# Delay in general

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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 concepts of minimum delay and maximum delay have been defined for two-length wavelets. For example, take the minimum-delay wavelet (7, 3), in which the coefficient 7 occurs at time index 0 and the coefficient 3 occurs at time index 1. The reverse is the maximum-delay wavelet (3, 7), in which the coefficient 3 occurs at time index 0 and the coefficient 7 occurs at time index 1. Now let us define these concepts for wavelets of greater length. To do so, take a group of dipoles, each with the minimum-delay wavelet on the left (Table 3).

The convolution of the minimum-delay wavelets in the group (i.e., the convolution of the wavelets on the left) yields a minimum-delay wavelet

 {\displaystyle {\begin{aligned}&(2,1)*(i,0.5)*(-i,0.5)\\&=(2,1)*(1,0,0.25)\\&=(2,1,0.5,0.25).\end{aligned}}} (30)

We shall now define maximum delay in general. The convolution of the maximum-delay wavelets in the group of dipoles (i.e., the convolution of the wavelets on the right [Table 3]) yields a maximum-delay wavelet. Thus, (0.25, 0.5, 1, 2) is a maximum-delay wavelet. We see that the maximum-delay wavelet is the reverse of the minimum-delay wavelet (2, 1, 0.5, 0.25), as we would expect. The reverse of a finite-length minimum-delay wavelet is thus a maximum-delay wavelet.

Let us now define mixed delay in general. The convolution of one wavelet from each dipole in the group, chosen so that we have a mixture of minimum-delay and maximum-delay wavelets, gives a mixed-delay wavelet. Thus,

Table 3.Three dipoles, each with the minimum-delay wavelet on the left.
Dipoles Minimum delay Maximum delay
Dipole 1 (2, 1) (1, 2)
Dipole 2 (i, 0.5) (0.5, –i)
Dipole 3 (–i, 0.5) (0.5, i)

 {\displaystyle {\begin{aligned}&\left({1,\ 2}\right)*\left(i{,\ 0.5}\right)*\left(-i{,\ 0.5}\right)=\left({1,2}\right)*\left({l,\ 0,0.25}\right)=\left({1,2,0.25,0.5}\right)\end{aligned}}} (31)

is a mixed-delay wavelet.

The inverse of the Z-transform of the two-term equal-delay wavelet ${\displaystyle \left({1\ ,\ }-e^{i\theta }\right)}$ is

 {\displaystyle {\begin{aligned}&{\frac {1}{1-e^{\iota \theta }Z}}=1+e^{i\theta }Z+e^{2i\theta }Z^{2}+e^{3i\theta }Z^{3}+\dots .\end{aligned}}} (32)

Thus, the inverse is the discrete form of the complex exponential for positive time. Similarly, the inverse of the Z-transform of the two-term equal-delay wavelet ${\displaystyle \left({1\ ,\ }-e^{-i\theta }\right)}$ is

 {\displaystyle {\begin{aligned}{\frac {1}{1-e^{-i\theta }Z}}=1+e^{-1\theta }Z+e^{-2i\theta }Z^{2}+e^{-{3}_{i}\theta }Z^{3}+\dots .\end{aligned}}} (33)

Thus, the Z-transform of the causal sine (with parameter ${\displaystyle \theta }$) is

 {\displaystyle {\begin{aligned}&u_{n}\mathrm {sin} \theta _{n}={\frac {u_{n}}{2i}}\left[e^{i\theta n}-e^{-i\theta n}\right],\end{aligned}}} (34)

where ${\displaystyle u_{n}}$ = 0,1,2,3,... is the unit step, and which is

 ${\displaystyle {\begin{array}{l}{\frac {1}{2i}}\left({{\frac {1}{1-e^{i\theta }Z}}-{\frac {1}{1-e^{-i\theta }Z}}}\right)={\frac {1}{2i}}{\frac {(1-e^{-i\theta }Z)-(1-e^{i\theta }Z)}{(1-e^{i\theta }Z)(1-e^{i\theta }Z)}}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;={\frac {Z\sin \theta }{1-2\cos \theta Z+Z^{2}}}.\\\end{array}}}$ (35)

We note that the denominator is the Z-transform of the convolution of the equal-delay wavelet ${\displaystyle \left({1\ ,\ }-e^{i\theta }\right)}$ with the equal-delay wavelet ${\displaystyle \left({1\ ,\ }-e^{-i\theta }\right)}$. This convolution gives the wavelet ${\displaystyle \left({1\ ,\ }-{2\ cos\ }\theta {,\ 1}\right)}$, whose inverse is seen to have the Z-transform

 {\displaystyle {\begin{aligned}{\frac {1}{1-2\;\cos \theta \;z+Z^{2}}}.\end{aligned}}} (36)

However, this expression is equal to ${\displaystyle {1/}\left(Z{\rm {\ sin\ }}\theta \right)}$ times the Z-transform of the causal sine.

Thus, we divide the causal sine by ${\displaystyle {\rm {\ sin\ }}\theta }$ and advance the result by one time unit to obtain the inverse of the wavelet ${\displaystyle \left({1\ }{,\ }-{2\ \ cos\ }\theta {,\ 1}\right)}$. In other words, this inverse is the sinusoid ${\displaystyle u_{n\;}\sin n/\theta \sin \theta }$. For example, let us take the case when the frequency is f = 12.5 Hz. Then the parameter ${\displaystyle \theta }$ is ${\displaystyle \theta =2\pi f\Delta t=2\pi \left({12.5}\right)\left({0.004}\right)=\pi {/10})}$. The one-sided sinusoid for this frequency is shown in Figure 12.

Figure 12.  One-sided sinusoid for 12.5 Hz.

For the damping factor ${\displaystyle \alpha }$ where ${\displaystyle {|}\alpha {|<}1}$, let us form the wavelet

 {\displaystyle {\begin{aligned}&\left({1,\ }-\alpha e^{i\theta }\right)*\left({l,\ }-\alpha e^{-i\theta }\right)=\left({1,\ }-2\alpha {\rm {\ cos\ }}\theta {,\ }{\alpha }^{2}\right).\end{aligned}}} (37)

Its inverse is a damped one-sided sinusoid. Figure 13 shows a damped 12.5-Hz sinusoid for ${\displaystyle \alpha =0.97.}$.

Figure 13.  One-sided damped sinusoid for 12.5 Hz for ${\displaystyle \alpha =0.97}$.

How many wavelets can be generated from N dipoles? If every two-length wavelet in a group of N dipoles is distinct, there are ${\displaystyle {2}^{N}}$ possible (N + 1)-length wavelets that can be generated by taking one two-length wavelet from each dipole. Of course, if some of the two-length wavelets are not distinct, some of the (N + 1)-length wavelets will not be distinct either. In any case, all the (N + 1)-length wavelets generated in this way from one group of dipoles will have the same autocorrelation. These (N + 1)-length wavelets, so generated, are said to belong to a suite of wavelets all with the same autocorrelation.

Now let us show that the minimum-delay concept is unique. Of the ${\displaystyle {2}^{\rm {N}}}$ possible (N + 1)-length wavelets in the suite, one is a minimum-delay wavelet, another is a maximum-delay wavelet, and the others are mixed-delay wavelets. However, these ${\displaystyle {2}^{N}\left(N{+\ 1}\right)}$-length wavelets are not the only wavelets in the suite because other wavelets exist that have greater length but the same autocorrelation. Thus, those other wavelets are also members of the suite. If we consider all the members of the suite, there is only one minimum-delay wavelet, which is the (N + 1)-length minimum-delay wavelet that we have found. Thus, the minimum-delay concept is unique.

In contrast, the maximum-delay concept is a relative one, because a wavelet is a maximum-delay wavelet with respect to its length. In our example, the wavelet (0.25, 0.5, 1, 2) is the maximum-delay four-length wavelet of the suite. By delaying this wavelet by one time unit, we obtain the wavelet (0, 0.25, 0.5, 1, 2), which is the maximum-delay five-length wavelet of the suite. Similarly, (0, 0, 0.25, 0.5, 1, 2) is the maximum-delay six-length wavelet of the suite, and so forth. All the other wavelets of the suite - that is, the wavelets other than the minimum-delay wavelet and the maximum-delay wavelets - are mixed-delay wavelets.

The Z-transform of a filter’s impulse response is called the filter’s transfer function. A stable causal time-invariant linear filter is said to be (strictly) a minimum-delay filter (or a minimum-phase filter) if the zeros (or roots) of its transfer function lie outside the unit circle.

The transfer function of a causal FIR filter is the polynomial

 {\displaystyle {\begin{aligned}&H\left(Z\right)=h_{0}+h_{1}Z+\dots +h_{m}Z^{m}.\end{aligned}}} (38)

This polynomial is called a minimum-delay polynomial if the filter is minimum delay. It is called a maximum-delay polynomial if the filter is maximum delay. It is called a mixed-delay polynomial if the filter is mixed delay.

The inverse of a minimum-delay filter A(Z) is ${\displaystyle A^{-{\rm {l}}}\left(Z\right)={\rm {l}}lA\left(Z\right)}$, which also is a minimum-delay filter. An important type of stable causal IIR (infinite-impulse-response) filter is the recursive filter, which has its transfer function given by the rational function ${\displaystyle H\left(Z\right)=B\left(Z\right)/A\left(Z\right)}$ with the provision that the denominator polynomial A(Z) is a minimum-delay polynomial (i.e., all the roots of the polynomial lie outside of the unit circle). This provision ensures that the recursive filter is stable (Treitel and Robinson, 1964[1]). The locations of the zeros of the numerator polynomial B(Z) do not need to be specified for the recursive filter to be stable. However, if it is further specified that the numerator polynomial B(Z) is a minimum-delay polynomial, the recursive filter is a minimum-delay filter.

## References

1. Treitel, S., and E. A. Robinson, 1964, The stability of digital filters: IEEE Transactions on Geoscience Electronics, GE-2, 6–18.

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