On the Probability Density Function and Stability Properties for a Cross-Product Frequency-Locked Loop

Tsung-Yu Chiou

Abstract:	The frequency locked-loop (FLL) has received new attention for modern Global Navigation Satellite Systems (GNSS) receivers, especially for its performance under severe noise interference and high dynamic environments. It has been shown that an FLL is more robust to interference and dynamics than a phase locked-loop (PLL). Therefore, it is beneficial to use an FLL as a fallback tracking loop when the primary PLL is unable to maintain carrier tracking in hostile environments. Besides tracking, it is to also crucial to preserve the Bit Error Rate (BER) of the data demodulation and probability of losing locks when receivers operate an FLL. These characteristics rely on the probability density function (PDF) of the frequency estimate made by the frequency discriminator in the FLL. However, the PDF has not been determined as of yet. The purpose of this paper is to solve the PDF and evaluate the stability region, which is essential for determining the statistics of loss of lock. In this work, a nonlinear model was developed and the corresponding Fokker-Planck equation (FPE) for the cross-product FLL was derived. The PDF of the frequency estimate was then solved by applying the Crank-Nicolson numerical method.
Published in:	Proceedings of the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2007) September 25 - 28, 2007 Fort Worth Convention Center Fort Worth, TX
Pages:	739 - 748
Cite this article:	Chiou, Tsung-Yu, "On the Probability Density Function and Stability Properties for a Cross-Product Frequency-Locked Loop," Proceedings of the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2007), Fort Worth, TX, September 2007, pp. 739-748.
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