Scaling problem from Fourier continuous transformate via IFFT

Dear mates,

I am having a problem comparing the impulse response of a system calculated through impulse() with the one obtained by applying ifft to a theorical fourier transformate evaluated in N points. I am getting an error between maximums of the order of 1000. I attach the code, hope someone could bring some light on this:

    %%Variable initialization
  syms w 
  s = tf('s');
  sh = 0.01;
  Ts = 0.001; % sampling period
  Fs = 1/Ts; % sampling frequency
  nElementos = 2^16; % Number of elementos for ifft

%%System parameters
% masa suspendida
ms=175;ks=200000;cs=40000;
l0muelle=90; % longitud inicial del muelle en mm
RH0=45; % Altura al suelo en reposo del vehiculo en mm
% masa no suspendida
mu=20;ku=200000;cu=1;
l0neumatico=30; % longitud inicial del neumatico en mm

%%System characterization via Laplace transformate applied to Diff Eqs
% System
t11=(ms*(s^2)+cs*s+ks);
t12=(-cs*s-ks);
t21=t12/(s*cu+ku);
t22=(mu*(s^2)+(cs+cu)*s+(ks+ku))/(s*cu+ku);
T=[t11,t12;t21,t22];
M=T^-1;
X=M*[0;1];
Xu=X(2);

t = linspace(0,20,20000);
%%Sprung Mass impulse response
%         [y,t]=step(sh*Xs,t);

          y = impulse(Xu,t);
          yf1=1000*y; % Scaling to mm
          hy=plot(t,yf1,'b');
          hold on;
          title('Time Response')
          xlabel('Time (s)') % x-axis label
          ylabel('Ride Height Sprung Mass (mm)') % y-axis label

%%System characterization via Fourier transformate (substituing s=jw in Laplace transf)
t11=(ms*((1i*w)^2)+cs*1i*w+ks);
t12=(-cs*1i*w-ks);
t21=t12/(1i*w*cu+ku);
t22=(mu*((1i*w)^2)+(cs+cu)*1i*w+(ks+ku))/(1i*w*cu+ku);
T=[t11,t12;t21,t22];
M=T^-1;
X=M*[0;1];
Xu=X(2);

wdiscreta = 2*pi*Fs*linspace(0,1,nElementos); % angular velocity vector
XuDiscreta = double(subs(Xu, w, wdiscreta))*Ts; % substituing w in continuous fourier transformate
xUFFT = ifft(XuDiscreta, nElementos)/Ts; % Applying ifft
yf=1000*(xUFFT); % Scaling to mm
plot(linspace(0,length(xUFFT)*Ts,length(xUFFT)), real(yf),'r')
error = (max(real(yf))-max(yf1))/max(yf1) % Compute error for comparison purposes
max(real(yf1))/max(real(yf)) % Maximums cocient
plot(linspace(0,length(xUFFT)*Ts,length(xUFFT)), (max(yf1)/max(yf))*real(yf),'g')

Answer 1

[Ahmet Cecen]

Just a quick thought without inspecting the code, so this might not be the case, or the only problem: Check if the signal from the one approach has the same sampling rate as the other, since fft will ensure the average over all sampling points is maintained. So if 1 approach is computed over 1000 points, while the other over 5000, you can see that kind of scaling error.