Английская Википедия:Hilbert transform

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Шаблон:Short description In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, Шаблон:Math of a real variable and produces another function of a real variable Шаблон:Math. The Hilbert transform is given by the Cauchy principal value of the convolution with the function <math>1/(\pi t)</math> (see Шаблон:Slink). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° (Шаблон:Pi/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see Шаблон:Slink). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal Шаблон:Math. The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

Definition

The Hilbert transform of Шаблон:Mvar can be thought of as the convolution of Шаблон:Math with the function Шаблон:Math, known as the Cauchy kernel. Because 1/Шаблон:Mvar is not integrable across Шаблон:Math, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by Шаблон:Math). Explicitly, the Hilbert transform of a function (or signal) Шаблон:Math is given by

<math display="block">

\operatorname{H}(u)(t) = \frac{1}{\pi}\, \operatorname{p.v.} \int_{-\infty}^{+\infty} \frac{u(\tau)}{t - \tau}\,\mathrm{d}\tau,

</math>

provided this integral exists as a principal value. This is precisely the convolution of Шаблон:Mvar with the tempered distribution Шаблон:Math. Alternatively, by changing variables, the principal-value integral can be written explicitly as

<math display="block">

\operatorname{H}(u)(t) = \frac{2}{\pi}\, \lim_{\varepsilon \to 0} \int_\varepsilon^\infty \frac{u(t - \tau) - u(t + \tau)}{2\tau} \,\mathrm{d}\tau.

</math>

When the Hilbert transform is applied twice in succession to a function Шаблон:Mvar, the result is

<math display="block">

\operatorname{H}\bigl(\operatorname{H}(u)\bigr)(t) = -u(t),

</math>

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is <math>\operatorname{H}^3</math>. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of Шаблон:Math (see Шаблон:Slink below).

For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if Шаблон:Math is analytic in the upper half complex plane Шаблон:Math, and Шаблон:Math, then Шаблон:Math up to an additive constant, provided this Hilbert transform exists.

Notation

In signal processing the Hilbert transform of Шаблон:Math is commonly denoted by <math>\hat{u}(t)</math>. However, in mathematics, this notation is already extensively used to denote the Fourier transform of Шаблон:Math. 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.

History

The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions,Шаблон:SfnШаблон:Sfn 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.Шаблон:SfnШаблон:Sfn 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.Шаблон:Sfn Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.Шаблон:Sfn These results were restricted to the spaces [[Lp space|Шаблон:Math and Шаблон:Math]]. In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in <math>L^p(\mathbb{R})</math> (L^p space) for Шаблон:Math, that the Hilbert transform is a bounded operator on <math>L^p(\mathbb{R})</math> for Шаблон:Math, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.Шаблон:Sfn The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals.Шаблон:Sfn 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.

Relationship with the Fourier transform

The Hilbert transform is a multiplier operator.Шаблон:Sfn The multiplier of Шаблон:Math is Шаблон:Math, where Шаблон:Math is the signum function. Therefore:

<math display="block">\mathcal{F}\bigl(\operatorname{H}(u)\bigr)(\omega) = -i \sgn(\omega) \cdot \mathcal{F}(u)(\omega) ,</math>

where <math>\mathcal{F}</math> denotes the Fourier transform. Since Шаблон:Math, it follows that this result applies to the three common definitions of <math> \mathcal{F}</math>.

By Euler's formula, <math display="block">\sigma_\operatorname{H}(\omega) = \begin{cases}

~~i = e^{+i\pi/2}, & \text{for } \omega < 0,\\
~~                      0, & \text{for } \omega = 0,\\
 -i = e^{-i\pi/2}, & \text{for } \omega > 0.

\end{cases}</math>

Therefore, Шаблон:Math has the effect of shifting the phase of the negative frequency components of Шаблон:Math by +90° (Шаблон:Frac radians) and the phase of the positive frequency components by −90°, and Шаблон:Math has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation (i.e., a multiplication by −1).

When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of Шаблон:Math are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e., Шаблон:Math, because

<math display="block">\left(\sigma_\operatorname{H}(\omega)\right)^2 = e^{\pm i\pi} = -1 \quad \text{for } \omega \neq 0 .</math>

Table of selected Hilbert transforms

In the following table, the frequency parameter <math>\omega</math> is real.

Signal <math>u(t)</math>	Hilbert transform <math>\operatorname{H}(u)(t)</math>
<math>\sin(\omega t + \varphi)</math>	<math>\begin{array}{lll} \sin\left(\omega t + \varphi - \tfrac{\pi}{2}\right)=-\cos\left(\omega t + \varphi \right), \quad \omega > 0\\ \sin\left(\omega t + \varphi + \tfrac{\pi}{2}\right)=\cos\left(\omega t + \varphi \right), \quad \omega < 0 \end{array}</math>
<math> \cos(\omega t + \varphi) </math>	<math>\begin{array}{lll} \cos\left(\omega t + \varphi - \tfrac{\pi}{2}\right)=\sin\left(\omega t + \varphi\right), \quad \omega > 0\\ \cos\left(\omega t + \varphi + \tfrac{\pi}{2}\right)=-\sin\left(\omega t + \varphi\right), \quad \omega < 0 \end{array}</math>
<math> e^{i \omega t} </math>	<math>\begin{array}{lll} e^{i\left(\omega t - \tfrac{\pi}{2}\right)}, \quad \omega > 0\\ e^{i\left(\omega t + \tfrac{\pi}{2}\right)}, \quad \omega < 0 \end{array}</math>
<math> e^{-i \omega t} </math>	<math>\begin{array}{lll} e^{-i\left(\omega t - \tfrac{\pi}{2}\right)}, \quad \omega > 0\\ e^{-i\left(\omega t + \tfrac{\pi}{2}\right)}, \quad \omega < 0 \end{array}</math>
<math> 1 \over t^2 + 1 </math>	<math> t \over t^2 + 1 </math>
<math> e^{-t^2} </math>	<math> \frac{2}{\sqrt{\pi\,}} F(t) </math> (see Dawson function)
Sinc function <math> \sin(t) \over t </math>	<math> 1 - \cos(t)\over t </math>
Dirac delta function <math> \delta(t) </math>	<math> {1 \over \pi t} </math>
Characteristic function <math> \chi_{[a,b]}(t) </math>	<math>{ \frac{1}{\,\pi\,}\ln \left\vert \frac{t - a}{t - b}\right\vert }</math>

Notes

An extensive table of Hilbert transforms is available.Шаблон:Sfn Note that the Hilbert transform of a constant is zero.

Domain of definition

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 <math>L^p(\mathbb{R})</math> for Шаблон:Math.

More precisely, if Шаблон:Mvar is in <math>L^p(\mathbb{R})</math> for Шаблон:Math, then the limit defining the improper integral

<math display="block">\operatorname{H}(u)(t) = \frac{2}{\pi} \lim_{\varepsilon \to 0} \int_\varepsilon^\infty \frac{u(t - \tau) - u(t + \tau)}{2\tau}\,d\tau</math>

exists for almost every Шаблон:Mvar. The limit function is also in <math>L^p(\mathbb{R})</math> and is in fact the limit in the mean of the improper integral as well. That is,

<math display="block">\frac{2}{\pi} \int_\varepsilon^\infty \frac{u(t - \tau) - u(t + \tau)}{2\tau}\,\mathrm{d}\tau \to \operatorname{H}(u)(t)</math>

as Шаблон:Math in the Шаблон:Mvar norm, as well as pointwise almost everywhere, by the Titchmarsh theorem.Шаблон:Sfn

In the case Шаблон:Math, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally.Шаблон:Sfn In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an Шаблон:Math function does converge, however, in Шаблон:Math-weak, and the Hilbert transform is a bounded operator from Шаблон:Math to Шаблон:Math.Шаблон:Sfn (In particular, since the Hilbert transform is also a multiplier operator on Шаблон:Math, Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that Шаблон:Mvar is bounded on Шаблон:Math.)

Properties

Boundedness

If Шаблон:Math, then the Hilbert transform on <math>L^p(\mathbb{R})</math> is a bounded linear operator, meaning that there exists a constant Шаблон:Mvar such that

<math display="block">\left\|\operatorname{H}u\right\|_p \le C_p \left\|u\right\|_p </math>

for all Шаблон:Nowrap

The best constant <math>C_p</math> is given by <math display="block">C_p = \begin{cases}

 \tan \frac{\pi}{2p} & \text{for} ~ 1 < p \leq 2, \\[4pt] 
 \cot \frac{\pi}{2p} & \text{for} ~ 2 < p < \infty.

\end{cases}</math>

An easy way to find the best <math>C_p</math> for <math>p</math> being a power of 2 is through the so-called Cotlar's identity that <math> (\operatorname{H}f)^2 =f^2 +2\operatorname{H}(f\operatorname{H}f)</math> for all real valued Шаблон:Mvar. The same best constants hold for the periodic Hilbert transform.

The boundedness of the Hilbert transform implies the <math>L^p(\mathbb{R})</math> convergence of the symmetric partial sum operator <math display="block">S_R f = \int_{-R}^R \hat{f}(\xi) e^{2\pi i x\xi} \, \mathrm{d}\xi </math>

to Шаблон:Mvar in Шаблон:Nowrap

Anti-self adjointness

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between <math>L^p(\mathbb{R})</math> and the dual space Шаблон:Nowrap where Шаблон:Mvar and Шаблон:Mvar are Hölder conjugates and Шаблон:Math. Symbolically,

<math display="block">\langle \operatorname{H} u, v \rangle = \langle u, -\operatorname{H} v \rangle</math>

for <math>u \isin L^p(\mathbb{R})</math> and Шаблон:NowrapШаблон:Sfn

Inverse transform

The Hilbert transform is an anti-involution,Шаблон:Sfn meaning that

<math display="block">\operatorname{H}\bigl(\operatorname{H}\left(u\right)\bigr) = -u</math>

provided each transform is well-defined. Since Шаблон:Math preserves the space Шаблон:Nowrap this implies in particular that the Hilbert transform is invertible on Шаблон:Nowrap and that

<math display="block">\operatorname{H}^{-1} = -\operatorname{H}</math>

Complex structure

Because Шаблон:Math ("Шаблон:Math" is the identity operator) on the real Banach space of real-valued functions in Шаблон:Nowrap the Hilbert transform defines a linear complex structure on this Banach space. In particular, when Шаблон:Math, the Hilbert transform gives the Hilbert space of real-valued functions in <math>L^2(\mathbb{R})</math> the structure of a complex Hilbert space.

The (complex) eigenstates of the Hilbert transform admit representations as holomorphic functions in the upper and lower half-planes in the Hardy space [[H square|Шаблон:Math]] by the Paley–Wiener theorem.

Differentiation

Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:

<math display="block">\operatorname{H}\left(\frac{ \mathrm{d}u}{\mathrm{d}t}\right) = \frac{\mathrm d}{\mathrm{d}t}\operatorname{H}(u)</math>

Iterating this identity,

<math display="block">\operatorname{H}\left(\frac{\mathrm{d}^ku}{\mathrm{d}t^k}\right) = \frac{\mathrm{d}^k}{\mathrm{d}t^k}\operatorname{H}(u)</math>

This is rigorously true as stated provided Шаблон:Mvar and its first Шаблон:Mvar derivatives belong to Шаблон:NowrapШаблон:Sfn One can check this easily in the frequency domain, where differentiation becomes multiplication by Шаблон:Mvar.

Convolutions

The Hilbert transform can formally be realized as a convolution with the tempered distributionШаблон:Sfn

<math display="block">h(t) = \operatorname{p.v.} \frac{1}{ \pi \, t }</math>

Thus formally,

<math display="block">\operatorname{H}(u) = h*u</math>

However, a priori this may only be defined for Шаблон:Mvar 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 in Шаблон:Math. Alternatively, one may use the fact that h(t) is the distributional derivative of the function Шаблон:Math; to wit

<math display="block">\operatorname{H}(u)(t) = \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\pi} \left(u*\log\bigl|\cdot\bigr|\right)(t)\right)</math>

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 applied on only one of either of the factors:

<math display="block">\operatorname{H}(u*v) = \operatorname{H}(u)*v = u*\operatorname{H}(v)</math>

This is rigorously true if Шаблон:Mvar and Шаблон:Mvar are compactly supported distributions since, in that case,

<math display="block"> h*(u*v) = (h*u)*v = u*(h*v)</math>

By passing to an appropriate limit, it is thus also true if Шаблон:Math and Шаблон:Math provided that

<math display="block"> 1 < \frac{1}{p} + \frac{1}{q} </math>

from a theorem due to Titchmarsh.Шаблон:Sfn

Invariance

The Hilbert transform has the following invariance properties on <math>L^2(\mathbb{R})</math>.

• It commutes with translations. That is, it commutes with the operators Шаблон:Math for all Шаблон:Mvar in <math>\mathbb{R}.</math>
• It commutes with positive dilations. That is it commutes with the operators Шаблон:Math for all Шаблон:Math.
• It anticommutes with the reflection Шаблон:Math.

Up to a multiplicative constant, the Hilbert transform is the only bounded operator on Шаблон:Mvar² with these properties.Шаблон:Sfn

In fact there is a wider set of operators that commute with the Hilbert transform. The group <math>\text{SL}(2,\mathbb{R})</math> acts by unitary operators Шаблон:Math on the space <math>L^2(\mathbb{R})</math> by the formula

<math display="block">\operatorname{U}_{g}^{-1} f(x) = \frac{1}{ c x + d } \, f \left( \frac{ ax + b }{ cx + d } \right) \,,\qquad g = \begin{bmatrix} a & b \\ c & d \end{bmatrix} ~,\qquad \text{ for }~ a d - b c = \pm 1 . </math>

This unitary representation is an example of a principal series representation of <math>~\text{SL}(2,\mathbb{R})~.</math> In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space <math>H^2(\mathbb{R})</math> and its conjugate. These are the spaces of Шаблон:Math boundary values of holomorphic functions on the upper and lower halfplanes. <math>H^2(\mathbb{R})</math> and its conjugate consist of exactly those Шаблон:Math functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to Шаблон:Math, with Шаблон:Mvar being the orthogonal projection from <math>L^2(\mathbb{R})</math> onto <math>\operatorname{H}^2(\mathbb{R}),</math> and Шаблон:Math the identity operator, it follows that <math>\operatorname{H}^2(\mathbb{R})</math> and its orthogonal complement are eigenspaces of Шаблон:Math for the eigenvalues Шаблон:Math. In other words, Шаблон:Math commutes with the operators Шаблон:Mvar. The restrictions of the operators Шаблон:Mvar to <math>\operatorname{H}^2(\mathbb{R})</math> and its conjugate give irreducible representations of <math>\text{SL}(2,\mathbb{R})</math> – the so-called limit of discrete series representations.

Extending the domain of definition

Hilbert transform of distributions

It is further possible to extend the Hilbert transform to certain spaces of distributions Шаблон:Harv. Since the Hilbert transform commutes with differentiation, and is a bounded operator on Шаблон:Mvar, Шаблон:Mvar restricts to give a continuous transform on the inverse limit of Sobolev spaces:

<math display="block">\mathcal{D}_{L^p} = \underset{n \to \infty}{\underset{\longleftarrow}{\lim}} W^{n,p}(\mathbb{R})</math>

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 Шаблон:Mvar distributions. This is accomplished by the duality pairing:
For Шаблон:Nowrap define:

<math display="block">\operatorname{H}(u)\in \mathcal{D}'_{L^p} = \langle \operatorname{H}u, v \rangle \ \triangleq \ \langle u, -\operatorname{H}v\rangle,\ \text{for all} \ v\in\mathcal{D}_{L^p} .</math>

It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to Gel'fand and Shilov,Шаблон:Sfn but considerably more care is needed because of the singularity in the integral.

Hilbert transform of bounded functions

The Hilbert transform can be defined for functions in <math>L^\infty (\mathbb{R})</math> as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps <math>L^\infty (\mathbb{R})</math> to the Banach space of bounded mean oscillation (BMO) classes.

Interpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined. For instance, with Шаблон:Math, the integral defining Шаблон:Math diverges almost everywhere to Шаблон:Math. To alleviate such difficulties, the Hilbert transform of an Шаблон:Math function is therefore defined by the following regularized form of the integral

<math display="block">\operatorname{H}(u)(t) = \operatorname{p.v.} \int_{-\infty}^\infty u(\tau)\left\{h(t - \tau)- h_0(-\tau)\right\} \, \mathrm{d}\tau</math>

where as above Шаблон:Math and

<math display="block">h_0(x) = \begin{cases} 0 & \text{for} ~ |x| < 1 \\ \frac{1}{\pi \, x} & \text{for} ~ |x| \ge 1 \end{cases}</math>

The modified transform Шаблон:Math agrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund. Furthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.

A deep result of Fefferman's work is that a function is of bounded mean oscillation if and only if it has the form Шаблон:Nowrap for some Шаблон:Nowrap

Conjugate functions

The Hilbert transform can be understood in terms of a pair of functions Шаблон:Math and Шаблон:Math such that the function <math display="block">F(x) = f(x) + i\,g(x)</math> is the boundary value of a holomorphic function Шаблон:Math in the upper half-plane.Шаблон:Sfn Under these circumstances, if Шаблон:Mvar and Шаблон:Mvar are sufficiently integrable, then one is the Hilbert transform of the other.

Suppose that <math>f \isin L^p(\mathbb{R}).</math> Then, by the theory of the Poisson integral, Шаблон:Mvar admits a unique harmonic extension into the upper half-plane, and this extension is given by

<math display="block">u(x + iy) = u(x, y) = \frac{1}{\pi} \int_{-\infty}^\infty f(s)\;\frac{y}{(x - s)^2 + y^2} \; \mathrm{d}s</math>

which is the convolution of Шаблон:Mvar with the Poisson kernel

<math display="block">P(x, y) = \frac{ y }{ \pi\, \left( x^2 + y^2 \right) }</math>

Furthermore, there is a unique harmonic function Шаблон:Mvar defined in the upper half-plane such that Шаблон:Math is holomorphic and <math display="block">\lim_{y \to \infty} v\,(x + i\,y) = 0</math>

This harmonic function is obtained from Шаблон:Mvar by taking a convolution with the conjugate Poisson kernel

<math display="block">Q(x, y) = \frac{ x }{ \pi\, \left(x^2 + y^2\right) } .</math>

Thus <math display="block">v(x, y) = \frac{1}{\pi}\int_{-\infty}^\infty f(s)\;\frac{x - s}{\,(x - s)^2 + y^2\,}\;\mathrm{d}s .</math>

Indeed, the real and imaginary parts of the Cauchy kernel are <math display="block">\frac{i}{\pi\,z} = P(x, y) + i\,Q(x, y)</math>

so that Шаблон:Math is holomorphic by Cauchy's integral formula.

The function Шаблон:Mvar obtained from Шаблон:Mvar in this way is called the harmonic conjugate of Шаблон:Mvar. The (non-tangential) boundary limit of Шаблон:Math as Шаблон:Math is the Hilbert transform of Шаблон:Mvar. Thus, succinctly, <math display="block">\operatorname{H}(f) = \lim_{y \to 0} Q(-, y) \star f</math>

Titchmarsh's theorem

Titchmarsh's theorem (named for E. C. Titchmarsh who included it in his 1937 work) makes precise the relationship between the boundary values of holomorphic functions in the upper half-plane and the Hilbert transform.Шаблон:Sfn It gives necessary and sufficient conditions for a complex-valued square-integrable function Шаблон:Math on the real line to be the boundary value of a function in the Hardy space Шаблон:Math of holomorphic functions in the upper half-plane Шаблон:Mvar.

The theorem states that the following conditions for a complex-valued square-integrable function <math>F : \mathbb{R} \to \mathbb{C}</math> are equivalent:

• Шаблон:Math is the limit as Шаблон:Math of a holomorphic function Шаблон:Math in the upper half-plane such that <math display="block"> \int_{-\infty}^\infty |F(x + i\,y)|^2\;\mathrm{d}x < K </math>
• The real and imaginary parts of Шаблон:Math are Hilbert transforms of each other.
• The Fourier transform <math>\mathcal{F}(F)(x)</math> vanishes for Шаблон:Math.

A weaker result is true for functions of class Шаблон:Mvar for Шаблон:Math.Шаблон:Sfn Specifically, if Шаблон:Math is a holomorphic function such that

<math display="block">\int_{-\infty}^\infty |F(x + i\,y)|^p\;\mathrm{d}x < K </math>

for all Шаблон:Mvar, then there is a complex-valued function Шаблон:Math in <math>L^p(\mathbb{R})</math> such that Шаблон:Math in the Шаблон:Mvar norm as Шаблон:Math (as well as holding pointwise almost everywhere). Furthermore,

<math display="block">F(x) = f(x) - i\,g(x)</math>

where Шаблон:Mvar is a real-valued function in <math>L^p(\mathbb{R})</math> and Шаблон:Mvar is the Hilbert transform (of class Шаблон:Mvar) of Шаблон:Mvar.

This is not true in the case Шаблон:Math. In fact, the Hilbert transform of an Шаблон:Math function Шаблон:Mvar need not converge in the mean to another Шаблон:Math function. Nevertheless,Шаблон:Sfn the Hilbert transform of Шаблон:Mvar does converge almost everywhere to a finite function Шаблон:Mvar such that

<math display="block">\int_{-\infty}^\infty \frac{ |g(x)|^p }{ 1 + x^2 } \; \mathrm{d}x < \infty</math>

This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc.Шаблон:Sfn Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see Paley–Wiener theorem), as well as work by Riesz, Hille, and Tamarkin

Riemann–Hilbert problem

One form of the Riemann–Hilbert problem seeks to identify pairs of functions Шаблон:Math and Шаблон:Math such that Шаблон:Math is holomorphic on the upper half-plane and Шаблон:Math is holomorphic on the lower half-plane, such that for Шаблон:Mvar along the real axis, <math display="block">F_{+}(x) - F_{-}(x) = f(x)</math>

where Шаблон:Math is some given real-valued function of Шаблон:Nowrap The left-hand side of this equation may be understood either as the difference of the limits of Шаблон:Math 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 Шаблон:Math solve the Riemann–Hilbert problem <math display="block">f(x) = F_{+}(x) - F_{-}(x)</math>

then the Hilbert transform of Шаблон:Math is given byШаблон:Sfn <math display="block">H(f)(x) = -i \bigl( F_{+}(x) + F_{-}(x) \bigr) .</math>

Hilbert transform on the circle

Шаблон:See also For a periodic function Шаблон:Mvar the circular Hilbert transform is defined:

<math display="block">\tilde f(x) \triangleq \frac{1}{ 2\pi } \operatorname{p.v.} \int_0^{2\pi} f(t)\,\cot\left(\frac{ x - t }{2}\right)\,\mathrm{d}t</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 display="block">\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.Шаблон:Sfn

The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel Шаблон:Frac periodic. More precisely, for Шаблон:Math

<math display="block">\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 Шаблон:Math, which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map

<math display="block"> U\,f(x) = \frac{1}{(x + i)\,\sqrt{\pi}} \, f\left(C\left(x\right)\right) </math>

of Шаблон:Math onto <math>L^2 (\mathbb{R}).</math> The operator Шаблон:Mvar carries the Hardy space Шаблон:Math onto the Hardy space <math>H^2(\mathbb{R})</math>.Шаблон:Sfn

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 Шаблон:Math and Шаблон:Math are the low- and high-pass signals respectively.Шаблон:Sfn 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 Шаблон:Math is the narrow bandwidth "message" waveform, such as voice or music. Then by Bedrosian's theorem:Шаблон:Sfn

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

Analytic representation

Шаблон:Main article A specific type of 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:

A Fourier transform property indicates that this complex heterodyne operation can shift all the negative frequency components of Шаблон:Math 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.

Шаблон:Anchor Angle (phase/frequency) modulation

The form:

<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 frequency is  <math>\omega + \varphi_m^\prime(t).</math>  For sufficiently large Шаблон:Mvar, compared to Шаблон:Nowrap

<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)

Шаблон:Main article When Шаблон:Math in Шаблон:EquationNote 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:

<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). Finite-length 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 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>.

Discrete Hilbert transform

Файл:Bandpass discrete Hilbert transform filter.tif

Figure 1: Filter whose frequency response is bandlimited to about 95% of the Nyquist frequency

Файл:Highpass discrete Hilbert transform filter.tif

Figure 2: Hilbert transform filter with a highpass frequency response

Файл:DFT approximation to Hilbert filter.png

Figure 3.

Файл:Effect of circular convolution on discrete Hilbert transform.png

Figure 4. The Hilbert transform of Шаблон:Math is Шаблон:Math. This figure shows Шаблон:Math and two approximate Hilbert transforms computed by the MATLAB library function, Шаблон:Mono

Файл:Discrete Hilbert transforms of a cosine function, using piecewise convolution.svg

Figure 5. Discrete Hilbert transforms of a cosine function, using piecewise convolution

For a discrete function, Шаблон:Nowrap with discrete-time Fourier transform (DTFT), Шаблон:Nowrap and discrete Hilbert transform Шаблон:Nowrap the DTFT of <math>\hat u[n]</math> in the region Шаблон:Math is given by:

<math>\operatorname{DTFT} (\hat u) = U(\omega)\cdot (-i\cdot \sgn(\omega)).</math>

The inverse DTFT, using the convolution theorem, is:

<math>

\begin{align} \hat u[n] &= {\scriptstyle \mathrm{DTFT}^{-1}} (U(\omega))\ *\ {\scriptstyle \mathrm{DTFT}^{-1}} (-i\cdot \sgn(\omega))\\ &= u[n]\ *\ \frac{1}{2 \pi}\int_{-\pi}^{\pi} (-i\cdot \sgn(\omega))\cdot e^{i \omega n} \,\mathrm{d}\omega\\ &= u[n]\ *\ \underbrace{\frac{1}{2 \pi}\left[\int_{-\pi}^0 i\cdot e^{i \omega n} \,\mathrm{d}\omega - \int_0^\pi i\cdot e^{i \omega n} \,\mathrm{d}\omega \right]}_{h[n]}, \end{align} </math>

where

<math>h[n]\ \triangleq \

\begin{cases}

            0, & \text{for }n\text{ even}\\

\frac 2 {\pi n} & \text{for }n\text{ odd}, \end{cases}</math>

which is an infinite impulse response (IIR). When the convolution is performed numerically, an FIR approximation is substituted for Шаблон:Math, as shown in Figure 1. An FIR filter with an odd number of anti-symmetric coefficients is called Type III, which inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in this case in a bandpass filter shape. A Type IV design (even number of anti-symmetric coefficients) is shown in Figure 2. Since the magnitude response at the Nyquist frequency does not drop out, it approximates an ideal Hilbert transformer a little better than the odd-tap filter. However

• A typical (i.e. properly filtered and sampled) Шаблон:Math sequence has no useful components at the Nyquist frequency.
• The Type IV impulse response requires a Шаблон:Frac sample shift in the Шаблон:Math sequence. That causes the zero-valued coefficients to become non-zero, as seen in Figure 2. So a Type III design is potentially twice as efficient as Type IV.
• The group delay of a Type III design is an integer number of samples, which facilitates aligning <math>\hat u[n]</math> with <math>u[n],</math> to create an analytic signal. The group delay of Type IV is halfway between two samples.

The MATLAB function, Шаблон:Mono, convolves a u[n] sequence with the periodic summation:Шаблон:Efn-ua

<math>h_N[n]\ \triangleq \sum_{m=-\infty}^\infty h[n - mN]</math>Шаблон:SpacesШаблон:Efn-uaШаблон:Efn-ua

and returns one cycle (Шаблон:Mvar samples) of the periodic result in the imaginary part of a complex-valued output sequence. The convolution is implemented in the frequency domain as the product of the array  <math>{\scriptstyle \mathrm{DFT}} \left(u[n]\right)</math>  with samples of the Шаблон:Math distribution (whose real and imaginary components are all just 0 or Шаблон:Math). Figure 3 compares a half-cycle of Шаблон:Math with an equivalent length portion of Шаблон:Math. Given an FIR approximation for <math>h[n],</math> denoted by <math>\tilde{h}[n],</math> substituting <math>{\scriptstyle\mathrm{DFT}} \left(\tilde{h}[n]\right)</math> for the Шаблон:Math samples results in an FIR version of the convolution.

The real part of the output sequence is the original input sequence, so that the complex output is an analytic representation of Шаблон:Math. When the input is a segment of a pure cosine, the resulting convolution for two different values of Шаблон:Mvar is depicted in Figure 4 (red and blue plots). Edge effects prevent the result from being a pure sine function (green plot). Since Шаблон:Math is not an FIR sequence, the theoretical extent of the effects is the entire output sequence. But the differences from a sine function diminish with distance from the edges. Parameter Шаблон:Mvar is the output sequence length. If it exceeds the length of the input sequence, the input is modified by appending zero-valued elements. In most cases, that reduces the magnitude of the differences. But their duration is dominated by the inherent rise and fall times of the Шаблон:Math impulse response.

An appreciation for the edge effects is important when a method called overlap-save is used to perform the convolution on a long Шаблон:Math sequence. Segments of length Шаблон:Mvar are convolved with the periodic function:

<math>\tilde{h}_N[n]\ \triangleq \sum_{m=-\infty}^\infty \tilde{h}[n - mN].</math>

When the duration of non-zero values of <math>\tilde{h}[n]</math> is <math>M < N,</math> the output sequence includes Шаблон:Math samples of <math>\hat u.</math> Шаблон:Math outputs are discarded from each block of Шаблон:Mvar, and the input blocks are overlapped by that amount to prevent gaps.

Figure 5 is an example of using both the IIR hilbert(·) function and the FIR approximation. In the example, a sine function is created by computing the Discrete Hilbert transform of a cosine function, which was processed in four overlapping segments, and pieced back together. As the FIR result (blue) shows, the distortions apparent in the IIR result (red) are not caused by the difference between Шаблон:Math and Шаблон:Math (green and red in Figure 3). The fact that Шаблон:Math is tapered (windowed) is actually helpful in this context. The real problem is that it's not windowed enough. Effectively, Шаблон:Math, whereas the overlap-save method needs Шаблон:Math.

Number-theoretic Hilbert transform

The number theoretic Hilbert transform is an extensionШаблон:Sfn of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization of discrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.Шаблон:Sfn

See also

• Analytic signal
• Harmonic conjugate
• Hilbert spectroscopy
• Hilbert transform in the complex plane
• Hilbert–Huang transform
• Kramers–Kronig relation
• Riesz transform
• Single-sideband signal
• Singular integral operators of convolution type

References

• Шаблон:Cite journal
• Шаблон:Cite report
• Шаблон:Springer
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite web; also http://www.fuchs-braun.com/media/d9140c7b3d5004fbffff8007fffffff0.pdf

• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Springer
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Citation
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite web; also https://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book

Further reading

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite conference
• Шаблон:Cite journal
• Шаблон:Cite journal

External links

• Derivation of the boundedness of the Hilbert transform
• Mathworld Hilbert transform — Contains a table of transforms
• Шаблон:MathWorld
• Шаблон:Cite web an entry level introduction to Hilbert transformation.