Hilbert transform

In mathematics, signal processing and control theory, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the function formula_1 (see ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency. (see ) The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal , that describes a physically realizable 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. 
The Hilbert Transform of a function is a one sided integration of finite differences of that function. This may seem useless at first glance, but it adds an orientation and causality and removes a constant bias. When it is applied more often, it behaves like a multiplication with the imaginary unit, which is reflected by the phase shift in the frequency domain. It is technically used to create an envelope representation of signal, by creating an analytic signal as an intermediate step. Since it is strongly related to analytic functions (or more precisely Holomorphic function for complex mappings), it is a fundamental concept in control theory as it naturally stabilizes a function. It includes (very) advanced/subtle mathematical concepts/calculus methods and is therefore often avoided explicitly in engeneering, by replacing it by schemes, that implicitly use it. Typically, one needs it, if these schemes are extended or prodcue counterintuitive results. However, if it (and its relations to other concepts, physics and in particular boundary value problems) is well understood, it is a very powerful tool and serves as a link between many disciplines.
Definition.
The Hilbert transform of is a convolution of with the function (i.e. a Hilbert Transform of the dirac delta), known as the Cauchy kernel. Because is not integrable across , the integral defining the convolution does not always converge (it only converges if the input function is analytic, i.e. if it has no singularities). Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by ). Explicitly, the Hilbert transform of a function (or signal) is given by
formula_2
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
formula_3
which is an integration over central differences. This reformulation is only valid, if the convergence of towards the boundary value at zero from the left and right leads to the same value. It has to converge quicker or equal to linear towards this value from both sides, i.e. dominating the singularity of the denominator. If this is not the case, one has to remove this (and maybe other finitely many) singulariti(es) to ensure convergence before the integration can be applied. It is an inherent part of the transformation and is called Analytic continuation. It is implicitly part of the first definition, due to the Cauchy Principal value. Due to the integration over finite differences, a bias is removed, i.e. for its Fourier Transform it holds . After applying it, the result of the transformation is an analytic function and the value of the bias cannot be recovered.
Both definitions include a physical damping factor in the denominator counteracting the integration, that ensures conservation of energy of the transformation. The second definition is more natural/intuitive and provides more insight as it describes its purpose more precisely. The factor formula_4 already reflects the relationship to a 90 degree phase shift in the frequency domain. When the Hilbert transform is applied twice in succession to a (analytic) function , the result is negative :
formula_5
Since is analytic, the convergence of the integrals is guaranteed. 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 (see , below). Due to this hermetian symmetry, the absence of a bias can be imediatly deduced. This property is crucial for the understanding of the whole concept. The following important porperty allows for a simplified handling as a formal convolution (provided fullfills the aforementioned convergence properties):
formula_6 and therefore formula_7.
For an analytic function in the upper half-plane (i.e. the causal region), the Hilbert transform links its real and imaginary part to satisfy the Cauchy–Riemann equations, that are the underlying invariance conditions for all these relations. That is, if is analytic in the upper half complex plane , and , then up to an additive constant, i.e. bias. This linkage ensures, that all mappings of all analytic functions (parametrized by ) onto the upper complex plane are conformal (i.e. preserving angles between arbitrary analytic signals at their crossings) and is necessary to satisfy the Fundamental theorem of calculus.
Notation.
In signal processing the Hilbert transform of is commonly denoted by formula_8. However, in mathematics, this notation is already extensively used to denote the Fourier transform of . Occasionally, the Hilbert transform may be denoted by formula_9. 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 formula_10 (Lp space) for , that the Hilbert transform is a bounded operator on formula_10 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. One has to be careful, since there are several definitions/variants, dependend on the classes of functions that can be applied. This is due to the fact, that it was stated before the formal developement of distributions was put on a solid theoretical basis.
Relationship with the Fourier transform.
The Hilbert transform is a multiplier operator. The multiplier of is , where is the signum function. Therefore:
formula_12
where formula_13 denotes the Fourier transform. Since , it follows that this result applies to the three common definitions of formula_14.
By Euler's formula,
formula_15
Therefore, has the effect of shifting the phase of the negative frequency components of by +90° ( radians) 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 (i.e., a multiplication by −1).
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
formula_16
Table of selected Hilbert transforms.
In the following table, the frequency parameter formula_17 is real.
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 formula_10 for .
More precisely, if is in formula_10 for , then the limit defining the improper integral
formula_20
exists for almost every . The limit function is also in formula_10 and is in fact the limit in the mean of the improper integral as well. That is,
formula_22
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 . (In particular, since the Hilbert transform is also a multiplier operator on , Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that is bounded on .)
Properties.
Boundedness.
If , then the Hilbert transform on formula_10 is a bounded linear operator, meaning that there exists a constant such that
formula_24
for all 
The best constant formula_25 is given by
formula_26
An easy way to find the best formula_25 for formula_28 being a power of 2 is through the so-called Cotlar's identity that formula_29 for all real valued . The same best constants hold for the periodic Hilbert transform.
The boundedness of the Hilbert transform implies the formula_10 convergence of the symmetric partial sum operator 
formula_31
to in 
Anti-self adjointness.
The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between formula_10 and the dual space where and are Hölder conjugates and . Symbolically,
formula_33
for formula_34 and 
Inverse transform.
The Hilbert transform is an anti-involution, meaning that
formula_35
provided each transform is well-defined. Since preserves the space this implies in particular that the Hilbert transform is invertible on and that
formula_36
Complex structure.
Because ("" is the identity operator) 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 formula_37 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 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:
formula_38
Iterating this identity,
formula_39
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
formula_40
Thus formally,
formula_41,
and therefore with formula_42 being the dirac delta distribution:
formula_43.
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 (which are distributions "a fortiori") are dense in . Alternatively, one may use the fact that "h"("t") is the distributional derivative of the function ; to wit
formula_44
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:
formula_45
This is rigorously true if and are compactly supported distributions since, in that case,
formula_46
By passing to an appropriate limit, it is thus also true if and provided that
formula_47
from a theorem due to Titchmarsh.
Invariance.
The Hilbert transform has the following invariance properties on formula_37.
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 formula_50 acts by unitary operators on the space formula_37 by the formula
formula_52
This unitary representation is an example of a principal series representation of formula_53 In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space formula_54 and its conjugate. These are the spaces of boundary values of holomorphic functions on the upper and lower halfplanes. formula_54 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 formula_37 onto formula_57 and the identity operator, it follows that formula_58 and its orthogonal are eigenspaces of for the eigenvalues . In other words, commutes with the operators . The restrictions of the operators to formula_58 and its conjugate give irreducible representations of formula_50 – 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 , restricts to give a continuous transform on the inverse limit of Sobolev spaces:
formula_61
The Hilbert transform can then be defined on the dual space of formula_62, denoted formula_63, consisting of distributions. This is accomplished by the duality pairing:
For define:
formula_64
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, 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 formula_65 as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps formula_65 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 , the integral defining diverges almost everywhere to . To alleviate such difficulties, the Hilbert transform of an function is therefore defined by the following regularized form of the integral
formula_67
where as above and
formula_68
The modified transform agrees with the original transform 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 for some 
Conjugate functions.
The Hilbert transform can be understood in terms of a pair of functions and such that the function
formula_69
is the boundary value of a holomorphic function in the upper half-plane. Under these circumstances, if and are sufficiently integrable, then one is the Hilbert transform of the other.
Suppose that formula_70 Then, by the theory of the Poisson integral, admits a unique harmonic extension into the upper half-plane, and this extension is given by
formula_71
which is the convolution of with the Poisson kernel
formula_72
Furthermore, there is a unique harmonic function defined in the upper half-plane such that is holomorphic and
formula_73
This harmonic function is obtained from by taking a convolution with the "conjugate Poisson kernel"
formula_74
Thus
formula_75
Indeed, the real and imaginary parts of the Cauchy kernel are
formula_76
so that is holomorphic by Cauchy's integral formula.
The function obtained from in this way is called the harmonic conjugate of . The (non-tangential) boundary limit of as is the Hilbert transform of . Thus, succinctly,
formula_77
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. It gives necessary and sufficient conditions for a complex-valued square-integrable function on the real line to be the boundary value of a function in the Hardy space of holomorphic functions in the upper half-plane .
The theorem states that the following conditions for a complex-valued square-integrable function formula_78 are equivalent:
These three points describe a physical realizable (analytic) signal: I.e. it is limited in energy and it has a starting point (i.e. no negative frequencies).
A weaker result is true for functions of class for . Specifically, if is a holomorphic function such that
formula_81
for all , then there is a complex-valued function in formula_10 such that in the norm as (as well as holding pointwise almost everywhere). Furthermore,
formula_83
where is a real-valued function in formula_10 and is the Hilbert transform (of class ) of .
This is not true in the case . In fact, the Hilbert transform of an function need not converge in the mean to another function. Nevertheless, the Hilbert transform of does converge almost everywhere to a finite function such that
formula_85
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 (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 and such that is holomorphic on the upper half-plane and is holomorphic on the lower half-plane, such that for along the real axis,
formula_86
where is some given real-valued function of The left-hand side of this equation may be understood either as the difference of the limits of 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 solve the Riemann–Hilbert problem
formula_87
then the Hilbert transform of is given by
formula_88
Hilbert transform on the circle.
For a periodic function the circular Hilbert transform is defined:
formula_89
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, 
formula_90
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 periodic. More precisely, for 
formula_91
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 , which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map
formula_92
of onto formula_93 The operator carries the Hardy space onto the Hardy space formula_54.
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
formula_95
where and are the low- and high-pass signals respectively. 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":
formula_96
where is the narrow bandwidth "message" waveform, such as voice or music. Then by Bedrosian's theorem:
formula_97
Analytic representation.
A specific type of conjugate function is:
formula_98
known as the "analytic representation" of formula_99 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 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:
formula_100
is called angle modulation, which includes both phase modulation and frequency modulation. The instantaneous frequency is  formula_101  For sufficiently large , compared to 
formula_102
and:
formula_103
Single sideband modulation (SSB).
When in   is also an analytic representation (of a message waveform), that is:
formula_104
the result is single-sideband modulation:
formula_105
whose transmitted component is:
formula_106
Causality.
The function formula_107 presents two challenges to practical implementation as a convolution:
Discrete Hilbert transform.
For a discrete function, with discrete-time Fourier transform (DTFT), and discrete Hilbert transform the DTFT of formula_110 in the region is given by:
formula_111
The inverse DTFT, using the convolution theorem, is:
formula_112
where
formula_113
which is an infinite impulse response (IIR). When the convolution is performed numerically, an FIR approximation is substituted for , 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
The MATLAB function, , convolves a u[n] sequence with the periodic summation:
and returns one cycle ( 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  formula_117  with samples of the distribution (whose real and imaginary components are all just 0 or ). "Figure 3" compares a half-cycle of with an equivalent length portion of . Given an FIR approximation for formula_118 denoted by formula_119 substituting formula_120 for the 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 . When the input is a segment of a pure cosine, the resulting convolution for two different values of is depicted in "Figure 4" (red and blue plots). Edge effects prevent the result from being a pure sine function (green plot). Since 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 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 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 sequence. Segments of length are convolved with the periodic function:
formula_121
When the duration of non-zero values of formula_122 is formula_123 the output sequence includes samples of formula_124 outputs are discarded from each block of , 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 and (green and red in "Figure 3"). The fact that is tapered ("windowed") is actually helpful in this context. The real problem is that it's not windowed enough. Effectively, , whereas the overlap-save method needs .
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.