Signal Processing Output LTI System

Hi! I'm triying to calculate the output signal of a LTI system. I have a gaussian pulse as an input and the transfer function in frequency domain of a filter. I want to compare both outputs by using the convolution of the input signal and impulse response (time domain solution) and using the product of the input spectrum and the transfer function (frequency domain solution). But the problem is that I get differents solutions. Any help??

Main code

%% Ultra short pulse at the input
c=299792458; %light speed
width=100e-15; %pulse width
T = width/(2*(sqrt(log(2)))); %period
lambda0=1550e-9; %carrier frequency
w0=(2*pi*c)/lambda0;
tstep=T/100;  % time step (s)
Fs=1/tstep %sampling frequency

t1=-2*T+tstep:tstep:2*T;  %axis time
tconv=-2*T:tstep/3.75:2*T+tstep/3.75;%convolution axis time
L=length(t1);
A1=sin(w0*t1).*exp(-2*log(2)*t1.^2/(width/2).^2);  % Gaussian intensity modulated sine pulse

A2=sin(w0*tconv).*exp(-2*log(2).*tconv.^2/(width/2).^2);%
y1=exp(-2*log(2)*t1.^2/(width/2).^2); % Gaussian intensity envelope

figure(1)
plot(t1,A1,'r','linewidth',1)
hold on
plot(t1,y1,'b','linewidth',2)
xlabel('t')
ylabel('E(t)')
legend('E(t)','Envelope')

%Spectrum of the gaussian input pulse
n = 10000;
Y=fft(A1,n);
fshift = 1:1/(tstep*n):2.5e14;
lambda=c./fshift;
powershift=(abs(Y(1:3*n/20+2)./L));

figure (2) %Hertzs plot
plot(fshift,powershift);
xlim([0, 2.5e14])
xlabel('frequency(Hz)')
ylabel('Magnitude |X(f)|')

figure(3) %Lambda plot
plot(lambda,powershift); 
xlim([1.2e-6, 1.8e-6])
xlabel('lambda(m)')
ylabel('Magnitude |X(f)|')

%Filter's transfer function
K=0.08;
L2=2*pi*5.7172e-06;
ng=4.18;

H12_2=-(sqrt(K^2.*exp(-j*2*pi*L2*ng./lambda))./(1-(1-K).*exp(-j*2*pi*L2*ng./lambda))); %Transfer function
output_frequency_abs=abs(H12_2).*powershift;
output_frequency=H12_2.*Y(1:3*n/20+2);

figure(4)
subplot(1,2,1)
plot(lambda,output_frequency_abs); 
xlim([1.2e-6, 1.8e-6])
xlabel('lambda(m)')
ylabel('Magnitude |X(f)|')

% Impulse Response time domain
Impulse_Response_time=(ifft((H12_2),L));
figure (5)
plot(real(Impulse_Response_time)); 
%Output time domain
output_time=ifft(output_frequency);
figure(6)
plot(real(output_time));

Convolution=conv(A1,Impulse_Response_time,'full');

output_freq_2=fft(Convolution,n);
powershift_2=(abs(output_freq_2(1:3*n/20+2)./L));
figure(4)
subplot(1,2,2)
plot(lambda,powershift_2);
xlim([1.2e-6, 1.8e-6])
xlabel('lambda(m)')
ylabel('Magnitude |X(f)|')

figure(7)
plot(real(Convolution));

Answer 1

[Haseeb Abbas]

%% Ultra short pulse at the input

c=299792458; %light speed

width=100e-15; %pulse width

T = width/(2*(sqrt(log(2)))); %period

lambda0=1550e-9; %carrier frequency

w0=(2*pi*c)/lambda0;

tstep=T/100; % time step (s)

Fs=1/tstep %sampling frequency

t1=-2*T+tstep:tstep:2*T; %axis time

tconv=-2*T:tstep/3.75:2*T+tstep/3.75;%convolution axis time

L=length(t1);

A1=sin(w0*t1).*exp(-2*log(2)*t1.^2/(width/2).^2); % Gaussian intensity modulated sine pulse

A2=sin(w0*tconv).*exp(-2*log(2).*tconv.^2/(width/2).^2);%

y1=exp(-2*log(2)*t1.^2/(width/2).^2); % Gaussian intensity envelope

figure(1)

plot(t1,A1,'r','linewidth',1)

hold on

plot(t1,y1,'b','linewidth',2)

xlabel('t')

ylabel('E(t)')

legend('E(t)','Envelope')

%Spectrum of the gaussian input pulse

n = 10000;

Y=fft(A1,n);

fshift = 1:1/(tstep*n):2.5e14;

lambda=c./fshift;

powershift=(abs(Y(1:3*n/20+2)./L));

figure (2) %Hertzs plot

plot(fshift,powershift);

xlim([0, 2.5e14])

xlabel('frequency(Hz)')

ylabel('Magnitude |X(f)|')

figure(3) %Lambda plot

plot(lambda,powershift);

xlim([1.2e-6, 1.8e-6])

xlabel('lambda(m)')

ylabel('Magnitude |X(f)|')

%Filter's transfer function

K=0.08;

L2=2*pi*5.7172e-06;

ng=4.18;

H12_2=-(sqrt(K^2.*exp(-j*2*pi*L2*ng./lambda))./(1-(1-K).*exp(-j*2*pi*L2*ng./lambda))); %Transfer function

output_frequency_abs=abs(H12_2).*powershift;

output_frequency=H12_2.*Y(1:3*n/20+2);

figure(4)

subplot(1,2,1)

plot(lambda,output_frequency_abs);

xlim([1.2e-6, 1.8e-6])

xlabel('lambda(m)')

ylabel('Magnitude |X(f)|')

% Impulse Response time domain

Impulse_Response_time=(ifft((H12_2),L));

figure (5)

plot(real(Impulse_Response_time));

%Output time domain

output_time=ifft(output_frequency);

figure(6)

plot(real(output_time));

Convolution=conv(A1,Impulse_Response_time,'full');

output_freq_2=fft(Convolution,n);

powershift_2=(abs(output_freq_2(1:3*n/20+2)./L));

figure(4)

subplot(1,2,2)

plot(lambda,powershift_2);

xlim([1.2e-6, 1.8e-6])

xlabel('lambda(m)')

ylabel('Magnitude |X(f)|')

figure(7)

plot(real(Convolution));