Gaussian examples

The filter (1+Z)/2 is a running average of two adjacent time points. Applying this filter N times yields the filter (1+Z)^N/2^N. The coefficients of the filter (1+Z)^N are generally known as Pascal's triangle. For large N the coefficients tend to a mathematical limit known as a Gaussian function, $\exp (-\alpha (t-t_0)^2)$, where $\alpha$and t₀ are constants that we will not determine here. We will not prove it here, but this Gaussian-shaped signal has a Fourier transform that also has a Gaussian shape, $\exp ( -\beta \omega^2)$.The Gaussian shape is often called a ``bell shape.'' Figure 7 shows an example for $N \approx 15$.Note that, except for the rounded ends, the bell shape seems a good fit to a triangle function.

 

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Figure 7 A Gaussian approximated by many powers of (1+Z).

Curiously, the filter (.75+.25Z)^N also tends to the same Gaussian but with a different t₀. A mathematical theorem says that almost any polynomial raised to the N-th power yields a Gaussian.

In seismology we generally fail to observe the zero frequency. Thus the idealized seismic pulse cannot be a Gaussian. An analytic waveform of longstanding popularity in seismology is the second derivative of a Gaussian, also known as a ``Ricker wavelet.''

Starting from the Gaussian and multiplying by (1-Z)²=1-2Z+Z² produces this old, favorite wavelet, shown in Figure 8.

 

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Figure 8 Ricker wavelet.