Appendix L: Design of Hilbert transforms
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 . Let us take the simplest possible case. We construct a signal that has a frequency spectrum that is made up of two spikes, each of strength 0.5. One spike is at a point on the positive side of the frequency axis. This spike can be represented as The other spike is at the mirror point on the negative side of the frequency axis. This second spike can be represented as . The frequency spectrum for all other points is zero. Because the frequency spectrum does not vanish for all negative points, the signal is not analytic. However, we will not give up. We will construct an analytic signal whose real part is the given signal .
The signal is called the in-phase signal. Now we will construct the quadrature signal . 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 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 of the quadrature signal.
The quadrature spectrum is made up of two spikes, each of strength 0.5. One spike is at a point on the positive side of the frequency axis. This spike can be represented as . The other spike is at the mirror point - on the negative side of the frequency axis. This second spike can be represented as . The frequency spectrum for all other points is zero.
We next claim that the complex signal , whose real part is the in-phase signal and whose imaginary part is the quadrature signal , which is the sought-after analytic signal. We must verify that its frequency spectrum vanishes for negative frequency. We have
We see that the two positive-frequency spikes add together to give a unit spike at positive frequency , and the two negative-frequency spikes sum to give nothing at negative frequency . Therefore, the complex signal is analytic, as we wanted.
Let us now look at the above example in the time domain. The frequency spectrum tells us that the in-phase signal is the cosine wave given by
The frequency spectrum tells us that the quadrature signal is the sine wave given by
The analytic signal is then
In the sum , we observe that the negative-frequency components of and 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 to the sine-wave output . This filter has a special name. It is called the Hilbert transformer filter . In other words, the discrete-time Hilbert transformer is a filter with a frequency response
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 is said to exhibit Hermitian symmetry about the origin if . 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,
It follows that a signal is real-valued if and only if its spectrum 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 . 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 to be the real part of a complex signal . We then construct another real-valued signal and consider this new signal to be the imaginary part of the complex signal . Thus, we can write
It is important to remember that both signals and are real. The new signal will be created in a special way to meet the requirements that the complex signal be an analytic signal. All the information we need is contained in the real and imaginary parts for positive angular frequencies . Because the new signal is also real, we need only to know its spectrum for the same positive angular frequencies. Let us represent its spectrum by . Let us do the simplest thing. For a given positive angular frequency , treat as a complex vector and rotate it by -90°. We call the result . That is, we let
Thus, the filter is has the angular frequency response
Note that the real part of the complex sequence is merely the original sequence and that the imaginary part of the complex sequence is the signal . The signal is obtained by passing through a linear filter with the impulse response . The filter is called the ideal discrete-time Hilbert transform. Because and are both real, the filter is also real. By Hermitian symmetry, we have
The discrete-time Hilbert transformer can be regarded as an all-pass filter with the constant phase lead of at all negative angular frequencies and with the constant phase lead of at all positive angular frequencies. The result of passing through the Hilbert transformer is the Hilbert transform .
The coefficients of the Hilbert transform, as given by the inverse Fourier transform, are
We see that 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