# Translations:Phase rotation/30/en

Because the carrier frequency ${\displaystyle {\omega }_{0}}$ lies to the right of the passband of the envelope spectrum, it follows that the shifted envelope spectrum ${\displaystyle Z\left(\omega \right)=S\left(\omega -\ {\omega }_{0}\right)}$ vanishes for negative values of frequency ${\displaystyle \omega }$. Hence, ${\displaystyle Z\left(\omega \right)}$ is the frequency spectrum of an analytic signal ${\displaystyle z_{n}}$. The inverse Fourier transform of ${\displaystyle Z\left(\omega \right)}$ gives ${\displaystyle z_{n}=s_{n}e^{i{\omega }_{0}n}}$. The real part of ${\displaystyle z_{n}}$ is the modulated wavelet ${\displaystyle x_{n}=s_{n}{\rm {\ cos\ }}{\omega }_{0}n}$, and the imaginary part ${\displaystyle y_{n}=s_{n}{\rm {\ sin\ }}{\omega }_{0}n}$ is the Hilbert transform of the modulated wavelet. We see that the envelope wavelet ${\displaystyle {\rm {s}}_{n}}$ gives the instantaneous amplitude and that ${\displaystyle {\omega }_{0}n}$ gives the instantaneous phase.