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I was learning the decomposition of a signal into one even signal and one odd signal:
$$x(t) = x_{e}(t)+x_{o}(t)$$ with $$x_{e}(t) = \frac{1}{2}\cdot [x(t)+x(-t)]$$ and $$x_{o}(t) = \frac{1}{2}\cdot [x(t)-x(-t)]$$
But why do we do that? How does it help us?
Assuming $x(t)$ is purely real, and also the even and odd components, one reason is that the Fourier Transform of an even function is purely real and has even symmetry. And the Fourier Transform of an odd function is purely imaginary and has odd symmetry.
Following Robert's answer, decomposing a (complicated) function into a combination of (simpler) functions is useful in many cases. Here "simpler" is related to showing more symmetry: an even function is symmetric, an odd one is anti-symmetric. And they have intricate properties, related to sums, products, etc. One quite-interesting property is that the derivative of odd functions are even, and the derivative of even functions are odd.
As Fourier transforms are hinged on cosines (even) and sines (odd), knowing that part of a function $f$ is odd (resp. even) can help compute very fast some terms of its Fourier transform $F$ (or Fourier series).
Such oddity knowledge is also useful in several problems: in computing (faster) integrals, when using convexity, etc. For instance, there is a nice result by Cantoni and Butler in : Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. For a symmetric centrosymmetric matrix of order $N$ (such as the ones for autocovariance), their are $⌈N/2⌉$ symmetric and $⌊N/2⌋$ skew symmetric (aka odd) eigenvectors. This observation has helped shape other standard signal processing transformations (DCT - discrete cosine transform, LOT - lapped orthogonal transform, GernLOT etc.). As eigenvalue or singular decomposition are known to be optimal in certain contexts, but heavy to compute, it is handy to have a precomputed approximation whose transform is fast. Indeed, this was quite a topic some decades ago:
