Hilbert transform

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. 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 functions. It is a basic tool in Fourier analysis, and provides a concrete means for realizing the conjugate of a given function or Fourier series. Furthermore, in harmonic analysis, it is an example of a singular integral operator, and of a Fourier multiplier. The Hilbert transform is also important in the field of signal processing where it is used to derive the analytic representation of a signal "u"("t").

The Hilbert transform was originally defined for periodic functions, 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 kernel, for functions defined on the real line R (the 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 transforms of functions on the real line.

Introduction

The Hilbert transform can be thought of as the convolution of "u"("t") with the function "h"("t") = 1/"πt". 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:$H(u)(t)\; =\; ext\{p.v.\}\; int\_\{-infty\}^\{infty\}u(\; au)\; h(t-\; au),\; d\; au$ provided this integral exists as a principal value. This is precisely the convolution of "u" with the tempered distribution p.v. 1/"πt" (harvtxt|Schwartz|1950; 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

:$H(u)(t)\; =\; -frac\{1\}\{pi\}lim\_\{epsilondownarrow\; 0\}int\_\{epsilon\}^infty\; frac\{u(t+\; au)-u(t-\; au)\}\{\; au\},d\; au.$

When the Hilbert transform is applied twice in succession to a function "u", the result is minus "u":

:$H(H(u))(t)\; =\; -u(t),,$

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is −"H".

In signal processing the Hilbert transform of "u"("t") is commonly denoted by $widehat\; u(t).,$ 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 $ilde\{u\}(t)$. Furthermore, many sources define the Hilbert transform as the negative of the one defined here.

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 an 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.

History

The Hilbert transform arose in Hilbert's 1905 work on a problem posed by Riemann concerning analytic functions (harvtxt|Kress|1983; harvtxt|Bitsadze|2001) which has come to be known as the Riemann-Hilbert problem. Hilbert's work was mainly concerned the Hilbert transform for functions defined on the circle harv|Hilbert|1953. Some of his earlier work related to the Discrete Hilbert Transform date back to lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation harv|Hardy|Littlewood|Polya|1952. Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case harv|Hardy|Littlewood|Polya|1952. These results were restricted to the spaces "L"² and ℓ². In 1928, Marcel Riesz proved that the Hilbert transform can be defined for "u" in "L"^p(R) for 1<p<∞, that the Hilbert transform is a bounded operator on "L"^p(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 integrals 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.

Relationship with the Fourier transform

As mentioned before, the Hilbert transform is a multiplier operator. The symbol of "H" is σ_H("ω")=-"i"sgn("ω") where sgn is the signum function. Therefore:

:$mathcal\{F\}(H(u))(omega)\; =\; (-i\; mathrm\{sgn\}(omega))cdot\; mathcal\{F\}(u)(omega).$where $mathcal\{F\}$ denotes the Fourier transform. Since sgn("x") = sgn("2πx"), it follows that this result applies to the three common definitions of $mathcal\{F\}.$

By Euler's formula,:

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.

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:

:$[sigma\_H(omega)]\; ^2\; =\; e^\{pm\; ipi\}\; =\; -1.$

Table of selected Hilbert transforms

This complex heterodyne operation shifts all the frequency components of "u"_"m"("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.

While the analytic representation of a signal is not necessarily an analytic function, "u"_"a"("t") is given by the boundary values of an analytic function in the upper half-plane.

Phase/Frequency modulation

The form:

:$u(t)\; =\; Acdot\; cos(omega\; t\; +\; phi\_m(t)),$

is called phase (or frequency) modulation. The instantaneous frequency is $omega\; +\; phi\_m^prime(t).$ For sufficiently large $omega,$ compared to $phi\_m^prime$:

:$widehat\{u\}(t)\; approx\; Acdot\; sin(omega\; t\; +\; phi\_m(t)),,$

and:

:$u\_a(t)\; approx\; A\; cdot\; e^\{i(omega\; t\; +\; phi\_m(t))\}.$

Single sideband modulation (SSB)

When $u\_m(t)$ in EquationNote|Eq.1 is also an analytic representation (of a message waveform), that is:

:$u\_m(t)\; =\; m(t)\; +\; icdot\; widehat\{m\}(t),$

the result is single-sideband modulation:

:$u\_a(t)\; =\; (m(t)\; +\; icdot\; widehat\{m\}(t))\; cdot\; e^\{i(omega\; t\; +\; phi)\},$

whose transmitted component is:

:

Causality

The function "h" with "h"("t") = 1/(π "t") is a 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 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 transforms

There are two objects of study which are considered discrete Hilbert transforms. The Discrete Hilbert transform of practical interest is described as follows.If the signal "u"("t") is bandlimited, then "H"("u")("t") is bandlimited in the same way. Consequently, both these signals can be sampled according to the sampling theorem, resulting in the discrete signals "u" ["n"] and H("u") ["n"] . The relation between the two discrete signals is then given by the convolution:

:$H(u)\; [n]\; =\; h\; [n]\; *\; u\; [n]\; ,$

where

:

which is non-causal and has infinite duration. In practice, a shortened and time-shifted approximation is used. The usual filter design tradeoffs apply (e.g., filter-order and latency vs. frequency-response). Also notice, that $h\; [n]\; ,$ is not just a sampled version of the "Hilbert filter" $h(t),$, defined above. Rather it is a sequence with this discrete-time Fourier transform:

:

We note that a sequence similar to $h\; [n]\; ,$ can be generated by sampling σ_"H"(ω) and computing the inverse discrete Fourier transform. The larger the transform (i.e., more samples per $2\; pi$ radians), the better the agreement (for a given value of the abscissa, n). The figure shows the comparison for a 512-point transform. (Due to odd-symmetry, only half the sequence is actually plotted.)
But that is not the actual point, because it is easier and more accurate to generate $h\; [n]\; ,$ directly from the formula. The point is that many applications choose to avoid the convolution by doing the equivalent frequency-domain operation: simple multiplication of the signal transform with σ_"H"(ω), made even easier by the fact that the real and imaginary components are 0 and ±1 respectively. The attractiveness of that approach is only apparent when the actual Fourier transforms are replaced by samples of the same, i.e., the DFT, which is an approximation and introduces some distortion. Thus, after transforming back to the time-domain, those applications have indirectly generated (and convolved with) "not" $h\; [n]\; ,$, but the DFT approximation to it, which is shown in the figure.

Notes on "fast convolution":
*Implied in the technique described above is the concept of dividing a long signal into segments of arbitrary size. The signal is filtered piecewise, and the outputs are subsequently pieced back together.
*The segment size is an important factor in controlling the amount of distortion. As the size increases, the DFT becomes more dense and is a better approximation to the underlying Fourier transform. In the time-domain, the same distortion is manifested as "aliasing", which results in a type of convolution called circular. It is as if the same segment is repeated periodically and filtered, resulting in distortion that is worst at either or both edges of the original segment. Increasing the segment size reduces the number of edges in the pieced-together result and therefore reduces overall distortion.
*Another mitigation strategy is to simply discard the most badly distorted output samples, because data loss can be avoided by overlapping the input segments. When the filter's impulse response is less than the segment length, this can produce a distortion-free (non-circular) convolution (Overlap-discard method). That requires an FIR filter, which the Hilbert transform is not. So yet another technique is to design an FIR approximation to a Hilbert transform filter. That moves the source of distortion from the convolution to the filter, where it can be readily characterized in terms of imperfections in the frequency response.
*Failure to appreciate or correctly apply these concepts is probably one of the most common mistakes made by non-experts in the digital signal processing field.

The other Discrete Hilbert transform is defined by :$b\_n=sum\_\{m=-infty\}^infty\; frac\{a\_m\}\{n-m\}qquad\; m\; eq\; n$. Hilbert showed that for "a"_n in ℓ² the sequence "b"_n is also in ℓ². An elementary proof of this fact can be found in harv|Grafakos|1994. The discrete Hilbert transform was used by E. C. Titchmarsh to give alternate proofs of the results of M. Riesz in the continuous case (harvtxt|Titchmarsh|1926; harvtxt|Hardy|Littlewood|Polya|1952).

See also

* Analytic signal
* Single-sideband signal
* Harmonic conjugate
* Kramers–Kronig relation
* Hilbert-Huang transform

References

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* citation | author=Bracewell, R.| title=The Fourier Transform and Its Applications| edition=2nd ed | publisher=McGraw-Hill | year=1986.
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* | first=C. |last=Fefferman|author-link=Charles Fefferman|title=Characterizations of bounded mean oscillation|journal= Bull. Amer. Math. Soc. |volume= 77 |year=1971|pages= 587–588
url=http://www.ams.org/bull/1971-77-04/S0002-9904-1971-12763-5/home.html|DOI= 10.1090/S0002-9904-1971-12763-5 .
*|first=C.|last= Fefferman|first2= E.M. |last2=Stein|title=H^p spaces of several variables" |journal=Acta Math. |volume= 129 |year=1972|pages= 137–193 |doi=10.1007/BF02392215.
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External links

* [http://www.ee.byu.edu/ee/class/ee444/ComBook/ComBook/node19.html Hilbert transform]
* [http://cyvision.if.sc.usp.br/visao/courses/hilbert/hilbert.htm another exposition] — [http://cyvision.if.sc.usp.br/visao/courses/hilbert/hproprie.htm Hilbert transform properties]
* [http://mathworld.wolfram.com/HilbertTransform.html Mathworld Hilbert transform] — Contains a table of transforms
* [http://ccrma-www.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html Analytic Signals and Hilbert Transform Filters]
* [http://www.complextoreal.com/tcomplex.htm Easy Fourier Analysis] hints to compute Hilbert transform in Time domain.
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