摘 要
在已知信號下的頻率同步量測問題通常都可以簡化成弦波頻率估測的問題，因此在訊號處理領域裡，單一複數弦波在白色高斯雜訊下的頻率估測是一個非常重要的問題。其相關的應用包括了陣列訊號處理，頻譜估測，數位通訊系統中的載波時脈回復，都卜勒變化率的估測，以及其他許多關於雷達與聲納系統方面的應用。
一般來講，以相位資料做為基礎的頻率估測器往往會產生一個臨界值效應。也就是在資料長度固定時，若估測器的訊雜比低於某一個值時，則估測器的性能會急遽地下降，使得估測器的變異數無法達到CRLB。
在本篇論文中，我們提出了兩種使用於白色高斯雜訊下的改良式頻率估測方法，此兩種方法主要是以Kim的方法做為基礎然後再予以延伸，而它們主要是根據相關資料的相位差來得到所要估測的頻率值。這兩種估測器在估測運算方面會比最佳的最大相似性估測器還要更有效率，並且在高訊雜比條件之下可以達到與最大相似性估測器一樣的效能。除此之外，當估測的弦波頻率很高時，此兩種估測器除了是一個無偏差的估測器外，估測器的臨界值也會遠比Kay估測方法以及Kim估測方法還要來的低。

Abstract
Under known signals environments, the problem of frequency estimation can be regarded as that of sinusoidal frequency estimation. Therefore, the frequency estimation of a single complex sinusoid signal in a white Gaussian noise channel is an important problem in the field of signal processing. Some of the applications include array signal processing, spectral estimation, carrier and clock synchronization for digital communications, Doppler rate estimation, and many others in radar and sonar systems.
Frequency estimations based on the information of phase have threshold effects. While the length of the observation data is fixed, the performance of the estimator will be degraded and the variance will not achieve Cramer-Rao lower bound under the condition that signal-to-noise ratio (SNR) is below a certain threshold.
In this thesis, two modified frequency estimation methods are proposed in additive white Gaussian noise channels. These two methods, estimating the frequency value by linearly combining the phase difference of correlated data, are basically extended from Kim’s method. These estimators have lower complexity than optimal maximum likelihood estimator and attain as good performance at moderately high SNR’s. These two methods, at high frequency values, yield a considerably lower variance threshold than Kay’s method and Kim’s method and remain unbiased.

目 錄
誌 謝 i
摘 要 ii
Abstract iii
目 錄 iv
圖目錄 vi
第一章 序言 1
1.1 研究背景 1
1.2 各章提要 2
第二章 單載波頻率估測演算法 3
2.1 頻率估測演算法簡介 3
第三章 改良式Kim估測器方法 18
3.1 第一種改良式相關性演算法 18
3.2 第二種改良式相關性演算法 21
3.3 改良式估測器之通式、變異數以及均值公式推導 23
3.3.1 改良式估測器變異數的推導 23
3.3.2 改良式估測器通式的推導 29
3.3.3 改良式估測器均值的推導 35
第四章 估測器效能模擬與比較 46
4.1 估測器效能與弦波頻率的關係 46
4.2 改良式估測器的效能模擬 49
第五章 結論 65
相關文獻 66

相關文獻
[1] S. M. Kay, “Fundamentals of Statistical Signal Processing: Estimation Theory,” Prentice Hall, New Jersey, 1993.

[2] L. Palmer, “Coarse frequency estimation using the discrete Fourier transform (Corresp.),” IEEE Transactions on Information Theory, pp.104-109, January 1974.

[3] D. Rife and R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Transactions on Information Theory, pp.591-598, September 1974.

[4] S. Tretter, “Estimating the frequency of a noisy sinusoid by linear regression (Corresp.),” IEEE Transactions on Information Theory, pp.832-835, November 1985.

[5] S. M. Kay, “Statistically/computationally efficient frequency estimation,” IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.2292-2295, 1988.

[6] S. M. Kay, “A Fast and Accurate Single Frequency Estimator,” IEEE Transactions on Acoustics Speech and Signal Processing, pp.1987-1990, December 1989.

[7] M.P. Fitz, “Planar filtered techniques for burst mode carrier synchronization,” IEEE Global Telecommunications Conference, pp.365-369, 1991.

[8] M.P. Fitz, “Further results in the fast estimation of a single frequency,” IEEE Transactions on Communications, pp.862-864, February-April 1994.

[9] M. Luise and R. Reggiannini, “Carrier frequency recovery in all-digital modems for burst-mode transmissions,” IEEE Transactions on Communications, pp.1169-1178, February 1995.

[10] D. Kim, M.J. Narasimha and D.C. Cox, “An improved single frequency estimator,” IEEE Signal Processing Letters, pp.212-214, July 1996.