Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/91834
標題: 以相位修正的離散傅立葉轉換做複數弦波信號的頻率估測之分析與改良
Analysis and Improvement on Discrete Fourier Transform for Frequency Estimation of Complex Sinusoid
作者: Sheng-Kai Wen
溫勝凱
關鍵字: Frequency estimation
discrete Fourier transform
Bias reduction
頻率估測
離散傅立葉轉換
降低偏差
引用: [1]M. A. Richards, Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005. [2]C. L. Phillips, J. M. Parr, and E. A. Riskin, Signals, Systems, and Transforms, 3rd ed., Pearson Education, 2003. [3]D. Rife, R. Boorstyn, ‘‘Single tone parameter estimation from discrete-time observations”, IEEE Tran. Information Theory, vol.20, pp. 591-598, Sep. 1974. [4]J.-R. Liao, C.-M. Chen, “Phase correction of discrete Fourier transform coefficients to reduce frequency estimation bias of single tone complex sinusoid,” Signal Processing, vol.94, pp.108-117, Jan. 2014. [5]J.-R. Liao, S.- Lo, “Analysis of Estimation Bias for Direct Frequency Estimators of Complex Sinusoid,” Ninth International Conference on Signal-Image Technology & Internet-Based Systems (SITIS 2013), Kyoto, Japan, pp.85-92, Dec. 2013. [6]C. Yang, G. Wei, ‘‘A noniterative frequency estimator with rational combination of three spectrum lines,” IEEE Trans. Signal Process., vol.59, no. 10, pp. 5065-5070, Oct. 2011. [7]G. Wei, C. Yang, F.-J. Chen, ‘‘Closed-form frequency estimator based on narrow-band approximation under noisy environment,” Signal Process., vol. 91, no. 4, pp. 841-851, Apr. 2011. [8]B. G. Quinn, ‘‘Estimating frequency by interpolation using Fourier coefficients,” IEEE Trans. Signal Process., vol. 42, no. 5, pp. 1264-1268, May 1994. [9]M. D. Macleod, ‘‘Fast nearly ML estimation of the parameters of real or complex single tones or resolved multiple tones,” IEEE Trans. Signal Process., vol. 46, no. 1, pp. 141-148, Jan. 1998. [10]E. Jacobsen, P. Kootsookos, ‘‘Fast, accurate frequency estimators,” IEEE Signal Process. Mag., vol. 24, no. 3, pp. 123-125, May 2007. [11]C. Candan, ‘‘A method for fine resolution frequency estimation from three DFT samples,” IEEE Signal Process. Lett., vol. 18, no. 6, pp. 351-354, Jun. 2011. [12]C. Candan, S. Koc, ‘‘Beamspace approach for detection of the number of coherent sources,” in: Proceedings of the IEEE Radar Conference, 2012, pp. 913–918. [13]“The 3 roots of third degree equation,” http://www.josechu.com/ecuaciones_polinomicas/cubica_solucion.htm
摘要: 在白雜訊觀察下複數弦波的參數估測通常為以下兩個階段。第一階段為粗略估測,是應用輸入長度N的N點離散傅立葉轉換(Discrete Fourier Transform,DFT)來搜尋。第二階段為精細搜尋,其搜尋是實施在第一階段決定的峰值附近,經由兩階段的搜尋找到一個內插頻率位置,透過這個內插的頻率估測出弦波頻率。 四種常見的精細頻率估測器分別為: Quinn估測器、Jacobsen估測器、MacLeod估測器與Candan估測器。Quinn方法為最先被提出的頻率估測法,其特點是使用3個DFT樣本內插處理頻率估測,這個方法是非常直接、簡單而且有效的。Candan方法為近來最佳的以DFT內插法達到精確估測頻率的方法。它推導Jacobsen公式並提出一個偏差修正,這個修正項在高訊號雜訊比(Signal-to-noise ratio,SNR)值是很有效的。 相位修正法與多項式根修正法均為降低估測頻率偏差的有效方法,相位修正法是在估測器所估測出的頻率後乘上一個相位修正項,對估測頻率進行修正;多項式根修正法是使用泰勒展開式將估測頻率近似到一個三階多項式,並且對此多項式求根,此根能夠精準的估測出複數弦波的頻率。另外,我們發現Quinn估測器與Candan估測器經由相位修正後在不同的頻率範圍內有最好的估計,我們找出一個臨界值並提出一個混合式估測器,在這個臨界值前後使用不同的頻率估測器達到最好的估測。 本論文以理論分析方式將多項式根修正法應用到相位修正法當中,我們將四種精細搜尋的頻率估測器當中結合兩種修正方法,實驗結果表明,相位修正法的確能有效的降低估測頻率偏差,它能夠移除原估測頻率中造成偏差的相位項;多項式根修正法有精準的估測頻率,降低了原本估測器所產生的偏差。結合兩種修正法的改良方式,能夠得到最低的估測頻率偏差,有最好的估測頻率品質。
The frequency estimation of a complex sinusoid under the influence of white noise is usually divided into two stages. The first stage is called “coarse search”. It searches the maximum magnitude of the Discrete Fourier Transform (DFT) coefficients. The second stage is called “fine search”. It searches around the peak magnitude found from the coarse search. A popular way of fine search is to interpolate the DFT coefficients around the peak magnitude. There are four commonly used interpolators in fine search. They are proposed by Quinn, Jacobsen, MacLeod, and Candan. These interpolators exhibit estimation bias because of the approximation in the derivation. In this regard, Candan proposed a constant scaling factor to reduce the bias and has shown that their estimator provides the lowest bias among these estimators. Two methods, phase correction and polynomial root correction, have also been proposed to reduce the estimation bias. Phase correction method multiplies a phase correction term to each DFT coefficient to eliminate bias caused by phase inconsistency. On the other hand, polynomial root correction uses Taylor series expansion to derive a third-order polynomial equation for the estimated frequency. Then, a more accurate estimation can be obtained from the root of the third-order polynomial. In this thesis, we apply the polynomial root correction method to the phase correction method. In the experiments, we show that the two methods combined can achieve the best performance as compared to the previous methods.
URI: http://hdl.handle.net/11455/91834
文章公開時間: 2015-07-16
Appears in Collections:通訊工程研究所

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