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fitKT.m

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+function [T90,eps,tn,zeta,sigma,fn] = fitKT(t,y,guessEnvelop,guessKT,varargin)
+% [T90,eps,tn,zeta,sigma,fn] = fitKT(t,y,fs,guessEnvelop,guessKT,
+% varargin) fit the nonstationary Kanai傍ajimi model to ground acceleration
+% records. 
+% 
+% 
+% INPUTS 
+% 
+% y: size: [ 1 x N ] : aceleration record
+% t: size: [ 1 x N ] : time vector
+% guessEnvelop: [1 x 3 ]: first guess for envelop function
+% guessKT: [1 x 3 ]: first guess for Kanai傍ajimi spectrum
+% varargin:
+%      'F3DB'        - cut off frequency for the low pass filter
+%      'TolFun'      - Termination tolerance on the residual sum of squares.
+%                      Defaults to 1e-8.
+%      'TolX'        - Termination tolerance on the estimated coefficients
+%                      BETA.  Defaults to 1e-8.
+%      'dataPlot'        - 'yes': shows the results of the fitting process
+% 
+% OUTPUTS 
+% 
+% sigma: [1 x 1 ]: Fitted standard deviation of the excitation.
+% fn: [1 x 1 ]:  Fitted dominant frequency of the earthquake excitation (Hz).
+% zeta: [1 x 1 ]: Fitted bandwidth of the earthquake excitation.
+% f: [ 1 x M ]: Fitted frequency vector for the Kanai-tajimi spectrum.
+% T90: [1 x 1 ]: Fitted value at 90 percent of the duration.
+% eps: [1 x 1 ]: Fitted normalized duration time when ground motion achieves peak.
+% tn: [1 x 1 ]: Fitted duration of ground motion.
+% 
+% 
+% EXAMPLE: this requires the function seismSim.m
+% 
+% f = linspace(0,40,2048);
+% zeta = 0.3;
+% sigma = 0.3;
+% fn =5;
+% eta = 0.3;
+% eps = 0.4;
+% tn = 30;
+% [y,t] = seismSim(sigma,fn,zeta,f,eta,eps,tn);
+% guessEnvelop=[0.33,0.43,50];
+% guessKT = [1,1,5];
+% [eta,eps,tn,zeta,sigma,fn] = fitKT(t,y,guessEnvelop,guessKT,...
+% 'dataPlot','yes');
+% 
+% see also seismSim.m
+% Author: Etienne Cheynet - modified: 23/04/2016
+
+%% inputParser
+p = inputParser();
+p.CaseSensitive = false;
+p.addOptional('f3DB',0.05);
+p.addOptional('tolX',1e-8);
+p.addOptional('tolFun',1e-8);
+p.addOptional('dataPlot','no');
+p.parse(varargin{:});
+tolX = p.Results.tolX ;
+tolFun = p.Results.tolFun ;
+f3DB = p.Results.f3DB ;
+dataPlot = p.Results.dataPlot ;
+% check number of input
+narginchk(4,8)
+
+%% Get envelop parameters
+dt = median(diff(t));
+
+h1=fdesign.lowpass('N,F3dB',8,f3DB,1/dt);
+d1 = design(h1,'butter');
+Y = filtfilt(d1.sosMatrix,d1.ScaleValues, abs(hilbert(y)));
+Y = Y./max(abs(Y));
+
+options=optimset('Display','off','TolX',tolX,'TolFun',tolFun);
+coeff1 = lsqcurvefit(@(para,t) Envelop(para,t),guessEnvelop,t,Y,[0.01,0.01,0.1],[3,3,100],options);
+
+eps = coeff1(1);
+T90 = coeff1(2);
+tn = coeff1(3);
+
+
+
+
+
+%% Get stationnary perameters for the spectrum
+E =Envelop(coeff1,t);
+x = y./E; % there may be better solution than this one, but I don't have better idea right now.
+x(1)=0;
+
+% calculate the PSD
+[PSD,freq]=pmtm(x,7/2,numel(t),1/median(diff(t)));%%
+coeff2 = lsqcurvefit(@(para,t) KT(para,freq),guessKT,freq,PSD,[0.01,0.01,1],[5,5,100],options);
+zeta = coeff2(1);
+sigma = coeff2(2);
+fn = coeff2(3);
+
+
+
+%% dataPLot (optional)
+if strcmpi(dataPlot,'yes'),
+    spectra = KT(coeff2,freq);
+
+    figure
+    subplot(211)
+    plot(t,y./max(abs(y)),'k',t,Envelop(coeff1,t),'b',t,Y,'r')
+    legend('original data','Fitted envelop','measured envelop')
+    title([' T_{90} = ',num2str(coeff1(2),3),'; \epsilon = ',...
+        num2str(coeff1(1),3),'; t_{n} = ',num2str(coeff1(3),3)]);
+    xlabel('time (s)')
+    ylabel('ground acceleration (m/s^2)')
+    axis tight
+
+    subplot(212)
+    plot(freq,PSD,'b',freq,spectra,'r')
+    legend('Measured','Fitted envelop')
+    legend('original data','Fitted Kanai-Tajimi spectrum')
+    title([' \zeta = ',num2str(coeff2(1),3),';  \sigma = ',...
+        num2str(coeff2(2),3),';  f_{n} = ',num2str(coeff2(3),3)]);
+    xlabel('freq (Hz)')
+    ylabel('Acceleration spectrum (m^2/s)')
+    axis tight
+    set(gcf,'color','w')
+end
+
+% 
+%% NESTED FUNCTIONS
+% 
+
+    function E = Envelop(para,t)
+        eps0 = para(1);
+        eta0 = para(2);
+        tn0 = para(3);
+        b = -eps0.*log(eta0)./(1+eps0.*(log(eta0)-1));
+        c = b./eps0;
+        a = (exp(1)./eps0).^b;
+        E = a.*(t./tn0).^b.*exp(-c.*t./tn0);
+    end
+    function S = KT(para,freq)
+        zeta0 = para(1);
+        sigma0 = para(2);
+        omega0 = 2*pi.*para(3);
+        w =2*pi*freq; 
+        s0 = 2*zeta0*sigma0.^2./(pi.*omega0.*(4*zeta0.^2+1));
+        A = omega0.^4+(2*zeta0*omega0*w).^2;
+        B = (omega0.^2-w.^2).^2+(2*zeta0*omega0.*w).^2;
+        S = s0.*A./B; % single sided PSD
+    end
+end
+

seismSim.m

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+function [y,t] = seismSim(sigma,fn,zeta,f,T90,eps,tn)
+% [y,t] = seismSim(sigma,fn,zeta,f,T90,eps,tn) generate one time series
+%  corresponding to acceleration record from a seismometer. 
+% The function requires 7 inputs, and gives 2 outputs. The time series is 
+% generated in two steps: First, a stationary process is created based on 
+% the Kanai-Tajimi spectrum, then an envelope function is used to transform
+% this stationary time series into a non-stationary record. For more 
+% information, see [1-3].
+% 
+% References
+% 
+%  [1] Lin, Y. K., & Yong, Y. (1987). Evolutionary Kanai-Tajimi earthquake models. Journal of engineering mechanics, 113(8), 1119-1137.
+%  [2] Rofooei, F. R., Mobarake, A., & Ahmadi, G. (2001). 
+%  Generation of artificial earthquake records with a nonstationary 
+%  Kanai-Tajimi model. Engineering Structures, 23(7), 827-837.
+%  [3] Guo, Y., & Kareem, A. (2016). 
+%  System identification through nonstationary data using Time-requency Blind
+%  Source Separation. Journal of Sound and Vibration, 371, 110-131.
+% 
+%  
+% INPUTS 
+% 
+% sigma: [1 x 1 ]: standard deviation of the excitation.
+% fn: [1 x 1 ]: dominant frequency of the earthquake excitation (Hz).
+% zeta: [1 x 1 ]:  bandwidth of the earthquake excitation.
+% f: [ 1 x M ]: frequency vector for the Kanai-tajimi spectrum.
+% T90: [1 x 1 ]: value at 90 percent of the duration.
+% eps: [1 x 1 ]: normalized duration time when ground motion achieves peak.
+% tn: [1 x 1 ]: duration of ground motion.
+% 
+%  
+% OUTPUTS 
+% 
+% y: size: [ 1 x N ] : Simulated aceleration record
+% t: size: [ 1 x N ] : time
+% 
+%  
+% EXAMPLE:
+%  
+% f = linspace(0,40,2048);
+% zeta = 0.3;
+% sigma = 0.9;
+% fn =5;
+% T90 = 0.3;
+% eps = 0.4;
+% tn = 30;
+% [y,t] = seismSim(sigma,fn,zeta,f,T90,eps,tn);
+% figure
+% plot(t,y);axis tight
+% xlabel('time(s)');
+% ylabel('ground acceleration (m/s^2)')
+% 
+% see also fitKT.m
+% Author: Etienne Cheynet - modified: 23/04/2016
+
+%% Initialisation
+w = 2*pi.*f;
+fs = f(end); dt = 1/fs;
+f0= median(diff(f));
+Nfreq = numel(f);
+t = 0:dt:dt*(Nfreq-1);
+
+%% Generation of the spectrum S
+fn = fn *2*pi; % transformation in rad;
+s0 = 2*zeta*sigma.^2./(pi.*fn.*(4*zeta.^2+1));
+A = fn.^4+(2*zeta*fn*w).^2;
+B = (fn.^2-w.^2).^2+(2*zeta*fn.*w).^2;
+S = s0.*A./B; % single sided PSD
+
+
+%% Time series generation - Monte Carlo simulation
+A = sqrt(2.*S.*f0);
+B =cos(w'*t + 2*pi.*repmat(rand(Nfreq,1),[1,Nfreq]));
+x = A*B; % stationary process
+
+%% Envelop function E
+b = -eps.*log(T90)./(1+eps.*(log(T90)-1));
+c = b./eps;
+a = (exp(1)./eps).^b;
+E = a.*(t./tn).^b.*exp(-c.*t./tn);
+
+%% Envelop multiplied with stationary process to get y
+y = x.*E;
+
+end