FID-FT

\(ClearAll["Global`*"]; 
\ w = {1000, 2000, 3000, 7000, 8000, 9000}; 
    
s = {1, 2, 1, 2, 4, 2}; \ tau = 0.0025; \ a = 100; 

    loren[i_, f_] := a^2\ \ s[\([i]\)]/\((a^2 + \((f - w[\([i]\)])\)^2)\); 

    f[x_] = 0; 

    For[i = 1, i <= Length[w], \(i++\), 
aux = loren[i, x]; 

      f[x_] = f[x] + aux\ ]; 
Plot[f[x], {x, 100, 10000}, PlotRange -> All]; 
    
g[t_] = 
      E^\((\(-t\)/tau)\)\ 
        Sum[\((s[\([i]\)]/\((2\ Pi)\))\)\ Cos[2\ Pi\ w[\([i]\)]\ t], 
          \ {i, \ 1, 
Length[w]}]; 

    Plot[g[t], {t, 0.00001, 0.007}, PlotRange -> {\(-1\), 1}, Frame -> True, 
      Axes -> False]\)

\(ClearAll["Global`*"]; 
\ w = {1000, 2000, 3000, 7000, 8000, 9000}; 
    
s = {1, 1, 1, 1, 1, 1}; \ tau = 0.0025; \ a = 100; 

    loren[i_, f_] := a^2\ \ s[\([i]\)]/\((a^2 + \((f - w[\([i]\)])\)^2)\); 

    f[x_] = 0; 

    For[i = 1, i <= Length[w], \(i++\), 
aux = loren[i, x]; 

      f[x_] = f[x] + aux\ ]; 
Plot[f[x], {x, 100, 10000}, PlotRange -> All]; 
    
g[t_] = 
      E^\((\(-t\)/tau)\)\ 
        Sum[\((s[\([i]\)]/\((2\ Pi)\))\)\ Cos[2\ Pi\ w[\([i]\)]\ t], 
          \ {i, \ 1, 
Length[w]}]; 

    Plot[g[t], {t, 0.00001, 0.007}, PlotRange -> {\(-1\), 1}, Frame -> True, 
      Axes -> False]\)