The Hilbert transform (HT) changes the phase of all the components by -90 degrees for positive frequencies and by +90 degrees for negative frequencies.
The calculation has a per-mutating effect for the real and imaginary parts of the positive frequencies of the initial signal. The real part is changed in sign, the properties of symmetry and anti-symmetry of the Fourier transform being conserved.

The example shows an initial signal (in blue), for which is possible to set the starting time, with its envelope (in black), for which we can set the damping.