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| In [[mathematics]] and in [[signal processing]], the '''Hilbert transform''' is a [[linear operator]] which takes a function, ''u''(''t''), and produces a function, ''H''(''u'')(''t''), with the same [[Domain (mathematics)|domain]]. The Hilbert transform is named after [[David Hilbert]], who first introduced the operator in order to solve a special case of the [[Riemann–Hilbert problem]] for [[holomorphic function]]s. It is a basic tool in [[Fourier analysis]], and provides a concrete means for realizing the [[harmonic conjugate]] of a given function or [[Fourier series]]. Furthermore, in [[harmonic analysis]], it is an example of a [[Singular integral|singular integral operator]], and of a [[Multiplier (Fourier analysis)|Fourier multiplier]]. The Hilbert transform is also important in the field of signal processing where it is used to derive the [[Analytic signal|analytic representation]] of a signal ''u''(''t'').
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| The Hilbert transform was originally defined for [[periodic function]]s, or equivalently for functions on the [[circle]], in which case it is given by [[convolution]] with the ''Hilbert kernel''. More commonly, however, the Hilbert transform refers to a convolution with the ''[[Cauchy integral formula|Cauchy kernel]]'', for functions defined on the [[real line]] '''R''' (the [[boundary (topology)|boundary]] of the [[upper half-plane]]). The Hilbert transform is closely related to the [[Paley–Wiener theorem]], another result relating holomorphic functions in the upper half-plane and [[Fourier transform]]s of functions on the real line.
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| [[Image:Hilbert transform.svg|thumb|300px|The Hilbert transform, in red, of a [[square wave]], in blue]]
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| == Introduction ==
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| The Hilbert transform of ''u'' can be thought of as the [[convolution]] of ''u''(''t'') with the function ''h''(''t'') = 1/(π''t''). Because ''h''(''t'') is not [[integrable]] the integrals defining the convolution do not converge. Instead, the Hilbert transform is defined using the [[Cauchy principal value]] (denoted here by p.v.) Explicitly, the Hilbert transform of a function (or signal) ''u''(''t'') is given by
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| :<math>H(u)(t) = \text{p.v.} \int_{-\infty}^{\infty}u(\tau) h(t-\tau)\, d\tau = \frac{1}{\pi} \ \text{p.v.} \int_{-\infty}^{\infty} \frac{u(\tau)}{t-\tau}\, d\tau</math>
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| provided this integral exists as a principal value. This is precisely the convolution of ''u'' with the [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] p.v. 1/π''t'' (due to {{harvtxt|Schwartz|1950}}; see {{harvtxt|Pandey|1996|loc=Chapter 3}}). Alternatively, by changing variables, the principal value integral can be written explicitly {{harv|Zygmund|1968|loc=§XVI.1}} as
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| :<math>H(u)(t) = -\frac{1}{\pi}\lim_{\varepsilon\rightarrow 0}\int_{\varepsilon}^\infty \frac{u(t + \tau) - u(t - \tau)}{\tau}\,d\tau</math>
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| When the Hilbert transform is applied twice in succession to a function ''u'', the result is negative ''u'':
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| :<math>H(H(u))(t) = -u(t)</math>
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| provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is −''H''. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of ''u''(''t'') (see '''[[Hilbert transform#Relationship with the Fourier transform|Relationship with the Fourier transform]]''', below).
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| For an [[analytic function]] in [[upper half-plane]] the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if ''f''(''z'') is analytic in the plane Im ''z'' > 0 and ''u''(''t'') = Re ''f''(''t'' + 0·''i'' ) then Im ''f''(''t'' + 0·''i'' ) = ''H''(''u'')(''t'') up to an additive constant, provided this Hilbert transform exists. | |
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| ===Notation===
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| {{Unreferenced section|date=December 2011}}
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| In [[signal processing]] the Hilbert transform of ''u''(''t'') is commonly denoted by <math>\widehat u(t).\,</math> However, in mathematics, this notation is already extensively used to denote the Fourier transform of ''u''(''t''). Occasionally, the Hilbert transform may be denoted by <math>\tilde{u}(t)</math>. Furthermore, many sources define the Hilbert transform as the negative of the one defined here.
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| == History ==
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| The Hilbert transform arose in Hilbert's 1905 work on a problem posed by Riemann concerning analytic functions ({{harvtxt|Kress|1989}}; {{harvtxt|Bitsadze|2001}}) which has come to be known as the [[Riemann–Hilbert problem]]. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle ({{harvnb|Khvedelidze|2001}}; {{harvnb|Hilbert|1953}}). Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation {{harv|Hardy|Littlewood|Polya|1952|loc=§9.1}}. Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case {{harv|Hardy|Littlewood|Polya|1952|loc=§9.2}}. These results were restricted to the spaces [[Lp space|''L''<sup>2</sup> and ℓ<sup>2</sup>]]. In 1928, [[Marcel Riesz]] proved that the Hilbert transform can be defined for ''u'' in [[Lp space|''L<sup>p</sup>''('''R''')]] for 1 ≤ ''p'' ≤ ∞, that the Hilbert transform is a [[bounded operator]] on ''L<sup>p</sup>''('''R''') for the same range of ''p'', and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform {{harv|Riesz|1928}}. The Hilbert transform was a motivating example for [[Antoni Zygmund]] and [[Alberto Calderón]] during their study of [[singular integral]]s {{harv|Calderón|Zygmund|1952}}. Their investigations have played a fundamental role in modern harmonic analysis. Various generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.
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| == Relationship with the Fourier transform ==
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| The Hilbert transform is a [[Multiplier (Fourier analysis)|multiplier operator]] {{harv|Duoandikoetxea|2000|loc=Chapter 3}}. The symbol of ''H'' is σ<sub>''H''</sub>(ω) = −''i'' sgn(ω) where sgn is the [[sign function|signum function]]. Therefore:
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| :<math>\mathcal{F}(H(u))(\omega) = (-i\,\operatorname{sgn}(\omega)) \cdot \mathcal{F}(u)(\omega)</math>
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| where <math>\mathcal{F}</math> denotes the [[Fourier transform]]. Since sgn(''x'') = sgn(2π''x''), it follows that this result applies to the three common definitions of <math> \mathcal{F}</math>.
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| By [[Euler's formula]],
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| :<math>\sigma_H(\omega) = \begin{cases}
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| i = e^{+\frac{i\pi}{2}}, & \mbox{for } \omega < 0\\
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| 0, & \mbox{for } \omega = 0\\
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| -i = e^{-\frac{i\pi}{2}}, & \mbox{for } \omega > 0
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| \end{cases}</math>
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| Therefore ''H''(''u'')(''t'') has the effect of shifting the phase of the [[negative frequency]] components of ''u''(''t'') by +90° (π/2 radians) and the phase of the positive frequency components by −90°. And ''i''·''H''(''u'')(''t'') has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.
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| When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of ''u''(''t'') are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e., ''H''(''H''(''u'')) = −''u'', because:
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| :<math>\big(\sigma_H(\omega)\big)^2 = e^{\pm i\pi} = -1 \qquad \text{for } \omega \neq 0</math>
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| == Table of selected Hilbert transforms ==
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| {| class="wikitable"
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| |-
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| ! Signal<br> <math>u(t)\,</math> !! Hilbert transform<ref group="fn">Some authors (e.g., Bracewell) use our −''H'' as their definition of the forward transform. A consequence is that the right column of this table would be negated.
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| </ref><br> <math>H(u)(t)</math>
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| |-
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| | align="center"| <math>\sin(t)</math> <ref group="fn" name="ex02">The Hilbert transform of the ''sin'' and ''cos'' functions can be defined in a distributional sense, if there is a concern that the integral defining them is otherwise conditionally convergent. In the periodic setting this result holds without any difficulty.
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| </ref> || align="center"| <math>-\cos(t)</math>
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| |-
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| | align="center"| <math>\cos(t)</math> <ref group="fn" name="ex02"/> || align="center"| <math>\sin(t)\,</math>
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| |-
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| | align="center"| <math> \exp \left( i t \right) </math> || align="center"| <math> - i \exp \left( i t \right) </math>
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| |-
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| | align="center"| <math> \exp \left( -i t \right) </math> || align="center"| <math> i \exp \left( -i t \right) </math>
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| |-
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| | align="center"| <math>1 \over t^2 + 1</math> || align="center"| <math>t \over t^2 + 1</math>
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| |-
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| | align="center"| '''[[Sinc function]]''' <br /> <math>\sin(t) \over t</math>|| align="center"| <math> 1 - \cos(t)\over t</math>
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| |-
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| | align="center"| '''[[Rectangular function]]''' <br /> <math> \sqcap(t)</math>|| align="center"| <math>{1 \over \pi} \log \left | {t + {1 \over 2} \over t - {1 \over 2}} \right |</math>
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| |-
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| | align="center"| '''[[Dirac delta function]]''' <br /><math>\delta(t) \, </math> || align="center"| <math> {1 \over \pi t}</math>
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| |-
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| | align="center"|'''[[Indicator function|Characteristic Function]]''' <br /> <math>\chi_{[a,b]}(t) \,</math> || align="center"| <math>\frac{1}{\pi}\log \left \vert \frac{t - a}{t - b}\right \vert </math>
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| |}
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| ;Notes
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| <references group="fn" />
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| An extensive table of Hilbert transforms is available ({{harvnb|King|2009}}).
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| Note that the Hilbert transform of a constant is zero.
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| ==Domain of definition==
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| It is by no means obvious that the Hilbert transform is well-defined at all, as the improper integral defining it must converge in a suitable sense. However, the Hilbert transform is well-defined for a broad class of functions, namely those in [[Lp space|''L<sup>p</sub>''('''R''')]] for 1< ''p'' <∞.
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| More precisely, if ''u'' is in ''L<sup>p</sup>''('''R''') for 1<''p''<∞, then the limit defining the improper integral
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| :<math>H(u)(t) = -\frac{1}{\pi}\lim_{\epsilon\downarrow 0}\int_\epsilon^\infty \frac{u(t + \tau) - u(t - \tau)}{\tau}\,d\tau</math>
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| exists for [[almost every]] ''t''. The limit function is also in ''L''<sup>''p''</sup>('''R'''), and is in fact the limit in the mean of the improper integral as well. That is,
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| :<math>-\frac{1}{\pi}\int_\epsilon^\infty \frac{u(t + \tau) - u(t - \tau)}{\tau}\,d\tau\to H(u)(t)</math>
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| as ε→0 in the ''L''<sup>''p''</sup>-norm, as well as pointwise almost everywhere, by the [[#Titchmarsh.27s theorem|Titchmarsh theorem]] {{harv|Titchmarsh|1948|loc=Chapter 5}}.
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| In the case ''p''=1, the Hilbert transform still converges pointwise almost everywhere, but may fail to be itself integrable even locally {{harv|Titchmarsh|1948|loc=§5.14}}. In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an ''L''<sup>1</sup> function does converge, however, in ''L''<sup>1</sup>-weak, and the Hilbert transform is a bounded operator from ''L''<sup>1</sup> to ''L''<sup>1,w</sup> {{harv|Stein|Weiss|1971|loc=Lemma V.2.8}}. (In particular, since the Hilbert transform is also a multiplier operator on ''L''<sup>2</sup>, [[Marcinkiewicz interpolation]] and a duality argument furnishes an alternative proof that ''H'' is bounded on ''L<sup>p</sup>''.)
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| == Properties ==
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| ===Boundedness===
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| If 1<''p''<∞, then the Hilbert transform on ''L''<sup>''p''</sup>('''R''') is a [[bounded linear operator]], meaning that there exists a constant ''C<sub>p</sup>'' such that
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| :<math>\|Hu\|_p \le C_p\| u\|_p</math>
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| for all ''u''∈''L''<sup>''p''</sup>('''R'''). This theorem is due to {{harvtxt|Riesz|1928|loc=VII}}; see also {{harvtxt|Titchmarsh|1948|loc=Theorem 101}}.
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| The best constant ''C<sub>p</sub>'' is given by
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| :<math>C_p = \begin{cases}
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| \tan \frac{\pi}{2p} & \text{for } 1 < p \leq 2\\
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| \cot \frac{\pi}{2p} & \text{for } 2 < p < \infty
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| \end{cases}</math>
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| This result is due to {{harv|Pichorides|1972}}; see also {{harvtxt|Grafakos|2004|loc=Remark 4.1.8}}. The same best constants hold for the periodic Hilbert transform.
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| The boundedness of the Hilbert transform implies the ''L''<sup>''p''</sup>('''R''') convergence of the symmetric partial sum operator
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| :<math>S_R f = \int_{-R}^{R}\hat{f}({\xi})e^{2\pi i x\xi}\,d\xi</math>
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| to ''f'' in ''L<sup>p</sup>''('''R'''), see for example {{harv|Duoandikoetxea|2000|p=59}}.
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| ===Anti-self adjointness===
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| The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between ''L''<sup>''p''</sup>('''R''') and the dual space ''L''<sup>''q''</sup>('''R'''), where ''p'' and ''q'' are [[Hölder conjugate]]s and 1 < ''p'',''q'' < ∞. Symbolically,
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| :<math>\langle Hu, v \rangle = \langle u, -Hv \rangle</math>
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| for ''u'' ∈ ''L<sup>p</sup>''('''R''') and ''v'' ∈ ''L''<sup>''q''</sup>('''R''') {{harv|Titchmarsh|1948|loc=Theorem 102}}.
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| ===Inverse transform===
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| The Hilbert transform is an anti-involution {{harv|Titchmarsh|1948|p=120}}, meaning that
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| :<math>H(H(u)) = -u</math>
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| provided each transform is well-defined. Since ''H'' preserves the space ''L<sup>p</sup>''('''R'''), this implies in particular that the Hilbert transform is invertible on ''L<sup>p</sup>''('''R'''), and that
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| :<math>H^{-1} = -H</math>
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| ===Differentiation===
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| Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:
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| :<math>H\left(\frac{du}{dt}\right) = \frac{d}{dt}H(u)</math>
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| Iterating this identity,
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| :<math>H\left(\frac{d^ku}{dt^k}\right) = \frac{d^k}{dt^k}H(u)</math>
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| This is rigorously true as stated provided ''u'' and its first ''k'' derivatives belong to ''L<sup>p</sup>''('''R''') {{harv|Pandey|1996|loc=§3.3}}. One can check this easily in the frequency domain, where differentiation becomes multiplication by ω.
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| ===Convolutions===
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| The Hilbert transform can formally be realized as a [[convolution]] with the [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] {{harv|Duistermaat|Kolk|2010|p=211}}
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| :<math>h(t) = \text{p.v. }\frac{1}{\pi t}</math>
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| Thus formally,
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| :<math>H(u) = h*u</math>
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| However, ''a priori'' this may only be defined for ''u'' a distribution of [[compact support]]. It is possible to work somewhat rigorously with this since compactly supported functions (which are distributions ''a fortiori'') are [[dense (topology)|dense]] in ''L<sup>p</sup>''. Alternatively, one may use the fact that ''h''(''t'') is the [[distributional derivative]] of the function log|''t''|/π; to wit
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| :<math>H(u)(t) = \frac{d}{dt}\left(\frac{1}{\pi} (u*\log|\cdot|)(t)\right)</math>
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| For most operational purposes the Hilbert transform can be treated as a convolution. For example, in a formal sense, the Hilbert transform of a convolution is the convolution of the Hilbert transform on either factor:
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| :<math>H(u*v) = H(u)*v = u*H(v)</math>
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| This is rigorously true if ''u'' and ''v'' are compactly supported distributions since, in that case,
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| :<math> h*(u*v) = (h*u)*v = u*(h*v)</math>
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| By passing to an appropriate limit, it is thus also true if ''u'' ∈ ''L''<sup>''p''</sup> and ''v'' ∈ ''L''<sup>''r''</sup> provided
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| :<math>1 < \frac{1}{p} + \frac{1}{r}</math> | |
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| a theorem due to {{harvtxt|Titchmarsh|1948|loc=Theorem 104}}.
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| ===Invariance===
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| The Hilbert transform has the following invariance properties on ''L''<sup>2</sup>('''R''').
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| * It commutes with translations. That is, it commutes with the operators ''T''<sub>''a''</sub>ƒ(''x'') = ƒ(''x'' + ''a'') for all ''a'' in '''R'''
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| * It commutes with positive dilations. That is it commutes with the operators ''M''<sub>λ</sub>ƒ(''x'') = ƒ(λ''x'') for all λ > 0.
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| * It [[Anticommutativity|anticommutes]] with the reflection ''R''ƒ(''x'') = ƒ(−x).
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| Up to a multiplicative constant, the Hilbert transform is the only bounded operator on ''L''<sup>2</sup> with these properties {{harv|Stein|1970|loc=§III.1}}.
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| In fact there is a larger group of operators commuting with the Hilbert transform. The group SL(2,'''R''') acts by unitary operators ''U''<sub>''g''</sub> on the space ''L''<sup>2</sup>('''R''') by the formula
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| :<math>\displaystyle{U_{g}^{-1}f(x) = (cx + d)^{-1} f\left({ax + b \over cx + d}\right),\,\,\,g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}}</math>
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| This [[unitary representation]] is an example of a [[principal series representation]] of SL(2,'''R'''). In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, [[Hardy space]] ''H''<sup>2</sup>('''R''') and its conjugate. These are the spaces of ''L''<sup>2</sup> boundary values of holomorphic functions on the upper and lower halfplanes. ''H''<sup>2</sup>('''R''') and its conjugate consist of exactly those ''L''<sup>2</sup> functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to ''H'' = -''i'' (2''P'' - ''I''), with ''P'' being the orthogonal projection from ''L''<sup>2</sup>('''R''') onto ''H''<sup>2</sup>('''R'''), it follows that ''H''<sup>2</sup>('''R''') and its orthogonal are eigenspaces of ''H'' for the eigenvalues ± ''i''. In other words ''H'' commutes with the operators ''U''<sub>''g''</sub>. The restrictions of the operators ''U''<sub>''g''</sub> to ''H''<sup>2</sup>('''R''') and its conjugate give irreducible representations of SL(2,'''R''')—the so-called [[limit of discrete series representation]]s.<ref>See:
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| *{{harvnb|Bargmann|1947}}
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| *{{harvnb|Lang|1985}}
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| *{{harvnb|Sugiura|1990}}
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| </ref>
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| ==Extending the domain of definition==
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| ===Hilbert transform of distributions===
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| It is further possible to extend the Hilbert transform to certain spaces of [[distribution (mathematics)|distributions]] {{harv|Pandey|1996|loc=Chapter 3}}. Since the Hilbert transform commutes with differentiation, and is a bounded operator on ''L<sup>p</sup>'', ''H'' restricts to give a continuous transform on the [[inverse limit]] of [[Sobolev spaces]]:
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| :<math>\mathcal{D}_{L^p} = \underset{n\to\infty}{\underset{\longleftarrow}{\lim}} W^{n,p}(\mathbb{R})</math>
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| The Hilbert transform can then be defined on the dual space of <math>\mathcal{D}_{L^p}</math>, denoted <math>\mathcal{D}_{L^p}'</math>, consisting of ''L<sup>p</sup>'' distributions. This is accomplished by the duality pairing: for <math>u\in \mathcal{D}'_{L^p}</math>, define <math>H(u)\in \mathcal{D}'_{L^p}</math> by
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| :<math>\langle Hu, v \rangle \overset{\mathrm{def}}{=} \langle u, -Hv\rangle</math>
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| for all <math>v\in\mathcal{D}_{L^p}</math>.
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| It is possible to define the Hilbert transform on the space of [[tempered distributions]] as well by an approach due to {{harvtxt|Gel'fand|Shilov|1967}}{{Page needed|date=December 2011}}, but considerably more care is needed because of the singularity in the integral.
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| === Hilbert transform of bounded functions ===
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| The Hilbert transform can be defined for functions in ''L''<sup>∞</sup>('''R''') as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps ''L''<sup>∞</sup>('''R''') to the [[Banach space]] of [[bounded mean oscillation]] (BMO) classes.
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| Interpreted naively, the Hilbert transform of a bounded function is clearly ill-defined. For instance, with ''u'' = sgn(''x''), the integral defining ''H''(''u'') diverges almost everywhere to ±∞. To alleviate such difficulties, the Hilbert transform of an ''L''<sup>∞</sup>-function is therefore defined by the following [[regularization (physics)|regularized]] form of the integral
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| :<math>H(u)(t) = \text{p.v.} \int_{-\infty}^\infty u(\tau)\left\{h(t - \tau)- h_0(-\tau)\right\}\,d\tau</math>
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| where as above ''h''(''x'') = 1/π''x'' and
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| :<math>h_0(x) = \begin{cases} 0&\mathrm{if\ }|x| < 1\\ \frac{1}{\pi x} &\mathrm{otherwise} \end{cases}</math>
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| The modified transform ''H'' agrees with the original transform on functions of compact support by a general result of {{harvtxt|Calderón|Zygmund|1952}}; see {{harvtxt|Fefferman|1971}}. The resulting integral, furthermore, converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation. | |
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| A [[deep result]] of {{harvtxt|Fefferman|1971}} and {{harvtxt|Fefferman|Stein|1972}} is that a function is of bounded mean oscillation if and only if it has the form ''ƒ'' + ''H''(''g'') for some ''ƒ'', ''g'' ∈ ''L''<sup>∞</sup>('''R''').
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| ==Conjugate functions==
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| The Hilbert transform can be understood in terms of a pair of functions ''f''(''x'') and ''g''(''x'') such that the function
| |
| :<math>F(x) = f(x) + ig(x)</math>
| |
| is the boundary value of a [[holomorphic function]] ''F''(''z'') in the upper half-plane {{harv|Titchmarsh|1948|loc=Chapter V}}. Under these circumstances, if ''f'' and ''g'' are sufficiently integrable, then one is the Hilbert transform of the other.
| |
| | |
| Suppose that ''f'' ∈ ''L''<sup>''p''</sup>('''R'''). Then, by the theory of the [[Poisson integral]], ''f'' admits a unique harmonic extension into the upper half-plane, and this extension is given by
| |
| | |
| :<math>u(x + iy) = u(x, y) = \frac{1}{\pi}\int_{-\infty}^\infty f(s)\frac{y}{(x - s)^2 + y^2}\,ds</math>
| |
| | |
| which is the convolution of ''f'' with the [[Poisson kernel]]
| |
| | |
| :<math>P(x, y) = \frac{1}{\pi}\frac{y}{x^2 + y^2}</math>
| |
| | |
| Furthermore, there is a unique harmonic function ''v'' defined in the upper half-plane such that ''F''(''z'') = ''u''(''z'') + i''v''(''z'') is holomorphic and
| |
| :<math>\lim_{y \to \infty} v(x + iy) = 0</math>
| |
| | |
| This harmonic function is obtained from ''f'' by taking a convolution with the '''conjugate Poisson kernel'''
| |
| | |
| :<math>Q(x, y) = \frac{1}{\pi}\frac{x}{x^2 + y^2}</math>
| |
| | |
| Thus
| |
| :<math>v(x, y) = \frac{1}{\pi}\int_{-\infty}^\infty f(s)\frac{x - s}{(x - s)^2 + y^2}\,ds</math>
| |
| | |
| Indeed, the real and imaginary parts of the Cauchy kernel are
| |
| :<math>\frac{i}{\pi z} = P(x, y) + iQ(x, y)</math>
| |
| | |
| so that ''F'' = ''u'' + i''v'' is holomorphic by the [[Cauchy integral theorem]].
| |
| | |
| The function ''v'' obtained from ''u'' in this way is called the [[harmonic conjugate]] of ''u''. The (non-tangential) boundary limit of ''v''(''x'',''y'') as ''y'' → 0 is the Hilbert transform of ''f''. Thus, succinctly,
| |
| :<math>H(f) = \lim_{y \to 0}Q(-, y) \star f</math>
| |
| | |
| === Titchmarsh's theorem ===
| |
| A theorem due to [[Edward Charles Titchmarsh]] makes precise the relationship between the boundary values of holomorphic functions in the upper half-plane and the Hilbert transform {{harv|Titchmarsh|1948|loc=Theorem 95}}. It gives necessary and sufficient conditions for a complex-valued [[square-integrable]] function ''F''(''x'') on the real line to be the boundary value of a function in the [[Hardy space]] ''H''<sup>2</sup>(''U'') of holomorphic functions in the upper half-plane ''U''.
| |
| | |
| The theorem states that the following conditions for a complex-valued square-integrable function ''F'' : '''R''' → '''C''' are equivalent:
| |
| | |
| * ''F''(''x'') is the limit as ''z'' → ''x'' of a holomorphic function ''F''(''z'') in the upper half-plane such that
| |
| | |
| ::<math>\int_{-\infty}^\infty |F(x + iy)|^2\,dx < K</math>
| |
| | |
| * −Im(''F'') is the Hilbert transform of Re(''F''), where Re(''F'') and Im(''F'') are real-valued functions with ''F'' = Re(''F'') + i Im(''F'').
| |
| | |
| * The [[Fourier transform]] <math>\mathcal{F}(F)(x)</math> vanishes for ''x'' < 0.
| |
| | |
| A weaker result is true for functions of class [[Lp space|''L''<sup>''p''</sup>]] for ''p'' > 1 {{harv|Titchmarsh|1948|loc=Theorem 103}}. Specifically, if ''F''(''z'') is a holomorphic function such that
| |
| | |
| :<math>\int_{-\infty}^\infty |F(x + iy)|^p\,dx < K </math>
| |
| | |
| for all ''y'', then there is a complex-valued function ''F''(''x'') in ''L<sup>p</sup>''('''R''') such that ''F''(''x'' + i''y'') → ''F''(''x'') in the ''L<sup>p</sup>'' norm as ''y'' → 0 (as well as holding pointwise [[almost everywhere]]). Furthermore,
| |
| | |
| :<math>F(x) = f(x) - i g(x)</math>
| |
| | |
| where ƒ is a real-valued function in ''L''<sup>''p''</sup>('''R''') and ''g'' is the Hilbert transform (of class ''L<sup>p</sup>'') of ƒ.
| |
| | |
| This is not true in the case ''p'' = 1. In fact, the Hilbert transform of an ''L''<sup>1</sup> function ƒ need not converge in the mean to another ''L''<sup>1</sup> function. Nevertheless {{harv|Titchmarsh|1948|loc=Theorem 105}}, the Hilbert transform of ƒ does converge almost everywhere to a finite function ''g'' such that
| |
| | |
| :<math>\int_{-\infty}^\infty \frac{|g(x)|^p}{1 + x^2}\,dx < \infty</math>
| |
| | |
| This result is directly analogous to one by [[Andrey Kolmogorov]] for Hardy functions in the disc {{harv|Duren|1970|loc=Theorem 4.2}}.
| |
| | |
| === Riemann–Hilbert problem ===
| |
| One form of the [[Riemann–Hilbert problem]] seeks to identify pairs of functions ''F''<sub>+</sub> and ''F''<sub>−</sub> such that ''F''<sub>+</sub> is [[holomorphic function|holomorphic]] on the upper half-plane and ''F''<sub>−</sub> is holomorphic on the lower half-plane, such that for ''x'' along the real axis,
| |
| :<math>F_+(x) - F_-(x) = f(x)</math>
| |
| | |
| where ''f''(''x'') is some given real-valued function of ''x'' ∈ '''R'''. The left-hand side of this equation may be understood either as the difference of the limits of ''F''<sub>±</sub> from the appropriate half-planes, or as a [[hyperfunction]] distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.
| |
| | |
| Formally, if ''F''<sub>±</sub> solve the Riemann–Hilbert problem
| |
| :<math>f(x) = F_+(x) - F_-(x)</math>
| |
| | |
| then the Hilbert transform of ''f''(''x'') is given by
| |
| :<math>H(f)(x) = \frac{1}{i}(F_+(x) + F_-(x))</math> {{harv|Pandey|1996|loc=Chapter 2}}.
| |
| | |
| == Hilbert transform on the circle ==
| |
| {{see also|Hardy space}}
| |
| For a periodic function ''f'' the circular Hilbert transform is defined as
| |
| | |
| :<math>\tilde f(x) = \frac{1}{2\pi}\text{ p.v.}\int_0^{2\pi}f(t)\cot\left(\frac{x - t}{2}\right)\,dt</math>
| |
| | |
| The circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series. The kernel,
| |
| :<math>\scriptstyle \cot\left(\frac{x - t}{2}\right)</math> is known as the '''Hilbert kernel''' since it was in this form the Hilbert transform was originally studied {{harv|Khvedelidze|2001}}.
| |
| | |
| The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel 1/''x'' periodic. More precisely, for ''x''≠0
| |
| | |
| :<math>\frac{1}{2}\cot\left(\frac{x}{2}\right) = \frac{1}{x} + \sum_{n=1}^\infty \left(\frac{1}{x + 2n\pi} + \frac{1}{x - 2n\pi} \right)</math>
| |
| | |
| Many results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.
| |
| | |
| Another more direct connection is provided by the Cayley transform ''C''(''x'') = (''x'' – ''i'') / (''x'' + ''i''), which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map
| |
| | |
| :<math>\displaystyle{Uf(x) = \pi^{-\frac{1}{2}} (x + i)^{-1} f(C(x))}</math>
| |
| | |
| of ''L''<sup>2</sup>('''T''') onto ''L''<sup>2</sup>('''R'''). The operator ''U'' carries the Hardy space ''H''<sup>2</sup>('''T''') onto the Hardy space ''H''<sup>2</sup>('''R''').<ref>{{harvnb|Rosenblum|Rovnyak|1997|p=92}}</ref>
| |
| | |
| == Hilbert transform in signal processing ==
| |
| | |
| === Bedrossian's theorem ===
| |
| '''Bedrossian'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>H(f_{LP}(t) f_{HP}(t)) = f_{LP}(t) H(f_{HP}(t))</math>
| |
| | |
| where ''f<sub>LP</sub>'' and ''f<sub>HP</sub>'' are the low- and high-pass signals respectively {{harv|Schreier|Scharf|2010|loc=14}}.
| |
| | |
| Amplitude modulated signals are modeled as the product of a [[bandlimited]] "message" waveform, ''u''<sub>''m''</sub>(''t''), and a sinusoidal "carrier":
| |
| | |
| :<math>u(t) = u_m(t) \cdot \cos(\omega t + \phi)</math>
| |
| | |
| When ''u<sub>m</sub>''(''t'') has no frequency content above the carrier frequency, <math>\frac{\omega}{2\pi}\text{ Hz,}</math> then by Bedrossian's theorem:
| |
| | |
| :<math>H(u)(t) = u_m(t) \cdot \sin(\omega t + \phi)</math> {{harv|Bedrossian|1962}}
| |
| | |
| === Analytic representation ===
| |
| {{main|analytic signal}}
| |
| {{Unreferenced section|date=December 2011}}
| |
| In the context of signal processing, the conjugate function interpretation of the Hilbert transform, discussed above, gives the analytic representation of a signal ''u''(''t''):
| |
| | |
| :<math>u_a(t) = u(t) + i\cdot H(u)(t)</math>
| |
| | |
| which is a [[holomorphic function]] in the upper half plane.
| |
| | |
| For the narrowband model [above], the analytic representation is:
| |
| | |
| :{|
| |
| |<math>u_a(t)</math>
| |
| |<math>= u_m(t) \cdot \cos(\omega t + \phi) + i\cdot u_m(t) \cdot \sin(\omega t + \phi)</math>
| |
| |-
| |
| |
| |
| |<math>= u_m(t) \cdot \left[\cos(\omega t + \phi) + i\cdot \sin(\omega t + \phi)\right]</math>
| |
| |}
| |
| {{NumBlk|:::|<math>= u_m(t) \cdot e^{i(\omega t + \phi)}\,</math> (by [[Euler's formula]])|{{EquationRef|Eq.1}}}}
| |
| | |
| This complex [[heterodyne]] operation shifts all the frequency components of ''u''<sub>''m''</sub>(''t'') above 0 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.
| |
| | |
| === Phase/frequency modulation ===
| |
| {{Unreferenced section|date=December 2011}}
| |
| The form:
| |
| | |
| :<math>u(t) = A\cdot \cos(\omega t + \phi_m(t))</math>
| |
| | |
| is called [[phase modulation|phase (or frequency) modulation]]. The [[Instantaneous phase#Instantaneous frequency|instantaneous frequency]] is <math>\omega + \phi_m^\prime(t).</math> For sufficiently large ω, compared to <math>\phi_m^\prime</math>:
| |
| | |
| :<math>H(u)(t) \approx A \cdot \sin(\omega t + \phi_m(t))</math>
| |
| | |
| and:
| |
| | |
| :<math>u_a(t) \approx A \cdot e^{i(\omega t + \phi_m(t))}</math>
| |
| | |
| === Single sideband modulation (SSB) ===
| |
| {{Unreferenced section|date=December 2011}}
| |
| {{Main|Single-sideband_modulation}}
| |
| When ''u''<sub>''m''</sub>(''t'') in {{EquationNote|Eq.1}} is <u>also</u> an analytic representation (of a message waveform), that is:
| |
| | |
| :<math>u_m(t) = m(t) + i \cdot \widehat{m}(t)</math>
| |
| | |
| the result is [[single-sideband]] modulation:
| |
| | |
| :<math>u_a(t) = (m(t) + i\cdot \widehat{m}(t)) \cdot e^{i(\omega t + \phi)}</math>
| |
| | |
| whose transmitted component is:
| |
| | |
| :<math>\begin{align}
| |
| u(t) &= \operatorname{Re}\{u_a(t)\}\\
| |
| &= m(t)\cdot \cos(\omega t + \phi) - \widehat{m}(t)\cdot \sin(\omega t + \phi)
| |
| \end{align}</math>
| |
| | |
| ===Causality===
| |
| {{Unreferenced section|date=December 2011}}
| |
| The function ''h'' with ''h''(''t'') = 1/(π''t'') is a [[causal filter|non-causal filter]] and therefore cannot be implemented as is, if ''u'' is a time-dependent signal. If ''u'' is a function of a non-temporal variable (e.g., spatial) the non-causality might not be a problem. The filter is also of infinite [[support (mathematics)|support]] which may be a problem in certain applications. Another issue relates to what happens with the zero frequency (DC), which can be avoided by assuring that ''s'' does not contain a DC-component.
| |
| | |
| A practical implementation in many cases implies that a finite support filter, which in addition is made causal by means of a suitable delay, is used to approximate the computation. The approximation may also imply that only a specific frequency range is subject to the characteristic phase shift related to the Hilbert transform. See also [[quadrature filter]].
| |
| | |
| == Discrete Hilbert transform ==
| |
| [[Image:Bandpass discrete Hilbert transform filter.tif|thumb|400px|right|Figure 1: Filter whose frequency response is bandlimited to about 95% of the Nyquist frequency]]
| |
| [[Image:Highpass discrete Hilbert transform filter.tif|thumb|400px|right|Figure 2: Hilbert transform filter with a highpass frequency response]]
| |
| [[Image:DFT approximation to Hilbert filter.png|thumb|400px|right|Figure 3.]]
| |
| [[Image:Effect of circular convolution on discrete Hilbert transform.png|thumb|400px|right|Figure 4. The Hilbert transform of cos(wt) is sin(wt). This figure shows the difference between sin(wt) and an approximate Hilbert transform computed by the MATLAB library function, hilbert(·)]]
| |
| For a discrete function, u[n], with [[discrete-time Fourier transform]] (DTFT), U(ω), the Hilbert transform is given by:
| |
| | |
| :<math>H(u)[n] = \scriptstyle{DTFT}^{-1} \displaystyle \{U(\omega)\cdot \sigma_H(\omega)\}</math>
| |
| | |
| where:
| |
| | |
| :<math>\sigma_H(\omega)\ \stackrel{\mathrm{def}}{=}\
| |
| \begin{cases}
| |
| e^{+i\pi/2}, & -\pi < \omega < 0 \\
| |
| e^{-i\pi/2}, & 0 < \omega < \pi\\
| |
| 0, & \omega = -\pi, 0, \pi
| |
| \end{cases}</math>
| |
| | |
| And by the [[Convolution theorem#Functions of a discrete variable... sequences|convolution theorem]], an equivalent formulation is:
| |
| | |
| :<math>H(u)[n] = u[n] * h[n]</math>
| |
| | |
| where:
| |
| | |
| :<math>h[n]\ \stackrel{\mathrm{def}}{=}\ \scriptstyle{DTFT}^{-1} \big \{\displaystyle \sigma_H(\omega)\big \} =
| |
| \begin{cases}
| |
| 0, & \mbox{for }n\mbox{ even}\\
| |
| \frac2{\pi n} & \mbox{for }n\mbox{ odd}
| |
| \end{cases}</math>
| |
| | |
| When the convolution is performed numerically, an [[finite impulse response|FIR]] approximation is substituted for ''h''[''n''], as shown in '''Figure 1''', and we see rolloff of the passband at the low and high ends (0 and Nyquist), resulting in a bandpass filter. The high end can be restored, as shown in '''Figure 2''', by an FIR that more closely resembles samples of the smooth, continuous-time ''h''(''t''). But as a practical matter, a properly-sampled ''u''[''n''] sequence has no useful components at those frequencies. As the impulse response gets longer, the low end frequencies are also restored.<ref>Hilbert studied the discrete transform:
| |
| | |
| :<math>\frac{1}{n} * u[n] = \sum_{m=-\infty}^\infty \frac{u(m)}{n-m}\qquad m \neq n</math>
| |
| | |
| and showed that for ''u''(''n'') in ℓ<sup>2</sup> the sequence ''H''(''u'')[''n''] is also in ℓ<sup>2</sup> (see [[Hilbert's inequality]]). An elementary proof of this fact can be found in {{harv|Grafakos|1994}}. This transform was used by E. C. Titchmarsh to give alternate proofs of the results of M. Riesz in the continuous case ({{harvnb|Titchmarsh|1926}}; {{harvnb|Hardy|Littlewood|Polya|1952|loc=¶314}}), but it is not used for pragmatic signal processing.</ref>
| |
| | |
| With an FIR approximation to ''h''[''n''], a method called '''[[Overlap-save method|overlap-save]]''' is an efficient way to perform the convolution on a long ''u''[''n''] sequence. Sometimes the array FFT{''h''[''n'']} is replaced by corresponding samples of σ<sub>''H''</sub>(ω). That has the effect of convolving with the [[periodic summation]]:<ref>see [[Convolution theorem#Functions of a discrete variable... sequences|Convolution Theorem]]</ref>
| |
| | |
| :<math>h_N[n]\ \stackrel{\text{def}}{=}\ \sum_{m=-\infty}^{\infty} h[n - mN]</math>
| |
| | |
| Figure 3 compares a half-cycle of ''h<sub>N</sub>''[''n''] with an equivalent length portion of ''h''[''n'']. The difference between them and the fact that they are not shorter than the segment length (''N'') are sources of distortion that are managed (reduced) by increasing the segment length and overlap parameters.
| |
| | |
| The popular [[MATLAB]] function, '''[http://www.mathworks.com/help/toolbox/signal/ref/hilbert.html;jsessionid=67ed4e69e9729363548abed31054 hilbert(u,N)]''', returns an approximate discrete Hilbert transform of ''u''[''n''] in the imaginary part of the complex output sequence. The real part is the original input sequence, so that the complex output is an [[Analytic signal|analytic representation]] of ''u''[''n'']. Similar to the discussion above, hilbert(u, N) only uses samples of the sgn(ω) distribution and therefore convolves with ''h<sub>N</sub>''[''n'']. Distortion can be managed by choosing ''N'' larger than the actual ''u''[''n''] sequence and discarding an appropriate number of output samples. An example of this type of distortion is shown in Figure 4.
| |
| | |
| == See also ==
| |
| * [[Analytic signal]]
| |
| * [[Harmonic conjugate]]
| |
| * [[Hilbert spectroscopy]]
| |
| * [[Hilbert transform in the complex plane]]
| |
| * [[Hilbert–Huang transform]]
| |
| * [[Kramers–Kronig relation]]
| |
| * [[Single sideband|Single-sideband signal]]
| |
| * [[Singular integral operators of convolution type]]
| |
| | |
| == Notes ==
| |
| {{reflist}}
| |
| | |
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| year=1985|ISBN= 0-387-96198-4}}
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| * {{citation|first=S.|last=Pichorides|title=On the best value of the constants in the theorems of Riesz, Zygmund, and Kolmogorov|journal=Studia Mathematica| volume=44| year=1972| pages=165–179}}
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| *{{citation|title=Statistical signal processing of complex-valued data: the theory of improper and noncircular signals|last1=Schreier|first1=P.|first2=L.|last2=Scharf|publisher=Cambridge University Press|year=2010}}
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| * {{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=Princeton University Press|year=1970|isbn=0-691-08079-8}}.
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| * {{citation|first1=Elias|last1=Stein|authorlink1=Elias Stein|first2=Guido|last2=Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=0-691-08078-X}}.
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| *{{citation|title=Unitary Representations and Harmonic Analysis: An Introduction|volume=44|series=North-Holland Mathematical Library|first=Mitsuo |last=Sugiura|edition=2nd|publisher=Elsevier|year= 1990|isbn=0444885935}}
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| * {{citation|last=Titchmarsh|first=E|authorlink=Edward Charles Titchmarsh|title= Reciprocal formulae involving series and integrals|journal=Mathematische Zeitschrift|pages= 321–347|volume=25|issue=1|year=1926|doi=10.1007/BF01283842}}.
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| * {{citation|last=Titchmarsh|first=E|authorlink=Edward Charles Titchmarsh|title=Introduction to the theory of Fourier integrals|isbn=978-0-8284-0324-5|year=1948|edition=2nd|publication-date=1986|publisher=Clarendon Press|publication-place=Oxford University}}.
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| * {{citation|title=Trigonometric series|first=Antoni|last=Zygmund|authorlink=Antoni Zygmund|publisher=Cambridge University Press|year=1968|publication-date=1988|isbn=978-0-521-35885-9|edition=2nd}}.
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| == External links ==
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| * [http://arxiv.org/abs/0909.1426 Derivation of the boundedness of the Hilbert transform]
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| * [http://mathworld.wolfram.com/HilbertTransform.html Mathworld Hilbert transform] — Contains a table of transforms
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| * [http://ccrma-www.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html Analytic Signals and Hilbert Transform Filters]
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| * {{mathworld|title=Titchmarsh theorem|urlname=TitchmarshTheorem}}
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| * [http://w3.msi.vxu.se/exarb/mj_ex.pdf Mathias Johansson, "The Hilbert transform"] a student level summary to Hilbert transformation. {{Dead link|date=July 2013}} [https://web.archive.org/web/20120205214945/http://w3.msi.vxu.se/exarb/mj_ex.pdf (via www.archive.org)]
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| * [http://www.geol.ucsb.edu/faculty/toshiro/GS256_Lecture3.pdf GS256 Lecture 3: Hilbert Transformation], an entry level introduction to Hilbert transformation. {{Dead link|date=July 2013}} [https://web.archive.org/web/20120227061333/http://www.geol.ucsb.edu/faculty/toshiro/GS256_Lecture3.pdf (via www.archive.org)]
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| {{DEFAULTSORT:Hilbert Transform}}
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| [[Category:Harmonic functions]]
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| [[Category:Integral transforms]]
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| [[Category:Signal processing]]
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| [[Category:Singular integrals]]
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