Digital Signal Processing/Hilbert Transform

Definition

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

$h(t)=\frac{1}{(\pi t)}$

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:

$\hat{x}(t)=\mathscr{H}\{x\}(t)=(h*x)(t)=\frac{1}{\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{x(\tau )}{t-\tau }d\tau .$

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