Acceleration to velocity via frequency domain integration

Hello everyone,

I'm trying to integrate an acceleration signal in frequency domain, in order to obtain 'velocity vs time' curve.

To verify the frequency-domain integration I apply the method on an ideal acceleration signal that represents a simple linear motion. The same signal is also calculated using time-domain integration via "cumtrapz" command. I attach an image presenting my results for the both methods. As you can see, velocity curves have approximately the same shape however, the major difference is that the FD Integration curve is shifted down along the vertical axis. My results in FD integration are obviously unreasonable, although it should be easy to get proper results for ideal signals.

Any ideas regarding what is wrong and how to proceed?

Here is my code, with the valuable help of those related topics: Link1, Link2 and Link3.

I attach "Acceleration_Signal.mat" to open in matlab workspace before running the code.

dt=0.005;                   % sampling every 0.005 sec
fs=1/dt;                    % sampling frequency in Hz
len=5.995;                  % signal length in sec
N1=ceil(len/dt);            % signal length in number of samples
T=(0:dt:N1*dt);             % time series in sec

ax=Acceleration_Signal;     %Ideal acceleration signal

%TIME-DOMAIN INTEGRATION
vx=cumtrapz(T,ax);          

figure(1)
subplot(2,1,1),plot(T,ax,'LineWidth',1);title('Acceleration in m/s^2','fontsize',14);
subplot(2,1,2),plot(T,vx,'LineWidth',1);title('Velocity in m/s','fontsize',14);
sgtitle('TIME-DOMAIN INTEGRATION');

%FREQUENCY-DOMAIN INTEGRATION

AX = fft(ax);               %Fourier transformation of acceleration signal
df = 1/(N1*dt);             %Frequency resolution

  
f = df*(0:N1);                
w = 2*pi*f*sqrt(-1);        %Calculate omega

for i = 1:length(AX)
    if f(i)~=0 
        VX(i) = AX(i)/w(i);
    else
        VX(i) = 0;
    end      
end

v= ifft(VX,'symmetric');                %Inverse Fourier transformation to get velocity signal
   
figure(2)
subplot(2,1,1),plot(T,ax,'LineWidth',1);title('Acceleration in m/s^2','fontsize',14);
subplot(2,1,2),plot(T,v,'LineWidth',1);title('Velocity in m/s','fontsize',14);
sgtitle('FREQUENCY-DOMAIN INTEGRATION');

Thank you ,

George Papazetis