Inmathematics andsignal processing, theHilbert transform is a specificsingular integral that takes a function,u(t) of a real variable and produces another function of a real variableH(u)(t). The Hilbert transform is given by theCauchy principal value of theconvolution with the function${\displaystyle 1/(\pi t)}$ (see§ Definition). The Hilbert transform has a particularly simple representation in thefrequency domain: It imparts aphase 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§ Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component of theanalytic representation of a real-valued signalu(t). The Hilbert transform was first introduced byDavid Hilbert in this setting, to solve a special case of theRiemann–Hilbert problem for analytic functions.

The Hilbert transform ofu can be thought of as theconvolution ofu(t) with the functionh(t) =⁠1/πt⁠, known as theCauchy kernel. Because 1/t is notintegrable acrosst = 0, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using theCauchy principal value (denoted here byp.v.). Explicitly, the Hilbert transform of a function (or signal)u(t) is given by

$${\displaystyle \mathrm{H}(u)(t)=\frac{1}{\pi }\mathrm{p}.\mathrm{v}.{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{u(\tau )}{t-\tau }\mathrm{d}\tau ,}$$

provided this integral exists as a principal value. This is precisely the convolution ofu with thetempered distributionp.v.⁠1/πt⁠.^[1] Alternatively, by changing variables, the principal-value integral can be written explicitly^[2] as

$${\displaystyle \mathrm{H}(u)(t)=\frac{2}{\pi }\underset{\epsilon \to 0}{lim}{\int }_{\epsilon }^{\mathrm{\infty }}\frac{u(t-\tau )-u(t+\tau )}{2\tau }\mathrm{d}\tau .}$$

When the Hilbert transform is applied twice in succession to a functionu, the result is

$${\displaystyle \mathrm{H}(\mathrm{H}(u))(t)=-u(t),}$$

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is${\displaystyle -\mathrm{H}}$. This fact can most easily be seen by considering the effect of the Hilbert transform on theFourier transform ofu(t) (see§ Relationship with the Fourier transform below).

For ananalytic function in theupper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, iff(z) is analytic in the upper half complex plane{z : Im{z} > 0}, andu(t) = Re{f (t + 0·i)}, thenIm{f(t + 0·i)} = H(u)(t) up to an additive constant, provided this Hilbert transform exists.

Insignal processing the Hilbert transform ofu(t) is commonly denoted by${\displaystyle \hat{u}(t)}$.^[3] However, in mathematics, this notation is already extensively used to denote the Fourier transform ofu(t).^[4] Occasionally, the Hilbert transform may be denoted by${\displaystyle \tilde{u}(t)}$. Furthermore, many sources define the Hilbert transform as the negative of the one defined here.^[5]

The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions,^[6][7] which has come to be known as theRiemann–Hilbert problem. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.^[8][9] Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave inGöttingen. The results were later published byHermann Weyl in his dissertation.^[10] Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.^[11] These results were restricted to the spacesL² andℓ². In 1928,Marcel Riesz proved that the Hilbert transform can be defined foru in${\displaystyle L^p(\mathbb{R})}$ (L^p space) for1 <p < ∞, that the Hilbert transform is abounded operator on${\displaystyle L^p(\mathbb{R})}$ for1 <p < ∞, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.^[12] The Hilbert transform was a motivating example forAntoni Zygmund andAlberto Calderón during their study ofsingular integrals.^[13] 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.

The Hilbert transform is amultiplier operator.^[14] The multiplier ofH isσ_H(ω) = −i sgn(ω), wheresgn is thesignum function. Therefore:

$${\displaystyle \mathcal{F}(\mathrm{H}(u))(\omega )=-i\mathrm{sgn}(\omega )\cdot \mathcal{F}(u)(\omega ),}$$

where${\displaystyle \mathcal{F}}$ denotes theFourier transform. Sincesgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of${\displaystyle \mathcal{F}}$.

ByEuler's formula,

Therefore,H(u)(t) has the effect of shifting the phase of thenegative frequency components ofu(t) by +90° (π⁄2 radians) and the phase of the positive frequency components by −90°, andi·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation (i.e., a multiplication by −1).

When the Hilbert transform is applied twice, the phase of the negative and positive frequency components ofu(t) are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e.,H(H(u)) = −u, because

$${\displaystyle {\left({\sigma }_{\mathrm{H}}(\omega )\right)}^2=e^{\pm i\pi }=-1\text{for }\omega \ne 0.}$$

In the following table, thefrequency parameter${\displaystyle \omega }$ is real.

Signal ${\displaystyle u(t)}$	Hilbert transform[fn 1] ${\displaystyle \mathrm{H}(u)(t)}$
${\displaystyle \sin (\omega t+\phi )}$[fn 2]
${\displaystyle \cos (\omega t+\phi )}$[fn 2]
${\displaystyle e^{i\omega t}}$
${\displaystyle e^{-i\omega t}}$
$\frac{{\displaystyle 1}}{t^2+1}$	$\frac{{\displaystyle t}}{t^2+1}$
${\displaystyle e^{-t^2}}$	${\displaystyle \frac{2}{\sqrt{\pi }}F(t)}$ (seeDawson function)
Sinc function $\frac{{\displaystyle \sin (t)}}{t}$	$\frac{{\displaystyle 1-\cos (t)}}{t}$
Dirac delta function ${\displaystyle \delta (t)}$	${\displaystyle \frac{1}{\pi t}}$
Characteristic function ${\displaystyle {\chi }_{[a,b]}(t)}$	${\displaystyle \frac{1}{\pi }\ln \left|\frac{t-a}{t-b}\right|}$

Notes

An extensive table of Hilbert transforms is available.^[15]Note that the Hilbert transform of a constant is zero.

It is by no means obvious that the Hilbert transform is well-defined at all, as theimproper 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${\displaystyle L^p(\mathbb{R})}$ for1 <p < ∞.

More precisely, ifu is in${\displaystyle L^p(\mathbb{R})}$ for1 <p < ∞, then the limit defining the improper integral

$${\displaystyle \mathrm{H}(u)(t)=\frac{2}{\pi }\underset{\epsilon \to 0}{lim}{\int }_{\epsilon }^{\mathrm{\infty }}\frac{u(t-\tau )-u(t+\tau )}{2\tau }d\tau }$$

exists foralmost everyt. The limit function is also in${\displaystyle L^p(\mathbb{R})}$ and is in fact the limit in the mean of the improper integral as well. That is,

$${\displaystyle \frac{2}{\pi }{\int }_{\epsilon }^{\mathrm{\infty }}\frac{u(t-\tau )-u(t+\tau )}{2\tau }\mathrm{d}\tau \to \mathrm{H}(u)(t)}$$

asε → 0 in theL^p norm, as well as pointwise almost everywhere, by theTitchmarsh theorem.^[16]

In the casep = 1, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally.^[17] In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of anL¹ function does converge, however, inL¹-weak, and the Hilbert transform is a bounded operator fromL¹ toL^1,w.^[18] (In particular, since the Hilbert transform is also a multiplier operator onL²,Marcinkiewicz interpolation and a duality argument furnishes an alternative proof thatH is bounded onL^p.)

If1 <p < ∞, then the Hilbert transform on${\displaystyle L^p(\mathbb{R})}$ is abounded linear operator, meaning that there exists a constantC_p such that

$${\displaystyle {\Vert \mathrm{H}u\Vert }_p\le C_p{\Vert u\Vert }_p}$$

for all${\displaystyle u\in L^p(\mathbb{R})}$.^[19]

The best constant${\displaystyle C_p}$ is given by^[20]

An easy way to find the best${\displaystyle C_p}$ for${\displaystyle p}$ being a power of 2 is through the so-called Cotlar's identity that${\displaystyle (\mathrm{H}f{)}^2=f^2+2\mathrm{H}(f\mathrm{H}f)}$ for all real valuedf. The same best constants hold for the periodic Hilbert transform.

The boundedness of the Hilbert transform implies the${\displaystyle L^p(\mathbb{R})}$ convergence of the symmetric partial sum operator$${\displaystyle S_{R}f={\int }_{-R}^R\hat{f}(\xi )e^{2\pi ix\xi }\mathrm{d}\xi }$$

tof in${\displaystyle L^p(\mathbb{R})}$.^[21]

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between${\displaystyle L^p(\mathbb{R})}$ and the dual space${\displaystyle L^q(\mathbb{R})}$, wherep andq areHölder conjugates and1 <p,q < ∞. Symbolically,

$${\displaystyle ⟨\mathrm{H}u,v⟩=⟨u,-\mathrm{H}v⟩}$$

for${\displaystyle u\in L^p(\mathbb{R})}$ and${\displaystyle v\in L^q(\mathbb{R})}$.^[22]

The Hilbert transform is ananti-involution,^[23] meaning that

$${\displaystyle \mathrm{H}(\mathrm{H}\left(u\right))=-u}$$

provided each transform is well-defined. SinceH preserves the space${\displaystyle L^p(\mathbb{R})}$, this implies in particular that the Hilbert transform is invertible on${\displaystyle L^p(\mathbb{R})}$, and that

$${\displaystyle {\mathrm{H}}^{-1}=-\mathrm{H}}$$

BecauseH² = −I ("I" is theidentity operator) on the realBanach space ofreal-valued functions in${\displaystyle L^p(\mathbb{R})}$, the Hilbert transform defines alinear complex structure on this Banach space. In particular, whenp = 2, the Hilbert transform gives theHilbert space of real-valued functions in${\displaystyle L^2(\mathbb{R})}$ the structure of acomplex Hilbert space.

The (complex)eigenstates of the Hilbert transform admit representations asholomorphic functions in the upper and lower half-planes in theHardy spaceH² by thePaley–Wiener theorem.

Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:

$${\displaystyle \mathrm{H}\left(\frac{\mathrm{d}u}{\mathrm{d}t}\right)=\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{H}(u)}$$

Iterating this identity,

$${\displaystyle \mathrm{H}\left(\frac{{\mathrm{d}}^{k}u}{\mathrm{d}{t}^k}\right)=\frac{{\mathrm{d}}^k}{\mathrm{d}{t}^k}\mathrm{H}(u)}$$

This is rigorously true as stated providedu and its firstk derivatives belong to${\displaystyle L^p(\mathbb{R})}$.^[24] One can check this easily in the frequency domain, where differentiation becomes multiplication byω.

The Hilbert transform can formally be realized as aconvolution with thetempered distribution^[25]

$${\displaystyle h(t)=\mathrm{p}.\mathrm{v}.\frac{1}{\pi t}}$$

Thus formally,

$${\displaystyle \mathrm{H}(u)=h\ast u}$$

However,a priori this may only be defined foru a distribution ofcompact support. It is possible to work somewhat rigorously with this since compactly supported functions (which are distributionsa fortiori) aredense inL^p. Alternatively, one may use the fact thath(t) is thedistributional derivative of the functionlog|t|/π; to wit

$${\displaystyle \mathrm{H}(u)(t)=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\pi }\left(u\ast \log |\cdot |\right)(t)\right)}$$

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 ononly one of either of the factors:

$${\displaystyle \mathrm{H}(u\ast v)=\mathrm{H}(u)\ast v=u\ast \mathrm{H}(v)}$$

This is rigorously true ifu andv are compactly supported distributions since, in that case,

$${\displaystyle h\ast (u\ast v)=(h\ast u)\ast v=u\ast (h\ast v)}$$

By passing to an appropriate limit, it is thus also true ifu ∈L^p andv ∈L^q provided that

from a theorem due to Titchmarsh.^[26]

The Hilbert transform has the following invariance properties on${\displaystyle L^2(\mathbb{R})}$.

• It commutes with translations. That is, it commutes with the operatorsT_af(x) =f(x +a) for alla in${\displaystyle \mathbb{R}.}$
• It commutes with positive dilations. That is it commutes with the operatorsM_λ f (x) =f (λ x) for allλ > 0.
• Itanticommutes with the reflectionR f (x) =f (−x).

Up to a multiplicative constant, the Hilbert transform is the only bounded operator onL² with these properties.^[27]

In fact there is a wider set of operators that commute with the Hilbert transform. The group${\displaystyle \text{SL}(2,\mathbb{R})}$ acts by unitary operatorsU_g on the space${\displaystyle L^2(\mathbb{R})}$ by the formula

Thisunitary representation is an example of aprincipal series representation of${\displaystyle \text{SL}(2,\mathbb{R}).}$ In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces,Hardy space${\displaystyle H^2(\mathbb{R})}$ and its conjugate. These are the spaces ofL² boundary values of holomorphic functions on the upper and lower halfplanes.${\displaystyle H^2(\mathbb{R})}$ and its conjugate consist of exactly thoseL² functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal toH = −i (2P − I), withP being the orthogonal projection from${\displaystyle L^2(\mathbb{R})}$ onto${\displaystyle {\mathrm{H}}^2(\mathbb{R}),}$ andI theidentity operator, it follows that${\displaystyle {\mathrm{H}}^2(\mathbb{R})}$ and its orthogonal complement are eigenspaces ofH for the eigenvalues±i. In other words,H commutes with the operatorsU_g. The restrictions of the operatorsU_g to${\displaystyle {\mathrm{H}}^2(\mathbb{R})}$ and its conjugate give irreducible representations of${\displaystyle \text{SL}(2,\mathbb{R})}$ – the so-calledlimit of discrete series representations.^[28]

It is further possible to extend the Hilbert transform to certain spaces ofdistributions (Pandey 1996, Chapter 3). Since the Hilbert transform commutes with differentiation, and is a bounded operator onL^p,H restricts to give a continuous transform on theinverse limit ofSobolev spaces:

$${\displaystyle {\mathcal{D}}_{L^p}=\underset{n\to \mathrm{\infty }}{\underset{⟵}{lim}}{W}^{n,p}(\mathbb{R})}$$

The Hilbert transform can then be defined on the dual space of${\displaystyle {\mathcal{D}}_{L^p}}$, denoted${\displaystyle {\mathcal{D}}_{L^p}^{\prime }}$, consisting ofL^p distributions. This is accomplished by the duality pairing:
For${\displaystyle u\in {\mathcal{D}}_{L^p}^{\prime }}$, define:

$${\displaystyle \mathrm{H}(u)\in {\mathcal{D}}_{L^p}^{\prime }=⟨\mathrm{H}u,v⟩\triangleq ⟨u,-\mathrm{H}v⟩,\text{for all}v\in {\mathcal{D}}_{L^p}.}$$

It is possible to define the Hilbert transform on the space oftempered distributions as well by an approach due to Gel'fand and Shilov,^[29] but considerably more care is needed because of the singularity in the integral.

The Hilbert transform can be defined for functions in${\displaystyle L^{\mathrm{\infty }}(\mathbb{R})}$ as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps${\displaystyle L^{\mathrm{\infty }}(\mathbb{R})}$ to theBanach space ofbounded mean oscillation (BMO) classes.

Interpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined. For instance, withu = sgn(x), the integral definingH(u) diverges almost everywhere to±∞. To alleviate such difficulties, the Hilbert transform of anL^∞ function is therefore defined by the followingregularized form of the integral

$${\displaystyle \mathrm{H}(u)(t)=\mathrm{p}.\mathrm{v}.{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}u(\tau )\left\{h(t-\tau )-h_0(-\tau )\right\}\mathrm{d}\tau }$$

where as aboveh(x) =⁠1/πx⁠ and

The modified transformH agrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.^[30] Furthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.

Adeep result of Fefferman's work^[31] is that a function is of bounded mean oscillation if and only if it has the formf + H(g) for some${\displaystyle f,g\in L^{\mathrm{\infty }}(\mathbb{R})}$.

The Hilbert transform can be understood in terms of a pair of functionsf(x) andg(x) such that the function$${\displaystyle F(x)=f(x)+ig(x)}$$is the boundary value of aholomorphic functionF(z) in the upper half-plane.^[32] Under these circumstances, iff andg are sufficiently integrable, then one is the Hilbert transform of the other.

Suppose that${\displaystyle f\in L^p(\mathbb{R}).}$ Then, by the theory of thePoisson integral,f admits a unique harmonic extension into the upper half-plane, and this extension is given by

$${\displaystyle u(x+iy)=u(x,y)=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(s)\frac{y}{(x-s{)}^2+y^2}\mathrm{d}s}$$

which is the convolution off with thePoisson kernel

$${\displaystyle P(x,y)=\frac{y}{\pi \left(x^2+y^2\right)}}$$

Furthermore, there is a unique harmonic functionv defined in the upper half-plane such thatF(z) =u(z) +i v(z) is holomorphic and$${\displaystyle \underset{y\to \mathrm{\infty }}{lim}v(x+iy)=0}$$

This harmonic function is obtained fromf by taking a convolution with theconjugate Poisson kernel

$${\displaystyle Q(x,y)=\frac{x}{\pi \left(x^2+y^2\right)}.}$$

Thus$${\displaystyle v(x,y)=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(s)\frac{x-s}{(x-s{)}^2+y^2}\mathrm{d}s.}$$

Indeed, the real and imaginary parts of the Cauchy kernel are$${\displaystyle \frac{i}{\pi z}=P(x,y)+iQ(x,y)}$$

so thatF =u +i v is holomorphic byCauchy's integral formula.

The functionv obtained fromu in this way is called theharmonic conjugate ofu. The (non-tangential) boundary limit ofv(x,y) asy → 0 is the Hilbert transform off. Thus, succinctly,$${\displaystyle \mathrm{H}(f)=\underset{y\to 0}{lim}Q(-,y)\star f}$$

Titchmarsh's theorem (named forE. 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.^[33] It gives necessary and sufficient conditions for a complex-valuedsquare-integrable functionF(x) on the real line to be the boundary value of a function in theHardy spaceH²(U) of holomorphic functions in the upper half-planeU.

The theorem states that the following conditions for a complex-valued square-integrable function${\displaystyle F:\mathbb{R}\to \mathbb{C}}$ are equivalent:

• F(x) is the limit asz →x of a holomorphic functionF(z) in the upper half-plane such that
• The real and imaginary parts ofF(x) are Hilbert transforms of each other.
• TheFourier transform${\displaystyle \mathcal{F}(F)(x)}$ vanishes forx < 0.

A weaker result is true for functions of classL^p forp > 1.^[34] Specifically, ifF(z) is a holomorphic function such that

for ally, then there is a complex-valued functionF(x) in${\displaystyle L^p(\mathbb{R})}$ such thatF(x +i y) →F(x) in theL^p norm asy → 0 (as well as holding pointwisealmost everywhere). Furthermore,

$${\displaystyle F(x)=f(x)+ig(x)}$$

wheref is a real-valued function in${\displaystyle L^p(\mathbb{R})}$ andg is the Hilbert transform (of classL^p) off.

This is not true in the casep = 1. In fact, the Hilbert transform of anL¹ functionf need not converge in the mean to anotherL¹ function. Nevertheless,^[35] the Hilbert transform off does converge almost everywhere to a finite functiong such that

This result is directly analogous to one byAndrey Kolmogorov for Hardy functions in the disc.^[36] Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (seePaley–Wiener theorem), as well as work by Riesz, Hille, and Tamarkin^[37]

One form of theRiemann–Hilbert problem seeks to identify pairs of functionsF₊ andF₋ such thatF₊ isholomorphic on the upper half-plane andF₋ is holomorphic on the lower half-plane, such that forx along the real axis,$${\displaystyle F_{+}(x)-F_{-}(x)=f(x)}$$

wheref(x) is some given real-valued function of${\displaystyle x\in \mathbb{R}}$. The left-hand side of this equation may be understood either as the difference of the limits ofF_± from the appropriate half-planes, or as ahyperfunction distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.

Formally, ifF_± solve the Riemann–Hilbert problem$${\displaystyle f(x)=F_{+}(x)-F_{-}(x)}$$

then the Hilbert transform off(x) is given by^[38]$${\displaystyle H(f)(x)=-i(F_{+}(x)+F_{-}(x)).}$$

For a periodic functionf the circular Hilbert transform is defined:

$${\displaystyle \tilde{f}(x)\triangleq \frac{1}{2\pi }\mathrm{p}.\mathrm{v}.{\int }_0^{2\pi }f(t)\cot \left(\frac{x-t}{2}\right)\mathrm{d}t}$$

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,$${\displaystyle \cot \left(\frac{x-t}{2}\right)}$$is known as theHilbert kernel since it was in this form the Hilbert transform was originally studied.^[8]

The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel1⁄x periodic. More precisely, forx ≠ 0

$${\displaystyle \frac{1}{2}\cot \left(\frac{x}{2}\right)=\frac{1}{x}+\sum _{n=1}^{\mathrm{\infty }}\left(\frac{1}{x+2n\pi }+\frac{1}{x-2n\pi }\right)}$$

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 transformC(x) = (x –i) / (x +i), which carries the real line onto the circle and the upper half plane onto theunit disk. It induces a unitary map

$${\displaystyle Uf(x)=\frac{1}{(x+i)\sqrt{\pi }}f\left(C\left(x\right)\right)}$$

ofL²(T) onto${\displaystyle L^2(\mathbb{R}).}$ The operatorU carries the Hardy spaceH²(T) onto the Hardy space${\displaystyle H^2(\mathbb{R})}$.^[39]

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

$${\displaystyle \mathrm{H}\left(f_{\text{LP}}(t)\cdot f_{\text{HP}}(t)\right)=f_{\text{LP}}(t)\cdot \mathrm{H}\left(f_{\text{HP}}(t)\right),}$$

wheref_LP andf_HP are the low- and high-pass signals respectively.^[40] A category of communication signals to which this applies is called thenarrowband signal model. A member of that category is amplitude modulation of a high-frequency sinusoidal "carrier":

$${\displaystyle u(t)=u_m(t)\cdot \cos (\omega t+\phi ),}$$

whereu_m(t) is the narrow bandwidth "message" waveform, such as voice or music. Then by Bedrosian's theorem:^[41]

A specific type ofconjugate function is:

$${\displaystyle u_a(t)\triangleq u(t)+i\cdot H(u)(t),}$$

known as theanalytic representation of${\displaystyle u(t).}$ The name reflects its mathematical tractability, due largely toEuler's formula. Applying Bedrosian's theorem to the narrowband model, the analytic representation is:^[42]

Eq.1

A Fourier transform property indicates that this complexheterodyne operation can shift all the negative frequency components ofu_m(t) above 0 Hz. In that case, the imaginary part of the result is a Hilbert transform of the real part. This is an indirect way to produce Hilbert transforms.

The form:^[43]

$${\displaystyle u(t)=A\cdot \cos (\omega t+{\phi }_m(t))}$$

is calledangle modulation, which includes bothphase modulation andfrequency modulation. Theinstantaneous frequency is  ${\displaystyle \omega +{\phi }_m^{\mathrm{\prime }}(t).}$  For sufficiently largeω, compared to${\displaystyle {\phi }_m^{\mathrm{\prime }}}$:

$${\displaystyle \mathrm{H}(u)(t)\approx A\cdot \sin (\omega t+{\phi }_m(t))}$$and:$${\displaystyle u_a(t)\approx A\cdot e^{i(\omega t+{\phi }_m(t))}.}$$

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

$${\displaystyle u_m(t)=m(t)+i\cdot \hat{m}(t)}$$

the result issingle-sideband modulation:

$${\displaystyle u_a(t)=(m(t)+i\cdot \hat{m}(t))\cdot e^{i(\omega t+\phi )}}$$

whose transmitted component is:^[44][45]

The function${\displaystyle h(t)=1/(\pi t)}$ presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0):

• Its duration is infinite (technicallyinfinitesupport). Finite-lengthwindowing reduces the effective frequency range of the transform; shorter windows result in greater losses at low and high frequencies. See alsoquadrature filter.
• It is anon-causal filter. So a delayed version,${\displaystyle h(t-\tau ),}$ is required. The corresponding output is subsequently delayed by${\displaystyle \tau .}$ When creating the imaginary part of ananalytic signal, the source (real part) must also be delayed by${\displaystyle \tau }$.

For a discrete function,${\displaystyle u[n],}$ withdiscrete-time Fourier transform (DTFT),${\displaystyle U(\omega )}$, anddiscrete Hilbert transform${\displaystyle \hat{u}[n],}$ the DTFT of${\displaystyle \hat{u}[n]}$ in the region−π < ω <π is given by:

${\displaystyle \mathrm{DTFT}(\hat{u})=U(\omega )\cdot (-i\cdot \mathrm{sgn}(\omega )).}$

The inverse DTFT, using theconvolution theorem, is:^[46][47]

where

which is an infinite impulse response (IIR).

Practical considerations^[48]

Method 1: Direct convolution of streaming${\displaystyle u[n]}$ data with an FIR approximation of${\displaystyle h[n],}$ which we will designate by${\displaystyle \tilde{h}[n].}$ Examples of truncated${\displaystyle h[n]}$ are shown in figures 1 and 2.Fig 1 has an odd number of anti-symmetric coefficients and is called Type III.^[49] This type inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in a bandpass filter shape.^[50][51] A Type IV design (even number of anti-symmetric coefficients) is shown inFig 2.^[52][53] It has a highpass frequency response.^[54] Type III is the usual choice.^[55][56] for these reasons:

• A typical (i.e. properly filtered and sampled)${\displaystyle u[n]}$ sequence has no useful components at the Nyquist frequency.
• The Type IV impulse response requires a${\displaystyle \frac{1}{2}}$ sample shift in the${\displaystyle h[n]}$ sequence. That causes the zero-valued coefficients to become non-zero, as seen inFigure 2. So a Type III design is potentially twice as efficient as Type IV.
• The group delay of a Type III design is an integer number of samples, which facilitates aligning${\displaystyle \hat{u}[n]}$ with${\displaystyle u[n]}$ to create ananalytic signal. The group delay of Type IV is halfway between two samples.

The abrupt truncation of${\displaystyle h[n]}$ creates a rippling (Gibbs effect) of the flat frequency response. That can be mitigated by use of a window function to taper${\displaystyle \tilde{h}[n]}$ to zero.^[57]

Method 2: Piecewise convolution. It is well known that direct convolution is computationally much more intensive than methods likeoverlap-save that give access to the efficiencies of the Fast Fourier transform via the convolution theorem.^[58] Specifically, thediscrete Fourier transform (DFT) of a segment of${\displaystyle u[n]}$ is multiplied pointwise with a DFT of the${\displaystyle \tilde{h}[n]}$ sequence. An inverse DFT is done on the product, and the transient artifacts at the leading and trailing edges of the segment are discarded. Over-lapping input segments prevent gaps in the output stream. An equivalent time domain description is that segments of length${\displaystyle N}$ (an arbitrary parameter) are convolved with the periodic function:

${\displaystyle {\tilde{h}}_N[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\tilde{h}[n-mN].}$

When the duration of non-zero values of${\displaystyle \tilde{h}[n]}$ is the output sequence includes${\displaystyle N-M+1}$ samples of${\displaystyle \hat{u}.}$ ${\displaystyle M-1}$ outputs are discarded from each block of${\displaystyle N,}$ and the input blocks are overlapped by that amount to prevent gaps.

Method 3: Same as method 2, except the DFT of${\displaystyle \tilde{h}[n]}$ is replaced by samples of the${\displaystyle -i\mathrm{sgn}(\omega )}$ distribution (whose real and imaginary components are all just${\displaystyle 0}$ or ${\displaystyle \pm 1.}$) That convolves${\displaystyle u[n]}$ with aperiodic summation:^[A]

${\displaystyle h_N[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}h[n-mN],}$   ^[B][C]

for some arbitrary parameter,${\displaystyle N.}$${\displaystyle h[n]}$ is not an FIR, so the edge effects extend throughout the entire transform. Deciding what to delete and the corresponding amount of overlap is an application-dependent design issue.

Fig 3 depicts the difference between methods 2 and 3. Only half of the antisymmetric impulse response is shown, and only the non-zero coefficients. The blue graph corresponds to method 2 where${\displaystyle h[n]}$ is truncated by a rectangular window function, rather than tapered. It is generated by a Matlab function,hilb(65). Its transient effects are exactly known and readily discarded. The frequency response, which is determined by the function argument, is the only application-dependent design issue.

The red graph is${\displaystyle h_{512}[n],}$ corresponding to method 3. It is the inverse DFT of the${\displaystyle -i\mathrm{sgn}(\omega )}$ distribution. Specifically, it is the function that is convolved with a segment of${\displaystyle u[n]}$ by theMATLAB function,hilbert(u,512).^[61] The real part of the output sequence is the original input sequence, so that the complex output is ananalytic representation of${\displaystyle u[n].}$

When the input is a segment of a pure cosine, the resulting convolution for two different values of${\displaystyle N}$ is depicted inFig 4 (red and blue plots). Edge effects prevent the result from being a pure sine function (green plot). Since${\displaystyle h_N[n]}$ is not an FIR sequence, the theoretical extent of the effects is the entire output sequence. But the differences from a sine function diminish with distance from the edges. Parameter${\displaystyle N}$ is the output sequence length. If it exceeds the length of the input sequence, the input is modified by appending zero-valued elements. In most cases, that reduces the magnitude of the edge distortions. But their duration is dominated by the inherent rise and fall times of the${\displaystyle h[n]}$ impulse response.

Fig 5 is an example of piecewise convolution, using both methods 2 (in blue) and 3 (red dots). 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${\displaystyle h[n]}$ and${\displaystyle h_N[n]}$ (green and red inFig 3). The fact that${\displaystyle h_N[n]}$ is tapered (windowed) is actually helpful in this context. The real problem is that it's not windowed enough. Effectively,${\displaystyle M=N,}$ whereas the overlap-save method needs

The number theoretic Hilbert transform is an extension^[62] of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization ofdiscrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.^[63]

• Analytic signal
• Harmonic conjugate
• Hilbert spectroscopy
• Hilbert transform in the complex plane
• Hilbert–Huang transform
• Kramers–Kronig relations
• Riesz transform
• Single-sideband modulation
• Singular integral operators of convolution type
• Stieltjes transformation

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• Derivation of the boundedness of the Hilbert transform
• Mathworld Hilbert transform — Contains a table of transforms
• Weisstein, Eric W."Titchmarsh theorem".MathWorld.
• "GS256 Lecture 3: Hilbert Transformation"(PDF). Archived fromthe original(PDF) on 2012-02-27. an entry level introduction to Hilbert transformation.

• Harmonic functions
• Integral transforms
• Signal processing
• Singular integrals
• Schwartz distributions

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