Power Spectrum estimation (reprint)

Power spectrum estimation is to estimate the signal's power with the change of frequency, the actual use of filter, signal recognition (analysis of the frequency of the signal), signal separation, system identification. Spectral estimation technology is an important part of modern signal processing, including spatial spectral estimation, high-order spectral estimation and so on. Wiener Filter, Kalman filter, can be used for adaptive filtering, signal waveform prediction (aircraft track pre-judgment in Fire control system). If I add a signal waveform to the noise. Is it possible to completely filter out the signal waveforms I have added? If you know some information, using a reference signal waveform, you can use adaptive filtering (the initial part of the signal is slightly distorted).

Power spectrum estimation is one of the main contents of digital signal processing, which mainly studies the characteristics of signal in frequency domain, and aims to extract useful signals in frequency domain based on finite data. The following is a brief review of the development process of spectral estimation:
The British scientist Newton first gave the concept of "spectrum". Later, in 1822, French engineer Fourier presented the famous theory of Fourier harmonic analysis. The theory is still the theoretical basis for signal analysis and signal processing.
After the Fourier series is put forward, it is applied when people observe the periodic phenomena in nature. At the end of 19th century, Schuster proposed to use the amplitude squared of Fourier series as a measure of power in the function and named it "periodic graph" (periodogram). This is the earliest reference to classical spectral estimation, and this formulation is still being used, but it is now using fast Fourier transform (FFT) to calculate the discrete Fourier transform (DFT), using the amplitude squared of the DFT as a measure of the power in the signal.
　　The poor variance performance of periodic graphs encourages people to study additional analytical methods. In 1927, Yule proposed to use linear regression equation to simulate a time series. Yule's work has actually become the most important method in modern spectral estimation--the basis of spectral estimation of parametric model method.
　　 Walker uses Yule's analytical method to study the decay sine time series, and to get the Yule-walker equation, it can be said that Yule and Walker are pioneers in developing autoregressive models.
In 1930, the famous control theory expert, Wiener, for the first time in his writings, precisely defined aStochastic Processautocorrelation function and power spectral density, and the spectral analysis is based on the statistical characteristics of stochastic processes, i.e., "power spectral density is the Fourier transform of the second-order statistic autocorrelation function of stochastic process", which isWiener-khintchine theorem。 The theorem defines power spectral density as a continuous function of frequency, rather than as a function previously defined as a discrete harmonic frequency.
In 1949, Tukey based on the wiener-khintchine theorem, a self-correlation method for spectral estimation of finite-length data was proposed, i.e. usingFinite length dataThe autocorrelation function is estimated, and then the spherical Fourier transform of the autocorrelation function is obtained to estimate the spectrum. 1958,Blackman and TukeyThe self-correlation spectral estimation method is discussed in the published monograph on classical spectral estimation, soThe self-correlation method is also called BT method。 Both the periodic graph method and the Autocorrelation method can be realized by the fast Fourier transform algorithm, and the physical concept is clear, so it is still more commonly used spectral estimation method.
　　 In 1948, Bartlett first proposed the calculation of power spectra by using autoregressive model coefficients . Both autoregressive models and linear predictions using the Toeplitz matrix structure proposed in 1911 , Levinson has proposed a fast calculation method for understanding Yule-walker in 1947 according to the characteristics of the matrix. These work has laid a good theoretical foundation for the development of modern spectral estimation.
In the 1965, the FFT algorithm proposed by Cooley and Tukey also promoted the rapid development of spectral estimation.
Modern spectral estimation is mainly aimed atPoor resolution and variance performance of classical spectral estimationQuestions raised. The modern spectral estimation can be broadly divided intoThere are two kinds of spectral estimation and nonparametric model spectral estimation, the former AR model, MA model, ARMA Model, Prony index model, etc., the latter has the least variance method, multi-component music method and so on.

Arma Spectral Estimationlinear systems can be described by a linear difference equation, which is the autoregressive----sliding average model (autoregression----moving average,arma). : Any rational power spectral density can be accurately approximated by the power spectral density of an arma stochastic process. ?? ARMA model Definition If discrete stochastic process {x (n)} obeys linear difference equation x (n) +ai*x (n-i) =e (n) +bj*e (n-j)i=1,2,... p;j=1,2,... q;e (n) is a discrete white noise, then {x (N)} is an Arma process, which is called an ARMA model. coefficients and respectively called Autoregressive (AR) parameters and sliding average (MA) parameters, and P and Q are respectively called AR Order and MA order. Obviously, the ARMA model describes a time-invariant linear system. The ARMA process, which has the AR order p and Ma order Q, usually plays an arma (P,Q).

Power Spectrum estimation (reprint)