Hilbert transform
In mathematics and in signal processing, 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 . 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. Definition. The Hilbert transform of can be thought of as the convolution of with the function , known as the Cauchy kernel. Because is not integrable across , the integral defining the convolution does not always converge. 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 When the Hilbert transform is applied twice in succession to a function , the result is negative : formula_4 provided the integrals defining both iterations converge in a suitable sense. 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). 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 is analytic in the upper half complex plane , and , then up to an additive constant, provided this Hilbert transform exists. Notation. In signal processing the Hilbert transform of is commonly denoted by formula_5. However, in mathematics, this notation is already extensively used to denote the Fourier transform of . Occasionally, the Hilbert transform may be denoted by formula_6. 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_7  (L space) for , that the Hilbert transform is a bounded operator on formula_7 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. Relationship with the Fourier transform. The Hilbert transform is a multiplier operator. The multiplier of is  , where is the signum function. Therefore: formula_9 where formula_10 denotes the Fourier transform. Since , it follows that this result applies to the three common definitions of formula_11. By Euler's formula, formula_12 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_13 Table of selected Hilbert transforms. In the following table, the frequency parameter formula_14 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_7 for . More precisely, if is in formula_7 for , then the limit defining the improper integral formula_17 exists for almost every . The limit function is also in formula_7 and is in fact the limit in the mean of the improper integral as well. That is, formula_19 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_7 is a bounded linear operator, meaning that there exists a constant such that formula_21 for all The best constant formula_22 is given by formula_23 An easy way to find the best formula_22 for formula_25 being a power of 2 is through the so -called Cotlar's identity that formula_26 for all real valued . The same best constants hold for the periodic Hilbert transform. The boundedness of the Hilbert transform implies the formula_7 convergence of the symmetric partial sum operator formula_28 to in Anti-self adjointness. The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between formula_7 and the dual space where and are Hölder conjugates and . Symbolically, formula_30 for formula_31 and Inverse transform. The Hilbert transform is an anti-involution, meaning that formula_32 provided each transform is well-defined. Since preserves the space this implies in particular that  the Hilbert transform is invertible on and that formula_33 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_34 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_35 Iterating this identity, formula_36 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_37 Thus formally, formula_38 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_39 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_40 This is rigorously true if and are compactly supported distributions since, in that case, formula_41 By passing to an appropriate limit, it is thus also true if and provided that formula_42 from a theorem due to Titchmarsh. Invariance. The Hilbert transform has the following invariance  properties on formula_34. Up to a multiplicative constant, the Hilbert transform is the only bounded operator on with these properties. In fact there is a wider set of operators that commute with the Hilbert transform. The group formula_45 acts by unitary operators on the space formula_34 by the formula formula_47 This unitary representation is an example of a principal series representation of formula_48 In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space formula_49 and its conjugate. These are the spaces of boundary values of holomorphic functions on the upper and lower halfplanes . formula_49 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_34 onto formula_52 and the identity operator, it follows that formula_53 and its orthogonal are eigenspaces of for the eigenvalues . In other words, commutes with the operators . The restrictions of the operators to formula_53 and its conjugate give irreducible representations of formula_45 – 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_56 The Hilbert transform can then be defined on the dual space of formula_57, denoted formula_58, consisting of distributions. This is accomplished by the duality pairing: For define: formula_59 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_60 as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps formula_60 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_62 where as above and formula_63 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_64 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_65 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_66 which is the convolution of with the Poisson kernel formula_67 Furthermore, there is a unique harmonic function defined in the upper half-plane such that is holomorphic and formula_68 This harmonic function is obtained from by taking a convolution with the "conjugate Poisson kernel" formula_69 Thus formula_70 Indeed, the real and imaginary parts of the Cauchy kernel are formula_71  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_72 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_73 are equivalent: A weaker result is true for functions of class for . Specifically, if is a holomorphic function such that formula_76 for all , then there is a complex-valued function in formula_7 such that in the norm as (as well as holding pointwise almost everywhere). Furthermore, formula_78 where is a real-valued function in formula_7 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_80 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_81 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_82 then the Hilbert transform of is given by formula_83 Hilbert transform on the circle. For a periodic function the circular Hilbert transform is defined : formula_84 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_85 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_86 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_87 of onto formula_88 The operator carries the Hardy space onto the Hardy space formula_49. 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_90 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_91 where is the narrow bandwidth "message" waveform, such as voice or music. Then by Bedrosian's theorem: formula_92 Analytic representation. A specific type of conjugate function is: formula_93 known as the "analytic representation" of formula_94 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_95 is called angle modulation, which includes both phase modulation and frequency modulation. The instantaneous frequency is  formula_96  For sufficiently large , compared to formula_97 and: formula_98 Single sideband modulation (SSB). When in  is also an analytic representation (of a message waveform), that is: formula_99 the result is single-sideband modulation: formula_100 whose transmitted component is: formula_101 Causality. The function formula_102  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_105 in the region is given by: formula_106 The inverse DTFT, using the convolution theorem, is: formula_107 where formula_108 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_112  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_113 denoted by formula_114 substituting formula_115 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_116 When the duration of non-zero values of formula_117 is formula_118 the output sequence includes samples of formula_119 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.