[Wiley Series in Probability and Statistics] Random Data (Analysis and Measurement Procedures) || Statistical Errors in Advanced Estimates

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<ul><li><p>C H A P T E R 9 </p><p>Statistical Errors in Advanced Estimates </p><p>This chapter continues the development from Chapter 8 on statistical errors in random data analysis. Emphasis is now on frequency domain properties of joint sample records from two different stationary (ergodic) random processes. The advanced parameter estimates discussed in this chapter include magnitude and phase estimates of cross-spectral density functions, followed by various quantities contained in single-input/output problems and multiple-input/output problems, as covered in Chapters 6 and 7. In particular, statistical error formulas are developed for </p><p>Frequency response function estimates (gain and phase) </p><p>Coherence function estimates </p><p>Coherent output spectrum estimates </p><p>Multiple coherence function estimates </p><p>Partial coherence function estimates </p><p>9.1 CROSS-SPECTRAL DENSITY FUNCTION ESTIMATES </p><p>Consider the cross-spectral density function between two stationary (ergodic) Gaussian random processes {*(/)} and (y(r)}, as defined in Equation (5.66). Speci-fically, given a pair of sample records x(t) and y(r) of unlimited length T, the one-sided cross-spectrum is given by </p><p>G{f) = lim lE{X*(f,T)Y(fJ)} (9.1) </p><p>Random Data: Analysis and Measurement Procedures, Fourth Edition. By Julius S. Bendat and Allan G. Piersol Copyright 2010 John Wiley &amp; Sons, Inc. </p><p>289 </p></li><li><p>290 STATISTICAL ERRORS IN ADVANCED ESTIMATES </p><p>where X(f, T) and Y(f, T) are the finite Fourier transforms of x(t) and y(i), respectively-that is, </p><p>tT X(f)=X(f,T) x{t)e-j2,rf'dt </p><p>y(t)e-j27rftdt </p><p>(9.2) </p><p>Y(f) = y(f,T) "0 </p><p>It follows that a "raw" estimate (no averages) of the cross-spectrum for a finite record length is given by </p><p>Gxy = 2,[x\f)Y(f)] (9.3) </p><p>and will have a resolution bandwidth of </p><p>Be~Af = j (9.4) </p><p>meaning that spectral components will be estimated only at the discrete frequencies </p><p>/ * = ! * = 0 , 1 , 2 , . . . (9.5) </p><p>As discussed in Section 8.5.1, the resolution Be(1/7) in Equation (9.4) establishes the potential resolution bias error in an analysis. It will be assumed in this chapter, however, that is sufficiently long to make the resolution bias error of the cross-spectrum estimates negligible. </p><p>As for the autospectrum estimates discussed in Section 8.5.4, the "raw" cross-spectrum estimate given by Equation (9.3) will have an unacceptably large random error for most applications. In practice, the random error is reduced by computing an ensemble of estimates from nd different (distinct, disjoint) subrecords, each of length T, and averaging the results to obtain a final "smooth" estimate given by </p><p>2 " d </p><p>ndTj^ </p><p>It follows that the minimum total record length required to compute the cross-spectrum estimate is Tr ndT. Special cases of Equation (9.6) produce the autospectra estimates Gxx(f) and Gyy(f) by letting x(t) = y(t). </p><p>The quantities X(f) and Y(f) in Equation (9.2) can be broken down into real and imaginary parts as follows: </p><p>X(f) = XK(f)-jXi(f) Y(f) = YR{f)-jYi{f) (9-7) </p><p>where </p><p>x(t)cos2Trfidt X,(f) = </p><p>x(t)sin2vftdt (9.8) </p><p>Yn(f) = \ y(t)cos27rftdt Y,(f) = \ y(t)sm27Tfldt (9.9) Jo Jo </p><p>If x(t) and y(t) are normally distributed with zero mean values, the quantities in Equations (9.8) and (9.9) will be normally distributed with zero mean values. From </p></li><li><p>C R O S S - S P E C T R A L D E N S I T Y F U N C T I O N E S T I M A T E S 291 </p><p>Equation (9.3), omitting the frequency/to simplify the notation, "raw" estimates are then given by </p><p>G = I (X2R + 2,) pi = \{Y</p><p>2 + Yj) (9.10) </p><p>Gxy = Cxy-jQxy = \Gxy\e~j''xy (9.11) </p><p>where </p><p>2 - 2 CXy = j{XRYR + XiY,) QXy = j{XRYI-XiYR) (9.12) </p><p>fo|2 = + t a n ^ - g y c ^ (9.13) </p><p>For values computed only at / = d e f i n e d in Equation (9.5), one can verify from Equations (9.8) and (9.9) that </p><p>E[XRX,} = E[YRY,) = 0 </p><p>E[X2R] = E[X2} = ( r / 4 ) G </p><p>E\Y2}=E{Y2] = (T/4)Gyy (9.14) </p><p>E[XRYR] = E[X,Y,\ = (r /4)Cq, </p><p>E[XRY,] = -E[X,YR] = [T/A)Qxy </p><p>Hence, E[Cxy] = and E[Qxy} Qxy with </p><p>[] Gxx E[Gyy] = Gyy (9.15) </p><p>E[GV] = EiC^-jElQ^] = Cty-jQ^ = Gv (9.16) </p><p>The Gaussian assumption for any four variables au a2, a3, and with zero mean values gives from Equation (3.73) </p><p>[\,4\ = [\2\[,4\ + [\3][2^} + E[a\a/\\E[a2a3] (9.17) </p><p>Repeated application of this formula shows that </p><p> , </p><p>= ^E[XR + 2XJXl+X*}=2Gl (9.18) </p><p>where the following relationships are used: </p><p>E[XR]=3{E[XR])2 = 3{T/A)2GXX </p><p>E[X2X2} = E[X2]E[X2R] = ( / 4 )2 ( % (9.19) </p><p>E[X^)=3{E[X2})2 = 3{T/A)2G2XX </p></li><li><p>292 STATISTICAL ERRORS IN ADVANCED ESTIMATES </p><p>Similarly, one can verify that </p><p>E[Gyy\ = 2G2</p><p>y </p><p>= \(GxxGyy + 3C2</p><p>xy-Q%) </p><p>(9.20) </p><p>(9.21) </p><p>(9.22) </p><p>9.1.1 Variance Formulas </p><p>Basic variance error formulas will now be calculated. By definition, for any unbiased "raw" estimate A, where E[A] =A, its variance from Equation (8.3) is </p><p>Hence </p><p>Var[A] = E[A}-A2 </p><p>Vax[Gxx) = G2</p><p>xx V a r [ G w ] = G yy </p><p>Var[Cxy]=l-(GxxGyy + C</p><p>2</p><p>xy-Q2</p><p>xy) </p><p>V a r [ G ^ ] = I ( G G w + </p><p>Observe also from Equations (9.13), (9.21) and (9.22) that </p><p>-2 - 2 . [ | G ^ ] = [ C J + E[Q] = G ^ + |G 'xy\ </p><p>(9.23) </p><p>(9.24) </p><p>(9.25) </p><p>(9.26) </p><p>(9.27) </p><p>Hence </p><p>*y\ Var[|G^|J = G G W = 2 (9.28) </p><p>where y2^ is the ordinary coherence function defined by </p><p>y2 Jxyl GxxGyy </p><p>(9.29) </p><p>In accordance with Equation (4.9), the corresponding variance errors for "smooth" estimates of all of the above "raw" estimates will be reduced by a factor of nd when averages are taken over nd statistically independent "raw" quantities. To be specific, </p><p>V a r [ G j = G2Jnd Var[G w ] = G2</p><p>yy/nd </p><p>Var[|G^|] = [G^2/y2nd </p><p>(9.30) </p><p>(9.31) </p></li><li><p>CROSS-SPECTRAL DENSITY FUNCTION ESTIMATES 293 </p><p>CJf) </p><p>\jxy(f)\\/n~d </p><p>[G^Gyy^ + ClifhQlyif)]1'2 </p><p>Cv(f)y/SQ </p><p>Q*y(f) [Gxx(f)Gyy(f) + ( 4 (f)-%,(/)] '/ 2 </p><p>Qxy(f)y/2n~d </p><p>From Equation (8.9), the normalized rms errors (which are here the same as the normalized random errors) become </p><p>4 4 , ] = ^ = [ 0 * 1 = ^ </p><p>'^- (933) The quantity ly^l is the positive square root of y1 . Note that for the cross-spectrum magnitude estimate IGy varies inversely with ly^l and approaches (l/y/h~d) as y1^ approaches one. </p><p>A summary is given in Table 9.1 on the main normalized random error formulas for various spectral density estimates. The number of averages nd represents nd distinct (nonoverlapping) records, which are assumed to contain statistically different in-formation from record to record. These records may occur by dividing a long stationary ergodic record into nd parts, or they may occur by repeating an experiment nd times under similar conditions. With the exception of autospectrum estimates, all error formulas are functions of frequency. Unknown true values of desired quantities are replaced by measured values when one applies these results to evaluate the random errors in actual measured data. </p><p>9.1.2 Covariance Formulas </p><p>A number of basic covariance error formulas will now be derived. By the definition in Equation (3.34), for two unbiased "raw" estimates A and B, where E[A] = A and E[B] = B, their covariance is </p><p>Cov(A, ) = [()()] = E[{A-A)(B-B)} = E[AB]-AB (9 .34) </p><p>where the increments AA (AA) and AB = (B B). The estimates A and are said to be uncorrelated when Cov(A, B) 0. </p><p>Table 9.1 Normalized Random Errors for Spectral Estimates </p><p>Estimate Normalized Random Error, e </p><p>Gif), Gyy{f) J _ </p><p>\Gxy(f)\ 1 </p></li><li><p>294 STATISTICAL ERRORS IN ADVANCED ESTIMATES </p><p>The application of Equation (9.17) shows that </p><p>EiCtyQty] = ^E[(XRYR +X,Y,)(XRY,-XIYR)} = 2 C ^ (9.35) </p><p>Hence, by Equation (9.34), at every frequency fk </p><p>C O V ( Q , Qxy) = CtyQxy </p><p>Similarly, one can verify that </p><p>(9.36) </p><p>Also, </p><p>Hence, </p><p>Cov(G, Cty) = G^Ciy Cov(G, 4 y ) = G &amp; y </p><p>C O V ( G W , Cxy) = GyyCxy COV (Gyy , Q^) = GyyQ^y </p><p>EiG^Gty] = ^E[(X2R+XJ)(Y2</p><p>R + Y2)] = G^Gyy + \Gryl2 </p><p>C o v ( G x l , G x y ) = \Gxyl2 = ylyG^Gyy </p><p>The following covariance results will now be proved. </p><p>CO\(G, ) = 0 COV(Oyy, ) = 0 </p><p>&gt;(|&lt; ^) = </p><p>(9.37) </p><p>(9.38) </p><p>(9.39) </p><p>(9.40) (9.41) </p><p>In words, is uncorrelated with G^, Gyy, and |G^,| at every frequency fk. To prove Equation (9.40), note that </p><p>^xy </p><p>as illustrated below. Then, differential increments of both sides yield </p><p>CxyAQty-QxyACxy se^dxyAdry </p><p>C2 </p><p>^xy where </p><p>Thus </p><p>sec 2 0 = C2 </p><p>*~xy </p><p>A a CxyAQxy QxyACxy M w </p><p>I Gey I </p><p>(9.42) </p></li><li><p>CROSS-SPECTRAL DENSITY FUNCTION ESTIMATES 295 </p><p>Now, one can set </p><p> Qxy &amp;Qxy Qxy~Qxy ACxy = Cxy Cxy </p><p>Then, using Equation (9.37), </p><p> &gt; ( , ^ ) = [ ( ) ( ^ ) ] </p><p>w T ^ C o v ( G , G ^ ) - - ^ _ C o v ( G e , C x y ) = 0 </p><p>\Gxy\ \Gxy\ </p><p>Similarly </p><p>C o v i G j y , ^ ) = 0 This proves Equation (9.40). </p><p>To prove Equation (9.41), note that </p><p>icy 2 = </p><p>Taking differential increments of both sides yields </p><p>iGrylAlG^I CxyACxy + QxyAQxy Thus </p><p>Now, Equations (9.42) and (9.43) give </p><p>3(\\,) = [{\\)()] </p><p>J - j E^AC^ + ^ ^ ^ ^ - ^ ^ ) ] </p><p>* { ( ^ - ^ ^ , ^ ) </p><p>+ C ^ G ^ ( V a r [ f 2 j - V a r [ C ^ ) } = 0 </p><p>This proves Equation (9.41). From Equation (9.43), one can also derive the following covariance results: </p><p>Cov(G, |G^|) = [(AG)(A|G^|)] G j G ^ I (9.44) </p><p>C o v ( G w , |G^|) = [(AG W )(A|G^|)] Gyy I Gjy I </p><p>C o v ( 4 y , \GV\) = [ ( , ) ( | &gt;M ~ </p><p>C ^ l G ^ | ( l + y 2 y ) </p><p>Q x y l ^ K l + y j ) ,2 (9-45) </p><p>2y: </p></li><li><p>296 STATISTICAL ERRORS IN ADVANCED ESTIMATES </p><p>Equation (9.43) offers a way to derive an approximate formula for Var[|G^|] as follows: </p><p>Var[|G^|] = (A |G|) 1 </p><p>'xy\ </p><p>{CtyACty + QtyAQty) </p><p> {% Var(C^) + 2 C ^ Cov(C^, Q^) + Q% Var(f i^)} </p><p>"T GwOyv --\G*y\ 2y</p><p>2 </p><p>I XV </p><p>(9.46) </p><p>This approximate formula from Ref. 1, based on differential increments, is inferior to the direct derivation producing the more exact formula of Equation (9.28), namely, </p><p>Var | G r Jxy\ </p><p>fxy (9.47) </p><p>Note that the variance from Equation (9.47) will always be greater than the variance from Equation (9.46) because </p><p> l+yl </p><p>y2 Ixy </p><p>&gt; 2y2 IXV </p><p>for all yly &lt; 1 </p><p>Consider the coherence function defined by </p><p>Jxy\ "^xy f~&lt; (- </p><p>^xx^yy </p><p>Logarithms of both sides give </p><p>(9.48) </p><p>log y x y = 2 l og |G^ | - log G - l o g G yy </p><p>Then, differential increments of both sides show that </p><p>Ay% 2A\Gxy\ AGxx AG-yy </p><p>ixy Jyy </p><p>where </p><p>A)?xy 7xy 7xy </p><p>AGxx = Gxx Gxx </p><p>A\Gxy\ - \Gxy\-Gxy </p><p>^(jyy (jyy (jyy </p></li><li><p>CROSS-SPECTRAL DENSITY FUNCTION ESTIMATES 297 </p><p>Cov(C^, |G^|)= ^ 2 </p><p>CovL, |G^|) = xy, l^xyi; </p><p>Thus </p><p>^ - * { ^ - ^ } Now, using Equations (9.40) and (9.41), it follows directly from Equation (9.49) that </p><p>C o v ( % , ^ ) = 0 (9.50) </p><p>In words, the estimates y2^ and are uncorrelated at every frequency fk. A summary is given in Table 9.2 of some of the main covariance formulas derived </p><p>in this section. </p><p>9.1.3 Phase Angle Estimates </p><p>A formula for the variance of "raw" phase estimates (expressed in radians) can be derived from Equation (9.42) as follows: </p><p>Var[0] = \(^)2] * - ^ E U C ^ Q ^ - ^ A C ^ ) 2 </p><p>L J I G^ I L </p><p>1 {c% Var(^)-2Cxy!2xy Cov(Cxy, Qxy) + 0% Var(Cxy)} (9.51) </p><p>2y2 xy </p><p>Table 9.2 Covariance Formulas for Various Estimates </p><p>C0V (C A 7 ,G^,) = CxyQxy </p><p>COV(G, Cxy) = GxjGxy </p><p>Cov{GJZ,Qxy) = GQXy </p><p>Cov(G,G w ) = \Gxyf = y^G^G^ Cov(G,fl^) = 0 </p><p>Cov(|G v | ,e v) = 0 </p><p>C o v ( ^ , e v ) = 0 </p><p>Cov(Gxx,|Gxy|) = G|Gxy| </p></li><li><p>298 STATISTICAL ERRORS IN ADVANCED ESTIMATES </p><p>For "smooth" estimates of from nd averages, one obtains </p><p>sA[h]=imjp^ (9.52) This gives the standard deviation of the phase estimate ^,. Here, one should not use the normalized random error because may be zero. The result in Equation (9.52) for the cross-spectrum phase angle, like the result in Equation (9.33) for the cross-spectrum magnitude estimate, varies with frequency. Independent of nd for any nd &gt; 1, note that s.d.[r)^] approaches zero as y2^ approaches one. </p><p>Example 9.1. Illustration of Random Errors in Cross-Spectral Density Estimate. Consider a pair of sample records x(t) and y(i), which represent stationary Gaussian random processes with zero mean values. Assume that the coherence function between the two random processes at a frequency of interest is y2^ = 0.25. If the cross-spectral density function Gxy(f) is estimated using nd = 100 averages, determine the normalized random error in the magnitude estimate and the standard deviation of the phase estimate at the frequency of interest. </p><p>From Equation (9.33), the normalized random error in the magnitude of the cross-spectral density function estimate will be </p><p>41^1]=^^ = 0.20 Note that this is twice as large as the normalized random errors in the autospectra estimates given by Equation (9.32). The standard deviation of the phase estimate is given by Equation (9.52) as </p><p>(0 75 ) 1 , / 2 s . d . [ 0 j = v ' ;. = 0.12rad </p><p>or about 7. </p><p>9.2 SINGLE-INPUT/OUTPUT MODEL ESTIMATES </p><p>Consider the single-input/single-output linear model of Figure 9.1 where </p><p>X(f) = Fourier transform of measured input signal x(t), assumed noise free </p><p>Y(f) = Fourier transform of measured output signal y(i) = v(f) + n{t) </p><p>X(f) Hx {f) j ( V(f) ' v </p><p>Yin </p><p>Figure 9.1 Single-input/single-output linear model. </p></li><li><p>S I N G L E - I N P U T / O U T P U T M O D E L E S T I M A T E S 299 </p><p>V(f) = Fourier transform of computed output signal v(r) </p><p>N(f) = Fourier transform of computed output noise n{f) </p><p>Hxyif) = frequency response function of optimum constant-parameter linear </p><p>system estimating y(t) from x{t) </p><p>Assume that x(t) and y{t) are the only records available for analysis and that they are representative members of zero mean value Gaussian random processes. Data can be either stationary random data or transient random data. For definiteness here, stationary data will be assumed and spectral results will be expressed using one-sided spectra. Normalized error formulas are the same for one-sided or two-sided spectra. </p><p>From Section 6.1.4, the following equations apply to this model in Figure 9.1 to estimate various quantities of interest. The optimum frequency response function estimate is </p><p>Gxx{j) </p><p>where Go-(/) and G^f) are "smooth" estimates of the input autospectral density function and the input/output cross-spectral density function, respectively. The associated ordinary coherence function estimate is </p><p>f (/) = . \G*yU)\2 ( 9 . 5 4 ) </p><p>where Gyy(f) is a "smooth" estimate of the output autospectral density function. The coherent output spectrum estimate is </p><p>Gw(/) = l * M / ) | 2 G ( / ) = llyifiGyyif) (9.55) </p><p>The noise output spectrum estimate is </p><p>Gm(f) = [l-fv(f)]Gyy(f) (9.56) </p><p>It is also known from Equation (6.61) that </p><p>Gyy(f) = G(f) + Gm(f) (9.57) </p><p>since v(r) and n(t) are uncorrelated with Gvn(f) = 0 when Hxy(f) is computed by Equation (9.53). </p></li><li><p>300 STATISTICAL ERRORS IN ADVANCED ESTIMATES </p><p>In polar form, the frequency response function can be expressed as </p><p>where </p><p>HM = \HM)\e-^U) (9-58) </p><p>IG (f)\ \HM)\ = - = system gain factor estimate (9.59) </p><p>GM) </p><p>) = tan" 1 QM) </p><p>CM) system phase factor estimate (9.60) </p><p>This quantity ) is the same as the phase angle &amp;M) m Gxy(f),u ^ also the phase that would be assigned to y^f) when is defined as the complex-valued function given by </p><p>Uf) = \%{)\*-*() = + \/y2Jf)e^f) (9.61) </p><p>Note that all of the above "smooth" estimates can be computed from the original computed "smooth" estimates of G^f), Gyy(f) and Gxy(f). </p><p>Random error formulas for all of these quantities will be derived in terms of the unknown true coherence function y2^ ( / ) and the number of indepen...</p></li></ul>