Autocorrelation - book

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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 7 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The autocorrelation of a signal ${\displaystyle b=\{\ldots ,b_{-2},\;b_{-1},\;b_{0},\;b_{1},\,b_{2},\;\ldots }$ is defined to be

 {\displaystyle {\begin{aligned}&r_{k}=\sum _{j=-\infty }^{\infty }{b_{j+k}}b_{j}^{*}.\end{aligned}}} (45)

This formula gives the autocorrelation coefficient ${\displaystyle r_{k}}$ for each integer k. For a finite-length causal wavelet ${\displaystyle b=\left\{b_{0}{,\ }b_{\rm {l}}{,\ }b_{2}{,\ .\ .\ .}b_{N}\right\}}$, the expression becomes

${\displaystyle r_{k}=\sum _{j=0}^{\infty }{b_{+k}}b_{j}^{*}=r_{-k}^{*}\mathrm {for} k=0,1,\dots ,N}$

 {\displaystyle {\begin{aligned}&r_{k}=0\mathrm {\ for\ } k<-N\mathrm {\ and\ } k>N\end{aligned}}} (46)

The superscript asterisk indicates the complex conjugate. The superscript asterisk can be omitted except in cases in which the wavelet coefficients are complex.

Let us examine wavelets, all of which have the same autocorrelation. Figures 14 through 17 depict representative members of a suite of wavelets, all with the same autocorrelation.

Figure 14.  (left) The minimum-delay wavelet of a given suite of wavelets, all with the same autocorrelation. (right) The Energy buildup of this wavelet.
Figure 15.  A mixed-delay wavelet of the given suite of wavelets all with the same autocorrelation (right) The energy buildup of this wavelet.
Figure 16.  The maximum-delay wavelet of the given suite of wavelets all with the same autocorrelation. (right) The Energy buildup of this wavelet.
Figure 17.  Joint plot of the energy-buildup curves of the three wavelets shown in Figures 14 – 16.
Table 4. Energy buildups for a minimum-delay wavelet, a mixed-delay four-length wavelet, and a maximum-delay four-length wavelet.
Energy buildup
Wavelet type ${\displaystyle p_{0}}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle p_{3}}$
Minimum-delay wavelet 4 5 5.25 5.3125
Mixed-delay four-length wavelet 1 5 5.0625 5.3125
Maximum-delay four-length wavelet 0.0625 0.3125 1.3125 5.3125

Next we shall show that the autocorrelation is the convolution of a signal with its zero-point reverse. To autocorrelate a wavelet, we merely convolve the wavelet with its reverse. For example, let us find the autocorrelation of the wavelet ${\displaystyle \left\{b_{0}{,\ }b_{\rm {l}}{,\ }b_{2}\right\}}$. Its reverse is the anticausal wavelet ${\displaystyle b^{R}=\{b_{2}^{*},b_{1}^{*},b_{0}^{*}\}}$. Therefore, the folding table for such an operation is

 {\displaystyle {\begin{aligned}{\begin{array}{*{20}c}{\begin{array}{l}\\\left.{\begin{array}{l}b_{0}\\b_{1}\\b_{2}\\\end{array}}\right|\\\\\end{array}}&{\begin{array}{l}{\underline {b_{2}^{*}\;\;\;\;\;\;\;b_{1}^{*}\;\;\;\;\;\;b_{0}^{*}}}\\b_{0}b_{2}^{*}\;\;\;\;b_{0}b_{1}^{*}\;\;\;b_{0}b_{0}^{*}\\b_{1}b_{2}^{*}\;\;\;\;b_{1}b_{1}^{*}\;\;\;b_{1}b_{0}^{*}\\b_{2}b_{2}^{*}\;\;\;\;b_{2}b_{1}^{*}\;\;b_{2}b_{0}^{*}\\\\\end{array}}\\\end{array}}.\end{aligned}}} (47)

We see that the autocorrelation ${\displaystyle r_{0}}$ for time index 0 is equal to the sum on the main southwest-northeast diagonal:

 {\displaystyle {\begin{aligned}&r_{0}=b_{0}b_{0}^{*}+b_{1}b_{1}^{*}+b_{2}b_{2}^{*}.\end{aligned}}} (48)

For each summand, the first index minus the second index is 0. The value ${\displaystyle r_{0}}$ gives the Energy of the wavelet. Working up from the diagonal, we get the ${\displaystyle r_{k}}$ for the negative ks; that is,

 {\displaystyle {\begin{aligned}&r_{-{l}}=b_{1}b_{2}+b_{0}b_{1}^{*}.\end{aligned}}} (49)

For each summand, the first index minus the second index is –1. In addition, we see that

 {\displaystyle {\begin{aligned}&r_{-2}=b_{0}b_{2}^{*}.\end{aligned}}} (50)

The first index minus the second index is –2. Working down from the diagonal, we get the ${\displaystyle r_{k}}$ for positive k; that is,

 {\displaystyle {\begin{aligned}&r_{l}=b_{2}b_{1}^{*}+b_{1}b_{0}^{*}.\end{aligned}}} (51)

For each summand, the first index minus the second is 1. In addition, we see that

 {\displaystyle {\begin{aligned}r_{2}=b_{2}b_{0}^{*}.\end{aligned}}} (52)

The first index minus the second index is 2. The wavelet ${\displaystyle b=\left\{b_{0}{,\ }b_{1}{\ ,\ }b_{2}\right\}}$ is called the component signal of the autocorrelation. In other words, an autocorrelation is given by the convolution of a component signal and its zero-point reverse. A component signal for an autocorrelation is not unique.

Now let us show that a wavelet and its reverse have the same autocorrelation. Convolution is described as a folding operation between two signals. The autocorrelation of a signal is defined as the convolution of the signal with its zero-point reverse. Consider the wavelet (2, 1) in which the value 2 occurs at time 0. Its zero-point reverse is (1, 2) in which the value 2 occurs at time 0, so its autocorrelation table is

 {\displaystyle {\begin{aligned}&\ \ \ {\underline {1}}\,\,\,{\underline {2}}\\&2|2\,\,\,\,4.\\&1|1\,\,\,\,2\end{aligned}}} (53)

Its autocorrelation is

 {\displaystyle {\begin{aligned}&r_{-1}=2,\ r_{0}=1+4=5,\ r_{1}=2.\end{aligned}}} (54)

On the other hand, consider the wavelet (1, 2), in which the 1 occurs at time 0. Its zero-point reverse is (2, 1), in which the 1 occurs at time 0. The autocorrelation table is

 {\displaystyle {\begin{aligned}&\ \ \ {\underline {2}}\,\,\,{\underline {1}}\\&1|2\,\,\,\,1\ ,\\&2|4\,\,\,\,2\end{aligned}}} (55)

and its autocorrelation is the same. Thus, the wavelet (2, 1) has the same autocorrelation as does its reverse (1, 2). This is always so. That is, the autocorrelation of any wavelet is the same as the autocorrelation of the reverse wavelet.

Now we shall illustrate an all-pass filter. The foregoing example illustrates something further. Let ${\displaystyle \left(b_{0}{,\ }b_{1}\right)}$ be a two-length wavelet. Then the only other two-length wavelet with the same autocorrelation as that of ${\displaystyle \left\{b_{0}{,\ }b_{1}\right\}}$ is its (midpoint) reverse ${\displaystyle \left(b_{1}^{*}{\ ,\ }b_{0}^{*}\right)}$. (Of course, when we speak about the wavelet ${\displaystyle \left\{b_{0}{,\ }b_{1}\right\}}$ and its reverse ${\displaystyle \left(b_{1}^{*}{,\ }b_{0}^{*}\right)}$, we mean the equivalence class of ${\displaystyle \left\{cb_{O}{,\ }cb_{1}\right\}}$ and ${\displaystyle \left\{c^{*}{\ }b_{1}^{*}{,\ }c^{*}{\ }b_{0}^{*}\right\}}$ for any constant c where ${\displaystyle cc^{*}=1}$. Here, we take c = 1.) Thus, the only other two-length wavelet with the same autocorrelation as that of (2, 1) is (1, 2). It turns out that there are wavelets of greater length with the same autocorrelation, but there is no other wavelet of length 2 with the same autocorrelation. Figure 18 shows the wavelet (2, 1) and a wavelet of greater length, both of which have the same autocorrelation. The wavelet of greater length is obtained by convolving the wave-let (2, 1) with the all-pass filter shown in the figure.

Figure 18.  Two wavelets with the same autocorrelation. Because the autocorrelation is symmetric, only one side is shown.

A causal all-pass filter is defined as a causal filter with an amplitude spectrum equal to unity for all frequencies. The transfer function of a causal all-pass filter is

 {\displaystyle {\begin{aligned}&P\left(Z\right)={\frac {A^{R}\left(Z\right)}{A\left(Z\right)}},\end{aligned}}} (56)

where A(Z) is the Z-transform of a finite-length minimum-delay wavelet a, and ${\displaystyle {\rm {A}}^{R}\left(Z\right)}$ is the Z-transform of the reverse ${\displaystyle a^{R}}$. The amplitude spectrum of this all-pass filter is unity because both a and its reverse ${\displaystyle a^{R}}$ have the same amplitude spectrum. This reverse is, of course, maximum delay. Because a causal all-pass wavelet has a unit amplitude spectrum, it follows that its inverse is equal to its reverse; that is, ${\displaystyle p^{-{1}}=p^{R}}$, or

 {\displaystyle {\begin{aligned}&P^{-1}\left(Z\right)=A\left(Z\right)/A^{R}\left(Z\right)={\left(A^{R}\left(Z\right)/A\left(Z\right)\right)}^{R}=P^{R}\left(Z\right).\end{aligned}}} (57)

Thus the inverse is noncausal and one-sided to the past.

Generally, a causal all-pass wavelet has infinite length. However, in two special cases, a causal all-pass wavelet reduces to a finite-length wavelet. In the first special case, A(Z) is a constant. As a result, the transfer function of the causal all-pass wavelet is equal to one, so the causal all-pass wavelet is the unit spike {1, 0, 0, 0, …}, where the 1 occurs at time index 0. Such a causal all-pass wavelet is called trivial because the unit spike convolved with any signal does not change the signal. In the second special case, ${\displaystyle A^{R}\left(Z\right)=Z^{n}}$ where n is a positive integer. In this case, ${\displaystyle P\left(Z\right)=Z^{n}}$, so the causal all-pass wavelet produces a pure delay of n time units.