It is obvious that in the field of mathematics the term Hilbert transform can be defined as a transform that takes a function of a real variable and produces another function of a real variable.
Concept of Hilbert transform
The fast and easily way to grasp the concept regarding this transform is that, let's say we have the signal constructed by a cosine function, then we can extract the waveform of the sine function from our cosine function, simply create a sine wave out of cosine wave through the Hilbert transform.
The Hilbert transform is important in signal processing, at which the analytic representation of a real-valued signal, say x(t) , is produced.
The Hilbert transform was first introduced by David Hilbert in 1905 for the purpose that a special case of the Riemann–Hilbert problem for analytic functions can be solved.
The analytic signal produced by the Hilbert transform is useful in many signal analysis applications, while implementing the Hilbert transform enables the creation of an analytic signal based on some original real-valued signal.
Analytic signal
What is an Analytic signal?
Analytic signal is the complex time signal that is constructed by real valued input signal x (t), and its Hilbert transform. See figure 1
X (t) = x (t) + j h(t)
Where,
X (t) = Analytic signal of the input signal, x (t)
x (t) = input signal
h(t) = Hilbert transform of input signal
In polar coordinate representation,
X(t) = A(t)ejqt
Where,
A(t) = Amplitude of analytic signal
q = phase of analytic signal
Figure 1: Complex seismic signal, X(t)
The analytic signal can be used to extract the instantaneous attributes of the given signal in time series. The instantaneous attributes correspond at any point in a given time. For example instantaneous amplitudes and frequency are important seismic attributes in seismic signal analysis.
Lets explain briefly various basic terms related to analytic signal which is produced by Hilbert transform
1. Instantaneous phase
Recall, Analytic signal, X (t) = x (t) + j h(t)
If x(t) = real part of input signal, j h(t) = y (t) = imaginary part of analytic signal
Then q = tan-1 [y(t)/x(t)]
Where, q = instantaneous phase
Figure 2: instantaneous phase
2. Instantaneous frequency
Derivative of the phase (angular) component of the analytic signal
w (t) = dq/dt
Figure 3: instantaneous frequency
3. Instantaneous amplitude
A(t) = √[x(t)2 + y(t)2]
Figure 4: instantaneous amplitude
How is the Hilbert transform performed?
Hilbert transform can be performed in different modes depending on the nature of the signal itself, however when it comes to issues of seismic signal processing and analysis the following two (2) modes are basically and frequently applied. This is because converts between these two modes (time domain to frequency domain and vice - versa is true) can easily be performed through the aid of Fourier transform
1. Time domain
This involves the convolving of the function x(t) in the time domain with the standard function, 1/Ï€t. This means that Hilbert transform is a convolution operation in the time domain.
H(t) = (1/Ï€t)* x(t)
H(t) = 1/Ï€ −∞∫ +∞(x(k)/t - k) dk
Also,
H(t) = 1/Ï€ −∞∫ +∞(x(t - k)/k) dk
With the equation above, this implies that Hilbert transform of a signal x(t) is the Linear operation
The inverse of Hilbert transform can be defined as,
x (t) = - 1/Ï€ −∞∫ +∞(H(k)/t - k) dk
Where,
x(t) and H(t) are Hilbert transform pairs
2. Frequency domain
This involves the shifting of phases of the frequency components by - π/2 radians. You have to note that it is the only phase component of a signal that changes while the amplitude and the energy density spectrum remain unaffected. In frequency domain, Hilbert transform is the multiplication operation of the fourier transform of the two functions (in time domain).
F = (1/Ï€t) = - jsgn(w)
Where,
sgn(w) = Signum function in frequency domain, w = angular frequency
sgn(w) = { 1; w >0 -1; w < 0
One can consider this transform in Linear device as multiplication of signal x(t) which is passing through this device having a transfer function, - jsgn (w) which can be expressed as
F(H (t)) = x(w). - jsgn (w)
Where,
F(H (t)) = Fourier transform of Hilbert transform
jsgn (w) = transfer function, Signum function in frequency domain
Linear device = Hilbert transformer, Filter
Properties of Hilbert transform
> The Hilbert transform of Hilbert transform of function (double Hilbert transform of function) yields the original function which has the opposite sign. Such as H(x(t)) = - x (t)
> The real valued signal and its Hilbert transform have the same amplitude spectrum and energy density.
> The real valued signal and its Hilbert transform are orthogonal, such as phase shifted by π/2 radian.
> The origin signal and its Hilbert transform have the same autocorrelation function.
As described previously, Hilbert transform in seismic signal processing and analysis is mostly used to extract the instantaneous seismic attributes such as phase, frequency, amplitudes of complex seismic trace.
However this transform has many interesting applications in other fields beyond seismic geophysics as listed here down,
Practical applications of Hilbert transform
Other examples of Hilbert transform applications include,
> Sampling of narrowband signals in telecommunications (mostly using Hilbert filters).
> Medical imaging.
> Array processing for direction of Arrival.
> System response analysis.
Thank you for your time!
References
Nabighian, M.N., (1984), Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations: Geophysics, v.49, no.6, p.780-786.
Nabighian, M.N., and Hansen, R.O., (2001), Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform: Geophysics, v.66, no.6, p.1805-1810.
Naoki Saito (2014), Basics of Analytic Signals, Department of Mathematics, University of California, Davis.
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