4 How to determine Hilbert transform of a signal?

input \(x\left ( t\right ) \). Find \(\hat {x}\left ( t\right ) \) which is Hilbert transform of \(x\left ( t\right ) \) defined as \(\hat {x}\left ( t\right ) =x\left ( t\right ) \otimes \frac {1}{\pi t}\)

An easy way is to first find \(\hat {G}\left ( f\right ) \) which is the Fourier transform of \(\hat {x}\left ( t\right ) \) and then inverse it to find \(\hat {x}\left ( t\right ) \)

\[ \hat {G}\left ( f\right ) =-j\ sgn\left ( f\right ) \ G\left ( f\right ) \]

Where \(G\left ( f\right ) \) is Fourier transform of \(x\left ( t\right ) \)