Correlation measures the similarity between two signals. It is used to detect patterns or signal alignment, especially in noise.
\[ f^*(x) \star f(x) \iff |F(s)|^2 \]
\[ f^*(x) \star g(x) = \int_{-\infty}^{\infty} f^*(u - x)g(u)du \]
\[ \int_{-\infty}^{\infty} f(x)g^*(x)dx = \int_{-\infty}^{\infty} F(s)G^*(s)ds \]
\[ \frac{d}{dx} \left[f(x) * g(x)\right] = \frac{df(x)}{dx} * g(x) = \frac{dg(x)}{dx} * f(x) \]