含噪实信号频率估计算法研究
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摘要
对被噪声污染的正弦信号进行参数估计是一个十分重要的课题，它在雷达、声纳、通信、语音信号处理、生物医学工程、检测等领域中有很大的应用价值。频率是正弦信号最重要的参数和最本质的特征，频率估计的研究是信号处理领域的一个经典课题。在许多实际工程应用中，获取的信号样点值都是实数，如语音信号。由于实正弦信号在频谱上存有“负频率”，其自相关的频率受到非零相位的影响，致使实正弦信号的频率估计相对复正弦信号更困难。而且也使得复正弦信号的频率估计方法不能直接应用于实正弦信号，特别是基于时域的频率估计方法。高斯白噪声是自然界最常见的噪声，因而如何从混有高斯白噪声的实信号样点中提取信号的频率一直以来备受关注。
频率估计算法可分为基于原信号本身的时域特性和基于变换域的特征。基于变换域的频率估计方法通常是考虑信号的频域特性，其在估计性能方面有一定的优势，但实现较为复杂，而时域方法相对简单，特别适用于计算量需求小，快速实时处理实正弦信号的场合。衡量频率估计算法性能的主要指标是频率估计的均方误差，它在样点序列固定长度和信噪比下有一个理论界，称为Cramer Rao Bound(CRB界)。对于短序列在时域的频率估计，寻求估计性能能够逼近CRB且计算量小的频率估计算法是实信号频率估计研究中的难点。由于正弦信号的自相关含有信号的频率信息，且去除部分噪声影响，所以，本文基于信号自相关的频率估计这一主题展开研究，主要学术贡献在于：
(1)提出了基于扩展自相关的频率估计算法，并详细推导了高斯白噪声对算法性能带来的影响，得到了闭合的理论表达式。由于正弦信号的自相关仍然是同频率的正弦信号，可以利用少量的自相关系数快速地获得频率估计值，但是此类估计器性能需要进一步的提升，因而研究者们提出改进自相关的定义和利用多步自相关来提升频率估计性能，然而前者性能提升不多，后者计算量增多。本文在性能和计算量上折中考虑提出了基于扩展自相关的频率估计算法。该算法借鉴频域中频率估计常用的粗估计、精估计，在基于自相关的粗估计频率基础上对自相关进行泰勒级数展开，运用最小二乘均方逼近得到精估计频率。该算法在两步估计中都充分挖掘自相关的信息，利用多个自相关系数含有的频率信息，使得该估计器与其它时域有闭合解的算法相比，在0dB信噪比时就能逼近CRB界，且计算量比基于多步自相关的频率估计算法有明显优势。成果在SignalProcessing等刊物发表。
(2)提出了基于自相关相位补偿的频率估计算法。深入分析了正弦信号自相关的非零相位特性，而经典的基于自相关的频率估计算法多忽略了此相位而带来了近似误差。为此，本文提出了基于自相关相位补偿的频率估计算法。该算法一方面利用多个自相关含有的频率信息，同时对自相关的相位进行了补偿。通过仿真分析深入讨论了多个自相关系数的选取，仿真结果也验证了该算法明显优于传统的Pisarenko谐波分解（PisarenkoHarmonic Decomposer，PHD)算法。与其它计算量相当的算法相比，在中高信噪比，特别对于短序列具有明显的性能优势，能更逼近于CRB。成果在IEICE Transactions刊物发表。
(3)提出了基于加窗自相关序列的Pisarenko谐波分解(PHD)改进算法。传统的PHD算法只简单使用0、1这两个低序号的自相关系数来估计频率，致使其性能不佳。而使用高序号的自相关可以提升频率估计性能，但是会带来频率估计模糊问题和边界问题。为此，本文研究利用高序号自相关来进行频率估计的方法，确保其能进一步提升频率估计性能，同时避免频率模糊问题和频率边界问题。另外，高斯白噪声的自相关集中在0序号自相关系数处，而高序号的自相关系数因为参与计算的信号样点数减少而误差增大。因此，本文还研究讨论自相关序列的加窗选择问题。在此基础上，提出基于加窗自相关序列的PHD改进算法，先利用高序号自相关系数来计算频率粗估计，然后充分利用加窗自相关序列的信息，包括低序号和高序号自相关的信息，计算一个频率调节因子来对频率粗估计进行微调。文中通过理论分析证明了算法的性能可以逼近CRB界，同时该性能界从理论上指导加窗自相关序列的选取。仿真分析表明其估计性能明显优于传统的PHD算法和其它基于时域自相关的算法，避免了使用高序号自相关带来的频率估计模糊问题和边界问题。成果在《电子学报》英文版等刊物发表。
Parameter estimation of a tone in noise is important in many fields such as radar, sonar,communications, speech signal processing, biomedical engineering, control and measurement.Frequency is the most important parameter and the most essential feature, so its estimation isa classic issue in the field of signal processing. In many practical engineering applications, thesignal samples have real values, such as voice signals. However, fequency estimation ofa real sinusoid is more difficult relatively than the complex one, for the formerincludes "negative frequency" in the spectrum, and the phase of its correlation affects theaccuracy of frequency estimation. So that, the method of frequency estimationfor the complex sinusoid can not be directly applied to the real one, especially for thetime-domain algorithms. White Gaussian noise is a common noise in the nature, so thefrequency estimation of the real sinusoid embedded in white Gaussian noise hasbeen received extensive attention.
The method of the frequency estimation can be divided into two categories: based ontime-domain and transform domain, such as frequency-domain. For the methods based onfrequency-domain, they have advantages in the estimation performance, but they arecomputationally demanding. While, the methods based on time-domain are relativelysimple, and they are especially suitable for applications where the real-time estimation isrequired. The mean square error is often used as a measure of inaccuracy for the frequencyestimation, and there is an ideal theoretic bound named Cramer Rao Bound (CRB), whichdepends on the length of signal and SNR. For the short sequence, efficient time-domainestimators which approach the CRB and have less computation demanding as well are the keypoint in the study of frequency estimation. Since the sample correlation sequence has thesame frequency as the original signal, but with less noise effect, this dissertation has madedeep research on the correlation-based estimator and the major contributions are as thefollowing:
(1)A closed-form expanded correlation method for real sinusoid frequency estimation isproposed and the effects of White Gaussian noise on the performance is derived, so that aclosed-form theoretical performances bound is abtained. Since the sample correlationsequence has the same frequency as the original signal, a few correlation coefficients can beexploited to estimate the frequency qulickly but the performance is inefficient. To improvethe estimation formance, many researchers proposed a lot of modified correlations andmultiple-stage correlaton, however, the former is not very effective and the latter increases the amount of computation. We take the performance and complexity into consideration andpropose an expanded correlation method. The mothod makes full use of the multiplecorrelaition lags and extends the idea of a coarse search and a fine search of frequencyestimation in the frequency-domain to the time-domain. Firstly, the modified covariance(MC) method based on multiple correlation lags is applied to provide a coarse frequencyestimate. Then, a closed-form adjustment term based on a least square cost function is derivedto get the fine frequency estimate. Simulation results show that the performance of theproposed algorithm, when compared with several existing closed-form time-domainestimators, is closer to the Cramer-Rao Bound (CRB) at0dB. Moreover, the proposed methodhas lower computation complexity than other autocorrelation-based approaches, which alsouse multiple autocorrelation lags. The research productions are published in Signal Processing,etc.
(2)An real single-tone frequency estimator based on phase compensation of multiplecorrelation lags is proposed. For the limited-length single sinusoid, its correlation has anon-zero phase, which is always neglected for correlaton-based methods. So we propose touse Taylor series to expand the correlation at the coarse estimated frequency to exploitmultiple correlation information and take the phase of the correlation into consideration. Theselection of multiple correlation lags is discussed deeply by experiments. Simulation resultsshow that this new method outperforms the Pisarenko harmonic decomposer stimator.Moreover, when compared with other existing considerable computational estimator, themean square frequency error of the proposed method is closer to the CRB for certain SNRrange, especially when the signal length is very short. The research productions are publishedin IEICE Transactions, etc.
(3)A modified PHD method based on windowed correlation sequence is investigated.The PHD method only utilizes correlation lags1and2to estimate the frequency. It has beenproved using higher correlation lags can improve the estimtation performance but leads tofrequency ambiguity and edge frequency. So, we research on a method, which exploiteshigher correlation lags and can improve performance with avoiding frequency ambiguity andedge frequency. In addition, the noise effect on the correlation concentrates around lag0,and the correlation with higher lag is inaccurate for its computation involves less availablesamples, thus we consider adding a rectangle window to the correlaton. Based on thewindowed correlation sequence, an adjustment term is derived to add to the coase frequency,which is obtained by PHD method with higher correlation lags. Theoretical analysis showsthat the algorithm can approach the CRB, and it gives guidance on how to select a windowed correlation sequence. Simulation experiments illustrate the performance of our proposedmethod is generally superior to the PHD and other correlation-based methods, and it can solvethe problem about frequency ambiguity and edge frequency. The research productions arepublished in Chinese Electronic Journal, etc.
频率估计算法可分为基于原信号本身的时域特性和基于变换域的特征。基于变换域的频率估计方法通常是考虑信号的频域特性，其在估计性能方面有一定的优势，但实现较为复杂，而时域方法相对简单，特别适用于计算量需求小，快速实时处理实正弦信号的场合。衡量频率估计算法性能的主要指标是频率估计的均方误差，它在样点序列固定长度和信噪比下有一个理论界，称为Cramer Rao Bound(CRB界)。对于短序列在时域的频率估计，寻求估计性能能够逼近CRB且计算量小的频率估计算法是实信号频率估计研究中的难点。由于正弦信号的自相关含有信号的频率信息，且去除部分噪声影响，所以，本文基于信号自相关的频率估计这一主题展开研究，主要学术贡献在于：
(1)提出了基于扩展自相关的频率估计算法，并详细推导了高斯白噪声对算法性能带来的影响，得到了闭合的理论表达式。由于正弦信号的自相关仍然是同频率的正弦信号，可以利用少量的自相关系数快速地获得频率估计值，但是此类估计器性能需要进一步的提升，因而研究者们提出改进自相关的定义和利用多步自相关来提升频率估计性能，然而前者性能提升不多，后者计算量增多。本文在性能和计算量上折中考虑提出了基于扩展自相关的频率估计算法。该算法借鉴频域中频率估计常用的粗估计、精估计，在基于自相关的粗估计频率基础上对自相关进行泰勒级数展开，运用最小二乘均方逼近得到精估计频率。该算法在两步估计中都充分挖掘自相关的信息，利用多个自相关系数含有的频率信息，使得该估计器与其它时域有闭合解的算法相比，在0dB信噪比时就能逼近CRB界，且计算量比基于多步自相关的频率估计算法有明显优势。成果在SignalProcessing等刊物发表。
(2)提出了基于自相关相位补偿的频率估计算法。深入分析了正弦信号自相关的非零相位特性，而经典的基于自相关的频率估计算法多忽略了此相位而带来了近似误差。为此，本文提出了基于自相关相位补偿的频率估计算法。该算法一方面利用多个自相关含有的频率信息，同时对自相关的相位进行了补偿。通过仿真分析深入讨论了多个自相关系数的选取，仿真结果也验证了该算法明显优于传统的Pisarenko谐波分解（PisarenkoHarmonic Decomposer，PHD)算法。与其它计算量相当的算法相比，在中高信噪比，特别对于短序列具有明显的性能优势，能更逼近于CRB。成果在IEICE Transactions刊物发表。
(3)提出了基于加窗自相关序列的Pisarenko谐波分解(PHD)改进算法。传统的PHD算法只简单使用0、1这两个低序号的自相关系数来估计频率，致使其性能不佳。而使用高序号的自相关可以提升频率估计性能，但是会带来频率估计模糊问题和边界问题。为此，本文研究利用高序号自相关来进行频率估计的方法，确保其能进一步提升频率估计性能，同时避免频率模糊问题和频率边界问题。另外，高斯白噪声的自相关集中在0序号自相关系数处，而高序号的自相关系数因为参与计算的信号样点数减少而误差增大。因此，本文还研究讨论自相关序列的加窗选择问题。在此基础上，提出基于加窗自相关序列的PHD改进算法，先利用高序号自相关系数来计算频率粗估计，然后充分利用加窗自相关序列的信息，包括低序号和高序号自相关的信息，计算一个频率调节因子来对频率粗估计进行微调。文中通过理论分析证明了算法的性能可以逼近CRB界，同时该性能界从理论上指导加窗自相关序列的选取。仿真分析表明其估计性能明显优于传统的PHD算法和其它基于时域自相关的算法，避免了使用高序号自相关带来的频率估计模糊问题和边界问题。成果在《电子学报》英文版等刊物发表。
Parameter estimation of a tone in noise is important in many fields such as radar, sonar,communications, speech signal processing, biomedical engineering, control and measurement.Frequency is the most important parameter and the most essential feature, so its estimation isa classic issue in the field of signal processing. In many practical engineering applications, thesignal samples have real values, such as voice signals. However, fequency estimation ofa real sinusoid is more difficult relatively than the complex one, for the formerincludes "negative frequency" in the spectrum, and the phase of its correlation affects theaccuracy of frequency estimation. So that, the method of frequency estimationfor the complex sinusoid can not be directly applied to the real one, especially for thetime-domain algorithms. White Gaussian noise is a common noise in the nature, so thefrequency estimation of the real sinusoid embedded in white Gaussian noise hasbeen received extensive attention.
The method of the frequency estimation can be divided into two categories: based ontime-domain and transform domain, such as frequency-domain. For the methods based onfrequency-domain, they have advantages in the estimation performance, but they arecomputationally demanding. While, the methods based on time-domain are relativelysimple, and they are especially suitable for applications where the real-time estimation isrequired. The mean square error is often used as a measure of inaccuracy for the frequencyestimation, and there is an ideal theoretic bound named Cramer Rao Bound (CRB), whichdepends on the length of signal and SNR. For the short sequence, efficient time-domainestimators which approach the CRB and have less computation demanding as well are the keypoint in the study of frequency estimation. Since the sample correlation sequence has thesame frequency as the original signal, but with less noise effect, this dissertation has madedeep research on the correlation-based estimator and the major contributions are as thefollowing:
(1)A closed-form expanded correlation method for real sinusoid frequency estimation isproposed and the effects of White Gaussian noise on the performance is derived, so that aclosed-form theoretical performances bound is abtained. Since the sample correlationsequence has the same frequency as the original signal, a few correlation coefficients can beexploited to estimate the frequency qulickly but the performance is inefficient. To improvethe estimation formance, many researchers proposed a lot of modified correlations andmultiple-stage correlaton, however, the former is not very effective and the latter increases the amount of computation. We take the performance and complexity into consideration andpropose an expanded correlation method. The mothod makes full use of the multiplecorrelaition lags and extends the idea of a coarse search and a fine search of frequencyestimation in the frequency-domain to the time-domain. Firstly, the modified covariance(MC) method based on multiple correlation lags is applied to provide a coarse frequencyestimate. Then, a closed-form adjustment term based on a least square cost function is derivedto get the fine frequency estimate. Simulation results show that the performance of theproposed algorithm, when compared with several existing closed-form time-domainestimators, is closer to the Cramer-Rao Bound (CRB) at0dB. Moreover, the proposed methodhas lower computation complexity than other autocorrelation-based approaches, which alsouse multiple autocorrelation lags. The research productions are published in Signal Processing,etc.
(2)An real single-tone frequency estimator based on phase compensation of multiplecorrelation lags is proposed. For the limited-length single sinusoid, its correlation has anon-zero phase, which is always neglected for correlaton-based methods. So we propose touse Taylor series to expand the correlation at the coarse estimated frequency to exploitmultiple correlation information and take the phase of the correlation into consideration. Theselection of multiple correlation lags is discussed deeply by experiments. Simulation resultsshow that this new method outperforms the Pisarenko harmonic decomposer stimator.Moreover, when compared with other existing considerable computational estimator, themean square frequency error of the proposed method is closer to the CRB for certain SNRrange, especially when the signal length is very short. The research productions are publishedin IEICE Transactions, etc.
(3)A modified PHD method based on windowed correlation sequence is investigated.The PHD method only utilizes correlation lags1and2to estimate the frequency. It has beenproved using higher correlation lags can improve the estimtation performance but leads tofrequency ambiguity and edge frequency. So, we research on a method, which exploiteshigher correlation lags and can improve performance with avoiding frequency ambiguity andedge frequency. In addition, the noise effect on the correlation concentrates around lag0,and the correlation with higher lag is inaccurate for its computation involves less availablesamples, thus we consider adding a rectangle window to the correlaton. Based on thewindowed correlation sequence, an adjustment term is derived to add to the coase frequency,which is obtained by PHD method with higher correlation lags. Theoretical analysis showsthat the algorithm can approach the CRB, and it gives guidance on how to select a windowed correlation sequence. Simulation experiments illustrate the performance of our proposedmethod is generally superior to the PHD and other correlation-based methods, and it can solvethe problem about frequency ambiguity and edge frequency. The research productions arepublished in Chinese Electronic Journal, etc.
引文
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