Re: Laplace Transform vs Fourier transform

On Oct 27, 8:39 pm, "fisico32" <marcoscipio...@xxxxxxxxx> wrote:
Hello Forum,

everyone is familiar with the Fourier transform and its importance.
Its independent variable, the angular frequency w, is a measurable
physical quantity.

What about the Laplace transform? What is, conceptually, the advantage of
taking the Laplace transform of a signal instead of the Fourier transform?
I know that by looking at its zeros and poles we can assess the stability
of the system for example.... is that not possible with the FT alone?

Does the Laplace transform show information on the system that the FT does
not?

thanks
fisico32

The fourier transform is a special case of the laplace transform (as I
understand it)

The Laplace transform correlates a given waveform with every possible
(exponential x sinusoidal) wave.

The Fourier transform correlates it with every possible sinusoidal
wave.

Here is the one point of the Laplace transrom:

If you correlate a signal with a "decaying" exponential, and the
resulting correlation is unbounded, then you have a stability problem.
(if you have pole in right hand plane it means that your correlation
became infinite with an exponentially decaying sinusoid)

It is OK to correlate your system with an exponentially "increasing"
sinusoidal wave and have an unbounded result.

So you take your signal and correlate it with every possible
exponentially increasing and exponentially decreasing sinusoidal wave
(sinusoidal means both exponential cosines and exponential sines) and
see what happens.

One thing to note, when you get a pole in the S plane, that really
sets the boundary of instability. There will be an infinite number of
poles to the left of the leftmost "official" pole.

The fourier transform only correlates the f(t) with pure (non
exponential) sinusoids.
This is what we typically want to know about when thinking of the
frequency response of a circuit (or system)
The fourier transform is especially useful because of the convolution
theorem, that says multiplication in the frequency domain is equiv to
convolution in the time domain. The response of the circuit is the
impulse response convolved with the sinusoidal input signal. This is
a piece of cake when you take the fourier transform of the system to
immediately get the frequency response.

In conclusion

Fourier transform is good tool for frequency response

Laplace transform is good tool for stability analysis.



PS

Laplace also is a good tool for solving differential equations becuase
the S-plane is a mapping of every possible solution to an ordinary
differential equation.


caveat: this may all be BS so do your own due diligence :-)

.

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