Ni G R 2008 Acta Phys. Sin. 57 6683 (in Chinese) [谢振华, 许录平, 倪广仁 2008 57 6683]
[9] Su Z, Xu L P, Wang T 2011 Journal of Astronautics. 32 1256 (in Chinese) [苏哲, 许录平, 王婷 2011 宇航学报 32 1256]
[10] Emadzadeh A A, Speyer J L, Golshan A 2009 AIAA Guidance, Navigation, and Control Conferenc,Chicago, August 10–13, 2009 p5974
[11] Sun H F, Bao W M, Fang H Y, Li X P 2014 Acta Phys. Sin. 63 069701 (in Chinese) [孙海峰, 包为民, 方海燕, 李小平 2014 63 069701]
[12] Emadzadeh A A, Speyer J L 2010 IEEE Transactions on Signal Processing 58 4484
[13] Jahoda K, Swank J H, Cues A B, Stark M J, Strohmayer T, Zhang W 1996 Proc. SPIE 2808, EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy VII, United States, August 4, 1996 p59
[14] Kay S M (translated by Huang J G) 1994 Modern Spectral Estimation: Theory and Ap-plication (Beijing: Science Press) pp326-331 (in Chinese) [Key S M 著 (黄建国 译) 1994 现代谱估计:原理与应用(北京: 科学出版社) 第326–331页]
[15] Rawley L, Stinebring D, Taylor J 1986 Proceedings of the Eighteenth Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Washington, December 1986 p453
[16] Emadzadeh A A 2009 Ph. D. Dissertation (Los Angeles: University of California)
[17] Li J X, Ke X Z 2011 Chinese Astronomy and Astrophysics 35 19
[18] Nguyen D T, Renaux A, Boyer R 2014 IEEE Transactions on Aerospace and Electronic Systems 50 786
[19] Golshan A R, Sheikh S I Proceedings of the 63rd Annual Meeting of the Institute of Navigation Cambridge, April 23-25, 2007, p413
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