Fourier vs. Laplace transforms

Question

Electronics books often use Laplace to analyze circuits, while in physics we use Fourier, most of the times... if not always: from complex impedances to electromagnetism, quantum mechanics, Green functions, etc etc.

Various sources maintain that Laplace is somehow necessary to properly analyze circuits. I still cannot see why, exactly. Can you point me towards a single well-defined and relevant example/exercise (no generic suggestions like... "stability of amplifiers") that can be solved using Laplace transform and that cannot be solved using Fourier?

Maybe it's me, but I have a hard time in finding examples where Fourier analysis doesn't lead to the same conclusions (of course with some $i\omega$ instead of $s$ in the final equations). Surely in some cases there could be convergence issues maybe, but I think non-converging integrals never scared physicists and most of the times they can be easily tamed in one limit or another.

Note I have nothing against Laplace, I even like it. I am concerned with time investment and homogeneity of mathematical methods.

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EDIT. Few examples. In physics one would typically use $Z=1/i\omega C$ for capacitors, not $1/sC$. I could write the response of a RC low pass as $1/(1+i\omega RC)$ using Fourier or $1/(1+sRC)$ using Laplace. I think this applies to any response function. I see Laplace automatically implements initial conditions in linear ODEs but... is there anything different/more insightful with respect to:

1. Using a characteristic polynomial (in $k$, $s$, $i\omega$, whatever) to find the homogeneous solutions, in case one is interested in transients

2. Use Fourier to deal with the response function for what concerns the non-homogeneous forcing terms?

Or about stability? But this can be deduced from the location of the poles of the response function, in Fourier just like in Laplace. Pls don't get me wrong here, I am perfectly fine with using Laplace approach. I am just wondering about the practical necessity/advantage of it. This seems just a purely cultural thing, completely replaceable by Fourier analysis. Maybe necessity is there and I don't see it.

Answer 1

Fourier and Laplace transforms are so closely related that the substitution $s \Leftrightarrow i \omega$ usually works in practical cases to turn one into the other (may need to adjust normalization). They both turn linear differential equations into algebra. You may do control theory in the $\omega$ domain and spectroscopy in the $s$ domain if you wish.

So, which do you choose? The $\omega$ domain has the advantage of closer connection to the physical. Things like spectroscopes are well modeled in this domain. Complex frequency $s$ is more abstract, and thus a barrier to comprehension in a physical problem. On the other hand, for things like circuit theory and control theory, $s$ has the advantage of putting problems into the domain of polynomials with real coefficients. That's well-trodden mathematical territory, so the tools are often sharper.

But you don't need to make a hard choice. As I noted above, switching domains is usually trivial.

Answer 2

The thing is the transform that we use depends on the domain of the application (I am talking about engineering or physics). If you have harmonic signals (exists for all times, no envelope, for exemple) propagating in the circuit or media, it is advisable to use Fourier transform, because it is adapted to those systems. If, however you want to calculate impulses propagation, it is useful to use Laplace transform, because it is simplifies a calculation. As for those systems there is good book were this approach really shines ["Circuits à contstantes réparties", "Electronique des impulsions", Georges Metzger, Jean-Paul Vabre, 1966, Masson et Cie ], it is in french, so there might be an english book, I do not know.

Also, Laplace transform is very useful in control theory or feedback. Where you usually have an input and output, and feedback controller. What happens that there is an input signal that can has any shape (a step for example) and Laplace transform is useful in order to calculate the output.

Another thing that in the control theory one talks about transfer functions which are Laplace transforms of differential equations of the systems. By looking to the transfer function, you can deduce a lot of things about the system stability etc.

Answer 3

There is an area where Fourier Transforms dominate and Laplace transforms are not useful and it is among the most important applications, namely spectrum analysis of stationary stochastic processes. Stationarity requires that the waveforms (signals) to extend from $-\infty$ to $+\infty$ and time dependent transients are to be excluded. It is true that causal signals can be made stationary by randomizing their starting instants and calculate their spectrum by first calculating a Laplace transform of a representative and then to convert it to a Fourier transform but that intermediate step is artificial without any significance; the meaningful operation is the Fourier transform for spectral analysis.

Answer 4

Fourier is used where we need to know spectral component or extract information from a function. While Laplace is use for response of a system, that include real part as damping or loss and imaginary part as phase. So Laplace is more general but when an impulse response is already in hand, we are interested in how system reach to steady state if there is discontinuity in source term, which is often then we use green function and solve it with Fourier transform.

Answer 5

If you allow complex frequencies then Fourier transform and Laplace transform are the same thing, just with different parametrisations and minor issues such as contour selection etc. If you only allow real frequencies, that’s a different story; you cannot Fourier transform a causal response function that grows polynomially in time because they are not $L^1$ functions (unless Schwarzian distributions are introduced), although the Laplace transform is perfectly well-defined.