# Appendix L: Design of Hilbert transforms

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 12 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 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 ${\displaystyle x_{n}}$. Let us take the simplest possible case. We construct a signal ${\displaystyle x_{n}}$ that has a frequency spectrum ${\displaystyle X\left(\omega \right)}$ that is made up of two spikes, each of strength 0.5. One spike is at a point ${\displaystyle \Omega }$ on the positive side of the frequency ${\displaystyle \omega }$ axis. This spike can be represented as ${\displaystyle {0.5}{\delta }_{\Omega }}$ The other spike is at the mirror point ${\displaystyle {\rm {-}}\Omega }$ on the negative side of the frequency ${\displaystyle \omega }$ axis. This second spike can be represented as ${\displaystyle 0.5\delta _{{\rm {-}}\Omega }}$. The frequency spectrum for all other points is zero. Because the frequency spectrum does not vanish for all negative points, the signal ${\displaystyle x_{n}}$ is not analytic. However, we will not give up. We will construct an analytic signal ${\displaystyle z_{n}}$ whose real part is the given signal ${\displaystyle x_{n}}$.

The signal ${\displaystyle x_{n}}$ is called the in-phase signal. Now we will construct the quadrature signal ${\displaystyle y_{n}}$. 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 ${\displaystyle X\left(\omega \right)}$ 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 ${\displaystyle {\rm {y}}\left(\omega \right)}$ of the quadrature signal.

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

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

 {\displaystyle {\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 ${\displaystyle \Omega }$, and the two negative-frequency spikes sum to give nothing at negative frequency ${\displaystyle \Omega }$. Therefore, the complex signal ${\displaystyle z_{n}}$ is analytic, as we wanted.

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

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

 ${\displaystyle {\begin{array}{l}y_{n}=-0.5ie^{i\Omega n}+0.5ie^{-i\Omega n}\\\;\;\;\;=-0.5i(\cos \Omega n+i{\rm {sin}}\Omega n)+0.5i(\cos \Omega n-i\sin \Omega n)=\sin(\Omega n).\\\end{array}}}$ (L3)

The analytic signal is then

 {\displaystyle {\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 ${\displaystyle x_{n}+iy_{n}}$, we observe that the negative-frequency components of ${\displaystyle x_{n}}$ and ${\displaystyle y_{n}}$ 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 ${\displaystyle {\rm {\ cos\ }}\left(\omega n\right)}$ to the sine-wave output ${\displaystyle {\rm {\ sin\ }}\left(\omega n\right)}$. This filter has a special name. It is called the Hilbert transformer filter ${\displaystyle h_{n}}$. In other words, the discrete-time Hilbert transformer is a filter with a frequency response

 ${\displaystyle {\begin{array}{l}H(\omega )=i=e^{i\pi /2}\;\;\;\;\;\;\;{\rm {for}}\;\;\;\;-\pi \leq \omega <0\\H(\omega )=i-ie^{-i\pi /2}\;\;\;\,{\rm {for}}\;\;\;\;\,\,\,\,0\leq \omega <\pi .\\\end{array}}}$ (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 ${\displaystyle X\left(\omega \right)}$ is said to exhibit Hermitian symmetry about the origin if ${\displaystyle \left(-\omega \right)=X\left(\omega \right)}$. 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,

 ${\displaystyle {\begin{array}{l}\int \limits _{-\pi }^{\pi }{X(\omega )\;d\omega }=\int \limits _{-\pi }^{0}{X(\omega )\;d\omega +\int \limits _{0}^{\pi }{X(\omega )\;d\omega =\int \limits _{0}^{\pi }{[X(-\omega )+X(\omega )]\;d\omega }}}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\,=\int \limits _{0}^{\pi }{[X^{*}(\omega )+X(\omega )]\;d\omega =2\int \limits _{0}^{\pi }{{\mathop {\rm {Re}} \nolimits }[X(\omega )]\;d\omega .}}\\\end{array}}}$ (L6)

It follows that a signal ${\displaystyle x_{n}}$ is real-valued if and only if its spectrum ${\displaystyle X\left(\omega \right)}$ 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 ${\displaystyle \omega ={0}}$. 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 ${\displaystyle x_{n}}$ to be the real part of a complex signal ${\displaystyle z_{n}}$. We then construct another real-valued signal ${\displaystyle y_{n}}$ and consider this new signal to be the imaginary part of the complex signal ${\displaystyle z_{n}}$. Thus, we can write

 {\displaystyle {\begin{aligned}z_{n}=x_{n}+iy_{n}.\end{aligned}}} (L7)

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

 {\displaystyle {\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 ${\displaystyle h_{n}}$ is has the angular frequency response

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

 {\displaystyle {\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 ${\displaystyle \pi /2}$ at all negative angular frequencies and with the constant phase lead of ${\displaystyle -\pi /2}$ at all positive angular frequencies. The result of passing ${\displaystyle x_{n}}$ through the Hilbert transformer is the Hilbert transform ${\displaystyle y_{n}}$.

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

 ${\displaystyle h_{n}={\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{H(\omega )\;e^{i\omega n}\;d\omega =\left\langle {\begin{array}{l}0\;\;\;\;\;\;\;{\rm {for}}\;{\rm {(positive}}\;{\rm {or}}\;{\rm {negative)}}\;{\rm {even}}\;{\rm {integers}}\\{\frac {2}{n\pi }}\;\;\;\;\,{\rm {for}}\,{\rm {(positive}}\;{\rm {or}}\;{\rm {negative)}}\,\,{\rm {odd}}\,{\rm {integers}}{\rm {.}}\\\end{array}}\right.}}$ (L11)

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