Hilbert transform


In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u of a real variable and produces another function of a real variable H. This linear operator is given by convolution with the function :
the improper integral being understood in the principal value sense. The Hilbert transform has a particularly simple representation in the frequency domain: it imparts a phase shift of -90° to every Fourier component of a function. For example, the Hilbert transform of, where ω > 0, is.
The Hilbert transform is important in signal processing, where it derives the analytic representation of a real-valued signal u. Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations. 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.

Introduction

The Hilbert transform of u can be thought of as the convolution of u with the function h = 1/, known as the Cauchy kernel. Because h is not integrable, 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 u is given by
provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/t. 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 u, the result is negative u:
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.
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 f is analytic in the plane Im z > 0, and u = Re f, then Im f = H up to an additive constant, provided this Hilbert transform exists.

Notation

In signal processing the Hilbert transform of u is commonly denoted by . However, in mathematics, this notation is already extensively used to denote the Fourier transform of u. 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 L2 and ℓ2. In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in Lp for 1 ≤ p < ∞, that the Hilbert transform is a bounded operator on Lp for 1 < 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

The Hilbert transform is a multiplier operator. The multiplier of H is σH = −i sgn, where sgn is the signum function. Therefore:
where denotes the Fourier transform. Since sgn = sgn, it follows that this result applies to the three common definitions of.
By Euler's formula,
Therefore, H has the effect of shifting the phase of the negative frequency components of u by +90° and the phase of the positive frequency components by −90°. And i·H 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 are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e., H = −u, because

Table of selected Hilbert transforms

In the following table, the frequency parameter is real.
Signal
Hilbert transform

Sinc function
Rectangular 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 Lp for 1 < p < ∞.
More precisely, if u is in Lp for 1 < p < ∞, then the limit defining the improper integral
exists for almost every t. The limit function is also in Lp and is in fact the limit in the mean of the improper integral as well. That is,
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 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 L1 function does converge, however, in L1-weak, and the Hilbert transform is a bounded operator from L1 to L1,w.

Properties

Boundedness

If 1 < p < ∞, then the Hilbert transform on Lp is a bounded linear operator, meaning that there exists a constant Cp such that
for all u∈Lp. This theorem is due to ; see also.
The best constant Cp is given by
This result is due to ; see also. An easy way to find the best for p being a power of 2 is through the so-called Cotlar's identity that for all real valued f. The same best constants hold for the periodic Hilbert transform.
The boundedness of the Hilbert transform implies the Lp convergence of the symmetric partial sum operator
to f in Lp, see for example.

Anti-self adjointness

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

Inverse transform

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

Complex structure

Because H2=−Id on the real Banach space of real-valued functions in Lp, the Hilbert transform defines a linear complex structure on this Banach space. In particular, when p=2, the Hilbert transform gives the Hilbert space of real-valued functions in L2 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 H2 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 u and its first k derivatives belong to Lp. One can check this easily in the frequency domain, where differentiation becomes multiplication by ω.

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 u a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported functions are dense in Lp. Alternatively, one may use the fact that h is the distributional derivative of the function log|t|/; 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 on either factor:
This is rigorously true if u and v are compactly supported distributions since, in that case,
By passing to an appropriate limit, it is thus also true if u ∈ Lp and v ∈ Lr provided
a theorem due to.

Invariance

The Hilbert transform has the following invariance properties on L2.
Up to a multiplicative constant, the Hilbert transform is the only bounded operator on L2 with these properties.
In fact there is a larger group of operators commuting with the Hilbert transform. The group SL acts by unitary operators Ug on the space L2 by the formula
This unitary representation is an example of a principal series representation of SL. In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space H2 and its conjugate. These are the spaces of L2 boundary values of holomorphic functions on the upper and lower halfplanes. H2 and its conjugate consist of exactly those L2 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, with P being the orthogonal projection from L2 onto H2, it follows that H2 and its orthogonal are eigenspaces of H for the eigenvalues ± i. In other words, H commutes with the operators Ug. The restrictions of the operators Ug to H2 and its conjugate give irreducible representations of SL—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. 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:
The Hilbert transform can then be defined on the dual space of, denoted, consisting of Lp distributions. This is accomplished by the duality pairing:
For, define:
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∞ as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps L∞ to the Banach space of bounded mean oscillation classes.
Interpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined. For instance, with u = sgn, the integral defining H 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
where as above h = 1/x and
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 ƒ + H for some ƒ, g ∈ L∞.

Conjugate functions

The Hilbert transform can be understood in terms of a pair of functions f and g such that the function
is the boundary value of a holomorphic function F 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. 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
which is the convolution of f with the Poisson kernel
Furthermore, there is a unique harmonic function v defined in the upper half-plane such that F = u + iv is holomorphic and
This harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel
Thus
Indeed, the real and imaginary parts of the Cauchy kernel are
so that F = u + iv is holomorphic by Cauchy's integral formula.
The function v obtained from u in this way is called the harmonic conjugate of u. The boundary limit of v as y → 0 is the Hilbert transform of f. Thus, succinctly,

Titchmarsh's theorem

Titchmarsh's theorem, named for Edward Charles 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. It gives necessary and sufficient conditions for a complex-valued square-integrable function F on the real line to be the boundary value of a function in the Hardy space H2 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:
A weaker result is true for functions of class Lp for p > 1. Specifically, if F is a holomorphic function such that
for all y, then there is a complex-valued function F in Lp such that F → F in the Lp norm as y → 0. Furthermore,
where ƒ is a real-valued function in Lp and g is the Hilbert transform 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
This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc. Although usually called Titchmarsh's theorem, the result aggregates much work of others, including
Hardy, Paley and Wiener 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 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,
where f 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
then the Hilbert transform of f is given by

Hilbert transform on the circle

For a periodic function f the circular Hilbert transform is defined:
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,
The Hilbert kernel can be obtained by making the Cauchy kernel 1/x periodic. More precisely, for x ≠ 0
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 = / , which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map
of L2 onto L2. The operator U carries the Hardy space H2 onto the Hardy space H2.

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
where fLP and fHP are the low- and high-pass signals respectively.
Amplitude modulated signals are modeled as the product of a bandlimited "message" waveform, um, and a sinusoidal "carrier":
When um has no frequency content above the carrier frequency, then by Bedrosian's theorem:

Analytic representation

In the context of signal processing, the conjugate function interpretation of the Hilbert transform, discussed above, gives the analytic representation of a signal u:
which is a holomorphic function in the upper half plane.
For the narrowband model, the analytic representation is:
This complex heterodyne operation shifts all the frequency components of um 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.

Angle (phase/frequency) modulation

The form:
is called angle modulation, which includes both phase modulation and frequency modulation. The instantaneous frequency is For sufficiently large ω, compared to :
and:

Single sideband modulation (SSB)

When um in is also an analytic representation, that is:
the result is single-sideband modulation:
whose transmitted component is:

Causality

The function h with h = 1/ 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 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, 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

For a discrete function, with discrete-time Fourier transform, and discrete Hilbert transform the DTFT of in the region − < ω < is given by:
The inverse DTFT, using the convolution theorem, is:
where:
which is an infinite impulse response. When the convolution is performed numerically, an FIR approximation is substituted for h, 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 is shown in Figure 2. Since the magnitude response at Nyquist does not drop out, it approximates an ideal Hilbert transformer a little better than the odd-tap filter. However:
The MATLAB function, , convolves a u sequence with the periodic summation:
and returns one cycle 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 with samples of the −i•sgn distribution. Figure 3 compares a half-cycle of hN with an equivalent length portion of h. Given an FIR approximation for denoted by substituting for the −i•sgn 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 u. When the input is a segment of a pure cosine, the resulting convolution for two different values of N is depicted in Figure 4. Edge effects prevent the result from being a pure sine function. Since hN 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 N 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 h 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 u sequence. Segments of length N are convolved with the periodic function:
When the duration of non-zero values of is M < N, the output sequence includes N − M + 1 samples of M-1 outputs are discarded from each block of N, 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 shows, the distortions apparent in the IIR result are not caused by the difference between h and hN. The fact that hN is tapered is actually helpful in this context. The real problem is that it's not windowed enough. Effectively, M = N, whereas the overlap-save method needs M < N.

Number-theoretic Hilbert transform

The number theoretic Hilbert transform is an extension 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.