${\displaystyle \text{f}\left(t\right)+\text{j g}\left(t\right)=\text{f}\left(t\right)-\frac{\text{j}}{\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{\text{f}\left(t\right)}{\tau -t}}\text{ d}\tau }$

NOTE 1 – The real part f(t) of an analytic signal is the opposite of the Hilbert transform of the imaginary part g(t).

NOTE 2 – If it exists, the complex Fourier transform of an analytic signal is zero for all negative frequencies so that, for instance, the analytic signal can be used to represent a single sideband modulated signal.