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

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