The Hilbert Transform - Part 1

This is part one of a multi-part series that defines The Hilbert Transform and its significance. 1. Definition The Hilbert transform of a signal is defined as \begin{eqnarray} \mathscr{H}[g(t)] = g(t) * \frac{1}{\pi t} = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{g(\tau)}{t-\tau} d\tau = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{g(t-\tau)}{\tau} d\tau \end{eqnarray} The Hilbert transform of is the convolution of with the signal […]