Two sided laplace transform

Hello, is there any function that generates two sided laplace transform?

Answer 1

[Paul]

No, there is not, at least not directly.

However, the two-sided Laplace transform of f(t) can be developed by

1. expressing f(t) as the sum of a strictly left-sided signal and a strictly right-sided signal
2. computing the two-sided transform of each part. The two-sided transform of the strictly right-sided signal is the same as its one-sided transform. The two-sided transform of the strictly left-sided signal can be found from the unilateral transform using the time reversal property.
3. summing the transforms after verifying their respective regions of convergence intersect

Example

syms a b t real
f(t) = exp(-a*t)*heaviside(t) - exp(-b*t)*heaviside(-t);

1. define right- and left-sided parts

f_r(t) =  exp(-a*t)*heaviside(t);
f_l(t) = -exp(-b*t)*heaviside(-t);

2. two-sided transforms

syms s
F_r(s) = laplace(f_r(t),t,s)

F_r(s) = 

The ROC of F_r(s) is Re(s) > -a

F_l(s) = subs(laplace(f_l(-t),t,s),s,-s)

F_l(s) = 

The ROC of F_l(s) is Re(s) < -b

3. The two-sided transform of f(t) is

F(s) = F_l(s) + F_r(s)

F(s) = 

[num,den] = numden(F(s));
F(s) = num/den

F(s) = 

The ROC of F(s) is -a < Re(s) < -b, as long as -b > -a.

If -b < -a, then the two-sided transform of f(t) does not exist.