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I am a beginner . We convert a signal in time domain to frequency domain by applying Fourier transform on the signal to obtain frequency and phase spectrum.

So,whether the job of Fourier transform is just to convert signal from time domain to frequency domain only (and Whose importance is limited by Heisenberg's uncertainty principle)?

Is it used just to compute only phase and magnitude spectrum in which both spectrums are localised only in frequency domain ?

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    $\begingroup$ A Fourier transform is just one of a very large class of linear operators acting on spaces of functions. The mathematicians have developed a very extensive theory of these spaces and the theory of linear operators that are well-defined on them. In physical terms I have to warn you to mistake the uncertainty principle for a simple mathematical relationship of Fourier transformed data. It's anything but that. The dead giveaway is that the uncertainty principle defines an actual absolute physical scale which separates classical physics from quantum mechanics, but the Fourier transform does not. $\endgroup$
    – CuriousOne
    May 7, 2015 at 15:19
  • $\begingroup$ Asking what the Fourier transformation has to offer aside from transforming signals is kind of like asking the car manufacturer why their cars can't fly, teleport or time travel. Don't you think that going all the way to frequency domain land is enough operation an operator can offer? I mean, compare it to something like addition. How could you live with an operator as lame as that? =) $\endgroup$
    – Name
    May 7, 2015 at 15:59

3 Answers 3

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By Fourier transforming a signal you indeed obtain frequency magnitude and phase information, and that can be very useful in a analysing experimental results.

Another, related, extremely useful application of the Fourier Transform is in terms of solving differential equations. It turns out that many systems can be analysed pretty with differential equations whose solutions are sums of sines and cosines (i.e. waves and oscillatory/vibratory behaviour). So it is in fact useful to Fourier Transform the whole equation and solve it in frequency space to obtain the Fourier coefficients.

The Heisenberg uncertainty principle is connected to a certain property of the Fourier Transform, which states that the width of a signal in time space multiplied by the width of its Fourier transform is always larger or equal than $2\pi$. This is true for all signals and their Fourier transform.

In QM, the probability amplitude for momentum turns out to be (proportional to) the Fourier transform of the probability amplitude for position. Their widths are the uncertainties, and from there you get the Heisenberg Uncertainty Principle.

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    $\begingroup$ However, the Heisenberg Uncertainty Principle is a particular case of the expression $\Delta_{\psi}A \cdot \Delta_{\psi}B \ge \frac{1}{2}\left|<\psi |[A,B]| \psi > \right|$, with $A,B$ operators and $\Delta_{\psi}C=[<\psi|C^2|\psi> - <\psi|C|\psi>^2]^{\frac{1}{2}}$ the uncertainty of the operator $C$. Taking $A$ and $B$ as position and momemtum operators you find the Heisenberg's Principle. This can be demostrated in Hilbert spaces without the use of the FT. Also you can use the expression to compute other uncertainty relations. $\endgroup$
    – Victor Buendía
    May 8, 2015 at 13:04
  • $\begingroup$ That's true, and it was a delight to see the HUP be a special case of a more comprehensive mathematical theorem that can be derived from the mathematical foundations of QM. But that does not add much to the understanding of a beginner. In wave mechanics, the HUP does appear because of the shape of the Schrodinger equation, and this particular theorem take on a more manifest and explicit shape in the FT. @V_Programmer $\endgroup$
    – Andrea
    May 8, 2015 at 13:43
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The Fourier Transform has many applications and you can find it in many places of maths and physics.

I will show a pair of well-know examples; in Quantum Mechanics, the position space and the momentum space are linked by a Fourier Transform:

$$\Psi(\vec{r},t)=\frac{1}{(2\pi \hbar)^{3/2}}\int_{\mathbb{R}^3} e^{i\vec{p}\cdot\vec{r}/\hbar}\Phi(\vec{p},t)d\vec{p}$$ $$\Phi(\vec{p},t)=\frac{1}{(2\pi \hbar)^{3/2}}\int_{\mathbb{R}^3} e^{-i\vec{p}\cdot\vec{r}/\hbar}\Psi(\vec{r},t)d\vec{r}$$

You can see that this is, in fact, a way to change the basis we are using.

Also, in optics, it can be used in coherence theory and in diffraction. Here's an example of a calculus of a coherence factor, which is the FT of the intensity received on a plane (with coordinates $\xi,\eta$):

$$\gamma _{12}(t=0,x,y) = \frac{\int _{S}I(\xi,\eta)e^{ik\left(\xi x /R-\eta y/R\right)}d\xi d\eta}{\int _{S}I(\xi,\eta)d\xi d\eta}$$

Fourier Transform is in fact very important in optics, and as I've said it is used in coherence theory and diffraction.

And of course, the application to change to frequency domain is really useful. Also, it can be used to solve some differential partial differential equations (incluided Maxwell equations in dielectric media, using frequency-time change).

Hope the examples are useful =)

PS. As someone pointed out in the comment, the FT is not linked with Heisenberg Uncertainty Principle. When applied to waves, FT shows that you can't have a 100% monochromatic wave -you must have a $\Delta \omega$ width.

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    $\begingroup$ Even the frequency width definition by Fourier transformation is misleading. A periodic function is determined by three parameters: an amplitude, a frequency and a phase. Within the limits of the sampling theorem one can determine all three parameters with exactly three classical measurements to an arbitrary precision without any uncertainty. If, on the other hand, the spectrum of the wave is not monochromatic to begin with (i.e. if it's not a perfect periodic function), then $\omega$" itself is merely an approximation and one needs the entire spectrum to reconstruct the function. $\endgroup$
    – CuriousOne
    May 7, 2015 at 15:56
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Expanding on the answer by @AndreaDiBiagio about the use of Fourier transforms to solve differential equations, specific applications I encounter occur in simulations of fluid dynamics, magneto-fluid dynamics (MFD), and plasma physics.

Specifically, I have used Fast Fourier Transforms (FFTs) to significantly speed up the solution of a poisson problem for the pressure in an MFD code I wrote. In the below p is the pressure, b is a source term, and I'll restrict things to one dimension.

$$\frac{\partial^{2} p}{\partial x^{2}} = b$$ $$F\Big[\frac{\partial^{2} p}{\partial x^{2}}\Big] = -\omega^{2}F[p]$$ $$p = F^{-1}\Bigg[-\frac{F\Big[\frac{\partial^{2} p}{\partial x^{2}}\Big]}{\omega^{2}}\Bigg] $$

The second equation can be verified by integration by parts (with the definition of the Fourier transform of a function; see my reference below).

As mentioned by @CuriousOne in a comment, there are many linear operators that can act on a function. I'd recommend Farlow's "Partial Differential Equations for Scientists and Engineers" if you want an accessible but slightly more formal introduction.

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