Translations:Appendix L: Design of Hilbert transforms/23/en

It is important to remember that both signals $x_{n}$ and $y_{n}$ are real. The new signal $y_{n}$ will be created in a special way to meet the requirements that the complex signal $z_{n}$ be an analytic signal. All the information we need is contained in the real and imaginary parts $X_{R}\left(\omega \right){\;{\rm {and\;}}}X_{I}\left(\omega \right)$ for positive angular frequencies $0\leq \omega {<}\pi$. Because the new signal $y_{n}$ is also real, we need only to know its spectrum for the same positive angular frequencies. Let us represent its spectrum by $Y\left(\omega \right)$. Let us do the simplest thing. For a given positive angular frequency $\omega$, treat $X\left(\omega \right)$ as a complex vector and rotate it by -90°. We call the result ${\rm {Y}}\left(\omega \right)$. That is, we let