Hilbert transform

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 1/t is not integrable across 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 p.v.). Explicitly, the Hilbert transform of a function (or signal) u(t) is given by

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

provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/πt.[1] Alternatively, by changing variables, the principal-value integral can be written explicitly[2] as

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

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

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

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is ${\displaystyle \operatorname {H} ^{3}}$. 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 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 f(z) is analytic in the upper half complex plane {z : 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.

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 the Riemann–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 in Göttingen. The results were later published by Hermann 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 spaces L² and ℓ². In 1928, Marcel Riesz proved that the Hilbert transform can be defined for u in ${\displaystyle L^{p}(\mathbb {R} )}$ (L^p space) for 1 < p < ∞, that the Hilbert transform is a bounded operator on ${\displaystyle L^{p}(\mathbb {R} )}$ for 1 < 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 for Antoni Zygmund and Alberto Calderón during their study of singular 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 a multiplier operator.[14] The multiplier of H is σ_H(ω) = −i sgn(ω), where sgn is the signum function. Therefore:

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

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

By Euler's formula,

$${\displaystyle \sigma _{\operatorname {H} }(\omega )={\begin{cases}~~i=e^{+i\pi /2},&{\text{for }}\omega <0,\\~~0,&{\text{for }}\omega =0,\\-i=e^{-i\pi /2},&{\text{for }}\omega >0.\end{cases}}}$$

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 (i.e., a multiplication by −1).

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

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

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

Signal ${\displaystyle u(t)}$	Hilbert transform[fn 1] ${\displaystyle \operatorname {H} (u)(t)}$
${\displaystyle \sin(\omega t+\varphi )}$ [fn 2]	${\displaystyle {\begin{array}{lll}\sin \left(\omega t+\varphi -{\tfrac {\pi }{2}}\right)=-\cos \left(\omega t+\varphi \right),\quad \omega >0\\\sin \left(\omega t+\varphi +{\tfrac {\pi }{2}}\right)=\cos \left(\omega t+\varphi \right),\quad \omega <0\end{array}}}$
${\displaystyle \cos(\omega t+\varphi )}$ [fn 2]	${\displaystyle {\begin{array}{lll}\cos \left(\omega t+\varphi -{\tfrac {\pi }{2}}\right)=\sin \left(\omega t+\varphi \right),\quad \omega >0\\\cos \left(\omega t+\varphi +{\tfrac {\pi }{2}}\right)=-\sin \left(\omega t+\varphi \right),\quad \omega <0\end{array}}}$
${\displaystyle e^{i\omega t}}$	${\displaystyle {\begin{array}{lll}e^{i\left(\omega t-{\tfrac {\pi }{2}}\right)},\quad \omega >0\\e^{i\left(\omega t+{\tfrac {\pi }{2}}\right)},\quad \omega <0\end{array}}}$
${\displaystyle e^{-i\omega t}}$	${\displaystyle {\begin{array}{lll}e^{-i\left(\omega t-{\tfrac {\pi }{2}}\right)},\quad \omega >0\\e^{-i\left(\omega t+{\tfrac {\pi }{2}}\right)},\quad \omega <0\end{array}}}$
${\displaystyle 1 \over t^{2}+1}$	${\displaystyle t \over t^{2}+1}$
${\displaystyle e^{-t^{2}}}$	${\displaystyle {\frac {2}{\sqrt {\pi \,}}}F(t)}$ (see Dawson function)
Sinc function ${\displaystyle \sin(t) \over t}$	${\displaystyle 1-\cos(t) \over t}$
Dirac delta function ${\displaystyle \delta (t)}$	${\displaystyle {1 \over \pi t}}$
Characteristic function ${\displaystyle \chi _{[a,b]}(t)}$	${\displaystyle {{\frac {1}{\,\pi \,}}\ln \left\vert {\frac {t-a}{t-b}}\right\vert }}$

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 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 ${\displaystyle L^{p}(\mathbb {R} )}$ for 1 < p < ∞.

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

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

exists for almost every t. 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,

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

as ε → 0 in the L^p norm, as well as pointwise almost everywhere, by the Titchmarsh theorem.[16]

In the case p = 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 an L¹ function does converge, however, in L¹-weak, and the Hilbert transform is a bounded operator from L¹ to L^1,w.[18] (In particular, since the Hilbert transform is also a multiplier operator on L², Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that H is bounded on L^p.)

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

$${\displaystyle F(x)=f(x)+i\,g(x)}$$

is the boundary value of a holomorphic function F(z) in the upper half-plane.[32] Under these circumstances, if f and g 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 the Poisson integral, f admits a unique harmonic extension into the upper half-plane, and this extension is given by

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

which is the convolution of f with the Poisson kernel

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

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

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

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

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

Thus

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

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

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

so that F = u + i v is holomorphic by Cauchy's integral formula.

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

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

Titchmarsh's theorem

Titchmarsh's theorem (named for E. C. Titchmarsh who included it in his 1937 work) makes precise the relationship between the boundary values of holomorphic functions in the upper half-plane and the Hilbert transform.[33] It gives necessary and sufficient conditions for a complex-valued square-integrable function F(x) on the real line to be the boundary value of a function in the Hardy space H²(U) of holomorphic functions in the upper half-plane U.

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 as z → x of a holomorphic function F(z) in the upper half-plane such that
  $${\displaystyle \int _{-\infty }^{\infty }|F(x+i\,y)|^{2}\;\mathrm {d} x<K}$$

  
• The real and imaginary parts of F(x) are Hilbert transforms of each other.
• The Fourier transform ${\displaystyle {\mathcal {F}}(F)(x)}$ vanishes for x < 0.

A weaker result is true for functions of class L^p for p > 1.[34] Specifically, if F(z) is a holomorphic function such that

$${\displaystyle \int _{-\infty }^{\infty }|F(x+i\,y)|^{p}\;\mathrm {d} x<K}$$

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

$${\displaystyle F(x)=f(x)-i\,g(x)}$$

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

This is not true in the case p = 1. In fact, the Hilbert transform of an L¹ function f need not converge in the mean to another L¹ function. Nevertheless,[35] the Hilbert transform of f does converge almost everywhere to a finite function g such that

$${\displaystyle \int _{-\infty }^{\infty }{\frac {|g(x)|^{p}}{1+x^{2}}}\;\mathrm {d} x<\infty }$$

This result is directly analogous to one by Andrey 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 (see Paley–Wiener theorem), as well as work by Riesz, Hille, and Tamarkin[37]

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 half-plane and F₋ is holomorphic on the lower half-plane, such that for x along the real axis,

$${\displaystyle F_{+}(x)-F_{-}(x)=f(x)}$$

where f(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 of 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 F_± solve the Riemann–Hilbert problem

$${\displaystyle f(x)=F_{+}(x)-F_{-}(x)}$$

then the Hilbert transform of f(x) is given by[38]

$${\displaystyle H(f)(x)=-i{\bigl (}F_{+}(x)+F_{-}(x){\bigr )}.}$$

For a periodic function f the circular Hilbert transform is defined:

$${\displaystyle {\tilde {f}}(x)\triangleq {\frac {1}{2\pi }}\operatorname {p.v.} \int _{0}^{2\pi }f(t)\,\cot \left({\frac {x-t}{2}}\right)\,\mathrm {d} t}$$

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 the Hilbert 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 kernel 1⁄x periodic. More precisely, for x ≠ 0

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

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 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

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

of L²(T) onto ${\displaystyle L^{2}(\mathbb {R} ).}$ The operator U carries the Hardy space H²(T) onto the Hardy space ${\displaystyle H^{2}(\mathbb {R} )}$.[39]

Figure 1: Filter whose frequency response is bandlimited to about 95% of the Nyquist frequency

Figure 2: Hilbert transform filter with a highpass frequency response

Figure 3.

Figure 4. The Hilbert transform of cos(ωt) is sin(ωt). This figure shows sin(ωt) and two approximate Hilbert transforms computed by the MATLAB library function, hilbert()

Figure 5. Discrete Hilbert transforms of a cosine function, using piecewise convolution

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

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

The inverse DTFT, using the convolution theorem, is:[46]

${\displaystyle {\begin{aligned}{\hat {u}}[n]&={\scriptstyle \mathrm {DTFT} ^{-1}}(U(\omega ))\ *\ {\scriptstyle \mathrm {DTFT} ^{-1}}(-i\cdot \operatorname {sgn}(\omega ))\\&=u[n]\ *\ {\frac {1}{2\pi }}\int _{-\pi }^{\pi }(-i\cdot \operatorname {sgn}(\omega ))\cdot e^{i\omega n}\,\mathrm {d} \omega \\&=u[n]\ *\ \underbrace {{\frac {1}{2\pi }}\left[\int _{-\pi }^{0}i\cdot e^{i\omega n}\,\mathrm {d} \omega -\int _{0}^{\pi }i\cdot e^{i\omega n}\,\mathrm {d} \omega \right]} _{h[n]},\end{aligned}}}$

where

${\displaystyle h[n]\ \triangleq \ {\begin{cases}0,&{\text{for }}n{\text{ even}}\\{\frac {2}{\pi n}}&{\text{for }}n{\text{ odd}},\end{cases}}}$

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 anti-symmetric coefficients is called Type III, which inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in this case in a bandpass filter shape. A Type IV design (even number of anti-symmetric coefficients) is shown in Figure 2. Since the magnitude response at the Nyquist frequency does not drop out, it approximates an ideal Hilbert transformer a little better than the odd-tap filter. However

• 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 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 ${\displaystyle {\hat {u}}[n]}$ with ${\displaystyle u[n],}$ to create an analytic signal. The group delay of Type IV is halfway between two samples.

The MATLAB function, hilbert(u,N),[47] convolves a u[n] sequence with the periodic summation:[upper-alpha 1]

${\displaystyle h_{N}[n]\ \triangleq \sum _{m=-\infty }^{\infty }h[n-mN]}$   [upper-alpha 2][upper-alpha 3]

and returns one cycle (N samples) of the periodic result in the imaginary part of a complex-valued output sequence. The convolution is implemented in the frequency domain as the product of the array  ${\displaystyle {\scriptstyle \mathrm {DFT} }\left(u[n]\right)}$  with samples of the −i sgn(ω) distribution (whose real and imaginary components are all just 0 or ±1). Figure 3 compares a half-cycle of h_N[n] with an equivalent length portion of h[n]. Given an FIR approximation for ${\displaystyle h[n],}$ denoted by ${\displaystyle {\tilde {h}}[n],}$ substituting ${\displaystyle {\scriptstyle \mathrm {DFT} }\left({\tilde {h}}[n]\right)}$ 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 zero-valued elements. In most cases, that reduces the magnitude of the differences. But their duration is dominated by the inherent rise and fall times of the h[n] impulse response.

An appreciation for the edge effects is important when a method called overlap-save is used to perform the convolution on a long u[n] sequence. Segments of length N are convolved with the periodic function:

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

When the duration of non-zero values of ${\displaystyle {\tilde {h}}[n]}$ is ${\displaystyle M<N,}$ the output sequence includes N − M + 1 samples of ${\displaystyle {\hat {u}}.}$ M − 1 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 Figure 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 M < N.

The number theoretic Hilbert transform is an extension[50] 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.[51]

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

Wikimedia Commons has media related to Hilbert transform.

• 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 from the original (PDF) on 2012-02-27. an entry level introduction to Hilbert transformation.