Editing Hilbert transform (section)

=== Bedrosian's theorem ===
'''Bedrosian's theorem''' states that the Hilbert transform of the product of a low-pass and a high-pass signal with non-overlapping spectra is given by the product of the low-pass signal and the Hilbert transform of the high-pass signal, or

<math display="block">\operatorname{H}\left(f_\text{LP}(t)\cdot f_\text{HP}(t)\right) = f_\text{LP}(t)\cdot \operatorname{H}\left(f_\text{HP}(t)\right),</math>

where {{math|''f''<sub>LP</sub>}} and {{math|''f''<sub>HP</sub>}} are the low- and high-pass signals respectively.{{sfn|Schreier|Scharf|2010|loc=14}}  A category of communication signals to which this applies is called the ''narrowband signal model.''  A member of that category is amplitude modulation of a high-frequency sinusoidal "carrier":

<math display="block">u(t) = u_m(t) \cdot \cos(\omega t + \varphi),</math>

where {{math|''u''<sub>''m''</sub>(''t'')}} is the narrow bandwidth "message" waveform, such as voice or music.  Then by Bedrosian's theorem:{{sfn|Bedrosian|1962}}

<math display="block">\operatorname{H}(u)(t) = 
\begin{cases}
+u_m(t) \cdot \sin(\omega t + \varphi) & \text{if } \omega > 0 \\
-u_m(t) \cdot \sin(\omega t + \varphi) & \text{if } \omega < 0
\end{cases}</math>