Hilbert transform

The Hilbert transform of {{mvar|u}} can be thought of as the convolution of {{math|u(t)}} with the function {{math|1=h(t) = {{sfrac|1|{{pi}}t}}}}, known as the Cauchy kernel. Because 1/{{mvar|t}} is not integrable across {{math|1=t = 0}}, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by {{math|p.v.}}). Explicitly, the Hilbert transform of a function (or signal) {{math|u(t)}} is given by

\operatorname{H}(u)(t) = \frac{1}{\pi}\, \operatorname{p.v.} \int_{-\infty}^{+\infty} \frac{u(\tau)}{t - \tau}\,\mathrm{d}\tau,

provided this integral exists as a principal value. This is precisely the convolution of {{mvar|u}} with the tempered distribution {{math|p.v. {{sfrac|1|{{pi}}t}}}}.Due to {{harvnb|Schwartz|1950}}; see {{harvnb|Pandey|1996|loc=Chapter 3}}. Alternatively, by changing variables, the principal-value integral can be written explicitly{{harvnb|Zygmund|1968|loc=§XVI.1}}. as

\operatorname{H}(u)(t) = \frac{2}{\pi}\, \lim_{\varepsilon \to 0} \int_\varepsilon^\infty \frac{u(t - \tau) - u(t + \tau)}{2\tau} \,\mathrm{d}\tau.

When the Hilbert transform is applied twice in succession to a function {{mvar|u}}, the result is

\operatorname{H}\bigl(\operatorname{H}(u)\bigr)(t) = -u(t),

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is

$-\backslash operatorname\{H\}$. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of {{math|u(t)}} (see {{slink|#Relationship with the Fourier transform}} below).

For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if {{math|f(z)}} is analytic in the upper half complex plane {{math|1={z : Im{z} > 0}}}, and {{math|1=u(t) = Re{f (t + 0·i)} }}, then {{math|1= Im{f(t + 0·i)} = H(u)(t)}} up to an additive constant, provided this Hilbert transform exists.

In signal processing the Hilbert transform of {{math|u(t)}} is commonly denoted by $\backslash hat\{u\}(t)$.E.g., {{harvnb|Brandwood|2003|p=87}}. However, in mathematics, this notation is already extensively used to denote the Fourier transform of {{math|u(t)}}.E.g., {{harvnb|Stein|Weiss|1971}}. Occasionally, the Hilbert transform may be denoted by $\backslash tilde\{u\}(t)$. Furthermore, many sources define the Hilbert transform as the negative of the one defined here.E.g., {{harvnb|Bracewell|2000|p=359}}.

The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions,{{sfn|Kress|1989}}{{sfn|Bitsadze|2001}} which has come to be known as the Riemann–Hilbert problem. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.{{sfn|Khvedelidze|2001}}{{sfn|Hilbert|1953}} Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation.{{sfn|Hardy|Littlewood|Pólya|1952|loc=§9.1}} Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.{{sfn|Hardy|Littlewood|Pólya|1952|loc=§9.2}} These results were restricted to the spaces Lp space. In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in $L^p(\backslash mathbb\{R\})$ (L^p space) for {{math|1 < p < ∞}}, that the Hilbert transform is a bounded operator on $L^p(\backslash mathbb\{R\})$ for {{math|1 < p < ∞}}, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.{{sfn|Riesz|1928}} The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals.{{sfn|Calderón|Zygmund|1952}} Their investigations have played a fundamental role in modern harmonic analysis. Various generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.

The Hilbert transform is a multiplier operator.{{sfn|Duoandikoetxea|2000|loc=Chapter 3}} The multiplier of {{math|H}} is {{math|1=σ_H(ω) = −i sgn(ω)}}, where {{math|sgn}} is the signum function. Therefore:

$$\backslash mathcal\{F\}\backslash bigl(\backslash operatorname\{H\}(u)\backslash bigr)(\backslash omega)\; =\; -i\; \backslash sgn(\backslash omega)\; \backslash cdot\; \backslash mathcal\{F\}(u)(\backslash omega)\; ,$$

where $\backslash mathcal\{F\}$ denotes the Fourier transform. Since {{math|1=sgn(x) = sgn(2{{pi}}x)}}, it follows that this result applies to the three common definitions of $\backslash mathcal\{F\}$.

By Euler's formula,

$$\backslash sigma\_\backslash operatorname\{H\}(\backslash omega)\; =\; \backslash begin\{cases\}$$

~~i = e^{+i\pi/2} & \text{if } \omega < 0\\

~~ 0 & \text{if } \omega = 0\\

-i = e^{-i\pi/2} & \text{if } \omega > 0

\end{cases}

Therefore, {{math|H(u)(t)}} has the effect of shifting the phase of the negative frequency components of {{math|u(t)}} by +90° ({{frac|{{pi}}|2}} radians) and the phase of the positive frequency components by −90°, and {{math|i·H(u)(t)}} has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation (i.e., a multiplication by −1).

When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of {{math|u(t)}} are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e., {{math|1=H(H(u)) = −u}}, because

$$\backslash left(\backslash sigma\_\backslash operatorname\{H\}(\backslash omega)\backslash right)^2\; =\; e^\{\backslash pm\; i\backslash pi\}\; =\; -1\; \backslash quad\; \backslash text\{for\; \}\; \backslash omega\; \backslash neq\; 0\; .$$

In the following table, the frequency parameter $\backslash omega$ is real.

Notes

An extensive table of Hilbert transforms is available.{{sfn|King|2009b}}

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 the improper integral defining it must converge in a suitable sense. However, the Hilbert transform is well-defined for a broad class of functions, namely those in $L^p(\backslash mathbb\{R\})$ for {{math|1 < p < ∞}}.

More precisely, if {{mvar|u}} is in $L^p(\backslash mathbb\{R\})$ for {{math|1 < p < ∞}}, then the limit defining the improper integral

$$\backslash operatorname\{H\}(u)(t)\; =\; \backslash frac\{2\}\{\backslash pi\}\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash int\_\backslash varepsilon^\backslash infty\; \backslash frac\{u(t\; -\; \backslash tau)\; -\; u(t\; +\; \backslash tau)\}\{2\backslash tau\}\backslash ,d\backslash tau$$

exists for almost every {{mvar|t}}. The limit function is also in $L^p(\backslash mathbb\{R\})$ and is in fact the limit in the mean of the improper integral as well. That is,

$$\backslash frac\{2\}\{\backslash pi\}\; \backslash int\_\backslash varepsilon^\backslash infty\; \backslash frac\{u(t\; -\; \backslash tau)\; -\; u(t\; +\; \backslash tau)\}\{2\backslash tau\}\backslash ,\backslash mathrm\{d\}\backslash tau\; \backslash to\; \backslash operatorname\{H\}(u)(t)$$

as {{math|ε → 0}} in the {{mvar|L^p}} norm, as well as pointwise almost everywhere, by the Titchmarsh theorem.{{sfn|Titchmarsh|1948|loc=Chapter 5}}

In the case {{math|1=p = 1}}, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally.{{sfn|Titchmarsh|1948|loc=§5.14}} In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an {{math|L¹}} function does converge, however, in {{math|L¹}}-weak, and the Hilbert transform is a bounded operator from {{math|L¹}} to {{math|L^1,w}}.{{sfn|Stein|Weiss|1971|loc=Lemma V.2.8}} (In particular, since the Hilbert transform is also a multiplier operator on {{math|L²}}, Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that {{mvar|H}} is bounded on {{math|L^p}}.)

If {{math|1 < p < ∞}}, then the Hilbert transform on $L^p(\backslash mathbb\{R\})$ is a bounded linear operator, meaning that there exists a constant {{mvar|C_p}} such that

$$\backslash left\backslash |\backslash operatorname\{H\}u\backslash right\backslash |\_p\; \backslash le\; C\_p\; \backslash left\backslash |u\backslash right\backslash |\_p$$

for all {{nowrap|$u\; \backslash isin\; L^p(\backslash mathbb\{R\})$.}}This theorem is due to {{harvnb|Riesz|1928|loc=VII}}; see also {{harvnb|Titchmarsh|1948|loc=Theorem 101}}.

The best constant $C\_p$ is given byThis result is due to {{harvnb|Pichorides|1972}}; see also {{harvnb|Grafakos|2004|loc=Remark 4.1.8}}.

$$C\_p\; =\; \backslash begin\{cases\}$$

\tan \frac{\pi}{2p} & \text{if} ~ 1 < p \leq 2 \\[4pt]

\cot \frac{\pi}{2p} & \text{if} ~ 2 < p < \infty

\end{cases}

An easy way to find the best $C\_p$ for $p$ being a power of 2 is through the so-called Cotlar's identity that $(\backslash operatorname\{H\}f)^2\; =f^2\; +2\backslash operatorname\{H\}(f\backslash operatorname\{H\}f)$ for all real valued {{mvar|f}}. The same best constants hold for the periodic Hilbert transform.

The boundedness of the Hilbert transform implies the $L^p(\backslash mathbb\{R\})$ convergence of the symmetric partial sum operator

$$S\_R\; f\; =\; \backslash int\_\{-R\}^R\; \backslash hat\{f\}(\backslash xi)\; e^\{2\backslash pi\; i\; x\backslash xi\}\; \backslash ,\; \backslash mathrm\{d\}\backslash xi$$

to {{mvar|f}} in {{nowrap|$L^p(\backslash mathbb\{R\})$.}}See for example {{harvnb|Duoandikoetxea|2000|p=59}}.

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between $L^p(\backslash mathbb\{R\})$ and the dual space {{nowrap|$L^q(\backslash mathbb\{R\})$,}} where {{mvar|p}} and {{mvar|q}} are Hölder conjugates and {{math|1 < p, q < ∞}}. Symbolically,

$$\backslash langle\; \backslash operatorname\{H\}\; u,\; v\; \backslash rangle\; =\; \backslash langle\; u,\; -\backslash operatorname\{H\}\; v\; \backslash rangle$$

for $u\; \backslash isin\; L^p(\backslash mathbb\{R\})$ and {{nowrap|$v\; \backslash isin\; L^q(\backslash mathbb\{R\})$.}}{{sfn|Titchmarsh|1948|loc=Theorem 102}}

The Hilbert transform is an anti-involution,{{sfn|Titchmarsh|1948|p=120}} meaning that

$$\backslash operatorname\{H\}\backslash bigl(\backslash operatorname\{H\}\backslash left(u\backslash right)\backslash bigr)\; =\; -u$$

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

$$\backslash operatorname\{H\}^\{-1\}\; =\; -\backslash operatorname\{H\}$$

Because {{math|1=H² = −I}} ("{{math|I}}" is the identity operator) on the real Banach space of real-valued functions in {{nowrap|$L^p(\backslash mathbb\{R\})$,}} the Hilbert transform defines a linear complex structure on this Banach space. In particular, when {{math|1=p = 2}}, the Hilbert transform gives the Hilbert space of real-valued functions in $L^2(\backslash mathbb\{R\})$ the structure of a complex Hilbert space.

The (complex) eigenstates of the Hilbert transform admit representations as holomorphic functions in the upper and lower half-planes in the Hardy space H square by the Paley–Wiener theorem.

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

$$\backslash operatorname\{H\}\backslash left(\backslash frac\{\; \backslash mathrm\{d\}u\}\{\backslash mathrm\{d\}t\}\backslash right)\; =\; \backslash frac\{\backslash mathrm\; d\}\{\backslash mathrm\{d\}t\}\backslash operatorname\{H\}(u)$$

Iterating this identity,

$$\backslash operatorname\{H\}\backslash left(\backslash frac\{\backslash mathrm\{d\}^ku\}\{\backslash mathrm\{d\}t^k\}\backslash right)\; =\; \backslash frac\{\backslash mathrm\{d\}^k\}\{\backslash mathrm\{d\}t^k\}\backslash operatorname\{H\}(u)$$

This is rigorously true as stated provided {{mvar|u}} and its first {{mvar|k}} derivatives belong to {{nowrap|$L^p(\backslash mathbb\{R\})$.}}{{sfn|Pandey|1996|loc=§3.3}} One can check this easily in the frequency domain, where differentiation becomes multiplication by {{mvar|ω}}.

The Hilbert transform can formally be realized as a convolution with the tempered distribution{{sfn|Duistermaat|Kolk|2010|p=211}}

$$h(t)\; =\; \backslash operatorname\{p.v.\}\; \backslash frac\{1\}\{\; \backslash pi\; \backslash ,\; t\; \}$$

Thus formally,

$$\backslash operatorname\{H\}(u)\; =\; h*u$$

However, a priori this may only be defined for {{mvar|u}} a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported functions (which are distributions a fortiori) are dense in {{math|L^p}}. Alternatively, one may use the fact that h(t) is the distributional derivative of the function {{math|1=log{{!}}t{{!}}/π}}; to wit

$$\backslash operatorname\{H\}(u)(t)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\backslash left(\backslash frac\{1\}\{\backslash pi\}\; \backslash left(u*\backslash log\backslash bigl|\backslash cdot\backslash bigr|\backslash right)(t)\backslash right)$$

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 on only one of either of the factors:

$$\backslash operatorname\{H\}(u*v)\; =\; \backslash operatorname\{H\}(u)*v\; =\; u*\backslash operatorname\{H\}(v)$$

This is rigorously true if {{mvar|u}} and {{mvar|v}} are compactly supported distributions since, in that case,

$$h*(u*v)\; =\; (h*u)*v\; =\; u*(h*v)$$

By passing to an appropriate limit, it is thus also true if {{math|u ∈ L^p}} and {{math|v ∈ L^q}} provided that

from a theorem due to Titchmarsh.{{sfn|Titchmarsh|1948|loc=Theorem 104}}

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

• It commutes with translations. That is, it commutes with the operators {{math|1=T_a f(x) = f(x + a)}} for all {{mvar|a}} in $\backslash mathbb\{R\}.$
• It commutes with positive dilations. That is it commutes with the operators {{math|1=M_λ f (x) = f (λ x)}} for all {{math|λ > 0}}.
• It anticommutes with the reflection {{math|1=R f (x) = f (−x)}}.

Up to a multiplicative constant, the Hilbert transform is the only bounded operator on {{mvar|L}}² with these properties.{{sfn|Stein|1970|loc=§III.1}}

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

This unitary representation is an example of a principal series representation of $~\backslash text\{SL\}(2,\backslash mathbb\{R\})~.$ In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space $H^2(\backslash mathbb\{R\})$ and its conjugate. These are the spaces of {{math|L²}} boundary values of holomorphic functions on the upper and lower halfplanes. $H^2(\backslash mathbb\{R\})$ and its conjugate consist of exactly those {{math|L²}} functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to {{math|1=H = −i (2P − I)}}, with {{mvar|P}} being the orthogonal projection from $L^2(\backslash mathbb\{R\})$ onto $\backslash operatorname\{H\}^2(\backslash mathbb\{R\}),$ and {{math|I}} the identity operator, it follows that $\backslash operatorname\{H\}^2(\backslash mathbb\{R\})$ and its orthogonal complement are eigenspaces of {{math|H}} for the eigenvalues {{math|±i}}. In other words, {{math|H}} commutes with the operators {{mvar|U_g}}. The restrictions of the operators {{mvar|U_g}} to $\backslash operatorname\{H\}^2(\backslash mathbb\{R\})$ and its conjugate give irreducible representations of $\backslash text\{SL\}(2,\backslash mathbb\{R\})$ – the so-called limit of discrete series representations.See {{harvnb|Bargmann|1947}}, {{harvnb|Lang|1985}}, and {{harvnb|Sugiura|1990}}.

It is further possible to extend the Hilbert transform to certain spaces of distributions {{harv|Pandey|1996|loc=Chapter 3}}. Since the Hilbert transform commutes with differentiation, and is a bounded operator on {{mvar|L^p}}, {{mvar|H}} restricts to give a continuous transform on the inverse limit of Sobolev spaces:

$$\backslash mathcal\{D\}\_\{L^p\}\; =\; \backslash underset\{n\; \backslash to\; \backslash infty\}\{\backslash underset\{\backslash longleftarrow\}\{\backslash lim\}\}\; W^\{n,p\}(\backslash mathbb\{R\})$$

The Hilbert transform can then be defined on the dual space of $\backslash mathcal\{D\}\_\{L^p\}$, denoted $\backslash mathcal\{D\}\_\{L^p\}\text{'}$, consisting of {{mvar|L^p}} distributions. This is accomplished by the duality pairing:

For {{nowrap|$u\backslash in\; \backslash mathcal\{D\}\text{'}\_\{L^p\}$,}} define:

$$\backslash operatorname\{H\}(u)\backslash in\; \backslash mathcal\{D\}\text{'}\_\{L^p\}\; =\; \backslash langle\; \backslash operatorname\{H\}u,\; v\; \backslash rangle\; \backslash \; \backslash triangleq\; \backslash \; \backslash langle\; u,\; -\backslash operatorname\{H\}v\backslash rangle,\backslash \; \backslash text\{for\; all\}\; \backslash \; v\backslash in\backslash mathcal\{D\}\_\{L^p\}\; .$$

It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to Gel'fand and Shilov,{{sfn|Gel'fand|Shilov|1968}} but considerably more care is needed because of the singularity in the integral.

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

Interpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined. For instance, with {{math|1=u = sgn(x)}}, the integral defining {{math|H(u)}} diverges almost everywhere to {{math|±∞}}. To alleviate such difficulties, the Hilbert transform of an {{math|L^∞}} function is therefore defined by the following regularized form of the integral

$$\backslash operatorname\{H\}(u)(t)\; =\; \backslash operatorname\{p.v.\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\; u(\backslash tau)\backslash left\backslash \{h(t\; -\; \backslash tau)-\; h\_0(-\backslash tau)\backslash right\backslash \}\; \backslash ,\; \backslash mathrm\{d\}\backslash tau$$

where as above {{math|1=h(x) = {{sfrac|1|πx}}}} and

$$h\_0(x)\; =\; \backslash begin\{cases\}$$

0 & \text{if} ~ |x| < 1 \\

\frac{1}{\pi \, x} & \text{if} ~ |x| \ge 1

\end{cases}

The modified transform {{math|H}} agrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.{{harvnb|Calderón|Zygmund|1952}}; see {{harvnb|Fefferman|1971}}. Furthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.

A deep result of Fefferman's work{{harvnb|Fefferman|1971}}; {{harvnb|Fefferman|Stein|1972}} is that a function is of bounded mean oscillation if and only if it has the form {{nowrap| {{math|f + H(g)}} }} for some {{nowrap|$f,g\; \backslash isin\; L^\backslash infty\; (\backslash mathbb\{R\})$.}}

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

$$F(x)\; =\; f(x)\; +\; i\backslash ,g(x)$$

is the boundary value of a holomorphic function {{math|F(z)}} in the upper half-plane.{{sfn|Titchmarsh|1948|loc=Chapter V}} Under these circumstances, if {{mvar|f}} and {{mvar|g}} are sufficiently integrable, then one is the Hilbert transform of the other.

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

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

which is the convolution of {{mvar|f}} with the Poisson kernel

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

Furthermore, there is a unique harmonic function {{mvar|v}} defined in the upper half-plane such that {{math|1=F(z) = u(z) + i v(z)}} is holomorphic and

$$\backslash lim\_\{y\; \backslash to\; \backslash infty\}\; v\backslash ,(x\; +\; i\backslash ,y)\; =\; 0$$

This harmonic function is obtained from {{mvar|f}} by taking a convolution with the conjugate Poisson kernel

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

Thus

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

Indeed, the real and imaginary parts of the Cauchy kernel are

$$\backslash frac\{i\}\{\backslash pi\backslash ,z\}\; =\; P(x,\; y)\; +\; i\backslash ,Q(x,\; y)$$

so that {{math|1=F = u + i v}} is holomorphic by Cauchy's integral formula.

The function {{mvar|v}} obtained from {{mvar|u}} in this way is called the harmonic conjugate of {{mvar|u}}. The (non-tangential) boundary limit of {{math|v(x,y)}} as {{math|y → 0}} is the Hilbert transform of {{mvar|f}}. Thus, succinctly,

$$\backslash operatorname\{H\}(f)\; =\; \backslash lim\_\{y\; \backslash to\; 0\}\; Q(-,\; y)\; \backslash star\; f$$

Titchmarsh's theorem (named for E. C. Titchmarsh who included it in his 1937 work) makes precise the relationship between the boundary values of holomorphic functions in the upper half-plane and the Hilbert transform.{{sfn|Titchmarsh|1948|loc=Theorem 95}} It gives necessary and sufficient conditions for a complex-valued square-integrable function {{math|F(x)}} on the real line to be the boundary value of a function in the Hardy space {{math|H²(U)}} of holomorphic functions in the upper half-plane {{mvar|U}}.

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

• {{math|F(x)}} is the limit as {{math|z → x}} of a holomorphic function {{math|F(z)}} in the upper half-plane such that
• The real and imaginary parts of {{math|F(x)}} are Hilbert transforms of each other.
• The Fourier transform $\backslash mathcal\{F\}(F)(x)$ vanishes for {{math|x < 0}}.

A weaker result is true for functions of class {{mvar|L^p}} for {{math|p > 1}}.{{sfn|Titchmarsh|1948|loc=Theorem 103}} Specifically, if {{math|F(z)}} is a holomorphic function such that

for all {{mvar|y}}, then there is a complex-valued function {{math|F(x)}} in $L^p(\backslash mathbb\{R\})$ such that {{math|F(x + i y) → F(x)}} in the {{mvar|L^p}} norm as {{math|y → 0}} (as well as holding pointwise almost everywhere). Furthermore,

$$F(x)\; =\; f(x)\; +\; i\backslash ,g(x)$$

where {{mvar|f}} is a real-valued function in $L^p(\backslash mathbb\{R\})$ and {{mvar|g}} is the Hilbert transform (of class {{mvar|L^p}}) of {{mvar|f}}.

This is not true in the case {{math|1=p = 1}}. In fact, the Hilbert transform of an {{math|L¹}} function {{mvar|f}} need not converge in the mean to another {{math|L¹}} function. Nevertheless,{{sfn|Titchmarsh|1948|loc=Theorem 105}} the Hilbert transform of {{mvar|f}} does converge almost everywhere to a finite function {{mvar|g}} such that

This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc.{{sfn|Duren|1970|loc=Theorem 4.2}} Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see Paley–Wiener theorem), as well as work by Riesz, Hille, and Tamarkinsee {{harvnb|King|2009a|loc=§ 4.22}}.

One form of the Riemann–Hilbert problem seeks to identify pairs of functions {{math|F₊}} and {{math|F₋}} such that {{math|F₊}} is holomorphic on the upper half-plane and {{math|F₋}} is holomorphic on the lower half-plane, such that for {{mvar|x}} along the real axis,

$$F\_\{+\}(x)\; -\; F\_\{-\}(x)\; =\; f(x)$$

where {{math|f(x)}} is some given real-valued function of {{nowrap|$x\; \backslash isin\; \backslash mathbb\{R\}$.}} The left-hand side of this equation may be understood either as the difference of the limits of {{math|F_±}} from the appropriate half-planes, or as a hyperfunction distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.

Formally, if {{math|F_±}} solve the Riemann–Hilbert problem

$$f(x)\; =\; F\_\{+\}(x)\; -\; F\_\{-\}(x)$$

then the Hilbert transform of {{math|f(x)}} is given by{{sfn|Pandey|1996|loc=Chapter 2}}

$$H(f)(x)\; =\; -i\; \backslash bigl(\; F\_\{+\}(x)\; +\; F\_\{-\}(x)\; \backslash bigr)\; .$$

{{see also|Hardy space}}

For a periodic function {{mvar|f}} the circular Hilbert transform is defined:

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

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,

$$\backslash cot\backslash left(\backslash frac\{\; x\; -\; t\; \}\{2\}\backslash right)$$

is known as the Hilbert kernel since it was in this form the Hilbert transform was originally studied.{{sfn|Khvedelidze|2001}}

The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel {{frac|1|{{mvar|x}}}} periodic. More precisely, for {{math|x ≠ 0}}

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

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 transform {{math|1=C(x) = (x – i) / (x + i)}}, which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map

$$U\backslash ,f(x)\; =\; \backslash frac\{1\}\{(x\; +\; i)\backslash ,\backslash sqrt\{\backslash pi\}\}\; \backslash ,\; f\backslash left(C\backslash left(x\backslash right)\backslash right)$$

of {{math|L²(T)}} onto $L^2\; (\backslash mathbb\{R\}).$ The operator {{mvar|U}} carries the Hardy space {{math|H²(T)}} onto the Hardy space $H^2(\backslash mathbb\{R\})$.{{sfn|Rosenblum|Rovnyak|1997|p=92}}

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

$$\backslash operatorname\{H\}\backslash left(f\_\backslash text\{LP\}(t)\backslash cdot\; f\_\backslash text\{HP\}(t)\backslash right)\; =\; f\_\backslash text\{LP\}(t)\backslash cdot\; \backslash operatorname\{H\}\backslash left(f\_\backslash text\{HP\}(t)\backslash right),$$

where {{math|f_LP}} and {{math|f_HP}} are the low- and high-pass signals respectively.{{sfn|Schreier|Scharf|2010|loc=14}} A category of communication signals to which this applies is called the narrowband signal model. A member of that category is amplitude modulation of a high-frequency sinusoidal "carrier":

$$u(t)\; =\; u\_m(t)\; \backslash cdot\; \backslash cos(\backslash omega\; t\; +\; \backslash varphi),$$

where {{math|u_m(t)}} is the narrow bandwidth "message" waveform, such as voice or music. Then by Bedrosian's theorem:{{sfn|Bedrosian|1962}}

$$\backslash operatorname\{H\}(u)(t)\; =$$

\begin{cases}

+u_m(t) \cdot \sin(\omega t + \varphi) & \text{if } \omega > 0 \\

-u_m(t) \cdot \sin(\omega t + \varphi) & \text{if } \omega < 0

\end{cases}

{{main article|analytic signal}}

A specific type of conjugate function is:

$$u\_a(t)\; \backslash triangleq\; u(t)\; +\; i\backslash cdot\; H(u)(t),$$

known as the analytic representation of $u(t).$ The name reflects its mathematical tractability, due largely to Euler's formula. Applying Bedrosian's theorem to the narrowband model, the analytic representation is:{{harvnb|Osgood|page=320}}

{{Equation box 1

|cellpadding= 0 |border= 0 |background colour=white

|indent=: |equation={{NumBlk|| $\backslash begin\{align\}$

u_a(t) & = u_m(t) \cdot \cos(\omega t + \varphi) + i\cdot u_m(t) \cdot \sin(\omega t + \varphi), \quad \omega > 0 \\

& = u_m(t) \cdot \left[\cos(\omega t + \varphi) + i\cdot \sin(\omega t + \varphi)\right], \quad \omega > 0 \\

& = u_m(t) \cdot e^{i(\omega t + \varphi)}, \quad \omega > 0.\,

\end{align}

| {{EquationRef|Eq.1}}

}}

}}

A Fourier transform property indicates that this complex heterodyne operation can shift all the negative frequency components of {{math|u_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:{{harvnb|Osgood|page=320}}

$$u(t)\; =\; A\; \backslash cdot\; \backslash cos(\backslash omega\; t\; +\; \backslash varphi\_m(t))$$

is called angle modulation, which includes both phase modulation and frequency modulation. The instantaneous frequency is  $\backslash omega\; +\; \backslash varphi\_m^\backslash prime(t).$  For sufficiently large {{mvar|ω}}, compared to {{nowrap|$\backslash varphi\_m^\backslash prime$:}}

$$\backslash operatorname\{H\}(u)(t)\; \backslash approx\; A\; \backslash cdot\; \backslash sin(\backslash omega\; t\; +\; \backslash varphi\_m(t))$$

and:

$$u\_a(t)\; \backslash approx\; A\; \backslash cdot\; e^\{i(\backslash omega\; t\; +\; \backslash varphi\_m(t))\}.$$

{{Main article|Single-sideband modulation}}

When {{math|u_m(t)}} in {{EquationNote|Eq.1}} is also an analytic representation (of a message waveform), that is:

$$u\_m(t)\; =\; m(t)\; +\; i\; \backslash cdot\; \backslash widehat\{m\}(t)$$

the result is single-sideband modulation:

$$u\_a(t)\; =\; (m(t)\; +\; i\; \backslash cdot\; \backslash widehat\{m\}(t))\; \backslash cdot\; e^\{i(\backslash omega\; t\; +\; \backslash varphi)\}$$

whose transmitted component is:{{harvnb|Franks|1969|p=88}}{{harvnb|Tretter|1995|p=80 (7.9)}}

$$\backslash begin\{align\}$$

u(t) &= \operatorname{Re}\{u_a(t)\}\\

&= m(t)\cdot \cos(\omega t + \varphi) - \widehat{m}(t)\cdot \sin(\omega t + \varphi)

\end{align}

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

• Its duration is infinite (technically infinite support). Finite-length windowing reduces the effective frequency range of the transform; shorter windows result in greater losses at low and high frequencies. See also quadrature filter.
• It is a non-causal filter. So a delayed version, $h(t-\backslash tau),$ is required. The corresponding output is subsequently delayed by $\backslash tau.$ When creating the imaginary part of an analytic signal, the source (real part) must also be delayed by $\backslash tau$.

File:Bandpass discrete Hilbert transform filter.tif

File:Highpass discrete Hilbert transform filter.tif

File:DFT approximation to Hilbert filter.png

File:Effect of circular convolution on discrete Hilbert transform.png

File:Discrete Hilbert transforms of a cosine function, using piecewise convolution.svg

For a discrete function, $u[n],$ with discrete-time Fourier transform (DTFT), $U(\backslash omega)$, and discrete Hilbert transform $\backslash widehat\; u[n],$ the DTFT of $\backslash widehat\; u[n]$ in the region {{math|1=−π < ω < π}} is given by:

:$\backslash operatorname\{DTFT\}\; (\backslash widehat\; u)\; =\; U(\backslash omega)\backslash cdot\; (-i\backslash cdot\; \backslash sgn(\backslash omega)).$

The inverse DTFT, using the convolution theorem, is:{{harvnb|Carrick|Jaeger|harris|2011|p=2}}{{harvnb|Rabiner|Gold|1975|p=71 (Eq 2.195)}}

:$$

\begin{align}

\widehat u[n] &= {\scriptstyle \mathrm{DTFT}^{-1}} (U(\omega))\ *\ {\scriptstyle \mathrm{DTFT}^{-1}} (-i\cdot \sgn(\omega))\\

&= u[n]\ *\ \frac{1}{2 \pi}\int_{-\pi}^{\pi} (-i\cdot \sgn(\omega))\cdot e^{i \omega n} \,\mathrm{d}\omega\\

&= u[n]\ *\ \underbrace{\frac{1}{2 \pi}\left[\int_{-\pi}^0 i\cdot e^{i \omega n} \,\mathrm{d}\omega - \int_0^\pi i\cdot e^{i \omega n} \,\mathrm{d}\omega \right]}_{h[n]},

\end{align}

where

:$h[n]\backslash \; \backslash triangleq\; \backslash$

\begin{cases}

0, & \text{if }n\text{ even}\\

\frac 2 {\pi n} & \text{if }n\text{ odd}

\end{cases}

which is an infinite impulse response (IIR).

Practical considerations{{harvnb|Oppenheim|Schafer|Buck |1999 |p=794-795}}

Method 1: Direct convolution of streaming $u[n]$ data with an FIR approximation of $h[n],$ which we will designate by $\backslash tilde\; h[n].$ Examples of truncated $h[n]$ are shown in figures 1 and 2. Fig 1 has an odd number of anti-symmetric coefficients and is called Type III.{{harvnb|Isukapalli|p=14}} This type inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in a bandpass filter shape.{{harvnb|Isukapalli|p=18}}{{harvnb|Rabiner|Gold|1975|p=172 (Fig 3.74)}} A Type IV design (even number of anti-symmetric coefficients) is shown in Fig 2.{{harvnb|Isukapalli|p=15}}{{harvnb|Rabiner|Gold|1975|p=173 (Fig 3.75)}} It has a highpass frequency response.{{harvnb|Isukapalli|p=18}} Type III is the usual choice.{{harvnb|Carrick|Jaeger|harris|2011|p=3}}{{harvnb|Rabiner|Gold|1975|p=175}} for these reasons:

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

The abrupt truncation of $h[n]$ creates a rippling (Gibbs effect) of the flat frequency response. That can be mitigated by use of a window function to taper $\backslash tilde\; h[n]$ to zero.{{harvnb|Carrick|Jaeger|harris|2011|p=3}}

Method 2: Piecewise convolution. It is well known that direct convolution is computationally much more intensive than methods like overlap-save that give access to the efficiencies of the Fast Fourier transform via the convolution theorem.{{harvnb|Rabiner|Gold|1975|p=59 (2.163)}} Specifically, the discrete Fourier transform (DFT) of a segment of $u[n]$ is multiplied pointwise with a DFT of the $\backslash tilde\; h[n]$ 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 $N$ (an arbitrary parameter) are convolved with the periodic function:

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

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

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

|see {{slink|Convolution_theorem#Periodic_convolution|nopage=y}}, Eq.4b}}

:$h\_N[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\backslash infty\; h[n\; -\; mN],${{spaces|3}}{{efn-ua

|1=A closed form version of $h\_N[n]$ for even values of $N$ is:{{harvnb|Johansson|p=24}}

h_N[n] = \begin{cases}

\frac{2}{N} \cot(\pi n/N) & \text{for }n\text{ odd},\\

0 & \text{for }n\text{ even}.

\end{cases}

}}{{efn-ua

|A closed form version of $h\_N[n]$ for odd values of $N$ is:{{harvnb|Johansson|p=25}}

$$h\_N[n]\; =\; \backslash frac\{1\}\{N\}\; \backslash left(\backslash cot(\backslash pi\; n/N)\; -\; \backslash frac\{\backslash cos(\backslash pi\; n)\}\{\backslash sin(\backslash pi\; n/N)\}\backslash right),$$

}}

for some arbitrary parameter, $N.$ $h[n]$ 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 $h[n]$ 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 $h\_\{512\}[n],$ corresponding to method 3. It is the inverse DFT of the $-i\; \backslash operatorname\{sgn\}(\backslash omega)$ distribution. Specifically, it is the function that is convolved with a segment of $u[n]$ by the MATLAB function, hilbert(u,512).{{cite web |author=MathWorks |title=hilbert – Discrete-time analytic signal using Hilbert transform |url=http://www.mathworks.com/help/toolbox/signal/ref/hilbert.html |access-date=2021-05-06 |work=MATLAB Signal Processing Toolbox Documentation}} The real part of the output sequence is the original input sequence, so that the complex output is an analytic representation of $u[n].$

When the input is a segment of a pure cosine, the resulting convolution for two different values of $N$ is depicted in Fig 4 (red and blue plots). Edge effects prevent the result from being a pure sine function (green plot). Since $h\_N[n]$ 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 $N$ 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 $h[n]$ 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 $h[n]$ and $h\_N[n]$ (green and red in Fig 3). The fact that $h\_N[n]$ is tapered (windowed) is actually helpful in this context. The real problem is that it's not windowed enough. Effectively, $M=N,$ whereas the overlap-save method needs

The number theoretic Hilbert transform is an extension{{sfn|Kak|1970}} of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization of discrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.{{sfn|Kak|2014}}

• 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

{{notelist-ua}}

{{reflist|25em}}

{{sfn whitelist |CITEREFBitsadze2001 |CITEREFKhvedelidze2001}}

{{Refbegin}}

• {{cite journal

|last = Bargmann |first = V. |author-link=Valentine Bargmann

|date = 1947

|title = Irreducible unitary representations of the Lorentz group

|journal = Ann. of Math.

|volume = 48 |issue = 3 |pages = 568–640

|doi = 10.2307/1969129 |jstor = 1969129

}}

• {{cite report

|last1 = Bedrosian

|first1 = E.

|date = December 1962

|title = A product theorem for Hilbert transforms

|publisher = Rand Corporation

|id = RM-3439-PR

|url = http://www.rand.org/content/dam/rand/pubs/research_memoranda/2008/RM3439.pdf

}}

• {{springer

|last = Bitsadze |first = A. V.

|date = 2001

|title = Boundary value problems of analytic function theory

|id = b/b017400

}}

• {{cite book

|last = Bracewell |first=R. |author-link=Ronald N. Bracewell

|date = 2000

|title = The Fourier Transform and Its Applications |edition = 3rd

|publisher = McGraw–Hill

|isbn = 0-07-116043-4

}}

• {{cite book

|last = Brandwood |first = David

|date = 2003

|title = Fourier Transforms in Radar and Signal Processing

|publisher = Artech House |location = Boston

|isbn = 9781580531740

}}

• {{cite journal

|last1 = Calderón |first1 = A. P. |author-link1 = Alberto Calderón

|last2 = Zygmund |first2 = A. |author-link2 = Antoni Zygmund

|date = 1952

|title = On the existence of certain singular integrals

|journal = Acta Mathematica

|volume = 88 |issue = 1 |pages = 85–139

|doi = 10.1007/BF02392130 |doi-access = free

}}

• {{cite conference

|last1 = Carrick

|first1 = Matt

|last2 = Jaeger

|first2 = Doug

|last3 = harris

|first3 = fred

|title = Design And Application Of A Hilbert Transformer In A Digital Receiver

|publisher = Proceedings of the SDR 11 Technical Conference and Product Exposition, Wireless Innovation Forum

|date = 2011

|location = Chantilly, VA

|url = https://www.wirelessinnovation.org/assets/Proceedings/2011/2011-1b-carrick.pdf

|access-date = 2024-06-05

}}

• {{cite book

|last = Duoandikoetxea |first = J.

|date = 2000

|title = Fourier Analysis

|publisher = American Mathematical Society

|isbn = 0-8218-2172-5

}}

• {{cite book

|last1 = Duistermaat |first1 = J. J. |author-link=Hans Duistermaat

|last2 = Kolk |first2 = J. A. C.

|date = 2010

|title = Distributions

|publisher = Birkhäuser

|isbn = 978-0-8176-4672-1

|doi = 10.1007/978-0-8176-4675-2

}}

• {{cite book

|last = Duren |first = P.

|date = 1970

|title = Theory of H^p Spaces

|publisher = Academic Press |place = New York, NY

}}

• {{cite journal

|last = Fefferman

|first = C.

|author-link = Charles Fefferman

|date = 1971

|title = Characterizations of bounded mean oscillation

|journal = Bulletin of the American Mathematical Society

|volume = 77

|issue = 4

|pages = 587–588

|mr = 0280994

|doi = 10.1090/S0002-9904-1971-12763-5

|doi-access = free

|url = https://www.ams.org/bull/1971-77-04/S0002-9904-1971-12763-5/home.html

}}

• {{cite journal

|last1 = Fefferman |first1 = C.

|last2 = Stein |first2 = E. M.

|date = 1972

|title = H^p spaces of several variables

|journal = Acta Mathematica

|volume = 129 |pages = 137–193

|mr = 0447953

|doi = 10.1007/BF02392215 |doi-access = free

}}

• {{cite book

|last =Franks

|first =L.E.

|title =Signal Theory

|editor=Thomas Kailath

|publisher =Prentice Hall

|series =Information theory

|date =September 1969

|location =Englewood Cliffs, NJ

|isbn =0138100772

}}

• {{cite book

|last1 = Gel'fand |first1 = I. M. |author-link1 = Israel Gelfand

|last2 = Shilov |first2 = G. E.

|date = 1968

|title = Generalized Functions

|volume = 2 |pages = 153–154

|publisher = Academic Press

|isbn = 0-12-279502-4

}}

• {{cite book

|last = Grafakos |first = Loukas

|date = 2004

|title = Classical and Modern Fourier Analysis |pages = 253–257

|publisher = Pearson Education

|isbn = 0-13-035399-X

}}

• {{cite book

|last1 = Hardy |first1 = G. H. |author-link1 = G. H. Hardy

|last2 = Littlewood |first2 = J. E. |author-link2 = J. E. Littlewood

|last3 = Pólya |first3 = G. |author-link3 = G. Pólya

|date = 1952

|title = Inequalities

|publisher = Cambridge University Press |place = Cambridge, UK

|isbn = 0-521-35880-9

}}

• {{cite book

|last = Hilbert

|first = David

|author-link = David Hilbert

|date = 1953

|orig-year = 1912

|title = Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen

|trans-title = Framework for a General Theory of Linear Integral Equations

|language = de

|place = Leipzig & Berlin, DE (1912); New York, NY (1953)

|publisher = B.G. Teubner (1912); Chelsea Pub. Co. (1953)

|isbn = 978-3-322-00681-3

|oclc = 988251080

|url = https://archive.org/details/grundzugeallg00hilbrich/page/n7

|via = archive.org

|access-date = 2020-12-18

}}

• {{cite web

| last = Isukapalli

| first = Yogananda

| title = Types of linear phase FIR filters

| url = https://web.ece.ucsb.edu/~yoga/courses/DSP/P10_Linear_phase_FIR.pdf

| page = 18

| access-date = 2024-06-08

}}

• {{cite web

|last = Johansson

|first = Mathias

|title = The Hilbert transform, Masters Thesis

|url = http://w3.msi.vxu.se/exarb/mj_ex.pdf

|archive-url = https://web.archive.org/web/20120205214945/http://w3.msi.vxu.se/exarb/mj_ex.pdf

|archive-date = 2012-02-05

}}; also http://www.fuchs-braun.com/media/d9140c7b3d5004fbffff8007fffffff0.pdf {{Webarchive|url=https://web.archive.org/web/20210501230529/http://www.fuchs-braun.com/media/d9140c7b3d5004fbffff8007fffffff0.pdf |date=2021-05-01 }}

• {{cite journal

|last = Kak |first = Subhash |author-link = Subhash Kak

|date = 1970

|title = The discrete Hilbert transform

|journal = Proc. IEEE

|volume = 58 |issue = 4 |pages = 585–586

|doi = 10.1109/PROC.1970.7696

}}

• {{cite journal

|last = Kak |first = Subhash |author-link = Subhash Kak

|date = 2014

|title = Number theoretic Hilbert transform

|journal = Circuits, Systems and Signal Processing

|volume = 33 |issue = 8 |pages = 2539–2548

|doi = 10.1007/s00034-014-9759-8

|arxiv = 1308.1688 |s2cid = 21226699

}}

• {{springer

|last = Khvedelidze |first = B. V.

|date = 2001

|title = Hilbert transform

|id = H/h047430

}}

• {{cite book

|last = King |first = Frederick W.

|date = 2009a

|title = Hilbert Transforms |volume = 1

|publisher = Cambridge University Press |place = Cambridge, UK

}}

• {{cite book

|last = King |first = Frederick W.

|date = 2009b

|title = Hilbert Transforms |volume = 2 |pages = 453

|publisher = Cambridge University Press |place = Cambridge, UK

|isbn = 978-0-521-51720-1 }}

• {{cite book

|last = Kress |first = Rainer

|date = 1989

|title = Linear Integral Equations |page = 91

|publisher = Springer-Verlag |place = New York, NY

|isbn = 3-540-50616-0

}}

• {{cite book

|last = Lang |first = Serge |author-link = Serge Lang

|date = 1985

|title = SL(2,$\backslash mathbb\{R\}$)

|series = Graduate Texts in Mathematics |volume = 105

|publisher = Springer-Verlag |place = New York, NY

|isbn = 0-387-96198-4

}}

• {{Cite book

|last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim

|last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer

|last3=Buck |first3=John R.

|title=Discrete-time signal processing

|year=1999

|publisher=Prentice Hall |location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd

|chapter=11.4.1

|url-access=registration |url=https://archive.org/details/discretetimesign00alan

|quote=The exactness of the phase of type III and IV FIR systems is a compelling motivation for their use in approximating Hilbert transformers.

}}

• {{Citation

| last =Osgood

| first =Brad

| title =The Fourier Transform and its Applications

| publisher =Stanford University

| date =

| year =

| url =https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf

| access-date =2021-04-30

}}

• {{cite book

|last = Pandey |first = J. N.

|date = 1996

|title = The Hilbert transform of Schwartz distributions and applications

|publisher = Wiley-Interscience

|isbn = 0-471-03373-1

}}

• {{cite journal

|last = Pichorides |first = S. |author-link=Stylianos Pichorides

|date = 1972

|title = On the best value of the constants in the theorems of Riesz, Zygmund, and Kolmogorov

|journal = Studia Mathematica

|volume = 44 |issue = 2 |pages = 165–179

|doi = 10.4064/sm-44-2-165-179

|doi-access = free

}}

• {{Cite book

| last1=Rabiner|first1=Lawrence R.

| last2=Gold|first2=Bernard

| title=Theory and application of digital signal processing

| year=1975

| publisher=Prentice-Hall

| location=Englewood Cliffs, N.J.

| isbn=0-13-914101-4

| url=https://archive.org/details/theoryapplicatio00rabi

}}

• {{cite journal

|last = Riesz |first = Marcel |author-link = Marcel Riesz

|title = Sur les fonctions conjuguées

|date = 1928

|journal = Mathematische Zeitschrift

|volume = 27 |issue = 1 |pages = 218–244

|doi = 10.1007/BF01171098 |s2cid = 123261514

|language = fr

}}

• {{cite book

|last1 = Rosenblum |first1 = Marvin

|last2 = Rovnyak |first2 = James

|date = 1997

|title = Hardy classes and operator theory

|publisher = Dover

|isbn = 0-486-69536-0

}}

• {{cite book

|last = Schwartz |first = Laurent |author-link = Laurent Schwartz

|date = 1950

|title = Théorie des distributions

|publisher = Hermann |place = Paris, FR

}}

• {{cite book

|last1 = Schreier |first1 = P.

|last2 = Scharf |first2 = L.

|date = 2010

|title = Statistical signal processing of complex-valued data: The theory of improper and noncircular signals

|publisher = Cambridge University Press |place = Cambridge, UK

}}

• {{cite web

|last = Smith |first = J. O.

|date = 2007

|title = Analytic Signals and Hilbert Transform Filters, in Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications

|edition = 2nd

|access-date = 2021-04-29

|url = https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html

}}; also https://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html

• {{cite book

|last = Stein |first = Elias |author-link = Elias Stein

|date = 1970

|title = Singular integrals and differentiability properties of functions

|publisher = Princeton University Press

|isbn = 0-691-08079-8

|url = https://archive.org/details/singularintegral0000stei

|url-access = registration

}}

• {{cite book

|last1 = Stein |first1 = Elias |author-link1 = Elias Stein

|last2 = Weiss |first2 = Guido

|date = 1971

|title = Introduction to Fourier Analysis on Euclidean Spaces

|publisher = Princeton University Press

|isbn = 0-691-08078-X

|url = https://archive.org/details/introductiontofo0000stei

|url-access = registration

}}

• {{cite book

|last = Sugiura |first = Mitsuo

|date = 1990

|title = Unitary Representations and Harmonic Analysis: An Introduction

|edition = 2nd

|series = North-Holland Mathematical Library |volume = 44

|publisher = Elsevier

|isbn = 0444885935

}}

• {{cite book

|last = Titchmarsh |first = E. |author-link = Edward Charles Titchmarsh

|orig-year = 1948 |date = 1986

|title = Introduction to the theory of Fourier integrals |edition = 2nd

|isbn = 978-0-8284-0324-5

|publisher = Clarendon Press |place = Oxford, UK

|ref = {{sfnRef|Titchmarsh|1948}}

}}

• {{cite book

|last=Tretter

|first=Steven A.

|title=Communication System Design Using DSP Algorithms

|editor=R.W.Lucky

|publisher=Springer

|date=1995

|location=New York

|isbn=0306450321

}}

• {{cite book

|last = Zygmund |first = Antoni |author-link = Antoni Zygmund

|orig-year = 1968 |date = 1988

|title = Trigonometric Series |edition = 2nd

|publisher = Cambridge University Press |place = Cambridge, UK

|isbn = 978-0-521-35885-9

|ref = {{sfnRef|Zygmund|1968}}

}}

{{Refend}}

{{Refbegin}}

• {{cite book|last=Benedetto|first=John J.|date=1996|title=Harmonic Analysis and its Applications|publisher=CRC Press|location=Boca Raton, FL|isbn=0849378796|url=https://books.google.com/books?id=_SCeYgvPgoYC&q=%22Hilbert+transform%22}}
• {{cite book|author1=Carlson|author2=Crilly|author3=Rutledge|name-list-style=amp|date=2002|title=Communication Systems|publisher=McGraw-Hill |edition=4th|isbn=0-07-011127-8}}
• {{cite conference|last1=Gold|first1=B.|last2=Oppenheim|first2=A. V.|last3=Rader|first3=C. M.|date=1969|title=Theory and Implementation of the Discrete Hilbert Transform|book-title=Proceedings of the 1969 Polytechnic Institute of Brooklyn Symposium|location=New York}}
• {{cite journal|last=Grafakos|first=Loukas|date=1994|title=An elementary proof of the square summability of the discrete Hilbert transform|journal=American Mathematical Monthly|volume=101|issue=5|pages=456–458|doi=10.2307/2974910|jstor=2974910|publisher=Mathematical Association of America}}
• {{cite journal|last=Titchmarsh|first=E.|author-link=Edward Charles Titchmarsh|date=1926|title=Reciprocal formulae involving series and integrals|journal=Mathematische Zeitschrift|volume=25|issue=1|pages=321–347|doi=10.1007/BF01283842|s2cid=186237099}}

{{Refend}}

{{Commons category}}

• [https://arxiv.org/abs/0909.1426 Derivation of the boundedness of the Hilbert transform]
• [http://mathworld.wolfram.com/HilbertTransform.html Mathworld Hilbert transform] — Contains a table of transforms
• {{MathWorld |title = Titchmarsh theorem |urlname = TitchmarshTheorem}}
• {{cite web |url = http://www.geol.ucsb.edu/faculty/toshiro/GS256_Lecture3.pdf |title = GS256 Lecture 3: Hilbert Transformation |archive-url = https://web.archive.org/web/20120227061333/http://www.geol.ucsb.edu/faculty/toshiro/GS256_Lecture3.pdf |archive-date = 2012-02-27 }} an entry level introduction to Hilbert transformation.

{{DEFAULTSORT:Hilbert Transform}}

Category:Harmonic functions

Category:Integral transforms

Category:Signal processing

Category:Singular integrals

Category:Schwartz distributions