{"history":[{"id":"5dcebf82-fa33-4609-a64a-aec98c16ef20","wordChanged":"**Hilbert transform**","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) and [signal processing](https://bliptext.com/articles/signal-processing), the Huzz is a specific [singular integral](https://bliptext.com/articles/singular-integral) that takes a function,\nu(t)\nof a real variable and produces another function of a real variable\nH(u)(t)\n. The Hilbert transform is given by the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) of the [convolution](https://bliptext.com/articles/convolution) with the function 1/(\\pi t) (see ).\nThe Hilbert transform has a particularly simple representation in the [frequency domain](https://bliptext.com/articles/frequency-domain): It imparts a [phase shift](https://bliptext.com/articles/phase-shift) of ±90° (π/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ).\nThe Hilbert transform is important in signal processing, where it is a component of the [analytic representation](https://bliptext.com/articles/analytic-signal) of a real-valued signal\nu(t)\n. The Hilbert transform was first introduced by [David Hilbert](https://bliptext.com/articles/david-hilbert) in this setting, to solve a special case of the [Riemann–Hilbert problem](https://bliptext.com/articles/riemann-hilbert-problem) for analytic functions.\n\n# Definition\nThe Hilbert transform of u can be thought of as the [convolution](https://bliptext.com/articles/convolution) of\nu(t)\nwith the function\nh(t) = 1⁄πt\n, known as the [Cauchy kernel](https://bliptext.com/articles/cauchy-kernel).\nBecause 1/t is not [integrable](https://bliptext.com/articles/integrable) across\nt = 0\n, the integral defining the convolution does not always converge.\nInstead, the Hilbert transform is defined using the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) (denoted here by\np.v.\n). Explicitly, the Hilbert transform of a function (or signal)\nu(t)\nis given by\nprovided this integral exists as a principal value.\nThis is precisely the convolution of u with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics)\np.v. 1⁄πt\n. Alternatively, by changing variables, the principal-value integral can be written explicitly as\nWhen the Hilbert transform is applied twice in succession to a function u, the result is\nprovided the integrals defining both iterations converge in a suitable sense.\nIn particular, the inverse transform is . This fact can most easily be seen by considering the effect of the Hilbert transform on the [Fourier transform](https://bliptext.com/articles/fourier-transform) of\nu(t)\n(see below).\nFor an [analytic function](https://bliptext.com/articles/analytic-function) in the [upper half-plane](https://bliptext.com/articles/upper-half-plane), the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values.\nThat is, if\nf(z)\nis analytic in the upper half complex plane\n{z : Im{z} > 0}\n, and\nu(t) = Re{f (t + 0·i)}\n, then\nIm{f(t + 0·i)} = H(u)(t)\nup to an additive constant, provided this Hilbert transform exists.\n\n## Notation\nIn [signal processing](https://bliptext.com/articles/signal-processing) the Hilbert transform of\nu(t)\nis commonly denoted by \\hat{u}(t).\nHowever, in mathematics, this notation is already extensively used to denote the Fourier transform of\nu(t)\n. Occasionally, the Hilbert transform may be denoted by.\nFurthermore, many sources define the Hilbert transform as the negative of the one defined here.\n\n# History\nThe 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](https://bliptext.com/articles/riemann-hilbert-problem).\nHilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.\nSome of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in [Göttingen](https://bliptext.com/articles/g-ttingen).\nThe results were later published by Hermann Weyl in his dissertation.\nSchur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.\nThese results were restricted to the spaces [[Lp space|\nL2\nand\nℓ2\n]]. In 1928, [Marcel Riesz](https://bliptext.com/articles/marcel-riesz) proved that the Hilbert transform can be defined for u in ([Lp space](https://bliptext.com/articles/lp-space)) for\n1 < p < ∞\n, that the Hilbert transform is a [bounded operator](https://bliptext.com/articles/bounded-operator) on for\n1 < p < ∞\n, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.\nThe Hilbert transform was a motivating example for [Antoni Zygmund](https://bliptext.com/articles/antoni-zygmund) and [Alberto Calderón](https://bliptext.com/articles/alberto-calder-n) during their study of [singular integrals](https://bliptext.com/articles/singular-integral).\nTheir investigations have played a fundamental role in modern harmonic analysis.\nVarious generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.\n\n# Relationship with the Fourier transform\nThe Hilbert transform is a [multiplier operator](https://bliptext.com/articles/multiplier-fourier-analysis).\nThe multiplier of\nH\nis\nσH(ω) = −i sgn(ω)\n, where\nsgn\nis the [signum function](https://bliptext.com/articles/sign-function).\nTherefore:\nwhere \\mathcal{F} denotes the [Fourier transform](https://bliptext.com/articles/fourier-transform).\nSince\nsgn(x) = sgn(2πx)\n, it follows that this result applies to the three common definitions of \\mathcal{F}.\nBy [Euler's formula](https://bliptext.com/articles/euler-s-formula),\nTherefore,\nH(u)(t)\nhas the effect of shifting the phase of the [negative frequency](https://bliptext.com/articles/negative-frequency) components of\nu(t)\nby +90° (π/2 radians) and the phase of the positive frequency components by −90°, and\ni·H(u)(t)\nhas 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).\nWhen the Hilbert transform is applied twice, the phase of the negative and positive frequency components of\nu(t)\nare respectively shifted by +180° and −180°, which are equivalent amounts.\nThe signal is negated; i.e.,\nH(H(u)) = −u\n, because\n\n# Table of selected Hilbert transforms\nIn the following table, the [frequency](https://bliptext.com/articles/frequency) parameter \\omega is real.\n**Notes**\nAn extensive table of Hilbert transforms is available.\nNote that the Hilbert transform of a constant is zero.\n\n# Domain of definition\nIt 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.\nHowever, the Hilbert transform is well-defined for a broad class of functions, namely those in for\n1 < p < ∞\n.\nMore precisely, if u is in for\n1 < p < ∞\n, then the limit defining the improper integral\nexists for [almost every](https://bliptext.com/articles/almost-every) t. The limit function is also in and is in fact the limit in the mean of the improper integral as well.\nThat is,\nas\nε → 0\nin the Lp norm, as well as pointwise almost everywhere, by the [Titchmarsh theorem](https://bliptext.com/articles/).\nIn the case\np = 1\n, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally.\nIn particular, convergence in the mean does not in general happen in this case.\nThe Hilbert transform of an\nL1\nfunction does converge, however, in\nL1\n-weak, and the Hilbert transform is a bounded operator from\nL1\nto\nL1,w\n. (In particular, since the Hilbert transform is also a multiplier operator on\nL2\n, [Marcinkiewicz interpolation](https://bliptext.com/articles/marcinkiewicz-interpolation) and a duality argument furnishes an alternative proof that H is bounded on\nLp\n.)\n\n# Properties\n\n## Boundedness\nIf\n1 < p < ∞\n, then the Hilbert transform on is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator), meaning that there exists a constant Cp such that\nfor all.\nThe best constant C_p is given by\nAn easy way to find the best C_p 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.\nThe boundedness of the Hilbert transform implies the convergence of the symmetric partial sum operator\nto f in.\n\n## Anti-self adjointness\nThe Hilbert transform is an anti-[self adjoint](https://bliptext.com/articles/self-adjoint) operator relative to the duality pairing between and the dual space, where p and q are [Hölder conjugates](https://bliptext.com/articles/h-lder-conjugate) and\n1 < p, q < ∞\n. Symbolically,\nfor and.\n\n## Inverse transform\nThe Hilbert transform is an [anti-involution](https://bliptext.com/articles/anti-involution), meaning that\nprovided each transform is well-defined.\nSince\nH\npreserves the space, this implies in particular that the Hilbert transform is invertible on , and that\n\n## Complex structure\nBecause\nH2 = −I\n(\"\nI\n\" is the [identity operator](https://bliptext.com/articles/identity-operator)) on the real [Banach space](https://bliptext.com/articles/banach-space) of real-valued functions in, the Hilbert transform defines a [linear complex structure](https://bliptext.com/articles/linear-complex-structure) on this [Banach space](https://bliptext.com/articles/banach-space). In particular, when\np = 2\n, the Hilbert transform gives the Hilbert space of real-valued functions in the structure of a complex Hilbert space.\nThe (complex) [eigenstates](https://bliptext.com/articles/eigenstate) of the Hilbert transform admit representations as [holomorphic functions](https://bliptext.com/articles/holomorphic-function) in the upper and lower half-planes in the [Hardy space](https://bliptext.com/articles/hardy-space) [[H square|\nH2\n]] by the [Paley–Wiener theorem](https://bliptext.com/articles/paley-wiener-theorem).\n\n## Differentiation\nFormally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:\nIterating this identity,\nThis is rigorously true as stated provided u and its first k derivatives belong to.\nOne can check this easily in the frequency domain, where differentiation becomes multiplication by ω.\n\n## Convolutions\nThe Hilbert transform can formally be realized as a [convolution](https://bliptext.com/articles/convolution) with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics)\nThus formally,\nHowever, a priori this may only be defined for u a distribution of [compact support](https://bliptext.com/articles/compact-support).\nIt is possible to work somewhat rigorously with this since compactly supported functions (which are distributions a fortiori) are [dense](https://bliptext.com/articles/dense-topology) in\nLp\n. Alternatively, one may use the fact that h(t) is the [distributional derivative](https://bliptext.com/articles/distributional-derivative) of the function\nt\nFor most operational purposes the Hilbert transform can be treated as a convolution.\nFor 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:\nThis is rigorously true if u and v are compactly supported distributions since, in that case,\nBy passing to an appropriate limit, it is thus also true if\nu ∈ Lp\nand\nv ∈ Lq\nprovided that\nfrom a theorem due to Titchmarsh.\n\n## Invariance\nThe Hilbert transform has the following invariance properties on.\nTa f(x) = f(x + a)\nfor all a in \\mathbb{R}.\nMλ f (x) = f (λ x)\nfor all\nλ > 0\n.\nR f (x) = f (−x)\n.\nUp to a multiplicative constant, the Hilbert transform is the only bounded operator on L2 with these properties.\nIn fact there is a wider set of operators that commute with the Hilbert transform.\nThe group acts by unitary operators\nUg\non the space by the formula\nThis [unitary representation](https://bliptext.com/articles/unitary-representation) is an example of a [principal series representation](https://bliptext.com/articles/principal-series-representation) of In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, [Hardy space](https://bliptext.com/articles/hardy-space) and its conjugate.\nThese are the spaces of\nL2\nboundary values of holomorphic functions on the upper and lower halfplanes.\nand its conjugate consist of exactly those\nL2\nfunctions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively.\nSince the Hilbert transform is equal to\nH = −i (2P − I)\n, with P being the orthogonal projection from onto and\nI\nthe [identity operator](https://bliptext.com/articles/identity-operator), it follows that and its orthogonal complement are eigenspaces of\nH\nfor the eigenvalues\n±i\n. In other words,\nH\ncommutes with the operators Ug.\nThe restrictions of the operators Ug to and its conjugate give irreducible representations of – the so-called [limit of discrete series representations](https://bliptext.com/articles/limit-of-discrete-series-representation).\n\n# Extending the domain of definition\n\n## Hilbert transform of distributions\nIt is further possible to extend the Hilbert transform to certain spaces of [distributions](https://bliptext.com/articles/distribution-mathematics).\nSince the Hilbert transform commutes with differentiation, and is a bounded operator on Lp, H restricts to give a continuous transform on the [inverse limit](https://bliptext.com/articles/inverse-limit) of [Sobolev spaces](https://bliptext.com/articles/sobolev-spaces):\nThe Hilbert transform can then be defined on the dual space of, denoted , consisting of Lp distributions.\nThis is accomplished by the duality pairing:\nFor, define:\nIt is possible to define the Hilbert transform on the space of [tempered distributions](https://bliptext.com/articles/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.\n\n## Hilbert transform of bounded functions\nThe Hilbert transform can be defined for functions in as well, but it requires some modifications and caveats.\nProperly understood, the Hilbert transform maps to the [Banach space](https://bliptext.com/articles/banach-space) of [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation) (BMO) classes.\nInterpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined.\nFor instance, with\nu = sgn(x)\n, the integral defining\nH(u)\ndiverges almost everywhere to\n±∞\n. To alleviate such difficulties, the Hilbert transform of an\nL∞\nfunction is therefore defined by the following [regularized](https://bliptext.com/articles/regularization-physics) form of the integral\nwhere as above\nh(x) = 1⁄πx\nand\nThe modified transform\nH\nagrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.\nFurthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.\nA [deep result](https://bliptext.com/articles/deep-result) of Fefferman's work is that a function is of bounded mean oscillation if and only if it has the form f + H(g) for some.\n\n# Conjugate functions\nThe Hilbert transform can be understood in terms of a pair of functions\nf(x)\nand\ng(x)\nsuch that the function\nis the boundary value of a [holomorphic function](https://bliptext.com/articles/holomorphic-function)\nF(z)\nin the upper half-plane.\nUnder these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.\nSuppose that Then, by the theory of the [Poisson integral](https://bliptext.com/articles/poisson-integral), f admits a unique harmonic extension into the upper half-plane, and this extension is given by\nwhich is the convolution of f with the [Poisson kernel](https://bliptext.com/articles/poisson-kernel)\nFurthermore, there is a unique harmonic function v defined in the upper half-plane such that\nF(z) = u(z) + i v(z)\nis holomorphic and\nThis harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel\nThus\nIndeed, the real and imaginary parts of the Cauchy kernel are\nso that\nF = u + i v\nis holomorphic by [Cauchy's integral formula](https://bliptext.com/articles/cauchy-s-integral-formula).\nThe function v obtained from u in this way is called the [harmonic conjugate](https://bliptext.com/articles/harmonic-conjugate) of u. The (non-tangential) boundary limit of\nv(x,y)\nas\ny → 0\nis the Hilbert transform of f. Thus, succinctly,\n\n## Titchmarsh's theorem\nTitchmarsh's theorem (named for [E. C. Titchmarsh](https://bliptext.com/articles/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.\nIt gives necessary and sufficient conditions for a complex-valued [square-integrable](https://bliptext.com/articles/square-integrable) function\nF(x)\non the real line to be the boundary value of a function in the [Hardy space](https://bliptext.com/articles/hardy-space)\nH2(U)\nof holomorphic functions in the upper half-plane U.\nThe theorem states that the following conditions for a complex-valued square-integrable function are equivalent:\nF(x)\nis the limit as\nz → x\nof a holomorphic function\nF(z)\nin the upper half-plane such that\nF(x)\nare Hilbert transforms of each other.\nx < 0\n.\nA weaker result is true for functions of class Lp for\np > 1\n. Specifically, if\nF(z)\nis a holomorphic function such that\nfor all y, then there is a complex-valued function\nF(x)\nin such that\nF(x + i y) → F(x)\nin the Lp norm as\ny → 0\n(as well as holding pointwise [almost everywhere](https://bliptext.com/articles/almost-everywhere)).\nFurthermore,\nwhere f is a real-valued function in and g is the Hilbert transform (of class Lp) of f.\nThis is not true in the case\np = 1\n. In fact, the Hilbert transform of an\nL1\nfunction f need not converge in the mean to another\nL1\nfunction.\nNevertheless, the Hilbert transform of f does converge almost everywhere to a finite function g such that\nThis result is directly analogous to one by [Andrey Kolmogorov](https://bliptext.com/articles/andrey-kolmogorov) for Hardy functions in the disc.\nAlthough usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see [Paley–Wiener theorem](https://bliptext.com/articles/paley-wiener-theorem)), as well as work by Riesz, Hille, and Tamarkin\n\n## Riemann–Hilbert problem\nOne form of the [Riemann–Hilbert problem](https://bliptext.com/articles/riemann-hilbert-problem) seeks to identify pairs of functions\nF+\nand\nF−\nsuch that\nF+\nis [holomorphic](https://bliptext.com/articles/holomorphic-function) on the upper half-plane and\nF−\nis holomorphic on the lower half-plane, such that for x along the real axis,\nwhere\nf(x)\nis some given real-valued function of.\nThe left-hand side of this equation may be understood either as the difference of the limits of\nF±\nfrom the appropriate half-planes, or as a [hyperfunction](https://bliptext.com/articles/hyperfunction) distribution.\nTwo functions of this form are a solution of the Riemann–Hilbert problem.\nFormally, if\nF±\nsolve the Riemann–Hilbert problem\nthen the Hilbert transform of\nf(x)\nis given by\n\n# Hilbert transform on the circle\nFor a periodic function f the circular Hilbert transform is defined:\nThe circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series.\nThe kernel,\nis known as the **Hilbert kernel** since it was in this form the Hilbert transform was originally studied.\nThe Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel 1/x periodic.\nMore precisely, for\nx ≠ 0\nMany results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.\nAnother more direct connection is provided by the Cayley transform\nC(x) = (x – i) / (x + i)\n, which carries the real line onto the circle and the upper half plane onto the unit disk.\nIt induces a unitary map\nof\nL2(T)\nonto The operator U carries the Hardy space\nH2(T)\nonto the Hardy space.\n\n# Hilbert transform in signal processing\n\n## Bedrosian's theorem\n**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\nwhere\nfLP\nand\nfHP\nare the low- and high-pass signals respectively.\nA 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\":\nwhere\num(t)\nis the narrow bandwidth \"message\" waveform, such as voice or music.\nThen by Bedrosian's theorem:\n\n## Analytic representation\nA specific type of [conjugate function](https://bliptext.com/articles/) is**:**\nknown as the analytic representation of u(t).\nThe name reflects its mathematical tractability, due largely to [Euler's formula](https://bliptext.com/articles/euler-s-formula).\nApplying Bedrosian's theorem to the narrowband model, the analytic representation is**:**\nA Fourier transform property indicates that this complex [heterodyne](https://bliptext.com/articles/heterodyne) operation can shift all the negative frequency components of\num(t)\nabove 0 Hz. In that case, the imaginary part of the result is a Hilbert transform of the real part.\nThis is an indirect way to produce Hilbert transforms.\n\n## Angle (phase/frequency) modulation\nThe form:\nis called [angle modulation](https://bliptext.com/articles/angle-modulation), which includes both [phase modulation](https://bliptext.com/articles/phase-modulation) and [frequency modulation](https://bliptext.com/articles/frequency-modulation).\nThe [instantaneous frequency](https://bliptext.com/articles/instantaneous-phase) is For sufficiently large ω, compared to :\nand:\n\n## Single sideband modulation (SSB)\nWhen\num(t)\nin is also an analytic representation (of a message waveform), that is:\nthe result is [single-sideband](https://bliptext.com/articles/single-sideband) modulation:\nwhose transmitted component is:\n\n## Causality\nThe function presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0):\n\n# Discrete Hilbert transform\n[[File:Effect of circular convolution on discrete Hilbert transform.png|thumb|400px|right|Figure 4. The Hilbert transform of\ncos(ωt)\nis\nsin(ωt)\n. This figure shows\nsin(ωt)\nand two approximate Hilbert transforms computed by the MATLAB library function, hilbert]]\nFor a discrete function, u[n], with [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT), U(\\omega), and **discrete Hilbert transform** the DTFT of in the region\n−π < ω < π\nis given by**:**\nThe inverse DTFT, using the [convolution theorem](https://bliptext.com/articles/convolution-theorem), is**:**\nwhere\nwhich is an infinite impulse response (IIR).\n**Practical considerations**\n**Method 1:** Direct convolution of streaming u[n] data with an FIR approximation of h[n], which we will designate by Examples of truncated h[n] are shown in figures 1 and 2.\n**Fig 1** has an odd number of anti-symmetric coefficients and is called Type III.\nThis type inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in a bandpass filter shape.\nA Type IV design (even number of anti-symmetric coefficients) is shown in **Fig 2**.\nIt has a highpass frequency response.\nType III is the usual choice.\nfor these reasons**:**\nThe abrupt truncation of h[n] creates a rippling (Gibbs effect) of the flat frequency response.\nThat can be mitigated by use of a window function to taper \\tilde h[n] to zero.\n**Method 2:** Piecewise convolution.\nIt is well known that direct convolution is computationally much more intensive than methods like **[overlap-save](https://bliptext.com/articles/overlap-save-method)** that give access to the efficiencies of the Fast Fourier transform via the convolution theorem.\nSpecifically, the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT) of a segment of u[n] is multiplied pointwise with a DFT of the \\tilde h[n] sequence.\nAn inverse DFT is done on the product, and the transient artifacts at the leading and trailing edges of the segment are discarded.\nOver-lapping input segments prevent gaps in the output stream.\nAn equivalent time domain description is that segments of length N (an arbitrary parameter) are convolved with the periodic function**:**\nWhen the duration of non-zero values of is M < N, the output sequence includes N-M+1 samples of \\widehat u. M-1 outputs are discarded from each block of N, and the input blocks are overlapped by that amount to prevent gaps.\n**Method 3**:**** Same as method 2, except the DFT of is replaced by samples of the distribution (whose real and imaginary components are all just 0 or \\pm 1.) That convolves u[n] with a [periodic summation](https://bliptext.com/articles/periodic-summation)**:**\nfor some arbitrary parameter, N. h[n] is not an FIR, so the edge effects extend throughout the entire transform.\nDeciding what to delete and the corresponding amount of overlap is an application-dependent design issue.\n**Fig 3** depicts the difference between methods 2 and 3.\nOnly half of the antisymmetric impulse response is shown, and only the non-zero coefficients.\nThe blue graph corresponds to method 2 where h[n] is truncated by a rectangular window function, rather than tapered.\nIt is generated by a Matlab function, **hilb(65)**.\nIts transient effects are exactly known and readily discarded.\nThe frequency response, which is determined by the function argument, is the only application-dependent design issue.\nThe red graph is h_{512}[n], corresponding to method 3.\nIt is the inverse DFT of the distribution.\nSpecifically, it is the function that is convolved with a segment of u[n] by the [MATLAB](https://bliptext.com/articles/matlab) function, **hilbert(u,512)**.\nThe real part of the output sequence is the original input sequence, so that the complex output is an [analytic representation](https://bliptext.com/articles/analytic-signal) of u[n].\nWhen the input is a segment of a pure cosine, the resulting convolution for two different values of N is depicted in **Fig 4** (red and blue plots).\nEdge effects prevent the result from being a pure sine function (green plot).\nSince h_N[n] is not an FIR sequence, the theoretical extent of the effects is the entire output sequence.\nBut the differences from a sine function diminish with distance from the edges.\nParameter N is the output sequence length.\nIf it exceeds the length of the input sequence, the input is modified by appending zero-valued elements.\nIn most cases, that reduces the magnitude of the edge distortions.\nBut their duration is dominated by the inherent rise and fall times of the h[n] impulse response.\n**Fig 5** is an example of piecewise convolution, using both methods 2 (in blue) and 3 (red dots).\nA 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.\nAs the FIR result (blue) shows, the distortions apparent in the IIR result (red) are not caused by the difference between h[n] and h_N[n] (green and red in **Fig 3**).\nThe fact that h_N[n] is tapered (windowed) is actually helpful in this context.\nThe real problem is that it's not windowed enough.\nEffectively, M=N, whereas the overlap-save method needs M < N.\n# [Figure 3. | upload.wikimedia.org/wikipedia/commons/e/e8/DFT///approximation///to///Hilbert///filter.png]\n# [Figure 5. Discrete Hilbert transforms of a cosine function, using piecewise convolution | upload.wikimedia.org/wikipedia/commons/1/1c/Discrete///Hilbert///transforms///of///a///cosine///function%2C///using///piecewise///convolution.svg]\n\n# Number-theoretic Hilbert transform\nThe number theoretic Hilbert transform is an extension of the discrete Hilbert transform to integers modulo an appropriate prime number.\nIn this it follows the generalization of [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) to number theoretic transforms.\nThe number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.\n\n# Page citations","wordIndex":7,"createdAt":"2025-03-08T02:32:52.154Z","user":{"id":"E6iXdKCEcr2dpx7wlFc1v38vYpQIgHKW","name":"DynamicArrays","image":"https://cdn.discordapp.com/avatars/887802393054572584/bd9d90fedebd723bbc46de9db3ffba71.png"}}],"article":{"id":"94431492-7df9-466b-b932-6a22b3b78613","title":"Hilbert transform","slug":"hilbert-transform","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) and [signal processing](https://bliptext.com/articles/signal-processing), the Huzz is a specific [singular integral](https://bliptext.com/articles/singular-integral) that takes a function,\nu(t)\nof a real variable and produces another function of a real variable\nH(u)(t)\n. The Hilbert transform is given by the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) of the [convolution](https://bliptext.com/articles/convolution) with the function 1/(\\pi t) (see ).\nThe Hilbert transform has a particularly simple representation in the [frequency domain](https://bliptext.com/articles/frequency-domain): It imparts a [phase shift](https://bliptext.com/articles/phase-shift) of ±90° (π/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ).\nThe Hilbert transform is important in signal processing, where it is a component of the [analytic representation](https://bliptext.com/articles/analytic-signal) of a real-valued signal\nu(t)\n. The Hilbert transform was first introduced by [David Hilbert](https://bliptext.com/articles/david-hilbert) in this setting, to solve a special case of the [Riemann–Hilbert problem](https://bliptext.com/articles/riemann-hilbert-problem) for analytic functions.\n\n# Definition\nThe Hilbert transform of u can be thought of as the [convolution](https://bliptext.com/articles/convolution) of\nu(t)\nwith the function\nh(t) = 1⁄πt\n, known as the [Cauchy kernel](https://bliptext.com/articles/cauchy-kernel).\nBecause 1/t is not [integrable](https://bliptext.com/articles/integrable) across\nt = 0\n, the integral defining the convolution does not always converge.\nInstead, the Hilbert transform is defined using the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) (denoted here by\np.v.\n). Explicitly, the Hilbert transform of a function (or signal)\nu(t)\nis given by\nprovided this integral exists as a principal value.\nThis is precisely the convolution of u with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics)\np.v. 1⁄πt\n. Alternatively, by changing variables, the principal-value integral can be written explicitly as\nWhen the Hilbert transform is applied twice in succession to a function u, the result is\nprovided the integrals defining both iterations converge in a suitable sense.\nIn particular, the inverse transform is . This fact can most easily be seen by considering the effect of the Hilbert transform on the [Fourier transform](https://bliptext.com/articles/fourier-transform) of\nu(t)\n(see below).\nFor an [analytic function](https://bliptext.com/articles/analytic-function) in the [upper half-plane](https://bliptext.com/articles/upper-half-plane), the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values.\nThat is, if\nf(z)\nis analytic in the upper half complex plane\n{z : Im{z} > 0}\n, and\nu(t) = Re{f (t + 0·i)}\n, then\nIm{f(t + 0·i)} = H(u)(t)\nup to an additive constant, provided this Hilbert transform exists.\n\n## Notation\nIn [signal processing](https://bliptext.com/articles/signal-processing) the Hilbert transform of\nu(t)\nis commonly denoted by \\hat{u}(t).\nHowever, in mathematics, this notation is already extensively used to denote the Fourier transform of\nu(t)\n. Occasionally, the Hilbert transform may be denoted by.\nFurthermore, many sources define the Hilbert transform as the negative of the one defined here.\n\n# History\nThe 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](https://bliptext.com/articles/riemann-hilbert-problem).\nHilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.\nSome of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in [Göttingen](https://bliptext.com/articles/g-ttingen).\nThe results were later published by Hermann Weyl in his dissertation.\nSchur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.\nThese results were restricted to the spaces [[Lp space|\nL2\nand\nℓ2\n]]. In 1928, [Marcel Riesz](https://bliptext.com/articles/marcel-riesz) proved that the Hilbert transform can be defined for u in ([Lp space](https://bliptext.com/articles/lp-space)) for\n1 < p < ∞\n, that the Hilbert transform is a [bounded operator](https://bliptext.com/articles/bounded-operator) on for\n1 < p < ∞\n, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.\nThe Hilbert transform was a motivating example for [Antoni Zygmund](https://bliptext.com/articles/antoni-zygmund) and [Alberto Calderón](https://bliptext.com/articles/alberto-calder-n) during their study of [singular integrals](https://bliptext.com/articles/singular-integral).\nTheir investigations have played a fundamental role in modern harmonic analysis.\nVarious generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.\n\n# Relationship with the Fourier transform\nThe Hilbert transform is a [multiplier operator](https://bliptext.com/articles/multiplier-fourier-analysis).\nThe multiplier of\nH\nis\nσH(ω) = −i sgn(ω)\n, where\nsgn\nis the [signum function](https://bliptext.com/articles/sign-function).\nTherefore:\nwhere \\mathcal{F} denotes the [Fourier transform](https://bliptext.com/articles/fourier-transform).\nSince\nsgn(x) = sgn(2πx)\n, it follows that this result applies to the three common definitions of \\mathcal{F}.\nBy [Euler's formula](https://bliptext.com/articles/euler-s-formula),\nTherefore,\nH(u)(t)\nhas the effect of shifting the phase of the [negative frequency](https://bliptext.com/articles/negative-frequency) components of\nu(t)\nby +90° (π/2 radians) and the phase of the positive frequency components by −90°, and\ni·H(u)(t)\nhas 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).\nWhen the Hilbert transform is applied twice, the phase of the negative and positive frequency components of\nu(t)\nare respectively shifted by +180° and −180°, which are equivalent amounts.\nThe signal is negated; i.e.,\nH(H(u)) = −u\n, because\n\n# Table of selected Hilbert transforms\nIn the following table, the [frequency](https://bliptext.com/articles/frequency) parameter \\omega is real.\n**Notes**\nAn extensive table of Hilbert transforms is available.\nNote that the Hilbert transform of a constant is zero.\n\n# Domain of definition\nIt 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.\nHowever, the Hilbert transform is well-defined for a broad class of functions, namely those in for\n1 < p < ∞\n.\nMore precisely, if u is in for\n1 < p < ∞\n, then the limit defining the improper integral\nexists for [almost every](https://bliptext.com/articles/almost-every) t. The limit function is also in and is in fact the limit in the mean of the improper integral as well.\nThat is,\nas\nε → 0\nin the Lp norm, as well as pointwise almost everywhere, by the [Titchmarsh theorem](https://bliptext.com/articles/).\nIn the case\np = 1\n, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally.\nIn particular, convergence in the mean does not in general happen in this case.\nThe Hilbert transform of an\nL1\nfunction does converge, however, in\nL1\n-weak, and the Hilbert transform is a bounded operator from\nL1\nto\nL1,w\n. (In particular, since the Hilbert transform is also a multiplier operator on\nL2\n, [Marcinkiewicz interpolation](https://bliptext.com/articles/marcinkiewicz-interpolation) and a duality argument furnishes an alternative proof that H is bounded on\nLp\n.)\n\n# Properties\n\n## Boundedness\nIf\n1 < p < ∞\n, then the Hilbert transform on is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator), meaning that there exists a constant Cp such that\nfor all.\nThe best constant C_p is given by\nAn easy way to find the best C_p 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.\nThe boundedness of the Hilbert transform implies the convergence of the symmetric partial sum operator\nto f in.\n\n## Anti-self adjointness\nThe Hilbert transform is an anti-[self adjoint](https://bliptext.com/articles/self-adjoint) operator relative to the duality pairing between and the dual space, where p and q are [Hölder conjugates](https://bliptext.com/articles/h-lder-conjugate) and\n1 < p, q < ∞\n. Symbolically,\nfor and.\n\n## Inverse transform\nThe Hilbert transform is an [anti-involution](https://bliptext.com/articles/anti-involution), meaning that\nprovided each transform is well-defined.\nSince\nH\npreserves the space, this implies in particular that the Hilbert transform is invertible on , and that\n\n## Complex structure\nBecause\nH2 = −I\n(\"\nI\n\" is the [identity operator](https://bliptext.com/articles/identity-operator)) on the real [Banach space](https://bliptext.com/articles/banach-space) of real-valued functions in, the Hilbert transform defines a [linear complex structure](https://bliptext.com/articles/linear-complex-structure) on this [Banach space](https://bliptext.com/articles/banach-space). In particular, when\np = 2\n, the Hilbert transform gives the Hilbert space of real-valued functions in the structure of a complex Hilbert space.\nThe (complex) [eigenstates](https://bliptext.com/articles/eigenstate) of the Hilbert transform admit representations as [holomorphic functions](https://bliptext.com/articles/holomorphic-function) in the upper and lower half-planes in the [Hardy space](https://bliptext.com/articles/hardy-space) [[H square|\nH2\n]] by the [Paley–Wiener theorem](https://bliptext.com/articles/paley-wiener-theorem).\n\n## Differentiation\nFormally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:\nIterating this identity,\nThis is rigorously true as stated provided u and its first k derivatives belong to.\nOne can check this easily in the frequency domain, where differentiation becomes multiplication by ω.\n\n## Convolutions\nThe Hilbert transform can formally be realized as a [convolution](https://bliptext.com/articles/convolution) with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics)\nThus formally,\nHowever, a priori this may only be defined for u a distribution of [compact support](https://bliptext.com/articles/compact-support).\nIt is possible to work somewhat rigorously with this since compactly supported functions (which are distributions a fortiori) are [dense](https://bliptext.com/articles/dense-topology) in\nLp\n. Alternatively, one may use the fact that h(t) is the [distributional derivative](https://bliptext.com/articles/distributional-derivative) of the function\nt\nFor most operational purposes the Hilbert transform can be treated as a convolution.\nFor 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:\nThis is rigorously true if u and v are compactly supported distributions since, in that case,\nBy passing to an appropriate limit, it is thus also true if\nu ∈ Lp\nand\nv ∈ Lq\nprovided that\nfrom a theorem due to Titchmarsh.\n\n## Invariance\nThe Hilbert transform has the following invariance properties on.\nTa f(x) = f(x + a)\nfor all a in \\mathbb{R}.\nMλ f (x) = f (λ x)\nfor all\nλ > 0\n.\nR f (x) = f (−x)\n.\nUp to a multiplicative constant, the Hilbert transform is the only bounded operator on L2 with these properties.\nIn fact there is a wider set of operators that commute with the Hilbert transform.\nThe group acts by unitary operators\nUg\non the space by the formula\nThis [unitary representation](https://bliptext.com/articles/unitary-representation) is an example of a [principal series representation](https://bliptext.com/articles/principal-series-representation) of In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, [Hardy space](https://bliptext.com/articles/hardy-space) and its conjugate.\nThese are the spaces of\nL2\nboundary values of holomorphic functions on the upper and lower halfplanes.\nand its conjugate consist of exactly those\nL2\nfunctions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively.\nSince the Hilbert transform is equal to\nH = −i (2P − I)\n, with P being the orthogonal projection from onto and\nI\nthe [identity operator](https://bliptext.com/articles/identity-operator), it follows that and its orthogonal complement are eigenspaces of\nH\nfor the eigenvalues\n±i\n. In other words,\nH\ncommutes with the operators Ug.\nThe restrictions of the operators Ug to and its conjugate give irreducible representations of – the so-called [limit of discrete series representations](https://bliptext.com/articles/limit-of-discrete-series-representation).\n\n# Extending the domain of definition\n\n## Hilbert transform of distributions\nIt is further possible to extend the Hilbert transform to certain spaces of [distributions](https://bliptext.com/articles/distribution-mathematics).\nSince the Hilbert transform commutes with differentiation, and is a bounded operator on Lp, H restricts to give a continuous transform on the [inverse limit](https://bliptext.com/articles/inverse-limit) of [Sobolev spaces](https://bliptext.com/articles/sobolev-spaces):\nThe Hilbert transform can then be defined on the dual space of, denoted , consisting of Lp distributions.\nThis is accomplished by the duality pairing:\nFor, define:\nIt is possible to define the Hilbert transform on the space of [tempered distributions](https://bliptext.com/articles/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.\n\n## Hilbert transform of bounded functions\nThe Hilbert transform can be defined for functions in as well, but it requires some modifications and caveats.\nProperly understood, the Hilbert transform maps to the [Banach space](https://bliptext.com/articles/banach-space) of [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation) (BMO) classes.\nInterpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined.\nFor instance, with\nu = sgn(x)\n, the integral defining\nH(u)\ndiverges almost everywhere to\n±∞\n. To alleviate such difficulties, the Hilbert transform of an\nL∞\nfunction is therefore defined by the following [regularized](https://bliptext.com/articles/regularization-physics) form of the integral\nwhere as above\nh(x) = 1⁄πx\nand\nThe modified transform\nH\nagrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.\nFurthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.\nA [deep result](https://bliptext.com/articles/deep-result) of Fefferman's work is that a function is of bounded mean oscillation if and only if it has the form f + H(g) for some.\n\n# Conjugate functions\nThe Hilbert transform can be understood in terms of a pair of functions\nf(x)\nand\ng(x)\nsuch that the function\nis the boundary value of a [holomorphic function](https://bliptext.com/articles/holomorphic-function)\nF(z)\nin the upper half-plane.\nUnder these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.\nSuppose that Then, by the theory of the [Poisson integral](https://bliptext.com/articles/poisson-integral), f admits a unique harmonic extension into the upper half-plane, and this extension is given by\nwhich is the convolution of f with the [Poisson kernel](https://bliptext.com/articles/poisson-kernel)\nFurthermore, there is a unique harmonic function v defined in the upper half-plane such that\nF(z) = u(z) + i v(z)\nis holomorphic and\nThis harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel\nThus\nIndeed, the real and imaginary parts of the Cauchy kernel are\nso that\nF = u + i v\nis holomorphic by [Cauchy's integral formula](https://bliptext.com/articles/cauchy-s-integral-formula).\nThe function v obtained from u in this way is called the [harmonic conjugate](https://bliptext.com/articles/harmonic-conjugate) of u. The (non-tangential) boundary limit of\nv(x,y)\nas\ny → 0\nis the Hilbert transform of f. Thus, succinctly,\n\n## Titchmarsh's theorem\nTitchmarsh's theorem (named for [E. C. Titchmarsh](https://bliptext.com/articles/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.\nIt gives necessary and sufficient conditions for a complex-valued [square-integrable](https://bliptext.com/articles/square-integrable) function\nF(x)\non the real line to be the boundary value of a function in the [Hardy space](https://bliptext.com/articles/hardy-space)\nH2(U)\nof holomorphic functions in the upper half-plane U.\nThe theorem states that the following conditions for a complex-valued square-integrable function are equivalent:\nF(x)\nis the limit as\nz → x\nof a holomorphic function\nF(z)\nin the upper half-plane such that\nF(x)\nare Hilbert transforms of each other.\nx < 0\n.\nA weaker result is true for functions of class Lp for\np > 1\n. Specifically, if\nF(z)\nis a holomorphic function such that\nfor all y, then there is a complex-valued function\nF(x)\nin such that\nF(x + i y) → F(x)\nin the Lp norm as\ny → 0\n(as well as holding pointwise [almost everywhere](https://bliptext.com/articles/almost-everywhere)).\nFurthermore,\nwhere f is a real-valued function in and g is the Hilbert transform (of class Lp) of f.\nThis is not true in the case\np = 1\n. In fact, the Hilbert transform of an\nL1\nfunction f need not converge in the mean to another\nL1\nfunction.\nNevertheless, the Hilbert transform of f does converge almost everywhere to a finite function g such that\nThis result is directly analogous to one by [Andrey Kolmogorov](https://bliptext.com/articles/andrey-kolmogorov) for Hardy functions in the disc.\nAlthough usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see [Paley–Wiener theorem](https://bliptext.com/articles/paley-wiener-theorem)), as well as work by Riesz, Hille, and Tamarkin\n\n## Riemann–Hilbert problem\nOne form of the [Riemann–Hilbert problem](https://bliptext.com/articles/riemann-hilbert-problem) seeks to identify pairs of functions\nF+\nand\nF−\nsuch that\nF+\nis [holomorphic](https://bliptext.com/articles/holomorphic-function) on the upper half-plane and\nF−\nis holomorphic on the lower half-plane, such that for x along the real axis,\nwhere\nf(x)\nis some given real-valued function of.\nThe left-hand side of this equation may be understood either as the difference of the limits of\nF±\nfrom the appropriate half-planes, or as a [hyperfunction](https://bliptext.com/articles/hyperfunction) distribution.\nTwo functions of this form are a solution of the Riemann–Hilbert problem.\nFormally, if\nF±\nsolve the Riemann–Hilbert problem\nthen the Hilbert transform of\nf(x)\nis given by\n\n# Hilbert transform on the circle\nFor a periodic function f the circular Hilbert transform is defined:\nThe circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series.\nThe kernel,\nis known as the **Hilbert kernel** since it was in this form the Hilbert transform was originally studied.\nThe Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel 1/x periodic.\nMore precisely, for\nx ≠ 0\nMany results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.\nAnother more direct connection is provided by the Cayley transform\nC(x) = (x – i) / (x + i)\n, which carries the real line onto the circle and the upper half plane onto the unit disk.\nIt induces a unitary map\nof\nL2(T)\nonto The operator U carries the Hardy space\nH2(T)\nonto the Hardy space.\n\n# Hilbert transform in signal processing\n\n## Bedrosian's theorem\n**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\nwhere\nfLP\nand\nfHP\nare the low- and high-pass signals respectively.\nA 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\":\nwhere\num(t)\nis the narrow bandwidth \"message\" waveform, such as voice or music.\nThen by Bedrosian's theorem:\n\n## Analytic representation\nA specific type of [conjugate function](https://bliptext.com/articles/) is**:**\nknown as the analytic representation of u(t).\nThe name reflects its mathematical tractability, due largely to [Euler's formula](https://bliptext.com/articles/euler-s-formula).\nApplying Bedrosian's theorem to the narrowband model, the analytic representation is**:**\nA Fourier transform property indicates that this complex [heterodyne](https://bliptext.com/articles/heterodyne) operation can shift all the negative frequency components of\num(t)\nabove 0 Hz. In that case, the imaginary part of the result is a Hilbert transform of the real part.\nThis is an indirect way to produce Hilbert transforms.\n\n## Angle (phase/frequency) modulation\nThe form:\nis called [angle modulation](https://bliptext.com/articles/angle-modulation), which includes both [phase modulation](https://bliptext.com/articles/phase-modulation) and [frequency modulation](https://bliptext.com/articles/frequency-modulation).\nThe [instantaneous frequency](https://bliptext.com/articles/instantaneous-phase) is For sufficiently large ω, compared to :\nand:\n\n## Single sideband modulation (SSB)\nWhen\num(t)\nin is also an analytic representation (of a message waveform), that is:\nthe result is [single-sideband](https://bliptext.com/articles/single-sideband) modulation:\nwhose transmitted component is:\n\n## Causality\nThe function presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0):\n\n# Discrete Hilbert transform\n[[File:Effect of circular convolution on discrete Hilbert transform.png|thumb|400px|right|Figure 4. The Hilbert transform of\ncos(ωt)\nis\nsin(ωt)\n. This figure shows\nsin(ωt)\nand two approximate Hilbert transforms computed by the MATLAB library function, hilbert]]\nFor a discrete function, u[n], with [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT), U(\\omega), and **discrete Hilbert transform** the DTFT of in the region\n−π < ω < π\nis given by**:**\nThe inverse DTFT, using the [convolution theorem](https://bliptext.com/articles/convolution-theorem), is**:**\nwhere\nwhich is an infinite impulse response (IIR).\n**Practical considerations**\n**Method 1:** Direct convolution of streaming u[n] data with an FIR approximation of h[n], which we will designate by Examples of truncated h[n] are shown in figures 1 and 2.\n**Fig 1** has an odd number of anti-symmetric coefficients and is called Type III.\nThis type inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in a bandpass filter shape.\nA Type IV design (even number of anti-symmetric coefficients) is shown in **Fig 2**.\nIt has a highpass frequency response.\nType III is the usual choice.\nfor these reasons**:**\nThe abrupt truncation of h[n] creates a rippling (Gibbs effect) of the flat frequency response.\nThat can be mitigated by use of a window function to taper \\tilde h[n] to zero.\n**Method 2:** Piecewise convolution.\nIt is well known that direct convolution is computationally much more intensive than methods like **[overlap-save](https://bliptext.com/articles/overlap-save-method)** that give access to the efficiencies of the Fast Fourier transform via the convolution theorem.\nSpecifically, the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT) of a segment of u[n] is multiplied pointwise with a DFT of the \\tilde h[n] sequence.\nAn inverse DFT is done on the product, and the transient artifacts at the leading and trailing edges of the segment are discarded.\nOver-lapping input segments prevent gaps in the output stream.\nAn equivalent time domain description is that segments of length N (an arbitrary parameter) are convolved with the periodic function**:**\nWhen the duration of non-zero values of is M < N, the output sequence includes N-M+1 samples of \\widehat u. M-1 outputs are discarded from each block of N, and the input blocks are overlapped by that amount to prevent gaps.\n**Method 3**:**** Same as method 2, except the DFT of is replaced by samples of the distribution (whose real and imaginary components are all just 0 or \\pm 1.) That convolves u[n] with a [periodic summation](https://bliptext.com/articles/periodic-summation)**:**\nfor some arbitrary parameter, N. h[n] is not an FIR, so the edge effects extend throughout the entire transform.\nDeciding what to delete and the corresponding amount of overlap is an application-dependent design issue.\n**Fig 3** depicts the difference between methods 2 and 3.\nOnly half of the antisymmetric impulse response is shown, and only the non-zero coefficients.\nThe blue graph corresponds to method 2 where h[n] is truncated by a rectangular window function, rather than tapered.\nIt is generated by a Matlab function, **hilb(65)**.\nIts transient effects are exactly known and readily discarded.\nThe frequency response, which is determined by the function argument, is the only application-dependent design issue.\nThe red graph is h_{512}[n], corresponding to method 3.\nIt is the inverse DFT of the distribution.\nSpecifically, it is the function that is convolved with a segment of u[n] by the [MATLAB](https://bliptext.com/articles/matlab) function, **hilbert(u,512)**.\nThe real part of the output sequence is the original input sequence, so that the complex output is an [analytic representation](https://bliptext.com/articles/analytic-signal) of u[n].\nWhen the input is a segment of a pure cosine, the resulting convolution for two different values of N is depicted in **Fig 4** (red and blue plots).\nEdge effects prevent the result from being a pure sine function (green plot).\nSince h_N[n] is not an FIR sequence, the theoretical extent of the effects is the entire output sequence.\nBut the differences from a sine function diminish with distance from the edges.\nParameter N is the output sequence length.\nIf it exceeds the length of the input sequence, the input is modified by appending zero-valued elements.\nIn most cases, that reduces the magnitude of the edge distortions.\nBut their duration is dominated by the inherent rise and fall times of the h[n] impulse response.\n**Fig 5** is an example of piecewise convolution, using both methods 2 (in blue) and 3 (red dots).\nA 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.\nAs the FIR result (blue) shows, the distortions apparent in the IIR result (red) are not caused by the difference between h[n] and h_N[n] (green and red in **Fig 3**).\nThe fact that h_N[n] is tapered (windowed) is actually helpful in this context.\nThe real problem is that it's not windowed enough.\nEffectively, M=N, whereas the overlap-save method needs M < N.\n# [Figure 3. | upload.wikimedia.org/wikipedia/commons/e/e8/DFT///approximation///to///Hilbert///filter.png]\n# [Figure 5. Discrete Hilbert transforms of a cosine function, using piecewise convolution | upload.wikimedia.org/wikipedia/commons/1/1c/Discrete///Hilbert///transforms///of///a///cosine///function%2C///using///piecewise///convolution.svg]\n\n# Number-theoretic Hilbert transform\nThe number theoretic Hilbert transform is an extension of the discrete Hilbert transform to integers modulo an appropriate prime number.\nIn this it follows the generalization of [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) to number theoretic transforms.\nThe number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.\n\n# Page citations","current_revision":"5dcebf82-fa33-4609-a64a-aec98c16ef20","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T20:03:36.945Z","updated_at":"2025-03-08T02:32:52.170Z","search_vector":"'hilbert':1 'transform':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) and [signal processing](https://bliptext.com/articles/signal-processing), the **Hilbert transform** is a specific [singular integral](https://bliptext.com/articles/singular-integral) that takes a function,\nu(t)\nof a real variable and produces another function of a real variable\nH(u)(t)\n. The Hilbert transform is given by the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) of the [convolution](https://bliptext.com/articles/convolution) with the function 1/(\\pi t) (see ).\nThe Hilbert transform has a particularly simple representation in the [frequency domain](https://bliptext.com/articles/frequency-domain): It imparts a [phase shift](https://bliptext.com/articles/phase-shift) of ±90° (π/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ).\nThe Hilbert transform is important in signal processing, where it is a component of the [analytic representation](https://bliptext.com/articles/analytic-signal) of a real-valued signal\nu(t)\n. The Hilbert transform was first introduced by [David Hilbert](https://bliptext.com/articles/david-hilbert) in this setting, to solve a special case of the [Riemann–Hilbert problem](https://bliptext.com/articles/riemann-hilbert-problem) for analytic functions.\n\n# Definition\nThe Hilbert transform of u can be thought of as the [convolution](https://bliptext.com/articles/convolution) of\nu(t)\nwith the function\nh(t) = 1⁄πt\n, known as the [Cauchy kernel](https://bliptext.com/articles/cauchy-kernel).\nBecause 1/t is not [integrable](https://bliptext.com/articles/integrable) across\nt = 0\n, the integral defining the convolution does not always converge.\nInstead, the Hilbert transform is defined using the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) (denoted here by\np.v.\n). Explicitly, the Hilbert transform of a function (or signal)\nu(t)\nis given by\nprovided this integral exists as a principal value.\nThis is precisely the convolution of u with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics)\np.v. 1⁄πt\n. Alternatively, by changing variables, the principal-value integral can be written explicitly as\nWhen the Hilbert transform is applied twice in succession to a function u, the result is\nprovided the integrals defining both iterations converge in a suitable sense.\nIn particular, the inverse transform is . This fact can most easily be seen by considering the effect of the Hilbert transform on the [Fourier transform](https://bliptext.com/articles/fourier-transform) of\nu(t)\n(see below).\nFor an [analytic function](https://bliptext.com/articles/analytic-function) in the [upper half-plane](https://bliptext.com/articles/upper-half-plane), the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values.\nThat is, if\nf(z)\nis analytic in the upper half complex plane\n{z : Im{z} > 0}\n, and\nu(t) = Re{f (t + 0·i)}\n, then\nIm{f(t + 0·i)} = H(u)(t)\nup to an additive constant, provided this Hilbert transform exists.\n\n## Notation\nIn [signal processing](https://bliptext.com/articles/signal-processing) the Hilbert transform of\nu(t)\nis commonly denoted by \\hat{u}(t).\nHowever, in mathematics, this notation is already extensively used to denote the Fourier transform of\nu(t)\n. Occasionally, the Hilbert transform may be denoted by.\nFurthermore, many sources define the Hilbert transform as the negative of the one defined here.\n\n# History\nThe 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](https://bliptext.com/articles/riemann-hilbert-problem).\nHilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.\nSome of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in [Göttingen](https://bliptext.com/articles/g-ttingen).\nThe results were later published by Hermann Weyl in his dissertation.\nSchur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.\nThese results were restricted to the spaces [[Lp space|\nL2\nand\nℓ2\n]]. In 1928, [Marcel Riesz](https://bliptext.com/articles/marcel-riesz) proved that the Hilbert transform can be defined for u in ([Lp space](https://bliptext.com/articles/lp-space)) for\n1 0\n.\nR f (x) = f (−x)\n.\nUp to a multiplicative constant, the Hilbert transform is the only bounded operator on L2 with these properties.\nIn fact there is a wider set of operators that commute with the Hilbert transform.\nThe group acts by unitary operators\nUg\non the space by the formula\nThis [unitary representation](https://bliptext.com/articles/unitary-representation) is an example of a [principal series representation](https://bliptext.com/articles/principal-series-representation) of In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, [Hardy space](https://bliptext.com/articles/hardy-space) and its conjugate.\nThese are the spaces of\nL2\nboundary values of holomorphic functions on the upper and lower halfplanes.\nand its conjugate consist of exactly those\nL2\nfunctions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively.\nSince the Hilbert transform is equal to\nH = −i (2P − I)\n, with P being the orthogonal projection from onto and\nI\nthe [identity operator](https://bliptext.com/articles/identity-operator), it follows that and its orthogonal complement are eigenspaces of\nH\nfor the eigenvalues\n±i\n. In other words,\nH\ncommutes with the operators Ug.\nThe restrictions of the operators Ug to and its conjugate give irreducible representations of – the so-called [limit of discrete series representations](https://bliptext.com/articles/limit-of-discrete-series-representation).\n\n# Extending the domain of definition\n\n## Hilbert transform of distributions\nIt is further possible to extend the Hilbert transform to certain spaces of [distributions](https://bliptext.com/articles/distribution-mathematics).\nSince the Hilbert transform commutes with differentiation, and is a bounded operator on Lp, H restricts to give a continuous transform on the [inverse limit](https://bliptext.com/articles/inverse-limit) of [Sobolev spaces](https://bliptext.com/articles/sobolev-spaces):\nThe Hilbert transform can then be defined on the dual space of, denoted , consisting of Lp distributions.\nThis is accomplished by the duality pairing:\nFor, define:\nIt is possible to define the Hilbert transform on the space of [tempered distributions](https://bliptext.com/articles/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.\n\n## Hilbert transform of bounded functions\nThe Hilbert transform can be defined for functions in as well, but it requires some modifications and caveats.\nProperly understood, the Hilbert transform maps to the [Banach space](https://bliptext.com/articles/banach-space) of [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation) (BMO) classes.\nInterpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined.\nFor instance, with\nu = sgn(x)\n, the integral defining\nH(u)\ndiverges almost everywhere to\n±∞\n. To alleviate such difficulties, the Hilbert transform of an\nL∞\nfunction is therefore defined by the following [regularized](https://bliptext.com/articles/regularization-physics) form of the integral\nwhere as above\nh(x) = 1⁄πx\nand\nThe modified transform\nH\nagrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.\nFurthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.\nA [deep result](https://bliptext.com/articles/deep-result) of Fefferman's work is that a function is of bounded mean oscillation if and only if it has the form f + H(g) for some.\n\n# Conjugate functions\nThe Hilbert transform can be understood in terms of a pair of functions\nf(x)\nand\ng(x)\nsuch that the function\nis the boundary value of a [holomorphic function](https://bliptext.com/articles/holomorphic-function)\nF(z)\nin the upper half-plane.\nUnder these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.\nSuppose that Then, by the theory of the [Poisson integral](https://bliptext.com/articles/poisson-integral), f admits a unique harmonic extension into the upper half-plane, and this extension is given by\nwhich is the convolution of f with the [Poisson kernel](https://bliptext.com/articles/poisson-kernel)\nFurthermore, there is a unique harmonic function v defined in the upper half-plane such that\nF(z) = u(z) + i v(z)\nis holomorphic and\nThis harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel\nThus\nIndeed, the real and imaginary parts of the Cauchy kernel are\nso that\nF = u + i v\nis holomorphic by [Cauchy's integral formula](https://bliptext.com/articles/cauchy-s-integral-formula).\nThe function v obtained from u in this way is called the [harmonic conjugate](https://bliptext.com/articles/harmonic-conjugate) of u. The (non-tangential) boundary limit of\nv(x,y)\nas\ny → 0\nis the Hilbert transform of f. Thus, succinctly,\n\n## Titchmarsh's theorem\nTitchmarsh's theorem (named for [E. C. Titchmarsh](https://bliptext.com/articles/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.\nIt gives necessary and sufficient conditions for a complex-valued [square-integrable](https://bliptext.com/articles/square-integrable) function\nF(x)\non the real line to be the boundary value of a function in the [Hardy space](https://bliptext.com/articles/hardy-space)\nH2(U)\nof holomorphic functions in the upper half-plane U.\nThe theorem states that the following conditions for a complex-valued square-integrable function are equivalent:\nF(x)\nis the limit as\nz → x\nof a holomorphic function\nF(z)\nin the upper half-plane such that\nF(x)\nare Hilbert transforms of each other.\nx 1\n. Specifically, if\nF(z)\nis a holomorphic function such that\nfor all y, then there is a complex-valued function\nF(x)\nin such that\nF(x + i y) → F(x)\nin the Lp norm as\ny → 0\n(as well as holding pointwise [almost everywhere](https://bliptext.com/articles/almost-everywhere)).\nFurthermore,\nwhere f is a real-valued function in and g is the Hilbert transform (of class Lp) of f.\nThis is not true in the case\np = 1\n. In fact, the Hilbert transform of an\nL1\nfunction f need not converge in the mean to another\nL1\nfunction.\nNevertheless, the Hilbert transform of f does converge almost everywhere to a finite function g such that\nThis result is directly analogous to one by [Andrey Kolmogorov](https://bliptext.com/articles/andrey-kolmogorov) for Hardy functions in the disc.\nAlthough usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see [Paley–Wiener theorem](https://bliptext.com/articles/paley-wiener-theorem)), as well as work by Riesz, Hille, and Tamarkin\n\n## Riemann–Hilbert problem\nOne form of the [Riemann–Hilbert problem](https://bliptext.com/articles/riemann-hilbert-problem) seeks to identify pairs of functions\nF+\nand\nF−\nsuch that\nF+\nis [holomorphic](https://bliptext.com/articles/holomorphic-function) on the upper half-plane and\nF−\nis holomorphic on the lower half-plane, such that for x along the real axis,\nwhere\nf(x)\nis some given real-valued function of.\nThe left-hand side of this equation may be understood either as the difference of the limits of\nF±\nfrom the appropriate half-planes, or as a [hyperfunction](https://bliptext.com/articles/hyperfunction) distribution.\nTwo functions of this form are a solution of the Riemann–Hilbert problem.\nFormally, if\nF±\nsolve the Riemann–Hilbert problem\nthen the Hilbert transform of\nf(x)\nis given by\n\n# Hilbert transform on the circle\nFor a periodic function f the circular Hilbert transform is defined:\nThe circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series.\nThe kernel,\nis known as the **Hilbert kernel** since it was in this form the Hilbert transform was originally studied.\nThe Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel 1/x periodic.\nMore precisely, for\nx ≠ 0\nMany results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.\nAnother more direct connection is provided by the Cayley transform\nC(x) = (x – i) / (x + i)\n, which carries the real line onto the circle and the upper half plane onto the unit disk.\nIt induces a unitary map\nof\nL2(T)\nonto The operator U carries the Hardy space\nH2(T)\nonto the Hardy space.\n\n# Hilbert transform in signal processing\n\n## Bedrosian's theorem\n**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\nwhere\nfLP\nand\nfHP\nare the low- and high-pass signals respectively.\nA 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\":\nwhere\num(t)\nis the narrow bandwidth \"message\" waveform, such as voice or music.\nThen by Bedrosian's theorem:\n\n## Analytic representation\nA specific type of [conjugate function](https://bliptext.com/articles/) is**:**\nknown as the analytic representation of u(t).\nThe name reflects its mathematical tractability, due largely to [Euler's formula](https://bliptext.com/articles/euler-s-formula).\nApplying Bedrosian's theorem to the narrowband model, the analytic representation is**:**\nA Fourier transform property indicates that this complex [heterodyne](https://bliptext.com/articles/heterodyne) operation can shift all the negative frequency components of\num(t)\nabove 0 Hz. In that case, the imaginary part of the result is a Hilbert transform of the real part.\nThis is an indirect way to produce Hilbert transforms.\n\n## Angle (phase/frequency) modulation\nThe form:\nis called [angle modulation](https://bliptext.com/articles/angle-modulation), which includes both [phase modulation](https://bliptext.com/articles/phase-modulation) and [frequency modulation](https://bliptext.com/articles/frequency-modulation).\nThe [instantaneous frequency](https://bliptext.com/articles/instantaneous-phase) is For sufficiently large ω, compared to :\nand:\n\n## Single sideband modulation (SSB)\nWhen\num(t)\nin is also an analytic representation (of a message waveform), that is:\nthe result is [single-sideband](https://bliptext.com/articles/single-sideband) modulation:\nwhose transmitted component is:\n\n## Causality\nThe function presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0):\n\n# Discrete Hilbert transform\n[[File:Effect of circular convolution on discrete Hilbert transform.png|thumb|400px|right|Figure 4. The Hilbert transform of\ncos(ωt)\nis\nsin(ωt)\n. This figure shows\nsin(ωt)\nand two approximate Hilbert transforms computed by the MATLAB library function, hilbert]]\nFor a discrete function, u[n], with [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT), U(\\omega), and **discrete Hilbert transform** the DTFT of in the region\n−π < ω < π\nis given by**:**\nThe inverse DTFT, using the [convolution theorem](https://bliptext.com/articles/convolution-theorem), is**:**\nwhere\nwhich is an infinite impulse response (IIR).\n**Practical considerations**\n**Method 1:** Direct convolution of streaming u[n] data with an FIR approximation of h[n], which we will designate by Examples of truncated h[n] are shown in figures 1 and 2.\n**Fig 1** has an odd number of anti-symmetric coefficients and is called Type III.\nThis type inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in a bandpass filter shape.\nA Type IV design (even number of anti-symmetric coefficients) is shown in **Fig 2**.\nIt has a highpass frequency response.\nType III is the usual choice.\nfor these reasons**:**\nThe abrupt truncation of h[n] creates a rippling (Gibbs effect) of the flat frequency response.\nThat can be mitigated by use of a window function to taper \\tilde h[n] to zero.\n**Method 2:** Piecewise convolution.\nIt is well known that direct convolution is computationally much more intensive than methods like **[overlap-save](https://bliptext.com/articles/overlap-save-method)** that give access to the efficiencies of the Fast Fourier transform via the convolution theorem.\nSpecifically, the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT) of a segment of u[n] is multiplied pointwise with a DFT of the \\tilde h[n] sequence.\nAn inverse DFT is done on the product, and the transient artifacts at the leading and trailing edges of the segment are discarded.\nOver-lapping input segments prevent gaps in the output stream.\nAn equivalent time domain description is that segments of length N (an arbitrary parameter) are convolved with the periodic function**:**\nWhen the duration of non-zero values of is M < N, the output sequence includes N-M+1 samples of \\widehat u. M-1 outputs are discarded from each block of N, and the input blocks are overlapped by that amount to prevent gaps.\n**Method 3**:**** Same as method 2, except the DFT of is replaced by samples of the distribution (whose real and imaginary components are all just 0 or \\pm 1.) That convolves u[n] with a [periodic summation](https://bliptext.com/articles/periodic-summation)**:**\nfor some arbitrary parameter, N. h[n] is not an FIR, so the edge effects extend throughout the entire transform.\nDeciding what to delete and the corresponding amount of overlap is an application-dependent design issue.\n**Fig 3** depicts the difference between methods 2 and 3.\nOnly half of the antisymmetric impulse response is shown, and only the non-zero coefficients.\nThe blue graph corresponds to method 2 where h[n] is truncated by a rectangular window function, rather than tapered.\nIt is generated by a Matlab function, **hilb(65)**.\nIts transient effects are exactly known and readily discarded.\nThe frequency response, which is determined by the function argument, is the only application-dependent design issue.\nThe red graph is h_{512}[n], corresponding to method 3.\nIt is the inverse DFT of the distribution.\nSpecifically, it is the function that is convolved with a segment of u[n] by the [MATLAB](https://bliptext.com/articles/matlab) function, **hilbert(u,512)**.\nThe real part of the output sequence is the original input sequence, so that the complex output is an [analytic representation](https://bliptext.com/articles/analytic-signal) of u[n].\nWhen the input is a segment of a pure cosine, the resulting convolution for two different values of N is depicted in **Fig 4** (red and blue plots).\nEdge effects prevent the result from being a pure sine function (green plot).\nSince h_N[n] is not an FIR sequence, the theoretical extent of the effects is the entire output sequence.\nBut the differences from a sine function diminish with distance from the edges.\nParameter N is the output sequence length.\nIf it exceeds the length of the input sequence, the input is modified by appending zero-valued elements.\nIn most cases, that reduces the magnitude of the edge distortions.\nBut their duration is dominated by the inherent rise and fall times of the h[n] impulse response.\n**Fig 5** is an example of piecewise convolution, using both methods 2 (in blue) and 3 (red dots).\nA 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.\nAs the FIR result (blue) shows, the distortions apparent in the IIR result (red) are not caused by the difference between h[n] and h_N[n] (green and red in **Fig 3**).\nThe fact that h_N[n] is tapered (windowed) is actually helpful in this context.\nThe real problem is that it's not windowed enough.\nEffectively, M=N, whereas the overlap-save method needs M < N.\n# [Figure 3. | upload.wikimedia.org/wikipedia/commons/e/e8/DFT///approximation///to///Hilbert///filter.png]\n# [Figure 5. Discrete Hilbert transforms of a cosine function, using piecewise convolution | upload.wikimedia.org/wikipedia/commons/1/1c/Discrete///Hilbert///transforms///of///a///cosine///function%2C///using///piecewise///convolution.svg]\n\n# Number-theoretic Hilbert transform\nThe number theoretic Hilbert transform is an extension of the discrete Hilbert transform to integers modulo an appropriate prime number.\nIn this it follows the generalization of [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) to number theoretic transforms.\nThe number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.\n\n# Page citations","revisionCount":1},"hasMore":false,"nextCursor":null}