Bir boyutlu işaretlerin güç spektrum kestirim yöntemleri ve bilgisayar uygulamaları
Power spectral density estimation techniques and their computer simulations
- Tez No: 22072
- Danışmanlar: DOÇ. DR. AHMET KAYRAN
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 103
Özet
ÖZET Bu çalışmada son yıllarda güç spektrum kestirim alanında geliştirilmiş olan yöntemler incelenmiştir. Güç spektrum kestirim problemi genel olarak gözlemlenen sonlu sayıda işaret değerine dayanarak, işarete ait güç s spektrumunun kestirilmesi olarak tanımlanabilir. Güç spektrum kestirim yöntemleri, parametrik ve parametrik olmayan adı altında iki ana grupta sınıflandırılabilir. Her yöntem önce kısaca açıklanmış, ve bilgisayar uygulamaları gerçekleştirilmiştir. Son olarak da tüm yöntemler bilgisayar uygulamalarının ışığı altında birbirleriyle karşılaştırılmıştır. İncelenen güç spektrum kestirim yöntemleri parametrik olmayan periodogram, Black man -Tukey ve Capon, parametrik AR Yule-Walker, AR Burg, AR Kovar yans, MA Dur bin, ARMA Yule-Walker, ARMA En küçük kareler, ARMA Mayne- Firoozan yöntemleridir. - v -
Özet (Çeviri)
SUMMARY POWER SPECTRAL DENSITY ESTIMATION TECHNIQUES AND THEIR COMPUTER SIMULATIONS A summary of many of the new techniques developed in the last two decades for spectrum anaylsis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the the various spectrum analysis approaches. The general problem of spectral estimation is that of determining the spectral content of a random process based on a finite set of observations from that process. Formally the power spectral density , which will be denoted by P Cf>, of a complex wide sense stationary random process xCn> is defined as oo -I P = \ r e"J2nfk XX / XX J 00 where r is the autocorrelation function of x is XX defi ned as = EC x*x ] and E is the expectation operator CI 3. In practice we have only the contiguous observations < xC0>, x,... x > with which to determine the PSD. Since the PSD depends on an infinite number of autocorrelation values, determination of the PSD is in general an impossible task. Based on the N contiguous observations < x, x,.. » x >of a single realization of a random process, it is desired to estimate the PSD for -1/2 < f < 1/2..The remaining 10 power spectrum estimation methods may - vibe grouped into two broad categories; namely, parametric and nonoaramslrlc. and nonpar ameir i c In parametric methods of spectral analysis a model is assumed in the formulation of the problem, and the requirement is to estimate the parameters of the model from observations of the given process for a limited duration of time. This model may take on a variety of different forms. For example, it may be used a model in the form of a rational function for the power spectrum of the time series. Accordingly one may be distinguish an autoregressive model represented by an all pole rational function, a moving average model represented by an all zero rational function, and an autoregressive moving average model represented by a rational transfer function with both poles and zeros. Parametric methods discussed in this tutorial are AR Yule, AR Burg, AR Covariance, MA Durbin, ARMA Least Squares, ARMA Modified Yule Walker, ARMA Mayne- Firoozan power spectrum estimation methods [23. In nonparametric methods of spectral analysis differ from parametric ones in that no specific model is presupposed in formulating the estimation problem. Here again there are numerous nonparametric approaches that have been developed to estimate the power spectrum of a given time series. Nonparametric methods discussed in this tutorial are Periodogram, Blackman-Tukey, Minimum Variance power spectrum estimation methods. One highly popular method is to use the periodogram. It uses the formulation for spectrum estimation. Many of the problems of the periodogram PSD estimation technique can be traced to the assumptions made about the data outside the measurement intervall. The finite data sequence may be viewed as being obtained by windowing an infinite length sample sequence with a boxcar function. This multiplication of the actual time series by a window function means the overall transform is the convolution of the desired transform with the transform of the window function. The problem then becomes one of choosing a suitable window that satisfies some conflicting requirements. Most used window function is the boxcar function. Another widely used method is the Blackman-Tukey approach, which involves taking the Fourier transform of a set of windowed autocorrelation estimates of the time series. Equation is used for spectrum estimation. We must be careful, however, to ensure that the window chosen will always lead to a nonnegative spectral estimate. Only the Bartlett and Parzen windows have nonnegative Fourier transforms. Therefore only both can be used. - vii -In general the spectral estimates P and P, ^ r per bt are not identical. However, if the biased autocorrelation estimate is used and as many lags as data samples < M = N-l >are computed, then the BT estimate and the periodogram estimate yield identical numerical results. Thus the periodogram can be viewed as a special case of BT estimate. The third nonpar ametric method discussed is the minimum variance spectrum estimator. Here the parameters are chosen so that the input signal would be undistorted at the filter output and the variance of the output process is minimized. This procedure uses the equation using a biased autocorrelation estimator. The estimate of the white noise variance is given by . The Yule -Walker method has been found to produce poorer resolution spectral estimates than the other estimators to be described. Second is the AR Burg method described. In contrast to the AR Yule-Walker and covariance methods, which estimate the AR parameters directly, the Burg method estimates the reflection coefficients and then uses the Levinson recursion to obtain the AR parameter estimates. The reflection coefficient estimates are obtained by minimizing estimates of the prediction error power for different order predictors in a recursive manner. Third method is the AR Covariance method. The only difference between the AR Yule-Walker and the covariance method is the range of summation in the prediction error power estimate. Used equations are and . Next method is the MA Dur bin PSD method. Here, in Durbin's method first one have to fit a large order AR model using the AR Yule method using the data sequence. For an AR model order of L, where q . Then using the AR parameter estimates obtained from first step as input data, one have use the AR Yule method with order q to find MA parameters as given by . Since the MA PSD is based on an all zero model of the data, it is not possible to use it to estimate PSD' s with sharp peaks. The last three PSD estimation methods are ARMA methods. The first approach uses the Yule-Walker equations. The ARMA Yule-Walker can be solved in an efficient manner - viii -using an extension of the Levinson recursion Once the AR parameter estimates have been obtained, the data are filtered with the estimate of A to generate an approximate MA process. Then MA Dur bin method can be employed to estimate the MA parameters. As an ARMA PSD estimator is next the ARMA Least Squares estimator discussed. In an attempt to reduce the variance of the ARMA Yule-Walker method Cadzow developed a new technique, namely ARMA Least Squares. There is information in the autocorrelation samples at higher order samples. To use this information, assume that the highest sample of the autocorrelation that can be accurately estimated is r . Assuming that M-q > p, there will be more equations than unknowns. Equations used are . And last ARMA Mayne-Firoozan method is discussed. A class of suboptimal ARMA estimators have been proposed which rely on estimation of the driving white noise.y. If 4» were known, then one have knowledge of the input as well as the output, this means ARMA parameters could be estimated as the solution of a set of linear equations. In practice, 4> is estimated from x as the output of a large order prediction error filter. Equations used here are - . As a test case is a process including three sinusoids used. x = 2cos + 2cos + 2cos + z 1 2 3 where f = 0.05, f = 0.40, and f = 0.42. z(n) is a 12 3 complex autoregressi ve process. z = -az + v All necessary parameters including data samples are in a FORTRAN subroutine. All comparisons which are drawn between the various spectral estimators are qualitative in nature and should only be used as an informal guideline in choosing a particular method for an application. It is usually prudent to apply several spectral estimators to the data set to serve as a consistency check of the result. It is apparent that there is no unique solution to the spectral estimation problem. All computer programs for these 10 PSD estimators are FORTRAN subroutines. Main programs for each technique are also in FORTRAN written. Each main program computes - ix -the required PSD values and writes these in an ASCII file, namely *. txt. Then a PASCAL graphic program reads from this ASCII file the PSD values and plots the power spectral density graphic. All the programs are developed on a 80286 Processor computer with 1 Mb RAM and Hercules Graphic Card. Printouts are made directly from the screen with prtscr.com. x -
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