How to compute the Laplace transform of a discrete signal?

Question

Assume I have a discrete random signal, $f(t)$ for which I want to calculate the laplace transform.

How can I do it in matlab without using sym variables, for example consider I have this discrete signal f(t):

>> t=linspace(0,1000, 10000);
>> f=t.*cos(t);

Is there a way to calculate the Laplace numerically?

After thinking more about the problem I came up with this approach:

t=linspace(0,1000, 10000);
f=t.*cos(t);
syms s;                                
F_s = symfun(sum(f.*exp(-s*t)), s);
ilaplace(F_s)

Though I am not sure it's plausible.

Answer 1

This requires use of the MATLAB symbolic toolbox

>> syms x
>> f = x * cos(x);

>> t = linspace(0, 1000, 1000); % Or whatever values you want to evaluate the Laplace Transform over
>> L = double(laplace(f, t)); % Simulataneously compute the Transform and convert it from 'syms' to double

I guess this isn't exactly what you're asking for, as it requires keeping your function $f$ as a "symbolic expression". I dont know of any function which performs a "discrete Laplace tranform" though.

EDIT: Actually, I dont believe there is such thing as a Laplace Transform for discrete functions. However, the Laplace Transform is just a specific case of the z-Transform when $z = e^s$, which is definitely defined for discrete signals. Unfortunately I can't say much more about this relation between the two transforms, but hopefully this gives you a little more information about how to proceed from here.

Answer 2

This comes from my Junior Level Linear Systems course and your question reminded me of:

Lathi, Bhagwandas Pannalal. Signal, Systems, and Controls. Intext, 1973.

in his last chapter (7) where he combines discrete and continuous time elements in a hybrid system, the discrete time elements are modeled as continuous time $\delta(t)$ functions followed by a zero order hold. He gives a table (Table 7.4) of "equivalences", and apologizes that $G(s)$ and $G(z)$ is an abuse of notation.

but I don't know how you it's going to help you if all you have is a sequence of numbers, and no functional form, because the Z transform of a finite sequence (necessary to compute) corresponds to a Z transform of a FIR filter that is all zeros.

You could also try fitting an AR or ARMA model to your data and then you have a functional form but it would need to be a very good fit. Any residuals would get you back to the FIR filter Z transform. The Z transform is linear so adding one to another would be OK. The Bilinear transform would get you back to a zero state one sided Laplace.

The 2 approaches FIR and ARMA, will not give the same Z transform and by extension the same Laplace. You need to decide what you want to do with the Laplace and choose accordingly.

Answer 3

Since your original function is discrete, you can either model that function as a sum of weighted, shifted Kronecker delta functions, and the apply the formula for the Laplace transform.

Or you could simply find the z-transform, and then apply some kind of discrete-to-continuous transformation on the z-transform (example the Bilinear transform), to come to a laplace transform.