數位訊號處理技術漸漸成為新世代無線通訊系統中不可或缺的關鍵性技術，而數位訊號處理演算法的種類相當的眾多。在本篇論文中，提出兩種演算法，動差法及馬可夫蒙地卡羅法來估測訊號頻率。動差法的優點在於計算方便，實現簡單，而且不需要事先知道未知變數的分布情形；而其缺點則是解非唯一，所以並不能保證所求得的解有良好的特性。本文中用動差法在三種不同環境下，訊號振幅已知條件下，及未知訊號振幅，高雜訊比條件下，還有未知訊號振幅，低雜訊比條件下，估測訊號頻率，並做經過Matlab模擬分析，結果證實再三種環境下的動差法估計試驗均能得估計出理想的結果模擬分析。

本文另外的一種估測方法為馬可夫蒙地卡羅法，馬可夫蒙地卡羅法是經過多次重複的遞迴氏抽取樣本，而架構出一長串的馬可夫鏈 (Markov chain)，以求得一個近似的分配。馬可夫蒙地卡羅演算法，大多利用Metropolis-Hastings演算法及Gibbs sampler兩種演算法，來進行模式的參數抽樣和估測。此估測法的優點是一次能處理多個變數的估測，本文中將針對訊號的振幅，頻率，及雜訊的變異數來做估測，且所求出的解保證有良好的特性；相對於動差法不必事先知道機率分布，此估測法的缺點則是必須事先知道未知變數的事前機率分布，一旦所假設的事前機率錯誤，將會得到相當不理想的結果。

In this paper we address two frequency estimation algorithms, Moment Method and Markov Chain Monte Carlo method (MCMC). The advantage of Moment Method estimator is easy to determine and simple to implement. It don’t need the distribution of parameters. The disadvantage is that its solution is not only. Hence, the solution is not surely with good property. In this paper, we produce the Moment Method estimator with three cases, known signal amplitude, unknown signal amplitude and high SNR, unknown signal amplitude and lower SNR.

Markov Chain Monte Carlo method builds a series of Markov chain with repeating sampling to get an approximate distribution. Markov Chain Monte Carlo method uses two algorithms, Metropolis-Hastings algorithm and Gibbs sampler.
The advantage of MCMC is that it is a demonstrated way to deal with more parameter numbers and the result has surely good property. The disadvantage is that MCMC must generate a probability distribution function for parameters. Hence, we will get the bad result with wrong assumption of probability distribution function.

[1] Steven M. Kay., Fundamentals of Statistical Signal Processing: Detection theory, pp.289-307, Prentice-Hall, 1993.

[2] Xiaodong Wang, Rong Chen, Jun S. Liu, “Monte Carlo Bayesian Signal Processing for Wireless Communications”, Journal of VLSI Signal Processing 30, pp.89-105, 2002.

[3] Xiaodong Wang, Rong Chen, “Adaptive Bayesian Multiuse Detection for Synchronous CDMA with Gaussian and Impulsive Noise”, IEEE Transactions on Signal Processing, vol.47, NO. 7, July 2000.

[4] Kay, S. M., “A Fast and Accurate Single Frequency Estimator by Linear Prediction”, IEEE Transactions on Signal Processing., vol.37, pp. 1987-1990, Dec. 1989.

[5] Lank, G. W., Reed, I. S. and Pollon, G. E., “A Semicoherent Detection and Soppler Estimation Satistics”, IEEE Transactions on Aerospace and Electronic Systems, vol.9, pp.151-165, 1973.

[6] Signal Processing, Special Issue on Markov Chain Monte Carlo (MCMC) Methods for Signal Processing, vol.81, no.1, 2001.

[7] P.J.Green, “Revisible jump Markov Chain Monte Carlo computation and Bayesian model determination”, Biometrika, vol.82, pp.711-732. 1985.

[8] Rider, P.R., “Estimating the Parameters of mixed Poisson, Binomial, and Weibull Distributions by the Method of Moment”, Bull. Int. Statist. Inst., vol.38, pp.1-8, 1961.

[9] W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications”, Biometrika, vol.57, pp.97-109, 1970.

[10] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images”, IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp.721-741, Nov.1984.

[11] IEEE Trans. Sig. Proc., Special Issue on Monte Carlo Methods for Statistical Signal Processing, 2002.

[12] Harry L. Van Trees, “Detection, Estimation, and Modulation Theory”, John Wiley and Sons, 2001.

[13] A. Doucet, A. Logothetis, and V. Krishnamurthy, “Stochastic sampling algorithms for state estimation in jump Markov linear systems”, IEEE Trans. Automat. Contr., vol.45, no.2, pp.188-202, 2002.

[14] J. Besag, P. Green, D. Higdon, and K. Mengersen, “Bayesian computation and stochastic systems”, Statist. Sci., vol. 10, pp.3-66, 1995.

[15] Peter J. Green, “Markov chain Monte Carlo in action: a tutorial”

[16] Carlin, B. P. and S. Chib, “Bayesian model choice via Markov chain Monte Carlo method”, Journal of the Royal Statistical society, Series B, Methodological 57, 473-484, 1995.

[17] Tierney, L., “A note on Metropolis-Hastings kernels for general state spaces”, Annals of Applied Probability 8, 1-9, 1998.