How to properly scale the lorentzian curve to the data

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Yogesh 2024-10-16

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https://ww2.mathworks.cn/matlabcentral/answers/2160140-how-to-properly-scale-the-lorentzian-curve-to-the-data

关闭： Torsten 2024-10-16

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RMS=[];

R=[];

P_threshold=[];

L=5; %[m] Fiber length

V1=[1];

P0=4.7960;

F1=0;

count=0;

count1=0;

%% general constants

n=1.45; %Index of refraction

eps0=8.854e-12; % [F/m] Vacuum permittivity

mu0 = 4*pi*1e-7;%[H/m] Vacuum permeability

c=2.9979e8; % [m/sec] Speed of light

Z0=sqrt(mu0/eps0); %[Ohm] Vacuum impedance

dt = 6e-12; dz=dt*c/n; %Spacial and Temporal step sizes.

N=round(L/dz); % Fiber length discretization

z=0:dz:(dz*N-dz);

T=20*2*L*n/c; %time taken for 10 round trips

Nt=round(T/dt);

%% material characteristics

lambdaL=1.064e-6; % [m] pump wavelength

WL=2*pi*c/lambdaL; %[rad/sec] pump frequency

GAMMAb=2*pi/17.5e-9; %[1/t] phonons decay rate(zeringue)

A=7.85e-11; %[m^2] fiber's effective area

Va=5960; %[m/s] Velocity of sound

Omega=2*n*Va*WL/c; %[rad/sec] (zeringue)

WS = WL-Omega;

gammaE=1.95; %electrostrictive coefficient

kT=300*1.3806504e-23; %[Joule] at room temprature.

roh0=2201; % [kg/m^3] mean density for SiO2, bulk (http://www.virginiasemi.com/pdf/generalpropertiesSi62002.pdf) .

Q=2*kT*roh0*GAMMAb/(Va^2*A); %Noise std.

g0=(gammaE^2*WL^2)/n/c^3/Va/roh0/GAMMAb; %[m/W] (Jenkins)

sigma=(WL*gammaE)/2/n/roh0/c; %(derivation)

q=2*WL*n/c; %derivation

I1_0=P0/A; %[W/m^2] Pump intensity

LAMBDA=gammaE*Omega/c/Z0/2/Va^2; %derivation

kappa=g0*GAMMAb*n/2/Z0/LAMBDA; %derivation

Nz=N;

%% Coefficients

G=g0*I1_0*L; %Gain factor

Gth=21; %Agrawal, pp.361

Elt=repelem(0,Nz);

Elt(1)=sqrt(P0*Z0/A/2/n);

Est=repelem(0,Nz);

%%

t = (-Nt/2:1:Nt/2-1)*dt;

nu = (-Nt/2:1:Nt/2-1)*1/T;

vpi = 3;

RL=50;

V=1;

fsine=2e9;

sine = V*sin(2*pi*fsine*t);

%% White noise modulation

np = 22; % power of RF White noise signal in dBm

np_l = 10^(np/10)*1e-3; % power of RF White noise signal in Watts

nbw = 0.5e9; % bandiwdth in Hz of the RF WNS

rng('default');

rng(2);

n1 = randn(1, Nt); % generate a white gaussian noise in time domain which is delta correlated in time domain

s=nbw*sinc(pi*t*nbw);

n_LPF_T=conv(n1,s,'same');

n_lpf = sqrt(np_l*T*RL)*(n_LPF_T).*(1./sqrt(trapz(t, abs(n_LPF_T).^2))); %normalize the noise volatge signal

%%

phi=n_lpf;

ES_0t=sqrt(I1_0/2/n/c/eps0)*exp((1i*phi)); % Original signal in time

Power=trapz(t,2*n*c*eps0*A*abs(ES_0t).^2)/T; % Area under the curve in time domain

FFt_EL0t=fftshift(abs(fft(ES_0t))); % fourier transform of the original signal

Power_FFt=T*trapz(nu,2*n*c*eps0*A*(FFt_EL0t/Nt).^2); % Area under the curve in frequency domain

% figure;plot(nu,2*n*c*eps0*A*(FFt_EL0t/Nt).^2)

%% ---------------------------------

%% Fitting with Lorentzian model

%% ---------------------------------

y = 2*n*c*eps0*A*(FFt_EL0t/Nt).^2;

[up, ~] = envelope(y, round(0.01*size(nu/max(nu), 2)), 'peak');

format long g

x = nu;

xm = max(nu)

xm =

83332316172.4138

x_m=std(x);

x_axis=(x)/xm;

% b1= sqrt(1/(max(y)/min(y)-1));

% b1=(3343175379.31034/xm)^2;

% b1=1/((max(x)-min(x))/10/xm)^2;

% b1=(abs(x(dsearchn(y',max(y)/10)))/xm)^2;

c1=x(dsearchn(y',max(y)));

% b1=((xm-c)^2/((max(y)/min(y))-1));

a1=max(y);

b1=abs(x(dsearchn(y',max(y)/2)));

tol=0.0001;

y_lor=a1./(((x-c1)/b1).^2+1);

% Fitting model

fo = fitoptions('Method', 'NonlinearLeastSquares',...

'Lower', [tol*a1, tol*b1,tol-c1 ],...

'Upper', [ a1, b1, tol+c1]);

% fo = fitoptions('Method', 'NonlinearLeastSquares' );

ft = fittype('(a)./(((x-c)/b).^2+1)', ...

'independent', {'x'}, ...

'coefficients', {'a','b', 'c'}, ...

'options', fo);

%% Fit curve to data

[yfit, gof] = fit(x', y', ft, 'StartPoint', [a1, b1,c1])

yfit =

General model: yfit(x) = (a)./(((x-c)/b).^2+1) Coefficients (with 95% confidence bounds): a = 0.001715 (0.001703, 0.001727) b = 1.148e+09 (1.136e+09, 1.159e+09) c = 0.0001 (fixed at bound)

gof = struct with fields:

sse: 0.0053242226215525 rsquare: 0.483756138064637 dfe: 161222 adjrsquare: 0.483752935996297 rmse: 0.000181725532855836

figure

plot(yfit, x, y), grid on, grid minor, xlim([-12e9, 12e9])

xlabel('\nu'), ylabel('y')

How do I increase the amplitude of the curve to properly match the data...

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