Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function :
the improper integral being understood in the principal value sense. The Hilbert transform has a particularly simple representation in the frequency domain: it imparts a phase shift of 90° to every Fourier component of a function. For example, the Hilbert transform of , where ω > 0, is .
The Hilbert transform is important in signal processing, where it derives the analytic representation of a realvalued signal u(t). Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complexvalued function u + iv admits an extension to the complex upper halfplane satisfying the Cauchy–Riemann equations. The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.
Introduction
The Hilbert transform of u can be thought of as the convolution of u(t) with the function h(t) = 1/(πt), known as the Cauchy kernel. Because h(t) is not integrable, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by p.v.). Explicitly, the Hilbert transform of a function (or signal) u(t) is given by
provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/πt (due to Schwartz (1950); see Pandey (1996, Chapter 3)). Alternatively, by changing variables, the principal value integral can be written explicitly (Zygmund 1968, §XVI.1) as
When the Hilbert transform is applied twice in succession to a function u, the result is negative u:
provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is −H. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of u(t) (see Relationship with the Fourier transform below).
For an analytic function in upper halfplane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if f(z) is analytic in the plane Im z > 0, and u(t) = Re f(t + 0·i ), then Im f(t + 0·i ) = H(u)(t) up to an additive constant, provided this Hilbert transform exists.
Notation
In signal processing the Hilbert transform of u(t) is commonly denoted by (e.g., Brandwood 2003, p. 87). However, in mathematics, this notation is already extensively used to denote the Fourier transform of u(t) (e.g., Stein & Weiss 1971). Occasionally, the Hilbert transform may be denoted by . Furthermore, many sources define the Hilbert transform as the negative of the one defined here (e.g., Bracewell 2000, p. 359).
History
The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions (Kress (1989); Bitsadze (2001)), which has come to be known as the Riemann–Hilbert problem. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle (Khvedelidze 2001; Hilbert 1953). Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation (Hardy, Littlewood & Pólya 1952, §9.1). Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case (Hardy, Littlewood & Pólya 1952, §9.2). These results were restricted to the spaces L^{2} and ℓ^{2}. In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in L^{p}(R) for 1 ≤ p < ∞, that the Hilbert transform is a bounded operator on L^{p}(R) for 1 < p < ∞, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform (Riesz 1928). The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals (Calderón & Zygmund 1952). Their investigations have played a fundamental role in modern harmonic analysis. Various generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.
Relationship with the Fourier transform
The Hilbert transform is a multiplier operator (Duoandikoetxea 2000, Chapter 3). The multiplier of H is σ_{H}(ω) = −i sgn(ω), where sgn is the signum function. Therefore:
where denotes the Fourier transform. Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of .
By Euler's formula,
Therefore, H(u)(t) has the effect of shifting the phase of the negative frequency components of u(t) by +90° (π/2 radians) and the phase of the positive frequency components by −90°. And i·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.
When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of u(t) are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e., H(H(u)) = −u, because
Table of selected Hilbert transforms
In the following table, the frequency parameter is real.
Signal  Hilbert transform[fn 1] 

[fn 2]  
[fn 2]  
(see Dawson function)  
Sinc function  
Rectangular function  
Dirac delta function  
Characteristic Function 
 Notes
 Some authors (e.g., Bracewell) use our −H as their definition of the forward transform. A consequence is that the right column of this table would be negated.
 The Hilbert transform of the sin and cos functions can be defined by taking the principal value of the integral at infinity. This definition agrees with the result of defining the Hilbert transform distributionally.
An extensive table of Hilbert transforms is available (King 2009b). Note that the Hilbert transform of a constant is zero.
Domain of definition
It is by no means obvious that the Hilbert transform is welldefined at all, as the improper integral defining it must converge in a suitable sense. However, the Hilbert transform is welldefined for a broad class of functions, namely those in L^{p}(R) for 1 < p < ∞.
More precisely, if u is in L^{p}(R) for 1 < p < ∞, then the limit defining the improper integral
exists for almost every t. The limit function is also in L^{p}(R) and is in fact the limit in the mean of the improper integral as well. That is,
as ε→0 in the L^{p}norm, as well as pointwise almost everywhere, by the Titchmarsh theorem (Titchmarsh 1948, Chapter 5).
In the case p = 1, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally (Titchmarsh 1948, §5.14). In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an L^{1} function does converge, however, in L^{1}weak, and the Hilbert transform is a bounded operator from L^{1} to L^{1,w} (Stein & Weiss 1971, Lemma V.2.8). (In particular, since the Hilbert transform is also a multiplier operator on L^{2}, Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that H is bounded on L^{p}.)
Properties
Boundedness
If 1 < p < ∞, then the Hilbert transform on L^{p}(R) is a bounded linear operator, meaning that there exists a constant C_{p} such that
for all u∈L^{p}(R). This theorem is due to Riesz (1928, VII); see also Titchmarsh (1948, Theorem 101). The best constant C_{p} is given by
This result is due to (Pichorides 1972); see also Grafakos (2004, Remark 4.1.8). The same best constants hold for the periodic Hilbert transform.
The boundedness of the Hilbert transform implies the L^{p}(R) convergence of the symmetric partial sum operator
to f in L^{p}(R), see for example (Duoandikoetxea 2000, p. 59).
Antiself adjointness
The Hilbert transform is an antiself adjoint operator relative to the duality pairing between L^{p}(R) and the dual space L^{q}(R), where p and q are Hölder conjugates and 1 < p,q < ∞. Symbolically,
for u ∈ L^{p}(R) and v ∈ L^{q}(R) (Titchmarsh 1948, Theorem 102).
Inverse transform
The Hilbert transform is an antiinvolution (Titchmarsh 1948, p. 120), meaning that
provided each transform is welldefined. Since H preserves the space L^{p}(R), this implies in particular that the Hilbert transform is invertible on L^{p}(R), and that
Complex structure
Because H^{2}=−Id on the real Banach space of realvalued functions in L^{p}(R), the Hilbert transform defines a linear complex structure on this Banach space. In particular, when p=2, the Hilbert transform gives the Hilbert space of realvalued functions in L^{2}(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 halfplanes in the Hardy space H^{2} by the Paley–Wiener theorem.
Differentiation
Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:
Iterating this identity,
This is rigorously true as stated provided u and its first k derivatives belong to L^{p}(R) (Pandey 1996, §3.3). One can check this easily in the frequency domain, where differentiation becomes multiplication by ω.
Convolutions
The Hilbert transform can formally be realized as a convolution with the tempered distribution (Duistermaat & Kolk 2010, p. 211)
Thus formally,
However, a priori this may only be defined for u a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported functions (which are distributions a fortiori) are dense in L^{p}. Alternatively, one may use the fact that h(t) is the distributional derivative of the function logt/π; to wit
For most operational purposes the Hilbert transform can be treated as a convolution. For example, in a formal sense, the Hilbert transform of a convolution is the convolution of the Hilbert transform on either factor:
This is rigorously true if u and v are compactly supported distributions since, in that case,
By passing to an appropriate limit, it is thus also true if u ∈ L^{p} and v ∈ L^{r} provided
a theorem due to Titchmarsh (1948, Theorem 104).
Invariance
The Hilbert transform has the following invariance properties on L^{2}(R).
 It commutes with translations. That is, it commutes with the operators T_{a}ƒ(x) = ƒ(x + a) for all a in R.
 It commutes with positive dilations. That is it commutes with the operators M_{λ}ƒ(x) = ƒ(λx) for all λ > 0.
 It anticommutes with the reflection Rƒ(x) = ƒ(−x).
Up to a multiplicative constant, the Hilbert transform is the only bounded operator on L^{2} with these properties (Stein 1970, §III.1).
In fact there is a larger group of operators commuting with the Hilbert transform. The group SL(2,R) acts by unitary operators U_{g} on the space L^{2}(R) by the formula
This unitary representation is an example of a principal series representation of SL(2,R). In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space H^{2}(R) and its conjugate. These are the spaces of L^{2} boundary values of holomorphic functions on the upper and lower halfplanes. H^{2}(R) and its conjugate consist of exactly those L^{2} functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to H = −i (2P − I), with P being the orthogonal projection from L^{2}(R) onto H^{2}(R), it follows that H^{2}(R) and its orthogonal are eigenspaces of H for the eigenvalues ± i. In other words, H commutes with the operators U_{g}. The restrictions of the operators U_{g} to H^{2}(R) and its conjugate give irreducible representations of SL(2,R)—the socalled limit of discrete series representations.[1]
Extending the domain of definition
Hilbert transform of distributions
It is further possible to extend the Hilbert transform to certain spaces of distributions (Pandey 1996, Chapter 3). Since the Hilbert transform commutes with differentiation, and is a bounded operator on L^{p}, H restricts to give a continuous transform on the inverse limit of Sobolev spaces:
The Hilbert transform can then be defined on the dual space of , denoted , consisting of L^{p} distributions. This is accomplished by the duality pairing:
For , define:
It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to Gel'fand & Shilov (1967), but considerably more care is needed because of the singularity in the integral.
Hilbert transform of bounded functions
The Hilbert transform can be defined for functions in L^{∞}(R) as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps L^{∞}(R) to the Banach space of bounded mean oscillation (BMO) classes.
Interpreted naïvely, the Hilbert transform of a bounded function is clearly illdefined. For instance, with u = sgn(x), the integral defining H(u) diverges almost everywhere to ±∞. To alleviate such difficulties, the Hilbert transform of an L^{∞}function is therefore defined by the following regularized form of the integral
where as above h(x) = 1/πx and
The modified transform H agrees with the original transform on functions of compact support by a general result of Calderón & Zygmund (1952); see Fefferman (1971). The resulting integral, furthermore, converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.
A deep result of Fefferman (1971) and Fefferman & Stein (1972) is that a function is of bounded mean oscillation if and only if it has the form ƒ + H(g) for some ƒ, g ∈ L^{∞}(R).
Conjugate functions
The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function
is the boundary value of a holomorphic function F(z) in the upper halfplane (Titchmarsh 1948, Chapter V). Under these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.
Suppose that f ∈ L^{p}(R). Then, by the theory of the Poisson integral, f admits a unique harmonic extension into the upper halfplane, and this extension is given by
which is the convolution of f with the Poisson kernel
Furthermore, there is a unique harmonic function v defined in the upper halfplane such that F(z) = u(z) + iv(z) is holomorphic and
This harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel
Thus
Indeed, the real and imaginary parts of the Cauchy kernel are
so that F = u + iv is holomorphic by Cauchy's integral formula.
The function v obtained from u in this way is called the harmonic conjugate of u. The (nontangential) boundary limit of v(x,y) as y → 0 is the Hilbert transform of f. Thus, succinctly,
Titchmarsh's theorem
Titchmarsh's theorem, named for Edward Charles Titchmarsh who included it in his 1937 work, makes precise the relationship between the boundary values of holomorphic functions in the upper halfplane and the Hilbert transform (Titchmarsh 1948, Theorem 95). It gives necessary and sufficient conditions for a complexvalued squareintegrable function F(x) on the real line to be the boundary value of a function in the Hardy space H^{2}(U) of holomorphic functions in the upper halfplane U.
The theorem states that the following conditions for a complexvalued squareintegrable function F : R → C are equivalent:
 F(x) is the limit as z → x of a holomorphic function F(z) in the upper halfplane such that
 The real and imaginary parts of F(x) are Hilbert transforms of each other.
 The Fourier transform vanishes for x < 0.
A weaker result is true for functions of class L^{p} for p > 1 (Titchmarsh 1948, Theorem 103). Specifically, if F(z) is a holomorphic function such that
for all y, then there is a complexvalued function F(x) in L^{p}(R) such that F(x + iy) → F(x) in the L^{p} norm as y → 0 (as well as holding pointwise almost everywhere). Furthermore,
where ƒ is a realvalued function in L^{p}(R) and g is the Hilbert transform (of class L^{p}) of ƒ.
This is not true in the case p = 1. In fact, the Hilbert transform of an L^{1} function ƒ need not converge in the mean to another L^{1} function. Nevertheless, (Titchmarsh 1948, Theorem 105), the Hilbert transform of ƒ does converge almost everywhere to a finite function g such that
This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc (Duren 1970, Theorem 4.2). Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see Paley–Wiener theorem) as well as work by Riesz, Hille, and Tamarkin (see section 4.22 in King (2009a)).
Riemann–Hilbert problem
One form of the Riemann–Hilbert problem seeks to identify pairs of functions F_{+} and F_{−} such that F_{+} is holomorphic on the upper halfplane and F_{−} is holomorphic on the lower halfplane, such that for x along the real axis,
where f(x) is some given realvalued function of x ∈ R. The lefthand side of this equation may be understood either as the difference of the limits of F_{±} from the appropriate halfplanes, or as a hyperfunction distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.
Formally, if F_{±} solve the Riemann–Hilbert problem
then the Hilbert transform of f(x) is given by
 (Pandey 1996, Chapter 2).
Hilbert transform on the circle
For a periodic function f the circular Hilbert transform is defined:
The circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series. The kernel,
 is known as the Hilbert kernel since it was in this form the Hilbert transform was originally studied (Khvedelidze 2001).
The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel 1/x periodic. More precisely, for x ≠ 0
Many results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.
Another more direct connection is provided by the Cayley transform C(x) = (x – i) / (x + i), which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map
of L^{2}(T) onto L^{2}(R). The operator U carries the Hardy space H^{2}(T) onto the Hardy space H^{2}(R).[2]
Hilbert transform in signal processing
Bedrosian's theorem
Bedrosian's theorem states that the Hilbert transform of the product of a lowpass and a highpass signal with nonoverlapping spectra is given by the product of the lowpass signal and the Hilbert transform of the highpass signal, or
where f_{LP} and f_{HP} are the low and highpass signals respectively (Schreier & Scharf 2010, 14).
Amplitude modulated signals are modeled as the product of a bandlimited "message" waveform, u_{m}(t), and a sinusoidal "carrier":
When u_{m}(t) has no frequency content above the carrier frequency, then by Bedrosian's theorem:
Analytic representation
In the context of signal processing, the conjugate function interpretation of the Hilbert transform, discussed above, gives the analytic representation of a signal u(t):
which is a holomorphic function in the upper half plane.
For the narrowband model (above), the analytic representation is:

(by Euler's formula)
(Eq.1)

This complex heterodyne operation shifts all the frequency components of 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.
Angle (phase/frequency) modulation
The form:
is called angle modulation, which includes both phase modulation and frequency modulation. The instantaneous frequency is For sufficiently large ω, compared to :
and:
Single sideband modulation (SSB)
When u_{m}(t) in Eq.1 is also an analytic representation (of a message waveform), that is:
the result is singlesideband modulation:
whose transmitted component is:
Causality
The function h with h(t) = 1/(πt) is a noncausal filter and therefore cannot be implemented as is, if u is a timedependent signal. If u is a function of a nontemporal variable (e.g., spatial) the noncausality might not be a problem. The filter is also of infinite support, which may be a problem in certain applications. Another issue relates to what happens with the zero frequency (DC), which can be avoided by assuring that s does not contain a DCcomponent.
A practical implementation in many cases implies that a finite support filter, which in addition is made causal by means of a suitable delay, is used to approximate the computation. The approximation may also imply that only a specific frequency range is subject to the characteristic phase shift related to the Hilbert transform. See also quadrature filter.
Discrete Hilbert transform
For a discrete function, with discretetime Fourier transform (DTFT), and discrete Hilbert transform the DTFT of in the region −π < ω < π is given by:
The inverse DTFT, using the convolution theorem, is:
where:
which is an infinite impulse response (IIR). When the convolution is performed numerically, an FIR approximation is substituted for h[n], as shown in Figure 1. An FIR filter with an odd number of antisymmetric coefficients is called Type III, which inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in this case in a bandpass filter shape. A Type IV design (even number of antisymmetric coefficients) is shown in Figure 2. Since the magnitude response at Nyquist does not drop out, it approximates an ideal Hilbert transformer a little better than the oddtap filter. However:
 A typical (i.e. properly filtered and sampled) u[n] sequence has no useful components at the Nyquist frequency.
 The Type IV impulse response requires a ½ sample shift in the h[n] sequence. That causes the zerovalued coefficients to become nonzero, as seen in Figure 2. So a Type III design is potentially twice as efficient as Type IV.
 The group delay of a Type III design is an integer number of samples, which facilitates aligning with to create an analytic signal. The group delay of Type IV is halfway between two samples.
The MATLAB function, hilbert(u,N), convolves a u[n] sequence with the periodic summation:[3]
and returns one cycle (N samples) of the periodic result in the imaginary part of a complexvalued output sequence. The convolution is implemented in the frequency domain as the product of the array with samples of the −i•sgn(ω) distribution (whose real and imaginary components are all just 0 or ±1). Figure 3 compares a halfcycle of h_{N}[n] with an equivalent length portion of h[n]. Given an FIR approximation for denoted by substituting for the −i•sgn(ω) samples results in an FIR version of the convolution.
The real part of the output sequence is the original input sequence, so that the complex output is an analytic representation of u[n]. When the input is a segment of a pure cosine, the resulting convolution for two different values of N is depicted in Figure 4 (red and blue plots). Edge effects prevent the result from being a pure sine function (green plot). Since 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 zerovalued elements. In most cases, that reduces the magnitude of the differences. But their duration is dominated by the inherent rise and fall times of the h[n] impulse response.
An appreciation for the edge effects is important when a method called overlapsave is used to perform the convolution on a long u[n] sequence. Segments of length N are convolved with the periodic function:
When the duration of nonzero values of is M < N, the output sequence includes N − M + 1 samples of M1 outputs are discarded from each block of N, and the input blocks are overlapped by that amount to prevent gaps.
Figure 5 is an example of using both the IIR hilbert() function and the FIR approximation. In the example, a sine function is created by computing the Discrete Hilbert transform of a cosine function, which was processed in four overlapping segments, and pieced back together. As the FIR result (blue) shows, the distortions apparent in the IIR result (red) are not caused by the difference between h[n] and 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 overlapsave method needs M < N.
Numbertheoretic Hilbert transform
The number theoretic Hilbert transform is an extension (Kak 1970) of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization of discrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences(Kak 2014).
See also
Notes
 See:
 Rosenblum & Rovnyak 1997, p. 92
 see Convolution Theorem
 For even values of N, an equivalent closed form is:
See http://www.rle.mit.edu/dspg/documents/HilbertComplete.pdf eq. (17), (18), and unlabeled eq. below (18).
References
 Bargmann, V. (1947), "Irreducible unitary representations of the Lorentz group", Ann. of Math., 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129
 Bedrosian, E. (December 1962), "A Product Theorem for Hilbert Transforms" (PDF), Rand Corporation Memorandum (RM3439PR)
 Benedetto, John J. (1996). Harmonic analysis and applications. Boca Raton, FL: CRC Press. ISBN 0849378796.
 Bitsadze, A.V. (2001) [1994], "Boundary value problems of analytic function theory", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104.
 Bracewell, R. (2000), The Fourier Transform and Its Applications (3rd ed.), McGraw–Hill, ISBN 0071160434.
 Calderón, A.P.; Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Mathematica, 88 (1): 85–139, doi:10.1007/BF02392130.
 Carlson, Crilly, and Rutledge (2002), Communication Systems (4th ed.), ISBN 0070111278CS1 maint: multiple names: authors list (link).
 Duoandikoetxea, J. (2000), Fourier Analysis, American Mathematical Society, ISBN 0821821725.
 Duistermaat, J.J.; Kolk, J.A.C. Kolk (2010), Distributions, Birkhäuser, doi:10.1007/9780817646752, ISBN 9780817646721.
 Duren, P. (1970), Theory of Spaces, New York: Academic Press.
 Fefferman, C. (1971), "Characterizations of bounded mean oscillation", Bull. Amer. Math. Soc., 77 (4): 587–588, doi:10.1090/S000299041971127635, MR 0280994.
 Fefferman, C.; Stein, E.M. (1972), "H^{p} spaces of several variables", Acta Math., 129: 137–193, doi:10.1007/BF02392215, MR 0447953.
 Gel'fand, I.M.; Shilov, G.E. (1967), Generalized Functions, Vol. 2, Academic Press.
 Grafakos, Loukas (1994), "An Elementary Proof of the Square Summability of the Discrete Hilbert Transform", American Mathematical Monthly, Mathematical Association of America, 101 (5): 456–458, doi:10.2307/2974910, JSTOR 2974910.
 Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, ISBN 013035399X.
 Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952), Inequalities, Cambridge: Cambridge University Press, ISBN 0521358809.
 Hilbert, David (1953), Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea Pub. Co.
 Kak, Subhash (1970), "The discrete Hilbert transform", Proc. IEEE, 58: 585–586
 Kak, Subhash (2014), "Number theoretic Hilbert transform", Circuits Systems Signal Processing, 33: 2539–2548
 Khvedelidze, B.V. (2001) [1994], "Hilbert transform", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104.
 King, Frederick W. (2009a), Hilbert Transforms, 1, Cambridge: Cambridge University Press.
 King, Frederick W. (2009b), Hilbert Transforms, 2, Cambridge: Cambridge University Press, p. 453, ISBN 9780521517201.
 Kress, Rainer (1989), Linear Integral Equations, New York: SpringerVerlag, p. 91, ISBN 3540506160.
 Lang, Serge (1985), SL(2,R), Graduate Texts in Mathematics, 105, SpringerVerlag, ISBN 0387961984
 Pandey, J.N. (1996), The Hilbert transform of Schwartz distributions and applications, WileyInterscience, ISBN 0471033731
 Pichorides, S. (1972), "On the best value of the constants in the theorems of Riesz, Zygmund, and Kolmogorov", Studia Mathematica, 44: 165–179
 Riesz, Marcel (1928), "Sur les fonctions conjuguées", Mathematische Zeitschrift, 27 (1): 218–244, doi:10.1007/BF01171098
 Rosenblum, Marvin; Rovnyak, James (1997), Hardy classes and operator theory, Dover, ISBN 0486695360
 Schwartz, Laurent (1950), Théorie des distributions, Paris: Hermann.
 Schreier, P.; Scharf, L. (2010), Statistical signal processing of complexvalued data: the theory of improper and noncircular signals, Cambridge University Press
 Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton University Press, ISBN 0691080798.
 Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, ISBN 069108078X.
 Sugiura, Mitsuo (1990), Unitary Representations and Harmonic Analysis: An Introduction, NorthHolland Mathematical Library, 44 (2nd ed.), Elsevier, ISBN 0444885935
 Titchmarsh, E (1926), "Reciprocal formulae involving series and integrals", Mathematische Zeitschrift, 25 (1): 321–347, doi:10.1007/BF01283842.
 Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.), Oxford University: Clarendon Press (published 1986), ISBN 9780828403245.
 Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 9780521358859.
External links
Wikimedia Commons has media related to Hilbert transform. 
 Derivation of the boundedness of the Hilbert transform
 Mathworld Hilbert transform — Contains a table of transforms
 Analytic Signals and Hilbert Transform Filters
 Weisstein, Eric W. "Titchmarsh theorem". MathWorld.
 Johansson, Mathias. "The Hilbert transform" (PDF). Archived from the original (PDF) on 20120205. a student level summary of the Hilbert transformation.
 "GS256 Lecture 3: Hilbert Transformation" (PDF). Archived from the original (PDF) on 20120227. an entry level introduction to Hilbert transformation.