Hilberttranszformáció

The Hilbert transform is a linear operator that takes a function and produces another function. It is named after the mathematician David Hilbert. In the context of a real-valued function of a single real variable, the Hilbert transform of $f(x)$, denoted as $\hat{f}(x)$ or $H\{f(x)\}$, is defined by the Cauchy principal value of the integral:

$\hat{f}(x) = H\{f(x)\} = \frac{1}{\pi} \text{p.v.} \int_{-\infty}^{\infty} \frac{f(t)}{x-t} dt$

The Hilbert transform is fundamental in signal processing, particularly for the analysis of analytic signals. An analytic signal $z(t)$ is a complex-valued signal constructed from a real-valued signal $f(t)$ and its Hilbert transform $\hat{f}(t)$ as $z(t) = f(t) + i\hat{f}(t)$. The analytic signal has the property that its Fourier transform is zero for negative frequencies, which simplifies many spectral analysis techniques.

The Hilbert transform has several important properties. It is a unitary operator, meaning it preserves the energy of the signal. It also has the property that applying it twice negates the original function: $H\{H\{f(t)\}\} = -f(t)$. This is related to a phase shift of $-\pi/2$ for positive frequencies and $+\pi/2$ for negative frequencies in the frequency domain. Applications of the Hilbert transform extend to areas such as image processing, communications, and the study of wave phenomena.