第1个回答 2011-03-29
功率谱估计是数字信号处理的主要内容之一，主要研究信号在频域中的各种特征，目的是根据有限数据在频域内提取被淹没在噪声中的有用信号。下面对谱估计的发展过程做简要回顾: 英国科学家牛顿最早给出了“谱”的概念。后来，1822年，法国工程师傅立叶提出了著名的傅立叶谐波分析理论。该理论至今依然是进行信号分析和信号处理的理论基础。
傅立叶级数提出后，首先在人们观测自然界中的周期现象时得到应用。19世纪末，Schuster提出用傅立叶级数的幅度平方作为函数中功率的度量，并将其命名为“周期图”（periodogram）。这是经典谱估计的最早提法，这种提法至今仍然被沿用，只不过现在是用快速傅立叶变换（FFT）来计算离散傅立叶变换（DFT），用DFT的幅度平方作为信号中功率的度量。 周期图较差的方差性能促使人们研究另外的分析方法。1927年，Yule提出用线性回归方程来模拟一个时间序列。Yule的工作实际上成了现代谱估计中最重要的方法——参数模型法谱估计的基础。 Walker利用Yule的分析方法研究了衰减正弦时间序列，得出Yule-Walker方程，可以说，Yule和Walker都是开拓自回归模型的先锋。 1930年，著名控制理论专家Wiener在他的著作中首次精确定义了一个随机过程的自相关函数及功率谱密度，并把谱分析建立在随机过程统计特征的基础上，即，“功率谱密度是随机过程二阶统计量自相关函数的傅立叶变换”，这就是Wiener—Khintchine定理。该定理把功率谱密度定义为频率的连续函数，而不再像以前定义为离散的谐波频率的函数。 1949年，Tukey根据Wiener—Khintchine定理提出了对有限长数据进行谱估计的自相关法，即利用有限长数据估计自相关函数，再对该自相关函数球傅立叶变换，从而得到谱的估计。1958年， Blackman和Tukey在出版的有关经典谱估计的专著中讨论了自相关谱估计法，所以自相关法又叫BT法。 周期图法和自相关法都可用快速傅立叶变换算法来实现，且物理概念明确，因而仍是目前较常用的谱估计方法。 1948年，Bartlett首次提出了用自回归模型系数计算功率谱。自回归模型和线性预测都用到了1911年提出的Toeplitz矩阵结构，Levinson曾根据该矩阵的特点于1947年提出了解Yule-Walker的快速计算方法。这些工作为现代谱估计的发展打下了良好的理论基础。 1965年，Cooley和Tukey提出的FFT算法，也促进了谱估计的迅速发展。 现代谱估计主要是针对经典谱估计的分辨率差和方差性能不好的问题而提出的。现代谱估计从方法上大致可分为参数模型谱估计和非参数模型谱估计两种，前者有AR模型、MA模型、ARMA模型、PRONY指数模型等；后者有最小方差方法、多分量的MUSIC方法等。 周期运动在功率谱中对应尖锋，混沌的特征是谱中出现"噪声背景"和宽锋。它是研究系统从分岔走向混沌的重要方法。 在很多实际问题中（尤其是对非线性电路的研究）常常只给出观测到的离散的时间序列X1, X2, X3,...Xn,那么如何从这些时间序列中提取前述的四种吸引子（零维不动点、一维极限环、二维环面、奇怪吸引子）的不同状态的信息呢？ 我们可以运用数学上已经严格证明的结论，即拟合。我们将N个采样值加上周期条件Xn+i=Xi，则自关联函数（即离散卷积）为 然后对Cj完成离散傅氏变换，计算傅氏系数。 Pk说明第k个频率分量对Xi的贡献，这就是功率谱的定义。当采用快速傅氏变换算法后，可直接由Xi作快速傅氏变换，得到系数 然后计算 ，由许多组{Xi}得一批{Pk'}，求平均后即趋近前面定义的功率谱Pk。 从功率谱上，四种吸引子是容易区分的，如图12 (a)，(b)对应的是周期函数，功率谱是分离的离散谱 (c)对应的是准周期函数，各频率中间的间隔分布不像(b)那样有规律。 (d)图是混沌的功率谱，表现为"噪声背景"及宽锋。 考虑到实际计算中，数据只能取有限个，谱也总以有限分辨度表示出来，从物理实验和数值计算的角度看，一个周期十分长的解和一个混沌解是难于区分的，这也正是功率谱研究的主要弊端。
Power spectrum estimation is the main content of digital signal processing, one of the main signal in the frequency domain of various features of the purpose in the frequency domain based on limited data lost in the noise in the extraction of the useful signal. The following estimates of the spectrum to do a brief review of the development process: British scientist Newton was first given a "spectrum" concept. Later, in 1822, proposed the famous French engineer Fourier Fourier harmonic analysis theory. The theory is still being used for signal analysis and signal processing theory.
Fourier series is proposed, first in one cycle of observing phenomena in nature are applied. The late 19th century, Schuster presented the magnitude of a Fourier series as a function of the square measure of power, and name it as the "periodogram" (periodogram). This is the earliest reference to the classical spectrum estimation, this formulation is still in use, but now is fast Fourier transform (FFT) to calculate the discrete Fourier transform (DFT), with the square of the magnitude of DFT as a measure of signal power . Periodogram poor performance prompted the variance of the other methods. In 1927, Yule made a linear regression equation to simulate a time series. Yule's work actually became the most important modern spectral estimation methods - parameter model based spectrum estimation method. Analysis using Yule Walker method of attenuation sinusoidal time series, obtained Yule-Walker equation, it can be said, Yule and Walker is a pioneer in developing self-regression model. In 1930, the famous Wiener control theory experts, the first time in his book accurately defines a random process the autocorrelation function and power spectral density, and the spectral analysis based on random process based on the statistical characteristics, namely, "the power spectral density is a random process of order statistics of the Fourier transform of the autocorrelation function, "which is the Wiener-Khintchine theorem. The theorem of the power spectral density is defined as the frequency of the continuous function, and not as previously defined for the discrete harmonic function of frequency. In 1949, Tukey proposed under the Wiener-Khintchine theorem of finite length data for the autocorrelation spectral estimation method, namely the use of finite length data to estimate the autocorrelation function, then the Fourier transform of the autocorrelation function of the ball, resulting in spectrum estimation. In 1958, Blackman and Tukey in the publication of the relevant classical spectrum estimation are discussed in the monograph autocorrelation spectrum estimation, the autocorrelation method also known as BT. Periodogram and autocorrelation are available to achieve the fast Fourier transform algorithm, and a clear physical concept, which is still the more commonly used method for spectral estimation. In 1948, Bartlett was the first time since the regression coefficients using the power spectrum. Since the regression model and linear prediction are used in the 1911's Toeplitz matrix structure, Levinson has been based on the characteristics of the matrix presented in 1947 to understand the rapid calculation of Yule-Walker method. The work was the development of modern spectral estimation theory has laid a good foundation. In 1965, Cooley and Tukey's FFT algorithm, but also promoted the rapid development of spectral estimation. Modern spectral estimation is mainly the resolution of the classical spectrum estimation deviation and variance problem of poor performance raised. Modern spectral estimation methods can be broadly divided from the parameter model spectrum estimation and two non-parametric model spectrum estimation, the former AR model, MA model, ARMA model, PRONY index models; which has a minimum variance method, multi-component MUSIC methods. Periodic motion of the corresponding point in the power spectrum front, chaos is characterized by a spectrum of a "background noise" and the wide front. It is to study the system from the bifurcation to chaos in important ways. In many practical problems (especially the research on nonlinear circuits) are often given only the observed discrete time series X1, X2, X3, ... Xn, then how to extract from these time series, the four aforementioned attractor (zero-dimensional fixed point, one-dimensional limit cycle, two-dimensional torus, strange attractors) of the different states of information? We can already strict mathematical proof of the conclusion that the fitting. Value of N samples we will add the periodic condition Xn + i = Xi, then the autocorrelation function (ie, discrete convolution) is then complete the Discrete Fourier Transform Cj calculated Fourier coefficients. K-Pk Notes the contribution of frequency components of Xi, which is the definition of the power spectrum. When using the fast Fourier transform algorithm, you can
the Xi as fast Fourier transformation, calculation of coefficients and then by many groups {Xi} be a group of {Pk '}, averaging closer to the front after the definition of the power spectrum Pk . From the power discrete spectrum (c) corresponds to quasi-periodic function, the frequency of the middle Unlike the interval distribution (b) as a regular. (D) map is chaotic power spectrum, expressed as "background noise" and the wide front. Taking into account the actual calculation, the data can only take a finite number, the total spectrum are represented by a limited degree of resolution, from the physical experiments and numerical point of view, the solution of a very long cycle and a chaotic solution is difficult to distinguish, and this is is the power spectrum of the major drawbacks