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:


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

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:


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

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