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SPECTRAL PARAMETER ESTIMATION for LINEAR SYSTEM IDENTIFICATION, by C I rtüii "inali*·'*!' ' *' EUR 4479 e i ■i$m\ SPECTRAL PARAMETI WÊà m ΐίίΒ-1« tum A.C. Ü léii**1 BE ii \>mm.m ' ä-'ÜBit-f , ,^. MWfj 1970 Blip Η ·■: Joint Nuclear Research Center Ispra Establishment - Italy Reactor Physics Department KesearcResearcnh KeacrorReactorss f tø pf IMI? Iiiiínílíi^'lliw^i»·*»!!!^ liillit, A A A K, f A this document was prepared under the sponsorship ot th of the European Communities^i.fttMä'iwi fiSt" «Hf Neither the Commission of the European Communities, its contractors nor ™ fiRá waëïiwmSm any person acting on their behalf : make any warranty or representation, express or implied, with respect to the accuracy, completeness or usefulness of the information contained in this document, or that the use of 'any information, apparatus, method or process disclosed in this document may not infringe privately owned rights; or fWJftJSû llåf , „ u , , , , 1»Ü1 froassumm the ean usy e liabilitof anyy witinformationh respect, tapparatuso the use, methoof, ord foor r damageprocesss discloseresultindg in this document. fi Aie ιΐϋΐ •sír'5¿i'}ll! .¡vit• This report is on sale at the addresses listed on cover page 4 fS'-iEuflfi'piîMLsfli'iiÎMiW'1 'J*' r'Urt^íSlTSfltislHiilMiMHjTÍ"^ ut the price of FF 9,45 FB 85- DM 6,20 Lit 1.060 FI. 6,20 ïah When ordering, please quote the EUR number and the title, which are indicated on the cover of each report 1 WMpttIcU *v*r 1 BE sfliilìl »l^ifie KS? ί MM Printed by Smeets, Brussels LuxembourouTg, , May 1970 -h Vlililí u#feeItó1 ψ4 K*l*LrV,;íld This document was reproduced on the basis of the best available copy »Av tfwi< ¡;KLí W* EUR 4479 e SPECTRAL PARAMETER ESTIMATION FOR LINEAR SYSTEM IDENTIFICATION, by C. LUCIA Commission of the European Communities Joint Nuclear Research Center - Ispra Establishment (Italy) Reactor Physics Department - Research Reactors Luxembourg, May 1970 - 58 Pages - 2 Figures - FB 85 The aim of this report is to examine the properties and statistical errors of the estimators of some spectral parameters of stationary, ergodic, linear processes (power and cross-power spectral density, Fourier transform, transfer functions). In addition, procedures and computing programmes are supplied for the estimation of these parameters by means of the statistical dynamics analyzer S.D.A. - Mod. 040. The case of deterministic signals is also taken into consideration. EUR 4479 e SPECTRAL PARAMETER ESTIMATION FOR LINEAR SYSTEM IDENTIFICATION, by C. LUCIA Commission of the European Communities Joint Nuclear Research Center - Ispra Establishment i(Italy) Reactor Physics Department - Research Reactors Luxembourg, May 1970 - 58 Pages - 2 Figures - FB 85 The aim of this report is to examine the properties ana statistical errors of the estimators of some spectral parameters of stationary, ergodic, linear processes (power and cross-power spectral density, Fourier transrorm, transfer functions). In addition, procedures and computing programmes are supplied for the estimation of these parameters by means of the statistical dynamics analyzer S.D.A. - Mod. 040. The case of deterministic signals is also taken into consideration. EUR 4479 e SPECTRAL PARAMETER ESTIMATION FOR LINEAR SYSTEM IDENTIFICATION, by C. LUCIA Commission of the European Communities Joint Nuclear Research Center - Ispra Establishment (Italy) Reactor Physics Department - Research Reactors Luxembourg, May 1970 - 58 Pages - 2 Figures - FB 85 The aim of this report is to examine the properties and statistical errors of the estimators of some spectral parameters of stationary, ergodic, linear processes (power and cross-power spectral density, Fourier transrorm, transfer functions). In addition, procedures and computing programmes are supplied for the estimation of these parameters by means of the statistical dynamics analyzer S.D.A. - Mod. 040. The case of deterministic signals is also taken into consideration. EUR 4479 e COMMISSION OF THE EUROPEAN COMMUNITIES SPECTRAL PARAMETER ESTIMATION FOR LINEAR SYSTEM IDENTIFICATION by AC. LUCIA 1970 Joint Nuclear Research Center Ispra Establishment - Italy Reactor Physics Department Research Reactors ABSTRACT The aim of this report is to examine the properties and statistical errors or the estimators of some spectral parameters of stationary, ergodic. linear processes (power and cross-power spectral density. Fourier transform, transfer functions). In addition, procedures and computing programmes are supplied for the estimation of these parameters by means of the statistical dynamics analyzer S.D.A. - Mod. 040. The case of deterministic signals is also taken into consideration. KEYWORDS MATHEMATICS PARAMETERS FOURIER TRANSFORMATIONS COMPUTERS STATISTICS SPECTRA CONTENTS pages INTRODUCTION 1.) SPECTRAL ANALYSIS OF RANDOM SIGNALS 1.1.) Power spectral density estimation 7 1.2.) Cross-power spectral density estimation -\ A 1.3.) Power and cross-power spectral density estimation with the statistical dynamics analyzer S.D.A 17 2.) SPECTRAL ANALYSIS OF APERIODIC SIGNALS 24 2.1.) Fourier transform estimation 25 2.2.) Fourier transform estimation with the statistical dyna• mics analyzer S.D.A. 26 2.3.) Energy and cross-energy spectral density estimation .... 3O 2.4.) Energy and cross-energy spectral density estimation with the statistical dynamics analyzer S.D.A 31 3.) SPECTRAL ANALYSIS OF PERIODIC SIGNALS 33 3.1.) Fourier transform estimation with the statistical dyna• mics analyzer S.D.A 34 3.2.) Power and cross-power spectral density estimation with the statistical dynamics analyzer S.D.A 37 APPENDIX 39 FIGURE CAPTIONS 56 LIST OF TABLES 56 REFERENCES 57 SPECTRAL PARAMETER ESTIMATION FOR LINEAR SYSTEM IDENTIFICATION INTRODUCTION *) The techniques for determining the power and cross-power spectral density, the Fourier transforms and the transfer functions assume considerable importance in the identification of a system or 12 3 process ) ) ). From these functions it is in fact often pos• sible to arrive at some basic parameters of the system or process un• der examination. For this reason it is interesting to know the characteristics of the estimators we use and the accuracy obtainable with them. This report studies the statistical properties of some continuous estimators in general use which allow direct determination of Fourier transforms, power and cross-power spectral densities and transfer functions. The determination of these functions is trea• ted categorically for the cases of random signals and of aperiodic and periodic signals. The estimators we deal with in this work also constitute the algorithms by which are obtained the determinations made on the statistical dynamics analyzer S.D.A. (a general purpose analyzer, 4 designed and built at the Euratom Joint Research Center of Ispra) ). In the sections 1.3; 2.3 and 3.2 and in the Appendix, in which processing procedures and computing programmes are given, re• ference is made exclusively to the S.D.A. Manuscript received on 11 March 1970 1.) SPECTRAL ANALYSIS OF RANDOM SIGNALS The random signals we will consider are assumed to be stationary and ergodic, even when this is not explicity stated. We know that in the case of random signals, no Fourier transform (of the classic type) exists, and that for this reason the power spectra are defined in terms of Fourier transforms of the corre­ 16 lation functions ) ) Τ 1 x Φ (ω) = lim 2T / (*) x(t+T)dt e­JWTdT (1) if _ T-* «e J _φ co Τ Φ (ω) ■—ƒ film ~ fy(t) y(t+r)at e-Jft,TdT (2) yy Τ 1 -JWT, #^(» 2π lim Jr / x(t) y(t-»r)dt (3) T-» ce J _τ β dr Nevertheless the validity of the direct determination (by the Fourier transform of the signals under examination) of the power and cross-power spectral density, is demonstrated ) ). The estimators by which the direct determinations can be carried out have the following expressions: .Τ -3<iì t 1 j_ c(t) θ dt (4) XX 2ir Τ 1 j_ -jut * (ω) 2ir T y(t) dt (5) yyv jCüt ^(¿ω) 1­ ifiítje^dt . fy(t)r dt (6) 2* Τ which are the estimators we­ are considering here; they are likewise 4 used in the S.D.A. for the spectral analysis of random signals ). 1.1.) Power spectral density estimation Let expression (4): Τ 1 ι d<üt Φx x (ω) c(t) e" dt (4) 2π Τ be the estimator of the power spectral density of a stationary, ergo­ dic random signal x(t). We will see what the properties of this estimator are, particularly as far as the bias and the variance of the estimation are concerned. The mathematical expectation of the estimator is: ■1­ IF Jwt Φrx x(ω ) x(t) e" dt (7) 2π Τ E Considering the square of the modulus of the integral at the right­ hand member of (7) as a double integral and remembering that: R (t­e) E c(t) x(e) xx (8) where R is the autocorrelation function of x(t), we can write ): Jw(t e) E Φ ν(ω ) ±-lfjí (t-e) e- - dt dö (9) χχ 2π Τ / / χχν ' 8 which gives finally ) ■[>»(·>] ­^¡j­^y^r) e­JwTdT (10) For an analysis time Τ tending to the infinite, wherever the auto­ correlation function can be integrated absolutely, we obtain from (10)ι -JUT lim E R (τ) e dr = rΦ x(ω) (H) Τ­*«· ƒ«(-)] Η. xx ' xx ' i.e. the estimator (4) is unbiased in the limit, but for a finite ana­ lysis time Τ , it is biased: that is, it contains a systematic error. Let us now see what the variance of the estimate is: E Φ' v(u) -E rψ x(ω ) var *„(-)] xx xx (12) Q where the mathematical expectation of the square of the estimate is ): J__ 1 2 E [>„<«: E r x(t) e-^at 2 2π Τ ] ] τ τ τ. τ = -L —ί ! [ ΓΕ Tx(t) χ(0) χ(τ,) χ(ξ)β'οω^+θ-η'ξ) Ί dt dø dr, ae 1*1* T?b h h Jo L J (13) the expression at the third member of (13) having been deduced by con• sidering the square of the integral as a double integral, and by using the property of interchangeability of the operation of finding the ma• thematical expectation and the operation of integration.
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