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With one common sign convention,
$$
Hf(y)=\frac1\pi\operatorname{PV}
\int_{-\infty}^{\infty}\frac{f(x)}{y-x}\,dx.
$$
Reversing the denominator reverses the sign.


Hilbert transform

{wiki=Hilbert_transform}

The Hilbert transform is a mathematical operation that takes a real-valued function and produces a related complex-valued function. It is widely used in signal processing, communication theory, and various fields of applied mathematics. The transform is particularly useful for analyzing signals and extracting their phase and amplitude characteristics.


 Hilbert transform

ID: hilbert-transform