Hilbert Transform

$$\begin{array}{rcll} y(t) &\isdef & (h_i \ast x)(t) & \hbox{(Hilbert transform of $x$)} \\ [20pt] h_i(t) &\isdef & \dfrac{1}{\pi t} & \hbox{(Hilbert transform \lq\lq kernel'')} \\ [20pt] H_i(\omega) &\isdef & \left\{\begin{array}{ll} -j, & \omega>0 \\ [5pt] \quad\! j, & \omega<0 \\ [5pt] \quad\! 0, & \omega=0 \\ \end{array} \right. & \hbox{(Hilbert frequency response)}\\ [30pt] x_a(t) &\isdef & x(t) + j y(t) & \hbox{(Analytic signal from $x$)}\\ [20pt] & = & \dfrac{1}{\pi}\displaystyle\int_0^{\infty} X(\omega)e^{j\omega t} d\omega & \hbox{(Note: Lower limit usually $-\infty$)}\\ [20pt] \emph{Proof:} & \\ [10pt] X_a(\omega) &=& X(\omega) + j Y(\omega) & \hbox{(By linearity of Fourier transform)}\\ [20pt] &=& X_++X_- & \hbox{(Apply frequency response)}\\ & & + j [-j X_+ + j X_-] \\ [20pt] &=& 2X_+(\omega) \end{array}$$