HilbertTransformationen

The Hilbert transform is a linear operator that takes a real-valued function of a single variable and produces another real-valued function of the same variable. It is defined for a function $f(x)$ as its convolution with $1/(\pi x)$, specifically $H(f)(x) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{f(t)}{x-t} dt$. This integral is understood in the sense of the Cauchy principal value.

The Hilbert transform has a close relationship with the Fourier transform. If $F(\omega)$ is the Fourier transform of $f(x)$, then the Fourier transform of $H(f)(x)$ is $-i \cdot \text{sgn}(\omega) F(\omega)$, where $\text{sgn}(\omega)$ is the sign function. This property highlights how the Hilbert transform affects different frequency components of a signal, specifically by introducing a phase shift of $-\pi/2$ for positive frequencies and $+\pi/2$ for negative frequencies.

The Hilbert transform finds numerous applications in signal processing, physics, and engineering. It is used in the construction of analytic signals, which are complex-valued functions whose real part is the original signal and whose imaginary part is its Hilbert transform. Analytic signals are crucial for analyzing the envelope and instantaneous frequency of modulated signals. Other applications include solving certain differential equations, in radar and communication systems for quadrature demodulation, and in geophysics for analyzing seismic waves.