Editing Hilbert transform (section)

== Hilbert transform in signal processing ==

=== 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>

=== 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.

=== {{anchor|Phase/frequency modulation}} Angle (phase/frequency) modulation ===
The form:<ref>{{harvnb|Osgood|page=320}}</ref>

<math display="block">u(t) = A \cdot \cos(\omega t + \varphi_m(t))</math>

is called [[angle modulation]], which includes both [[phase modulation]] and [[frequency modulation]]. The [[Instantaneous phase#Instantaneous frequency|instantaneous frequency]] is &nbsp;<math>\omega + \varphi_m^\prime(t).</math>&nbsp; For sufficiently large {{mvar|ω}}, compared to {{nowrap|<math>\varphi_m^\prime</math>:}}

<math display="block">\operatorname{H}(u)(t) \approx A \cdot \sin(\omega t + \varphi_m(t))</math>
and:
<math display="block">u_a(t) \approx A \cdot e^{i(\omega t + \varphi_m(t))}.</math>

=== Single sideband modulation (SSB) ===
{{Main article|Single-sideband modulation}}
When {{math|''u''<sub>''m''</sub>(''t'')}} in&nbsp;{{EquationNote|Eq.1}} is also an analytic representation (of a message waveform), that is:

<math display="block">u_m(t) = m(t) + i \cdot \widehat{m}(t)</math>

the result is [[single-sideband]] modulation:

<math display="block">u_a(t) = (m(t) + i \cdot \widehat{m}(t)) \cdot e^{i(\omega t + \varphi)}</math>

whose transmitted component is:<ref>{{harvnb|Franks|1969|p=88}}</ref><ref>{{harvnb|Tretter|1995|p=80 (7.9)}}</ref>

<math display="block">\begin{align}
  u(t) &= \operatorname{Re}\{u_a(t)\}\\
       &= m(t)\cdot \cos(\omega t + \varphi) - \widehat{m}(t)\cdot \sin(\omega t + \varphi)
\end{align}</math>

===Causality===
The function <math>h(t) = 1/(\pi t)</math> presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0):
* Its duration is infinite (technically ''infinite [[support (mathematics)|support]]'').  Finite-length ''[[Window function|windowing]]'' reduces the effective frequency range of the transform; shorter windows result in greater losses at low and high frequencies. See also [[quadrature filter]].
* It is a [[causal filter|non-causal filter]].  So a delayed version, <math>h(t-\tau),</math> is required.  The corresponding output is subsequently delayed by <math>\tau.</math>  When creating the imaginary part of an [[analytic signal]], the source (real part) must also be delayed by <math>\tau</math>.