In short, I need help by creating a function describing a decaying process in time. I know
$$ h(t) = e ^ {xt} * e ^ {xt} = te^{xt} $$
I use * as the convolution sign. if then the exponential functions are modulated with a white noise sequence,
$$ h(t) = e ^ {xt}w(t) * e ^ {xt}w(t-t_1) = t^{0.5}e^{xt}w'(t) $$
Here w'(t) is a non-gaussian white noise sequence with the same statistical properties as white noise.
My question is can I do something similar when the decay terms in the exponential are different: Without noise $$ h(t) = e ^ {x_1t} * e ^ {x_2t} = (e^{x_1t} - e^{x_2t})/(x1-x2) $$ With noise $$ h(t) = e ^ {x_1t}w(t) * e ^ {x_2t}w(t-t_1) = ?? $$ I hope my question is clear this way.
Thanks in advance