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figure_27.m
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figure_27.m
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% Code to generate Figure A2 of the echo statistics tutorial
%
% Same as Fig. A2, except on a log-log scale.
%
% Author: Kyungmin Baik | [email protected] | KRISS
clear all
close all
FigHandle = figure(1);
set(FigHandle, 'Position', [100, 100, 850, 900]);
subplot('Position', [3/48, 11/16, 10/24, 0.275])
subplot('Position', [27/48, 11/16, 10/24, 0.275])
subplot('Position', [3/48, 5.75/16, 10/24, 0.275])
subplot('Position', [27/48, 5.75/16, 10/24, 0.275])
subplot('Position', [3/48, 0.5/16, 10/24, 0.275])
subplot('Position', [27/48, 0.5/16, 10/24, 0.275])
X=logspace(-5,3,1e6);
subplot(321)
Gamma=0;
PDF=2*(1+Gamma)*X.*exp(-((1+Gamma)*(X.^2)+Gamma)).*besseli(0,2*sqrt(Gamma*(1+Gamma))*X);
loglog(X,PDF,'k-','LineWidth',2)
hold on
Gamma=0.5;
PDF=2*(1+Gamma)*X.*exp(-((1+Gamma)*(X.^2)+Gamma)).*besseli(0,2*sqrt(Gamma*(1+Gamma))*X);
loglog(X,PDF,'k-','LineWidth',1)
Gamma=1;
PDF=2*(1+Gamma)*X.*exp(-((1+Gamma)*(X.^2)+Gamma)).*besseli(0,2*sqrt(Gamma*(1+Gamma))*X);
loglog(X,PDF,'b-','LineWidth',2)
Gamma=2;
PDF=2*(1+Gamma)*X.*exp(-((1+Gamma)*(X.^2)+Gamma)).*besseli(0,2*sqrt(Gamma*(1+Gamma))*X);
loglog(X,PDF,'b-','LineWidth',2)
Gamma=5;
PDF=2*(1+Gamma)*X.*exp(-((1+Gamma)*(X.^2)+Gamma)).*besseli(0,2*sqrt(Gamma*(1+Gamma))*X);
loglog(X,PDF,'g-','LineWidth',2)
Gamma=10;
PDF=2*(1+Gamma)*X.*exp(-((1+Gamma)*(X.^2)+Gamma)).*besseli(0,2*sqrt(Gamma*(1+Gamma))*X);
loglog(X,PDF,'r-','LineWidth',2)
axis([1e-2 5e2 1e-3 1e3])
legend('\gamma=0 (Rayleigh)','\gamma=0.5','\gamma=1','\gamma=2','\gamma=5','\gamma=10')
subplot(322)
Alpha=0.5;
PDF=4*sqrt(Alpha)/gamma(Alpha)*((sqrt(Alpha)*X).^Alpha).*besselk(Alpha-1,2*sqrt(Alpha)*X);
loglog(X,PDF,'r--','LineWidth',2)
hold on
Alpha=1;
PDF=4*sqrt(Alpha)/gamma(Alpha)*((sqrt(Alpha)*X).^Alpha).*besselk(Alpha-1,2*sqrt(Alpha)*X);
loglog(X,PDF,'g--','LineWidth',2)
Alpha=2;
PDF=4*sqrt(Alpha)/gamma(Alpha)*((sqrt(Alpha)*X).^Alpha).*besselk(Alpha-1,2*sqrt(Alpha)*X);
loglog(X,PDF,'b--','LineWidth',2)
Alpha=5;
PDF=4*sqrt(Alpha)/gamma(Alpha)*((sqrt(Alpha)*X).^Alpha).*besselk(Alpha-1,2*sqrt(Alpha)*X);
loglog(X,PDF,'b--','LineWidth',1)
Alpha=100;
PDF=4*sqrt(Alpha)/gamma(Alpha)*((sqrt(Alpha)*X).^Alpha).*besselk(Alpha-1,2*sqrt(Alpha)*X);
loglog(X,PDF,'b','LineWidth',2)
legend('\alpha_K=0.5 (exponential)','\alpha_K=1','\alpha_K=2','\alpha_K=5','\alpha_K=\infty (Rayleigh)')
axis([1e-2 5e2 1e-3 1e3])
subplot(323)
nu=0.5;
PDF=nu*((2/nu*gamma(2/nu))^(nu/2))*(X.^(nu-1)).*exp(-((2/nu*gamma(2/nu))^(nu/2))*(X.^nu));
loglog(X,PDF,'r--','LineWidth',2)
hold on
nu=1;
PDF=nu*((2/nu*gamma(2/nu))^(nu/2))*(X.^(nu-1)).*exp(-((2/nu*gamma(2/nu))^(nu/2))*(X.^nu));
loglog(X,PDF,'g--','LineWidth',2)
nu=1.5;
PDF=nu*((2/nu*gamma(2/nu))^(nu/2))*(X.^(nu-1)).*exp(-((2/nu*gamma(2/nu))^(nu/2))*(X.^nu));
loglog(X,PDF,'b--','LineWidth',2)
nu=2;
PDF=nu*((2/nu*gamma(2/nu))^(nu/2))*(X.^(nu-1)).*exp(-((2/nu*gamma(2/nu))^(nu/2))*(X.^nu));
loglog(X,PDF,'k','LineWidth',2)
nu=3;
PDF=nu*((2/nu*gamma(2/nu))^(nu/2))*(X.^(nu-1)).*exp(-((2/nu*gamma(2/nu))^(nu/2))*(X.^nu));
loglog(X,PDF,'g','LineWidth',2)
nu=5;
PDF=nu*((2/nu*gamma(2/nu))^(nu/2))*(X.^(nu-1)).*exp(-((2/nu*gamma(2/nu))^(nu/2))*(X.^nu));
loglog(X,PDF,'r','LineWidth',2)
legend('\nu=0.5','\nu=1 (exponential)','\nu=1.5','\nu=2 (Rayleigh)','\nu=3','\nu=5')
axis([1e-2 5e2 1e-3 1e3])
subplot(324)
Sig=0.2;
PDF=1./(Sig*sqrt(2*pi)*X).*exp(-((log(X)+Sig^2).^2)/(2*(Sig^2)));
loglog(X,PDF,'r','LineWidth',2)
hold on
Sig=0.5;
PDF=1./(Sig*sqrt(2*pi)*X).*exp(-((log(X)+Sig^2).^2)/(2*(Sig^2)));
loglog(X,PDF,'g-','LineWidth',2)
Sig=1;
PDF=1./(Sig*sqrt(2*pi)*X).*exp(-((log(X)+Sig^2).^2)/(2*(Sig^2)));
plot(X,PDF,'b--','LineWidth',2)
Sig=1.5;
PDF=1./(Sig*sqrt(2*pi)*X).*exp(-((log(X)+Sig^2).^2)/(2*(Sig^2)));
loglog(X,PDF,'g--','LineWidth',2)
Sig=3;
PDF=1./(Sig*sqrt(2*pi)*X).*exp(-((log(X)+Sig^2).^2)/(2*(Sig^2)));
loglog(X,PDF,'r--','LineWidth',2)
% Add Rayleigh
loglog(X,2*X.*exp(-X.^2),'k','LineWidth',2)
legend('\sigma_{LN}=0.2','\sigma_{LN}=0.5','\sigma_{LN}=1','\sigma_{LN}=1.5','\sigma_{LN}=3','Rayleigh')
axis([1e-2 5e2 1e-3 1e3])
subplot(325)
m=0.5;
PDF=2*(m^m)/gamma(m)*(X.^(2*m-1)).*exp(-m*(X.^2));
loglog(X,PDF,'r--','LineWidth',2)
hold on
m=1;
PDF=2*(m^m)/gamma(m)*(X.^(2*m-1)).*exp(-m*(X.^2));
loglog(X,PDF,'k','LineWidth',2)
m=2;
PDF=2*(m^m)/gamma(m)*(X.^(2*m-1)).*exp(-m*(X.^2));
loglog(X,PDF,'b','LineWidth',2)
m=5;
PDF=2*(m^m)/gamma(m)*(X.^(2*m-1)).*exp(-m*(X.^2));
loglog(X,PDF,'g','LineWidth',2)
m=10;
PDF=2*(m^m)/gamma(m)*(X.^(2*m-1)).*exp(-m*(X.^2));
loglog(X,PDF,'r','LineWidth',2)
legend('m=0.5','m=1 (Rayleigh)','m=2','m=5','m=10')
axis([1e-2 5e2 1e-3 1e3])
subplot(326)
Rho=-2;
PDF=sqrt(2/((1-2*Rho)*(1-Rho)))*((1+Rho*X*sqrt(2/((1-2*Rho)*(1-Rho)))).^(-(1+Rho)/Rho));
loglog(X(PDF>0),PDF(PDF>0),'r','LineWidth',2)
hold on
Rho=-1;
PDF=sqrt(2/((1-2*Rho)*(1-Rho)))*((1+Rho*X*sqrt(2/((1-2*Rho)*(1-Rho)))).^(-(1+Rho)/Rho));
loglog(X(PDF>0),PDF(PDF>0),'g','LineWidth',2)
Rho=-0.5;
PDF=sqrt(2/((1-2*Rho)*(1-Rho)))*((1+Rho*X*sqrt(2/((1-2*Rho)*(1-Rho)))).^(-(1+Rho)/Rho));
loglog(X(PDF>0),PDF(PDF>0),'b--','LineWidth',2)
Rho=0;
PDF=sqrt(2/((1-2*Rho)*(1-Rho)))*exp(-X*sqrt(2/((1-2*Rho)*(1-Rho))));
loglog(X(PDF>0),PDF(PDF>0),'g--','LineWidth',2)
Rho=1/3;
PDF=sqrt(2/((1-2*Rho)*(1-Rho)))*((1+Rho*X*sqrt(2/((1-2*Rho)*(1-Rho)))).^(-(1+Rho)/Rho));
loglog(X(PDF>0),PDF(PDF>0),'r--','LineWidth',2)
% Add Rayleigh
loglog(X,2*X.*exp(-X.^2),'k','LineWidth',2)
legend('\rho=-2','\rho=-1','\rho=-1/2','\rho=0','\rho=1/3','Rayleigh')
axis([1e-2 5e2 1e-3 1e3])