Definition

The Hilbert transform is used to generate a complex signal from a real signal. The Hilbert transform is characterized by the impulse response:

${\displaystyle h(t)={1 \over (\pi t)}}$

The Hilbert Transform of a function x(t) is the convolution of x(t) with the function h(t), above. The Hilbert Transform is defined as such:

${\displaystyle {\widehat {x}}(t)={\mathcal {H}}\{x\}(t)=(h*x)(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {x(\tau )}{t-\tau }}\,d\tau .\,}$

We use the notation ${\displaystyle {\widehat {x}}(t)}$ to denote the Hilbert transformation of x(t).