Fourier transform vs Laplace transform. Why use one over the other?

We were discussing this in a comm systems class yesterday and no one could come up with a good explaination as to why we use Laplace transforms over Fourier and vice versa. Both take equations that are in the time domain and transform them into the frequency domain. It seems in Comm we use only Fourier analysis methods and in Circuit analysis and DE we onle use Laplace.

What are the situations that define which one you will use?

On a side note, what are Z transforms typically used for?

Fourier Transforms and Laplace Transforms are both frequency space transforms, but one has an imaginary argument and the other a real one. They’re associated with different physical situations that dictate where they are used. Fourier Transforms are useful if you want to see the frequency spectrum makeup of a signal – something you don’t get from the Laplace Transform. Laplace Transforms, on the other hand, are useful in solving rate equations. They come in useful if you’re taking a course in Diffie Q’s (Differential Equations).

Can’t help you on the Z-transforms, though.

IIRC, Fourier transforms only work for functions that converge to zero at infinity. Laplace transforms work for functions that diverge exponentially. Many real-world physics problems have exponential solutions.

Z transforms are a method for analyzing sampled data feedback (for example, servomechanism) systems.

In a nutshell,

The Laplace Transform is used for analyzing continuous time signals. It transforms a signal in the time-domain (function of t) to a signal in the “s” domain, where s = s + jw.

The Fourier Transform is a special case of the Laplace Transform in which s = 0, so that s = jw. It is used to analyze “stable” signals (non-exponential, etc.).

The Z-Transform is basically the same thing as the Laplace Transform, except it’s used for discrete (i.e. sampled) signals.