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In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). 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(t). Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u+iv 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.

Contents

IntroductionEdit

The Hilbert transform of u can be thought of as the convolution of u(t) with the function h(t) = 1/(πt), known as the Cauchy kernel. Because h(t) is not integrable, the integral defining the convolution does not always 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:

 

provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/πt (due to Schwartz (1950); see Pandey (1996, Chapter 3)). Alternatively, by changing variables, the principal value integral can be written explicitly (Zygmund 1968, §XVI.1) 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(t) (see Relationship with the Fourier transform below).

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 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.

NotationEdit

In signal processing the Hilbert transform of u(t) is commonly denoted by   (e.g., Brandwood 2003, pg 87). However, in mathematics, this notation is already extensively used to denote the Fourier transform of u(t) (e.g., Stein & Weiss 1971). Occasionally, the Hilbert transform may be denoted by  . Furthermore, many sources define the Hilbert transform as the negative of the one defined here (e.g., Bracewell 2000, pg 359).

HistoryEdit

The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions (Kress (1989); Bitsadze (2001)), 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 (Khvedelidze 2001; Hilbert 1953). 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 (Hardy, Littlewood & Pólya 1952, §9.1). Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case (Hardy, Littlewood & Pólya 1952, §9.2). 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 1 < p < ∞, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform (Riesz 1928). The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals (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 transformEdit

The Hilbert transform is a multiplier operator (Duoandikoetxea 2000, Chapter 3). The multiplier of H is σH(ω) = −i sgn(ω) where sgn is the signum function. Therefore:

 

where   denotes the Fourier transform. Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of  .

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:

 

Table of selected Hilbert transformsEdit

In the following table, the frequency parameter   is real.

Signal
 
Hilbert transform[fn 1]

 

  [fn 2]  
  [fn 2]  
   
   
   
(see Dawson function)
Sinc function
 
 
Rectangular function
 
 
Dirac delta function
 
 
Characteristic Function
 
 
Notes
  1. ^ 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.
  2. ^ a b The Hilbert transform of the sin and cos functions can be defined by taking the principal value of the integral at infinity. This definition agrees with the result of defining the Hilbert transform distributionally.

An extensive table of Hilbert transforms is available (King 2009b). Note that the Hilbert transform of a constant is zero.

Domain of definitionEdit

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 the limit defining the improper integral

 

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,

 

as ε→0 in the Lp-norm, as well as pointwise almost everywhere, by the Titchmarsh theorem (Titchmarsh 1948, Chapter 5).

In the case p = 1, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally (Titchmarsh 1948, §5.14). 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 (Stein & Weiss 1971, Lemma V.2.8). (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.)

PropertiesEdit

BoundednessEdit

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

 

for all uLp(R). This theorem is due to Riesz (1928, VII); see also Titchmarsh (1948, Theorem 101). The best constant Cp is given by

 

This result is due to (Pichorides 1972); see also Grafakos (2004, Remark 4.1.8). The same best constants hold for the periodic Hilbert transform.

The boundedness of the Hilbert transform implies the Lp(R) convergence of the symmetric partial sum operator

 

to f in Lp(R), see for example (Duoandikoetxea 2000, p. 59).

Anti-self adjointnessEdit

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,

 

for u ∈ Lp(R) and v ∈ Lq(R) (Titchmarsh 1948, Theorem 102).

Inverse transformEdit

The Hilbert transform is an anti-involution (Titchmarsh 1948, p. 120), meaning that

 

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

 

Complex structureEdit

Because H2=−Id on the real Banach space of real-valued functions in Lp(R), 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(R) 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 H2 by the Paley–Wiener theorem.

DifferentiationEdit

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(R) (Pandey 1996, §3.3). One can check this easily in the frequency domain, where differentiation becomes multiplication by ω.

ConvolutionsEdit

The Hilbert transform can formally be realized as a convolution with the tempered distribution (Duistermaat & Kolk 2010, p. 211)

 

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 (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

 

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 Titchmarsh (1948, Theorem 104).

InvarianceEdit

The Hilbert transform has the following invariance properties on L2(R).

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

Up to a multiplicative constant, the Hilbert transform is the only bounded operator on L2 with these properties (Stein 1970, §III.1).

In fact there is a larger group of operators commuting with the Hilbert transform. The group SL(2,R) acts by unitary operators Ug on the space L2(R) by the formula

 

This unitary representation is an example of a principal series representation of SL(2,R). In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space H2(R) and its conjugate. These are the spaces of L2 boundary values of holomorphic functions on the upper and lower halfplanes. H2(R) 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 (2P − I), with P being the orthogonal projection from L2(R) onto H2(R), it follows that H2(R) 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(R) and its conjugate give irreducible representations of SL(2,R)—the so-called limit of discrete series representations.[1]

Extending the domain of definitionEdit

Hilbert transform of distributionsEdit

It is further possible to extend the Hilbert transform to certain spaces of distributions (Pandey 1996, Chapter 3). 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   by

 

for all  .

It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to Gel'fand & Shilov (1967)[page needed], but considerably more care is needed because of the singularity in the integral.

Hilbert transform of bounded functionsEdit

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 naïvely, 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

 

where as above h(x) = 1/πx and

 

The modified transform H agrees with the original transform on functions of compact support by a general result of Calderón & Zygmund (1952); see Fefferman (1971). 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 Fefferman (1971) and Fefferman & Stein (1972) is that a function is of bounded mean oscillation if and only if it has the form ƒ + H(g) for some ƒ, g ∈ L(R).

Conjugate functionsEdit

The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function

 

is the boundary value of a holomorphic function F(z) in the upper half-plane (Titchmarsh 1948, Chapter V). 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

 

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(z) = u(z) + iv(z) 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 (non-tangential) boundary limit of v(x,y) as y → 0 is the Hilbert transform of f. Thus, succinctly,

 

Titchmarsh's theoremEdit

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 (Titchmarsh 1948, Theorem 95). 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
 
  • The real and imaginary parts of F(x) are Hilbert transforms of each other.
  • The Fourier transform   vanishes for x < 0.

A weaker result is true for functions of class Lp for p > 1 (Titchmarsh 1948, Theorem 103). Specifically, if F(z) is a holomorphic function such that

 

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,

 

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, (Titchmarsh 1948, Theorem 105), 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 (Duren 1970, Theorem 4.2). 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 (see section 4.22 in King (2009a)).

Riemann–Hilbert problemEdit

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(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

 

then the Hilbert transform of f(x) is given by

  (Pandey 1996, Chapter 2).

Hilbert transform on the circleEdit

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

 

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,

  is known as the Hilbert kernel since it was in this form the Hilbert transform was originally studied (Khvedelidze 2001).

The Hilbert kernel (for the circular Hilbert transform) 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(x) = (x – i) / (x + i), which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map

 

of L2(T) onto L2(R). The operator U carries the Hardy space H2(T) onto the Hardy space H2(R).[2]

Hilbert transform in signal processingEdit

Bedrosian's theoremEdit

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 (Schreier & Scharf 2010, 14).

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

 

When um(t) has no frequency content above the carrier frequency,   then by Bedrosian's theorem:

  (Bedrosian 1962)

Analytic representationEdit

In the context of signal processing, the conjugate function interpretation of the Hilbert transform, discussed above, gives the analytic representation of a signal u(t):

 

which is a holomorphic function in the upper half plane.

For the narrowband model (above), the analytic representation is:

 

    (by Euler's formula)

 

 

 

 

(Eq.1)

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.

Angle (phase/frequency) modulationEdit

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)Edit

When um(t) in  Eq.1 is also an analytic representation (of a message waveform), that is:

 

the result is single-sideband modulation:

 

whose transmitted component is:

 

CausalityEdit

The function h with h(t) = 1/(πt) 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 transformEdit

 
Figure 1: Filter whose frequency response is bandlimited to about 95% of the Nyquist frequency
 
Figure 2: Hilbert transform filter with a highpass frequency response
 
Figure 3.
 
Figure 4. The Hilbert transform of cos(ωt) is sin(ωt). This figure shows sin(ωt) and two approximate Hilbert transforms computed by the MATLAB library function, hilbert(·)
 
Figure 5. Discrete Hilbert transforms of a cosine function, using piecewise convolution

For a discrete function,     with discrete-time Fourier transform (DTFT),     and discrete Hilbert transform     the DTFT of     in the region  −π < ω <  π is given by:

 

The inverse DTFT, using the convolution theorem, is:

 

where:

 

which is an infinite impulse response (IIR). When the convolution is performed numerically, an FIR approximation is substituted for h[n], 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 Nyquist does not drop out, it approximates an ideal Hilbert transformer a little better than the odd-tap filter. However:

  • A typical (i.e. properly filtered and sampled) u[n] sequence has no useful components at the Nyquist frequency.
  • The Type IV impulse response requires a ½ sample shift in the h[n] sequence. That causes the zero-valued coefficients to become non-zero, as seen in Figure 2. So a Type III design is potentially twice as efficient as Type IV.
  • The group delay of a Type III design is an integer number of samples, which facilitates aligning   with   to create an analytic signal. The group delay of Type IV is halfway between two samples.

The MATLAB function, hilbert(u,N), convolves a u[n] sequence with the periodic summation:[3]

      [4]

and returns one cycle (N 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     with samples of the −i•sgn(ω) distribution (whose real and imaginary components are all just 0 or ±1). Figure 3 compares a half-cycle of hN[n] with an equivalent length portion of h[n]. Given an FIR approximation for    denoted by     substituting     for the −i•sgn(ω) 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[n]. When the input is a segment of a pure cosine, the resulting convolution for two different values of N is depicted in Figure 4 (red and blue plots). Edge effects prevent the result from being a pure sine function (green plot). Since hN[n] 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[n] 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[n] 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 (blue) shows, the distortions apparent in the IIR result (red) are not caused by the difference between h[n] and hN[n] (green and red in Fig 3). The fact that hN[n] is tapered (windowed) 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 transformEdit

The number theoretic Hilbert transform is an extension (Kak 1970) 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(Kak 2014).

See alsoEdit

NotesEdit

  1. ^ See:
  2. ^ Rosenblum & Rovnyak 1997, p. 92
  3. ^ see Convolution Theorem
  4. ^ For even values of N, an equivalent closed form is:    
    See http://www.rle.mit.edu/dspg/documents/HilbertComplete.pdf eq. (17), (18), and unlabeled eq. below (18).

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