f(t) =
Use the System Identification Toolbox.
The actual transfer function is probably a simple delay term or that includes one, for example or something similar, if both functions are essentially the same otherwise.
To illustrate —
syms k s t
f(t) = t*exp(-k*t)
F(s) = laplace(f,t,s)
clear k s t
s = tf('s');
HT = exp(-60*s);
figure
stepplot(HT)
grid
k1 = 0.1;
H1 = 0.01/(k1+s)^2
H2 = H1*HT
figure
impulseplot(H1)
hold on
impulseplot(H2)
hold off
grid
xlim([0 300])
This approximates your posted plot.
.