Hilbert transform

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In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u(t), to another function, H(u)(t), with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the Riemann-Hilbert problem for holomorphic functions. It is a basic tool in Fourier analysis, and provides a concrete means for realizing the conjugate of a given function or Fourier series. Furthermore, in harmonic analysis, it is an example of a singular integral operator, and of a Fourier multiplier. The Hilbert transform is also important in the field of signal processing where it is used to derive the analytic representation of a signal u(t).

The Hilbert transform was originally defined for periodic functions, or equivalently for functions on the circle, in which case it is given by convolution with the Hilbert kernel. More commonly, however, the Hilbert transform refers to a convolution with the Cauchy kernel, for functions defined on the real line R (the boundary of the upper half-plane). The Hilbert transform is closely related to the Paley-Wiener theorem, another result relating holomorphic functions in the upper half-plane and Fourier transforms of functions on the real line.

The Hilbert transform, in red, of a square wave, in blue

[edit] Introduction

The Hilbert transform can be thought of as the convolution of u(t) with the function h(t) = 1/πt. Because h(t) is not integrable the integrals defining the convolution do not 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

$H(u)(t) = \text{p.v.} \int_{-\infty}^{\infty}u(\tau) h(t-\tau)\, d\tau$

provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/πt (Schwartz (1950); Pandey (1996, Chapter 3)). Alternatively, by changing variables, the principal value integral can be written explicitly (Zygmund 1968, §XVI.1) as

$H(u)(t) = -\frac{1}{\pi}\lim_{\epsilon\downarrow 0}\int_{\epsilon}^\infty \frac{u(t+\tau)-u(t-\tau)}{\tau}\,d\tau.$

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

$H(H(u))(t) = -u(t),\,$

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

In signal processing the Hilbert transform of u(t) is commonly denoted by $\widehat u(t).\,$ However, in mathematics, this notation is already extensively used to denote the Fourier Transform of u(t). Occasionally, the Hilbert transform may be denoted by $\tilde{u}(t)$ . Furthermore, many sources define the Hilbert transform as the negative of the one defined here.

For an analytic function in 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 an 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.

[edit] History

The Hilbert transform arose in Hilbert's 1905 work on a problem posed by Riemann concerning analytic functions (Kress (1983); Bitsadze (2001)) which has come to be known as the Riemann-Hilbert problem. Hilbert's work was mainly concerned the Hilbert transform for functions defined on the circle (Hilbert 1953). Some of his earlier work related to the Discrete Hilbert Transform date back to lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation (Hardy, Littlewood & Polya 1952). Schur improved Hilberts results about the discrete Hilbert transform and to extended them to the integral case (Hardy, Littlewood & Polya 1952). 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 L^p(R) for 1<p<∞, the Hilbert transform is a bounded operator on L^p(R) for the same range of p, and that similar results held 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.

[edit] Relationship with the Fourier transform

As mentioned before, the Hilbert transform is a multiplier operator. The symbol of H is σ_H(ω)=-isgn(ω) where sgn is the signum function. Therefore:

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

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

By Euler's formula,

$\sigma_H(\omega) \, \ =\ \begin{cases} \ \ i = e^{+i\pi/2}, & \mbox{for } \omega < 0\\ \ \ \ \ 0, & \mbox{for } \omega = 0\\ \ \ -i = e^{-i\pi/2}. & \mbox{for } \omega > 0 \end{cases}$

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.

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:

$[\sigma_H(\omega)]^2 = e^{\pm i\pi} = -1.$

[edit] Table of selected Hilbert transforms

Signal $u(t)\,$	Hilbert transform² $H(u)(t)\,$
$\sin(t)\,$ ¹	$-\cos(t)\,$
$\cos(t)\,$ ¹	$\sin(t)\,$
$1 \over t^2 + 1$	$t \over t^2 + 1$
Sinc function $\sin(t) \over t$	$1- \cos(t)\over t$
Rectangular function $\sqcap(t)$	${1 \over \pi} \ln \left \| {t+{1 \over 2} \over t-{1 \over 2}} \right \|$
Dirac delta function $\delta(t) \,$	${1 \over \pi t}$
Characteristic Function $\chi_{[a,b]}(x) \,$	$\frac{1}{\pi}\log \left \vert \frac{x-a}{x-b}\right \vert \,$

Notes

¹ The Hilbert transform of the sin and cos functions can be defined in a distributional sense, if there is a concern that the integral defining them is otherwise conditionally convergent. In the periodic setting this result holds without any difficulty.

² 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.

[edit] Domain of definition

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(R) for 1<p<∞.

More precisely, if u is in L^p(R) for 1<p<∞, then limit defining the improper integral

$H(u)(t) = -\frac{1}{\pi}\lim_{\epsilon\downarrow 0}\int_\epsilon^\infty \frac{f(t+\tau)-f(t-\tau)}{\tau}\,d\tau$

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,

$-\frac{1}{\pi}\int_\epsilon^\infty \frac{f(t+\tau)-f(t-\tau)}{\tau}\,d\tau\to H(u)(t)$

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 fail to be itself integrable even locally (Titchmarsh 1948, §5.14). In particular, convergence in the mean does not in general happen in this case.

[edit] Properties

[edit] 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

$\|Hu\|_p \le C_p\| u\|_p$

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

$C_p=\begin{cases}\tan \frac{\pi}{2p} & \text{for } 1 < p\leq 2\\ \cot\frac{\pi}{2p} & \text{for } 2<p<\infty \end{cases}$

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.

[edit] Anti-self adjointness

The Hilbert transform is an anti-self 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,

$\langle Hu, v\rangle = \langle u, -Hv\rangle$

for u∈ L^p(R) and v∈L^q(R) (Titchmarsh 1948, Theorem 102).

[edit] Inverse transform

The Hilbert transform is an anti-involution, meaning that

$H(H(u)) = -u\,$

provided each transform is well-defined. Since H preserves the space L^p(R), this implies in particular that the Hilbert transform is invertible on L^p(R), and that

$H^{-1} = -H.\,$

[edit] Differentiation

Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative:

$H\left(\frac{du}{dt}\right) = \frac{d}{dt}H(u).$

Iterating this identity,

$H\left(\frac{d^ku}{dt^k}\right) = \frac{d^k}{dt^k}H(u).$

This is rigorously true as stated provided u and its first k derivatives belong to L^p(R) (Pandey 1996, §3.3).

[edit] Convolutions

The Hilbert transform can formally be realized as a convolution with the tempered distribution

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

Thus formally,

H (u) = h * u .

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.

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:

H (u * v) = H (u) * v = u * H (v).

This is rigorously true if u and 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 u∈L^p and v∈L^r provided

$1 < \frac{1}{p} + \frac{1}{r},$

a theorem due to Titchmarsh (1948, Theorem 104).

[edit] 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:

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

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

$\langle Hu,v\rangle \overset{\mathrm{def}}{=} \langle u, -Hv\rangle$

for all $v\in\mathcal{D}_{L^p}$ .

It is possible to define the Hilbert transform on the space of tempered distributions as well by generalizing the approach of (Gel'fand & Shilov 1984), but considerably more care is needed because of the singularity in the integral.

[edit] 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 naively, the Hilbert transform of a bounded function is clearly ill-defined. 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 modified form of the integral

$H(u)(t) = \text{p.v.}\int_{-\infty}^\infty u(\tau)\left\{h(t-\tau)- h_0(-\tau)\right\}\,d\tau$

where as above h(x) = 1/πx and

$h_0(x) = \begin{cases} 0&\mathrm{if\ }|x|<1\\ \frac{1}{x} &\mathrm{otherwise} \end{cases}$

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 f+H(g) for some f, g ∈ L^∞(R).

[edit] Hilbert transform on the circle

See also: Hardy space

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

$\tilde f(x)=\frac{1}{2\pi}\text{ p.v.}\int_0^{2\pi}f(t)\cot\frac{x-t}{2}\,dt.$

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 $\cot\frac{x-t}{2}$ 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 the Cauchy kernel 1/x periodic. More precisely, for x≠0

$\frac{1}{2}\cot\left(\frac{x}{2}\right)=\frac{1}{x}+\sum_{n=1}^\infty \frac{1}{x+2n\pi}-\frac{1}{2n\pi}.$

Many results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.

[edit] Hilbert transform in signal processing

[edit] Narrowband model

Amplitude modulated signals are modeled as the product of a bandlimited "message" waveform, u_m(t), and a sinusoidal "carrier":

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

When $u_m(t)\,$ has no frequency content above the carrier frequency, $\frac{\omega}{2\pi}$ Hz, then:

$\begin{align} \widehat{u}(t) \ &\stackrel{\mathrm{def}}{=}\ H(u)(t)\\ &= u_m(t) \cdot \sin(\omega t + \phi) \end{align}$

So, the Hilbert transform may be as simple as a circuit that produces a 90° phase shift at the carrier frequency. Furthermore:

$(\omega t + \phi)_{\mathrm{mod}\, 2 \pi}\,$	$= \operatorname{atan2}(\ \widehat{u}(t),\ u(t)\ )\,$
	$= \arg(u_a(t))\,$ (see next section)

from which one can reconstruct the carrier waveform. Then the message can be extracted from u(t) by coherent demodulation.

[edit] Analytic representation

The analytic representation of a signal is defined in terms of the Hilbert transform:

$u_a(t) = u(t) + i\cdot \widehat{u}(t).\,$

For the narrowband model [above], the analytic representation is:

$u_a(t)\,$

$= u_m(t) \cdot \cos(\omega t + \phi) + i\cdot u_m(t) \cdot \sin(\omega t + \phi)\,$

$= u_m(t) \cdot \left[\cos(\omega t + \phi) + i\cdot \sin(\omega t + \phi)\right]\,$

$= u_m(t) \cdot e^{i(\omega t + \phi)}\,$ (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.

While the analytic representation of a signal is not necessarily an analytic function, u_a(t) is given by the boundary values of an analytic function in the upper half-plane.

[edit] Phase/Frequency modulation

The form:

$u(t) = A\cdot \cos(\omega t + \phi_m(t))\,$

is called phase (or frequency) modulation. The instantaneous frequency is $\omega + \phi_m^\prime(t).$ For sufficiently large $ω,$ compared to $\phi_m^\prime$ :

$\widehat{u}(t) \approx A\cdot \sin(\omega t + \phi_m(t)),\,$

and:

$u_a(t) \approx A \cdot e^{i(\omega t + \phi_m(t))}.$

[edit] Single sideband modulation (SSB)

When $u m (t)$ in Eq.1 is also an analytic representation (of a message waveform), that is:

$u_m(t) = m(t) + i\cdot \widehat{m}(t),$

the result is single-sideband modulation:

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

whose transmitted component is:

$\begin{align} u(t) &= \operatorname{Re}\{u_a(t)\}\\ &= m(t)\cdot \cos(\omega t + \phi) - \widehat{m}(t)\cdot \sin(\omega t + \phi). \end{align}$

[edit] Causality

The function h with h(t) = 1/(π t) is a non-causal filter and therefore cannot be implemented as is, if u is a time-dependent signal. If u is a function of a non-temporal variable, e.g., spatial, the non-causality 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 DC-component.

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.

[edit] Discrete Hilbert transforms

There are two objects of study which are considered discrete Hilbert transforms. The Discrete Hilbert transform of practical interest is described as follows. If the signal u(t) is bandlimited, then H(u)(t) is bandlimited in the same way. Consequently, both these signals can be sampled according to the sampling theorem, resulting in the discrete signals u[n] and H(u)[n]. The relation between the two discrete signals is then given by the convolution:

$H(u)[n] = h[n] * u[n]\,$

where

$h[n]= \begin{cases} 0, & \mbox{for }n\mbox{ even},\\ \frac2{\pi n} & \mbox{for }n\mbox{ odd} \end{cases}$

which is non-causal and has infinite duration. In practice, a shortened and time-shifted approximation is used. The usual filter design tradeoffs apply (e.g., filter-order and latency vs. frequency-response). Also notice, that $h[n]\,$ is not just a sampled version of the Hilbert filter $h(t)\,$ , defined above. Rather it is a sequence with this discrete-time Fourier transform:

$\sigma_H(\omega) = \begin{cases} e^{+i\pi/2}, & -\pi < \omega < 0 \\ e^{-i\pi/2}, & 0 < \omega < \pi\\ 0 & \omega=-\pi, 0, \pi \end{cases}$

We note that a sequence similar to $h[n]\,$ can be generated by sampling σ_H(ω) and computing the inverse discrete Fourier transform. The larger the transform (i.e., more samples per $2π$ radians), the better the agreement (for a given value of the abscissa, n). The figure shows the comparison for a 512-point transform. (Due to odd-symmetry, only half the sequence is actually plotted.)
But that is not the actual point, because it is easier and more accurate to generate $h[n]\,$ directly from the formula. The point is that many applications choose to avoid the convolution by doing the equivalent frequency-domain operation: simple multiplication of the signal transform with σ_H(ω), made even easier by the fact that the real and imaginary components are 0 and ±1 respectively. The attractiveness of that approach is only apparent when the actual Fourier transforms are replaced by samples of the same, i.e., the DFT, which is an approximation and introduces some distortion. Thus, after transforming back to the time-domain, those applications have indirectly generated (and convolved with) not $h[n]\,$ , but the DFT approximation to it, which is shown in the figure.

Notes on fast convolution:

Implied in the technique described above is the concept of dividing a long signal into segments of arbitrary size. The signal is filtered piecewise, and the outputs are subsequently pieced back together.
The segment size is an important factor in controlling the amount of distortion. As the size increases, the DFT becomes more dense and is a better approximation to the underlying Fourier transform. In the time-domain, the same distortion is manifested as "aliasing", which results in a type of convolution called circular. It is as if the same segment is repeated periodically and filtered, resulting in distortion that is worst at either or both edges of the original segment. Increasing the segment size reduces the number of edges in the pieced-together result and therefore reduces overall distortion.
Another mitigation strategy is to simply discard the most badly distorted output samples, because data loss can be avoided by overlapping the input segments. When the filter's impulse response is less than the segment length, this can produce a distortion-free (non-circular) convolution (Overlap-discard method). That requires an FIR filter, which the Hilbert transform is not. So yet another technique is to design an FIR approximation to a Hilbert transform filter. That moves the source of distortion from the convolution to the filter, where it can be readily characterized in terms of imperfections in the frequency response.
Failure to appreciate or correctly apply these concepts is probably one of the most common mistakes made by non-experts in the digital signal processing field.

The other Discrete Hilbert transform is defined by

$b_n=\sum_{m=-\infty}^\infty \frac{a_m}{n-m}\qquad m\neq n$ .

Hilbert showed that for a_n in $\ell^2$ the sequence b_n is also in $\ell^2$ . An elementary proof of this fact can be found in (Grafakos 1994). The discrete Hilbert transform was used by E. C. Titchmarsh to give alternate proofs of the results of M. Riesz in the continuous case (Titchmarsh (1926); Hardy, Littlewood & Polya (1952)).

[edit] See also

[edit] References

Bitsadze, A.V. (2001), “Boundary value problems of analytic function theory”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .
Bracewell, R. (1986), The Fourier Transform and Its Applications (2nd ed ed.), McGraw-Hill .
Calderón, A.P. & Zygmund, A. (1952), “On the existence of certain singular integrals”, Acta Mathematica 88 (1): 85-139 .
Carlson, Crilly, and Rutledge (2002), Communication Systems (4th ed ed.) .
Fefferman, C. (1971), “Characterizations of bounded mean oscillation”, Bull. Amer. Math. Soc. 77: 587–588, MR 0280994, doi:10.1090/S0002-9904-1971-12763-5, <http://www.ams.org/bull/1971-77-04/S0002-9904-1971-12763-5/home.html> .
Fefferman, C. & Stein, E.M. (1972), “H^p spaces of several variables"”, Acta Math. 129: 137–193, MR 0447953, DOI 10.1007/BF02392215 .
Gel'fand, I.M. & Shilov, G.E. (1964), Generalized Functions, Academic Press .
Grafakos, Loukas (1994), “An Elementary Proof of the Square Summability of the Discrete Hilbert Transform”, American Mathematical Monthly 101 (5): 456–458 .
Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, ISBN 0-13-035399-X .
Hardy, G. H.; Littlewood, J. E. & Polya, G. (1952), Inequalities, Cambridge: Cambridge University Press, ISBN 0-521-35880-9 .
Hilbert, David (1953), Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea Pub. Co.
Khvedelidze, B.V. (2001), “Hilbert transform”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .
Kress, Rainer (1989), Linear Integral Equations, New York: Springer-Verlag, pp. 91, ISBN 3-540-50616-0 .
Pandey, J.N. (1996), The Hilbert transform of Schwartz distributions and applications, Wiley-Interscience
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 .
Schwartz, Laurent (1966), Théorie des distributions, Paris: Hermann .
Titchmarsh, E (1926), “Reciprocal formulae involving series and integrals”, Mathematische Zeitschrift 25 (1): 321–347 .
Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.) (published 1986), ISBN 978-0828403245 .
Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0521358859 .

[edit] External links

Hilbert transform
another exposition — Hilbert transform properties
Mathworld Hilbert transform — Contains a table of transforms
Analytic Signals and Hilbert Transform Filters
Easy Fourier Analysis hints to compute Hilbert transform in Time domain.