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Settings\stephen.r.chinn\Desktop\Est&tHomeworRRVSRecentWindowsAnalysis.ihfGraph3:wfr_FFT_realGraph4:wave0_HistGraph5:varCRLB,wVARIANCEGraph6:westGraph7:sdCRLB_dB,wSD_dBGraph8:wfrHelp BrowserIgor Reference.ihf 4Misc_EndpXOPState_Start`pANOVASupport7:sdCRLB_dB,wSD_dBData Browser5:varCRLB,wVARIANCEGizmoGraph3:wfr_FFT_real4XOPState_Endp.M9?]B?`=sdG M?varGwכ ?varGAgІ ?varG2&І ?omegap?V_FlagV_PeakLoc&)ؼe?V_PeakValqCWѿV_LeadingEdgeLoc{#/h?V_TrailingEdgeLoc"l,rz?V_PeakWidthR\5l?V_chisq8|i~@V_numNaNsV_numINFsV_npnts@V_nterms@V_nheldV_startRowV_endRow@V_startColV_endColV_startLayerV_endLayerV_startChunkV_endChunkSNRdBG@S_info~l{|{|pDATE=Thu, Apr 05, 2007;TIME=9:28:11 AM;FUNCTION=gauss;AUTODESTWAVE=fit_wsum;YDATA=root:wsum;XDATA=_calculated_,;"teststat() 0.951116 0.932094 teststat() 0.945936 0.927017 teststat() 0.854794 0.837698 teststat() 1.55999 1.5288 teststat() 0.847033 0.830092 0.830092 teststat() 0.723096 0.708634 0.708634 teststat() 0.762776 0.747521 0.747521 testfft() display wr testfft() testfft() display wr display fwr testfft() Display fwr ModifyGraph cmplxMode=1 ModifyGraph mode=4,marker=19 testfft() testfft() testfft() testfft() testfft() testfft() testfft() testfft() testfft() testfft() display fwreal testfft() testfft() V_npnts= 26; V_numNaNs= 0; V_numINFs= 0; V_avg= 5.47205; V_Sum= 142.273; V_sdev= 20.1959; V_rms= 20.5458; V_adev= 9.19362; V_skew= 3.80421; V_kurt= 15.0516; V_minloc= 1.13097; V_maxloc= 1.50796; V_min= -8.72916; V_max= 99.4167; V_startRow= 0; V_endRow= 25; print testfft() 1.50796 print testfft() 1.50796 print testfft2() 4.64956 print testfft2() -0.24 display fwrealc testfft2() print numpnts(fwrealc) 50 testsum() display wsum ModifyGraph mode=4,marker=19,msize=2 ModifyGraph mode=0 testsum() findest(4) findest(2) print findest(4) 1.50796 findest(2) print findest(2) 1.50796 print findest(1) 1.49226 print findest(1) 1.47655 print findest(1) 1.50796 print findest(1) 1.49226 print findest(1) 1.50796 print findest(1) 1.46869 print findest(1) 1.49226 print findest(1) 1.50011 print findest(5) 1.49226 print findest(3) 1.50011 print findest(0) 1.4844 print findest(-3) 1.47655 print GetEstStat(3) 0.0084998 print GetEstStat(0) 0.0105886 display west makeSDfunction() display wSDofSNR ModifyGraph log(left)=1 ModifyGraph log=0 wsdofsnr=10*log(wsdofsnr) makeCRLB() appendtograph wCRLBofSNR makeCRLB() makeCRLB() makeCRLB() makeCRLB() makeCRLB() makeSDfunction() display wsdofsnrdb modify log(left)=1 makeCRLB() appendtograph wcrlbofsnrdb makeSDfunction() teststat() 0.699229 0.685244 0.685244 print sdG 0.836199 teststat() 0.990594 0.990575 0.990575 teststat() 1.0027 1.00268 1.00268 teststat() 0.991061 0.991041 0.991041 teststat() 1.00063 1.00061 1.00061 ShowInfo HideInfo makecrlb() makeSDfunction() makeSDfunction() makeCRLB() AppendToGraph sdCRLB makeCRLB() makeCRLB() SetAxis left 0.001,1 makeCRLB() display sdCRLB_dB makeSDfunction() AppendToGraph wSDofSNRdB ModifyGraph height={Aspect,1} makeSDfunction() makeSDfunction() Rsum(1,1) makeSDfunction() makeSDfunction() teststat() 1.00259 1.00257 1.00257 Make/N=94/O wave0_Hist;DelayUpdate Histogram/B=4 wave0,wave0_Hist;DelayUpdate Display wave0_Hist Make/N=94/O wave0_Hist;DelayUpdate Histogram/P/B=4 wave0,wave0_Hist ShowInfo HideInfo findest(0) findest(10) ShowInfo HideInfo findest(10) makeSDfunction() ShowInfo makeSDfunction() makeSDfunction() makeSDfunction() HideInfo print omegap 1.5 findest(10) makeSDfunction() ShowInfo makeSDfunction() SetAxis/A SetAxis/A findest(20) findest(20) findest(20) findest(20) findest(20) SetAxis/A findest(10) FindPeak wsum V_Flag= 0; V_PeakLoc= 0.00265443; V_PeakVal= -0.274557; V_LeadingEdgeLoc= 0.00295234; V_TrailingEdgeLoc= 0.00645688; V_PeakWidth= 0.00350455; CurveFit/M=2/W=0 gauss, wsum/D Fit converged properly fit_wsum= W_coef[0]+W_coef[1]*exp(-((x-W_coef[2])/W_coef[3])^2) W_coef={1.8046,117.3,1.4996,0.04013} V_chisq= 83942.6; V_npnts= 1600; V_numNaNs= 0; V_numINFs= 0; V_startRow= 0; V_endRow= 1599; V_startCol= 0; V_endCol= 0; V_startLayer= 0; V_endLayer= 0; V_startChunk= 0; V_endChunk= 0; W_sigma={0.186,1.76,0.00049,0.000699} Coefficient values one standard deviation y0 =1.8046 0.186 A =117.3 1.76 x0 =1.4996 0.00049 width =0.04013 0.000699 findest(10) Fit converged properly fit_wsum= W_coef[0]+W_coef[1]*exp(-((x-W_coef[2])/W_coef[3])^2) W_coef={1.8979,118.09,1.4991,0.040455} V_chisq= 24049.5; V_npnts= 400; V_numNaNs= 0; V_numINFs= 0; V_startRow= 0; V_endRow= 399; V_startCol= 0; V_endCol= 0; V_startLayer= 0; V_endLayer= 0; V_startChunk= 0; V_endChunk= 0; W_sigma={0.399,3.77,0.00105,0.0015} Coefficient values one standard deviation y0 =1.8979 0.399 A =118.09 3.77 x0 =1.4991 0.00105 width =0.040455 0.0015 makeSDfunction() findest(10) findest(0) findest(-3) findest(-5) findest(-5) findest(-5) findest(-5) makeSDfunction() NewPanel ShowTools/A variable/G SNRdBG ValDisplay valdispSNR title="SNR(dB)",size={100,20},value=SNRdBG HideTools/A makeSDfunction() ShowTools/A HideTools/A makeSDfunction() getestsd(1) getestsd(1) print getestsd(1) 56.49 print 10*log( 56.49) 17.5197 display west getestsd(3) print sqrt(fvar(west, omegap)) 0.00153335 print 10*log(sqrt(fvar(west, omegap))) -28.1436 SetAxis/A makeSDfunction() makeSDfunction() RemoveFromGraph fit_wsum ModifyGraph mode=3,marker=19,msize=2 makeSDfunction() SetAxis left 1.45,1.55 makeSDfunction() makeSDfunction() makecrlb() display varcrlb modify log(left)=1 display varCRLB modify log(left)=1 makeVAR() appendtograph wVARofSNRdB makeVAR() makeVAR() makeCRLB() makeVAR() makeVAR() makeVAR() killwaves/A/Z makeCRLB() display varCRLB modify log(left)=1 display sdCRLB_dB ModifyGraph grid=1,gridStyle=3 ModifyGraph grid=1,gridStyle=3 makeVAR() makeVAR() appendtograph wVAR SetAxis/A makeVAR() RemoveFromGraph wvar makeVAR() display wvariance modify log(left)=1 appendtograph wVARIANCE appendtograph wSD_dB ModifyGraph lsize=2,lstyle(wSD_dB)=2 ModifyGraph lstyle(sdCRLB_dB)=2,lstyle(wSD_dB)=0 Label left "Standard Deviation (dB)";DelayUpdate Label bottom "SNR (dB)" Legend/C/N=text0/J/H=50/A=MC "\\s(wSD_dB) numerical estimate\r\\s(sdCRLB_dB) Cramer-Rao Lower Bound" TextBox/C/N=text1/F=0/H=50/A=MT/E "Problem 2 - Comparison of CRLB and numerical analysis" TextBox/C/N=text1 "Problem 2 - Comparison of CRLB and numerical analysis standard deviations" Label left "Variance";DelayUpdate Label bottom "SNR (dB)" TextBox/C/N=text0/F=0/H=50/A=MT/E "Problem 2 - Comparison of CRLB and numerical analysis VARIANCES" ModifyGraph lsize=2,lstyle(varCRLB)=2 Legend/C/N=text1/J/H=50/A=MT/E "\\s(wVARIANCE) numeric analysis\r\\s(varCRLB) Cramer-Rao Lower Bound" Legend/C/N=text1/J/A=MC/X=4.97/Y=37.72/E=0 Label left "Frequency";DelayUpdate Label bottom "Trial Number" ModifyGraph mode=2,lsize=4 ModifyGraph lsize=3 ModifyGraph lsize=4 TextBox/C/N=text0/F=0/H=50/A=MT/E "Problem 2 - 250 Trials of ML Frequency Estimator" makeVAR() ModifyGraph lsize=2 Label left "L(\\F'Symbol'w\\]0)";DelayUpdate Label bottom "\\F'Symbol'w\\]0" Label left "Likelihood function";DelayUpdate Label bottom "Frequency \\F'Symbol'w\\]0" TextBox/C/N=text0/F=0/H=50/A=MT/E "Problem 2 - Instantiation of likelihood function to be maximized, SNR = 10dB" makeVAR() makeVAR() makeVAR() SetAxis/A SetAxis left 1.48,1.52 TextBox/C/N=text0 "Problem 2 - 250 Trials of ML Frequency Estimator, SNR = 10dB" SetAxis/A/N=1 left ModifyGraph grid=2 ModifyGraph grid=2 TextBox/C/N=text0 "Problem 2 - Comparison of CRLB and numerical analysis VARIANCES\r\\JClog(linear data) vs SNR(dB)" FFTtest(sqrt(20)) display wfr FFT/OUT=1/PAD={4096}/DEST=wfr_FFT wfr display wfr_FFT ModifyGraph cmplxMode=1 ShowInfo print deltax(wfr_FFT) 0.000244141 setscale/P x, 0,deltax(wfr_FFT)*2*pi,"" wfr_FFT FFTtest(sqrt(20)) FFTtest(sqrt(20)) HideInfo FFTtest(sqrt(2)) FFTtest(sqrt(2)) display wfr_FFT_real FFTtest(sqrt(2)) FFTtest(sqrt(2)) FFTmax(sqrt(2)) print FFTmax(sqrt(2)) 1.50484 print FFTmax(sqrt(2)) 1.50637 print FFTmax(sqrt(2)) 1.50484 print FFTmax(sqrt(2)) 1.4941 print FFTmax(sqrt(2)) 1.50253 print FFTmax(sqrt(2)) 1.50484 makeVAR() makeVAR() appendtograph wfr_FFT_real RemoveFromGraph wsum ModifyGraph lsize=2 makeVAR() SetAxis/A/N=1 left ModifyGraph mode(wSD_dB)=4,marker(wSD_dB)=19,msize(wSD_dB)=2 ModifyGraph lstyle(sdCRLB_dB)=3,rgb(sdCRLB_dB)=(0,0,65280) ModifyGraph lstyle(sdCRLB_dB)=8 TextBox/C/N=text0 "Problem 2 - Instantiation of likelihood function for argmax, SNR = 10dB" TextBox/C/N=text0 "Problem 2 - Instantiation of argmax likelihood function, SNR = 10dB" makeVAR() makeVAR() makeVAR() makeVAR() makeVAR() Legend/C/N=text0/J "\\s(wSD_dB) numerical estimate\r\\s(sdCRLB_dB) Cramr-Rao Lower Bound" Rxwr2?::/&_@?6f83kg)4T@x(g?"V_Sb(r@Mn#@,;=F1O ?W6@qx9 B%F 0@C6 @̑/㿢v b+?`צ@(0Pv1ƿFD 7mI*@"w??b2Q@Pׂm@Ce7](oP @n@D[TH]ts>mP?%@58@J\! φ y(?e5@ NP))8}s?y?@ Vۓ6&c?^NKGfwr'?0909Q#o@dp 2A AK@>?qIR@SO@o@5x=KAXnx? 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DefaultFont "Arial" MoveWindow/P 256.5,74,919.5,680 Graph8() Graph4() MoveWindow/C 12,569,483.75,699.5 Graph6() Graph3() Graph5() Graph7() Window Graph7() : Graph PauseUpdate; Silent 1 // building window... Display /W=(48,92,537.75,512) sdCRLB_dB,wSD_dB ModifyGraph mode(wSD_dB)=4 ModifyGraph marker(wSD_dB)=19 ModifyGraph lSize=2 ModifyGraph lStyle(sdCRLB_dB)=8 ModifyGraph rgb(sdCRLB_dB)=(0,0,65280) ModifyGraph msize(wSD_dB)=2 ModifyGraph grid=1 ModifyGraph tick=2 ModifyGraph mirror=1 ModifyGraph minor=1 ModifyGraph lblMargin(left)=15,lblMargin(bottom)=6 ModifyGraph standoff=0 ModifyGraph gridStyle=3 ModifyGraph lblLatPos(left)=3 Label left "Standard Deviation (dB)" Label bottom "SNR (dB)" SetAxis/A/N=1 left Legend/N=text0/J/H=50/A=MC/X=20.56/Y=39.91 "\\s(wSD_dB) numerical estimate\r\\s(sdCRLB_dB) Cramr-Rao Lower Bound" TextBox/N=text1/F=0/H=50/A=MT/E "Problem 2 - Comparison of CRLB and numerical analysis standard deviations" EndMacro Window Graph5() : Graph PauseUpdate; Silent 1 // building window... Display /W=(24,59,513.75,478.25) varCRLB,wVARIANCE ModifyGraph lSize=2 ModifyGraph lStyle(varCRLB)=2 ModifyGraph grid=1 ModifyGraph log(left)=1 ModifyGraph tick=2 ModifyGraph mirror=1 ModifyGraph minor=1 ModifyGraph lblMargin(left)=15,lblMargin(bottom)=5 ModifyGraph standoff=0 ModifyGraph gridStyle=3 Label left "Variance" Label bottom "SNR (dB)" TextBox/N=text0/F=0/H=50/A=MT/E "Problem 2 - Comparison of CRLB and numerical analysis VARIANCES\r\\JClog(linear data) vs SNR(dB)" Legend/N=text1/J/H=50/A=MC/X=20.70/Y=14.43 "\\s(wVARIANCE) numeric analysis\r\\s(varCRLB) Cramer-Rao Lower Bound" EndMacro Window Graph3() : Graph PauseUpdate; Silent 1 // building window... Display /W=(335.25,290.75,949.5,700.25) wfr_FFT_real ModifyGraph lSize=2 ModifyGraph grid=2 ModifyGraph tick=2 ModifyGraph mirror=1 ModifyGraph minor=1 ModifyGraph lblMargin(left)=18,lblMargin(bottom)=9 ModifyGraph standoff=0 ModifyGraph lblLatPos(bottom)=-11 Label left "Likelihood function" Label bottom "Frequency \\F'Symbol'w\\]0" SetAxis/A/N=1 left TextBox/N=text0/F=0/H=50/A=MT/E "Problem 2 - Instantiation of argmax likelihood function, SNR = 10dB" EndMacro Window Graph6() : Graph PauseUpdate; Silent 1 // building window... Display /W=(398.25,44.75,951,460.25) west ModifyGraph mode=2 ModifyGraph marker=19 ModifyGraph lSize=4 ModifyGraph msize=2 ModifyGraph grid=2 ModifyGraph tick=2 ModifyGraph mirror=1 ModifyGraph minor=1 ModifyGraph lblMargin(left)=14,lblMargin(bottom)=12 ModifyGraph standoff=0 Label left "Frequency" Label bottom "Trial Number" SetAxis left 1.48,1.52 TextBox/N=text0/F=0/H=50/A=MT/E "Problem 2 - 250 Trials of ML Frequency Estimator, SNR = 10dB" EndMacro Window Graph4() : Graph PauseUpdate; Silent 1 // building window... Display /W=(5.25,42.5,399.75,251) wave0_Hist ModifyGraph tick=2 ModifyGraph mirror=1 ModifyGraph minor=1 ModifyGraph standoff=0 EndMacro Window Graph8() : Graph PauseUpdate; Silent 1 // building window... Display /W=(5.25,42.5,399.75,251) wfr ModifyGraph tick=2 ModifyGraph mirror=1 ModifyGraph minor=1 ModifyGraph standoff=0 EndMacro #pragma rtGlobals=1 // Use modern global access method. function makeCRLB() // Analytic CR lower bound formula from Fisher matrix variable K = 50 // observation vector length make/O/D/N=21 wA, wJ, varCRLB, sdCRLB_dB // steps of 1 dB, -10 to 10 setscale/P x, -10, 1,"" wA, wJ, varCRLB, sdCRLB_dB // 'x' variable is SNR(dB) wA = sqrt( 2 *(10^(x/10)) ) // linear amplitude in terms of SNR(dB); sigma = 1 wJ = 0.5 * (wA^2) * K * (K-1) * (2*K-1) / 6 // Fisher matrix (1x1) varCRLB = 1 / wJ // CRLB variance sdCRLB_dB = 10*log(sqrt(varCRLB)) // CRLB standard deviation expressed in dB end //-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- //--------------------------------------------NUMERICAL ANALYSIS------------------------------------------------------------------------------------------------------ function makeVAR() // make vectors of estimated variances and std. dev's. at different SNRs variable/G omegap = 1.5 // given frequency parameter (globally defined) make/O/D/N=21 wAmp, wVARIANCE, wSD_dB // signal amplitude, variance, standard deviation setscale/P x, -10, 1,"" wAmp, wVARIANCE, wsD_dB // increment 'x' in 1 dB steps, starting at -10 wAmp = sqrt( 2 *(10^(x/10)) ) // 'x' is SNR in dB; wAmp(x) is the corresponding sinusoid amplitude wVARIANCE = GetVAR(wAmp) // vector result for estimate of variance of frequency estimator // GetVAR() and its sub-functions do the heavy lifting wSD_dB = 10*log(sqrt(wVARIANCE)) // vector result for estimate of standard deviation of requency estimator, in dB end function GetVAR(A) // return the variance of the multiple trial results for one given signal amplitude variable A NVAR omegap variable Nt = 250 // number of trials make/O/D/N=(Nt) west west = FFTmax(A) // make a vector of frequency estimator instantiations with signal amplitude A DoUpdate // update graphs so progress can be visualized at each SNR return fvar(west, omegap) // evaluate and return an estimate of the variance of these instantiations; function below end function FFTmax(A) // create instantiation of R, and find its spectrum using FFT techniques variable A // 'A' is linear amplitude variable K = 50 // number of samples (points) in vector 'R' NVAR omegap // global frequency parameter for creating instantiations make/O/D/N=(K) wfr wfr = A * cos(omegap*p) + gnoise(1) // 'wr' is the instantiation vector; point number 'p'; standard deviation = 1; A varies with SNR // gnoise(1) is random GRV generator at each call, with standard deviation = 1 FFT/OUT=2/PAD={8192}/DEST=wfr_FFT_real wfr // use FFT to compute inner product of cos(omega*p) and instantiation, ' wfr ' // at all 'omega' simultaneously; transform result is complex // pad the 50-point input function with zeros to obtain fine frequency resolution // set OUT=2 mode for REAL output; use real part of the FFT to obtain the cos() transform setscale/P x, 0,deltax(wfr_FFT_real)*2*pi,"" wfr_FFT_real // modify Igor's FFT output frequency scale in Hz to radians wavestats/Q wfr_FFT_real // find the maximum location of the cos transform (likelihood function) return V_maxloc // and return it as function output end function fvar(inwave, mu) // returns estimate of VARIANCE of estimator trials; mu = estimator mean (here, omegap) wave inwave // vector of frequency estimator trials variable mu // input parameter for calculating original signal vector instantiations variable Ntr = numpnts(inwave) // extract number of trials from input vector make/O/D/N=(Ntr) w1, w2, wvar w1 = ((inwave-mu)^2) w2 = (inwave-mu) return sum(w1) / Ntr - (sum(w2)/Ntr)^2 // numerical variance of estimator trials (at a fixed SNR) end