# Fourier vs. Laplace transforms

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.

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.

• It's not a matter of discipline, physicists use Laplace transform as much as an engineer might for the same situations, like solving linear ODEs. You might be better off understanding just what the relation between the two transforms are and what they mean individually. Jan 21 at 8:22
• Mmmh. I know Laplace, I see the mathematical difference. I have seen very few physicists using it, and I have no example in my mind where I can't use Fourier instead. Solving (linear) ODEs can be done with Fourier and Green functions. I would say this is THE way to solve ODEs in physics, normally (maybe just transients depending on inizial conditions are less obvious?) So I still see no good counterexample. Maybe I just don't know it. That is what I am looking for: just one good counterexample.
– Ste
Jan 21 at 9:50
• I mean... also about transients, I have always seen people using characteristic polynomials (in k.. or i*omega... same stuff) to find the set of linearly indep homogeneous solutions... then choosing the right coeff to match the initial conditions is a triviality. Response to a forcing term seems much less obscure using Fourier and a response function. So I am fine with Laplace, why not: I only don't see how/where it is necessary. Or where it provides a superior insight. This s an opportunity question: is it necessary to know it? Is there a real advantage?
– Ste
Jan 21 at 10:17
• In electronics it's usually $Z=1/j \omega C$ because $i$ is a current. Jan 23 at 0:09
• I have usually seen electronics using Fourier Transforms and Controls using Laplace. The reason for Laplace in controls is simply to explicitly include the initial condition at time 0. Jan 25 at 18:35

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.

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 feedbak. 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.

• The Fourier and Laplace transform are interchangeable in a sense. All depends on the basis of the functions that you are using, for Fourier transform, it is harmonic functions, so if you work with these function, it is easier to calculate the rest. As for Laplace transform, it is increasing or decreasing exponents with oscillation. So it is kind of better for this signal. Jan 21 at 11:07
• In you example, you can take a harmonic signal and its Fourier transform is delta function., So if you multiply it with $H(\omega)$ you get the result instantly. In case of exponential decay you multiply $H(s)$ by $\frac{1}{a-s}$ which can be decomposed to the simplier fractions, that will give an exponents as for time response. Jan 21 at 11:07
• I should check my reference, if I remember correctly the author gives solution in both cases for Laplace and Fourier transform Jan 21 at 11:09
• The only problem with assessing things like stability with the Fourier transform is the polynomials you're dealing with have complex coefficients. That makes the math more difficult (but not impossible). Left and right half-planes become upper and lower. Jan 21 at 14:07
• @PierrePolovodov Z transforms are useful for discrete time systems, not only digital. The shaping filter in a delta-sigma modulator is an example of a sampled, discrete time analog system suitable for analysis in the Z domain. Jan 21 at 14:09

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.

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.

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.