Appendix L: Design of Hilbert transforms

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	12
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

We have spent a lot of time and effort worrying about whether a signal is causal. A causal signal is a signal that vanishes for negative time. Now we will worry about something else. No longer will we concern ourselves with whether the signal is causal; instead, we will worry about whether a signal is analytic. An analytic signal is a complex signal whose frequency spectrum vanishes for negative frequency.

Our starting point is a real-valued signal $x_{n}$. Let us take the simplest possible case. We construct a signal $x_{n}$ that has a frequency spectrum $X\left(\omega \right)$ that is made up of two spikes, each of strength 0.5. One spike is at a point $\Omega$ on the positive side of the frequency $\omega$ axis. This spike can be represented as ${0.5}{\delta }_{\Omega }$ The other spike is at the mirror point ${\rm {-}}\Omega$ on the negative side of the frequency $\omega$ axis. This second spike can be represented as $0.5\delta _{{\rm {-}}\Omega }$. The frequency spectrum for all other points is zero. Because the frequency spectrum does not vanish for all negative points, the signal $x_{n}$ is not analytic. However, we will not give up. We will construct an analytic signal $z_{n}$ whose real part is the given signal $x_{n}$.

The signal $x_{n}$ is called the in-phase signal. Now we will construct the quadrature signal $y_{n}$. The word quadrature refers to a phase difference of 90° between two waves of the same frequency, as in the color-difference signals of a television screen. Thus, we multiply the frequency spectrum $X\left(\omega \right)$ by the imaginary number -i on the positive-frequency side and by the imaginary number i on the negative-frequency side. The result is the frequency spectrum ${\rm {y}}\left(\omega \right)$ of the quadrature signal.

The quadrature spectrum is made up of two spikes, each of strength 0.5. One spike is at a point $\Omega$ on the positive side of the frequency $\omega$ axis. This spike can be represented as $-{0.5}i{\delta }_{\Omega }$. The other spike is at the mirror point -$\Omega$ on the negative side of the frequency $\omega$ axis. This second spike can be represented as ${0.5}i{\delta }_{-\Omega }$. The frequency spectrum for all other points is zero.

We next claim that the complex signal $z_{n}=x_{n}+iy_{n}$, whose real part is the in-phase signal $x_{n}$ and whose imaginary part is the quadrature signal $y_{n}$, which is the sought-after analytic signal. We must verify that its frequency spectrum $Z\left(\omega \right)$ vanishes for negative frequency. We have

${\begin{aligned}{\rm {Z}}\left(\omega \right)={X}\left(\omega \right)+i{\rm {Y}}\left(\omega \right)={0.5}{\delta }_{\Omega }+{0.5}{\delta }_{-\Omega }+i\left(-{0.5}i{\delta }_{\Omega }+{0.5}i{\delta }_{-\Omega }\right)={\delta }_{\Omega }.\end{aligned}}$

(L1)

We see that the two positive-frequency spikes add together to give a unit spike at positive frequency $\Omega$, and the two negative-frequency spikes sum to give nothing at negative frequency $\Omega$. Therefore, the complex signal $z_{n}$ is analytic, as we wanted.

Let us now look at the above example in the time domain. The frequency spectrum $X\left(\omega \right)$ tells us that the in-phase signal $x_{n}$ is the cosine wave given by

${\begin{aligned}x_{n}={0.5}e^{i\Omega n}+{0.5}e^{-i\Omega n}={\rm {cos\ }}\left(\Omega n\right).\end{aligned}}$

(L2)

The frequency spectrum $Y(\omega )$ tells us that the quadrature signal $x_{n}$ is the sine wave given by

${\begin{array}{l}y_{n}=-0.5ie^{i\Omega n}+0.5ie^{-i\Omega n}\\\;\;\;\;=-0.5i(\cos \Omega n+i{\rm {sin}}\Omega n)+0.5i(\cos \Omega n-i\sin \Omega n)=\sin(\Omega n).\\\end{array}}$

(L3)

The analytic signal is then

${\begin{aligned}z_{n}=x_{n}+iy_{n}={\rm {cos\ }}\left(\Omega n\right)+i{\rm {\ sin\ }}\left(\Omega n\right)=e^{i\Omega n}.\end{aligned}}$

(L4)

In the sum $x_{n}+iy_{n}$, we observe that the negative-frequency components of $x_{n}$ and $y_{n}$ cancel out, leaving only the positive-frequency components. In other words, the analytic signal has the property that all negative frequencies have been filtered out.

In the above development, we introduced a filter that multiplies the frequency spectrum of the input by the imaginary number -i for positive frequencies and by the imaginary number i for negative frequencies. The filter transforms the cosine-wave input ${\rm {\ cos\ }}\left(\omega n\right)$ to the sine-wave output ${\rm {\ sin\ }}\left(\omega n\right)$. This filter has a special name. It is called the Hilbert transformer filter $h_{n}$. In other words, the discrete-time Hilbert transformer is a filter with a frequency response

${\begin{array}{l}H(\omega )=i=e^{i\pi /2}\;\;\;\;\;\;\;{\rm {for}}\;\;\;\;-\pi \leq \omega <0\\H(\omega )=i-ie^{-i\pi /2}\;\;\;\,{\rm {for}}\;\;\;\;\,\,\,\,0\leq \omega <\pi .\\\end{array}}$

(L5)

The output of the filter is called the Hilbert transform of the input. The input to this filter is called the in-phase signal, and the output is called the quadrature signal.

Let us now give a more formal presentation of the Hilbert transform. A function $X\left(\omega \right)$ is said to exhibit Hermitian symmetry about the origin if $\left(-\omega \right)=X\left(\omega \right)$. It follows that the real part of such a function is symmetric about the origin, and the imaginary part is antisymmetric about the origin. Moreover, the integral of such a function is real; that is,

${\begin{array}{l}\int \limits _{-\pi }^{\pi }{X(\omega )\;d\omega }=\int \limits _{-\pi }^{0}{X(\omega )\;d\omega +\int \limits _{0}^{\pi }{X(\omega )\;d\omega =\int \limits _{0}^{\pi }{[X(-\omega )+X(\omega )]\;d\omega }}}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\,=\int \limits _{0}^{\pi }{[X^{*}(\omega )+X(\omega )]\;d\omega =2\int \limits _{0}^{\pi }{{\mathop {\rm {Re}} \nolimits }[X(\omega )]\;d\omega .}}\\\end{array}}$

(L6)

It follows that a signal $x_{n}$ is real-valued if and only if its spectrum $X\left(\omega \right)$ exhibits Hermitian symmetry. More specifically, the information of a real-valued time signal is contained completely within its frequency spectrum for positive frequencies, except for a possible ambiguity at $\omega ={0}$. Thus, there is a redundancy between the positive and negative portions of the angular frequency spectrum.

Now let us eliminate this redundancy by filling in the missing information as follows: We consider the real signal $x_{n}$ to be the real part of a complex signal $z_{n}$. We then construct another real-valued signal $y_{n}$ and consider this new signal to be the imaginary part of the complex signal $z_{n}$. Thus, we can write

${\begin{aligned}z_{n}=x_{n}+iy_{n}.\end{aligned}}$

(L7)

It is important to remember that both signals $x_{n}$ and $y_{n}$ are real. The new signal $y_{n}$ will be created in a special way to meet the requirements that the complex signal $z_{n}$ be an analytic signal. All the information we need is contained in the real and imaginary parts $X_{R}\left(\omega \right){\;{\rm {and\;}}}X_{I}\left(\omega \right)$ for positive angular frequencies $0\leq \omega {<}\pi$. Because the new signal $y_{n}$ is also real, we need only to know its spectrum for the same positive angular frequencies. Let us represent its spectrum by $Y\left(\omega \right)$. Let us do the simplest thing. For a given positive angular frequency $\omega$, treat $X\left(\omega \right)$ as a complex vector and rotate it by -90°. We call the result ${\rm {Y}}\left(\omega \right)$. That is, we let

${\begin{aligned}{\rm {y}}\left(\omega \right)=X\left(\omega \right)e^{-i\pi {/2}}=-iX\left(\omega \right)\mathrm {\;\;for\;\;} 0\leq \omega {<}\pi .\end{aligned}}$

(L8)

Thus, the filter $h_{n}$ is has the angular frequency response

${\begin{aligned}H\left(\omega \right)=e^{-i\pi {/2}}=-i\mathrm {\;\;\;for\;\;\;} 0\leq \omega {<}\pi .\end{aligned}}$

(L9)

Note that the real part of the complex sequence $z_{n}$ is merely the original sequence $x_{n}$ and that the imaginary part of the complex sequence $z_{n}$ is the signal $y_{n}$. The signal $y_{n}$ is obtained by passing $x_{n}$ through a linear filter with the impulse response $h_{n}$. The filter $h_{n}$ is called the ideal discrete-time Hilbert transform. Because $x_{n}$ and $y_{n}$ are both real, the filter $h_{n}$ is also real. By Hermitian symmetry, we have

${\begin{aligned}H\left(\omega \right)=H^{*}\left(-\omega \right)={\left(-i\right)}^{*}=i\mathrm {\;\;\;for\;\;\;} -\pi \leq \omega {<0}.\end{aligned}}$

(L10)

The discrete-time Hilbert transformer can be regarded as an all-pass filter with the constant phase lead of $\pi /2$ at all negative angular frequencies and with the constant phase lead of $-\pi /2$ at all positive angular frequencies. The result of passing $x_{n}$ through the Hilbert transformer is the Hilbert transform $y_{n}$.

The coefficients of the Hilbert transform, as given by the inverse Fourier transform, are

$h_{n}={\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{H(\omega )\;e^{i\omega n}\;d\omega =\left\langle {\begin{array}{l}0\;\;\;\;\;\;\;{\rm {for}}\;{\rm {(positive}}\;{\rm {or}}\;{\rm {negative)}}\;{\rm {even}}\;{\rm {integers}}\\{\frac {2}{n\pi }}\;\;\;\;\,{\rm {for}}\,{\rm {(positive}}\;{\rm {or}}\;{\rm {negative)}}\,\,{\rm {odd}}\,{\rm {integers}}{\rm {.}}\\\end{array}}\right.}$

(L11)

We see that $h_{n}$ is a doubly infinite antisymmetric impulse.

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Also in this chapter

• Interpretive processing
• Seismic attributes
• Instantaneous attributes
• Seismic sequence attribute map (SSAM)
• Coherence cube (C3)
• SSAM and C3
• Appendix M: Exercises

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