Hilbert transform

In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, of a real variable and produces another function of a real variable. The Hilbert transform is given by the Cauchy principal value of the convolution with the function . The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° to every frequency component of a function, the sign of the shift depending on the sign of the frequency. The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal. 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 can be thought of as the convolution of with the function, known as the Cauchy kernel. Because 1/ is not integrable across, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the Cauchy principal value. Explicitly, the Hilbert transform of a function is given by
provided this integral exists as a principal value. This is precisely the convolution of with the tempered distribution. Alternatively, by changing variables, the principal-value integral can be written explicitly as
When the Hilbert transform is applied twice in succession to a function, the result is
provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is
. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of .
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 is analytic in the upper half complex plane, and, then up to an additive constant, provided this Hilbert transform exists.

Notation

In signal processing the Hilbert transform of is commonly denoted by. However, in mathematics, this notation is already extensively used to denote the Fourier transform of. Occasionally, the Hilbert transform may be denoted by. 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, 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. 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. Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case. These results were restricted to the spaces and. In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in for, that the Hilbert transform is a bounded operator on for, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform. The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals. 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. The multiplier of is, where is the signum function. Therefore:
where denotes the Fourier transform. Since, it follows that this result applies to the three common definitions of.
By Euler's formula,
Therefore, has the effect of shifting the phase of the negative frequency components of by +90° and the phase of the positive frequency components by −90°, and has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.
When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e.,, because

Table of selected Hilbert transforms

In the following table, the frequency parameter is real.

Signal	Hilbert transform






Sinc function
Dirac delta function
Characteristic function

Notes
An extensive table of Hilbert transforms is available.
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 for.
More precisely, if is in for, then the limit defining the improper integral
exists for almost every. The limit function is also in and is in fact the limit in the mean of the improper integral as well. That is,
as in the norm, as well as pointwise almost everywhere, by the [|Titchmarsh theorem].
In the case, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally. In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an function does converge, however, in -weak, and the Hilbert transform is a bounded operator from to.

Properties

Boundedness

If, then the Hilbert transform on is a bounded linear operator, meaning that there exists a constant such that
for all
The best constant is given by
An easy way to find the best for being a power of 2 is through the so-called Cotlar's identity that for all real valued. The same best constants hold for the periodic Hilbert transform.
The boundedness of the Hilbert transform implies the convergence of the symmetric partial sum operator
to in

Anti-self adjointness

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between and the dual space where and are Hölder conjugates and. Symbolically,
for and

Inverse transform

The Hilbert transform is an anti-involution, meaning that
provided each transform is well-defined. Since preserves the space this implies in particular that the Hilbert transform is invertible on and that

Complex structure

Because on the real Banach space of real-valued functions in the Hilbert transform defines a linear complex structure on this Banach space. In particular, when, the Hilbert transform gives the Hilbert space of real-valued functions in the structure of a complex Hilbert space.
The eigenstates of the Hilbert transform admit representations as holomorphic functions in the upper and lower half-planes in the Hardy space H square| 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:
Iterating this identity,
This is rigorously true as stated provided and its first derivatives belong to One can check this easily in the frequency domain, where differentiation becomes multiplication by.

Convolutions

The Hilbert transform can formally be realized as a convolution with the tempered distribution
Thus formally,
However, a priori this may only be defined for a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported functions are dense in. Alternatively, one may use the fact that h is the distributional derivative of the function ; to wit
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:
This is rigorously true if and are compactly supported distributions since, in that case,
By passing to an appropriate limit, it is thus also true if and provided that
from a theorem due to Titchmarsh.

Invariance

The Hilbert transform has the following invariance properties on.

It commutes with translations. That is, it commutes with the operators for all in
It commutes with positive dilations. That is it commutes with the operators for all.
It anticommutes with the reflection.

Up to a multiplicative constant, the Hilbert transform is the only bounded operator on ² with these properties.
In fact there is a wider set of operators that commute with the Hilbert transform. The group acts by unitary operators on the space by the formula
This unitary representation is an example of a principal series representation of In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space and its conjugate. These are the spaces of boundary values of holomorphic functions on the upper and lower halfplanes. and its conjugate consist of exactly those functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to, with being the orthogonal projection from onto and the identity operator, it follows that and its orthogonal complement are eigenspaces of for the eigenvalues. In other words, commutes with the operators. The restrictions of the operators to and its conjugate give irreducible representations of – the so-called limit of discrete series representations.