I understand the mathematical concept behind Fourier Series that every periodic function can be represented as the sum of sines and cosines. But I can't understand its applications. Lets assume that the following image represents a signal. Suppose we have a detector that gets the signal on the right of the image. Signal composition

If we write the Fourier Series then we will get the image on the left i.e. we will decompose the signal. But how do we know that the signal we detect is a composition of the signals on the left? I mean if we had 7 signal producing devices and each of them produced only 1 of the signals on the left then we would get the right signal. But this is not the only way we could get the right signal. We could just have a single signal producing device that produces the signal on the right. How do we know then if the signals on the left side are constituents of the signal on the right side?


You are right, Fourier analysis does not help to determine whether a signal has one or several sources. But this is not why it is useful.

We use Fourier analysis in situations that are (a) governed by a linear PDE, such as the heat equation or the wave equation and (b) where there are specific and simple solutions that involve sines or cosines. Fourier analysis lets us represent boundary conditions or initial conditions as Fourier series, see how each component evolves separately in time and/or space, and then sum the resulting functions of time and/or space to get a general solution.


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