Iterated Hilbert Transform

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) = text{p.v.} int_{-infty}^{infty}u(tau) h(t-tau), dtau provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/πt (). Alternatively, by changing variables, the principal value integral can be written explicitly as

H(u)(t) = -frac{1}{pi}lim_{epsilondownarrow 0}int_{epsilon}^infty frac{u(t+tau)-u(t-tau)}{tau},dtau.

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 tilde{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 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 . 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 . Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case . These results were restricted to the spaces L2 and ℓ2. In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in Lp(R) for 1<p<∞, that the Hilbert transform is a bounded operator on Lp(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 . 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

As mentioned before, the Hilbert transform is a multiplier operator. The symbol of H is σH(ω)=-isgn(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,

sigma_H(omega) , = begin{cases}
i = e^{+ipi/2}, & mbox{for } omega < 0 0, & mbox{for } omega = 0 -i = e^{-ipi/2}. & mbox{for } omega > 0 end{cases}

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

Signal
u(t),
Hilbert transform2
H(u)(t),
sin(t), 1 -cos(t),
cos(t), 1 sin(t),
1 over t^2 + 1 t over t^2 + 1
Sinc function
sin(t) over t
1- cos(t)over t
Rectangular function
sqcap(t)
{1 over pi} ln left | {t+{1 over 2} over t-{1 over 2}} right |
Dirac delta function
delta(t) ,
{1 over pi t}
Characteristic Function
chi_{[a,b]}(x) ,
frac{1}{pi}log left vert frac{x-a}{x-b}right vert ,
Notes

1 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.

2 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.

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 Lp(R) for 1<p<∞.

More precisely, if u is in Lp(R) for 1<p<∞, then limit defining the improper integral

H(u)(t) = -frac{1}{pi}lim_{epsilondownarrow 0}int_epsilon^infty frac{f(t+tau)-f(t-tau)}{tau},dtau

exists for almost every t. The limit function is also in Lp(R), and is in fact the limit in the mean of the improper integral as well. That is,

-frac{1}{pi}int_epsilon^infty frac{f(t+tau)-f(t-tau)}{tau},dtauto H(u)(t)

as ε→0 in the Lp-norm, as well as pointwise almost everywhere, by the Titchmarsh theorem .

In the case p=1, the Hilbert transform still converges pointwise almost everywhere, but may fail to be itself integrable even locally . In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an L1 function does converge, however, in L1-weak, and the Hilbert transform is a bounded operator from L1 to L1,w . (In particular, since the Hilbert transform is also a multiplier operator on L2, Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that H is bounded on Lp.)

Properties

Boundedness

If 1<p<∞, then the Hilbert transform on Lp(R) is a bounded linear operator, meaning that there exists a constant Cp such that

|Hu|_p le C_p| u|_p

for all uLp(R). This theorem is due to ; see also . The best constant Cp is given by

C_p=begin{cases}tan frac{pi}{2p} & text{for } 1 < pleq 2
cotfrac{pi}{2p} & text{for } 2 This result is due to ; see also . The same best constants hold for the periodic Hilbert transform.

Anti-self adjointness

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between Lp(R) and the dual space Lq(R), where p and q are Hölder conjugates and 1<p,q<∞. Symbolically,

langle Hu, vrangle = langle u, -Hvrangle

for uLp(R) and vLq(R) .

Inverse transform

The Hilbert transform is an anti-involution, meaning that

H(H(u)) = -u,

provided each transform is well-defined. Since H preserves the space Lp(R), this implies in particular that the Hilbert transform is invertible on Lp(R), and that

H^{-1} = -H.,

Differentiation

Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative:

Hleft(frac{du}{dt}right) = frac{d}{dt}H(u).

Iterating this identity,

Hleft(frac{d^ku}{dt^k}right) = frac{d^k}{dt^k}H(u).

This is rigorously true as stated provided u and its first k derivatives belong to Lp(R) .

Convolutions

The Hilbert transform can formally be realized as a convolution with the tempered distribution

h(t) = p.v. frac{1}{pi t}.

Thus formally,

H(u) = hstar u.

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 in Lp. Alternatively, one may use the fact that h(t) is the distributional derivative of the function log|t|/π; to wit

H(u)(t) = frac{d}{dt}left(frac{1}{pi} (ustar log|cdot|)(t)right).

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:

H(u*v) = H(u)*v = u*H(v).

This is rigorously true if u and v are compactly supported distributions since, in that case,

h*(u*v) = (h*u)*v = u*(h*v).

By passing to an appropriate limit, it is thus also true if uLp and vLr provided

1 < frac{1}{p} + frac{1}{r},

a theorem due to .

Invariance

The Hilbert transform has the following invariance properties.

  • It commutes with translations. That is, it commutes with the operators Taƒ(x) = ƒ(x+a) for all a in Rn
  • It commutes with positive dilations. That is it commutes with the operators Mλƒ(x)=ƒ(λx) for all λ > 0.
  • It anticommutes wit the reflection Rƒ(x) = ƒ(−x).

Up to a multiplicative constant, the Hilbert transform is the only L2 bounded operator with these properties .

Extending the domain of definition

Hilbert transform of distributions

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

mathcal{D}_{L^p} = underset{ntoinfty}{underset{longleftarrow}{lim}} W^{n,p}(mathbb{R}).

The Hilbert transform can then be defined on the dual space of mathcal{D}_{L^p}, denoted mathcal{D}_{L^p}', consisting of Lp distributions. This is accomplished by the duality pairing: for uin mathcal{D}'_{L^p}, define H(u)in mathcal{D}'_{L^p} by

langle Hu,vrangle overset{mathrm{def}}{=} langle u, -Hvrangle

for all vinmathcal{D}_{L^p}.

It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to , 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 L(R) as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps L(R) to the Banach space of bounded mean oscillation (BMO) classes.

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-function is therefore defined by the following regularized form of the integral

H(u)(t) = p.v. int_{-infty}^infty u(tau)left{h(t-tau)- h_0(-tau)right},dtau

where as above h(x) = 1/πx and

h_0(x) = begin{cases} 0&mathrm{if }|x|<1 frac{1}{pi x} &mathrm{otherwise} end{cases}

The modified transform H agrees with the original transform on functions of compact support by a general result of ; see . The resulting integral, furthermore, converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.

A deep result of and is that a function is of bounded mean oscillation if and only if it has the form f+H(g) for some f, gL(R).

Conjugate functions

The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function
F(x) = f(x) + ig(x)
is the boundary value of a holomorphic function F(z) in the upper half-plane. Under these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.

Suppose that f ∈ Lp(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

u(x+iy) = u(x,y) = frac{1}{pi}int_{-infty}^infty f(s)frac{y}{(x-s)^2+y^2},ds

which is the convolution of f with the Poisson kernel

P(x,y) = frac{1}{pi}frac{y}{x^2+y^2}.

Furthermore, there is a unique harmonic function v defined in the upper half-plane such that F(z) = u(z) + iv(z) is holomorphic and

lim_{ytoinfty} v(x+iy) = 0.
This harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel

Q(x,y) = frac{1}{pi}frac{x}{x^2+y^2}.

Thus

v(x,y) = frac{1}{pi}int_{-infty}^infty f(s)frac{x-s}{(x-s)^2+y^2},ds.

Indeed, the real and imaginary parts of the Cauchy kernel are

frac{i}{pi z} = P(x,y) + iQ(x,y),
so that F = u + iv is holomorphic by Cauchy's 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,

H(f) = lim_{yto 0} Q(-,y)star f.

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 . 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 H2(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 : RC are equivalent:

  • F(x) is the limit as z → x of a holomorphic function F(z) in the upper half-plane such that

int_{-infty}^infty |F(x+iy)|^2,dx < K.

  • −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 mathcal{F}(F)(x) vanishes for x < 0.

A weaker result is true for functions of class Lp for p > 1 . Specifically, if F(z) is a holomorphic function such that

int_{-infty}^infty |F(x+iy)|^p,dx < K

for all y, then there is a complex-valued function F(x) in Lp(R) such that F(x + iy) → F(x) in the Lp norm as y → 0 (as well as holding pointwise almost everywhere). Furthermore,

F(x) = f(x) - i g(x),

where ƒ is a real-valued function in Lp(R) and g is the Hilbert transform (of class Lp) of ƒ.

This is not true in the case p = 1. In fact, the Hilbert transform of an L1 function ƒ need not converge in the mean to another L1 function. Nevertheless , the Hilbert transform of ƒ does converge almost everywhere to a finite function g such that

int_{-infty}^infty frac< infty.

This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc .

Riemann-Hilbert problem

One form of the Riemann-Hilbert problem seeks to identify pairs of functions F+ and F such that F+ is holomorphic on the upper half-plane and F is holomorphic on the lower half-plane, such that for x along the real axis,
F_+(x) - F_-(x) = f(x)
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± 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± solve the Riemann-Hilbert problem

f(x) = F_+(x) - F_-(x),
then the Hilbert transform of f(x) is given by
H(f)(x) = frac{1}{i}(F_+(x) + F_-(x)).

Hilbert transform on the circle

For a periodic function f the circular Hilbert transform is defined as

tilde f(x)=frac{1}{2pi}text{ p.v.}int_0^{2pi}f(t)cotfrac{x-t}{2},dt.

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 cotfrac{x-t}{2} is known as the Hilbert kernel since it was in this form the Hilbert transform was originally studied .

The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel 1/x periodic. More precisely, for x≠0

frac{1}{2}cotleft(frac{x}{2}right)=frac{1}{x}+sum_{n=1}^infty frac{1}{x+2npi}-frac{1}{2npi}.

Many results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.

Hilbert transform in signal processing

Narrowband model

Amplitude modulated signals are modeled as the product of a bandlimited "message" waveform, um(t), and a sinusoidal "carrier":

u(t) = u_m(t) cdot cos(omega t + phi),

When u_m(t), has no frequency content above the carrier frequency, frac{omega}{2pi} Hz, then:

begin{align} widehat{u}(t) &stackrel{mathrm{def}}{=} H(u)(t) &= u_m(t) cdot sin(omega t + phi) end{align}

So, the Hilbert transform may be as simple as a circuit that produces a 90° phase shift at the carrier frequency. Furthermore:

(omega t + phi)_{mathrm{mod}, 2 pi}, = operatorname{atan2}( widehat{u}(t), u(t) ),
= arg(u_a(t)),        (see next section)

from which one can reconstruct the carrier waveform. Then the message can be extracted from u(t) by coherent demodulation.

Analytic representation

The analytic representation of a signal is defined in terms of the Hilbert transform:

u_a(t) = u(t) + icdot widehat{u}(t).,

For the narrowband model [above], the analytic representation is:

u_a(t),   = u_m(t) cdot cos(omega t + phi) + icdot u_m(t) cdot sin(omega t + phi),
= u_m(t) cdot left[cos(omega t + phi) + icdot sin(omega t + phi)right],
{{NumBlk|:::|= u_m(t) cdot e^{i(omega t + phi)},   (by Euler's formula)}

This complex heterodyne operation shifts all the frequency components of um(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, ua(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   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:

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}

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

h[n]= begin{cases} 0, & mbox{for }nmbox{ even}, frac2{pi n} & mbox{for }nmbox{ odd} end{cases}

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:

sigma_H(omega) = begin{cases} e^{+ipi/2}, & -pi < omega < 0 e^{-ipi/2}, & 0 < omega < pi 0 & omega=-pi, 0, pi end{cases}

We note that a sequence similar to h[n], can be generated by sampling σH(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 σH(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 mneq n.
Hilbert showed that for an in ℓ2 the sequence bn is also in ℓ2. An elementary proof of this fact can be found in . The discrete Hilbert transform was used by E. C. Titchmarsh to give alternate proofs of the results of M. Riesz in the continuous case ().

See also

References

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