Editing Hilbert transform (section)

=== Analytic representation ===
{{main article|analytic signal}}
A specific type of [[#Conjugate functions|conjugate function]] is''':'''

<math display="block">u_a(t) \triangleq u(t) + i\cdot H(u)(t),</math>

known as the ''analytic representation'' of <math>u(t).</math>  The name reflects its mathematical tractability, due largely to [[Euler's formula]].  Applying Bedrosian's theorem to the narrowband model, the analytic representation is''':'''<ref>{{harvnb|Osgood|page=320}}</ref>

{{Equation box 1
|cellpadding= 0 |border= 0 |background colour=white
|indent=: |equation={{NumBlk|| <math>\begin{align}
  u_a(t) & = u_m(t) \cdot \cos(\omega t + \varphi) + i\cdot u_m(t) \cdot \sin(\omega t + \varphi), \quad \omega > 0 \\
         & = u_m(t) \cdot \left[\cos(\omega t + \varphi) + i\cdot \sin(\omega t + \varphi)\right], \quad \omega > 0 \\
         & = u_m(t) \cdot e^{i(\omega t + \varphi)}, \quad \omega > 0.\,
\end{align}</math>
 | {{EquationRef|Eq.1}}
 }}
}}

A Fourier transform property indicates that this complex [[heterodyne]] operation can shift all the negative frequency components of {{math|''u''<sub>''m''</sub>(''t'')}} above 0&nbsp;Hz. In that case, the imaginary part of the result is a Hilbert transform of the real part. This is an indirect way to produce Hilbert transforms.